Datasets:

sail

/

regmix-data-sample

like 2

Follow

Sea AI Lab 61

--- abstract: 'We report results of tests of the MISANS technique at the CG-1D beamline at High Flux Isotope Reactor (HFIR), Oak Ridge National Laboratory (ORNL). A chopper at 40Hz simulated a pulsed neutron source at the beamline. A compact turn-key MISANS module operating with the pulsed beam was installed and a well characterised MnSi sample was tested. The feasibility of application of high magnetic fields at the sample position was also explored. These tests demonstrate the great potential of this technique, in particular for examining magnetic and depolarizing samples, under extreme sample environments at pulsed sources, such as the Spallation Neutron Source (SNS) or the planned European Spallation Source (ESS).' address: - 'Forschungsneutronenquelle Heinz Maier-Leibnitz, Technische Universität München, Lichtenbergstr. 1, 85747 Garching, Germany' - 'Physik Department E21, Technische Universität München, James-Franck-Str., 85747 Garching, Germany' - 'Argonne National Laboratory, Materials Science Division, Argonne, IL 60439, USA' - 'Oak Ridge National Laboratory, Oak Ridge, TN 37831-6477, USA' - 'Technical University of Delft, Mekelweg 15, 2629JB Delft, Netherlands' author: - 'G. Brandl' - 'J. Lal' - 'J. Carpenter' - 'L. Crow' - 'L. Robertson' - 'R. Georgii' - 'P. Böni' - 'M. Bleuel' title: Tests of Modulated Intensity Small Angle Scattering in time of flight mode --- MIEZE ,spin echo ,HFIR ,MISANS ,ESS Introduction ============ MISANS, MIEZE (Modulation of Intensity with Zero Effort) in the Small Angle Neutron Scattering (SANS) geometry is a rather new technique to probe quasi-elastic scattering with extremely high energy resolution. The method is well understood [@Gaehler:89; @Hank:97; @Keller:93] and efforts are under way [@Georgii:11; @Brandl:11] to establish the technique as a standard tool for measurements of slow dynamics. The general trend of new neutron sources to be accelerator driven and thus to provide pulsed neutron beams raises the question how MISANS will perform in a pulsed mode. In earlier experiments the feasibility of MISANS on pulsed sources was demonstrated [@Bleuel:06; @Hayashida:07], however these tests were only using the direct beam and relative low MIEZE frequencies. The goal of this experiment was to show that a MIEZE can be set up easily at a new beamline and works well in the time-of-flight mode with samples. Therefore a compact turn-key MISANS setup from the FRM II in Munich [@Georgii:11] was installed at the HFIR in Oak Ridge, Tennessee, USA, at the beamline CG-1D [@CG1]. Setup of the Experiment ======================= ![\[fig:setup\] Sketch of the setup used for the MISANS experiment during the second beam session, viewed from the side. The chopper (Ch) provided a pulsed beam with a frequency of 40Hz and a pulse length of 0.14ms, corresponding to a chopper opening of $2^\circ$. The two polarizers (P$_1$ and P$_2$) were placed before and after the zero field region (hatched) of the MIEZE box, which contained the two MIEZE coils (C$_1$ and C$_2$). Three apertures (A$_1$ to A$_3$) were installed so that the beam is restricted to the sample (S), located inside a cryomagnet. The time-resolved detector (D) was placed at the correct distance to fulfill the MIEZE condition (eq. \[eq:miezecond\]). ](setup.eps){width="\linewidth"} The used setup is shown in Figure \[fig:setup\]: Two sets of MIEZE coils inside -metal shielding [@Arend:04; @Georgii:11] are placed between two polarizers. The polarizers are polarising solid state benders [@Krist:02], which polarise the transmitted beam, but keep the collimation intact [@Bleuel:07]. We used two approximately $10\times10$mm$^2$ apertures (A$_1$ and A$_3$ in Figure \[fig:setup\]) in about 3m distance from each other for a rough collimation and suppression of the reflected spin states from the polarizers. The frequencies in the RF coils ranged between 46 and 149kHz and matched the static fields: $$\omega = 2\pi \cdot f = \gamma B \quad\mathrm{with}\quad \gamma = 2\pi \cdot 2.913 \,\mathrm{kHz/G}.$$ In order to simplify the setup, the coupling coils were replaced by small guide fields and therefore all static fields were either parallel or anti-parallel to the strong magnetisation fields of the polarizers. This simplification of the setup removed the need for an adiabatic rotation of the neutron polarisation in the static fields by 90$^\circ$ as usual in MIEZE experiments and thus allowed a quicker adjustment of the setup in a new environment. The trade-off is that the MIEZE coils can only be operated in a $\pi/2$-mode, since the initial polarization is parallel to the static field of the $B_0$ coils. In this mode, each MIEZE coil must perform a $\pi/2$ flip to achieve the coherent splitting of two spin states that leads to the focused coherent overlap at the MIEZE point [@Ebisawa:04]. In contrast, in the usual MIEZE setup, which is preferred for all new developments, each MIEZE coil induces a $\pi$ flip, and in combination with a “bootstrap” arrangement a factor of four in time resolution is gained in comparison to our $\pi$/2 mode setup. For this time-of-flight MIEZE instrument, the RF amplitude needs to be modulated to match the condition $$\label{eq:rfmod} \gamma B \frac{l}{v} = \frac{\pi}{2} \quad\Rightarrow\quad B(\lambda) = \frac{\pi h}{2 m_n \gamma l} \frac{1}{\lambda},$$ where $h$ is Planck’s constant, $l$ is the RF coil thickness, $v$ the neutron velocity, $m_n$ its mass and $\lambda$ its wavelength. In a time-of-flight instrument, the wavelength of a neutron with time of flight $t$ at one component is given by $\lambda = ht/m_nx$, where $x$ is the component’s distance from the source. As the MIEZE module was originally developed for operation at a reactor source, the RF current was not modulated with the time of flight in the present setup. As detector we used a circular multi-channel-plate prototype [@Tremsin:08], with a diameter of 40mm and only about 15–20% detection efficiency for cold neutrons, but it provided sufficient time resolution and a very good spatial resolution (pixel size about 10m), both of which might become very useful in future tests at higher time resolution. The detector was always positioned according to the MIEZE condition $$\label{eq:miezecond} L_2 = \frac{L_1}{f_2 / f_1 - 1},$$ where $L_1$ is the distance between the coils, $L_2$ the distance between second coil and detector and $f_i$ are the RF frequencies. The chopper at CG-1D has an opening of 2$^\circ$ and was running at 40Hz, which results in a pulse length of 0.14ms. At the first MIEZE coil, positioned 0.6m from the chopper, this amounts to a wavelength spread of $\Delta\lambda/\lambda \approx 15\,\%$ at $\lambda=3.5$Å and 8% at $\lambda = 6.5$Å. At the detector position at 5.07m distance, the $\Delta\lambda/\lambda$ reduced to 1.7% and 0.9%, respectively. Results ======= We report the results from two separate beam sessions, where we implemented slightly different instrumental parameters. The goal in the first beam session was to set up the MIEZE devices for the first time at CG-1D, while the goal in the second beam session was to perform a first sample measurement. First session ------------- ![\[fig:twofreq\] Intensity modulation and resulting contrast for the direct beam wavelength spectrum for two different MIEZE frequencies, $\Delta f = 23$kHz and $\Delta f = 50$kHz. The contrast is calculated by binning the spectrum into 8 bins per oscillation and fitting a sinusoidal signal to the summation of 12 individual oscillations for each point. The black arrows indicate the Bragg absorption edges of Al. Note that for better clarity, the spectrum for $\Delta f = 23$kHz has been shifted by 0.5 on the intensity scale.](two_freq.eps){width="0.85\linewidth"} Figure \[fig:twofreq\] shows the result of the first test, a MIEZE modulation on the time-of-flight spectrum of the chopped beam at CG-1D with two different frequencies. The absorption edges of aluminum (marked with black arrows) and silicon (used in the solid-state polarizers) are clearly visible as dips in the Maxwellian shape of the spectra and were used to calibrate the time-of-flight to wavelength conversion. The RF frequencies were 46/69kHz and 99/149kHz, respectively, with distances $L_1 = 0.9$m and $L_2 = 1.8$m to fulfill the MIEZE condition. For calculating the MIEZE contrast in these and all other spectra, the spectrum is binned into 8 bins per MIEZE oscillation. Each oscillation spans a time-of-flight interval $t = 1/\Delta f$. To improve statistics, 8 to 15 individual oscillations are summed before a sinusoidal signal $$I(n) = B + A\sin(\frac{2\pi}{8}\cdot n + \varphi) \quad\mathrm{with~~}n=1,\dots,8$$ with a fixed phase $\varphi$ is fitted to the spectra. The contrast $A/B$ of the fitted signal is the MIEZE contrast. The maximum contrast in our tests was about 50%, which is most likely due to one of the MIEZE-coil sets having been damaged during the oversea transport causing static stray fields near the RF-coils and therefore interfering with the correct operation of the flippers. Note that for $\Delta f = 50$kHz, the effect of not ramping the RF current becomes clearly visible: the $\pi/2$ flip is achieved for neutrons of 3Å, which means that at the same current neutrons of 6Å will be flipped by $\pi$, which makes the contrast vanish completely in our MIEZE mode. If there was enough intensity at 9Å, one should see a considerable contrast again, in this case making $3\pi/2$ flips in the coils. Second session -------------- All the following measurements were performed in a second beam time, using only one set of RF frequencies of 99/124kHz. The frequency ratio was chosen differently because with eq. (\[eq:miezecond\]) it results in a longer distance between second coil and detector, and allowed us to insert a cryomagnet at the sample position. ![\[fig:amplscan\] Contrast as a function of wavelength in the direct beam for various output voltages applied at the RF generator. The solid lines are guides to the eye. The MIEZE frequency was $\Delta f = 25$kHz. Note that with eq. (\[eq:rfmod\]) half the RF field should flip twice the wavelength; this does not hold for the output voltage due to nonlinearities in the RF amplifier and circuit.](ampl_scan.eps){width="0.85\linewidth"} To elucidate the effect of RF current ramping, figure \[fig:amplscan\] shows the contrast as a function of wavelength in the direct beam for various output voltages in the RF generators. Note that on the one hand, towards very short wavelengths the initial beam polarization decreases rapidly, and therefore the maximum MIEZE contrast decreases. On the other hand, for large wavelengths the effect of stray fields becomes more pronounced, which again influences the maximum contrast. ![\[fig:linescan\] Intensity of the magnetic MnSi peaks as a function of position on the detector for various wavelengths. The detector was moved by 3cm out of the direct beam for fulfilling the Bragg condition and the intensity was summed over 1Å intervals around the given wavelength. The vertical arrows represent a calculation of the expected Bragg positions of the MnSi peak for the mean wavelength of the interval. ](linescan.eps){width="0.85\linewidth"} Without changing the MIEZE setup we moved the detector 3cm out of the direct beam in order to measure the neutrons scattered by the helical magnetic order in a MnSi sample placed inside the cryostat at zero field. MnSi is a weak itinerant ferromagnet that below $T_c = 28.9\,$K forms a magnetic spiral with a pitch $d \sim 180$Å, and magnetic satellite peaks become observable at $q= 2 \pi/d = 0.035\,$Å$^{-1}$ in the $\left\langle1\,1\,1\right\rangle$ directions [@Lebech:95]. We aligned the sample such that one of the peaks falls onto the detector in the small-angle scattering condition: see Figure \[fig:linescan\]. The vertical arrows on the plot are the calculated positions for the Bragg peak from the helical order using $$q \approx \frac{2\pi}{\lambda} \sin(2\vartheta) \quad\Rightarrow\quad 2\vartheta \approx \sin^{-1}\left( \frac{qht}{2\pi m_nL_0} \right),$$ with the position of the sample peak $q = 0.035$Å$^{-1}$, the Planck constant $h$, the neutron mass $m_n$, the time of flight $t$ between the chopper and the detector for a distance $L_0=5.066$m. The peak intensity moves further away from the direct beam for longer neutron wavelengths in order to maintain Bragg’s law. ![\[fig:mnsi\] The intermediate scattering function $S(q,\tau)$ of the MIEZE signal from MnSi at $q=0.035$Å$^{-1}$ is shown for $T=27.55$K as measured (blue) and after normalization (black) with the instrumental resolution function as determined at $T=2$K (green). The spin-echo times are calculated according to eq. (\[eq:miezetime\]).](mnsi.eps){width="0.85\linewidth"} For the data analysis, the part of the direct beam still visible on the detector has been subtracted in the form of a measurement at high temperatures, where no magnetic signal is present. The blue curve in Figure \[fig:mnsi\] shows the contrast measured in the neutron beam scattered by the MnSi sample at $T=27.55$K at the Bragg peak at $q=0.035$Å$^{-1}$ for different spin-echo times, corresponding to different wavelengths in the incoming beam according to $$\label{eq:miezetime} \tau_{\mathrm{MIEZE}} = \frac{m_n^2}{h^2}\, \Delta f\, L_S\, \lambda^3,$$ with the sample–detector distance $L_S=1.035$m. To obtain the intermediate scattering function $S(q,\tau)$, shown as the black curve in Figure \[fig:mnsi\], this data was normalized to the contrast measured analogously at $T=2$K (the green curve), serving as the reference measurement of the instrumental resolution [@Georgii:11]. It is expected that this $S(q,\tau)$ is equal to one, since the sample dynamics below $T_c$ are too slow to be observed at the present time resolution at the Bragg peak [@Georgii:11; @Pappas:09]. To finally assess the field compatibility of the technique beyond what had previously been demonstrated, we moved the detector back into the direct beam and looked at the contrast of the MIEZE signal as a function of a horizontal magnetic field at the sample position. ![\[fig:bfield\] Contrast of the MIEZE signal in the direct beam for various magnetic fields at the sample position. The small loss of $C$ with increasing field is due to the missing coupling coils and the insufficient -metal shielding. ](bfield.eps){width="0.85\linewidth"} Figure \[fig:bfield\] confirms that the MIEZE technique works with unshielded horizontal magnetic fields of up to 1T. The observed drop in contrast to about 70% of its zero-field value we attribute to the fact that we performed this experiment without coupling coils and with a low-quality -metal shielding around the MIEZE coils. Nevertheless the results show that MIEZE works even under unfavourable stray field conditions. Concluding Remarks ================== The effort to establish MISANS as a measurement technique for TOF applications is on the way and this paper documents important results along the path: the experiment establishes that MISANS can measure samples on pulsed neutron beamlines and the measurement of a MIEZE-signal in the direct beam with a 1T horizontal field at the sample region demonstrates that this is a most promising technique for measurements of slow dynamics in high magnetic fields. In the future, the setup could be extended to develop a MIEZE insert available as a standard option at several instruments at the SNS or at a dedicated MIEZE-SANS instrument at HFIR. It is also planned to propose a MISANS-type instrument for the European Spallation Source (ESS). Here, the instrument can benefit from the extended wavelength range, since the relaxed requirements on wavelength resolution typical for SANS are also acceptable for MIEZE-SANS. This means that similar gain factors can be expected. Acknowledgements ================ We acknowledge the support of Ian S. Anderson for making these tests possible at HFIR. We are grateful for the help of Mike Fleenor and the staff at HFIR during the beam times. We also acknowledge very helpful discussions with Wolfgang Häußler and the technical support of Reinhard Schwikowski at the FRM II. This work was funded by ONRL, the U.S. Department of Energy, BES-Materials Science, under Contract DE-AC02-06CH117, and by the German BMBF under “Mitwirkung der Zentren der Helmholtz Gemeinschaft und der Technischen Universität München an der Design-Update Phase der ESS, Förderkennzeichen 05E10WO1.” [15]{} \[1\][\#1]{} \[1\][`#1`]{} urlstyle \[1\][doi:\#1]{} \[1\][doi:]{}\[2\][\#2]{} , , , , () , ISSN , . , , , , , () , . , , Ph.D. thesis, , . , , , , , , , , , () , . , , , , , , () , . , , , , , , ,  () () , ISSN , . , , , , , , ,  () () , . , <http://neutrons.ornl.gov/instruments/HFIR/CG1/>, . , , , , , , ,  () () , ISSN , . , , , , () , . , , , , , ,  () () , . , , , , , , , , ,  () () , . , , , , , , , ,  () () , . , , , , , , , , , () . , , , , , , , , ,  () () , .

--- abstract: 'In this paper, we compare three methods to reconstruct galaxy cluster density fields with weak lensing data. The first method called FLens integrates an inpainting concept to invert the shear field with possible gaps, and a multi-scale entropy denoising procedure to remove the noise contained in the final reconstruction, that arises mostly from the random intrinsic shape of the galaxies. The second and third methods are based on a model of the density field made of a multi-scale grid of radial basis functions. In one case, the model parameters are computed with a linear inversion involving a singular value decomposition. In the other case, the model parameters are estimated using a Bayesian MCMC optimization implemented in the lensing software Lenstool. Methods are compared on simulated data with varying galaxy density fields. We pay particular attention to the errors estimated with resampling. We find the multi-scale grid model optimized with MCMC to provide the best results, but at high computational cost, especially when considering resampling. The SVD method is much faster but yields noisy maps, although this can be mitigated with resampling. The FLens method is a good compromise with fast computation, high signal to noise reconstruction, but lower resolution maps. All three methods are applied to the MACS J0717+3745 galaxy cluster field, and reveal the filamentary structure discovered in @ejlens:jauzac12. We conclude that sensitive priors can help to get high signal to noise, and unbiased reconstructions.' author: - | E. Jullo$^1$, S. Pires$^2$, M. Jauzac$^3$ & J.-P. Kneib$^{4,1}$\ $^1$Aix Marseille Université, CNRS, LAM (Laboratoire d’Astrophysique de Marseille) UMR 7326, 13388, Marseille, France\ $^2$Laboratoire AIM, CEA/DSM-CNRS, Université Paris 7 Diderot, IRFU/SAp-SEDI, Service d’Astrophysique, CEA Saclay, Orme des\ Merisiers, 91191 Gif-sur-Yvette, France\ $^3$Astrophysics and Cosmology Research Unit, School of Mathematical Sciences, University of KwaZulu-Natal, Durban 4041, South Africa\ $^4$LASTRO, Ecole polytechnique fédérale de Lausanne, Suisse bibliography: - 'ejlens.bib' date: Released 2013 Xxxxx XX title: Weak Lensing Galaxy Cluster Field Reconstruction --- \[firstpage\] Introduction ============ Galaxy redshift surveys such as SDSS @ejlens:york00 and N-body simulations of cosmic structure formation (for example the Millenium simulations @ejlens:springel05) have revealed a complicated network of matter, in which massive galaxy clusters are located at the nodes, filaments connect them to each others, and in-between extended regions with few galaxies and matter called voids fill about 80% of the volume of the Universe [@ejlens:pan12; @ejlens:bos12]. Galaxy clusters are of considerable cosmological interest, as they are the most recent structures to have formed at the largest angular scales. Taking advantage of this specificity, several cluster-related cosmological probes have been developed either based on cluster count statistics [@ejlens:berge08; @ejlens:pires09; @ejlens:shan12] or on the study of their physical properties (e.g. triaxialityÊ@ejlens:morandi11, bulleticity @ejlens:massey11 or gas mass fraction @ejlens:rapetti10). Filamentary structures surrounding galaxy clusters also happen to be of particular interest. On the one hand, they reveal cosmological voids and alike cluster count statistics, void number counts and sizes are effective cosmological probes (@ejlens:davis11 [@ejlens:higuchi12; @ejlens:krause13]). On the other hand, filaments funnel matter onto the galaxy clusters, and as such they play an important role in cluster and galaxy formation. Lensing has recently demonstrated its effectiveness at mapping filaments. For instances, @ejlens:heymans08 has uncovered a filamentary structure between the pair of clusters 901 and 902. In their analysis of the double cluster system Abell 222 and Abell 223, @ejlens:dietrich12 showed evidence for a possible dark matter filament connecting both clusters. Finally, in the COSMOS field [@ejlens:scoville07], @ejlens:massey07 uncovered a massive large-scale structure at redshift $z\sim0.73$ extending over about 1 degree in length. Recently, @ejlens:jauzac12 claimed another detection of a large-scale filament connected on one end to the massive cluster MACS J0717+3745, and vanishing into the cosmic web on the other end. They used a model made of a multi-scale grid of radial basis functions (RBF) and a Bayesian MCMC optimization algorithm implemented in the lensing software Lenstool to map its mass distribution and measure its size and density. In this paper, we study three methods of lensing map reconstruction, including the method used in @ejlens:jauzac12. The first method called FLens integrates an inpainting concept to invert the shear field with possible gaps, and a multi-scale entropy denoising procedure to remove the noise contained in the galaxies. The second and third method are based on the same model of multi-scale grid of RBFs, but in one case the parameters are estimated with Lenstool, and in the other case with a linear matrix inversion involving a singular value decomposition. We use simulated data and compare the reconstructed maps in terms of fidelity to the input map, sensitivity to the density of galaxies in the input weak lensing catalog. We also pay particular attention to the errors estimated either directly from the MCMC samples or the linear inversion theory, and the errors estimated with resampling. The outline of the paper is the following. In §\[sec:method\], we review the formalism of the different techniques. In §\[sec:simus\], we use simulations to compare the methods, focusing successively on the reconstructing maps, azimuthally averaged density profiles, errors and signal to noise maps. Finally in §\[sec:macs0717\], we compare the reconstructions obtained with the different methods applied to real data coming from HST observations of the massive galaxy cluster MACS J0717+3745. Throughout this paper, we compute cosmological distances to lensed galaxies assuming the Universe is flat and described by the $\Lambda$CDM model with $\Omega_mÊ= 0.3$ and $w = -1$. Methods {#sec:method} ======= Weak Lensing formalism ---------------------- Gravitational lensing i.e. the process by which light from distant galaxies is bent by the gravity of intervening mass in the Universe, is an ideal tool for mapping the mass distribution of lensed structures because it depends on the total matter distribution of the intervening structures. In lensing, the spin-2 shear field ${\bf \gamma_i}(\thetav)$ that is derived from the shapes of observed background galaxies, can be written in terms of the intervening lensing gravitational potential $\psi(\thetav)$ projected on the sky [@ejlens:bartelmann01]: $$\label{gamma} \begin{array}{l} \gamma_1(\theta)= \frac{1}{2}(\partial_1^2-\partial_2^2)\psi(\thetav)\\ \gamma_2(\thetav)=\partial_1\partial_2\psi(\thetav), \end{array}$$ where the partial derivatives $\partial_i$ are with respect to $\theta_i$. The convergence $\kappa(\thetav)$ can also be expressed in terms of the lensing potential $\psi(\thetav)$, $$\label{kappa} \kappa(\thetav)=\frac{1}{2}(\partial_1^2 + \partial_2^2) \psi(\thetav),$$ and is related to the mass density $\Sigma(\theta)$ projected along the line of sight by $$\label{sigma1} \kappa(\thetav) = \frac{\Sigma(\thetav)}{\Sigma_{crit}},$$ where the critical mass density $\Sigma_{crit}$ is given by $$\label{sigma2} \Sigma_{crit}=\frac{c^2}{4\pi G}\frac{D_{OS}}{D_{OL}D_{LS}},$$ where $G$ is Newton’s constant, $c$ the speed of light, and $D_{OS}$, $D_{OL}$, and $D_{LS}$ are the angular-diameter distances between the observer (O), the lens (L), and a galaxy source (S) at an arbitrary redshift. A new inverse method -------------------- If the shear field could be measured everywhere, the convergence field could be determined without error. In reality, we only have access to an estimator of the shear field at the random discrete locations of the background galaxies. The shear information is contained in the observed ellipticity of the background galaxies, but is overwhelmed by the intrinsic galaxy own ellipticity. Fortunately, we can assume that this intrinsic shape noise is random and Gaussian distributed. Therefore we can compute an unbiased estimate of the shear by binning the galaxies in a grid and average their ellipticities. ### The Kaiser & Squires inversion The weak lensing mass inversion problem consists in reconstructing the projected (normalized) mass distribution $\kappa(\theta)$ from the measured shear field $\gamma_i(\theta)$ in a grid. We invert Eq. (\[gamma\]) to find the lensing potential $\psi$ and then apply formula Eq. (\[kappa\]) to obtained $\kappa(\theta)$. This classical method is based on the pioneering work of @ejlens:kaiser93 [KS93]. In short, this corresponds to : $$\begin{aligned} \label{eqn_reckE} \tilde \kappa & = & \Delta^{-1}\left((\partial_1^2 - \partial_2^2) \gamma_1+ 2 \partial_1\partial_2 \gamma_2\right) \nonumber \\ & = & \frac{\partial_1^2 - \partial_2^2}{\partial_1^2 + \partial_2^2} \gamma_1+ \frac{2 \partial_1\partial_2}{\partial_1^2 + \partial_2^2}\gamma_2.\end{aligned}$$ Taking the Fourier transform of these equations, we obtain $$\hat{\kappa} = \hat P_1 \hat{\gamma_1} + \hat P_2 \hat{\gamma_2},$$ where the hat symbol denotes Fourier transforms and we have defined $k^2 \equiv k_1^2 + k_2^2$ and $$\begin{aligned} \hat{P_1}(\mathbf k) & = & \frac{k_1^2 - k_2^2}{k^2} \nonumber \\ \hat{P_2}(\mathbf k) & = & \frac{2 k_1 k_2}{k^2},\end{aligned}$$ with $\hat{P_1}(k_1,k_2) \equiv 0$ when $k_1^2 = k_2^2$, and $\hat{P_2}(k_1,k_2) \equiv 0$ when $k_1 = 0$ or $k_2 = 0$. Note that to recover $\kappa$ from both $\gamma_1$ and $\gamma_2$, there is a degeneracy when $k_1 = k_2 = 0$. Therefore, the mean value of $\kappa$ cannot be recovered from the shear maps. This is known as the mass-sheet degeneracy. This problem can be solved with additional information such as lensing magnification measurements for instance. In reality, the measured shear is noisy because only a finite number of galaxy ellipticities are averaged per pixel. The actual relation between the measured shear $\gamma_{ib}$ in pixel $b$ of area $A$ and the true convergence $\kappa$ is $$\label{eq_gamma} \gamma_{ib} = P_i * \kappa + n_i\;,$$ where the intrinsic galaxy shape noise contribution $n_i$ is Gaussian distributed with zero mean and width $\sigma_n \simeq \sigma_\epsilon/\sqrt{N_g}$. The average number of galaxies in a pixel $N_g = n_g\ A$ depends on the the average number of galaxies per square arcminute $n_g$. The ellipticity dispersion per galaxy $\sigma_\epsilon$ arises both from measurement errors and the dispersion in the intrinsic shape of galaxies. From the central limit theorem, we can assume to a good approximation that with $n_g \simeq 10$ galaxies per square arcminute, in pixels with area $A \gtrsim 1$ square arcminute the noise $n_i$ is Gaussian distributed and uncorrelated. The most important drawback of the KS93 method is that it requires a convolution of shears to be performed over the entire sky. As a result, if the field is small or irregularly-shaped, then the method can produce artifacts in the reconstructed matter distribution near the boundaries. ### The Seitz & Schneider inversion In @ejlens:seitz96, the authors propose a local inversion method that reduces these unwanted boundary effects. The convergence $\kappa$ is computed in real space (without Fourier transform) thanks to the kernel integration $$\kappa(\theta)-\kappa_0=\frac{1}{\pi}\int_{\theta' \in \Omega}K(\theta - \theta')\cdot {\bf \gamma}(\theta')\, d\theta',$$ where $\kappa_0$ stands for the mean value of $\kappa$. The kernel $K$ depends on the geometry of the domain $\Omega$. For $\Omega=\mathbb{R}^2$, it is given by $$K(\theta)=\left(\frac{\theta_2^2-\theta_1^2}{(\theta_1^2+\theta_2^2)^2} , \frac{-2\theta_1\theta_2}{(\theta_1^2+\theta_2^2)^2}\right) .$$ where we expressed the positions in complex coordinates $\theta = \theta_1 + i\theta_2$. For small irregularly-shaped fields, the authors propose to combine the derivatives of ${\gamma_i}$ $$\u=\left(\begin{array}{l} \partial_1\gamma_1+\partial_2\gamma_2 \\ \partial_1\gamma_2-\partial_2\gamma_1 \end{array}\right),$$ and then to apply the Helmholtz decomposition $\u=\nabla\kappa^{(E)}+\nabla\times\kappa^{(B)}$, in order to reconstruct the convergence $\kappa=\kappa^{(E)}$. This method reduces the unwanted boundary effects but whatever the formula, the reconstructed field is more noisy than that one obtained with a global inversion. Another point is that the reconstructed dark matter mass map still has a complex geometry that will complicate subsequent analyses. ### The FLens method {#sec:flens} [**Binning the shape catalogue**]{}\ As said previously, the shape catalogue is first binned into a regular grid, in which each pixel value is obtained by averaging the ellipticity of the galaxies it contains. The pixel size is a parameter defined by hand, so that all (or almost all) pixels contain at least one galaxy. Not doing so usually prevents mass inversion because of missing data. In general, the pixel size is adjusted to have about 10 galaxies per pixel. If we were having a method to deal with this missing data issue, there would be no particular limitation on the pixel size. However the increasing number of empty pixels would make the mass inversion step always more difficult. Ideally, it would be preferable to have about one galaxy per pixel on average. [**Dealing with missing data**]{}\ Missing data are common practice in weak lensing. They can be due to camera CCD defects, or bright stars that saturate the field of view. More specifically to cluster field reconstruction, the galaxies inside the Einstein radius are usually removed from the study because the weak lensing approximation does not hold there. In addition, depending on the pixel size and the regularity of the galaxy distribution, the amount of empty pixels can increase dramatically. As a result, the measured shear field is generally incomplete and the gaps in the data require proper handling. A solution that has been proposed by [@ejlens:pires09] to deal with missing data consists in filling-in judiciously the masked regions by performing an *inpainting* method simultaneously with a global inversion. Inpainting techniques are an extrapolation of the missing information using some priors on the solution. This new method uses a prior of sparsity in the solution introduced by [@ejlens:elad05]. It assumes that there exists a dictionary $\mathcal{D}$ (here the Discrete Cosine Transform) where the complete data are sparse and where the incomplete data are less sparse. The weak lensing inpainting problem consists of recovering a complete convergence map $\kappa$ from the incomplete measured shear field $\gamma_i^{obs}$. The solution is obtained by minimizing $$\min_{\kappa} \| \mathcal{D}^T \kappa \|_0 \textrm{ subject to } \sum_i \parallel \gamma_i^{obs} - M (P_i * \kappa) \parallel^2 \le \sigma,$$ noting $|| z ||_0$ the $l_0$ pseudo-norm, i.e. the number of non-zero entries in $z$ and $|| z ||$ the classical $l_2$ norm (i.e. $|| z || =\sum_k (z_k)^2$), where $\sigma$ stands for the standard deviation of the input shear map, and $M$ is the binary mask (i.e. $M_i = 1$ if we have information at pixel $i$, $M_i = 0$ otherwise).($\sigma=0$ is only used for noiseless data). If $\mathcal{D}^T \kappa$ is sparse enough, the $l_0$ pseudo-norm can also be replaced by the convex $l_1$ norm (i.e. $ || z ||_1 = \sum_k | z_k | $) [@ejlens:donoho_01]. The solution of such an optimization task can be obtained through an iterative thresholding algorithm called MCA [@ejlens:elad05] starting from the noisy $\kappa_0$ obtained with the KS93 method $$\kappa_{i+1} = \Delta_{\mathcal{D},\lambda_n}\left(\kappa_i + M[P_1*(\gamma_1^{obs}-P_1*\kappa_i)+P_2 * (\gamma_2^{obs}-P_2*\kappa_i)]\right), \label{eqn_mca}$$ where the nonlinear operator $\Delta_{\mathcal{D},\lambda}(Z)$ consists in: - decomposing the signal $Z$ on the dictionary $\mathcal{D}$ to derive the coefficients $\alpha = \mathcal{D}^T Z$. - threshold the coefficients with a hard-thresholding (${\tilde \alpha} = \alpha_i$ if $ | \alpha_i | > \lambda_i$ and $0$ otherwise). The threshold parameter $\lambda_i$ decreases with the iteration $i$. - reconstruct $\tilde Z$ from the thresholded coefficients ${\tilde \alpha}$. This method enables to reconstruct a complete convergence map $\kappa_n$. ![Illustration of the filtering of a raw in-painted convergence map with FLens.[]{data-label="fig:filter"}](filter){width="\linewidth"} However, this convergence map $\kappa_n$ obtained by inversion of the shear field is very noisy as shown in the left panel of Fig \[fig:filter\]. This noise originates from the shear measurement errors and the intrinsic galaxy shape noise, and grows inversely proportional to the number of galaxies per pixel.\ [**Dealing with noise in the Cluster reconstruction**]{}\ In this study, we use the MRLens (Multi-Resolution for weak Lensing) denoising method to denoise the reconstructed convergence map $\kappa$. The MRLens filter is based on the Bayesian theory that considers that some prior information can be used to improve the solution [@ejlens:starck06]. Bayesian filters search for a solution that maximizes the posterior probability $P(\kappa|\kappa_n)$ defined by the Bayes theorem : $$\begin{aligned} P(\kappa|\kappa_n)=\frac{P(\kappa_n|\kappa)\ P(\kappa)}{P(\kappa_n)}, \label{bayes}\end{aligned}$$ where : - $P(\kappa_n|\kappa)$ is the likelihood of obtaining the data $\kappa_n$ given a particular convergence distribution $\kappa$. - $P(\kappa_n)$ is the probability of having the data $\kappa_n$. This term, called evidence, is simply a constant that ensures that the posterior probability is correctly normalized. - $P(\kappa)$ is the prior probability of the estimated convergence map $\kappa$. This term codifies our expectations about the convergence distribution before acquisition of the data $\kappa_n$. Searching for a solution that maximizes posterior probability $P(\kappa|\kappa_n)$ is the same as searching for a solution that minimizes the following quantity $$\begin{aligned} \mathcal{Q} &=& - \log(P(\kappa|\kappa_n)), \\ \mathcal{Q}&=& - \log(P(\kappa_n|\kappa)) - \log(P(\kappa)). \label{qteinfo1}\end{aligned}$$ If the noise is uncorrelated and follows a Gaussian distribution, the likelihood term $P(\kappa_n|\kappa)$ can be written $$\begin{aligned} P(\kappa_n|\kappa) \propto \exp\ -\frac{\chi^2}{2}, \label{vraisemblance2}\end{aligned}$$ with the sum of squares of the residuals $$\begin{aligned} \chi^2 = \sum_{x,y} \frac{(\kappa_n(x,y)-\kappa(x,y))^2}{\sigma^2_{\kappa_n}}. \label{vraisemblance3}\end{aligned}$$ Eq \[qteinfo1\] can then be expressed as $$\begin{aligned} \mathcal{Q} = \frac{1}{2} \chi^2 - \log(P(\kappa)) = \frac{1}{2} \chi^2 - \beta H, \label{qteinfo2}\end{aligned}$$ where $\beta$ is a constant that can be seen as a parameter of regularization and $H$ represents the prior that is added to the solution. If we have no expectation about the distribution of the convergence field $\kappa$, the prior probability $P(\kappa)$ is uniform and searching for the maximum of the posterior $P(\kappa|\kappa_n)$ is equivalent to the well-known maximum likelihood search. This maximum likelihood method has been used by @ejlens:bartelmann96 and @ejlens:seljak98 to reconstruct weak lensing fields, but the solution has to be regularized in some way to prevent overfitting of the data. Choosing the prior is one of the most critical aspect in Bayesian analysis. An Entropic prior is frequently used but there are many definitions for Entropy [see @ejlens:gull84]. One currently in use is the Maximum Entropy Method (MEM) @ejlens:bridle98. A multi-scale maximum entropy prior has also been proposed by [@ejlens:marshall02] which uses the intrinsic correlation functions (ICF) with varying width. The MRLens filtering uses a prior based on the sparse representation of the data that consists in replacing the standard Entropy prior by a wavelet based prior [@ejlens:pantin96] . The entropy is now defined by $$H(I) = \sum_{j=1}^{J-1} \sum_{k,l} h(w_{j,k,l})\;,$$ where $J$ is the number of wavelet scales, and we set $\beta = 1$ in Eq. \[qteinfo2\]. In this approach, the information content of an image $I$ is viewed as sum of information at different scales $w_{j}$. The function $h$ defines the amount of information relative to a given wavelet coefficient [see @ejlens:starck06 for details on the choice of this function]. In [@ejlens:pantin96], it has been suggested to not apply the regularization on wavelet coefficients which are clearly detected (i.e. significant wavelet coefficients). The multi-scale entropy then becomes $$\begin{aligned} h_n(w_{j,k,l}) = {\bar M}(j,k,l) h(w_{j,k,l}) \end{aligned}$$ where ${\bar M}(j,k,l) = 1 - M(j,k,l)$, and $M$ is the multiresolution support [@ejlens:mur95_2]: $$\begin{aligned} M(j,k,l) = \left\{ \begin{array}{ll} \mbox{ 1 } & \mbox{ if } w_{j,k,l} \mbox{ is significant} \\ \mbox{ 0 } & \mbox{ if } w_{j,k,l} \mbox{ is not significant} \end{array} \right. \end{aligned}$$ This describes, in a Boolean way, whether the data contains information at a given scale $j$ and at a given position $(k,l)$. Commonly, in the case of Gaussian noise, $w_{j,k,l}$ is said to be significant if $| w_{j,k,l} | > k\sigma_j$, where $\sigma_j$ is the noise standard deviation at scale $j$, and $k$ is a constant, generally taken between 3 and 5. The False Discovery Rate method (FDR) offers an effective way to select this constant $k$ [@ejlens:benjamini95; @ejlens:miller01; @ejlens:hopkins02]. The FDR defined as the ratio $$\begin{aligned} FDR = \frac{V}{D}\end{aligned}$$ where $V$ is the number of pixels erroneously identified as pixels with signal, and $D$ is the number of pixels identified as pixels with signal, both truly and erroneously. This method requires to fix a rate $\alpha$ between 0 and 1. And it ensures that [*on average*]{}, the FDR will not be bigger than $\alpha$ $$\begin{aligned} E(FDR) \leq \frac{T}{V}.\alpha \leq \alpha\end{aligned}$$ The unknown factor $\frac{T}{V}$ is the proportion of truly noisy pixels. A complete description of the FDR method can be found in [@ejlens:miller01]. Here we apply the FDR method at each wavelet scale, which gives us a detection threshold $T_j$ per scale. We then consider a wavelet coefficient $w_{j,k,l}$ as significant if its absolute value is larger than $T_j$. This procedure is totally different from a $k\sigma$ thresholding, that only controls the ratio between the number of pixels erroneously identified over the total number of pixels in the map. The proposed filter called MRLens (Multi-Resolution for weak Lensing[^1]) outperforms other techniques (Gaussian, Wiener, MEM, MEM-ICF) in the reconstruction of dark matter. For this reason, it has also been used to reconstruct the dark matter mass map from the Hubble Space Telescope in the COSMOS field [@ejlens:massey07].\ [**Dealing with reduced shear**]{} In practice, the observed galaxy ellipticities, however, are induced not by the shear $\gamma$ but by the reduced shear $$g = \frac{\gamma}{1-\kappa}. \label{eq:redshear}$$ The distinction between the true and the reduced shear is negligible in the weak shear regime ($\kappa \approx 0$). However in galaxy cluster fields, as we focus on in the work, the weak shear regime is not perfectly satisfied, and the discrepancy in the reconstructions can be as high as 10 % if the reduced shear is not properly taken into account. In order to recover the true shear from the measured reduced shear, we consider an iterative algorithm. At the first iteration, we assume that the true shear is equal to the reduced shear. Then a convergence map is derived, and used along with Eq \[eq:redshear\] to compute a more accurate true shear for the next iteration. We found this procedure to effectively correct for the bias in the reconstruction, but found no improvement after three iterations. The multi-scale grid model -------------------------- ### RBF Model description Radial Basis Functions (RBFs) are commonly used to solve interpolation problems [see e.g. @ejlens:gentile12]. Let us consider an unknown function $f: \mathbb{R}^n \to \mathbb{R}$ probed at a set of locations $\xi \in \mathbb{R}^n$, and approximated by a function $s : \mathbb{R}^n \to \mathbb{R}$, a linear combination of translates of a set of RBFs $\phi_i$ $$\label{eq:rbf} s(\bold{x}) = \sum \lambda_i\ \phi_i(|| \cdot - \bold{x}||)\,.$$ with unknown real coefficients $\lambda_i$. Those coefficients are obtained by solving the linear system $f(\xi) = s(\xi)$. A unique solution exists if there are as many RBFs as data points and the RBF profiles are positive definite [@ejlens:buhmann03]. However in our case, since data points are noisy and we want to avoid overfitting, we arbitrarily restrict the number of RBFs to a few, thus practically compressing the data to a smaller basis set. In @ejlens:jullo09, we found that RBFs distributed on a hexagonal grid, and described by a Truncated Isothermal Mass Distribution (TIMD) [see e.g. @ejlens:kassiola93; @ejlens:kneib96; @ejlens:eliasdottir09] were giving good results. In our model, we approximate the true convergence field $\kappa$ with $$\label{eq:conv} \kappa(\theta) = \frac{1}{\Sigma_{crit}} \sum_i \sigma_i^2\ f(\ || \theta_i -\ \theta\ ||,\ s_i,\ t_i)$$ where the RBFs on grid nodes $\theta_i$ are described by $$\textstyle \label{eq:timd} f(R,s, t) = \frac{1}{2G} \frac{r_{cut}}{t- s} \left( \frac{1}{\sqrt{s^2 + R^2}} - \frac{1}{\sqrt{t^2 + R^2}} \right).$$ In the TIMD model, the scaling factor $\sigma_i^2$ is the velocity dispersion at the centre of the gravitational potential, and radii $s$ and $t$ mark 2 changes in the slope respectively from $\kappa \propto R^0$ to $\kappa \propto R^{-1}$ and $\kappa \propto R^{-3}$ respectively. In a similar manner, we approximate the true shear field with $$\begin{aligned} \gamma_1(\theta) = \sum \sigma_i^2\ \Gamma_1(\ || \theta_i -\ \theta\ ||,\ s_i,\ t_i)\\ \gamma_2(\theta) = \sum \sigma_i^2\ \Gamma_2(\ || \theta_i -\ \theta\ ||,\ s_i,\ t_i)\end{aligned}$$ where analytical expressions also exist for $\Gamma_1$ and $\Gamma_2$ [see Eq A8 in @ejlens:eliasdottir09]. Let us now consider a set of $M$ ellipticity measurements ordered in a vector $\bold{e} = [\bold{e_1},\ \bold{e_2}]^\dagger$, and a model made of $N$ RBFs distributed in the field with unknown weights $\sigma_i^2$ ordered in a vector $\bold{v} = [\sigma_1^2, \ldots, \sigma_N^2]$. In the weak lensing approximation, we can write the linear relation $$\label{eq:shearmat} \bold{e} = M_{\gamma v} \bold{v} + \bold{n}\;,$$ where $\bold{n}$ is the galaxy shape noise as in Eq \[eq\_gamma\], and the transform matrix $M_{\gamma v} = \left[ \Delta_1, \Delta_2 \right]^\dagger$ is a block-2 matrix. Its individual elements are the contribution of each unweighted RBF scaled by a ratio of angular diameter distances $$\begin{aligned} \label{eq:dshear1} \Delta_{1}^{(j,i)} &= &\frac{D_{LSi}}{D_{OSi}}\ \Gamma_{1}^i(|| \theta_i - \theta_j ||,\ s_i,\ t_i) , \\ \Delta_{2}^{(j,i)} &= &\frac{D_{LSi}}{D_{OSi}}\ \Gamma_{2}^i(|| \theta_i - \theta_j ||,\ s_i,\ t_i) . \end{aligned}$$ where subscript $j \inÊ[1, M]$ and $i\in [1, N]$ denote the rows and the columns of $M_{\gamma \nu}$ respectively. ### Comparison of TIMD and Gaussian filters ![Comparison between the TIMD profiles in convergence and shear spaces. In dashed-line, we also show the best-fit Gaussian profiles. The bottom panel shows that the TIMD profile in shear space is systematically broader than its equivalent in convergence space, in comparison to a self-similar Gaussian filter. []{data-label="fig:piemdgauss"}](piemd_gauss){width="\linewidth"} By construction, we use the same parameters for the RBFs ($\sigma^2_i,\ s_i,\ t_i$) in the convergence and shear spaces. However, the corresponding functions $f$, $\Gamma_1$ and $\Gamma_2$ have different profiles in these two spaces. In Fig \[fig:piemdgauss\], we actually show that the TIMD filter is sharper in convergence space than in shear space. In practice, this makes the TIMD filter very efficient at picking shear information far away for a given RBF, and concentrate it to produce high resolution convergence maps. For example from Fig \[fig:piemdgauss\] we see that if we use a TIMD filter of core radius $s= 20"$ (equivalent to a Gaussian filter of width $\sigma \simeq 30"$ in shear space), the reconstructed convergence field is smoothed similarly as with a Gaussian filter of width $\sigma \simeq 22"$. In contrast with the standard KS93 method, the size of the Gaussian filter is the same in shear and convergence space. ### Estimation of the RBFs weights [**Linear SVD inversion method**]{} Assuming the galaxy shape noise $\bold{n}$ is Gaussian distributed, we can write the sum of the squares of the residuals $$\label{eq:chi2} \chi^2 = (\bold{e} - 2 M_{\gamma v} \bold{v})^\dagger N_{e e}^{-1} (\bold{e} - 2 M_{\gamma v} \bold{v}),$$ where $N_{e e} \equiv\ < e e^\dagger >$ is the covariance matrix of the measured ellipticities. In this work, we assume this matrix is diagonal and its elements are $N_{e e}^{(i,j)} =(\sigma_m^2 + \sigma_{int}^2 )\ \delta_{ij}$ where $\delta_{ij}$ is the Kronecker symbol, $\sigma_m$ is the measurement uncertainty and $\sigma_{int}$ is the scatter in the distribution of the intrinsic shapes of the galaxies. Note also that we have a factor of $2$ in this equation because in Lenstool the ellipticity $e = \frac{a^2 - b^2}{a^2 + b^2}$ is computed as a function of the square of the major and minor axes [@ejlens:bartelmann01]. With Gaussian distributed errors, linear inversion theory tells us that an unbiased estimator of the RBF weights is $$\tilde{\bold{v}} = \left[ M_{\gamma v}^\dagger N_{e e}^{-1} M_{\gamma v} \right]^{-1} M_{\gamma v}^\dagger N_{e e}^{-1} \bold{e}$$ and their covariance is $$\label{eq:nvv} N_{v v} = \left[ M_{\gamma v}^\dagger N_{e e}^{-1} M_{\gamma v} \right]^{-1}$$ The convergence field is obtained by the matrix product $$\tilde{\kappa} = M_{\kappa v}\, \tilde{v}$$ and the corresponding covariance matrix $N_{\tilde{\kappa} \tilde{\kappa}}$ by $$\label{eq:svdconv} N_{\tilde{\kappa}Ê\tilde{ \kappa}} = M_{\kappa v} N_{v v} M_{\kappa v}^\dagger\;.$$ where the transform matrix $M_{\kappa v}$ is built from Eq \[eq:conv\] and \[eq:timd\]. In the following, we reconstruct the convergence field in grids of regularly spaced pixels. There are several ways of speeding the calculations in the expressions above. In particular, it happens that in our case, the transform matrix $ M_{\gamma v}$ is sufficiently sparse so that we can perform a singular value decomposition (SVD). Details of the SVD decomposition can be found in [@ejlens:vanderplas11; @ejlens:diego05].\ [**Bayesian MCMC optimisation**]{} ![Impact of different user defined nuisance parameters on the Lenstool reconstruction of a simulated convergence map. Parameter $q_0$ has the strongest impact on the reconstruction result. These reconstructions are without shape noise, and with a multi-scale grid of 575 RBFs. []{data-label="fig:alphacomp"}](alphacomp){width="\linewidth"} In this section, we describe the Bayesian Monte Carlo Markov Chain algorithm used to reconstruct the mass map in @ejlens:jauzac12. This algorithm called MassInf is also part of the Bayesys package [@ejlens:jullo07], but it is the first time we use it in Lenstool[^2]. It aims at inverting linear systems of equations in a Bayesian manner, i.e. with input priors. Based on our definition of the $\chi^2$ in Eq \[eq:chi2\], we define the likelihood of having a set of weights $\bold{v}$ given the measured ellipticities $\bold{e}$ as $$\label{eq:lhood} P(\bold{v}\, |\,Ê\bold{e}) = \frac{1}{Z_{L}} \exp{- \frac{ \chi^2}{2}}.$$ The normalization factor is given by $Z_{L} = \sqrt{(2 \pi)^{2M} \det N_{e e}}$. As a prior, we want the individual weights $\sigma_i^2$ to be positive, so that the final mass map is positive everywhere. This conducted us to assume they are described by a Poisson probability distribution function (pdf) $$\mathrm{Pr}(\sigma_i^2) = \exp(-\sigma_i^2 / q) / q\,,$$ where the normalization factor $q$ is a nuisance parameter with a pdf given by the following expression $$\pi(q) = q_0^2 q e^{-q/q_0}\,.$$ This expression has been chosen to be tractable analytically whilst keeping $q$ away from 0 and $\infty$. The parameter $q_0$ is fixed and seeded by the user. In our case, we found that $q_{0} = 10$ was giving good performances in terms of computation time, and reconstruction fidelity against the simulated data. In Fig \[fig:alphacomp\], we show that its exact value has little impact on the final reconstruction. In contrast to the standard Bayesys algorithm implemented in Lenstool, Massinf does not explore all the correlations between the parameters, but searches for the most relevant parameters (keeping the others fixed meanwhile), and explores their PDF individually, reproducing thus somehow the Gibbs sampling approach. It also makes use of an additional nuisance parameter called $n$, which is the number of RBFs the sampler estimates necessary to reproduce the data. We obtained good results with this number described by a geometric pdf $$\mathrm{Pr}(n) = (1 - c) c^{n-1} \quad \mathrm{where} \quad c = \frac{\alpha}{\alpha + 1}\,,$$ and parameter $\alpha = 2\%$ of the total number of RBFs. Again we show in Fig \[fig:alphacomp\] that this parameter has little impact on the reconstruction. Simulated filament study {#sec:simus} ======================== ![Simulated filamentary structure with 3 elliptical NFW clumps. The cross indicates the center of the field. Contour levels are in log scale between $10^{-4} < \kappa < 0.2$. []{data-label="fig:input3d"}](input3d){width="\linewidth"} We applied our reconstruction algorithms to a simulated mass map made of 3 NFW halos at redshift $z=0.5$. The field of view is $10 \times 10$ square arcminutes, and the 3 halos are located at (0, 0.5’), (-1’, 0) and (2’, 0) in equatorial coordinates. They form a 3’ long filamentary structure aligned along the right ascension axis. To emphasize the extended aspect of the structure we made the halos elliptical with an ellipticity $e = \frac{a^2 - b^2}{a^2 + b^2} = 0.4$. For each halo, the scale radius is $r_s = 300$ kpc (50”), and their concentration are $c = 3$ and $c=3.5$ for the halo central halo. This translates into masses $M_{200} = 1.4\times 10^{14} M_{\odot}$ and $M_{200} = 2.3\times 10^{14} M_{\odot}$ in a $\Lambda$CDM cosmology $(\Omega_m = 0.3, \Omega_\Lambda =0.7, H_0 = 70\ {\rm km.s^{-1}.Mpc^{-1}}, w_0 = -1)$.\ From this mass model, we generated a convergence map by setting the sources at redshift $z= 1.2$, which is reasonable for data coming from the Hubble Space Telescope, alike the COSMOS data. This convergence map is shown in Fig \[fig:input3d\]. We also produced reduced shear catalogs with sources taken randomly across the field of view, and to which we added a random intrinsic ellipticity drawn from a Gaussian pdf of width $\sigma_{\rm int} = 0.27$. Again, this is a reasonable value for data coming from HST [@ejlens:leauthaud07].\ Standard galaxy density catalog {#sec:model} ------------------------------- ![Convergence maps reconstructed with the three methods. Top panel reconstructions are made with 50 gals/arcmin$^2$, middle and bottom panels with 100 gals/arcmin$^2$. Bottom panel is obtain after resampling 100 times the shape noise of the input catalog. Globally, Lenstool and FLens reconstructions have a lower noise level than SVD reconstruction. Lenstool reconstructions have high resolution, but also contain spurious peaks, whereas FLens reconstructions have lower resolution, but no spurious peaks. Resampling is efficient at removing the spurious peaks in all 3 cases and increases the signal to noise of the SVD reconstructed peaks.[]{data-label="fig:comparison"}](comparison){width="\linewidth"} First, we compare the reconstruction obtained with a catalog containing 5,000 sources, i.e. with a density of 50 galaxies per square arcminute. Results are shown in the top panel of Fig. \[fig:comparison\]. At first, we note that Lenstool and FLens produce less noisy reconstructions than the SVD inversion method. The Lenstool reconstruction has high resolution, but also contains spurious peaks, whereas the FLens reconstruction has lower resolution, but no spurious peaks. In this simulation, the FLens map is 64x64 pixels, and the pixel size is 0.156’. To filter out the reconstructed noise, Flens uses a wavelet decomposition procedure that only keeps scales with $J>3$, i.e. structures larger than 8 pixels in size. As described in \[sec:flens\], this wavelet scales thresholding is controlled by the FDR method. If a scale is noise dominated, the detection threshold will be very high and the scale will be removed, thus degrading the resolution of the reconstructed map. This global estimation of the detection threshold per scale is more robust to the noise but less sensitive to small structures. A more local approach would increase the resolution and the detection of small structures, but would also increase the number of false detections. For the Lenstool and SVD inversion methods, we adjust the resolution of the grid-based reconstruction to the power spectrum of the input signal. Peaks can still be resolved by cutting high frequencies at $k > 10$ arcmin$^{-1}$ ($k = \frac{2\pi}{R}$). This translates into RBFs with core radius $s = 0.3'$. We choose an hexagonal grid of RBFs in order to limit high frequency noise at the junction between nearby RBFs. We can cover the whole FOV with a grid of 817 RBFs. The Lenstool reconstruction is less noisy than the SVD reconstruction essentially because of the priors implemented in Lenstool. High galaxy density catalog --------------------------- In order to increase the resolution of the FLens reconstruction, we produce a catalog with 10,000 sources, i.e. with a 100 galaxies per square arcminute. Results are shown in the middle panel of Fig. \[fig:comparison\]. By doubling the size of the catalog, we could decrease by 4 the pixel size (0.04’), and detect the halo on the right in the FLens reconstructed map. The Lenstool reconstruction still contains spurious peaks.\ Shape noise resampling ---------------------- In the two previous analysis, we observed some overfitting of the galaxy shape noise, especially with Lenstool and the SVD inversion, leading to spurious peaks.\ In order to mitigate this issue, we resample 100 times the intrinsic galaxy shape noise in the input catalog of 10,000 sources. We run Lenstool, FLens and the SVD reconstructions on each of 100 catalogs, and average the reconstructed convergence maps. The outcome of this procedure is presented in the bottom panel of Fig. \[fig:comparison\].\ We note that the spurious peaks have disappeared from the averaged maps, but also that the power in the peaks is globally less than in the original map.\ Reconstructed density profile ----------------------------- ![Comparison of the convergence profile recovered with FLens, Lenstool and the SVD inversion, assuming 100 galaxies per sq. arcmin., and resampling of the noise. Errors are given at 68.2% C.L.[]{data-label="fig:simu_profile"}](compprof){width="\linewidth"} It is a very common procedure in galaxy cluster studies to average the reconstructed mass maps azimuthally to produce a radial density profile. We perform this measurement for our three methods and compute the errors by taking the standard deviation of the 100 reconstructed maps. In Fig. \[fig:simu\_profile\], we show the comparison of the azimuthally averaged density profiles. The striking point of this figure is the amount of noise in the SVD reconstruction. The second point is the fact that the FLens density profile becomes negative at radius $R > 180$ arcsec and over-estimates the density at small radius. This is due to the fact that wavelets are compensated filters with null mean. In contrast, Lenstool reconstruction is unbiased, and contains the input profile in its 1$\sigma$ confidence contours. The correct normalization at large radius is due to the fact that Lenstool takes into account the redshifts of the lens and the individual sources in the fit.\ Errors on the reconstructed maps -------------------------------- ![Errors on the reconstructed convergence maps with the three methods. In theory SVD errors are independent of the underlying shear signal, but we still notice that locally they depend on the galaxy density. SVD errors have been divided by 4 to fit the colormap range.[]{data-label="fig:stdcomp"}](stdcomp){width="\linewidth"} We compute the errors of the reconstructed maps following 2 approaches. The Lenstool and the SVD inversion methods output an estimate of the error in each pixel, either by means the analysis of the MCMC samples, or the covariance matrix computed in Eq \[eq:svdconv\] respectively. Nonetheless to get rid of overfitting, we resample the galaxy shape noise in the input catalogs, and compute the variance of the pixels reconstructed both with Lenstool, Flens and the SVD inversion. Fig \[fig:stdcomp\] show that with the three methods, the errors scale with the input density field. It is worth noticing that the SVD error map also scales with the input signal, although the covariance matrix $N_{\kappa \kappa}$ does not directly depend on the ellipticity measurements $\bold{e}$. We have done some tests, and found that with a uniform distribution of galaxies, this effect vanishes. *Therefore, it seems this effect is due to lensing amplification*, which decreases the amount of galaxies in this region, and as a result increases the variance in the reconstruction. Finally, we have found that using RBFs with larger core radius increases the correlations between the RBF weights in $N_{vv}$, and decreases the resolution, as well as the overall signal to noise. In contrast, using RBFs with smaller core radius produces higher resolution but noisier reconstructions. We found that matching size of the RBFs to the grid resolution yields the best compromise.\ Errors on the reconstructed density profiles -------------------------------------------- ![Scaling of reconstructed noise as a function of reconstructed signal for different reconstruction methods. SVD inversion and Lenstool methods both provide a way to directly estimate errors on the reconstruction. This is what we call *Theroretical errors*. These errors are in good agreement with errors estimated with noise resampling.[]{data-label="fig:sigkok"}](sigkok_k){width="\linewidth"} We then focus on the estimated errors on the azimuthally averaged density profiles. In Fig \[fig:sigkok\], we find that the errors scale with the reconstructed density, in agreement with what we observed in the errors on the reconstructed maps. With this figure, we clearly see that the SVD inversion produces errors about 4 times larger than what can be achieved with Lenstool or FLens methods. Besides, it is reassuring to see that the errors estimated from the Lenstool MCMC samples or the covariance matrix $N_{\tilde{\kappa}Ê\tilde{\kappa}}$ agree with errors estimated after resampling. Regarding the bias between the reconstructed and the true convergence profiles, we note from Figure \[fig:simu\_profile\] that Lenstool bias is almost constant at less than 5% from the input values, whereas FLens and SVD biases increase with $\kappa$ and reach about 30% at $\kappa = 0.07$. Signal to noise estimates ------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![[**Top panel**]{} From left to right, probability distribution functions of the convergence reconstructed from 1000 noise maps, as obtained with Lenstool, FLens and the SVD inversion respectively. The dashed curve corresponds to Lenstool without the prior on positive convergence. [**Bottom panel**]{} Reconstructed convergence maps with 100 galaxies per sq. arc-minutes and noise resampling. Contours indicate the levels of confidence at 68.2%, 95.5%, 99.7% and 99.9%. []{data-label="fig:snmap"}](histo_all "fig:"){width="0.85\linewidth"} ![[**Top panel**]{} From left to right, probability distribution functions of the convergence reconstructed from 1000 noise maps, as obtained with Lenstool, FLens and the SVD inversion respectively. The dashed curve corresponds to Lenstool without the prior on positive convergence. [**Bottom panel**]{} Reconstructed convergence maps with 100 galaxies per sq. arc-minutes and noise resampling. Contours indicate the levels of confidence at 68.2%, 95.5%, 99.7% and 99.9%. []{data-label="fig:snmap"}](snmap "fig:"){width="\linewidth"} --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- It is a common procedure to compute the signal to noise by dividing the estimated signal by the variance of the noise. However in the top panel of Fig.\[fig:snmap\], we show that in our case, the pdf of the reconstructed noise is not necessarily Gaussian distributed. This is particularly evident for the Lenstool method. From each pdf, we therefore compute the threshold X, for which we have the probability of finding a value $x$, $P(x \le X)$ equals to 68.2%, 95.5%, 99.7% and 99.9%. We found that with 1000 realizations of noise, we had enough statistics to estimate up to only 4$\sigma$ level. In the bottom panel of Fig.\[fig:snmap\], we observe that the SVD inversion is more noisy than the Lenstool or the FLens methods. The 1$\sigma$ region of the confidence is larger with Lenstool and smaller with the SVD inversion. Globally, the regions of equal confidence are similar in size with Lenstool and FLens, especially at larger signal to noise. Application to MACSJ0717+3745 {#sec:macs0717} ============================= In this section, we apply our three methods to the real case of the galaxy cluster MACS J0717+3745, in which a filament was recently detected with Lenstool multi-scale grid reconstruction [@ejlens:jauzac12]. Modeling description -------------------- The analysis in @ejlens:jauzac12 was based on a mosaic of 18 multi-passband images obtained with the Advanced Camera for Surveys aboard the Hubble Space Telescope, covering an area of $\sim10\times 20$ square arcminute. The weak-lensing pipeline developed for the COSMOS survey, modified for the analysis of galaxy clusters, was used to produce a weak-lensing catalogue of roughly 52 galaxies per square arc-minutes. A uBV color diagram was used to distinguish the background sources from the foreground and cluster-member galaxies. Their redshift distribution was derived from photometric and spectroscopic redshifts obtained from Subaru and CFHT/WIRcam imaging in the same field [@ejlens:ma08]. Because they are in the strong lensing regime area, all the galaxies inside an elliptical region of 5 x 3 arc-minutes in size and 45$^\circ$-rotated centered on the cluster core were also removed from the catalog. The detail of the catalog construction is thoroughly described in @ejlens:jauzac12. In order to compute error bars on the reconstructions, we resampled the weak lensing catalog with a bootstrap strategy, i.e. each galaxy in the catalog can be removed or duplicated, in order to increase its weight in the reconstruction. We produced 50 of such bootstrapped catalogs. For the Lenstool and the SVD inversion methods, we built a grid of RBFs. In contrast to the model described above, in which all the RBFs had the same size, in Jauzac et al. we used a multi-scale grid with smaller RBF in regions where the cluster luminosity was brighter. First, we built a smoothed cluster luminosity map from the catalog of magnitudes in K-band of cluster member galaxies. Then, we computed a multi-scale grid of RBFs, making sure that the luminosity in each triangle was lower than a predefined threshold. As a result, we obtained a grid made of 468 RBFs, the smallest ones having a core radius $s = 26$ arcsec. Reconstructed maps ------------------  . \[fig:comp0717\] Fig \[fig:comp0717\] shows the reconstructed convergence maps of MACS J0717 obtained with the three methods. Globally, they all agree on the location of the cluster core, and the presence of an extension to the South-East. In the cluster core where data are missing, both the Lenstool and FLens reconstructions are smooth, whereas the SVD reconstruction is more clumpy. We attribute this difference to the priors assumed in both Lenstool and FLens. We also observe some disagreement on the exact shape of the filament. Lenstool reconstruction suggests MACS J0717 lies into an extended over-dense region. In contrast, FLens reconstruction shows that the cluster is compact and connected to a long filament. In both the FLens and the Lenstool reconstructions, the filament is detected at 95% C.L. The SVD reconstruction presents 2 filaments next to each other. Reconstructed density profile ----------------------------- ![Reconstructed convergence profiles obtained with the three methods. The FLens profile is in good agreement with the other profiles in the core, but deviates at large radius, were it becomes negative. The SVD reconstruction is not able to reproduce the central high density region. Without the positive prior on $\kappa$, we obtain a better agreement between Lenstool and FLens at large radius.[]{data-label="fig:sdens"}](sdens){width="\linewidth"} Fig.\[fig:sdens\] shows the corresponding radial convergence profiles obtained with the three methods. We took the coordinates $\alpha = 109.39102$ and $\delta = 37.746639$ as the central point of the azimuthal average. Based on the photometric redshift analysis performed in @ejlens:jauzac12, we assumed in the Lenstool reconstruction that the redshift of the weak lensing sources to be $z_s = 0.65$. As already observed in the simulations, the noise level estimated from bootstrap is about 4 times larger in the SVD reconstruction, especially close to the cluster center. The Lenstool method agrees with FLens at small radii, and with the SVD inversion at large radius. The FLens method predicts steeper radial profile between 500 kpc and 1000 kpc, and a bump at 3 Mpc, corresponding to the over-density in the filament. This feature is much less evident in the other reconstructions. Note as well that the convergence profiles derived from single catalog and bootstrap catalogs reconstructions with Lenstool agree together. Lenstool error estimates from the MCMC sampling are therefore reliable. Conclusion ========== Systematic errors in lensing map reconstruction, especially due to the reconstruction methods, is a concerning issue. With the current and forthcoming datasets, they start to dominate the error budget over the statistical errors. In this work, we have studied three methods of reconstruction of 10 arc-minutes scale structures, i.e. the environment of galaxy clusters. We limited our study to a toy-model structure in order to focus on the effect of priors. In a forthcoming paper, we will increase the level of complexity by using N-body simulations. The FLens method starts from a pixelated map of shear, with about one galaxy per pixel on average, and filter the noisy reconstructed convergence map by only keeping wavelet scales that contain non Gaussian signal. The Lenstool and the SVD inversion methods share the same underlying multi-scale grid model. The field is paved with a set of RBF, whose number density and size scale with the smoothed surface brightness of the cluster member galaxies. Lenstool uses a Bayesian MCMC sampler to estimate the weight of each RBF in the reconstruction, where the SVD inversion makes use of the linear formalism of the weak-lensing approximation to estimate the weights. The RBF shape is defined from the Truncated Isothermal Mass Distribution (TIMD), which can either give the shear for the inversion or the convergence for the reconstruction. So far with Lenstool, we have forced the density field and therefore the convergence to be positive everywhere. This assumption is valid here, because we consider the case of massive structures. Nonetheless in order to be exhaustive in this study, we also turned this prior off in Lenstool and redid all the computations. We found very similar results both for the simulated case and for MACSJ0717.\ From the simulations, we found the following : - All three methods can detect clusters and surrounding filaments in the convergence range $0.01 < \kappa < 1$, although with different levels of significance. - Doubling the galaxies number density from 50 to 100 per square arcminute allows to reduce the pixel size and increase the resolution of the FLens reconstruction. The resolution of Lenstool and the SVD inversion methods is more driven by the density of RBFs than by the galaxy density. However, the signal to noise per pixel increases with galaxy number density. - The error on the reconstructed convergence scales with the underlying signal, and depends on the method used for the reconstruction. The residual is offset from zero by a small amount, that decreases when we increase the grid resolution. - Thanks to the inpainting technique implemented in FLens, we could recover the shape of the cluster even in reasonably high density regions ($\kappa \sim 0.16$). - We compared these results to the forward fitting method presented in @ejlens:jauzac12 and implemented in <span style="font-variant:small-caps;">Lenstool</span>. The forward fitting method recovers the true density map with deviations less than 5% at $\kappa > 0.5$, and less than 20% at $0.5 > \kappa > 0.01$. In contrast to the other method, the redshift of the cluster and sources are used as a constraint to break the mass-sheet degeneracy. As a result no significant offset is found in the residual. - We found FLens to be more robust against shape noise than Lenstool or standard inversion methods. Resampling techniques increase the signal to noise of regions with low signal, but decrease signal to noise of regions with high signal. We applied the new method to the galaxy cluster MACSJ0717, and confirmed the presence of the filament at 3$\sigma$ C.L. We also repeated the Lenstool analysis previously done in @ejlens:jauzac12, but this time with a bootstrap of the input source catalog. The consistent results obtained with these two techniques give us more confidence in the detection of the structures around MACSJ0717. Without the prior of positive convergence applied, we obtained a very similar map and consistent signal to noise contours in Figure \[fig:comp0717\] , and a density profile in better agreement with FLens at large radius in Figure \[fig:sdens\]. To conclude, it is very encouraging to see that priors can significantly enhance the signal to noise in weak lensing reconstructions. FLens priors are strictly limited to the properties of the galaxy shape noise. In contrast, Lenstool priors enforce the mass-follows-light assumption to build the multi-scale grid. Ideally, the science goals condition the type of priors to choose. A weak lensing peak counting analysis to characterize dark energy might prefer limited priors in order to better compare to theory, whereas the exploration of the cosmic web might heavily rely on external priors coming from other observables, such as galaxy density, X-ray or SZ maps. The authors would like to thank J.-L. Starck for useful discussions. Computations have been performed at the Mésocentre d’Aix-Marseille Université. This work was supported by the European Research Council (ERC) grant SparseAstro (ERC-228261). JPK acknowledges support from the ERC advanced grant LIDA and from CNRS. \[lastpage\] [^1]: The MRLens denoising software is available at the following address: “http://irfu.cea.fr/Ast/mrlens software.php”. [^2]: Lenstool public package is available at the following address http://projects.lam.fr/projects/lenstool

--- abstract: 'We investigate a factor that can affect the number of links of a specific stock in a network between stocks created by the minimal spanning tree (MST) method, by using individual stock data listed on the S&P500 and KOSPI. Among the common factors mentioned in the arbitrage pricing model (APM), widely acknowledged in the financial field, a representative market index is established as a possible factor. We found that the correlation distribution, $\rho_{ij}$, of 400 stocks taken from the S&P500 index shows a very similar with that of the Korean stock market and those deviate from the correlation distribution of time series removed a nonlinearity by the surrogate method. We also shows that the degree distribution of the MSTs for both stock markets follows a power-law distribution with the exponent $\zeta \sim$ 2.1, while the degree distribution of the time series eliminated a nonlinearity follows an exponential distribution with the exponent, $\delta \sim 0.77$. Furthermore the correlation, $\rho_{iM}$, between the degree k of individual stock, $i$, and the market index, $M$, follows a power-law distribution, $\langle \rho_{iM}(k) \rangle \sim k^{\gamma}$, with the exponent $\gamma_{\textrm{S\&P500}} \approx 0.16$ and $\gamma_{\textrm{KOSPI}} \approx 0.14$, respectively. Thus, regardless of the markets, the indivisual stocks closely related to the common factor in the market, the market index, are likely to be located around the center of the network between stocks, while those weakly related to the market index are likely to be placed in the outside.' author: - Cheoljun Eom - Gabjin Oh - Seunghwan Kim title: Topological Properties of the Minimal Spanning Tree in the Korean and American Stock Markets --- Introduction ============ Recently, researchers from diverse disciplines are showing great interest in the topological properties of networks. In particular, the network observed in natural and social science shows a variety of properties different from those of random graphs [@watts99; @barabasi02] . The economic world, known as having the most complex structure among them, evolves through the nonlinear-interaction of the diverse heterogeneous agents. The stock market is a representative example. The stock prices of individual companies are formed by a complex evolution process of diverse information generated in the market and these have strong correlations with each other by the common factors in the market [@Farrell1974; @Ross1976; @Chen1986]. In other words, individual stocks are connected with each other and companies with the same properties tend to be grouped into a community. To investigate these properties, Mantegna.[*et al.*]{} proposed the minimal spanning tree (MST) method, which can be used to observe the grouping process of individual stocks transacted in the market, on the basis of the correlation of stocks [@Mantegna1999a]. Mantegna constructed the stock network visually using the MST method and found that this generated network between stocks has an economically significant grouping process [@Mantegna1999a; @Mantegna1999b]. The studies of the past several years showed that the degree distribution of the network created by the MST method follows a power-law distribution with the exponent $\xi$ $\approx 2$ [@Kim2002; @Vandewalle2000]. That is, most individual companies in the stock market have a small number of links with other stocks, while a few stocks have a great number of connections. However the KOSPI200 companies of the Korean stock market, one of the emerging market, does not follow a power-law distribution and for the American stock market, S&P500, the relation between market capitalization and $|q|$, the influence strength (IS), has a positive correlation, while the KOSPI200 has no correlation [@Jung2006]. Moreover, previous results showed that the network of individual stocks tends to gather around the companies of a homogeneous industrial group [@Bonanno2000a; @Bonanno2000b; @Bonanno2003; @Onnela2003]. Furthermore, the stocks forming a Markowitz efficient portfolio in the financial field are almost located at the outside of the network [@Onnela2003; @Onnela2003b]. However, until recently, studies on the possible factors important in determining the number of linkages with other stocks in the network between stocks were insufficient. In order to investigate the topological properties in the network of the stock market, we used the MST method introduced by Mantegna[*et al.*]{} We also consider the market index in terms of a possible factor that can affect the number of links of a specific stock with other stocks in the network created by the MST method. We used the data of 400 individual companies listed on the S&P500 index from January 1993 to May 2005, and 468 individual companies listed on the KOSPI from January 1991 to May 2003. We found that the correlation distribution, $\rho_{ij}$, of the 400 stocks in S&P500 index shows a very similar with that of the KOSPI and those deviate from the correlation distribution of time series removed a nonlinearity by the surrogate method introduced by J. Theiler [*et al.*]{} [@Theiler1992] for both stock markets. We also found that the degree distribution of the network, like those from previous research, follows a power-law distribution with the exponent $\zeta_{\textrm{S\&P500, KOSPI}} \approx 2.1$ for both the Korean and American stock markets. In order to observe the possible factor in determing the degree k on the MST network, we calculate the cross-correlation, $\rho_{iM}$, between a individual stock and market index for both stock markets. We also found that the cross-correlation, $\rho_{iM}$ between the market index and the companies with the degree k follows a power-law distribution, $\langle \rho_{iM}(k) \rangle \sim k^{\gamma}$, where the exponents are calculated to be $\gamma_{\textrm{S\&P500}} \sim$ 0.16, $\gamma_{\textrm{KOSPI}} \sim$ 0.14. In other words, individual stocks having many connections with other stocks in the network obtained by the MST method are more highly related to the market index than those having a comparatively small number of links. In the next section, we describe the financial data used in this paper. In Section 3, we introduce the methodology. In Section 4, we present the results in this investigation. Finally, we end with the summary. Data ==== We used 400 individual daily stocks data from January 1993 to May 2005 taken from individual stocks listed on the S&P500 index of the American stock market (from the Yahoo website) and 468 individual daily stocks data from January 1991 to May 2003 taken from individual stocks listed on the KOSPI of the Korean stock market (from the Korean Stock Exchange). In order to investigate the possible factors determined the number of links of an individual stock in the network, we used the $S\&P 500$ and KOSPI index with the same period as individual stocks, respectively. We used the normalized returns, $R_t$, by the standard deviation, $\sigma(r_t)$, after calculating the returns from the stock price, $P_t$, by the log-difference, $r_t \equiv \ln P_{t} - \ln P_{t-1}$, as in previous studies and defined as follow $${R_t} \equiv \frac{\ln{P_{t}}- \ln{P_{t-1}}}{\sigma(r_t)},\\ \label{e1}$$ where $\sigma(r_t)$ is the standard deviation of the return. Methodology {#sec:METHODOLOGY} =========== We make the network by using stocks listed on the S&P500 and KOSPI, respectively, through the MST method proposed by Mantegna [*et al.*]{} As the MST method makes the network based on the correlation of stocks, the cross-correlation of stocks listed on the S&P 500 and KOSPI stock markets, respectively, is calculated as follows $$\rho_{ij} = \frac{\langle R_{i}R_{j} \rangle - \langle R_{i}\rangle \langle R_{j}\rangle}{\sqrt{({\langle R_{i}^{2}\rangle}-{\langle R_{i}\rangle}^{2}) ({\langle R_{j}^{2}\rangle}-{\langle R_{j}\rangle}^{2}) }}, \\ \label{e2}$$ where $\langle .\rangle$ means the mean value of the whole period and the correlation lies within the range of $-1 \leq \rho_{ij} \leq +1$. If $\rho_{ij}$ is 1, two time series have a complete correlation and if $\rho_{ij}$ is -1, they have a complete anti-correlation. In the case where $\rho_{ij}$ is 0, the correlation of two time series is 0. On the basis of $\rho_{ij}$ calculated by Eq. \[e2\], the distance between nodes is calculated as follows $$\label{e3} d_{ij}= \sqrt{2(1-\rho_{i,j})}.$$ In order to find out the correlation between the number of connections of a individual stock with other stocks in the network and the market index, we investigated the correlation, $\rho_{iM}$, between the market index and individual stocks. Using the Eq. \[e4\], we calculated the correlation, $\rho_{iM}$, between returns of individual stocks, $R_i$, and the market index, $R_{M}$, and defined as follow $$\label{e4} \rho_{iM} = \frac{\langle R_{i}R_{M} \rangle - \langle R_{i}\rangle \langle R_{M}\rangle}{\sqrt{({\langle R_{i}^{2}\rangle}-{\langle R_{i}\rangle}^{2}) ({\langle R_{M}^{2}\rangle}-{\langle R_{M}\rangle}^{2}) }}. \\$$ Results {#sec:RESULTS} ======= ![The PDF of the cross-correlation $\rho_{ij}$ between the stocks listed on for the S&P500 and KOSPI stock markets. The red (circle), blue (square), green (diamond), and black (triangle) denotes the KOSPI, S&P500, KOSPI (surrogate), and S&P500 (surrogate) stock market, respectively.[]{data-label="fig1"}](f1.eps){height="8cm" width="8cm"} ![(a) MST structure composed of daily returns of 400 individual companies lised on the S&P500 index from 1993 to 2005, and (b) MST structure composed of the daily returns of 468 individual corporations listed on the KOSPI from 1991 to 2003. (c) shows the degree distribution of the MST structure composed of individual companies on the S&P500 index and KOSPI. The degree of both the S&P500 index and KOSPI follows a power law distribution with the exponent of $\zeta \sim 2.1$. The red (circle), blue (square), green (diamond), and black (triangle) denotes the KOSPI, S&P500, KOSPI (surrogate), S&P500 (surrogate), respectively.[]{data-label="fig2"}](f2_1.eps "fig:"){height="5cm" width="4cm"} ![(a) MST structure composed of daily returns of 400 individual companies lised on the S&P500 index from 1993 to 2005, and (b) MST structure composed of the daily returns of 468 individual corporations listed on the KOSPI from 1991 to 2003. (c) shows the degree distribution of the MST structure composed of individual companies on the S&P500 index and KOSPI. The degree of both the S&P500 index and KOSPI follows a power law distribution with the exponent of $\zeta \sim 2.1$. The red (circle), blue (square), green (diamond), and black (triangle) denotes the KOSPI, S&P500, KOSPI (surrogate), S&P500 (surrogate), respectively.[]{data-label="fig2"}](f2_2.eps "fig:"){height="5cm" width="4cm"} ![(a) MST structure composed of daily returns of 400 individual companies lised on the S&P500 index from 1993 to 2005, and (b) MST structure composed of the daily returns of 468 individual corporations listed on the KOSPI from 1991 to 2003. (c) shows the degree distribution of the MST structure composed of individual companies on the S&P500 index and KOSPI. The degree of both the S&P500 index and KOSPI follows a power law distribution with the exponent of $\zeta \sim 2.1$. The red (circle), blue (square), green (diamond), and black (triangle) denotes the KOSPI, S&P500, KOSPI (surrogate), S&P500 (surrogate), respectively.[]{data-label="fig2"}](f2_3.eps "fig:"){height="5cm" width="4cm"} ![(a) MST structure composed of daily returns of 400 individual companies lised on the S&P500 index from 1993 to 2005, and (b) MST structure composed of the daily returns of 468 individual corporations listed on the KOSPI from 1991 to 2003. (c) shows the degree distribution of the MST structure composed of individual companies on the S&P500 index and KOSPI. The degree of both the S&P500 index and KOSPI follows a power law distribution with the exponent of $\zeta \sim 2.1$. The red (circle), blue (square), green (diamond), and black (triangle) denotes the KOSPI, S&P500, KOSPI (surrogate), S&P500 (surrogate), respectively.[]{data-label="fig2"}](f2_4.eps "fig:"){height="5cm" width="4cm"} In this section, using the MST method we investigated the network properties of 400 individual stocks listed on the S&P500 index from January 1993 to May 2005, and 468 individual stocks listed on the KOSPI from January 1991 to May 2003, respectively. First, we analyze the distribution of the correlation matrix, $\rho_{ij}$, for individual stocks in the S&P500 index and KOSPI. In Fig. \[fig1\], we shows the distribution of the correlation matrix for both stock markets. The blue (square), red (circle), green (diamond), and black (triangle) indicates the S&P500, KOSPI, S&P500 (surrogate), KOSPI (surrogate) stock markets, respectively. We find that the correlation distribution, $\rho_{ij}$ between stocks in the S&P500 index shows a very similar with that between stocks in the KOSPI and those deviate from the correlation distribution of time series created by the surrogate method. In order to estimate the topology structure of both stock markets, we employ the MST method proposed by the Mantegna [*et al.*]{} Using the whole period data of the Korean and American stock markets, we present the network structure calculated by the MST method and its the degree distribution for both the stock markets. Fig. \[fig2\] shows the network structures between stocks generated by the MST method, using individual stock data lised on the American and Korean stock market, respectively and plot the degree distribution, $P(k)$, of the MST networks for both stock markets. In Fig. \[fig2\], (a) and (b) display the MST structure composed of daily returns of the individual companies lised on the S&P500 index and KOSPI, respectively and (c) and (d) show the degree distribution of the MST structure for both stock markets in the log-log and linear-log plot. The red (circle), blue (square), green (diamond), and black (triangle) indicates the KOSPI, S&P500, KOSPI (surrogate), S&P500 (surrogate), respectively and the notation (surrogate) denotes the corresponding surrogate data. We find that the degree distribution for both stock markets follows a power law distribution with the exponent $\zeta \sim 2.1$, while the degree distribution of MST network of the time series created by the surrogate method [@Theiler1992] does follows an exponential distribution with $\delta \sim 0.77$. Thus, as the results finding in the complex network such as the internet, WWW, protein-protein interaction, and so on, there is a scale-free network property. Therefore, the hubs existence in the financial markets means that there are a dominant company which gives many influence to the other stocks. In order to observe a possible factor determination the degree of an individual stock on the MST network, we calculated the correlation, $\rho_{iM}$, between an individual stock with the degree $k$ and the stock market index for both stock markets. ![Distribution of the correlation, $\rho_{iM}$, between an individual companies and market index for both the S&P500 index and KOSPI, respectively.[]{data-label="fig3"}](f3_1.eps){height="10cm" width="8cm"} In Fig. \[fig3\], we shows the distribution of the correlation, $\rho_{iM}$, between stocks in the S&P500 index and KOSPI, respectively. The blue (square) and red (circle) indicates the S&P500 and KOSPI. We find that the correlation, $\rho_{iM}$, between a stock with the degree k, the number of connected with other stocks, and the market index for both stock markets follows a power-law distribution, $\langle \rho_{iM}(k) \rangle \sim k^{\gamma}$, with the exponent $\gamma_{\textrm{S\&P500}} \approx 0.16$ and $\gamma_{\textrm{KOSPI}} \approx 0.14$, respectively. Thus, through these results, we found that the stocks having more links with other stocks are more highly correlated with the market index than those with relatively fewer connections. In other words, the stocks closely related to the market index have a larger number of links with other stocks and are likely to be located around the center of the network of the stocks. On the other hand, the stocks poorly related to the market index have fewer links with other stocks and tend to be placed at the outside of the network of the stocks. Our results suggest that the common factor such as the market index play a important rules in terms of determing the networks in the financial markets. Conclusions =========== In this paper, in order to examine possible factors capable of affecting the number of links that a specific stock has in relation to other stocks in the network between stocks created using the MST method, we carried out research using the market index, a representative among multiple common factors mentioned in the arbitrage pricing model (APM). We used 400 individual stocks listed on the S&P 500 index and 463 stocks listed on the KOSPI. We found that the correlation distribution, $\rho_{ij}$, between stocks in the S&P500 index shows a very similar with that between stocks lised on the KOSPI and those deviate from the correlation distribution of time series removed a nonlinearity by the surrogate method. We shows that the degree distribution in the network between stocks obtained by the MST method for both stock markets follows a power-law distribution with the exponent $\zeta_{\textrm{S\&P500, KOSPI}} \sim$ 2.1, while the degree distribution from the time series eliminated a nonlinearity follows an exponential distribution with the exponent, $\delta_{\textrm{S\&P500 (surrogate), KOSPI (surrogate)}} \sim 0.77$. In order to investigate a factor determining the degree k on the MST network, we used the market index for both stock markets. We found that in the degree distribution, the correlation, $\rho_{iM}$, between the degree k, the number of links, and the market index for both stock markets follows a power-law distribution, $\langle \rho_{iM}(k) \rangle \sim k^{\gamma}$, with the exponent $\gamma_{\textrm{S\&P500}} \approx 0.16$ and $\gamma_{\textrm{KOSPI}} \approx 0.14$, respectively. In other words, the stocks having the intimate relation with the market index have a larger number of links, while the stocks poorly related to the market index have fewer links. According to above finding results, we imply that the degree $k$ as most important quantity to describe the network topology, has a closely relation with the common factors such as market index. This work was supported by the Korea Research Foundation funded by the Korean Government (MOEHRD) (KRF-2006-332-B00152), and by a grant from the MOST/KOSEF to the National Core Research Center for Systems Bio-Dynamics (R15-2004-033), and by the Korea Research Foundation (KRF-2005-042-B00075), and by the Ministry of Science & Technology through the National Research Laboratory Project, and by the Ministry of Education through the program BK 21. [00]{} D. J. Watts, *Small Worlds: The Dynamics of Networks Between Order and Randomness* (Princeton Univ. Press, Princeton, NJ, USA, 1999) A.-L. Barabási, *Linked: The New Science of Networks* (Perseus Books Group, Cambridge, MA, USA, 2002). J. L. Farrell, J. Bus., 47 (1974) 186. S. A. Ross, J Econ. Theory, 13 (1976) 341. N. Chen, R. Roll S. A. Ross, J. Bus., 59 (1986) 383. R. N. Mantegna, Eur. Phys. J. B, 11 (1999) 193. R. N. Mantegna, Comput. Phys. Commun., 121 (1999) 153. H. J. Kim, I. M. Kim, Y. Lee B. Kahng, J. Korean Phys. Soc., 40 (2002) 1105. N. Vandewalle, F. Brisbois X. Tordoir, preprint available at arxiv.orgm, cond-mat/0009245 (2000). W.-S. Jung, S. Chae, J.-S. Yang, H.-T. Moon, Physica A, 361 (2006) 263. G. Bonanno, G. Caldarelli, F. Lillo, R. N. Mantegna, Phys. Rev. E, 68 (2003) 046130. G. Bonanno, F. Lillo, R. N. Mantegna, Quant. Financ., 1 (2000) 96. G. Bonanno, F. Lillo, R. N. Mantegna, Quant. Financ., 1 (2000) 96. J.-P. Onnela, A. Chakraborti, K. Kaski, Phys. Rev. E, 68 (2003) 056110. J.-P. Onnela, A. Chakraborti, K. Kaski, J. Kertész, Physica A, 324 (2003) 247. J. Theiler, S. Eubank, A. Lontin, B. Galdrikian J. D. Famer, Physica D, 58 (1992) 77.

--- abstract: 'We present a tight-binding calculation that, for the first time, accurately describes the structural, vibrational and elastic properties of amorphous silicon. We compute the interatomic force constants and find an unphysical feature of the Stillinger-Weber empirical potential that correlates with a much noted error in the radial distribution function associated with that potential. We also find that the intrinsic first peak of the radial distribution function is asymmetric, contrary to usual assumptions made in the analysis of diffraction data. We use our results for the normal mode frequencies and polarization vectors to obtain the zero-point broadening effect on the radial distribution function, enabling us to directly compare theory and a high resolution x-ray diffraction experiment.' author: - 'J. L. Feldman' - 'N. Bernstein' - 'D. A. Papaconstantopoulos' - 'M. J. Mehl' title: 'Tight-binding study of structure and vibrations of amorphous silicon' --- Amorphous silicon (a-Si) is a prototype for continuous-random-network covalent glasses that, with some hydrogen content, has technological applications as a relatively inexpensive electronic material. While the basic structure of a-Si is believed to be a four-fold-coordinated continuous random network, detailed information about network connectivity and defects is lacking. Atomic resolution structure is very difficult to determine directly, and experiments have relied on unusual or indirect probes such as variance coherence microscopy [@treac] and Raman spectroscopy [@beem; @vink1] as well as on more standard techniques such as diffraction [@fort; @laaz] and EXAFS [@wakagi; @new_exafs]. The experimental measurements suggest significant deviation from a continuous random network, including average coordination that is significantly less than 4 (e.g. Ref. ) and that unannealed samples may be paracrystalline [@treac]. Many empirical-potential simulations have been done, but it is not clear if empirical potentials are accurate enough to give reliable results for properties, such as coordination defects, that depend on bond breaking and bond formation. A number of simulations of a-Si structure have used electronic-structure based methods, which are generally among the most reliable for solid state systems (e.g. Refs. ). However, none have carefully compared the radial distribution function (RDF) to high resolution experiments [@laaz], and none included quantum-mechanical vibrational effects. Another important question concerns the vibrational properties of a-Si, which give us information about the structure and the interactions of atoms in the material. The vibrational density of states (VDOS) was measured experimentally using inelastic neutron scattering (INS) [@kama]. Empirical-potential simulations have been used to analyze vibrational properties in detail [@allen], but all show significant errors in the shape of the VDOS or in other properties. While the VDOS of a-Si has been simulated with electronic structure methods [@stich; @pbis; @nakdra], the underlying force constants themselves have not been analyzed. There have been many studies of force constants in crystalline Si, which shows unusual phonon dispersion and force constants that oscillate in magnitude as a function of distance [@kane; @rign]. We study the elastic constants, vibrational properties, and structure of a-Si using a tight-binding (TB) total-energy method. We find elastic constants and VDOS that are in good agreement with experiment, and qualitatively better than empirical-potential simulations. The structure has a sharp first-neighbor RDF peak that agrees very well with experiment when zero-point and thermal broadening is included. This peak is significantly non-Gaussian, calling into question the coordination-statistics analysis of previous diffraction experiments. We use the Naval Research Laboratory (NRL) TB method [@rec; @brn]. The non-orthogonal $sp^3$-basis TB model has been shown to accurately describe the elastic constants and phonon dispersion in crystalline Si and the electronic density of states for a highly defected amorphous model [@brn]. To generate the a-Si models we relax using TB-calculated forces a-Si models generated by other techniques. The NRL-TB model is used to calculate the energy of the structure and the atomic forces [@kirch]. The conjugate-gradient method is applied to find mechanical-equilibrium positions at a fixed volume, employing the criterion that components of atomic forces be less than $10^{-3}$ eV/[Å]{}. The relaxation procedure is carried out at several volumes to obtain results at zero pressure, but components of the stress tensor, generally of magnitude less than 0.8 GPa, remain. One model, which we denote TB1, is generated by relaxing (using TB) a 216 atom perfect continuous-random-network model [@woot] with periodic boundary conditions relaxed with a Keating interatomic potential [@www]. The TB-relaxed model is perfectly four-fold coordinated, with 1.3% lower density than the crystal, compared to 1.7% lower density measured experimentally [@laaz]. The bond-angle distribution has a RMS deviation of 11$^\circ$ from the average value of 109.2$^\circ$, in close agreement with relaxed [*ab initio*]{} calculation [@durdr] and analysis of experiment [@fort]. A second model, which we denote TB2, is generated by relaxing a structure from a molecular-dynamics simulation of the rapid quenching of liquid Si using the environment dependent interatomic potential [@edip]. The TB2 structure is slightly more dense than TB1, but still about 0.5% less dense than the crystal. The energy is 28 meV/atom lower than the TB1 energy, despite the presence of 6% 5-fold and 0.46% 3-fold coordinated atoms (corresponding to an average coordination of 4.05) [@model_energy]. The RMS bond-angle deviation is 12.5$^\circ$, although the distribution has wide, non-Gaussian wings; excluding 2% of the bond-angles reduces the RMS deviation to 10.4$^\circ$. We also show some comparisons with results using the Stillinger-Weber (SW) interatomic potential [@sw]. The SW potential, which includes radial and bond-angle terms, is one of the most often used potentials for simulations of Si. We use a structure (Ref. , Table II, model IV) generated by relaxing with SW the same starting structure as TB1. Finally, we note that while it is possible to use electronic structure methods to generate amorphous structures from procedures that are less dependent on the initial structure, it is very expensive computationally. The difficulty in fully annealing the structure seems to lead to a consistent overestimate of the width of the first-neighbor peak in the RDF [@stich; @pbis1]. -------------------------------------------------------------------------------------------------------------------- TB1 TB2 Exp./FP SW$^{(a)}$ ------------- ------------- ------------- ------------------------------------------------------------ ------------- c$_{ii}$ 16.31-16.45 15.06-16.00 13.8$^{(b)}$,17(2)$^{(c)}$ 11.94-13.11 c$_{jj}$ 5.68-5.84 5.26-5.56 4.8$^{(b)}$, 4.5$^{(a)}$ 2.54-3.21 c$_p^{(d)}$ 5.77 5.06 " & 2.62\ c$_{12}$ & 4.77 & 5.32 & & 6.69\ B & 8.73 & 8.99 & 5.9$^{(e)}$,8.25$^{(f)}$ & 8.52\ E & 14$^{(g)}$ & 13$^{(g)}$ & 12.4(3)$^{(a)}$ & 7$^{(g)}$\ & & & 11.7(5)-13.4(5)$^{(h)}$ &\  (a) Ref. ; (b) Ref. ; (c) Ref. ;\  (d) Defined here as (c$_{11}$-c$_{12}$)/2;  (e) Ref.\  (f) Ref. ; (g) based on values of c$_{12}$ and c$_p$;\  (h) Ref. . -------------------------------------------------------------------------------------------------------------------- : Selected elastic constants $c$, bulk modulus $B$ and Young’s modulus $E$ ($10^{11}$ dyn/cm$^2$). The index $i$ varies from 1 to 3, and $j$ from 4 to 6.[]{data-label="tab1"} The relaxed static lattice TB elastic constants $c_{ij}$ were obtained by the method of homogeneous deformation. The TB results [@elastic_isotropy] are compared in Table \[tab1\] with results of first-principles (FP) [@durdr] calculations, SW calculations, and several experiments on dense samples (a wider range of shear values are quoted in Table V of Ref. ). Although there is some deviation between the two TB structures it is small. While ultrasonic measurements of elastic properties are not available for a-Si, the Young’s modulus $E$ can be measured with a vibrating reed apparatus, and other elastic constants can be inferred from spectroscopic studies. Our TB results for both models are close to the experimental values, although our value of $c_{44}$ is likely 10–20% too large. The SW empirical potential results are significantly worse in comparison with experiment. The VDOS is calculated from a dynamical matrix approach. The matrix elements $\Phi_{\alpha\beta}(i,j) \equiv \Delta F_\alpha(i)/\Delta u_\beta(j)$ are calculated using the TB forces with a central-finite difference approach that eliminates all odd-order anharmonic terms in the potential energy [@explain]. The TB VDOS for structures TB1 and TB2 are compared with SW results and INS [@kama] measurements in Fig. \[gom\]. For both structures the TB calculation yields the overall shape very well; it exactly describes the low frequency TA peak, gives a slightly too small frequency of the LA peak (300 $cm^{-1}$) and about a 10$\%$ percent too high frequency of the high frequency TO peak. The TB results are a qualitative improvement over results based on the SW potential, as shown in the figure, and they are in good agreement with [*ab initio*]{} results for a 216 atom structural model [@nakdra]. The range of the effective interactions in the solid can give us information about the physics of the interactions, and can guide the development of approximations such as empirical potentials. In Fig. \[fcs\] we plot all of the cartesian force constants between pairs of atoms with interatomic distances less than 10 [Å]{}. The difference in range between the SW results and the TB results is easy to see: The SW interactions are large up to about 3.5 [Å]{}, and go to exactly zero at twice the SW cutoff of 3.75 [Å]{}. The TB interactions are already quite small at 2.8 [Å]{}, but do not go to zero even at 10 [Å]{}. This comparison of TB and SW leads to a view of interactions in the solid that is more subtle than the usual assumption that empirical potentials are short ranged and that the real interactions are long ranged: The SW potential interactions go to zero at a range that is too short, but at intermediate distances the interactions are too strong. We also note that the preponderance of force constants as a function of interatomic distance give a clear envelope function that has an oscillatory behavior which matches the RDF peak positions. This is qualitatively similar to the case of the crystal, even though the explanations for the oscillation in the crystal do not apply to the amorphous structure [@kane; @rign]. The problem with the SW potential is a direct consequence of the form of the potential. In the amorphous there are pairs of atoms in the second-neighbor peak with distances smaller than the SW cutoff. It is clear from the TB force constants that the effective interactions for these pairs is qualitatively different from the first-neighbor interactions. However, in the SW simulation these second-neighbor pairs interact through terms that are meant to describe the interactions of first-neighbor atoms. In particular, the two-body contribution has strong negative curvature at these distances, and the three-body terms include contributions from triplets with a vertex angle that does not correspond to an atom with two $sp^3$ orbitals in bonding configurations. These two types of contributions lead to the unphysically large force constants in the SW results at this range of distances. The range of incorrect force constants also coincides with the shoulder in the SW RDF that is not observed in our TB results or in the experimental measurements [@vink]. The distribution of force constants gives us information about the types of effective interactions between bonded atoms. Under the first peak of the RDF the largest positive cartesian force constants are twice the magnitude of the largest negative force constants for both SW and TB. This relation is consistent with an effective bond-stretching interaction for first-neighbors. We plot the results for the bond-stretching components in a plot as a function of $r$ (Fig. \[rdf\]a). The radial force constants decrease with increasing $r$ as one expects from a physically reasonable first-neighbor bonding potential. Pairs with large (small) interatomic bond stretching force constants will have small (large) relative mean square displacements, so these results clearly have an impact on the nature of the broadening of the RDF. Very little attention has been given in the literature to the shape of the first peak in the RDF $J(r)$ [@herrero]. This peak has been measured very carefully at $T=10$ K with x-ray diffraction, using high energy photons and high resolution, i.e., large $Q_\mathrm{max}$, by Laaziri [*et al.*]{} [@laaz]. They obtain a fit of their data to a Gaussian, with average coordination of 3.88${\pm .01}$ (3.79${\pm .01}$) for the annealed (unannealed) sample. In Fig. \[rdf\]a we plot the first peak of the static $J(r)$ for models TB1 and TB2, and the SW results. The TB static peak is asymmetric, and its width is significantly larger than the static-disorder estimate by Laaziri [*et al.*]{} In order to compare directly with the experimental $J(r)$ it is necessary to properly take account of the zero-point and thermal broadening. The quantity measured by the x-ray experiment is, in the small-displacement limit, $$J(r)=\frac{1}{N}\sum_{i,j=1}^N\frac{1}{\sqrt{2{\pi}U^r_{ij}}}exp(-(r_{ij}-r)^2/(2U^r_{ij})),$$ where U$^r_{ij}\equiv\langle({\bf \hat{r}}_{ij}\cdot {\bf u}_{ij}$)$^2\rangle$. Thus we need the mean-squared relative displacements, for pairs of atoms, along the interatomic vector direction. We calculate them within the harmonic approximation at $T=10$ K using our computed vibrational modes. Since $T=10$ K essentially corresponds to $T=0$ K for these considerations, what we obtain is the minimum measurable width for the first peak in the RDF of amorphous silicon. As seen in Fig. \[rdf\]b the results are in agreement with experiment, aside from a small skewing of the theoretical function to large $r$. Although it has not been observed in a-Si, this type of asymmetry has been observed in EXAFS of amorphous germanium [@ge_exafs]. Both the TB1 and TB2 models, despite the very different originating structure and differences in coordination defects, show nearly identical RDF first peaks. The good agreement with experiment of the broadened RDF suggests that our static peak width is correct, and that Laaziri [*et al.*]{} underestimate the static disorder contribution to the broadening. This may be caused by inaccuracy in the polycrystalline $J(r)$ that is used to estimate the dynamic broadening. In the experiments a lower $Q_\mathrm{max}$ (35 Å$^{-1}$) was used for the polycrystal than for the amorphous structure (55 Å$^{-1}$), although the former is expected to have a narrower first peak. Numerous other treatments using EXAFS or diffraction have not been considered here because they all use too low values of $Q_\mathrm{max}$ for obtaining reliable information on the first peak. The only other theoretical study of quantum effects in $J(r)$ is by Herrero [@herrero], who used the SW potential but treated the quantum-effects on the nuclear vibrations exactly. Our results using the SW potential are presented in Fig. \[rdf\]. The result for the amount of zero-point broadening is consistent with Herrero’s work, although due to differing approximations a direct comparison is not possible. We note that the Wooten model on which both the SW and TB1 models are based yields a [*static*]{} $J(r)$ (not shown) that is quite symmetric, and as broad as the [*experimental*]{} breadth. To conclude, we have shown that the NRL-TB method can reliably compute structural, vibrational, and elastic properties of a-Si. The results are nearly identical for two structural models, one with perfect four-fold coordination and one with several atomic percent coordination defects. We have presented the first discussion of force constants in a-Si, which has revealed limitations of the most frequently used empirical potential for silicon. Our calculated elastic constants fall within the range of experimental values for imperfect samples prepared under various conditions. We have also carefully studied the first peak in the radial distribution function. We observe a clear asymmetric peak in the case of the static quantity which is not observable experimentally. We have included the (essentially) zero-point broadening effects in $J(r)$ to obtain the experimentally measured quantity. Our two structural models, which have average coordinations of 4.00 and 4.05, respectively, reproduce the first peak in the experimental $J(r)$ (for the annealed sample) except for a slight asymmetry still present in the broadened result. We believe that such an asymmetry is expected on physical grounds and that perhaps it has been “missed” experimentally because of the challenging analysis required to obtain $J(r)$ from the diffraction results. This work was supported by the U.S. Office of Naval Research. We are also grateful to Dr. S. Roorda for a helpful communication and for sending us the x-ray data of Ref. [@laaz] for the radial distribution function. We thank Dr. S. Richardson for a helpful conversation. [99]{} M.M.J. Treacy, J.M. Gibson, and P.J. Keblinski, J. of Non-Cryst. Solids, [**231**]{}, 99 (1998). D. Beeman, R. Tsu, and M.F. Thorpe, Phys. Rev. B [**32**]{}, 874 (1985). R.L.C. Vink, G.T. Barkema, and W.F. van der Weg, Phys. Rev. B [**63**]{}, 115210 (2001) and references therein. J. Fortner and J.S. Lannin, Phys. Rev. B [**39**]{}, 5527 (1989). K. Laaziri, S. Kycia, S. Roorda, M. Chicoine, J.L. Robertson, J. Wang, and S.C. Moss, Phys. Rev. Lett., [**82**]{}, 3460 (1999) and references therein; ibid., Phys. Rev. B [**60**]{}, 13520 (1999). M.Wakagi, K. Ogata, and A. Nakano, Phys. Rev. B [**50**]{}, 10666 (1994-I); A. Filipponi, F. Evangelisti, M. Benfatto, S. Mobilio, and C.R. Natoli, Phys. Rev. B [**40**]{}, 9636 (1989). C.J. Glover, G.J. Foran, and M.C. Ridgway, Nuc. Instr. and Methods in Physics Res. B [**199**]{}, 195 (2003). E.g. I. Stich, R. Car, and M. Parrinello, Phys. Rev. B [**44**]{}, 11092 (1991-II) P. Biswas, Phys. Lett. A [**282**]{}, 294 (2001). M. Durandurdu and D.A. Drabold, Phys. Rev. B [**64**]{}, 014101 (2001). P. Klein, H. M. Urbassek, and T. Frauenheim, Comp. Mat. Sci. [**13**]{}, 252 (1999). P.B. Allen, J.L. Feldman, J. Fabian, and F. Wooten, Phil. Mag. B [**79**]{}, 1715 (1999). P. Biswas, Phys. Rev. B [**65**]{}, 125208 (2002). S.M. Nakhmanson and D.A. Drabold, J. of Non-Cryst. Solids, [**266-269**]{}, 156 (2000). E.O. Kane, Phys. Rev. B [**31**]{}, 7865 (1985). G.-M. Rignanese, J.-P. Michenaud, and X. Gonze, Phys. Rev. B [**53**]{}, 4488 (1996). R.E. Cohen, M.J. Mehl, and D.A. Papaconstantopoulos, Phys. Rev. B [**50**]{}, 14695 (1994); M.J. Mehl and D.A. Papaconstantopoulos, ibid., [**54**]{} 4519 (1996). N. Bernstein, M. J. Mehl, D. A. Papaconstantopoulos, N. I. Papanicolaou, M. Z. Bazant,and E. Kaxiras, Phys. Rev. B [**62**]{},4477 (2000); ibid., [**65**]{}, 249902 (2002) (E). F. Wooten, private communication. F. Wooten, K. Winer, and D. Weaire, Phys. Rev. Lett. [**54**]{},1392 (1985). F. Kirchhoff, M.J. Mehl, N.I. Papanicolaou, D.A. Papaconstantopoulos, and F.S. Khan, Phys. Rev. B [**63**]{}, 195101 (2001). M. Bazant, private communication; see also J.F. Justo, M.Z. Bazant, E. Kaxiras, V.V. Bulatov, and S Yip, Phys. Rev. B [**58**]{}, 2539 (1998). Although this energy difference suggests the possibility of energetically favorable coordination defects, the small system sizes and minimal annealing make this conclusion uncertain. J.L. Feldman, J.Q. Broughton, and F. Wooten, Phys. Rev. B [**43**]{}, 2152 (1991) and references therein. X. Zhang, J.D. Comins, A.G. Every, P.R. Stoddart, W. Pang, and T.E. Derry, Phys. Rev. B [**58**]{}, 13677 (1998). M. Grimsditch, W. Senn, G. Winterling, and M.H. Brodsky, Sol. State Commun. [**26**]{}, 229 (1978). K. Tanaka, Sol. State Commun. [**60**]{}, 295 (1986). Kuschnereit et al., Applied Phys. A (Mat. Sci. and Proc.) [**61**]{}, 269 (1995). F.H. Stillinger and T.A. Weber, Phys. Rev. B [**31**]{}, 5262 (1985). R.O Pohl, X. Liu, and E. Thompson, Rev. of Mod. Phys. [**74**]{}, 991 (2002). Due to finite-size effects, our results do not exactly satisfy the expected isotropy conditions on the elastic constants of an amorphous material. W.A. Kamitakahara, C.M. Soukoulis, H.R. Shanks, U. Bucheneau, and G.S. Grest, Phys. Rev. B [**36**]{}, 6539 (1987). We tacitly assume that the cell size is big enough that the force on an atom due to the displacement of a different atom and that of its periodic images can be ascribed solely to the closest displaced atom. R.L.C. Vink, G.T. Barkema, W.F. van der Weg, and N. Mousseau, J. Non-Cryst. Solids, [**282**]{}, 248 (2001). C. P. Herrero, Europhys. Lett. [**44**]{}, 734 (1998); C. P. Herrero, J. of Non-Cryst. Solids [**271**]{}, 18 (2000). M.C. Ridgway, C.J. Glover, K.M. Yu, G.J. Foran, C. Clerc, J.L. Hansen, and A.N. Larsen, Phys. Rev. B [**61**]{}, 12586 (2000).

--- abstract: 'We study maximum matchings and topologically-protected zero modes of slightly-diluted square and honeycomb lattice with compensated dilution $n_{\rm vac}$, [*i.e.*]{} exactly equal numbers of surviving sites on the two sublattices. Our approach is based on an alternate proof of the graph-theoretic identity between the number of such zero modes and the number of monomers in any maximum matching of the diluted lattice. Our proof shows that the corresponding wavefunctions can be chosen to live entirely within certain connected components ${\mathcal R}$ of the [*Dulmage-Mendelsohn decomposition*]{} of the lattice independent of the values of hopping amplitudes. This decomposition is argued to also control a factorization property of the partition function of the monomer-dimer model associated with the ensemble of maximum-matchings. In the $n_{\rm vac}\to 0$ limit, we find that the random geometry of these ${\mathcal R}$-type regions exhibits critical behaviour associated with incipient percolation of these regions. We obtain numerical estimates of the universal exponents and scaling functions that characterize this new universality class of incipient percolation in two dimensions. We outline implications of these results for the statistical mechanics of maximum matchings of such lattices, and for the quantum mechanics of a zero-energy particle hopping on such lattices. We also identify several promising lines of enquiry that are opened up by our work.' author: - Sounak Biswas - Ritesh Bhola - Md Mursalin Islam - Kedar Damle title: 'Geometry of maximum matchings and topologically-protected zero modes of slightly-diluted bipartite lattices: Incipient percolation in two dimensions' --- Introduction ============ Static impurities are an important feature of many physical systems, motivating the study of the effect of [*quenched*]{} disorder on various physical phenomena such as electronic conduction, diffusion of fluids, and magnetism. Paradigmatic examples include classical percolation theory which studies the random geometry of the disordered environment, and the theory of Anderson localization, which elucidates the effect of this disordered environment on the wavefunctions and spectrum of a quantum mechanical particle in this environment [@Broadbent_Hammersley; @Stauffer_Aharony_book; @Christensen_Moloney_book; @Anderson; @Abrahams_review]. Here, we study the geometry of topologically-protected (dependent only on the connectivity of the graph and insensitive to the actual values of the nonzero hopping amplitudes) zero energy states of a quantum-mechanical particle hopping with random hopping amplitudes on links of a slightly-diluted square or honeycomb lattice with compensated dilution $n_{\rm vac}$, [*i.e*]{} exactly equal numbers of surviving sites on the two sublattices. ------------------------------------------------------ ------------------------------------------------------ ------------------------------------------------------------ {width="0.66\columnwidth"} {width="0.66\columnwidth"} {width="0.66\columnwidth"} ------------------------------------------------------ ------------------------------------------------------ ------------------------------------------------------------ Our approach is based on an alternative proof of the graph-theoretic identity [@Longuet-Higgins; @Lovasz_Plummer] between the number of such topologically-protected zero modes and the number of monomers in any maximum matching of the diluted lattice. Our proof uses a graph-theoretic tool, namely the [*Dulmage-Mendelsohn decomposition*]{} [@Lovasz_Plummer; @Dulmage_Mendelsohn; @Pothen_Fan; @Kavitha] of a bipartite lattice into a unique set of non-overlapping connected subregions (see Fig. \[Rtypepicture\]) ${\mathcal R}_{i}$ ($i=1\dots N_{\mathcal R}$) and ${\mathcal P}_j$ ($j=1 \dots N_{\mathcal P}$). These have an intuitively appealing characterization from the point of view of the monomer-dimer model defined by the ensemble of maximum-matchings of the lattice: monomers of this monomer-dimer model are all confined to live in these ${\mathcal R}$-type regions, with each ${\mathcal R}$-type region ${{\mathcal R}}_i$ hosting the same fixed nonzero number ${{\mathcal I}}_i$ of monomers in any maximum matching. On the other hand, ${\mathcal P}$-type regions are parts of the lattice that are always perfectly matched in any maximum matching. Using this tool, we demonstrate that the wavefunctions of such topologically protected zero-energy states can be chosen to live entirely within individual ${\mathcal R}$-type regions ${{\mathcal R}}_i$ of this Dulmage-Mendelsohn decomposition, with ${{\mathcal I}}_i$ such linearly-independent zero modes coexisting in region ${{\mathcal R}}_i$. We also argue that the Dulmage-Mendelsohn decomposition implies a complete factorization of the partition function of the monomer-dimer model, with each ${\mathcal R}$-type and ${\mathcal P}$-type region independently contributing one factor to the partition function. This motivates our computational study of the random geometry of the Dulmage-Mendelsohn decomposition. At any nonzero $n_{\rm vac}$, we find a nonzero number density $n_{\mathcal R}$ ($n_{\mathcal P}$) of ${\mathcal R}$-type (${\mathcal P}$-type) regions in the thermodynamic limit (Fig. \[NRandNP\]). These ${\mathcal R}$-type (${\mathcal P}$-type) regions together contain a nonzero fraction $m_{\rm tot}$ ($m_{\mathcal P}$) of sites of the undiluted lattice, with the ${\mathcal R}$-type regions hosting a nonzero areal density $w$ of topologically-protected zero modes (Fig. \[MtotandW\]). In the small-$n_{\rm vac}$ limit, we find that most of the sites of the diluted lattice belong to ${\mathcal R}$-type regions, although $w$ goes rapidly to zero (Fig. \[MtotandW\]). In the range of system sizes ($L$) accessible to us, the typical size of ${\mathcal R}$-type regions is only limited by $L$ (Fig. \[Rgandxi\]) as $n_{\rm vac} \to 0$, and the dominant contribution to both $m_{\rm tot}$ and $w$ comes from such large ${\mathcal R}$-type regions (Fig. \[smallfraction\_mw\]). Indeed, the random geometry of the Dulmage-Mendelsohn decomposition in this $n_{\rm vac} \to 0$ limit is universal, and characterized by incipient percolation of ${\mathcal R}$-type regions. We obtain a numerical estimate of $\nu = 5.5 \pm 0.9$ for the correlation length exponent that characterizes this universality class, and place a bound $\eta \lesssim 0.08$ on the value of the corresponding anomalous exponent $\eta$. A particularly interesting aspect of this Dulmage-Mendelsohn percolation phenomenon is that this nontrivial behaviour is exhibited in the limit of vanishing vacancy density, raising the intriguing possibility of being accessible to a rigorous analysis in the $n_{\rm vac} \to 0^{+}$ limit. This critical phenomenon clearly has interesting implications in both settings: the statistical mechanics of maximum matchings of these diluted bipartite lattices and the quantum mechanics of a zero-energy particle hopping on the lattice. In the former setting, we argue that this geometric criticality can be thought of as [*incipient monomer percolation*]{} in the statistical mechanics of maximum-matchings, since these monomers only exist within ${\mathcal R}$-type regions, and the factorization property implies that they are uncorrelated between two different ${\mathcal R}$-type regions. In the latter context, we argue that the zero-energy Green function $G(x,x')$ is only nonzero when $x$ and $x'$ both belong to the same ${\mathcal R}$-type region, and this critical behaviour therefore corresponds to a [*wavefunction percolation*]{} phenomenon. ------------------------------------------------ ------------------------------------------------ {width="\columnwidth"} {width="\columnwidth"} ------------------------------------------------ ------------------------------------------------ Our results provide answers to interesting questions in three other contexts; indeed, the background and original motivation for our study came from these questions. The first of these is diluted undoped graphene: In earlier work [@Sanyal_Damle_Motrunich_PRL], it was noticed that an anomalous crossover in the low-energy density of single-particle states of the corresponding tight-binding model was controlled by a nonzero density of zero modes. These zero modes were argued to be topologically-protected and arising from the presence of so-called ${\mathcal R}_{A}$-type (${\mathcal R}_{B}$-type) regions having more $A$ ($B$) sublattice sites than $B$ ($A$) sublattice sites, but a boundary consisting of [*only*]{} $B$ ($A$) sites of the bipartite honeycomb lattice. The resulting local sublattice imbalance ${\mathcal I}$ was argued to result in ${\mathcal I}$ disorder-robust zero modes due to the specific structure of the boundary. The smallest nontrivial example of these, namely “${\mathcal R}_6$” regions involving six vacancies at specific relative positions, provided a rigorous lower-bound on the density of such disorder-robust zero modes. However, a calculation [@Sanyal_Damle_Motrunich_PRL] of the thermodynamic density of zero modes using multiprecision numerics yielded a value that was much greater in magnitude than this bound. The existence of this relatively large density of zero modes was also independently confirmed using the connection to maximum matchings to count the zero modes [@Weik_etal; @Footnote_Weik]. These results lead to the natural question: How does one characterize the vast majority of such ${\mathcal R}$-type regions, and what do the corresponding zero mode wavefunctions look like? The present work provides the answer: The ${\mathcal R}$-type regions introduced in Ref. [@Sanyal_Damle_Motrunich_PRL] are precisely the connected components ${\mathcal R}_i$ of the Dulmage-Mendelsohn decomposition of the diluted lattice, which can be obtained from [*any*]{} one maximum matching of the lattice. Our geometric study shows that an overwhelmingly large majority of zero modes (we will be more precise below) are associated with ${\mathcal R}$-type regions of typical spatial extent $\xi (n_{\rm vac}) \sim n_{\rm vac}^{-\nu}$ (with $\nu = 5.5 \pm 0.9$, as mentioned earlier). A second closely-related motivation comes from the fact that isolated non-magnetic impurities (vacancies) bind a $\pi$-flux and nucleate a zero-energy Majorana excitation in Kitaev’s exactly solvable honeycomb-lattice model of a Z$_2$ spin liquid  [@Kitaev_anyon; @Willans_Chalker_Moessner_PRB; @Udagawa], and in the exactly solvable Yao-Lee model of a SU(2) symmetric Z$_2$ spin liquid on the honeycomb lattice [@Yao_Lee; @Sanyal_Damle_Chalker_Moessner]. However, in the presence of a nonzero density of vacancies, most of these zero-energy states are lifted by their mixing, and merely contribute to a pile-up in the low-energy density of states [@Sanyal_Damle_Motrunich_PRL; @Willans_Chalker_Moessner_PRB; @Sanyal_Damle_Chalker_Moessner; @Hafner_etal_PRL; @Ostrovsky_etal_PRL]. The zero modes studied here represent exceptions to this, [*i.e.*]{} surviving zero-energy Majorana excitations whose existence is topologically protected in the presence of a nonzero density of nonmagnetic impurities and exchange-disorder. In the SU(2) symmetric spin liquid model of Yao and Lee [@Yao_Lee], Ref [@Sanyal_Damle_Chalker_Moessner] showed that these modes give rise to an impurity-induced Curie tail in the linear suscepbtibility. Although such a Curie tail is usually taken to signal the presence of local moments, the results obtained here on the spatial structure of these zero modes establish that this Curie tail is actually dominated by the response of moments which are spread out over regions of linear size $\xi(n_{\rm vac}) \sim n_{\rm vac}^{-\nu}$ that diverges in the small-$n_{\rm vac}$ limit. More broadly, our results are of interest in the context of the low-energy physics of Majorana networks [@Alicea_etal; @Laumann_Ludwig_Huse_Trebst] that can potentially be realized for instance by exploiting the localized Majorana fermion excitations of topological superconductors [@Read_Green; @Kitaev_chain]. Such networks are described by the generic quadratic Hamiltonian: $H_{\rm Majorana}= i\sum_{r r'} t_{r r'} \eta_r \eta_r'$, where $r$, $r'$ represent spatial locations of individual Majorana modes, $it_{r r'}$ characterizes the network of purely imaginary mixing amplitudes that couple these modes, and we have neglected quartic interaction terms that can become important in some cases [@Affleck_Rahmani_Pikulin; @Li_Franz]. When $t_{r r'}$ has a bipartite structure, for instance corresponding to nonzero amplitudes that couple nearest neighbours of a diluted bipartite lattice, this is a particular case (with real $t_{r r'}$ ) of the bipartite random hopping problem studied here. Interestingly, if this bipartite network is weakly perturbed by additional short-range mixing amplitudes that destroy the bipartite structure, each ${{\mathcal R}}$-type region ${{\mathcal R}}_i$ with an odd ${{\mathcal I}}_i$ is expected to host one Majorana mode that is perturbatively stable to the leading effects of the additional mixing amplitdues, while regions with even ${{\mathcal I}}_i$ do not host any such modes. This lends additional significance to the geometry of ${\mathcal R}$-type regions with odd ${\mathcal I}$. The rest of this article is organized as follows: In Sec. \[DM\], we provide a quick summary of of the standard Dulmage-Mendelsohn decomposition of a bipartite graph. In Sec. \[ZMGF\], we first use this decomposition to provide an alternate proof of the correspondence between the number of zero modes and the number of monomers alluded to earlier. This also enables us to prove the topologically-protected localization property of the zero-energy Green function of the hopping problem. In Sec. \[MD\], we discuss consequences for correlations in the monomer-dimer model, followed by a brief discussion in Sec. \[Majorana\] of implications for the perturbative stability of Majorana modes in bipartite Majorana networks. This sets the stage for a detailed account of the random geometry of Dulmage-Mendelsohn clusters in Sec. \[Geometry\]. This is split into a discussion of the basic picture in Sec. \[Basic\], followed by a detailed account of our finite-size scaling analysis in Sec. \[Scaling\], a discussion of the morphology of critical ${\mathcal R}$-type regions in Sec. \[Morphology\], and a summary in Sec. \[Odd\] of scaling properties of ${\mathcal R}$-type regions with odd monomer number ${\mathcal I}$. In Sec. \[Coulomb\], we provide a heuristic reinterpretation of our results in the language of fluctuating polarization fields commonly used to discuss large-length scale properties of bipartite dimer models. This is followed by a brief aside (Sec. \[Asides\]), in which we discuss how our ideas lead to a unified perspective on some interesting recent work on Penrose tilings, and are relevant to some aspects of the previous results on diluted quantum antiferromagnets. Finally, in Sec. \[Outlook\], we discuss several interesting lines of enquiry that are opened up by our results. -------------------------------------------------- ----------------------------------------------- {width="\columnwidth"} {width="\columnwidth"} -------------------------------------------------- ----------------------------------------------- The Dulmage-Mendelsohn decomposition {#DM} ==================================== The structural results of Dulmage and Mendelsohn [@Lovasz_Plummer; @Dulmage_Mendelsohn; @Pothen_Fan; @Kavitha] on bipartite graphs are simple and elegant, but perhaps not well-known to physicists. Here, we attempt to remedy this with a self-contained account, which also sets up our notation. With minor embellishments intended to better emphasize features of interest to us, what follows below is the version summarized in Ref. [@Kavitha]. Consider the monomer-dimer model on a bipartite graph, with the proviso that the number of monomers is restricted to the minimum possible value. Dulmage and Mendelsohn showed that this ensemble of maximum matchings defines a [*unique*]{} structural decomposition of the underlying graph, [*independent*]{} of the matching one starts with. To establish this, start with any one maximum matching, which has monomers at lattice sites $h_{k}$, with $k=1 \dots W$. Here, $W$ is the number of unmatched sites in any maximum matching. Now, one introduces the notion of an alternating path starting from unmatched vertices that host monomers: These are paths that begin at any unmatched vertex $h_k$, traverse any one of the unmatched (unoccupied by dimers) links emanating from it and subsequently go along an alternating sequence of matched and unmatched links of the lattice without visiting any site more than once or traversing any link more than once. Maximum matchings are characterized by the absence of alternating paths of odd length which start and end at unmatched sites (if there was such a path, one could add one more dimer to the system by switching the occupied and unoccupied links of this path, and the original matching would not have maximum cardinality). Next, one can classify sites into three groups, odd (o), even (e) and unreachable (u): Unreachable sites cannot be reached by such an alternating path starting from any monomer. Odd sites can be reached by an alternating path of odd length starting from some monomer. And even sites are those that can be reached by an alternating path of even length (including $0$) starting from some monomer (this class therefore includes the unmatched sites themselves). Using the fact that there are no alternating paths of odd length starting and ending at monomers in any maximum matching, it is fairly straightforward to see that this classification into odd, even and unreachable is independent of the maximum matching one starts with. Indeed, it represents a fundamental structural property of the graph itself. Also, even sites are never connected by a link of the lattice to other even sites or to unreachable sites. The connected components ${\mathcal R}_i$ and ${\mathcal P}_j$ of interest to us are now readily obtained: Color all odd and even sites red. Also color red all links that connect one odd and one even site. Color all unreachable sites blue. Also color blue all links between two unreachable sites. And delete all links between unreachable sites and odd sites (even sites are never neighbours of unreachable sites), as well as all links between two odd sites (two even sites never have a link between them). The graph now splits into $N_{\mathcal R} +N_{\mathcal P}$ connected components (due to the deletion of links described above): Red components ${\mathcal R}_i$ ($i=1 \dots N_{\mathcal R}$) and blue components ${\mathcal P}_j$ ($j=1 \dots N_{\mathcal P}$). It is easy to check that this decomposition is unique, no matter which maximum matching one starts with. One can also readily verify that the deleted links never host a dimer in any maximum matching; indeed, dimers only occupy links between two unreachable sites or between an odd site and an even site. Further, it is straightforward to see that no monomers live in the ${\mathcal P}$-type clusters. Indeed, each ${\mathcal P}_j$ always hosts a perfect matching, with its sites always being perfectly matched amongst themselves in any maximum matching. Additionally, one notes that the boundary sites of any ${\mathcal R}$-type region ([*i.e*]{} those connected to sites outside it by deleted links) are always of the odd type and always matched with an even site of the same ${\mathcal R}$-type region in any maximum matching. ----------------------------------------------------------- ----------------------------------------------------------- {width="\columnwidth"} {width="\columnwidth"} ----------------------------------------------------------- ----------------------------------------------------------- It is also clear that monomers of any maximum matching always live on even sites inside a ${\mathcal R}$-type region, with each ${\mathcal R}$-type region hosting the same fixed nonzero number of monomers in any maximum matching. Further, since all odd sites of a ${\mathcal R}$-type region are matched to an even site of the same region, this implies that the number of even sites in any ${\mathcal R}$-type region exceeds the number of odd sites [*in spite of all boundary sites being odd*]{}. Additionally, we note that these ${\mathcal R}$-type regions come in two “flavours” ${\mathcal R}_A$ and ${\mathcal R}_B$: ${\mathcal R}_A$-type ( ${\mathcal R}_B$-type) regions have all their even sites on the $A$ ($B$) sublattice and all their odd sites on the $B$ ($A$) sublattice. Finally we note that each ${\mathcal P}$-type region ${\mathcal P}_j$ itself can be uniquely decomposed into subregions defined to ensure that each overlap loop (in the ensemble of overlap loops obtained by superimposing any two perfect matchings of ${\mathcal P}_j$) lives entirely within a single subregion of ${\mathcal P}_j$. This corresponds to the so-called “fine decomposition” of Dulmage and Mendelsohn used by Pothen and Fan in their algorithm for block triangularization of matrices [@Dulmage_Mendelsohn; @Pothen_Fan]. As we will see below, this can play a potentially crucial role in determining the dimer correlation length of the monomer-dimer model in certain cases. Implications {#Consequences} ============ We now present three arguments that use this structural decomposition to provide information on i) topologically-protected zero mode wavefunctions and the corresponding zero-energy Green function, ii) on monomer and dimer correlation functions of the monomer-dimer model corresponding to the ensemble of maximum matchings, and iii) on the perturbative stability of Majorana modes in bipartite Majorana networks. Topologically-protected zero modes and zero-energy Green function {#ZMGF} ----------------------------------------------------------------- We first discuss topologically-protected zero modes of the hopping problem. By topologically-protected zero modes, we mean zero modes whose existence is robust to changes in the actual values of the hopping amplitudes, and depends only on the connectivity of the graph, [*i.e.*]{} on whether a given hopping amplitude is zero or nonzero. Such modes are disorder-robust, since their existence is unaffected by small randomness or modulations in the hopping amplitudes. In Ref. [@Sanyal_Damle_Motrunich_PRL], the nonzero density of zero modes of the tightbinding model for diluted graphene was argued to arise from the presence of so-called ${\mathcal R}_{A}$-type ( ${\mathcal R}_{B}$-type) regions of the diluted sample, having more $A$ ($B$) sites than $B$ ($A$) sites, but a boundary consisting of only $B$ ($A$) sites. From the foregoing summary of the Dulmage-Mendelsohn decomposition, it is now clear that these ${\mathcal R}$-type regions of Ref. [@Sanyal_Damle_Motrunich_PRL] may be identified precisely with the connected components ${\mathcal R}_i$ of the Dulmage-Mendelsohn decomposition of the diluted lattice. This allows us to now make a precise argument for the number and structure of such topologically-protected zero modes: Let the number of odd sites in any particular ${\mathcal R}$-type region be $N_o$ and the number of even sites be $N_e$, with their difference ${\mathcal I}= N_e -N_o$ being a positive number equal to the number of monomers hosted by this region in any maximum matching. It is easy to see that there are at least ${\mathcal I}$ linearly independent solutions of the zero-energy Schrodinger equation, with the property that the corresponding wavefunctions are only nonzero on even sites of this region. The reason is straightforward linear algebra: Writing down the Schrodinger equation for zero-energy wavefunctions of this form, we obtain a rectangular system of $N_0$ equations to be satisfied by the $N_e$ nonzero components of the wavefunction. This is because all boundary sites of a ${\mathcal R}$-type region are odd, and we are only solving for wavefunctions that are nonzero on even sites of that particular ${\mathcal R}$-type region. The minimum number of solutions of this set of equations equals $N_e -r_{\rm max}$, where $r_{\rm max}$ is the maximum rank of the corresponding rectangular matrix with $N_e$ columns and $N_o$ rows. Since $N_o < N_e$, $r_{\rm max} \leq N_o$, implying the existence at least $N_e -N_o \equiv {\mathcal I}$ linearly independent solutions. Since ${\mathcal I}$ of these are guaranteed to exist independent of the precise values of the nonzero hopping amplitudes, we see that each ${\mathcal R}$-type region ${\mathcal R}_i$ of the Dulmage-Mendelsohn decomposition contributes exactly ${\mathcal I}_i$ topologically-protected zero modes in the quantum mechanics of a particle hopping on the lattice. The total number of topologically-protected zero modes is thus $W=\sum_i {\mathcal I}_i$, [*i.e.*]{} exactly equal to the number of monomers in any maximum matching of the lattice. This provides an alternate route to the well-known graph-theoretic identity between the number of topologically-protected zero modes $W$ and the number of monomers in any maximum matching [@Longuet-Higgins; @Lovasz_Plummer]. In contrast to the standard “global” approaches [@Longuet-Higgins; @Lovasz_Plummer] that makes use of determinants to prove this, our “local” argument uses the structure of the Dulmage-Mendelsohn decomposition to provide a constructive proof. In the process, it uncovers a crucial aspect of the structure of the corresponding wavefunctions, namely, the fact that it is possible to choose a basis in the zero-energy eigenspace such that each zero-energy wavefunction of this basis lives entirely within one ${\mathcal R}$-type region, with ${\mathcal I}_i$ such basis functions co-existing in region ${\mathcal R}_i$ (for $i=1 \dots N_{\mathcal R}$). Next, we consider the zero-energy Green function $G(x,x') \equiv \sum_{\mu} \psi_\mu^*(x) \psi_\mu(x')$, where the sum on $\mu$ is over $W$ orthonormal zero-energy eigenfunctions that make up any choice of basis for the zero-energy eigenspace. By construction, the Green function defined in this manner is independent of this choice of basis. It is particularly convenient to evaluate it using the basis described above, [*i.e.*]{} consisting of ${\mathcal I}_i$ orthonormal wavefunctions that live entirely within each ${\mathcal R}$-type region ${\mathcal R}_i$ for $i= 1 \dots N_{\mathcal R}$. Employing this basis, we see that $G(x,x')$ in only nonzero if $x$ and $x'$ both belong to the same ${\mathcal R}$-type region. There is of course considerable residual freedom in choosing an orthonormal set to span this ${\mathcal I}_i$ dimensional subspace of wavefunctions that live within a single region ${\mathcal R}_i$. But again, the Green function is of course independent of these choices. Thus, our argument identifies a [*generic*]{} maximally-localized choice of zero-energy eigenbasis whose localization property is topological in nature and implies a corresponding topologically-protected localization property of the zero-energy Green function. However, the degree of localization of the Green function within any given ${\mathcal R}_i$ does depend on the actual values of the nonzero hopping amplitudes in that region and requires independent study. -------------------------------------------------- ---------------------------------------------------------- {width="\columnwidth"} {width="\columnwidth"} -------------------------------------------------- ---------------------------------------------------------- Monomer and dimer correlations {#MD} ------------------------------ Next, we turn to the statistical mechanics of the monomer-dimer model defined by the ensemble of maximum matchings. From the foregoing description of the Dulmage-Mendelsohn decomposition, it is not hard to see that this ensemble can be generated from any one maximum-matching by making all possible rearrangements of dimers (keeping their number fixed) [*independently*]{} within each connected component ${\mathcal R}_i$ and each connected component ${\mathcal P}_j$. To see that this is true, it suffices to note that odd sites are always matched to an even site in the same ${\mathcal R}$-type region, and unreachable sites are always matched with another unreachable site in the same ${\mathcal P}$-type region. Therefore all possible maximum matchings can be obtained from a single maximum matching by rearranging the dimers within each ${\mathcal R}$-type region and each ${\mathcal P}$-type region. The partition function $Z$ of the corresponding monomer-dimer model thus factorizes: $$\begin{aligned} Z&=&\left(\prod_i Z_{{\mathcal P}_i} \right ) \times \left(\prod_j Z_{{\mathcal R}_j} \right)\; . \label{factorization}\end{aligned}$$ This factorization is topological in nature, in the sense that it is independent of numerical values of the nonzero bond fugacities assigned to each link and only depends on the connectivity of the graph. It immediately implies that monomer and dimer correlations do not extend beyond individual connected components. The foregoing implies that the typical size of a ${\mathcal R}$-type region places an upper bound on the length scale over which correlations can propagate in the gas of monomers. For dimer correlations, the situation is more complicated, for the average two-point function of dimers has two contributions: One arising from dimer correlations within the ${\mathcal R}$-type regions and the other arising from perfectly-matched ${\mathcal P}$-type regions (each of which hosts a fully-packed dimer model in an irregular geometry). In the slightly-diluted case we study, it is not a priori obvious which of these dominates the long-distance behaviour of the average two-point dimer correlator. The issue is the following: Although the ${\mathcal P}$-type regions turn out to be typically much smaller than ${\mathcal R}$-type regions in the slightly-diluted samples studied here, there is no [*a priori*]{} guarantee that their contribution is negligible compared to that of ${\mathcal R}$-type regions. This is because the dimer correlation length in the ${\mathcal R}$-type regions can potentially be much smaller than the typical size of such a region, due to the presence of a coexisting gas of monomers. Indeed, this question becomes tricky if the dimer correlation length in ${\mathcal R}$-type regions is of the same order of magnitude as the typical size of ${\mathcal P}$-type regions. In this case, the fine decomposition of ${\mathcal P}$-type regions [@Dulmage_Mendelsohn; @Pothen_Fan] comes into play: As we have noted in the previous section, each ${\mathcal P}$-type region itself can be decomposed into subregions that are defined to ensure that overlap loops between two perfect matchings of ${\mathcal P}$ live entirely within a single subregion. This implies a further factorization of each $Z_{{\mathcal P}_i}$ in Eqn. \[factorization\], and means that dimer correlations cannot extend beyond these individual subregions identified by the fine decomposition. In this case, it is the typical size of these subregions that potentially play the key role in determining the dimer correlation length. Structure of surviving Majorana modes in bipartite networks {#Majorana} ----------------------------------------------------------- Our understanding of the spatial structure of topologically-protected zero mode wavefunctions also suggests some interesting consequences for nature of surviving Majorana fermion excitations in Majorana networks [@Alicea_etal; @Laumann_Ludwig_Huse_Trebst] modeled by the Hamiltonian: $$\begin{aligned} H_{\rm Majorana} &=& i\sum_{r r'} t_{r r'} \eta_r \eta_{r'}\; , \label{H_Majorana}\end{aligned}$$ where $r$, $r'$ represent the locations of individual Majorana modes, $it_{r r'}$ characterizes the bipartite network of purely imaginary mixing amplitudes that couple these modes, and we have neglected quartic interaction terms whose effects are assumed unimportant in our analysis below. If $it_{rr'}$ has a bipartite structure, say corresponding to nearest-neighbour links of a diluted bipartite lattice, the Dulmage-Mendelsohn decomposition immediately yields the structure of the surviving Majorana zero modes of the network, since this maps to a special case of the bipartite hopping problem, with real hopping amplitudes (via a basis change that multiplies the single-particle orbitals on the $B$ sublattice by a factor of $i=\sqrt{-1}$) [@Motrunich_Damle_Huse_PRB1]. Here we consider the perturbative effect of weak additional mixing amplitudes that couple nearby modes on the same sublattice. In other words, we write $it_{r r'} = it^{(0)}_{r r'} + i\delta t_{r r'}$, where $\delta t$ represents weak mixing between modes on nearby sites of the same sublattice of a bipartite lattice. To understand the effects of $\delta t$ on the surviving Majorana zero modes of the network to leading order in perturbation theory, we must project the perturbation $\delta t$ into the zero-energy subspace of $t^{(0)}_{r r'}$ and diagonalize the resulting problem. This is greatly facilitated by thinking in terms of the maximally-localized choice of basis described above, with each unperturbed zero-energy basis state living entirely within any one ${\mathcal R}$-type region, and ${\mathcal I}_i$ such basis states coexisting in region ${\mathcal R}_i$ for $i=1 \dots N_{\mathcal R}$. To see this, we consider for concreteness a $t^{0}_{r r'}$ whose nonzero elements correspond to nearest-neighbour links of a diluted honeycomb lattice, and a perturbation $\delta t$ which mixes next-nearest-neighbour sites of this lattice. We begin by noting that even sites (in the language of Sec. \[DM\]) of ${\mathcal R}_A$ (${\mathcal R}_B$) regions are all part of the $A$ ($B$) sublattice, and boundary sites of ${\mathcal R}_A$ (${\mathcal R}_B$) regions are all odd (again, in the language of Sec. \[DM\]), [*i.e.*]{} part of the $B$ ($A$) sublattice. Additionally, we recall that the basis states have nonzero amplitudes only on even sites of ${\mathcal R}$-type regions. As a result of this structure, the projection of $\delta t_{r r'}$ into the unperturbed zero-energy subspace spanned by this basis has a block-diagonal form in this basis if $\delta t_{r r'}$ is zero beyond next-nearest neighbour sites (in the sense of connectivity, not geometric distance). Indeed, when $\delta t_{r r'}$ does not extend beyond next-nearest neighbour sites, this projection decomposes into $N_{\mathcal R}$ independent blocks corresponding to the ${\mathcal R}$-type regions of the lattice, with each region ${\mathcal R}_i$ contributing a block of size ${\mathcal I}_i \times {\mathcal I}_i$ for $i= 1 \dots N_{\mathcal R}$. Further analysis depends crucially on the fact that the excitation spectrum has “chiral” symmetry which guarantees that each state at energy $\epsilon > 0$ has a partner at energy $-\epsilon$ [@Motrunich_Damle_Huse_PRB1]. This property is inherited from the original problem by each projected ${\mathcal I}_i \times {\mathcal I}_i$ block that determines the leading perturbative effects of $\delta t_{r r'}$. As a result of this symmetry, one can immediately conclude that every region ${\mathcal R}_i$ with odd ${\mathcal I}_i$ will host [*at least one*]{} Majorana zero mode that survives the leading-order effects of any perturbation $\delta t_{r r'}$ that does not extend beyond next-nearest neighbors. This lends additional significance to ${\mathcal R}$-type regions with odd imbalance ${\mathcal I}$. These robust Majorana modes hosted by such ${\mathcal R}$-type regions with odd ${\mathcal I}$ are also potentially stable to the leading effects of additional longer-range but weak mixing terms. However, this depends on the detailed morphology of the ${\mathcal R}$-type regions and the range of the perturbation. Finally, we note that the argument sketched here is somewhat analogous to established results in the one-dimensional case of Majorana wires, as exemplified by paradigmatic Kitaev chain [@Kitaev_chain]. In the thermodynamic limit of the original Kitaev chain, there is one Majorana mode at each end of the chain in the topological phase. Together, these two form a single complex fermion whose wavefunction is split between the two ends of the chain, and whose energy goes to zero exponentially in the thermodynamic limit. In multichannel generalizations of this situation, one finds that Majorana modes at one end of the wire can generically mix due to additional local mixing terms in the Hamiltonian, leading to a situation in which each end of the wire has either a single Majorana mode or none, depending on whether the number of original Majorana modes at each end was odd or even [@Potter_Lee]. ![The mean separation between vacancies, $l_{\rm vac} = 1/\sqrt{n_{\rm vac}}$, the mean distance between zero modes, $l_w = 1/\sqrt{w}$, the correlation length $\xi$ associated with the sample-averaged correlation function $C$, and $R_{\rm max}$, the sample-averaged radius of gyration of the largest ${\mathcal R}$-type region in a sample, plotted as functions of the vacancy density $n_{\rm vac}$. Note that $\xi$ and $R_{\rm max}$ track each other and both are much larger than the other two length scales in the small-$n_{\rm vac}$ limit, being limited only by the system size $L$ in this regime in the range of sizes accessible to our numerical work. See Sec.  for details. []{data-label="length-scalebasic"}](./plots/lengthscalesbasic.pdf){width="\columnwidth"} ![The probability $P$ that a sample with periodic boundary conditions has at least one ${\mathcal R}$-type region that wraps around at least one of the periodic directions tends towards $P=1$ in the small-$n_{\rm vac}$ limit. Note that curves for different sample sizes $L$ never cross each other at any nonzero $n_{\rm vac}$ accessible to our numerical work. See Sec.  for details.[]{data-label="Pw"}](./plots/Pw.pdf){width="\columnwidth"} Random geometry of ${\mathcal R}$-type regions: {#Geometry} =============================================== These observations on the behaviour of the zero-energy Green function, monomer and dimer correlators, and Majorana modes provide the key motivation for our computational study of the random geometry of the Dulmage-Mendelsohn decomposition of diluted bipartite lattices. This is what we turn to next. In our computational work, we focus on random dilution of $L\times L$ honeycomb (square) lattices consisting of $2L^2$ ($L^2$) sites, with periodic boundary conditions and and even $L$. To largely eliminate the possibility of subregions of the lattice disconnecting completely from the rest of the graph, we have chosen to impose a nearest and next-nearest exclusion constraint on the position of the vacancies. At the small vacancy concentrations $n_{\rm vac}$ that we study, this effectively eliminates any complications associated with the break up of the lattice into multiple geometrically disconnected components. Additionally we restrict ourselves to samples with compensated dilution, so that each random sample in our ensemble has an equal number of $A$ and $B$ sublattice sites. These two precautions ensure that our dataset is not “contaminated” by spurious effects associated with a global imbalance in the number of $A$ and $B$ sublattice sites or a similar imbalance at the level of individual disconnected components. However, we have also compared these results with corresponding results in an ensemble that has completely uncorrelated but compensated dilution. In this latter ensemble, especially in the honeycomb lattice case, the system typically has multiple disconnected components. Some fraction of these disconnected components, especially the smaller disconnected components, are of course imbalanced, and have additional trivial zero modes associated with this imbalance. However, the small values of dilution $n_{\rm vac}$ considered here place the sample deep within the percolated phase from the point of view of geometric percolation. As a result, the effects we focus on are dominated by the contribution of the largest geometric component, and are largely insensitive to this difference in correlations of the vacancy ensemble. The exception is some non-critical quantities (such as $N_{\mathcal R}$, the total number of ${\mathcal R}$-type regions) that depend sensitively on the lack of exclusion constraints amongst vacancies. In order to streamline the discussion, we therefore focus here on the vacancy ensemble with nearest-neighbour and next-nearest-neighbour exclusions on both lattices. Our tests of the efficiency of various maximum-matching algorithms described in Ref. [@Duff_Kaya_Ucar] suggest that the Breadth-First-Search (BFS) algorithm with pruning of search branches outperforms the others (including the Hopcroft-Karp algorithm which has the theoretical advantage in terms of worst-case complexity) in the regime of interest to us. To increase the computational efficiency of our matching code, we choose to use the maximum matching at a lower vacancy concentration to obtain an initial condition for the matching algorithm at the next higher concentration, and work out way up a grid of concentrations. This gives us one random sample at each concentration. This process is then repeated for many times to generate our ensemble. For most of the data shown, we use an ensemble consisting of at least $3000$ samples at the largest sizes, with smaller size data being obtained by averaging over a substantially larger number of samples. For the vacancy-ensemble without exclusion constraints, we have also tried to go in the reverse direction, [*i.e.*]{} start with a larger value of $n_{\rm vac}$ and randomly remove vacancies (add sites) to go to lower values of $n_{\rm vac}$. This is somewhat more efficient; however, the data shown here for the ensemble with exclusion amongst vacancies has been obtained from an ascending sequence of $n_{\rm vac}$. Once we have constructed a maximum matching of the diluted lattice corresponding to a given sample, we construct alternating paths starting from the unmatched sites. By using this network of alternating paths, we label each site odd (o), even (e) or unreachable (u) as defined in Sec. \[DM\]. Based on this labeling, we construct the Dulmage-Mendelsohn regions ${\mathcal R}_i$ ($i=1 \dots N_{\mathcal R}$) and ${\mathcal P}_j$ ($j=1 \dots N_{\mathcal P}$) using a simple burning algorithm. Below, we describe our results on the statistics of these regions in slightly-diluted honeycomb and square lattices. Basic picture {#Basic} ------------- We begin by asking some basic questions: How does $M_{\rm tot} = \sum_i^{N_{\mathcal R}} m_i$, the total “mass” (number of sites) contained in all ${\mathcal R}$-type regions, scale with $L$ and $n_{\rm vac}$ in the small-$n_{\rm vac}$ limit? How does the number $N_{\mathcal R}$ ($N_{\mathcal P}$) of ${\mathcal R}$-type (${\mathcal P}$-type) regions scale with system size $L$ and vacancy concentration $n_{\rm vac}$ in the small-$n_{\rm vac}$ limit? How does the total number of zero modes $W$ scale with system size $L$ and $n_{\rm vac}$ in this limit? Our answers to these questions are displayed in Fig. \[NRandNP\] and Fig. \[MtotandW\]. ![The probability $P$ that a $L \times L$ sample with periodic boundary conditions has at least one ${\mathcal R}$-type region that wraps around at least one of the periodic directions collapses onto a single scaling curve when plotted against the scaling variable $Ln_{\rm vac}^{\nu}$ with $\nu = 5.0 \pm 0.4$ ($\nu = 5.7 \pm 0.5$) for honeycomb (square) lattices of various large sizes $L$ in the limit of small dilution $n_{\rm vac}$. See Sec.  for details.[]{data-label="scalingP"}](./plots/Pwsc.pdf){width="\columnwidth"} From the behaviour of $n_{\mathcal R} \equiv N_{\mathcal R}/2L^2$ ($n_{\mathcal R} \equiv N_{\mathcal R}/L^2$) and $n_{\mathcal P} \equiv N_{\mathcal P}/2L^2$ ($n_{\mathcal P} \equiv N_{\mathcal P}/L^2$) for the honeycomb (square) lattice, we see that $n_{\mathcal R}$ and $n_{\mathcal P}$ both tend rapidly to a nonzero value in the thermodynamic limit on both lattices, with finite-size corrections that are not readily visible at the sizes used studied here. From the $n_{\rm vac}$ dependence of these quantities, we see that these densities decrease rapidly to zero as $n_{\rm vac} \to 0$ (Fig. \[NRandNP\]). From Fig. \[MtotandW\], we see that $m_{\mathrm tot} \equiv M_{\rm tot}/2L^2$ ($m_{\rm tot} \equiv M_{\mathrm tot}/L^2$)saturates rapidly to a nonzero value in the thermodynamic limit of the honeycomb (square) lattice, with finite-size corrections that are not visible in the size range studied. Moreover $m_{\rm tot}$ apparently tends towards the value $m_{\mathrm tot} = 1-n_{\rm vac}$ as $n_{\rm vac}$ goes to zero. In conjunction with the behaviour of $n_{\mathcal R}$, this implies that the entire sample is taken over by a few ${\mathcal R}$-type regions in this limit. This is, at first sight, a suprising counter-intuitive result, since $n_{\rm vac}=0$ corresponds to the pure square or honeycomb lattice, which admits a perfect matching. In other words [*at*]{} $n_{\rm vac} = 0$, the undiluted system has a single ${\mathcal P}$-type region in the language of the Dulmage-Mendelsohn decomposition. How are we to reconcile this counter-intuitive result for $m_{\rm tot}$ with the existence of perfect matchings at $n_{\rm vac}=0$? The answer has to do with the order of limits. A nonzero density of vacancies is naturally associated with the length scale $l_{\rm vac} = 1/\sqrt{n_{\rm vac}}$, corresponding to the typical distance between vacancies. If we take the thermodynamic limit at fixed nonzero $n_{\rm vac}$ and then take the limit $n_{\rm vac} \to 0$, we are studying the limit $L/l_{\rm vac} \to \infty$, $l_{\rm vac} \to \infty$. In this case, we see from our computations that $m_{\rm tot} \to 1$ in this $n_{\rm vac} \to 0$ limit. On the other hand, if we first send $n_{\rm vac}$ to zero while keeping $L$ fixed, and then take the thermodynamic limit, we are studying the limit $L \to \infty$, $l_{\rm vac} /L\to \infty$. This corresponds to the limiting case of the pure system for which $m_{\rm tot} \to 0$, since the entire sample has a perfect matching. From Fig. \[MtotandW\], we also see that the density of zero modes $w \equiv W/2L^2$ ($w \equiv W/L^2$) on the honeycomb (square) lattice saturates to a nonzero value in the thermodynamic limit, with finite-size corrections that are not readily visible in the range of sizes studied here. As expected, we also see that $w$ tends to zero as $n_{\rm vac} \to 0$. A nonzero $w$ defines a second length scale $l_w \equiv 1/\sqrt{w}$. This length scale $l_w$ is a measure of the “typical distance” between zero modes if one thinks of them as localized objects. From the $n_{\rm vac}$-dependence of $w$ in the small-$n_{\rm vac}$ limit, we see that $l_w/l_{\rm vac} \to \infty$ in this limit, since $w$ goes to zero very rapidly in the limit of small dilution. Thus, these two length-scales have a parametrically large separation $l_w \gg l_{\rm vac}$ at small $n_{\rm vac}$. In other words, zero modes in this limit are a cooperative effect of a very large number of vacancies. Naturally, one cannot take our earlier interpretation of $l_w$ too literally if the wavefunctions of individual modes are spread out over the whole ${\mathcal R}$-type region to which they belong (more precisely, if the localization length scale of the zero-energy Green function $G$ is comparable to the typical size of ${\mathcal R}$-type regions). Nevertheless, we see that the typical size of ${\mathcal R}$-type regions of a finite-size sample in the small-$n_{\rm vac}$ limit depends [*a priori*]{} on these two length scales $l_{\rm vac}$ and $l_w$, in addition to the sample size $L$. ![The sample-averaged radius of gyration $R_{\rm max}$ of the largest ${\mathcal R}$-type regions in a sample obeys a scaling form $LF_{R_{\rm max}}(Ln_{\rm vac}^{\nu})$ with $\nu = 4.9 \pm 0.4$ ($\nu = 5.2 \pm 0.4$) on the honeycomb (square) lattice, as is clear from the fact that plots of $R_{\rm max}/L$ for different sample sizes $L$ collapse on to a single curve when plotted against the scaling variable $Ln_{\rm vac}^\nu$. See Sec.  for details.[]{data-label="scalingRmax"}](./plots/R_scaled.pdf){width="\columnwidth"} We now come to a key observation that motivates much of our subsequent analysis. This has to do with the $n_{\rm vac}$-dependence of a suitably-defined length scale associated with the size of ${\mathcal R}$-type regions. Two natural definitions are possible. The first is simply $R_{\rm max}$, the mean radius of gyration of the largest ${\mathcal R}$-type region in a finite-size sample. Since ${\mathcal R}$-type regions come in two flavours, we use in practice the average of the radius of gyration of the largest ${\mathcal R}_A$ region and the largest ${\mathcal R}_B$ region. The second length scale $\xi$ admits a natural interpretation as a correlation length associated with a sample-averaged geometric correlation function $C(x, x')$ which gets a contribution of $+1$ from a sample if both $x$ and $x'$ are in the same ${\mathcal R}$-type region in that sample, and zero otherwise [@Stauffer_Aharony_book; @Christensen_Moloney_book]. In terms of $C$, we define $$\begin{aligned} \xi^2 &=& \frac{\sum_{x, x'} C(x,x') |x-x'|^2}{\sum_{x, x'} C(x,x') } \label{xidefn}\end{aligned}$$ In classical bond percolation, this correlation function and the associated correlation length map to corresponding properties of the $q$-state Potts model in the $q \to 1$ limit [@Stauffer_Aharony_book; @Christensen_Moloney_book]. Although we do not have such an interpretation in terms of a spin model, the considerations of Sec. \[Consequences\] tell us that $\xi$ is an upper bound on the correlation length of monomers in the monomer-dimer model, and the localization length of the zero energy Green function. By thinking in terms of contributions of each ${\mathcal R}$-type region to the double summation in Eqn. \[xidefn\], we see that $\xi$ is also the root mean square radius of gyration $R$ of ${\mathcal R}$-type regions, with the mean taken to be weighted by $m^2$, where $m$ is the mass of the ${\mathcal R}$-type region. Thus, we have the expression $$\begin{aligned} \xi^2 &=& \frac{\langle \sum_{i=1}^{N_{\mathcal R}} m_i^2 R_i^2 \rangle }{\langle \sum_{i=1}^{N_{\mathcal R}} m_i^2 \rangle} \;, \nonumber \\ &=& \frac{\sum_{m} m^2R^2_mN_m}{\sum_{m} m^2N_m} \;, \label{Definition_xi} \end{aligned}$$ where the angular brackets in the first expression indicate averaging over the ensemble of diluted samples, and $N(m)$ and $R^2(m)$ in the second expression are defined respectively to be the mean number of ${\mathcal R}$-type regions of mass $m$ and the mean square radius of gyration of such regions. In Fig. \[length-scalebasic\], we plot $R_{\rm max}$, $\xi$, $l_w$ and $l_{\rm vac}$ as a function of $n_{\rm vac}$ for the largest size studied ($L=26000$). It is clear from the displayed results that $R_{\rm max}$ and $\xi$ more or less track each other (as expected), with both these length scales dominating over $l_{\rm vac}$ and over $l_w$ in the small-$n_{\rm vac}$ limit. In Fig. \[Rgandxi\], we plot $R_{\rm max}$ and $\xi$ for different sizes as a function of $n_{\rm vac}$. For values of $L$ accessible to our computational method, we see that both these length scales appear to be limited only by the system size $L$ in the small $n_{\rm vac}$ limit. Thus, we conclude that the typical size of a ${\mathcal R}$-type region grows without bound as $n_{\rm vac}$ approaches $n_{\rm vac}=0$ on both the square and the honeycomb lattice. The presence of this diverging length scale suggests that the behaviour of slightly-diluted systems may be independent of lattice-scale details. ![The correlation length $\xi$ associated with the sample-averaged geometric correlation function $C(x,x')$ obeys a scaling form $LF_{\xi}(Ln_{\rm vac}^{\nu})$ with $\nu = 4.9 \pm 0.4$ ($\nu = 5.2 \pm 0.4$) on the honeycomb (square) lattice, as is clear from the fact that plots of $\xi/L$ for different sample sizes $L$ collapse on to a single curve when plotted against the scaling variable $Ln_{\rm vac}^\nu$. See Sec.  for details.[]{data-label="scalingxi"}](./plots/xi_scaled.pdf){width="\columnwidth"} Additional support for this conclusion comes from a study of the the $n_{\rm vac}$ dependence of $m_{\rm small}/m_{\rm tot}$, where $m_{\rm small}$ is the contribution to $m_{\rm tot}$ from “small” ${\mathcal R}$-type regions of absolute mass $m < m^*(n_{\rm vac})$. For the cutoff value of absolute mass, $m^*$, we choose $m^*(n_{\rm vac}) = V_{\rm small} /n_{\rm vac}$ to ensure that clusters of mass $m^*$ can be expected on average to be associated with some fixed “small” number of vacancies, $V_{\rm small}$. Another characterization of $m^*(n_{\rm vac})$ is that it corresponds to the expected number of sites in a region of linear size $l_{\rm small} = \sqrt{V_{\rm small}} \times l_{\rm vac}$ in the pure system. The results of this study with $V_{\rm small}=10000$, [*i.e.*]{} $l_{\rm small} = 100 \times l_{\rm vac}$, are shown in Fig. \[smallfraction\_mw\]. Analogous results for the $n_{\rm vac}$ dependence of ${\mathcal I}_{\rm small}/W$, where ${\mathcal I}_{\rm small}$ is the contribution to $W$ from these small ${\mathcal R}$-type of mass $m < m^*(n_{\rm vac})$, are also shown in Fig. \[smallfraction\_mw\]. From the data displayed in this figure, it is clear that the small-$n_{\rm vac}$ limit is dominated by the physics of large ${\mathcal R}$-type regions whose size diverges [*even when measured in units of $l_{\rm vac}$*]{}. Finally, we consider the probability $P$ that a $L \times L$ sample with periodic boundary conditions has at least one ${\mathcal R}$-type region that wraps around at least one cycle of the torus (Fig. \[Pw\]). In classical percolation theory, the analogous quantity provides a simple diagnostic of the percolation transition [@Cardy_conformalinvariancepercolation; @Langlands_etal]: If one plots $P$ as a function of concentration for various $L$, these curves cross each other at the critical concentration. In the percolated phase, $P$ is closer to $1$ for larger $L$, while the unpercolated phase is characterized by the opposite behaviour. In our case, it is clear that the corresponding curves for different $L$ all tend to $1$ as $n_{\rm vac} \to 0$, but there is no crossing point at any nonzero $n_{\rm vac}$ accessible to our numerics. All of the foregoing strongly suggests that the $n_{\rm vac} \to 0$ limit may be fruitfully thought of in terms of universality and critical scaling associated with an incipient percolation phenomenon, which is what we turn to next. Scaling and universality {#Scaling} ------------------------ With this motivation, we now present a finite-size scaling analysis of the structure of the Dulmage-Mendelsohn decomposition of slightly diluted square and honeycomb lattices. Before we begin, two general comments are perhaps useful by way of orientation: As in any such finite-size scaling analysis of numerical data, with all its inherent limitations of computationally accessible values of system size and control parameters, there are two valid approaches: One may try and find a single best-fit set of exponents that simultaneously achieves a reasonable scaling collapse of all quantities measured. In this approach, one may use the quality of this simultaneous scaling collapse to validate the underlying scaling hypothesis. Or one may try and find the values of exponents that give the best-looking collapse for each quantity separately, and use the size of the spread (for each quantity, and across different quantities) in their numerical values to test the underlying scaling hypothesis and set error bars on exponent estimates. Here, we choose the second approach since it seems to be more informative when studying a hitherto-unexplored critical phenomenon. Another important aspect of the analysis also deserves mention before we begin: Our computational resources do not allow a study of large-enough samples in the small-dilution regime with $n_{\rm vac} < 0.04$ ($n_{\rm vac} < 0.06$) on the honeycomb (square) lattice at present. As a result, we cannot definitively rule out the possibility that there is a critical point at a small nonzero $n_{\rm vac}^{\rm crit} \ll 0.04$ ($n_{\rm vac}^{\rm crit} \ll 0.06$) on the honeycomb (square) lattice, rather than a $n_{\rm vac}=0$ critical point on both lattices. Nevertheless, Occam’s razor dictates that we first explore to the fullest the simpler and more parsimonious description in terms of a $n_{\rm vac}=0$ critical point. With this in mind, we now present a scaling analysis that supports such an interpretation in terms of incipient percolation in the $n_{\rm vac} \to 0$ limit. ![The susceptibility $\chi$ associated with the sample-averaged geometric correlation function $C(x,x')$ grows rapidly in the $n_{\rm vac} \to 0$ limit, consistent with the evidence for a diverging correlation length presented in Fig. . See Sec.  for details. []{data-label="growthfig_chi"}](./plots/chi1.pdf){width="\columnwidth"} We begin by asking if the curves for the wrapping probabilities $P$ at different sizes obey the standard scaling form $$\begin{aligned} P & =& F_P(Ln_{\rm vac}^{\nu}) \; , \label{scalingform_P} \end{aligned}$$ where the scaling function $F_P(x)$ is expected to tend to a nonzero constant at $x=0$, and go to zero as $x \to \infty$ [@Stauffer_Aharony_book; @Christensen_Moloney_book]. In Fig. \[scalingP\], we see that this wrapping probability does indeed exhibit a reasonably convincing scaling collapse of the anticipated form. By varying $\nu$ we find that comparably good scaling can be obtained for $\nu = 5.0 \pm 0.4$ on the honeycomb lattice and $5.7 \pm 0.5$ on the square lattice. We note that the size of these ranges or the slight different between them does not immediately yield our error bar on $\nu$, since that the actual uncertainty in the value of $\nu$ will also have contributions from the differing ranges of $\nu$ that provide the best account of other quantities. Next we ask if the length scales $R_{\rm max}$ and $\xi$ obey finite-size scaling forms in the small-$n_{\rm vac}$ regime. Viewing $n_{\rm vac}$ as a measure of deviation from geometric criticality, we attempt a scaling collapse of the fractions $R_{\rm max}/L$, $\xi/L$ by plotting them as functions of the scaling variable $Ln_{\rm vac}^\nu$. In other words, we make the ansatz $$\begin{aligned} \xi &=& L F_{\xi} (Ln_{\rm vac}^{\nu})\;, \nonumber \\ R_{\rm max} &=& L F_{R_{\rm max}} (Ln_{\rm vac}^{\nu})\;, \label{scalingfns_xiRmax}\end{aligned}$$ where the scaling functions $F_{\xi}(x)$ and $F_{R_{\rm max}}(x)$ are expected to fall off as $\sim 1/x$ at large values of the scaling argument $x$, and go to a nonzero constant at $x=0$ [@Stauffer_Aharony_book; @Christensen_Moloney_book]. This is shown in Fig. \[scalingRmax\] and Fig. \[scalingxi\], which display the scaling collapse obtained for these quantities. We find that a reasonably good scaling can be obtained for a range of values of $\nu$, namely, $\nu = 4.9 \pm 0.4$ on the honeycomb lattice and $\nu = 5.2 \pm 0.4$ on the square lattice. We have checked that the large-$x$ behaviour of the scaling curves does indeed tend towards a $1/x$ fall-off at large $x$, and the small-$x$ limit is easily confirmed by eye. ![The susceptibility $\chi$ associated with the sample-averaged geometric correlation function $C(x,x')$ obeys a scaling form $L^{2-\eta}F_{\chi}(Ln_{\rm vac}^{\nu})$, as demonstrated by the fact that plots of $\chi/L^{2-\eta}$ for different sample sizes $L$ collapse on to a single curve when plotted against the scaling variable $x^{-1} = 1/Ln_{\rm vac}^\nu$. Comparably good scaling can be achieved for $\nu = 5.0 \pm 0.4$ ($\nu = 5.7 \pm 0.5$) on the honeycomb (square) lattice with $\eta = 0.0$. We choose to display this scaling behaviour as a function of $x^{-1}$ rather than $x$ since this provides a more stringent visual check on the quality of the collapse when the scaling function rises sharply to its $x=0$ value. This should be compared with the quality of the scaling collapse in Fig.  with $\eta = 0.04$. Similar scaling can be achieved for all $\eta \lesssim 0.08$ for slightly different ranges of $\nu$. See Sec.  for details.[]{data-label="scalingfig1_chi"}](./plots/chi_scaled1_inv.pdf){width="\columnwidth"} ![The susceptibility $\chi$ associated with the sample-averaged geometric correlation function $C(x,x')$ obeys a scaling form $L^{2-\eta}F_{\chi}(Ln_{\rm vac}^{\nu})$, as demonstrated by the fact that plots of $\chi/L^{2-\eta}$ for different sample sizes $L$ collapse on to a single curve when plotted against the scaling variable $x^{-1} = 1/Ln_{\rm vac}^\nu$. Comparably good scaling can be achieved for $\nu = 5.5 \pm 0.4$ ($\nu = 6.1 \pm 0.6$) on the honeycomb (square) lattice with $\eta=0.04$. We choose to display this scaling behaviour as a function of $x^{-1}$ rather than $x$ since this provides a more stringent visual check on the quality of the collapse when the scaling function rises sharply to its $x=0$ value. This should be compared with the quality of the scaling collapse in Fig.  with $\eta=0.0$. Similar scaling can be achieved for all $\eta \lesssim 0.08$ for slightly different ranges of $\nu$. See Sec.  for details.[]{data-label="scalingfig2_chi"}](./plots/chi_scaled2_inv.pdf){width="\columnwidth"} Next, we study the scaling of the susceptibility $\chi$, which is defined in terms of $C(x,x')$ in the usual way: $$\begin{aligned} \chi &=& \frac{1}{L^2}\sum_{x, x'} C(x,x') \nonumber \\ &=& \frac{1}{L^2}\langle \sum_{i=1}^{N_{\mathcal R}} m_i^2 \rangle \nonumber \\ &=& \frac{1}{L^2} \sum_m m^2 N_m \; . \label{chidefn}\end{aligned}$$ In Fig. \[growthfig\_chi\], we see that $\chi$ too appears to grow rapidly in the small-$n_{\rm vac}$ limit to a value that is set by some power of the system size $L$. As is standard in the finite-size scaling analysis of critical phenomena, we relate this power to the anomalous dimension $\eta$ and seek a scaling scaling collapse of the form $$\begin{aligned} \chi &=& L^{2-\eta} F_{\chi}(Ln_{\rm vac}^{\nu}) \; . \label{scalingfn_chi}\end{aligned}$$ When plotted as a function of the scaling variable $Ln_{\rm vac}^{\nu}$, all the data for $\chi/L^{2-\eta}$ is expected to collapse on to a single scaling curve for a suitable choice of $\eta$, and a value of $\nu$ consistent with our earlier results. Again, the expectation is that the scaling function falls off as $\sim 1/x^{2-\eta}$ in the limit of large scaling argument $x$. At small $x$, the conventional expectation is that the scaling function tends to a nonzero constant [@Stauffer_Aharony_book; @Christensen_Moloney_book]. Our test of this scaling hypothesis is displayed in Fig. \[scalingfig1\_chi\] and Fig. \[scalingfig2\_chi\]. These figures display as a function of the inverse scaling variable $x^{-1} = 1/Ln_{\rm vac}^{\nu}$ the putative scaling curves obtained for the best choice of $\nu$ for two slightly different values of $2-\eta$; this choice of independent variable provides a more stringent visual test of the quality of the scaling collapse when the scaling function rises relatively sharply to its $x=0$ value. As is clear from these, both sets of values give comparably convincing scaling collapse. Indeed, we find that it is possible to find comparably good scaling collapse for all $\eta \lesssim 0.08$ by slight adjustments in the value of $\nu$. The larger values of $\eta$ in this range require a slightly larger accompanying value of $\nu$ to produce good scaling collapse. For $\eta$ set to $\eta=0$, we find $\nu = 5.0 \pm 0.4$ ($\nu=5.7 \pm 0.5$) on the honeycomb (square) lattice, whereas for $\eta = 0.04$ we find $\nu = 5.5 \pm 0.4$ ($\nu=6.1 \pm 0.6$) on the honeycomb (square) lattice. This kind of uncertainty is the dominant contribution to our error estimates when we quote values for the critical exponents. Next we study the scaling of $m_{\rm max}$, the mean mass of the largest ${\mathcal R}$-type region in a finite sample (in practice, we take the average of the masses of the largest ${\mathcal R}_A$-type region and the largest ${\mathcal R}_B$-type region in each sample). In Fig. \[growthfig\_mmax\], we display the $L$ and $n_{\rm vac}$ dependence of $m_{\rm max}$. From this data, it is clear that $m_{\rm max}$ is indeed size-limited in the putative critical regime. A standard scaling argument, familiar in the context of classical percolation, relates the scaling dimension of $m_{\rm max}$ to that of $\chi$ to yield the scaling ansatz [@Stauffer_Aharony_book; @Christensen_Moloney_book] $$\begin{aligned} m_{\rm max} &=& L^{2-\eta/2} F_{m_{\rm max}}(Ln_{\rm vac}^{\nu}) \; . \label{scalingfn_mmax}\end{aligned}$$ ![The mean mass $m_{\rm max}$ of the largest ${\mathcal R}$-type regions in a sample grows rapidly in the $n_{\rm vac} \to 0$ limit, consistent with the evidence presented in Fig.  for a small-$n_{\rm vac}$ divergence in $R_{\rm max}$, the mean radius of gyration of these regions. See Sec.  for details. []{data-label="growthfig_mmax"}](./plots/m_max.pdf){width="\columnwidth"} We have tested this scaling ansatz by plotting $m_{\rm max}/L^{2-\eta/2}$ as a function of the inverse scaling variable $x^{-1} = 1/Ln_{\rm vac}^{\nu}$ in order to provide a stringent visual test of the quality of the scaling collapse obtained. Again, we find that reasonably good scaling is seen in a range of $\eta \lesssim 0.08$, with the best-fit values of $\nu$ increasing somewhat as we increase $\eta$ from $\eta=0$ to $\eta = 0.08$. By comparing the quality of collapse for various $\nu$ with $\eta$ fixed to a value in this viable range, we arrive for instance, at the estimate $\nu = 5.0 \pm 0.6$ ($\nu = 5.5 \pm 0.6$) for the honeycomb (square) lattice if $\eta=0$. and the estimate $\nu = 5.4 \pm 0.6$ ($\nu =5.9 \pm 0.5$) for the honeycomb (square) lattice if $\eta$ is set to $\eta=0.04$. Clearly these values fall nicely within the range established by our previous scaling analysis of $\chi$. ![The mean mass $m_{\rm max}$ of the largest ${\mathcal R}$-type regions in a sample obeys a scaling form $L^{2-\eta/2}F_{m_{\rm max}}(Ln_{\rm vac}^{\nu})$, as demonstrated by the fact that plots of $m_{\rm max}/L^{2-\eta/2}$ for different sample sizes $L$ collapse on to a single curve when plotted against the scaling variable $x^{-1} = 1/Ln_{\rm vac}^\nu$. Comparably good scaling can be achieved for $\nu = 5.0 \pm 0.6$ ($\nu = 5.5 \pm 0.6$) on the honeycomb (square) lattice with $\eta = 0.0$. We choose to display this scaling behaviour as a function of $x^{-1}$ rather than $x$ since this provides a more stringent visual check on the quality of the collapse when the scaling function rises sharply to its $x=0$ value. This should be compared with the quality of the scaling collapse in Fig.  with $\eta = 0.04$. Similar scaling can be achieved for all $\eta \lesssim 0.08$ for slightly different ranges of $\nu$. See Sec.  for details.[]{data-label="scalingfig1_mmax"}](./plots/m_max_scaled1_inv.pdf){width="\columnwidth"} ![The mean mass $m_{\rm max}$ of the largest ${\mathcal R}$-type regions in a sample obeys a scaling form $L^{2-\eta/2}F_{m_{\rm max}}(Ln_{\rm vac}^{\nu})$, as demonstrated by the fact that plots of $m_{\rm max}/L^{2-\eta}$ for different sample sizes $L$ collapse on to a single curve when plotted against the scaling variable $x^{-1} = 1/Ln_{\rm vac}^\nu$. Comparably good scaling can be achieved for $\nu = 5.4 \pm 0.6$ ($\nu = 5.9 \pm 0.5$) on the honeycomb (square) lattice with $\eta=0.04$. We choose to display this scaling behaviour as a function of $x^{-1}$ rather than $x$ since this provides a more stringent visual check on the quality of the collapse when the scaling function rises sharply to its $x=0$ value. This should be compared with the quality of the scaling collapse in Fig.  with $\eta=0.0$. Similar scaling can be achieved for all $\eta \lesssim 0.08$ for slightly different ranges of $\nu$. See Sec.  for details.[]{data-label="scalingfig2_mmax"}](./plots/m_max_scaled2_inv.pdf){width="\columnwidth"} Based on this scaling analysis of various quantities displayed here, and a similar analysis for another diverging length scale $\xi_2$ (see Appendix), we estimate the correlation length exponent to be $\nu = 5.5 \pm 0.9$, where the error bar reflects two things. First, it reflects the spread in the values of $\nu$ that provide the best scaling collapse for quantities such as $P$, $\xi$ and $R_{\rm max}$ whose scaling is independent of the anomalous dimension $\eta$. Second, it accounts for the fact that equivalently good fits can be obtained either with $\eta=0$ and a value of $\nu$ on the lower side of this range, or with small nonzero $\eta \lesssim 0.08$ and a corresponding value of $\nu$ on the higher side of this range. For $\eta$, we therefore conclude that $\eta \lesssim 0.08$, but are unable to pin the value down further. In particular, we emphasize that we cannot rule out the possibility that $\eta=0.0$. On the contrary, since the scaling behaviour of $P$, $\xi$ and $R_{\rm max}$ favours values of $\nu$ on the lower side of our estimated range, and these values of $\nu$ give a better account of the scaling if $\eta$ is $0$ or zero or close to zero, we must leave open for now the possibility that $\eta = 0$. In the Appendix, we also have displayed the scaling behaviour for a third length scale $\xi_2$, that is defined by analogy to the corresponding quantity in the theory of classical percolation. Additionally, in Sec. \[Odd\], we display the scaling behaviour of corresponding quantities defined by only taking into account ${\mathcal R}$-type regions with odd imbalance ${\mathcal I}$; this is motivated by the idea that such regions host at least one surviving Majorana mode that is perturbatively stable to the leading order effects of further neighbour mixing amplitudes in bipartite Majorana networks. As we will see there, the scaling picture does not change if one restricts attention to such odd regions alone. Morphology of critical ${\mathcal R}$-type regions {#Morphology} -------------------------------------------------- As we have argued in the foregoing, finite-size limited ${\mathcal R}$-type regions in the $n_{\rm vac} \to 0$ limit represent critical behaviour of the geometry of the Dulmage-Mendelsohn decomposition of such slightly-diluted lattices. This motivates a study of their basic morphology. To this end, we measure the ${\mathcal B}_{\rm max}$, defined to be the mean size of the boundary of the largest ${\mathcal R}$-type region in a finite-size sample, and and ${\mathcal I}_{\rm max}$, the mean number of zero modes in the largest ${\mathcal R}$-type region. We also measure ${\mathcal D}^{\rm nw}_{\rm max}$ (${\mathcal D}^{\rm w}_{\rm max}$), the mean number of defects adjacent to odd (even) sites (we use the appellations “odd” and “even” as defined in Sec. \[DM\]) of the largest ${\mathcal R}$-type region. The superscripts w and nw stand respectively for “wavefunction” and “nonwavefunction”, and remind us that the zero mode wavefunctions of a ${\mathcal R}$-type region have amplitude only on the even sites of the ${\mathcal R}$-type region. The underlying intuition is that each vacancy that is adjacent to even sites and counted in ${\mathcal D}^{\rm w}_{\rm max}$ “contributes” some zero modes to ${\mathcal R}_{\rm max}$, while each vacancy that is adjacent to odd sites and counted in ${\mathcal D}^{\rm nw}_{\rm max}$ “removes” some zero modes from ${\mathcal R}_{\rm max}$. In practice we use the average of each quantity over the largest ${\mathcal R}_A$ and ${\mathcal R}_B$ regions of a sample when computing the contribution of a sample to the ensemble average. Also, the boundary ${\mathcal B}$ of a ${\mathcal R}$-type region is defined as the number of surviving links it has to the rest of the graph. Similarly, each vacancy contributes to the corresponding ${\mathcal D}$ with a multiplicity equal to the number of deleted links that would have connected that vacant site to the ${\mathcal R}$-type region had the vacancy been absent. A plausible estimate for ${\mathcal B}_{\rm max}$ is as follows: In the classical theory of percolation, the analog of ${\mathcal B}_{\rm max}$ is expected to scale as $m_{\rm max}$. This is due to the fact that ${\mathcal B}_{\rm max}$ counts both the internal perimeter (due to voids or holes) and the actual external perimeter. As a result, the boundary ${\mathcal B}_{\rm max} $ of the critical cluster in classical percolation is dominated by the contribution of these voids. In our case, it is reasonable to assume that the analogue of voids are ${\mathcal P}$-type regions. From our earlier results, it is clear that the size of the typical ${\mathcal P}$-type regions remain very small in the $n_{\rm vac} \to 0$ limit. It is also clear that the number density $n_{\mathcal P}$ of these regions vanishes smoothly as $n_{\rm vac} \to 0$, as does $m_{\mathcal P}$, the mass density of ${\mathcal P}$-type regions per unit area. Assembling this information, we arrive at the estimate ${\mathcal B}_{\rm max} \sim n_{\mathcal P} \times \sqrt{m_{\mathcal P}/n_{\mathcal P}}$. Based on this argument, we expect ${\mathcal B}_{\rm max} \sim \sqrt{m_{\mathcal P}n_{\mathcal P}}$. In Fig. \[B\_perhole\], we see that this expectation is borne out by the data. The ratio ${\mathcal B}_{\rm max} / \sqrt{m_{\mathcal P}n_{\mathcal P}}$ is seen to converge very quickly to its thermodynamic limit (indeed, at the sizes we study, no finite-size effects are readily discernible), and has a very mild and nonsingular dependence on $n_{\rm vac}$ in the limit of small dilution. ![The sample-averaged boundary of the largest ${\mathcal R}$-type region scales with $m_{\max}$, the mean mass of such regions, with a proportionality constant whose $n_{\rm vac}$ dependence is accounted for by the behaviour of the expected contribution of “holes” corresponding to small ${\mathcal P}$-type regions that are engulfed by this large ${\mathcal R}$-type region. See Sec.  for details of the corresponding analysis.[]{data-label="B_perhole"}](./plots/bndry_max2.pdf){width="\columnwidth"} In Fig. \[I\_by\_wtimesm\], we plot ${\mathcal I}_{\rm max}/(wm_{\rm max})$ as a function of $n_{\rm vac}$ for various $L$. Two things become apparent from this. First, this ratio saturates to the thermodynamic limit at relatively small sizes compared to the range of sizes studied here. Second, it has a mild dependence on $n_{\rm vac}$, tending to a nonzero constant as $n_{\rm vac} \to 0$. In other words, the density of zero modes in the largest ${\mathcal R}$-type region differs from the globally averaged density $w$ by a nonsingular ${\mathcal O}(1)$ prefactor that has a mild dependence on the density $n_{\rm vac}$ of vacancies. ![The mean number of zero modes hosted by the largest ${\mathcal R}$-region scales with $m_{\rm max}$, with a proportionality constant whose $n_{\rm vac}$ dependence is accounted for by the behaviour of the globally-averaged density $w$ of zero modes.See Sec.  for details of the corresponding analysis.[]{data-label="I_by_wtimesm"}](./plots/Imbmax.pdf){width="\columnwidth"} Next, we consider ${\mathcal D}^{\rm nw}_{\rm max} + {\mathcal D}^{\rm w}_{\rm max}$, the mean number of vacancies adjacent to sites that belong to the largest ${\mathcal R}$-type region. If this region is not atypical in terms of the overall density of vacancies associated with it, one would expect $({\mathcal D}^{\rm nw}_{\rm max} + {\mathcal D}^{\rm w}_{\rm max} ) \sim n_{\rm vac} m_{\rm max}$. In Fig. \[Dplus\_by\_ndtimesm\], we see that the corresponding ratio saturates very quickly to the thermodynamic limit and has a very mild nonsingular dependence on $n_{\rm vac}$, thus confirming this basic picture. ![The mean number of vacancies associated with the largest ${\mathcal R}$-type region scales with $n_{\rm vac}m_{\rm max}$. See Sec.  for details of the corresponding analysis.[]{data-label="Dplus_by_ndtimesm"}](./plots/sumD.pdf){width="\columnwidth"} Turning to ${\mathcal D}^{\rm w}_{\rm max} - {\mathcal D}^{\rm nw}_{\rm max}$, we begin by making more explicit our intuitive picture for the zero modes and monomers hosted by ${\mathcal R}$-type regions: A local imbalance in the numbers of surviving sites on the two sublattices gives rise to these zero modes and mandates the existence of a corresponding density of monomers in this region. From this point of view, it is interesting to ask if $({\mathcal D}^{\rm w}_{\rm max} - {\mathcal D}^{\rm nw}_{\rm max}) \sim {\mathcal I}_{\rm max}$. And if this is the case, how does the corresponding ratio $({\mathcal D}^{\rm w}_{\rm max} - {\mathcal D}^{\rm nw}_{\rm max})/{\mathcal I}_{\rm max}$ scale with $n_{\rm vac}$ in the critical regime? In Fig. \[Dminus\_by\_Imax\], we display this ratio. As is clear from this figure, the two quantities are indeed proportional, and the ratio converges rapidly to the thermodynamic limit (again, at the sizes we study, the finite-size corrections to this ratio are not readily discernible). Interestingly, we find that the ratio is fairly large, and rises further to reach a finite limiting value in the $n_{\rm vac} \to 0$ limit. In other words, as $n_{\rm vac}$ becomes small, the local sublattice imbalance becomes more “inefficient” in terms of its ability to nucleate zero modes. Thus, our basic picture for the morphology of the largest ${\mathcal R}$-type regions that dominate the $n_{\rm vac} \to 0$ limit can be summarized in the following informal way: Such ${\mathcal R}$-type regions have a finite density of small holes, which are small ${\mathcal P}$-type regions themselves (Fig. \[B\_perhole\]). Indeed the density of these holes tracks $n_{\mathcal P}$, the density of ${\mathcal P}$-type regions. And the mean linear size of these holes tracks that of a typical ${\mathcal P}$-type region. The local density of zero modes in such regions is proportional to the globally averaged density $w$ of zero modes, with a proportionality constant that is close to unity and has a mild $n_{\rm vac}$ dependence in the $n_{\rm vac} \to 0$ limit (Fig. \[I\_by\_wtimesm\]). The total density of vacancies associated with such large ${\mathcal R}$-type regions is proportional to the globally averaged density of vacancies, with a nonsingular proportionality constant that has a mild $n_{\rm vac}$ dependence in the small-$n_{\rm vac}$ limit (Fig. \[Dplus\_by\_ndtimesm\]). However, the local sublattice imbalance in the number of vacancies on the two sublattices is proportional to the total number of zero modes hosted by the ${\mathcal R}$-type region, with a fairly large proportionality constant (of order $10$) that increases as $n_{\rm vac} $ goes to zero, but remains finite in this limit (Fig. \[Dminus\_by\_Imax\]). Thus, such ${\mathcal R}$-type regions are seeded by a local imbalance in the number of vacancies on the two sublattices. This imbalance is far in excess of central limit theorem expectations (which would predict that this imbalance scales as $\sqrt{n_{\rm vac} m_{\rm max}}$), and seeds a nonzero local density of zero modes. ![The mean sublattice imbalance in vacancies associated with the largest ${\mathcal R}$-type region is proportional to ${\mathcal I}_{\rm max}$ the mean number of zero modes hosted by such regions, with a proportionality constant that is fairly large and increases further to a nonzero value in the $n_{\rm vac} \to 0$ limit. See Sec.  for details of the corresponding analysis.[]{data-label="Dminus_by_Imax"}](./plots/diffDw.pdf){width="\columnwidth"} Scaling of surviving Majorana modes {#Odd} ----------------------------------- As we have noted earlier, ${\mathcal R}$-type regions that contain an odd number of zero modes acquire special significance from the point of view of identifiying Majorana degrees of freedom that survive the leading effects of next-nearest-neighbour couplings in bipartite Majorana networks: Such regions host one robust Majorana mode that survives these effects, whereas ${\mathcal R}$-type regions with an even number of zero modes are not expected to have any Majorana degrees of freedom that survive the effects of such next-nearest-neighbour couplings. In Fig. \[smallfraction\_nR\], we display the fraction of the total mass of such “odd” ${\mathcal R}$-type regions that is contained in small “odd” regions, with smallness defined as before: small regions have absolute mass $m < V/n_{\rm vac}$ (with $V=10000$). ![The fraction of total mass of odd ${\mathcal R}$-type regions ([*i.e.*]{} those with an odd imbalance ${\mathcal I}$) contributed by “small” ${\mathcal R}$-type regions of this type which have less than $10000$ vacancies associated with them scales rapidly to zero in the small-$n_{\rm vac}$ limit, implying that the physics of such regions is dominated by a diverging length scale in this limit. See Sec.  for details.[]{data-label="smallfraction_nR"}](./odd_plots/m_sh.pdf){width="\columnwidth"} Clearly, most of the total mass in odd ${\mathcal R}$-type regions is contained in large odd regions, again suggesting that the physics of such robust Majorana modes is controlled by a diverging length scale in the $n_{\rm vac} \to 0$ limit. Indeed, we have also studied the scaling behaviour of $R_{\rm max}^{\rm odd}$, $\xi^{\rm odd}$, $\chi^{\rm odd}$, and $m_{\rm max}^{\rm odd}$, defined by considering only the ${\mathcal R}$-type regions with an odd number of zero modes. Our basic conclusion is that restricting attention to such odd ${\mathcal R}$-type regions does not change the scaling picture. Examples of scaling behaviour of these quantities are shown in Fig. \[scalingfig\_Rmaxodd\], and Fig. \[scalingfig\_xiodd\], which display our results for $R_{\rm max}^{\rm odd}$ and $\xi^{\rm odd}$ respectively. Additional results for $\chi^{\rm odd}$ and $m_{\rm max}^{\rm odd}$ are relegated to the Appendix, since they show no discernible difference from their counterparts obtained without the restriction to odd imbalance. ![The sample-averaged radius of gyration $R_{\rm max}^{\rm odd}$ of the largest ${\mathcal R}$-type regions with odd imbalance ${\mathcal I}$ in a sample obeys a scaling form $LF_{R^{\rm odd}_{\rm max}}(Ln_{\rm vac}^{\nu})$ with $\nu = 4.9 \pm 0.4$ ($\nu = 5.2 \pm 0.4$) on the honeycomb (square) lattice. This behaviour is entirely analogous to that shown in Fig.  for $R_{\rm max}$. The small-$x$ downturn of the scaling function can be intuitively understood by noting that the ${\mathcal R}$-type regions that contribute to this quantity are not always the largest in the sample, and can sometimes be significantly smaller, particularly at small $n_{\rm vac}$. See Sec.  and Sec.  for details.[]{data-label="scalingfig_Rmaxodd"}](./odd_plots/R_scaled.pdf){width="\columnwidth"} ![The correlation length $\xi^{\rm odd}$ associated with the sample-averaged geometric correlation function $C^{\rm odd}(x,x')$ defined by considering only ${\mathcal R}$-type regions with odd imbalance ${\mathcal I}$ obeys a scaling form $LF_{\xi^{\rm odd}}(Ln_{\rm vac}^{\nu})$ with $\nu = 4.9 \pm 0.4$ ($\nu = 5.2 \pm 0.4$) on the honeycomb (square) lattice, as is clear from the fact that plots of $\xi^{\rm odd}/L$ for different sample sizes $L$ collapse on to a single curve when plotted against the scaling variable $Ln_{\rm vac}^\nu$. This behaviour is entirely analogous to that shown in Fig.  for $\xi$. See Sec.  and Sec.  for details.[]{data-label="scalingfig_xiodd"}](./odd_plots/xi_scaled.pdf){width="\columnwidth"} Summary of conclusions {#Summary} ---------------------- To summarize, we have provided fairly convincing evidence for a new universality class of incipient percolation in diluted two dimensional bipartite lattices such as the square and honeycomb lattice. This critical behaviour is exhibited by the random geometry of the Dulmage-Mendelsohn decomposition in the $n_{\rm vac} \to 0$ limit of the diluted lattice. This unusual critical phenomenon can legitimately be thought of as monomer percolation (from the point of view of the monomer-dimer model associated with the ensemble of maximum matchings of the lattice), wavefunction percolation (from the point of view of the zero-energy Green function of a particle hopping on this lattice), or Majorana percolation (from the perspective of robust Majorana zero modes of a bipartite Majorana network). From our scaling analysis of this putative $n_{\rm vac} = 0$ critical point, we obtain the estimate $\nu = 5.5 \pm 0.9$ for the correlation length exponent, and the bound $\eta \lesssim 0.08$ for the corresponding anomalous dimension that determines the critical behaviour of the geometric correlation function. Our results leave open the intriguing possibility that the associated wavefunction percolation and Majorana percolation phenomena are characterized by an independent set of additional exponents. In principle, this remains a possibility for the associated monomer percolation as well, and further computational work would be needed to settle these questions. Discussion ========== Heuristic reinterpretation of results {#Coulomb} ------------------------------------- At the risk of oversimplification, we present here a heuristic interpretation of these results. This is based on the coarse-grained Coulomb phenomenology of fully-packed dimers on the parent square and honeycomb lattices [@Youngblood_Axe; @Henley_review]. As is well-known, in this way of thinking, a fully-packed configuration corresponds to a divergence-free configuration of a polarization field ${\mathbf P}$. An unmatched site on the $A$ ($B$) sublattice corresponds to the location of a unit positive (negative) electric charge in this description. Consider now the undiluted square or honeycomb lattice with boundary conditions that allow a perfect matching. If we remove exactly one $A$ site (say the site $A_0$ at the origin) and look for a maximum matching, it is clear that any such a maximum matching leaves one $B$ site of the diluted lattice unmatched. The coarse-grained picture for this ensemble of maximum matchings then has a static unit charge $+1$ at the origin $A_0$, and a mobile unit negative charge that is attracted to the origin by an entropically generated logarithmic “Coulomb potential” $ V(r) = \eta_m \ln(r)$ of strength $\eta_m$ [@Rakala_Damle_Dhar]. At this coarse-grained level of description, one would say that this mobile charge can diffuse anywhere on the lattice, with the probability for being at distance $\vec{r}$ from the origin falling off in equilibrium as $1/r^{\eta_m}$ ($\eta_m$ is expected to be equal to $1/2$ for the square and honeycomb lattice dimer models). In other words, one would heuristically expect that there is a single Dulmage-Mendelsohn region ${\mathcal R}_{A_0}$ that spans the whole lattice. This can presumably be verified by a direct free-fermion (Pfaffian) calculation of the lattice level partition function with exactly two monomers fixed at two given locations. What about a nonzero density of vacancies? One may again think of each vacancy as being a static charge with a sign given by the sublattice. But it is no longer clear if the “screening cloud” provided by a mobile monomer on the other sublattice extends over all space. Indeed, in this heuristic language of fluctuating electrostatics, our results on the nontrivial geometry of ${\mathcal R}$-type regions of slightly-diluted samples corresponds to a phenomenon whereby groups of vacancies seed a “screening-cloud” of mobile charges that are all of like sign, and confined to a finite-region of the lattice (corresponding to an individual ${\mathcal R}$-type region). The typical size of this screening cloud grows as $\xi \sim n_{\rm vac}^{-\nu}$ in the limit of small $n_{\rm vac}$. A curious aspect of our computational results is that such a group of vacancies does not all correspond to static charges which all have the same sign, although vacancies with positive (negative) charge do outnumber those with negative (positive) charge if the screening cloud is made up of monomers of negative (positive) charge. An aside {#Asides} -------- In a brief aside, we use the perspective developed in our work to i) discuss earlier studies of diluted quantum antiferromagnets, and ii) comment on studies of the tight-binding model on Penrose tilings, and their connection with recent work on the classical dimer model on the same lattice. ### Diluted quantum antiferromagnets on bipartite lattices In a series of papers, Sandvik [@Sandvik_PRB2002] and Wang [@Wang_Sandvik_PRL2006; @Wang_Sandvik_PRB2010] reported on a detailed Quantum Monte Carlo (QMC) study of the $S=1/2$ Heisenberg antiferromagnets on the diluted square lattice. The basic conclusion was that long range antiferromagnetic order persists in the ground state all the way up to the classical percolation threshold of the diluted lattice. Indeed, it was argued that the critical percolating cluster at the geometric percolation transition has long range antiferromagnetic order in the ground state. As a result, the antiferromagnetic transition is driven by the underlying geometric transition, and occurs right at the percolation threshold of $n_{\rm vac}^* \approx 0.407$. These studies also explored the physics in the ordered phase in the vicinity of this transition. An interesting finding in this regard was the presence of weakly-interacting vacancy-induced local moments that affected the low energy physics as well as the quantum critical scaling. These moments were argued to arise from regions of the lattice with local sublattice imbalance. Motivated by this insight [@Wang_Sandvik_PRB2010], the closely-related physics of triplet excitations of such systems was modeled in terms of the spatial monomer distribution function of the corresponding monomer-dimer model. Our work provides us with a useful perspective on these results, since the ${\mathcal R}$-type regions of the Dulmage-Mendelsohn decomposition provide us with a precise characterization and construction of the monomer regions studied by Wang and Sandvik [@Wang_Sandvik_PRB2010]. Indeed, our work strongly suggests that it would be extremely interesting in follow-up work to repeat these QMC calculations at much smaller values of $n_{\rm vac}$, corresponding to the low-dilution limit of interest to us. In this small-$n_{\rm vac}$ limit, our results show that ${\mathcal R}$-type regions are very large in size, and the corresponding magnetic excitations are therefore likely to be quite different. In particular, it would be interesting to ask if there is a signature of the diverging length scale $\xi \sim n_{\rm vac}^{-\nu}$ in the magnetic response of slightly-diluted samples with very small $n_{\rm vac}$. Given the length scales of interest, this would be computationally very challenging, but perhaps achievable. Other closely related suggestions for follow-up work are described in Sec. \[Outlook\]. ### Tight-binding and dimer models on the Penrose tiling There is a fairly large body of work going back nearly four decades, whose focus has been the spectrum of tight-binding models defined on quasiperiodic lattices, most prominently the Penrose tiling made up of rhombii. For instance, Kohmoto and Sutherland [@Kohmoto_Sutherland] noted that the hopping Hamiltonian for a quantum mechanical particle hopping along links of the Penrose tiling had extensively degenerate zero-energy states with localized wavefunctions. In subsequent work, Arai and collaborators provided a partial characterization of these localized wavefunctions by identifying certain geometric motifs that supported such states [@Arai_etal]. Based on this, they also arrived at a conjecture for the density of such localized zero-energy states in the thermodynamic limit. In an insightful analysis [@Koga_Tsunetsugu], Koga and Tsunetsugu exploited the self-similar nature of the Penrose tiling to arrive at an essentially complete characterization of the geometric motifs that support such localized zero modes, and proved the conjecture of Arai and co-authors. Recognizing that local sublattice imbalance was an essential feature of these geometric motifs, Koga and Tsunetsugu also obtained a detailed characterization of the antiferromagnetic order that develops for infinitesimal onsite repulsion in the Hubbard model on this lattice. This local imbalance associated with these geometric motifs was also emphasized in very recent work [@Day-Roberts_Fernandes_Kamenev]. On the other hand, in the recent work of Flicker and collaborators [@Flicker_Simon_Parameswaran], essentially the same geometric motifs seem to arise as a by-product of their analysis of the density of monomers in any maximum matching of the Penrose tiling. Their result for the monomer density also corresponds exactly to the previously obtained density of zero modes of the hopping problem. Moreover, their characterization of regions accessible to monomers bears an uncanny resemblance to the earlier characterizations of the localized zero mode wavefunctions of the hopping problem. Using the perspective developed here, we see that this is no coincidence: Indeed, it becomes apparent that the results of Koga and Tsunetsugu amount to an essentially complete analytic characterization of the Dulmage-Mendelsohn decomposition of the Penrose tiling, which was independently rediscovered in the context of maximum matchings by Flicker and collaborators. Moreover, this perspective suggests some other natural questions that would be interesting to follow up on. These are discussed in Sec. \[Outlook\]. Outlook {#Outlook} ------- The foregoing results and their interpretation lead us to identify several natural and possibly interesting lines of enquiry. We conclude by listing some of these questions as suggestions for potentially fruitful follow-up studies. We begin with three questions concerning the results shown here in the $n_{\rm vac} \to 0$ limit in $d=2$. Our results imply that there is a hierarchy of growing length scales in this limit: $l_{\rm vac} \sim 1/\sqrt{n_{\rm vac}}$, $l_w \sim 1/\sqrt{w}$, and $\xi \sim 1/n_{\rm vac}^{\nu}$, with $\nu = 5.5 \pm 0.9$ ensuring that $l_{\rm vac} \ll l_w \ll \xi$. Our results imply that Dulmage-Mendelsohn clusters in finite-size systems with $l_w \ll L \ll \xi$ “look” critical. Given that the critical point of classical percolation in two dimensions has conformal invariance [@Cardy_conformalinvariancepercolation; @Langlands_etal], is there some sense in which a similar enlarged symmetry governs the behaviour of Dulmage-Mendelsohn clusters in finite-size samples that fall in this critical regime? Next, there are two interesting and closely related questions about the localization length of the zero-energy Green function $G$, and the monomer and dimer correlation lengths $\xi_M$ and $\xi_D$ in the associated monomer-dimer model. Clearly, the localization length $\xi_{\rm G}$ that controls the behaviour of $G$ is bounded from above by the scale $\xi$. But how does the ratio $\xi_{\rm G}/\xi$ scale in the critical regime? How much does this depend on the strength of the hopping disorder? Given that the localization length in the Gade-Wegner universality class of two-dimensional bipartite random hopping problems diverges in the zero-energy limit [@Evers_Mirlin; @Motrunich_Damle_Huse_GadeWegnerPRB], this line of enquiry is likely to yield interesting results. Similarly, does $\xi_M$ remain finite even as $\xi \to \infty$? How does it depend on the strength of the bond-disorder? Finally, what determines the dimer correlation length $\xi_D$ and how does it behave in the $n_{\rm vac} \to 0$ limit? Fourth, there is the natural question of generalizing to other diluted bipartite graphs in two dimension, most notably hyperbolic graphs similar to those studied recently in the context of circuit quantum electrodynamics [@Kollar_Fitzpatrick_Houck] and network theory [@Krioukov_etal]. The percolation theory of such graphs is a well-developed subject in the mathematical literature [@Benjamini_Schramm], and it would be interesting to explore the possible critical behaviour of Dulmage-Mendelsohn clusters in this setting. Fifth, there is the question of dimensionality. Classical percolation and Anderson localization phenomena on hypercubic lattices are both dimensionality dependent in interesting ways. For percolation, there is no percolated phase in $d=1$, while dimensions $d \geq 2$ exhibit a phase transition, with exponents that depend on dimensionality and take on mean-field values above the upper critical dimension $d_u=6$ [@Stauffer_Aharony_book; @Christensen_Moloney_book]. Given the close analogy of our Dulmage-Mendelsohn criticality with a classical percolation phenomenon, the third question concerns the random geometry of the Dulmage-Mendelsohn decomposition of a diluted cubic lattice or diamond lattice in $d=3$. Is there now a nonzero $n_{\rm vac}^{\rm crit}$ discernibly different from $n_{\rm vac} = 0$ below which the ${\mathcal R}$-type regions are in a percolated phase? If yes, what is the nature of the transiton at $n_{\rm vac}^{\rm crit}$? The sixth suggestion has to do with a natural generalization to random bipartite graphs. For such graphs, there is no notion of geometric distance between vertices, but the question of the distribution of sizes of the Dulmage-Mendelsohn clusters remains interesting. This is because recent work has already identified interesting algorithmic implications of the size of a maximum-matching in such graphs for some problems in computer science [@Frieze_Melsted]. Given our arguments about the factorization of the monomer-dimer partition function into factors associated with Dulmage-Mendelsohn clusters, it would also be interesting to study the size distribution of Dulmage-Mendelsohn clusters in this algorithmic context. Given that some results for such graphs can be obtained analytically, there is also the intriguing possibility of obtaining some exact results in this setting. The seventh is a natural question regarding the statistics of overlap loops. The dimer model on the undiluted square and honeycomb lattices has a useful coarse-grained description in terms of a compact scalar height field with a Gaussian action [@Youngblood_Axe_McCoy; @Youngblood_Axe; @Henley_review; @Kenyon_dimers]. Dimer correlations, and correlations of test-monomers are readily related to correlation functions of this Gaussian theory, which can be computed exactly. In a certain well-defined sense, the scaling limit of these dimer models, which governs the long-distance behaviour of correlation functions, has conformal invariance [@Kenyon_dimers]. The double dimer model [@Kenyon_doubledimers] consists of two independent copies of the dimer model, with partition function given by the square of the dimer model partition function. In such a double-dimer model, the interesting observables are “overlap loops” built by tracing closed paths that alternately go along dimers in one copy and then the other. This defines an ensemble of fully-packed loops, which has been argued to have conformal invariance in the scaling limit [@Kenyon_doubledimers]. In our case of diluted square or honeycomb lattices, the analog of the double-dimer model involves two copies of the monomer-dimer model associated with the ensemble of maximum matchings. An ensemble of overlap loops and strings can again be defined by a slight generalization of the usual definition above. The question then arises: What is the statistics of these overlap loops in the critical regime identified here. The next set of three suggestions in this list concerns the physics of SU(N) antiferromagnets in a certain large-$N$ limit [@Read_Sachdev] that has played a key role in the subsequent conceptual development of our understanding of quantum disordered phases of magnets. In these SU(N) magnets, one sublattice carries the fundamental representation and the other has SU(N) spins that transform under the complex-conjugate of the fundamental. Thinking in terms of the corresponding Hubbard model, each site has $N$ different fermion orbitals with the constraint that the total fermion number of $A$-sublattice sites is $1$, while that of $B$-sublattice sites is $N-1$. In the large-$N$ limit of this model on the pure square or honeycomb lattice, the physics is dominated by the subspace of singlet states spanned by any fully-packed configuration of nearest neighbour SU(N) singlet bonds, and reduces to a quantum dimer model on the lattice at leading order in $1/N$. At large but finite $N$, longer range valence bonds come into play. These models are amenable to explicit computational study, for example using Quantum Monte-Carlo methods that work in the basis of bipartite (but not necessarily nearest-neighbour) valence bonds [@Sandvik_Evertz; @Beach_Sandvik; @Mambrini_dimeraspects]. Such computational approaches have been used to study the physics of these systems as a function of $N$, finding a transition from quantum antiferromagnetism at $N=2$ to a valence-bond solid state above a threshold value of $N$ [@Beach_etal]. Clearly, the analogous large-$N$ limit of the diluted magnet will exhibit interesting effects associated with the presence of a finite-density of monomers in the corresponding maximum matchings, since these monomers are expected to be associated with SU(N) spinon degrees of freedom that cannot be quenched by short-ranged singlet bonds. Moreover, our results suggest that these effects may be crucially affected by the incipient percolation of Dulmage-Mendelsohn clusters. It would thus be interesting in the diluted case to revisit this large-$N$ limit, and to the physics at large but finite $N$ using these computational approaches. In this connection, it would aso be interesting to study resonating nearest-neighbour valence-bond wavefunctions [@Tang_Henley_Sandvik_nnRVB; @Albuquerque_Alet_nnRVB] for such SU(N) antiferromagnets: In the pure case, these are singlet wavefunctions which map to interesting loop ensembles that interpolate between the classical dimer model and the double-dimer model [@Damle_Dhar_Ramola_nnRVB; @Patil_Dasgupta_Damle_nnRVB; @Albuquerque_Alet_Moessner_3d]. In the diluted case, their generalizations will describe degenerate ground states with a nonzero spinon number [@Banerjee_Damle; @Banerjee_Damle_Alet; @Banerjee_Damle_Alet_SU3; @Sanyal_Banerjee_Damle; @Tang_Sandvik; @Sanyal_Banerjee_Damle_Sandvik], and map on to an ensemble of loops and strings closely related to the double-dimer model on the diluted lattice. It would be interesting to study the spinon localization properties of these wavefunctions on such slightly-diluted lattices, especially given the divergent size of the ${\mathcal R}$-regions studied here. Direct analogs of these questions are also potentially interesting in the context of Penrose tilings, since the results of Ref. [@Koga_Tsunetsugu] and Ref. [@Flicker_Simon_Parameswaran] provide us with an essentially complete [*analytic determination*]{} of the Dulmage-Mendelsohn decomposition of the Penrose tiling. Although there is no geometric criticality at play in this case, it would clearly be of interest to i) use these analytical results to explore the physics of the Read-Sachdev large-$N$ limit of SU(N) antiferromagnets, ii) perform a QMC study of the related physics at large but finite $N$, iii) to understand the nature of the SU(N) nearest-neighbour RVB wavefunctions mentioned above, and iv) to study the closely-related ensemble of overlap loops and strings defined by the double-dimer model. Next, we note that our results read in conjunction with those of Ref [@Koga_Tsunetsugu] suggest an interesting question about local-moment formation in the Hubbard model at the particle-hole symmetric chemical potential on the diluted square and honeycomb lattices. Localized states tied to the Fermi energy $\mu =0$ are expected to be intimately connected with the physics of local moment formation [@Milovanovic_Sachdev_Bhatt] in such situations. The question then arises: How is this physics affected by the large length scale $\xi$ associated with the size of the Dulmage-Mendelsohn region, and by the presence of ${\mathcal I}$ coexisting zero modes in each such region? Does the topologically-protected nature of the zero modes lead to these moments being relatively robust to perturbations that preserve the particle-hole symmetry? Our final suggestion has to do with the thermodynamic susceptibility of Kitaev’s honeycomb model with nonmagnetic vacancies. From the detailed analysis of vacancy-effects in Ref. [@Willans_Chalker_Moessner_PRB], it is clear that a vacancy-induced pile-up of low-but-non-zero energy Majorana fermion excitations is associated with a weak singularity in the low temperature susceptibility. By analogy with the results of Ref. [@Sanyal_Damle_Chalker_Moessner] on a SU(2) symmetric version [@Yao_Lee] of the Kitaev model, the topologically-protected zero-energy states studied here are potential sources of a stronger Curie-like singularity $\chi(T) \sim {\mathcal C}/T$ in the linear susceptibility. Indeed, it can be shown that this is the case. This leads to the natural question: How does the Curie coefficient ${\mathcal C}$ scale with $\xi$ in the small-$n_{\rm vac}$ limit of weakly-diluted samples? And does the topologically-protected nature of the zero-energy states endow this Curie term with some degree of protection against time-reversal invariant perturbations such as exchange disorder or a Heisenberg exchange term in the spin Hamiltonian? As one goes through this list of suggestions for follow-up studies, it is clear that our work opens up a number of potentially fruitful lines of enquiry. By the same token, it also becomes obvious that the elephant in the room throughout has been the bipartite nature of the underlying lattice. Are there natural generalizations to the nonbipartite case of any of the geometric questions studied here? Are the corresponding results in the small-dilution limit equally interesting? The answer to the first question turns out to be in the affirmative, and motivates our ongoing computational attempt to answer the second question. Acknowledgements {#Acknowledgements} ================ The work of SB formed a part of his Ph.D thesis submission at the Tata Institute of Fundamental Research (TIFR), while that of RB forms part of his mandatory pre-registration project submission DP-II at the TIFR. We thank T. Kavitha (School of Technology and Computer Science, TIFR) and A. Mondal (School of Mathematics, TIFR) for introducing us to the literature on graph decompositions. We acknowledge stimulating discussions with D. Sen on robustness of Majorana modes in various contexts, and D. Dhar and Mahan Mj. for pointers to the literature on percolation theory. We are grateful to S. Ramasesha and A. W. Sandvik for useful comments and constructive criticism in their referee reports on SB’s Ph.D thesis. We also thank S. Bera, R. Dandekar, D. Dhar, F. Evers, F. Flicker, S. A. Parameswaran, J. Radhakrishnan, K. Ramola, S. Ramasesha, A. W. Sandvik, and H. Tsunetsugu for stimulating discussions. KD gratefully acknowledges earlier collaborations with J. T. Chalker, R. Moessner, O. I. Motrunich, and S. Sanyal which provided the motivation for initiating this study. The work of RB, SB, and MMI was supported by a graduate-fellowship at the TIFR. During the completion of this manuscript, SB was supported by the European Research Council Grant 804213-TMCS while at Oxford University. All computations were performed using departmental computational resources of the Department of Theoretical Physics, TIFR. The work of KD was supported in part by a grant from the Infosys Foundation under the aegis of the Infosys-Chandrasekharan virtual center for Random Geometry at the TIFR. [*Statement of author contributions*]{}: RB and SB contributed equally to this work. SB and MMI implemented the various maximum matching algorithms discussed in Ref. [@Duff_Kaya_Ucar], and compared their relative performance in the present context. Using this, RB and SB wrote the codes to obtain the Dulmage-Mendelsohn decomposition and the zero-energy Green function, while RB and KD wrote the code to study the monomer-dimer model. RB and SB performed all the computations and developed all the data-analysis and visualization scripts. RB, SB and KD analyzed the data. KD conceived and directed this project, and wrote the manuscript using the results of this data analysis. [999]{} S. R. Broadbent and J. M. Hammersley, “Percolation processes I. Crystals and mazes”, Proceedings of the Cambridge Philosophical Society [**53**]{}, 629-641 (1957). “Introduction to Percolation Theory”, D. Stauffer and A. Aharony, Taylor and Francis Ltd (1992). “Complexity and Criticality”, K. Christensen and N. R. Moloney, Imperial College Press Advanced Physics Texts [**1**]{} (2005). P. W. Anderson, “Absence of diffusion in certain random lattices”, Phys. Rev. [**109**]{}, 1492 (1958) “Fifty years of Anderson Localization”, E. Abrahams (ed.), World Scientific Publishing (2010). H. C. Longuet-Higgins, “Some Studies in Molecular Orbital Theory I. Resonance Structures and Molecular Orbitals in Unsaturated Hydrocarbons”, J. Chem. Phys. [**18**]{}, 265 (1950). L. Lovasz and M. D. Plummer. “Matching Theory”, North-Holland, (Mathematics Studies 121, 1986). A. L. Dulmage and N. S. Mendelsohn, “Coverings of bipartite graphs”, Can. J. Math. [**10**]{}, 517 (1958). A. Pothen and C-J. Fan, “Computing the Block Triangular Form of a Sparse Matrix”, ACM Trans. Math. Softw. [**16**]{}, 303 (1990). R. W. Irving, T. Kavitha, K. Mehlhorn, D. Michail, and K. Paluch, “Rank-maximal matchings”, ACM Transactions on Algorithms [**2(4)**]{}, 602 (2006). S. Sanyal, K. Damle, and O. I. Motrunich, “Vacancy-Induced Low-Energy States in Undoped Graphene”, Phys. Rev. Lett. [**117**]{}, 116806 (2016). N. Weik, J. Schindler, S. Bera, G. .C Solomon, and F. Evers, “Graphene with vacancies: Supernumerary zero modes”, Phys. Rev. B [**94**]{}, 064204 (2016). We note parenthetically that the transcription of Longuet-Higgins’s result  presented in Ref.  unfortunately introduces a minor terminological confusion. Longuet-Higgins uses the term “supernumerary” to refer to zero modes that are not accounted for by the total number of monomers in a maximum matching. These supernumerary modes of Longuet-Higgins are not topological in nature, since they depend crucially on the actual values of the various hopping amplitudes, not merely on the connectivity of the underlying graph. The simplest example of a supernumerary zero mode in the sense of Longuet-Higgins is seeded on the honeycomb lattice by four vacancies arranged in the “four-triangle” motif of Ref. . However, Ref.  use “supernumerary” to refer to [*all*]{} zero modes of a compensated bipartite lattice (with equal numbers of $A$ and $B$ sublattice sites). This is at odds with the original usage of Longuet-Higgins. We steer clear of this terminology, making instead the more physical distinction between topologically-protected zero modes that survive the effects of hopping disorder, and fragile zero modes that do not. A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics [**321**]{}, 2 (2006). A. J. Willans, J. T. Chalker, and R. Moessner, Phys. Rev. B [**84**]{}, 115146 (2011) M. Udagawa, “Vison-Majorana complex zero-energy resonance in the Kitaev spin liquid”, Phys. Rev. B [**98**]{}, 220404(R) (2018). H. Yao and D.-H. Lee, “Fermionic Magnons, Non-Abelian Spinons, and the Spin Quantum Hall Effect from an Exactly Solvable Spin-1/2 Kitaev Model with SU(2) Symmetry”, Phys. Rev. Lett. [**107**]{}, 087205 (2011). S. Sanyal, K. Damle, J. T. Chalker, and R. Moessner, “Emergent moments and random singlet physics in a Majorana spin liquid”, arXiv:2006.16987 (unpublished). V. Hafner [*et al.*]{}, “Density of States in Graphene with Vacancies: Midgap Power Law and Frozen Multifractality”, Phys. Rev. Lett. [**113**]{}, 186802 (2014). P. M. Ostrovsky [*et al.*]{}, “Density of States in a Two-Dimensional Chiral Metal with Vacancies”, Phys. Rev. Lett. [**113**]{}, 186803 (2014). J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M. P. A. Fisher, “Non-Abelian statistics and topological quantum information processing in 1D wire networks”, Nat. Phys. [**7**]{}, 412 (2011). C. R. Laumann, A. W. W. Ludwig, D. A. Huse, and S. Trebst, “Disorder-induced Majorana metal in interacting non-Abelian anyon systems”, Phys. Rev. B [**85**]{}, 161301(R) (2012). N. Read and D. Green, “Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect”, Phys. Rev. B [**61**]{}, 10267 (2000). A. Kitaev, “Unpaired Majorana fermions in quantum wires”, Phys.Usp. [**44**]{}, 131 (2001); arXiv:cond-mat/0010440 (unpublished). I. Affleck, A. Rahmani. and D. Pikulin, “Majorana-Hubbard model on the square lattice” Phys. Rev. B [**96**]{} 125121 (2017). C. Li and M. Franz,“Majorana Hubbard model on the honeycomb lattice”, Phys. Rev. B [**98**]{}, 115123 (2018). O. Motrunich, K. Damle, and D. A. Huse, “Griffiths effects and quantum critical points in dirty superconductors without spin-rotation invariance: One-dimensional examples”, Phys. Rev. B [**63**]{}, 224204 (2001). A. C. Potter and P. A. Lee, “Multichannel Generalization of Kitaev’s Majorana End States and a Practical Route to Realize Them in Thin Films”, Phys. Rev. Lett. [**105**]{}, 227003 (2010). ‘ I. S Duff, K. Kaya, and B. Ucar, “Design, Implementation, and Analysis of Maximum Transversal Algorithms”, ACM Trans. Math. Softw. [**38**]{}, 13 (2011). R. Youngblood, J. D. Axe, and B. M. McCoy, “Correlations in ice-rule ferroelectrics”, Phys. Rev. B [**21**]{}, 5212 (1980). R. Youngblood and J. D. Axe, “Polarization fluctuations in ferroelectric models”, Phys. Rev. B [**23**]{}, 232 (1981). C. L. Henley, “The “Coulomb Phase” in frustrated systems”, Annu. Rev. Condens.Matter Phys. [**1**]{}, 179 (2010). G. Rakala, K. Damle, and D. Dhar, “Statistical properties of worm algorithms for two dimensional frustrated Ising models”,arXiv:1812.11861 (unpublished). A. W. Sandvik, “Classical percolation transition in the diluted two-dimensional $S=\frac{1}{2}$ Heisenberg antiferromagnet”, Phys. Rev. B [**66**]{}, 024418 (2002). L. Wang and A. W. Sandvik,“Low-Energy Dynamics of the Two-Dimensional $S=1/2$ Heisenberg Antiferromagnet on Percolating Clusters”, Phys. Rev. Lett. [**97**]{}, 117204 (2006). L. Wang and A. W. Sandvik, “Low-energy excitations of two-dimensional diluted Heisenberg quantum antiferromagnets”, Phys. Rev. B [**81**]{}, 054417 (2010). M. Kohmoto and B. Sutherland, “Electronic and vibrational modes on a Penrose lattice: Localized states and band structure”, Phys. Rev. B [**34**]{}, 3849 (1986). M. Arai, T. Tokihiro, T. Fujiwara, and M. Kohmoto, “Strictly Localized States on a Two-Dimensional Penrose Lattice”, Phys. Rev. B [**38**]{}, 1621 (1988). A. Koga and H. Tsunetsugu,“Antiferromagnetic order in the Hubbard model on the Penrose lattice”, Phys. Rev. B [**96**]{}, 214402 (2017). E. Day-Roberts, R. M. Fernandes, A. Kamenev, “Nature of Protected Zero Energy States in Penrose Quasicrystals”, arXiv:2004.12291 (unpublished). A. Jagannathan, R. Moessner, S. Wessel, “Inhomogeneous quantum antiferromagnetism on periodic lattices”, Phys. Rev. B [**74**]{}, 184410 (2006). A. Jagannathan, A. Szallas, S. Wessel, and M. Duneau, “Penrose quantum antiferromagnet”, Phys. Rev. B [**75**]{}, 212407 (2007). “Spin waves and local magnetizations on the Penrose tiling”, A. Szallas and A. Jagannathan, Phys. Rev. B [**77**]{}, 104427 (2008). S. Sanyal, A. Banerjee, K. Damle, and A. W. Sandvik, “Antiferromagnetic order in systems with doublet $S_{\rm tot} = 1/2$ ground states”, Phys. Rev. B [**86**]{}, 064418 (2012). F. Flicker, S. H. Simon, and S. A. Parameswaran, “Classical Dimers on Penrose Tilings”, Phys. Rev. X [**10**]{}, 011005 (2020). J. Cardy,“Conformal Invariance and Percolation”, arXiv:math-ph/0103018v2 (unpublished). R. Langlands, P. Pouliot, and Y. Saint-Aubin, “Conformal invariance in two-dimensional percolation”, Bull. of the Am. Math. Soc. [**30**]{}, 1 (1994). F. Evers and A. D. Mirlin, “Anderson Transitions”, Rev. Mod. Phys. [**80**]{}, 1355 (2008). O. Motrunich, K. Damle, and D. A. Huse, “Particle-hole symmetric localization in two dimensions”, Phys. Rev. B [**65**]{}, 064206 (2002). A. J. Kollar, M. Fitzpatrick, and A. A. Houck, “Hyperbolic lattices in circuit quantum electrodynamics”, Nature [**571**]{}, 45 (2019). D. Krioukov, F. Papadopoulos, M. Kitsak, A. Vahdat, and M. Boguna,“Hyperbolic geometry of complex networks”, Phys. Rev. E [**82**]{}, 036106 (2010). I. Benjamini and O. Schramm“Percolation in the Hyperbolic Plane”, J. Am. Math. Soc. [**14**]{}, 487 (2001).. A. Frieze, P. Melsted, “Maximum matchings in random bipartite graphs and the space utilization of Cuckoo Hash tables”, Rand. Struct. Algo. [**41**]{}, 334 (2012) R. Kenyon, “Dominos and the Gaussian free field”, Ann. Probab. [**29**]{}, 1128 (2001). R. Kenyon, “Conformal Invariance of Loops in the Double-Dimer Model”, Commun. Math. Phys. [**326**]{}, 477 (2014). North-Holland, Amsterdam N. Read and S. Sachdev, “Some features of the phase diagram of the square lattice SU(N) antiferromagnet”, Nuc. Phys. B [**316**]{}, 609 (1989). A. W. Sandvik and H. G. Evertz, “Loop updates for variational and projector quantum Monte Carlo simulations in the valence-bond basis”, Phys. Rev. B [**82**]{}, 024407 (2010). K. S. D. Beach and A. W. Sandvik, “Some formal results for the valence bond basis”, Nuc. Phys. B [**750**]{}, 142 (2006). M. Mambrini, “Hard-core dimer aspects of the SU(2) singlet wave function”, Phys. Rev. B [**77**]{}, 134430 (2008). K. S. D. Beach, F. Alet, M. Mambrini, and S. Capponi, “SU(N) Heisenberg model on the square lattice: A continuous-N quantum Monte Carlo study”, Phys. Rev. B [**80**]{}, 184401 (2009). Y. Tang, A. W. Sandvik, and C. L. Henley, “Properties of resonating-valence-bond spin liquids and critical dimer models”, Phys. Rev. B [**84**]{}, 174427 (2011). A. F. Albuquerque and F. Alet, “Critical correlations for short-range valence-bond wave functions on the square lattice”, Phys. Rev. B [**82**]{}, 180408(R) (2010). K. Damle, D. Dhar, and K. Ramola, “Resonating valence-bond wavefunctions and classical interacting dimer models”, Phys. Rev. Lett. [**108**]{}, 247216 (2012). A. F. Albuquerque, F. Alet, and R. Moessner, “Coexistence of Long-Range and Algebraic Correlations for Short-Range Valence-Bond Wave Functions in Three Dimensions”, Phys. Rev. Lett. [**109**]{}, 147204 (2012). P. Patil, I. Dasgupta, and K. Damle, “Resonating valence-bond physics on the honeycomb lattice”, Phys. Rev. B [**90**]{}, 245121 (2014). A. Banerjee and K. Damle, “Generalization of the singlet sector valence-bond loop algorithm to antiferromagnetic ground states with total spin Stot = 1/2”, J. Stat. Mech. [**2010**]{}, P08017 (2010). A. Banerjee, K. Damle, and F. Alet, “Impurity spin texture at a deconfined quantum critical point”, Phys. Rev. B [**82**]{}, 155139 (2010). A. Banerjee, K. Damle, and F. Alet, “Impurity spin texture at the critical point between Néel-ordered and valence-bond-solid states in two-dimensional SU(3) quantum antiferromagnets”, Phys. Rev. B 83, 235111 (2011). S. Sanyal, A. Banerjee, and K. Damle, “Vacancy-induced spin texture in a one-dimensional $S = 1/2$ Heisenberg antiferromagnet”, Phys. Rev. B [**84**]{}, 235129 (2011). Y. Tang and A. W. Sandvik, “Method to Characterize Spinons as Emergent Elementary Particles”,Phys. Rev. Lett. [**107**]{}, 157201 (2011). S. Sanyal, A. Banerjee, K. Damle, and A. W. Sandvik, “Antiferromagnetic order in systems with doublet $S_{\rm tot} = 1/2$ ground states”, Phys. Rev. B [**86**]{}, 064418 (2012). M. Milovanovic, S. Sachdev, and R. N. Bhatt, “Effective-field theory of local-moment formation in disordered metals”, Phys. Rev. Lett. [**63**]{}, 82 (1989). Taking our cue from the scaling theory of classical percolation [@Stauffer_Aharony_book; @Christensen_Moloney_book], we also define a third length scale $\xi_2$. This is given by the root mean square of the radii of gyration of the ${\mathcal R}$-type regions, with the mean being taken weighted by their mass $m$. Thus we have $$\begin{aligned} \xi^2_2 &=& \frac{\langle \sum_{i=1}^{N_{\mathcal R}} m_i R_i^2 \rangle }{\langle \sum_{i=1}^{N_{\mathcal R}} m_i \rangle} \;, \nonumber \\ &=& \frac{\sum_{m} mR^2_m N_m}{\sum_{m} m N_m} \;, \label{Definition_xi2} \end{aligned}$$ where the angular bracket represents the ensemble average over randomly generated vacancy configurations, and $N_m$ is the mean number of ${\mathcal R}$-type regions of mass $m$ in this ensemble. ![$\xi_2$ defined in Eqn.  obeys a scaling form $L^{1-\eta/4}F_{\xi_2}(Ln_{\rm vac}^{\nu})$, as demonstrated by the fact that plots of $\xi_2/L^{1-\eta/4}$ for different sample sizes $L$ collapse on to a single curve when plotted against the scaling variable $x^{-1} = 1/Ln_{\rm vac}^\nu$. Good scaling can be achieved for $\nu = 4.6 \pm 0.5$ ($\nu = 5.0 \pm 0.5$) on the honeycomb (square) lattice with $\eta = 0.0$. We choose to display this scaling behaviour as a function of $x^{-1}$ rather than $x$ since this provides a more stringent visual check on the quality of the collapse when the scaling function rises sharply to its $x=0$ value. This should be compared with the quality of the scaling collapse in Fig.  with $\eta = 0.04$. Similar scaling can be achieved for all $\eta \lesssim 0.08$ for slightly different ranges of $\nu$.[]{data-label="scalingfig1_xi_2"}](./plots/xi2_scaled1_inv.pdf){width="\columnwidth"} ![$\xi_2$ defined in Eqn.  obeys a scaling form $L^{1-\eta/4}F_{\xi_2}(Ln_{\rm vac}^{\nu})$, as demonstrated by the fact that plots of $\xi_2/L^{1-\eta/4}$ for different sample sizes $L$ collapse on to a single curve when plotted against the scaling variable $x^{-1} = 1/Ln_{\rm vac}^\nu$. Comparably good scaling can be achieved for $\nu = 4.8 \pm 0.5$ ($\nu = 5.1 \pm 0.5$) on the honeycomb (square) lattice with $\eta = 0.04$. We choose to display this scaling behaviour as a function of $x^{-1}$ rather than $x$ since this provides a more stringent visual check on the quality of the collapse when the scaling function rises sharply to its $x=0$ value. This should be compared with the quality of the scaling collapse in Fig.  with $\eta = 0.04$. Similar scaling can be achieved for all $\eta \lesssim 0.08$ for slightly different ranges of $\nu$.[]{data-label="scalingfig2_xi_2"}](./plots/xi2_scaled2_inv.pdf){width="\columnwidth"} Unlike $\xi$, $\xi_2$ does not admit a natural interpretation as the correlation length associated with some geometrical correlation function. However, one expects, based on a standard scaling argument familiar from the scaling theory of classical percolation [@Stauffer_Aharony_book; @Christensen_Moloney_book],, that $\xi_2$ to also diverge in the critical regime, specifically as $\xi^{1-\eta/4}$. Indeed, we do find that $\xi_2$ too also appears to grow rapidly in the small-$n_{\rm vac}$ limit to a size-limited value, consistent with this expectation. To analyse this further, we ask if our data for $\xi_2$ is consistent with the scaling ansatz [@Stauffer_Aharony_book; @Christensen_Moloney_book]: $$\begin{aligned} \xi_2 &=& L^{1-\eta/4} F_{\xi_2}(Ln_{\rm vac}^{\nu}) \; . \label{scalingfn_xi2}\end{aligned}$$ Analogous to the scaling of $\chi$ and $m_{\rm max}$, we find that our data for $\xi_2$ displays reasonably good scaling collapse for a range of $\eta \lesssim 0.08$, with the optimal value of $\nu$ increasing slightly as increase $\eta$ in this viable range. With $\eta$ set to $\eta=0$, we find that $\nu = 4.6 \pm 0.5$ ($\nu = 5.0 \pm 0.5$) gives the best scaling collapse on the honeycomb (square) lattice. With $\eta = 0.04$, the corresponding best-fit $\nu$ is given by $\nu = 4.8 \pm 0.5$ ($\nu = 5.1 \pm 0.5$) on the honeycomb (square) lattice. This is shown in Fig. \[scalingfig1\_xi\_2\] and Fig. \[scalingfig2\_xi\_2\]. As in the main text, we choose to depict the scaling collapse as a function of $x^{-1} = 1/Ln_{\rm vac}^{\nu}$ since this provides a more stringent visual check on the quality of collapse. We also note that the analogous quantity $\xi_{2}^{\rm odd}$ constructed using only clusters with odd imbalance ${\mathcal I}$ exhibits essentially identical scaling behaviour, and is therefore not displayed here. Finally, in Fig. \[scalingfig\_chiodd\] and Fig. \[scalingfig\_mmaxodd\], we demonstrate that our results for $\chi^{\rm odd}$ and $m_{\rm max}^{\rm odd}$ are not discernibly different from the corresponding results of Sec. \[Scaling\] for $\chi$ and $m_{\rm max}$ defined without the restriction to odd imbalance. Therefore, inclusion of these results in our analysis does not change our earlier estimates of the critical exponents $\eta$ and $\nu$. ![The susceptibility $\chi^{\rm odd}$ associated with the sample-averaged geometric correlation function $C^{\rm odd}(x,x')$ defined by only considering ${\mathcal R}$-type regions with odd imbalance ${\mathcal I}$ obeys a scaling form $L^{2-\eta}F_{\chi^{\rm odd}}(Ln_{\rm vac}^{\nu})$, as demonstrated by the fact that plots of $\chi^{\rm odd}/L^{2-\eta}$ for different sample sizes $L$ collapse on to a single curve when plotted against the scaling variable $x^{-1} = 1/Ln_{\rm vac}^\nu$. Good scaling can be achieved for $\nu = 5.0 \pm 0.4$ ($\nu = 5.7 \pm 0.5$) on the honeycomb (square) lattice with $\eta=0$. This is entirely analogous to the scaling of $\chi$ shown in Fig. . Indeed, there is hardly difference between the two that is discernible to the eye on the scale chosen to depict the data here. Similar scaling can be achieved for all $\eta \lesssim 0.08$ for slightly different ranges of $\nu$, again entirely analogous to the behaviour of $\chi$. See Sec.  and Sec. for details.[]{data-label="scalingfig_chiodd"}](./odd_plots/chi_scaled1_inv.pdf){width="\columnwidth"} ![The mean mass $m_{\rm max}^{\rm odd}$ of the largest ${\mathcal R}$-type regions with odd imbalance ${\mathcal I}$ in a sample obeys a scaling form $L^{2-\eta/2}F_{m_{\rm max}^{\rm odd}}(Ln_{\rm vac}^{\nu})$, as demonstrated by the fact that plots of $m^{\rm odd}_{\rm max}/L^{2-\eta}$ for different sample sizes $L$ collapse on to a single curve when plotted against the scaling variable $x^{-1} = 1/Ln_{\rm vac}^\nu$. Good scaling can be achieved for $\nu = 5.4 \pm 0.6$ ($\nu = 5.9 \pm 0.5$) on the honeycomb (square) lattice with $\eta=0.04$. This is entirely analogous to the scaling collapse of $m_{\rm max}$ in Fig. . Indeed, there is hardly difference between the two that is discernible to the eye on the scale chosen to depict the data here. Similar scaling can be achieved for all $\eta \lesssim 0.08$ for slightly different ranges of $\nu$. See Sec.  and Sec. for details.[]{data-label="scalingfig_mmaxodd"}](./odd_plots/m_max_scaled2_inv.pdf){width="\columnwidth"}

--- abstract: 'This paper presents a novel technique for process discovery. In contrast to the current trend, which only considers an event log for discovering a process model, we assume two additional inputs: an independence relation on the set of logged activities, and a collection of negative traces. After deriving an intermediate net unfolding from them, we perform a controlled folding giving rise to a Petri net which contains both the input log and all independence-equivalent traces arising from it. Remarkably, the derived Petri net cannot execute any trace from the negative collection. The entire chain of transformations is fully automated. A tool has been developed and experimental results are provided that witness the significance of the contribution of this paper.' author: - 'Hernán Ponce-de-León' - César Rodríguez - Josep Carmona - Keijo Heljanko - Stefan Haar bibliography: - 'mybib.bib' title: 'Unfolding-Based Process Discovery' --- Introduction ============ The derivation of process models from partial observations has received significant attention in the last years, as it enables eliciting evidence-based formal representations of the real processes running in a system [@AalstBook]. This discipline, known as *process discovery*, has similar premises as in *regression analysis*, i.e., only when moderate assumptions are made on the input data one can derive faithful models that represent the underlying system. Formally, a technique for process discovery receives as input an *event log*, containing the footprints of a process’ executions, and produces a model (, a Petri net) describing the real process. Many process discovery algorithms in the literature make strong implicit assumptions. A widely used one is *log completeness*, requiring every possible trace of the underlying system to be contained in the event log. This is hard to satisfy by systems with cyclic or infinite behavior, but also for systems that evolve continuously over time. Another implicit assumption is the lack of *noise* in the log, , traces denoting exceptional behavior that should not be contained in the derived process model. Finally, every discovery technique has a *representational bias*. For instance, the $\alpha$-algorithm [@Aalst11] can only discover Petri nets of a specific class (*structured workflow nets*). Few attempts have been made to remove the aforementioned assumptions. One promising direction is to relieve the discovery problem by assuming that more knowledge about the underlying system is available as input. On this line, the works in [@Ferreira2006; @Lamma2008; @Goedertier2009] are among the few that use domain knowledge in terms of *negative information*, expressed by traces which do not represent process behavior. In this paper we follow this direction, but additionally incorporate a crucial information to be used for the task of process discovery: when a pair of activities are [*independent*]{} of each other. One example could be the different tests that a patient should undergo in order to have a diagnosis: blood test, allergy test, and radiology test, which are independent each other. We believe that obtaining this coarse-grain independence information from a domain expert is an easy and natural step; however, if they are not available, one can estimate them from analysing the log with some of the techniques in the literature, e.g., the relations computed by the $\alpha$-algorithm [@AalstBook]. ![\[fig:flow\] Unfolding-based process discovery.](Figures/flow.pdf) The approach of this paper is summarized in . Starting from an event log and an independence relation on its set of activities, we conceptually construct a collection of *labeled partial orders* whose linearizations include both the sequences in the log as well those in the same Mazurkiewicz trace [@Mazurkiewicz86], , those obtained via successive permutations of independent activities. We then merge (the common prefixes of) this collection into an *event structure* which we next transform into an occurrence net representing the same behavior. Finally, we perform a controlled generalization by selectively folding the occurrence net into a Petri net. This step yields a net that (a) can execute all traces contained in the event log, and (b) generalizes the behavior of the log in a controlled manner, introducing no execution given in the collection of negative traces. The folding process is driven by a *folding equivalence relation*, which we synthesize using SMT. Different folding equivalences guarantee different properties about the final net. The paper proposes three different classes of equivalences and studies their properties. In particular we define a class of *independence-preserving* folding equivalences, guaranteeing that the natural independence relation in the final net will equal the one given by the expert. In summary, the main contributions of the paper are: - A general and efficient translation from prime event structures to occurrence nets (). - Three classes of folding equivalences of interest not only in process discovery but also in formal verification of concurrent systems (). - A method to synthesize folding equivalences using SMT (). - An implementation of our approach and experimental results witnessing its capacity to even rediscover the original model (). Remarkably, the discovery technique of this paper solves for the first time one of the foreseen operations in [@DumasG15], which advocates for the unified use of event structures to support process mining operations. Preliminaries {#sec:prelim} ============= #### Events: given an alphabet of actions $A$, several occurrences of a given action can happen on a run or execution. In this paper we consider a set $E$ of events representing the occurrence of actions in executions. Each event $e \in E$ has the form $e\eqdef\tup{a,H}$, where $a \in A$ and $H \subseteq E$ is a subset of events causing $e$ (its history). The label of an event is given by a function $\lambda \colon E \to A$ defined as $\lambda(\langle a,H \rangle) \eqdef a$. #### Labeled partial orders (lpos): we represent a labelled partial order by a pair $(E, \le)$, where ${\le} \subseteq E \times E$ is a reflexive, antisymmetric and transitive relation on the set $E$ of events. Two distinct events $e,e' \in E$ can be either ordered ($e \le e'$ or $e' \le e$) or concurrent ($e \not\le e'$ and $e' \not\le e$). Observe that all events are implicitly labelled by $\lambda$. #### Petri nets: a net consists of two disjoint sets $P$ and $T$ representing respectively places and transitions together with a set $F$ of flow arcs. The notion of state of the system in a net is captured by its markings. A marking is a multiset $M$ of places, , a map $M \colon P \to \nat$. We focus on the so-called safe nets, where markings are sets, , $M(p) \in \{0,1\}$ for all $p \in P$. A Petri net (PN) is a net together with an initial marking and a total function that labels its transitions over an alphabet $A$ of observable actions. Formally a PN is a tuple $\pnetdef$ where *(i)* $P \not = \emptyset$ is a set of places; *(ii)* $T \not = \emptyset$ is a set of transitions such that $P \cap T = \emptyset$; *(iii)* $F \subseteq (P \times T) \cup (T \times P)$ is a set of flow arcs; *(iv)* $\lambda \colon T \to A$ is a labeling mapping; and *(v)* $M_0 \subseteq P$ is an initial marking. Elements of $P \cup T$ are called the nodes of $\pnet$. For a transition $t \in T$, we call $\preset t \eqdef \{ p \mid (p,t) \in F \}$ the preset of $t$, and $\postset t \eqdef \{p \mid (t,p) \in F \}$ the postset of $t$. In figures, we represent as usual places by empty circles, transitions by squares, $F$ by arrows, and the marking of a place $p$ by black tokens in $p$. A transition $t$ is enabled in marking $M$, written $M \arrow t$, iff $\preset t \subseteq M$. This enabled transition can fire, resulting in a new marking $M' \eqdef (M \backslash \preset t) \cup \postset t$. This firing relation is denoted by $M \arrow t M'$. A marking $M$ is reachable from $M_0$ if there exists a firing sequence, i.e. transitions, $t_1, \dots, t_n$ such that $M_0 \arrow{t_1} \dots \arrow{t_n} M$. The set of reachable markings from $M_0$ is denoted by $\reach \pnet$. The set of co-enabled transitions is ${\coe \pnet} \eqdef \{ (t,t') \mid \exists M \in \reach \pnet \colon \preset t \subseteq M \land \preset{t'} \subseteq M \}$. The set of observations of a net is the image over $\lambda$ of its fireable sequences, , $\sigma \in \obs \pnet$ iff $M_0 \arrow{t_1} \dots \arrow{t_n} M$ and $\lambda(t_1) \dots \lambda(t_n) = \sigma$. #### Occurrence nets: occurrence nets can be seen as infinite Petri nets with a special acyclic structure that highlights conflict between transitions that compete for resources. Places and transitions of an occurrence net are usually called conditions and events. Formally, let $\netdef$ be a net, $<$ the transitive closure of $F$, and $\leq$ the reflexive closure of $<$. We say that transitions $t_1$ and $t_2$ are in structural conflict, written $t_1 \cfl[s] t_2$, if and only if $t_1 \not = t_2$ and $\preset{t_1} \cap \preset{t_2} \not = \emptyset$. Conflict is inherited along $<$, that is, the conflict relation $\cfl$ is given by $a \cfl b \Leftrightarrow \exists t_a,t_b \in T \colon t_a \cfl[s] t_b \land t_a \leq a \land t_b \leq b$. Finally, the concurrency relation $\bf co$ holds between nodes $a,b \in P \cup T$ that are neither ordered nor in conflict, i.e. $a\co b \Leftrightarrow \neg (a \leq b) \land \neg (a \cfl b) \land \neg (b \leq a)$. A net $\ondef$ is an occurrence net iff *(i)* $\leq$ is a partial order; *(ii)* for all $b \in B$, $\lvert \preset b \rvert \in \{0,1\}$; *(iii)* for all $x\in B \cup E$, the set $[x] := \{y \in E \mid y \leq x\}$ is finite; *(iv)* there is no self-conflict, i.e. there is no $x \in B \cup E$ such that $x\cfl x$. The initial marking $M_0$ of an occurrence net is the set of conditions with an empty preset, i.e. $\forall b \in B\colon b \in M_0 \Leftrightarrow \preset b = \emptyset$. Every $\leq$-closed and conflict-free set of events $C$ is called a configuration and generates a reachable marking defined as $\marking C \eqdef (M_0 \cup \postset C) \setminus \preset C$. We also assume a labeling function $\lambda \colon E \to A$ from events in $\beta$ to alphabet $A$. Conditions are of the form $\langle e , X \rangle$ where $e \in E$ is the event generating the condition and $X \subseteq E$ are the events consuming it. Occurrence nets are the mathematical form of the partial order unfolding semantics of a Petri net [@EsparzaRV02]; we use indifferently the terms occurrence net and unfolding. Conditions in an occurrence net can be removed by keeping the causal dependencies and introducing a conflict relation; the obtained object is an event structure [@NielsenPW81]. #### Event structures: an event structure is a tuple $\lesdef$ where $E$ is a set of events; $\leq\ \subseteq E \times E$ is a partial order (called causality) satisfying the property of finite causes, i.e. $\forall e \in E : \lvert [e] \rvert < \infty$ where $[e] := \{ e' \in E \mid e' \leq e \}$; ${\cfl} \subseteq E \times E$ is an irreflexive symmetric relation (called conflict) satisfying the property of conflict heredity, i.e. $\forall e,e',e'' \in E : e \cfl e' \land e' \leq e'' \Rightarrow e \cfl e''$. Note that in most cases one only needs to consider reduced versions of relations $\leq$ and $\cfl$, which we will denote $\leqi$ and $\dcfl$, respectively. Formally, $\leqi$ (which we call direct causality) is the transitive reduction of $\leq$, and $\dcfl$ (direct conflict) is the smallest relation inducing $\cfl$ through the property of conflict heredity. A configuration is a computation state represented by a set of events that have occurred; if an event is present in a configuration, then so must all the events on which it causally depends. Moreover, a configuration does not contain conflicting events. Formally, a configuration of $(E,{\leq},{\cfl})$ is a set $C \subseteq E$ such that $e \in C \Rightarrow (\forall e' \leq e : e' \in C)$, and $(e \in C \land e \cfl e') \Rightarrow e'\not\in C$. The set of configurations of $\les$ is denoted by $\Omega(\les)$. #### Mazurkiewicz traces: let $A$ be a finite alphabet of letters and $\ind \subseteq A \times A$ a symmetric and irreflexive relation called independence. The relation $\ind$ induces an equivalence relation $\equiv_\ind$ over $A^*$. Two words $\sigma$ and $\sigma'$ are equivalent ($\sigma \equiv_\ind \sigma'$) if there exists a sequence $\sigma_1 \dots \sigma_k$ of words such that $\sigma=\sigma_1, \sigma'=\sigma_k$ and for all $1\leq i \leq k$ there exists words $\sigma_i', \sigma_i''$ and letters $a_i,b_i$ satisfying $$\sigma_i=\sigma_i' a_i b_i \sigma_i'', \hspace{5mm} \sigma_{i+1}=\sigma_i' b_i a_i \sigma_i'', \hspace{4mm} \text{and } (a_i,b_i) \in \ind$$ Thus, two words are equivalent by $\equiv_\ind$ if one can be obtained from the other by successive commutation of neighboring independent letters. For a word $\sigma \in A^*$ the equivalence class of $\sigma$ under $\equiv_\ind$ is called a Mazurkiewicz trace [@Mazurkiewicz86]. We now describe the problem tackled in this paper, one of the main challenges in the [*process mining*]{} field [@AalstBook]. #### Process Discovery: a log ${\logs}$ is a finite set of traces over an alphabet $A$ representing the footprints of the real process executions of a system $\sys$ that is only (partially) visible through these runs. Process discovery techniques aim at extracting from a log ${\logs}$ a process model ${\model}$ (e.g., a Petri net) with the goal to elicit the process underlying in ${\sys}$. By relating the behaviors of ${\logs}$, $\obs {\model}$ and ${\sys}$, particular concepts can be defined [@BuijsDA14]. A log is *incomplete* if ${\sys} \backslash {\logs} \ne \emptyset$. A model ${\model}$ *fits* log ${\logs}$ if ${\logs} \subseteq \obs {\model}$. A model is *precise* in describing a log ${\logs}$ if $\obs {\model} \backslash {\logs}$ is small. A model ${\model}$ represents a *generalization* of log ${\logs}$ with respect to system ${\sys}$ if some behavior in ${\sys} \backslash {\logs}$ exists in $\obs {\model}$. Finally, a model ${\model}$ is *simple* when it has the minimal complexity in representing $\obs {\model}$, i.e., the well-known *Occam’s razor principle*. It is widely acknowledged that the size of a process model is the most important simplicity indicator. Let ${\cal U}^{\cal N}$ be the universe of nets, we define a function $\hat c: {\cal U}^{\cal N} \to \mathbb{N}$ to measure the simplicity of a net by counting the number of some of its elements, , its transitions and/or places. Independence-Preserving Discovery {#sec:discovery} ================================= Let $\sys$ be a system whose set of actions is $A$. Given two actions $a,b \in A$ and one state $s$ of $\sys$, we say that $a$ and $b$ *commute* at $s$ when - if $a$ can fire at $s$ and its execution reaches state $s'$, then $b$ is possible at $s$ iff it is possible at $s'$; and - if both $a$ and $b$ can fire at $s$, then firing $ab$ and $ba$ reaches the same state. Commutativity of actions at states identifies an equivalence relation in the set of executions of the system $\sys$; it is a *ternary* relation, relating two transitions with one state. Since asking the expert to provide the commutativity relation of $\sys$ would be difficult, we restrict ourselves to unconditional independence, , a conservative overapproximation of the commutativity relation that is a sole property of transitions, as opposed to transitions and states. An *unconditional independence* relation of $\sys$ is any *binary*, symmetric, and irreflexive relation $\ind \subseteq A \times A$ satisfying that if $a \ind b$ then $a$ and $b$ commute at *every reachable state* of $\sys$. If $a, b$ are not independent according to $\ind$, then they are dependent, denoted by $a \dep b$. In this section, given a log $\logs \subseteq A^*$, representing some behaviors of $\sys$, and an arbitrary unconditional independence $\ind$ of $\sys$, provided by the expert, we construct an occurrence net whose executions contain $\logs$ together with all sequences in $A^*$ which are $\equiv_{\ind}$-equivalent to some sequence in $\logs$. If commuting actions are not declared independent by the expert (, $\ind$ is smaller than it could be), then $\model$ will be more sequential than $\sys$; if some actions that did not commute are marked as independent, then $\model$ will not be a truthful representation of $\sys$. The use of expert knowledge in terms of an independence relation is a novel feature not considered before in the context of process discovery. We believe this is a powerful way to fight with the problem of log incompleteness in a practical way since it is only needed to observe in the log one trace representative of a class in $\equiv_\ind$ to include the whole set of traces of the class in the process model’s executions. Our final goal is to generate a Petri net that represents the behavior of the underlying system. We start by translating $\logs$ into a collection of partial orders whose shape depends on the specific definition of $\ind$. \[def:log2lpo\] Given a sequence $\sigma \in A^*$ and an independence relation $\ind \subseteq A \times A$, we associate to $\sigma$ a labeled partial order $\lpo \sigma$ inductively defined by: 1. If $\sigma = \varepsilon$, then let $\bot \eqdef \tup{\tau, \emptyset}$ and set $\lpo \sigma \eqdef (\set \bot, \emptyset)$. 2. If $\sigma = \sigma' a$, then let $\lpo{\sigma'} \eqdef (E',\leq')$ and let $e \eqdef \tup{a, H}$ be the single event such that $H$ is the unique $\subseteq$-minimal, causally-closed set of events in $E'$ satisfying that for any event $e' \in E'$, if $\lambda(e') \dep a$, then $e' \in H$. Then set $\lpo \sigma \eqdef (E,\leq)$ with $E \eqdef E' \cup \{ e \}$ and ${\leq} \eqdef {\leq'} \cup (H \times \{ e \})$. Since a system rarely generates a single observation, we need a compact way to model all the possible observations of the system. We represent all the partially ordered executions of a system with an event structure. \[def:lpo2es\] Given a set of partial orders $ S \eqdef \{ (E_i,\leq_i) \mid 1 \leq i \leq n \}$, we define $\es S \eqdef (E,\leq, \cfl)$ where: 1. $E \eqdef \bigcup\limits_{1 \leq i \leq n} E_i$, 2. ${\leq} \eqdef (\bigcup\limits_{1 \leq i \leq n} \leq_i)^*$, and 3. for $e \eqdef \langle a,H \rangle$ and $e' \eqdef \langle b,H' \rangle$, we have that $e \dcfl e'$ (read: $e$ and $e'$ are in direct conflict) iff $e' \not \in H, e \not \in H'$ and $a \dep b$. The conflict relation $\cfl$ is the smallest relation that includes $\dcfl$ and is inherited $\leq$, , for $e \cfl e'$ and $e \leq f$, $e' \leq f'$, one has $f \cfl f'$. Given a set of finite partial orders $S$, we now show that $S$ is included in the configurations of the event structure obtained by . This means that our event structure is a fitting representation of $\logs$. \[lemma:fit\] If $S$ is finite, then $S \subseteq \Omega(\es S)$. Since we want to produce a Petri net, we now need to “*attach conditions*” to the result of . Event structures and occurrence nets are conceptually very similar objects so this might seem very easy for the acquainted reader. However, this definition is crucial for the success of the subsequent folding step (), as we will be constrained to merge conditions in the preset and postset of an event when we merge the event. As a result, the conditions that we produce now should constraint as little as possible the future folding step. \[def:es2on\] Given an event structure $\lesdef$ we construct the occurrence net $\on \eqdef (B,E \backslash \{ \bot \},F)$ in two steps 1. Let $G \eqdef (V,A)$ be a graph where $V \eqdef E$ and $(e_1,e_2) \in A$ iff $e_1 \dcfl e_2$. For each clique (maximal complete subgraph) $K \eqdef \{e_1, \dots, e_n \}$ of $G$, let $C_K \eqdef [e_1] \cap \dots \cap [e_n]$ and $e_K \in \max (C_K)$. We add a condition $b$ to $B$ and set $b \in \postset{e_K}$ and $b \in \preset{e_i}$ for $i = 1 \dots n$. 2. For each $e \in E$, let $G_e \eqdef (V_e,A_e)$ be a graph where $V_e \eqdef \{ e' \in E \mid e \leqi e' \}$ and $(e_1,e_2) \in A_e$ iff $\lambda(e_1) \dep \lambda(e_2)$. For each clique $K_e := \{e_1, \dots, e_n \}$ of $G_e$, we add a condition $b$ to $B$ and set $b \in \postset{e}$ and $b \in \preset{e_i}$ for $i = 1 \dots n$. .1. adds a condition for every set of pairwise direct conflicting events; the condition is generated by some event $e_K$ which is in the past of every conflicting event and consumed by all of them; by the latter the conflict of the event structure is preserved in the occurrence net. For each event and its immediate successors, .2. adds conditions between them to preserve causality. To minimize the number of conditions, for the successor events having dependent labels only one condition is generated. This step does not introduce new conflicts in the occurrence net since the events have dependent labels and none is in the past of the other, then by they are also in conflict in the event structure. We note that Winskel already explained, in categorical terms, how to relate an event structure with an occurrence net [@Winskel84a]. However, his definition is of interest only in that context, while ours focus on a practical and efficient translation. Given a log $\logs$ and an independence relation $\ind$, the net obtained applying Definitions \[def:log2lpo\], \[def:lpo2es\] and \[def:es2on\], in this order, is denoted by $\onl$. Since every trace in $\logs$ is a linearization of some of the partial orders in the set $S$ obtained by and these partial orders are included by in the configurations of $\es S$ (which are the same as the configurations in $\onl$), the obtained net is fitting. \[prop:fit\] Let $\logs$ be a log and $\ind$ an independence relation, for every $\sigma \in \logs$ we have $\sigma \in \obs \onl$. Since every trace is a linearization of some partial order obtained by , by every trace is a linearization of the maximal configurations of the event structure; since causality and conflict are preserved by , their configurations coincide, the trace correspond to a sequential execution of the occurrence net and the result holds. It is worth noticing that the obtained net generalizes the behavior of the model, but in a controlled manner imposed by the independence relation. For instance, if $\logs \eqdef \set{ab}$ and $a \ind b$, then $ba \in \obs \onl$, even if this behavior was not present in the log. If the expert rightly declared $a$ and $b$ independent (, if they commute at all states of $\sys$), then necessarily $ba$ is a possible observation of $\sys$, even if it is not in $\logs$. The extra information provided by the expert allows us to generalize the discovered model in a provably sound manner, thus coping with the log incompleteness problem. The independence relation between labels gives rise to an arbitrary relation between transitions of a net (not necessarily an independence relation): \[def:indt\] Let $\ind \subseteq A \times A$ be an independence relation, $\pnetdef$ a net, and $\lambda \colon T \to A$. We define relation ${\indt \net} \subseteq T \times T$ between transitions of $\net$ as $$t \indt \net t' \Leftrightarrow \lambda(t) \ind \lambda(t').$$ In the next section we will define an approach to fold $\onl$ into a Petri net whose natural independence relation equals $\ind$. To formalize our approach we first need to define such natural independence. \[def:indu\] Let $\netdef$ be a net. We define the *natural independence* relation ${\indu \net} \subseteq T \times T$ on $\net$ as $$t \indu \net t' \Leftrightarrow \preset t \cap \preset{t'} = \emptyset \land \postset t \cap \preset{t'} = \emptyset \land \preset t \cap \postset{t'} = \emptyset.$$ In fact, one can prove that when $\net$ is safe, then $\indu \net$ is the notion of independence underlying the unfolding semantics of $\net$. In other words, the equivalence classes of $\equiv_{\indu \net}$ are in bijective correspondence with the configurations in the unfolding of $\net$. The following result shows that the natural independence on the discovered occurrence net corresponds to the relation provided by the expert, when both we restrict to the set of co-enabled transitions. \[prop:ind\_on\] Let $\onl$ be the occurrence net from the log $\logs$ with $\ind$ as the independence relation, then $${\indt \onl} \cap {\coe \onl} = {\indu \onl} \cap {\coe \onl}$$ Introducing Generalization {#sec:generalization} ========================== The construction described in the previous section guarantees that the unfolding obtained is fitting (see ). However, the difference between $\sys$ and $\logs$ may be significant (e.g., $\sys$ can contain cyclic behavior that can be instantiated an arbitrary number of times whereas only finite traces exist in $\logs$) and the unfolding may be poor in generalization. The goal of this section is to generalize $\onl$ in a way that the right patterns from $\sys$, partially observed in $\logs$ (, loops), are incorporated in the generalized model. To generalize, we fold the discovered occurrence net. This folding is driven by an equivalence relation $\sim$ on $E \cup B$ that dictates which events merge into the same transition, and analogously for conditions; events cannot be merged with conditions. We write $[x]_\sim \eqdef \{ x' \mid x \sim x' \}$ for the equivalence class of node $x$. For a set $X$, $[X]_\sim \eqdef \{ [x]_\sim \mid x \in X \}$ is a set of equivalence classes. \[def:foldednet\] Let $\on \eqdef (B,E,F)$ be an occurrence net and $\sim$ a equivalence relation on the nodes of $\on$. The folded Petri net ( $\sim$) is defined as $\on^\sim \eqdef (P_\sim,T_\sim,F_\sim,{M_0}_\sim)$ where $$\begin{aligned} P_\sim & \eqdef \{ [b]_\sim \mid b \in B \}, & F_\sim & \eqdef \{ ([x]_\sim,[y]_\sim) \mid (x,y) \in F \}, \\ T_\sim & \eqdef \{ [e]_\sim \mid e \in E \}, & {M_0}_\sim([b]_\sim) & \eqdef \lvert \{ b' \in [b]_\sim \mid \preset{b'} = \emptyset \} \rvert. \end{aligned}$$ Notice that the initial marking of the folded net is not necessarily safe. Safeness of the net depends on the chosen equivalence relation (see ). Language-Preserving Generalization {#sec:lpgeneralization} ---------------------------------- Different folding equivalences guarantee different properties on the folded net. From now on we focus our attention on three interesting classes of folding equivalences. The first preserves sequential executions of $\onl$. \[def:fold1\] Let $\on$ be an occurrence net; an equivalence relation $\sim$ is called a sequence preserving (SP) folding equivalence iff $e_1 \sim e_2$ implies $\lambda(e_1) = \lambda(e_2)$ and $[\preset{e_1}]_\sim = [\preset{e_2}]_\sim$ for all events $e_1,e_2 \in E$. From the definition above it follows that $e_1 \sim e_2$ implies $\forall b \in \preset{e_1}: \exists b' \in \preset{e_2}$ with $b \sim b'$. Since for every folded net obtained from a SP folding equivalence only equally labeled events are merged; we define then $\lambda([e]_\sim) \eqdef \lambda(e)$. \[ex:fold1\] Consider the log $\logs = \{ abc, bd \}$ and the independence relation $\ind = \emptyset$. shows the obtained unfolding $\onl$ (left) and three of its folded nets. The equivalence relation $\sim_1$ merges events labeled by $b$, but it does not merge their presets, i.e. is not a SP folding equivalence. It can be observed that $bd$ is not fireable in $\onl^{\sim_1}$. Whenever two events are merged, their preconditions need to be merged to preserved sequential executions. The equivalence relation $\sim_2$ does not only merge events labeled by $b$, but it also sets $p_1 \sim_2 p_2$ and is a SP folding equivalence. The folded net $\onl^{\sim_2}$ can replay every trace in the log $\logs$, but it also adds new traces of the form $a^*, a^*b, a^*bc, a^*bd, a^*bcd$ and $a^*bdc$. Given an unfolding, every SP folding equivalence generates a net that preserves its sequential executions. [theorem]{}[thetwo]{} \[the:fire\_seq\] Let $\on$ be an occurrence net and $\sim$ a SP folding equivalence, then every fireable sequence $M_0 \arrow{e_1} \dots \arrow{e_n} M_n$ from $\on$ generates a fireable sequence $[M_0]_\sim \arrow{[e_1]_\sim} \dots \arrow{[e_n]_\sim} [M_n]_\sim$ from $\on^\sim$. As a corollary of the result above and , the folded net obtained from $\onl$ with a SP folding equivalence is fitting. \[cor:fitting\] Let $\logs$ be a log, $\ind$ an independence relation and $\sim$ a SP folding equivalence, then for every $\sigma \in \logs$ we have $\sigma \in \obs \netl$. We saw in that every trace from $\logs$ can be replayed in $\onl^{\sim_2}$, but (as expected) the net accepts more traces. However this net also adds some independence between actions of the system: after firing $b$ the net puts tokens at $[p_3]_{\sim_2}$ and $[p_4]_{\sim_2}$ and the reached marking enables concurrently actions $c$ and $d$ which contradicts $c \dep d$ (the independence relation $\ind = \emptyset$ implies $c \dep d$). In order to avoid this extra independence, we now consider the following class of equivalences. \[def:fold2\] Let $\on$ be an occurrence net and $\ind$ an independence relation; an equivalence relation $\sim$ is called an independence preserving (IP) folding equivalence iff 1. $\sim$ is a SP folding equivalence, 2. $\lambda(e_1) \ind \lambda(e_2) \Leftrightarrow [\preset e_1]_\sim \cap [\preset{e_2}]_\sim = \emptyset \land [\preset e_1]_\sim \cap [\postset{e_2}]_\sim = \emptyset \land [\postset e_1]_\sim \cap [\preset{e_2}]_\sim = \emptyset$ for all events $e_1,e_2 \in E$. 3. $b_1 \co b_2$ implies $b_1 \not \sim b_2$ for all conditions $b_1,b_2 \in B$. IP folding equivalences not only preserve the sequential behavior of $\on$, but also ensure that $\on^\sim$ and $\on$ exhibit the same natural independence relation. The definition above differs from the folding equivalence definition given in [@FahlandA13]; they consider occurrence nets coming from an unfolding procedure which takes as an input a net. This procedure generates a mapping between conditions and events of the generated occurrence net and places and transitions in the original net. Such mapping is necessary to define their folding equivalence. In our setting, the occurrence net does not come from a given net and therefore the mapping is not available. The equivalence $\sim_2$ from is not an IP folding equivalence since the intersection of the equivalent classes of the preset of $c$ and $d$ is empty ($[\preset c]_{\sim_2} = \{[p_4]_{\sim_2}\}, [\preset d]_{\sim_2} = \{[p_3]_{\sim_2}\}$ and $\{[p_4]_{\sim_2}\} \cap \{[p_3]_{\sim_2}\} = \emptyset$), but $c$ and $d$ are not independent. Consider the equivalence relation $\sim_3$ which merges events labeled by $b$ and it sets $p_1 \sim_3 p_2$ and $p_3 \sim_3 p_4$; this relation is an IP folding equivalence. It can be observed in the net $\onl^{\sim_3}$ of that all the traces from the log can be replayed, but new independence relations are not introduced. The occurrence net $\onl$ is clearly safe. We show that $\netl$ is also safe when $\sim$ is an IP folding equivalence. In this work, we constraint IP equivalences to generate safe nets because their natural independence relation is well understood (), thus allowing us to assign a solid meaning to the class IP. It is unclear what is the natural unconditional independence of an unsafe net, and extending our definitions to such nets is subject of future work. \[prop:safe\] Let $\onl$ be the unfolding obtained from the log $\logs$ with $\ind$ as the independence relation and $\sim$ an IP folding equivalence. Then $\netl$ is safe. The unfolding $\onl$ is trivially safe since its initial marking puts one token in its minimal conditions and each condition contains only one event in its preset and that event cannot put more than one token in the condition. Suppose $\netl$ is not safe, by the above this is possible iff there exists $C \in \reach \onl$ and $b_1,b_2 \in C$ such that $b_1 \sim b_2$. If $b_1$ and $b_2$ belong to a reachable marking, then they must be concurrent and since $\sim$ is an IP folding equivalence they cannot be merged, which leads to a contradiction. Finally $\netl$ must be safe. shows that the structural relation between events of the unfolding and the relation generated by the independence given by the expert coincide (when we restrict to co-enabled events); the result also holds for the folded net when an IP folding equivalence is used. [theorem]{}[thethree]{} \[the:ip\] Let $\onl$ be the unfolding obtained from the log $\logs$ with $\ind$ as the independence relation and $\sim$ an IP folding equivalence, then ${\indt \netl} = {\indu \netl}$. Controlling Generalization via Negative Information --------------------------------------------------- We have shown that IP folding equivalences preserve independence. However, they could still introduce new unintended behaviour not present in $\sys$. In this section we limit this phenomena by considering *negative information*, denoted by traces that should not be allowed by the model. Concretely, we consider negative information which is also given in the form of sequences $\sigma \in \logs^- \subseteq A^*$. Negative information is often provided by an expert, but it can also be obtained automatically by recent methods [@BrouckeWVB14]. Very few techniques in the literature use negative information in process discovery [@Goedertier2009]. In this work, we assume a minimality criterion on the negative traces used: Let $\logs \eqdef \logs^+ \uplus \logs^-$ be a pair of positive and negative logs and $\ind$ the independence relation given by the expert. Any negative trace $\sigma \in \logs^-$ corresponds to the local configuration of some event $\esig$ in $\onl$. This assumption implies that each negative trace is of the form $\sigma' a$ where $\sigma'$ only contains the actions that are necessarily to fire $a$. If $a$ can happen without them, they should not be consider part of $\sigma$. By removing all events $\esig$ from $\onl$ (one for each negative trace $\sigma \in \logs^-$), we obtain a new occurrence net denoted by $\onls$. The goal of this section is to fold this occurrence net without re-introducing the negative traces in the generalization step. If the expert is unable to provide negative traces satisfying this assumption, the discovery tool can always let him/her choose $\esig$ from a visual representation of the unfolding. \[def:fold3\] Let $\ondef$ be an occurrence net and $\logs^-$ a negative log; an equivalence relation $\sim$ is called removal aware (RA) folding equivalence iff 1. $\sim$ is a SP folding equivalence, and 2. for every $\sigma \in \logs^-$ and $e' \in E$ we have $\lambda(e') = \lambda(\esig)$ implies $[\preset{e'}]_\sim \not \subseteq [\preset{\esig}]$. The folded net obtained from $\onls$ with a RA folding equivalence does not contain any of the negative traces. [theorem]{}[thefour]{} \[the:ra\] Let $\onls$ be the unfolding obtained from the log $\logs \eqdef \logs^+ \uplus \logs^-$ with $\ind$ as the independence relation after removing the corresponding event of each negative trace and $\sim$ a RA folding equivalence,[^1] then $$\obs \netls \cap \logs^- = \emptyset$$ Computing Folding Equivalences {#sec:computing} ============================== presents a discovery algorithm that generates fitting occurrence nets and defines three classes of folding criteria, SP, IP, and RA, that ensure various properties. This section proposes an approach to synthesize SP, IP and RA folding equivalences using SMT. SMT Encoding {#sec:sat} ------------ We use an SMT encoding to find folding equivalences generating a net $\on^\sim$ satisfying specific metric properties. Specifically, given a measure $\hat c$ (cf., ), decidable in polynomial time, and a number $k \in \nat$, we generate an SMT formula which is satisfiable iff there exists a folding equivalence $\sim$ such that $\hat c(\on^\sim) = k$. We consider the number of transitions in the folded net as the measure $\hat c$, however, theoretically, any other measure that can be computed in polynomial time could be used. As explained in simple functions like counting the number of nodes/arcs provide in practice reasonable results. Given an occurrence net $\ondef$, for every event $e \in E$ and condition $b \in B$ we have integer variables $v_e$ and $v_b$. The key intuition is that two events (conditions) whose variables have equal number are equivalent and will be merged into the same transition (place). The following formulas state, respectively, that every element of a set $X$ is related with at least one element of a set $Y$, and that every element of $X$ is not related with any element of $Y$: $$\ssc X Y \eqdef \bigwedge\limits_{x \in X}\bigvee\limits_{y \in Y} (v_x = v_y) \hspace{10mm} \disj X Y \eqdef \hspace{-2mm}\bigwedge\limits_{x \in X, y \in Y} \hspace{-2mm} (v_x \not = v_y)$$ We force any satisfying assignment to represent an SP folding equivalence () with the following two constraints: $$\phi_\on^{SP} \eqdef \phi_\on^{lab} \land \phi_\on^{pre}.$$ Formulas $\phi_\on^{lab}$ and $\phi_\on^{pre}$ impose that only equally labeled events should be equivalent and that if two events are equivalent, then their presets should generate the same equivalence class: $$\phi_\on^{lab} \eqdef \bigwedge\limits_{\substack{e,e' \in E \\ \lambda(e) \not = \lambda(e')}} \hspace{-2mm} (v_e \not = v_{e'}) \hspace{10mm} \phi_\on^{pre} \eqdef \bigwedge\limits_{e,e' \in E} (v_e = v_{e'} \Rightarrow (\ssc {\preset e} {\preset{e'}} \land \ssc {\preset{e'}} {\preset e}))$$ In addition to the properties encoded above, an IP folding equivalence () should satisfy some other restrictions: $$\phi_\on^{IP} \eqdef \phi_\on^{SP} \land \phi_\on^{ind} \land \phi_\on^{co}$$ where $\phi_\on^{ind}$ imposes that the presets and postsets of events with independent labels should generate equivalence classes that do not intersect and $\phi_\on^{co}$ forbids concurrent conditions to be merged: $$\phi_\on^{ind} \eqdef \bigwedge\limits_{e,e' \in E} \hspace{-2mm} (\lambda(e) \ind \lambda(e') \Leftrightarrow (\disj{\preset e}{\preset{e'}} \land \disj{\preset e}{\postset{e'}} \land \disj{\preset e}{\postset{e'}})) \hspace{8mm}\phi_\on^{co} \eqdef \bigwedge\limits_{\substack{b,b' \in B \\ b \co b'}} \hspace{-2mm} (v_b \not = v_{b'})$$ Given a negative log $\logs^-$, to encode a RA folding equivalence () we define: $$\phi_{\on, \logs^-}^{RA} \eqdef \phi_\on^{SP} \land (\hspace{-2mm}\bigwedge\limits_{\substack{\sigma \in \logs^-, e' \in E\\ \lambda(e') = \lambda(\esig)}} \hspace{-3mm} \neg \ssc {\preset{e'}} {\preset \esig})$$ where the right part of the conjunction imposes that for every $\esig$ generated by a negative trace and any other event with the same label, their presets cannot generate the same equivalence class. We now encode the optimality (the number of transitions) of the mined net. Given an occurrence net $\ondef$, each event $e \in E$ generates a transition $v_e$ in the folded net $\on^\sim$. To impose that the number of transitions in $\on^\sim$ should be at most $k \in \nat$, we define: $$\phi_{\on,k}^{MET} \eqdef \bigwedge\limits_{e \in E} (1 \le v_e \leq k)$$ To find an IP and RA folding equivalence that generates a net with at most $k$ transitions we propose the following encoding: $$\phi_{\on,\logs^-,k}^{OPT} \eqdef \phi_{\on}^{IP} \land \phi_{\on,\logs^-}^{RA} \land \phi_{\on,k}^{MET}$$ [theorem]{}[thefive]{} Let $\logs \eqdef \logs^+ \uplus \logs^-$ be a set of positive and negative logs, $\ind \subseteq A \times A$ and independence relation and $k \in \nat$. The formula $\phi_{\on,\logs^-,k}^{OPT}$ is satisfiable iff there exists an IP and RA folding equivalence $\sim$ such that $\on_{\logs, \ind, *}^\sim$ contains at most $k$ transitions. Finding an Optimal Folding Equivalence {#sec:opt_fold} -------------------------------------- explains how to compute a folding equivalence that generates a folded net with a bounded number of transitions; this section explain how to obtain the optimal folded net, i.e the one with minimal number of transitions satisfying the properties of and . Iterative calls to the SMT solver can be done for a binary search with $k$ between $min_k$ and $max_k$; since only equally labeled events can be merged by the folding equivalence, the minimal number of transitions in the folded net is $min_k \eqdef \lvert A \rvert$; in the worst case, when events cannot be merged, $max_k \eqdef \lvert E \rvert$. As a side remark, we have noted that the optimal folding equivalence can be encoded as a MaxSMT problem [@NieuwenhuisO06] where some clauses which are called hard must be true in a solution (in our case $\phi_{\on}^{IP}$ and $\phi_{\on,\logs^-}^{RA}$) and some soft clauses may not ($\phi_{\on,k}^{MET}$ for $\lvert A \rvert \leq k \leq \lvert E \rvert$); a MaxSMT solver maximizes the number of soft clauses that are satisfiable and thus it obtains the minimal $k$ generating thus the optimal folded net. Experiments {#sec:experiments} =========== As a proof of concept, we implemented our approach into a new tool called (Partial Order Discovery).[^2] It supports synthesis of SP and IP folding equivalences using a restricted form of our SMT encoding. In particular merges all events with equal label, in contrast to the encoding in which may in general yield more than one transition per log action. While this ensures a minimum (optimal as per ) number of folded transitions, the tool could sometimes not find a suitable equivalence (unsatisfiable SMT encoding). Since the number of transitions in the folded net is fixed, it turns out that the quality of the mined model increases as we increase the number of folded places, as we show below. Using we evaluate the ability of our approach to rediscover the original process model, given its independence relation and a set of logs. For this we have used standard benchmarks from the verification and process mining literature [@MCC; @WDHS08]. In our experiments, , we consider a set of original processes faithfully modelled as safe Petri nets. For every model $\sys$ we consider a log $\logs$, i.e. a subset of its traces. We extract from $\sys$ the (best) independence relation $\indu \sys$ that an expert could provide. We then provide $\logs$ and $\indu \sys$ to and find an SP folding equivalence with the largest number of places (“max. places”) and with 60% of the places of $\sys$ (last group of ), giving rise to two different mined models. All three models, original plus mined ones, have perfect fitness but varying levels of precision, i.e. traces of the model not present in the log. For the mined models, we report (“%Prec.”) on the ratio between their precision and the precision of the original model $\sys$. All precisions were estimated using the technique from [@AMCDA15]. All running times were below 10s. Additionally, we measure how much independence of the original model is preserved in the mined ones. For that, we define the ratios ${r_{{\sys} \subseteq {\model}}}\eqdef \nicefrac{|{\indu \sys} \cap {\indu \model}|}{|\indu \sys|}$ and ${r_{{\model} \subseteq {\sys}}}\eqdef \nicefrac{|{\indu \sys} \cap {\indu \model}|}{|\indu \model|}$. The closer ${r_{{\sys} \subseteq {\model}}}$ is to 1, the larger is the number of pairs in $\indu \sys$ also contained in $\indu \model$ (, the more independence was preserved), and conversely for ${r_{{\model} \subseteq {\sys}}}$ (the less independence was “*invented*”). Remark that ${\indu \sys} = {\indu \model}$ iff ${r_{{\sys} \subseteq {\model}}}= {r_{{\model} \subseteq {\sys}}}= 1$. In 7 out of the 11 benchmarks in our proof-of-concept tool rediscovers the original model or finds one with only minor differences. This is even more encouraging when considering that we only asked to find SP equivalences which, unlike IP, do not guarantee preservation of independence. In 9 out of 11 cases both ratios ${r_{{\sys} \subseteq {\model}}}$ and ${r_{{\model} \subseteq {\sys}}}$ are above 98%, witnessing that independence is almost entirely preserved. Concerning the precision, we observe that it is mostly preserved for these 9 models. We observe a clear correlation between the number of discovered places and the precision of the resulting model. The running times of on all benchmarks in were under few seconds. In and our tool could not increase the number of places in the folded net, resulting in a significant loss of independence and precision. We tracked the reason down to (a) the additional restrictions on the SMT encoding imposed by our implementation and (b) the algorithm for transforming event structures into unfoldings (, introducing conditions). We plan to address this in future work. This also prevented us from of employing IP equivalences instead of SP for these experiments: could find IP equivalences for only 5 out of 11 cases. Nonetheless, as we said before, in 9 out of 11 the found SP equivalences preserved at least 98% of the independence. Finally, we instructed to synthesize SP equivalences folding into an arbitrarily chosen low number of places (60% of the original). Here we observe a large reduction of precision and significant loss of independence (surprisingly only ${r_{{\sys} \subseteq {\model}}}$ drops, but not ${r_{{\model} \subseteq {\sys}}}$). This witnesses a strong dependence between the number of discovered places and the ability of our technique to preserve independence. Related Work ============ To the best of our knowledge, there is no technique in the literature that solves the particular problem we are considering in this paper: given a set of positive and negative traces and an independence relation on events, derive a Petri net that both preserves the independence relation and satisfies the quality dimensions enumerated in . However, there is related work that intersects partially with the techniques of this paper. We now report on it. Perhaps the closest work is [@FahlandA13], where the simplification of an initial process model is done by first unfolding the model (to derive an overfitting model) and then folding it back in a controlled manner, thus generalizing part of the behavior. The approach can only be applied for fitting models, which hampers its applicability unless alignment techniques [@AryaThesis] are used. The folding equivalences presented in this paper do not consider a model and therefore are less restrictive than the ones presented in [@FahlandA13]. [*Synthesis*]{} is a problem different from discovery: in synthesis, the underlying system is given and therefore one can assume $\sys = \logs$. Considering a synthesis scenario, Bergenthum [*et al.*]{} have investigated the synthesis of a p/t net from partial orders [@BergenthumDLM08]. The class of nets considered in this paper (safe Petri nets) is less expressive than p/t nets, which in practice poses no problems in the context of business processes. The algorithms in [@BergenthumDLM08] are grounded in the [*theory of regions*]{} and split the problem into two steps *(i)* the p/t net $\model$ is generated which, by construction, satisfies $\logs \subseteq \obs {\model}$, and *(ii)* it is checked whereas $\logs = \obs {\model}$. Actually, by avoiding *(ii)*, a discovery scenario is obtained where the generalization feature is not controlled, in contrast to the technique of this paper. With the same goal but relying on ad-hoc operators tailored to compose lpos (choice, sequentialization, parallel compositions and repetition), a discovery technique is presented in [@BergenthumDML09]. Since the operators may in practice introduce wrong generalizations, a domain expert is consulted for the legality of every extra run. Conclusions =========== A fresh look at process discovery is presented in this paper, which establishes theoretical basis for coping with some of the challenges in the field. By automating the folding of the unfolding that covers traces in the log but also combinations thereof derived from the input independence relation, problems like log incompleteness and noise may be alleviated. The approach has been implemented and the initial results show the potential of the technique in rediscovering a model, even for the simplest of the folding equivalences described in this paper. Next steps will focus on implementing the remaining folding equivalences, and in general improving the SMT constraints for computing folding equivalences. Also, incorporating the notion of trace frequency in the approach will be considered, to guide the technique to focus on principal behavior. This will allow to also test the tool in presence of incomplete or noisy logs. [^1]: Since refers to the events that generates the local configurations of the negative traces, the folding equivalence must be defined over the nodes of $\onl$ and not those of $\onls$. [^2]: Tool and benchmarks: <http://lipn.univ-paris13.fr/~rodriguez/exp/atva15/>.

--- abstract: 'Gutzwiller’s trace formula for the semiclassical density of states in a chaotic system diverges near bifurcations of periodic orbits, where it must be replaced with uniform approximations. It is well known that, when applying these approximations, complex predecessors of orbits created in the bifurcation (“ghost orbits”) can produce pronounced signatures in the semiclassical spectra in the vicinity of the bifurcation. It is the purpose of this paper to demonstrate that these ghost orbits themselves can undergo bifurcations, resulting in complex, nongeneric bifurcation scenarios. We do so by studying an example taken from the Diamagnetic Kepler Problem, viz. the period quadrupling of the balloon orbit. By application of normal form theory we construct an analytic description of the complete bifurcation scenario, which is then used to calculate the pertinent uniform approximation. The ghost orbit bifurcation turns out to produce signatures in the semiclassical spectrum in much the same way as a bifurcation of real orbits would.' address: - ' Institut für Theoretische Physik I, Ruhr-Universität Bochum, D-44780 Bochum, Germany' - ' Institut für Theoretische Physik und Synergetik, Universität Stuttgart, D-70550 Stuttgart, Germany' author: - 'T Bartsch, J Main and G Wunner' title: Significance of ghost orbit bifurcations in semiclassical spectra --- Introduction ============ Since its discovery in the early 1970s, Gutzwiller’s trace formula [@Gut67; @Gut90] has become a widely used tool for the interpretation of quantum mechanical spectra of systems whose classical counterpart exhibits chaotic behaviour. It represents the density of states of the quantum system as a sum over a smooth part and fluctuations from all periodic orbits of the classical system, where the contribution of a single periodic orbit reads $$\label{Gutzw} {\cal A}{_{\rm po}}=\frac{T{_{\rm po}}\rme^{\rmi(S{_{\rm po}}/\hbar-\frac{\pi}{2}\mu{_{\rm po}})}} {\sqrt{|\det(M{_{\rm po}}-I)|}} \;,$$ with $T{_{\rm po}}, S{_{\rm po}}, M{_{\rm po}}, \mu{_{\rm po}}$ denoting the orbital period, action, monodromy matrix, and Maslov index, respectively. This formula assumes that all periodic orbits can be regarded as isolated, which is the case, in particular, for completely hyperbolic systems. In the generic case of mixed regular-chaotic dynamics, however, the formula fails whenever bifurcations of periodic orbits occur, because close to a bifurcation periodic orbits approach one another arbitrarily closely. The failure of the formula manifests itself in divergences of the periodic orbit contributions (\[Gutzw\]). The generic cases of period-$m$-tupling bifurcations were studied by Ozorio de Almeida and Hannay [@Alm87; @Alm88], who derived uniform semiclassical approximations by taking into account all orbits involved in a bifurcation collectively. Their solutions were refined by Sieber and Schomerus [@Sie96; @Sch97a; @Sie98], who derived uniform approximations for all types of bifurcations of codimension one in generic Hamiltonian systems with two degrees of freedom. Their formulas smooth the divergence in Gutzwiller’s trace formula, and, in contrast to the approximations in [@Alm87; @Alm88], asymptotically approach the result of the trace formula (\[Gutzw\]) for isolated periodic orbit contributions as the distance from the bifurcation increases. As a characteristic feature, uniform approximations require the inclusion of complex “ghost orbits”. At the bifurcation points, new periodic orbits are born. However, before they come into being, the orbits possess predecessors – ghost orbits – in the complexified phase space. As was shown by Kuś et al. [@Kus93], some of these ghost orbits, which in the limit $\hbar\to 0$ yield exponentially small contributions, have to be included in Gutzwiller’s trace formula (\[Gutzw\]). As a result, in constructing a uniform approximation complete information about the bifurcation scenario including the ghost orbits is required. A closer inspection of the bifurcation scenarios encountered in practical applications of uniform approximations reveals that bifurcations of codimension two, although they cannot generically be observed if only one control parameter is varied, can nevertheless have an effect on semiclassical spectra, because in their neighbourhood two bifurcations of codimension one come close to each other, and therefore have to be treated collectively. Examples of that situation have been studied by Main and Wunner [@Mai97; @Mai98] as well as Schomerus and Haake [@Sch97b; @Sch97c]. The bifurcation scenarios described in the literature so far involve bifurcations of real orbits only. However, one should expect bifurcations of ghost orbits also to be possible and of particular importance for complicated bifurcation scenarios with codimension greater than one. It is the purpose of this paper to demonstrate that ghost orbit bifurcations do indeed occur and have a pronounced effect on semiclassical spectra. To this end, we present an example taken from the Diamagnetic Kepler Problem. It turns out that even the analysis of the period-quadrupling bifurcation of one of the shortest periodic orbits in that system requires the inclusion of a ghost bifurcation. The appearance of ghost orbit bifurcations represents an additional challenge for the construction of uniform approximations. It will turn out that normal form theory allows treating both ghost bifurcations and bifurcations of real orbits on an equal footing. Consequently, ghost bifurcations contribute to uniform approximations in much the same way as real bifurcations do. From these observations we conclude that the occurence of ghost bifurcations in systems with mixed regular-chaotic dynamics is not a very exotic but rather quite a common phenomenon. The organization of the paper is as follows: In we describe the bifurcation scenario of the example chosen in detail. provides the general form of uniform approximations. presents the normal form describing the bifurcation scenario in point and the discussion of how the ghost bifurcation can be included in the normal form. In section \[ParmSec\] we determine the normal form parameters to quantitatively describe the bifurcations, and in the uniform approximation is evaluated. The bifurcation scenario {#BifSec} ======================== As an example, we study the hydrogen atom in a magnetic field, which has been described in detail, e.g., in Refs.[@Fri89; @Has89; @Wat93]. We assume the nucleus fixed and regard the electron as a structureless point charge. If the magnetic field is directed along the $z$-axis, the nonrelativistic Hamiltonian describing the electron motion reads \[in atomic units, with $\gamma=B/(2.35\times 10^5\,{\rm T})$ the magnetic field strength\] $$H = \frac{1}{2}\bi{p}^2 +\frac{1}{2}\gamma L_z +\frac{1}{8}\gamma^2\left(x^2+y^2\right)-\frac{1}{r} = E \;.$$ Here, $r$ is the distance from the nucleus, and $L_z$ denotes the angular momentum along the field axis, which is conserved because of the rotational symmetry around that axis. In the following we restrict ourselves to the case where $L_z = 0$. To further simplify the Hamiltonian, we exploit its scaling property with respect to the magnetic field strength $\gamma$. In scaled coordinates and momenta $$\label{prScal} \tilde{\bi{r}} = \gamma^{2/3}\bi{r},\qquad \tilde{\bi{p}} = \gamma^{-1/3}\bi{p}$$ the Hamiltonian assumes the form $$\label{hamScal} \tilde{H} = \gamma^{-2/3} H = \frac{1}{2}\tilde{\bi{p}}^2 + \frac{1}{8} \left(\tilde{x}^2+\tilde{y}^2\right) - \frac{1}{\tilde{r}} = \tilde{E} \;.$$ Thus, the classical dynamics does not depend on the energy $E$ and field strength $\gamma$ separately, but only on the scaled energy $\tilde{E} = \gamma^{-2/3}E$. From the scaling prescriptions (\[prScal\]) and (\[hamScal\]) we derive the scaling laws for classical actions and times as $$\tilde{S} = \gamma^{1/3}S\;,\qquad \tilde{T}=\gamma T\;.$$ Due to the Coulomb potential, the Hamiltonian (\[hamScal\]) is singular at $\tilde r=0$. The equations of motion can be regularized by introducing semiparabolical coordinates $$\label{spKoord} \mu^2 = \tilde{r}+\tilde{z}\;,\qquad \nu^2 = \tilde{r}-\tilde{z} \; ,$$ and a new time parameter $\tau$ defined by $dt = 2 r\, d\tau$. Finally, the regularized Hamiltonian is obtained as [@Fri89; @Has89; @Wat93] $${\cal H}= \frac{1}{2}\left(p_\mu^2+p_\nu^2\right) - \tilde{E}\left(\mu^2+\nu^2\right) + \frac{1}{8}\mu^2\nu^2\left(\mu^2+\nu^2\right) \equiv 2\; ,$$ and Hamilton’s equations of motion read \[with primes denoting derivatives $\rmd/\rmd\tau$\], $$\begin{aligned} \label{HamEqs}\eqalign{ \mu' = p_\mu\;, \qquad& p_\mu' = 2\tilde{E}\mu -\frac{1}{4}\mu\nu^2(2\mu^2+\nu^2)\;, \\ \nu' = p_\nu\;, & p_\nu' = 2\tilde{E}\nu -\frac{1}{4}\mu^2\nu(\mu^2+2\nu^2)\;. }\end{aligned}$$ These equations are free of singularities and can easily be integrated numerically. When using them, we must keep in mind that the definition (\[spKoord\]) determines the semiparabolical coordinates $\mu, \nu$ up to a choice of sign only, giving a many-to-one coordinate system. Thus, if we integrate the equations of motion (\[HamEqs\]) until the trajectory closes in $(\mu,\nu)$-coordinates this may correspond to more than one period in the original configuration space. Furthermore, we have to identify orbits which can be transformed into one another by reflections at the coordinate axes. We now complexify our phase space by allowing coordinates and momenta to assume complex values. This extension allows us to look for ghost predecessors of real orbits born in a bifurcation [@Mai97]. At any given scaled energy $\tilde{E}$, there is a periodic orbit along the magnetic field axis. For sufficiently low negative $\tilde{E}$ it is stable, while, as $\tilde{E}\nearrow 0$, it loses and regains stability infinitely often [@Win87; @Fri89]. Stability is lost, for the first time, at $\tilde{E} =-0.391$. At this scaled energy, the so-called balloon orbit [@Mao92] is born as a new, stable periodic orbit. As the scaled energy increases further, the orbit exhibits all kinds of period-$m$-tupling bifurcations before finally turning unstable at $\tilde{E}=-0.291$. Here, we consider the period-quadrupling bifurcation of the balloon orbit at $\tilde{E}_c=-0.342025$. For $\tilde{E}>\tilde{E}_c$, two real satellite orbits of quadruple period exist. As $\tilde{E}\searrow\tilde{E}_c$, they collide with the balloon orbit and form an island-chain bifurcation as described in [@Alm87; @Alm88]. The real orbits are shown in figure \[reSat\] at scaled energy $\tilde E=-0.340$. The solid and dashed curves represent the stable and unstable satellite orbits, respectively. For comparison, the balloon orbit is shown as a dotted curve. Below $\tilde{E}_c$, no real satellite orbits exist. Instead, there are a stable and an unstable complex ghost satellite. The real and imaginary parts of the stable and unstable ghost orbits at scaled energy $\tilde E=-0.343$ are drawn as solid and dotted curves in figure \[ghSat1\]. Note that the imaginary parts are small compared to the real parts. As predicted by normal form theory (see ), both satellites coincide with their complex conjugates, whence the total number of orbits is conserved in the bifurcation. The orbits described so far form a generic type of period-quadrupling bifurcation as investigated by Sieber and Schomerus [@Sie98]. However, the classical periodic orbit search in the complexified phase space reveals the existence of an additional ghost orbit at scaled energies around $\tilde E_c$. The shape of this orbit is shown as the dashed curve in figure \[ghSat1\]. It is similar to the stable ghost satellite originating from the period-quadrupling of the balloon orbit. When following this ghost orbit to lower energies we find another bifurcation at $\tilde E_c'=-0.343605$, i.e., slightly below the bifurcation point $\tilde E_c=-0.342025$ of the period-quadrupling of the balloon orbit. At energy $\tilde E=\tilde E_c'$ the additional ghost orbit (dashed curve in figure \[ghSat1\]) collides with the stable ghost satellite of the period-quadrupling bifurcation (solid curve in figure \[ghSat1\]), and these two orbits turn into a pair of complex conjugate ghost orbits. Their shapes are presented at scaled energy $\tilde E=-0.344$ by the solid and dashed curves in figure \[ghSat2\]. The imaginary parts clearly exhibit the loss of conjugation symmetry described, if the aforementioned symmetry of the semiparabolical coordinate system is taken into account. The dotted curve in figure \[ghSat2\] is the unstable ghost satellite which already exists at higher energy $\tilde E>\tilde E_c'$ (see the dotted curve in figure \[ghSat1\]). It is important to note that the second bifurcation at $\tilde E=\tilde E_c'$ involves ghost orbits only. This kind of bifurcation has not been described in the literature so far; in particular, Meyer’s classification of codimension-one bifurcations [@Mey70] contains bifurcations of real orbits only and does not cover ghost bifurcations. The existence of the ghost orbit bifurcation implies that the results of Ref. [@Sie98] for the uniform semiclassical approximation for the generic period-quadrupling bifurcation cannot be applied to the more complicated bifurcation scenario considered here. As in cases described before by Main and Wunner [@Mai97; @Mai98] and Schomerus and Haake [@Sch97b; @Sch97c], the closeness of the two bifurcations requires the construction of a uniform approximation taking into account all orbits involved in the successive bifurcations collectively. Thus, the ghost bifurcation at $\tilde{E}_c'$ turns out to contribute to the semiclassical approximation in the same way as a real bifurcation would, as long as we do not go to the extreme semiclassical domain where the bifurcations can be regarded as isolated. We will demonstrate in that the techniques of normal form theory can be extended as to include the description of ghost bifurcations. The construction of the uniform approximation requires the knowledge of the periodic orbit parameters of all orbits participating in the bifurcation scenario. The numerically calculated parameters are shown in as functions of the scaled energy $\tilde E$. Part (a) of this figure displays the actions of the periodic orbits, where the action of four repetitions of the balloon orbit was chosen as a reference level ($\Delta S =0$). This kind of presentation exhibits the sequence of bifurcations more clearly than a plot of the actual action integrals. Around $\tilde{E}_c$, we recognize two almost parabolic curves, indicating the actions of the stable (upper curve) and unstable (lower curve) satellites, respectively. At $\tilde{E}_c$, the curves change from solid to dashed as the satellites become complex. Below $\tilde{E}_c$, the unstable ghost satellite does not undergo any further bifurcations in the energy range shown, whereas the stable satellite collides, at $\tilde{E}_c'$, with the additional ghost orbit, which can clearly be seen not to be involved in the bifurcation at $\tilde{E}_c$. Below $\tilde{E}_c'$, these two orbits are complex conjugates of each other. Thus, the real parts of their actions coincide, whereas the imaginary parts are different from zero and have opposite signs. Analogously, b displays the orbital periods. Here, no differences were taken, so that the fourth repetition of the balloon orbit, which is always real, shows up as a nearly horizontal solid line at $\tilde T\approx 5.84$. The other orbits can be identified with the help of the bifurcations they undergo, in the same way as discussed above. Finally, c presents the traces of monodromy matrices minus two. These quantities agree with $\det(M-I)$ for systems with two degrees of freedom. At $\tilde{E}_c$ and $\tilde{E}_c'$, they can be seen to vanish for the orbits involved in the bifurcations, causing the divergences of the periodic orbit amplitudes at the bifurcation points. The generic period-quadrupling bifurcation at $\tilde{E}_c$ can be described with the help of lowest order normal form theory presented in . As the ghost bifurcation is approached, this description fails. However, the influence of the additional ghost orbit can be taken into account by including higher order terms in the normal form as discussed in . The general form of the uniform approximation {#UnifSec} ============================================= Before we return to classical normal form theory in , we introduce, in this section, the basic formulas for the quantum density of states necessary for the construction of the uniform semiclassical approximation. The density of states of a quantum system with the Hamiltonian $\hat H$ can be expressed with the help of the Green’s function $G(E)=(E-\hat H)^{-1}$ as $$d(E) = -\frac{1}{\pi}{{\rm Im}\,}\Tr \, G(E) \; ,$$ where the trace of the Green’s function can be evaluated in the coordinate representation, $$\Tr \, G(E) = \int d\bi{x'}d\bi{x}\delta(\bi{x'}-\bi{x}) G(\bi{x'}\bi{x},E) \; .$$ The basic steps in the formulation of [*periodic orbit theory*]{} [@Gut67; @Gut90] are to replace the Green’s function $G(\bi{x'}\bi{x},E)$ with its semiclassical Van Vleck-Gutzwiller approximation and to carry out the integrals in stationary-phase approximation. For systems with two degrees of freedom the semiclassical approximation to the Green’s function reads $$\fl G(\bi{x'}\bi{x}, E) = \frac{1}{\rmi\hbar\sqrt{2\pi\rmi\hbar}} \sum_{\rm class.~traj.}\sqrt{D} \exp\left\{\frac{\rmi}{\hbar}S(\bi{x'}\bi{x}, E) -\rmi\frac{\pi}{2}\nu\right\} \; .$$ Here, the sum extends over all classical trajectories running from $\bi{x}$ to $\bi{x'}$ at energy $E$, $S$ is the action of a trajectory, $\nu$ its Maslov index, and $D$ is defined by the second derivatives of the action, $$D = \det \left( \begin{array}{cc} {\frac{\partial^2S}{\partial\bi{x'}\partial\bi{x}}} & {\frac{\partial^2S}{\partial\bi{x'}\partialE}} \\[.5ex] {\frac{\partial^2S}{\partialE\partial\bi{x}}} & {\frac{\partial^2S}{\partial E^2}} \end{array}\right) \; .$$ The contribution of a single orbit to the density of states can be evaluated by introducing coordinates parallel and perpendicular to the orbit. The integration along the orbit can then be performed in a straightforward fashion. Finally, Gutzwiller’s trace formula (\[Gutzw\]) for isolated periodic orbits is obtained by integrating over the coordinates perpendicular to the trajectory using the stationary-phase approximation. It is this last step which fails close to a bifurcation, where periodic orbits are not isolated. To find an expression for the density of states which is valid close to bifurcations, it is convenient to go over to a coordinate-momentum representation of the Green’s function. Close to a period-$m$-tupling bifurcation of a real orbit the uniform approximation takes the form [@Sie96] $$\fl d(E) = \frac{1}{2\pi^2m\hbar^2}\, {{\rm Re}\,}\int dy\,dp_y'\, {\frac{\partial \hat{S}}{\partial E}} \sqrt{\left|{\frac{\partial^2\hat{S}}{\partialy\partialp_y'}}\right|} \exp\left\{\frac{\rmi}{\hbar}\left(\hat{S}+yp_y'\right) -i\frac{\pi}{2}\hat{\nu}\right\} \; , \label{d_uni}$$ with $y$, $p_y'$, and $\hat S$ defined as follows. Let $(y,p_y)$ be the canonical variables in the Poincaré surface of section perpendicular to the bifurcating orbit, and $(y',p_y')$ the corresponding variables after the period-$m$ cycle of the orbit. The function $$\hat{S}(p_y' y,E) = S(y' y,E) -y'p_y'$$ is the Legendre transform of the action integral with respect to the final coordinate or, in other words, the generating function of the Poincaré map for $m$ periods of the bifurcating orbit in a coordinate-momentum representation. The function in the exponent in (\[d\_uni\]), $$f(y,p_y',E) = \hat{S}(y,p_y',E) +y p_y' \;,$$ is stationary at the fixed points of the $m$-traversal Poincaré map, that is, fixed points of $f$ correspond to classical periodic orbits. In the spirit of catastrophe theory, we now relate $f$ to a given ansatz function $\Phi$, which has the same distribution of stationary points, by a smooth invertible change of coordinates $\psi$ as $$f(y,p_y',E) = S_0(E) +\Phi(\psi(y,p_y';E);E) \;.$$ Here, $\psi$ is assumed to keep the origin fixed, $\psi(0,0;E)=(0,0)$, and $S_0(E)$ is the action of the central bifurcating orbit. Inserting this ansatz, we obtain $$\begin{aligned} \label{unifApprox}\eqalign{ \fl d(E) = \frac{1}{2\pi^2m\hbar^2}\, {{\rm Re}\,}\exp\left\{\frac{\rmi}{\hbar}S_0(E) -\rmi\frac{\pi}{2}\hat{\nu}\right\}\\ \times \int dY\,dP_Y' {\frac{\partial \hat{S}}{\partial E}} \sqrt{\left|{\frac{\partial^2\hat{S}}{\partialy\partialp_y'}}\right|} \sqrt{\frac{|{{\rm Hess}\,}\Phi|}{|{{\rm Hess}\,}f|}} \exp\left\{\frac{\rmi}{\hbar}\Phi(Y,P_Y')\right\}\;, }\end{aligned}$$ where ${{\rm Hess}\,}$ denotes the Hessian matrix and the coefficient $$X := {\frac{\partial \hat{S}}{\partial E}} \sqrt{\left|{\frac{\partial^2\hat{S}}{\partialy\partialp_y'}}\right|} \sqrt{\frac{|{{\rm Hess}\,}\Phi|}{|{{\rm Hess}\,}f|}}$$ is unknown. At a stationary point of $\Phi$, it can be shown to assume the value $$X \stackrel{sp}{=} \frac{\{m\}T}{\sqrt{|\Tr M-2|}}\sqrt{|{{\rm Hess}\,}\Phi|}\;,$$ where $T$ and $M$ denote the period and the monodromy matrix of the corresponding classical orbit, and the notation $\{m\}$ is meant to indicate that this factor does not occur at the satellite orbits. Normal-form description of the bifurcation {#NFSec} ========================================== To evaluate the uniform approximation , we need to find a suitable ansatz function $\Phi$. Normal form theory as developed by Birkhoff [@Bir27] and Gustavson [@Gus66] provides us with a systematic way to construct such ansatz functions. We will first investigate the lowest nontrivial order of the normal form expansion which describes generic bifurcations. Then, we will show that higher-order terms in the expansion can account for the more complicated bifurcation scenario studied here. The generic period-quadrupling bifurcation {#generic_bif} ------------------------------------------ The simplest normal form describing the generic period-quadrupling bifurcations reads [@Alm87] $$\label{NF2} \Phi = \varepsilon I + a I^2 + bI^2\cos(4\varphi) \;.$$ This normal form is expressed in terms of canonical (action-angle) polar coordinates $(I,\varphi)$, which are connected to Cartesian coordinates $(p,q)$ by $$\label{polKoord} p = \sqrt{2I}\cos\varphi\;,\qquad q=\sqrt{2I}\sin\varphi\;.$$ To establish the connection with classical periodic orbits, we need to determine the stationary points of the normal form and then evaluate its stationary values. The stationary points are given by the equations $$\begin{aligned} \eqalign{ 0 &{\stackrel{\textstyle !}{=}}{\frac{\partial \Phi}{\partial \varphi}} = -4I^2b\sin(4\varphi) \;,\\ 0 &{\stackrel{\textstyle !}{=}}{\frac{\partial \Phi}{\partial I}} = \varepsilon +2aI+2bI\cos(4\varphi) \;. }\end{aligned}$$ The first of these equations yields $$\begin{aligned} \eqalign{ \sin(4\varphi) = 0 \;, \\ \cos(4\varphi) = \sigma = \pm 1\;, } \label{stat_phi}\end{aligned}$$ so that the second equation reads $$\varepsilon + 2(a+\sigma b)I = 0 \; ,$$ with its solution $$I_{\sigma} = -\frac{\varepsilon}{2(a+\sigma b)} \; .$$ In addition, there is the central periodic orbit at $I=0$, which does not show up as a stationary point because the polar coordinate system (\[polKoord\]) is singular at the origin. To interprete these results, we observe that according to its definition (\[polKoord\]) the coordinate $I$ is positive for real orbits and that the action difference $\Phi(I,\varphi)$ is real for real $I, \varphi$. Therefore, negative real solutions $I$ correspond to ghost orbits which are symmetric with respect to complex conjugation and thus have real actions, whereas a complex $I$ indicates an asymmetric ghost orbit. Now, if $|a|>|b|$, the two solutions $I_{\pm}$ have the same sign, which changes at $\varepsilon=0$. So, the two satellite orbits change from two real orbits to two ghosts at the bifurcation point $\varepsilon=0$, forming an island-chain-bifurcation. If $|a|<|b|$, $I_+$ and $I_-$ have different signs, so that on either side of the bifurcation there is one real and one ghost satellite. At $\varepsilon=0$, the satellites collide with the central orbit, forming a touch-and-go-bifurcation. Generalization to nongeneric bifurcations {#ungeneric_bif} ----------------------------------------- To describe a sequence of two bifurcations, we need to include higher-order terms in the normal form. Here, we adopt the normal form $$\label{NF} \Phi = \varepsilon I + a I^2 + bI^2\cos(4\varphi) + cI^3(1+\cos(4\varphi))$$ given by Schomerus [@Sch98] as a variant of the normal form used by Sadovskií and Delos [@Sad96] to describe a sequence of bifurcations close to a period-quadrupling. It turns out to qualitatively describe the sequence of bifurcations encountered here for suitably chosen parameter values. The stationary points of $\Phi(I,\varphi)$ are given by the equations $$\begin{aligned} \eqalign{ 0 &{\stackrel{\textstyle !}{=}}{\frac{\partial \Phi}{\partial \varphi}} = -4I^2(b+cI)\sin(4\varphi) \;,\\ 0 &{\stackrel{\textstyle !}{=}}{\frac{\partial \Phi}{\partial I}} = \varepsilon +2aI+2bI\cos(4\varphi) +3cI^2(1+\cos(4\varphi)) \;. }\end{aligned}$$ As in , the first of these equations yields (see eq. \[stat\_phi\]) $$\begin{aligned} \sin(4\varphi) = 0 \;, \\ \cos(4\varphi) = \sigma = \pm 1 \; .\end{aligned}$$ Thus, the second equation reads $$\varepsilon + 2(a+\sigma b)I + 3cI^2(1+\sigma) = 0\;.$$ In solving this equation, we shall assume $c<0$. For $\sigma=-1$, the equation is linear. Its only solution reads $$I_{-1} = -\frac{\varepsilon}{2(a-b)}\;.$$ For $\sigma=+1$, however, we obtain a quadratic equation with two solutions: $$I_\pm = -c^{-1/3}\left(\delta\pm\sqrt{\eta+\delta^2}\right) \;,$$ where we have introduced the abbreviations $$\label{etaDeltaDef} \eta \equiv -\frac{\varepsilon}{6c^{1/3}}, \quad \delta \equiv \frac{a+b}{6c^{2/3}} \; .$$ Thus, we obtain three different stationary points corresponding to the three satellite orbits in addition to the central periodic orbit at $I=0$. The dependence of the solutions $I$ on $\varepsilon$ is shown schematically in figure \[IEtaFig\] for the case $|a|>|b|$ and $c<0$. Comparing the normal form results to the bifurcation scenario described in section \[BifSec\], we recognize the sequence of an island-chain-bifurcation at $\varepsilon=0$ and a ghost orbit bifurcation at some negative value of $\varepsilon$. Thus, our normal form correctly describes the bifurcation scenario under consideration, and we adopt it as an ansatz function in the uniform approximation (\[unifApprox\]). This correspondence allows us to identify stationary points with classical periodic orbits as follows: The $(\sigma=-1)$-solution describes the unstable satellite orbits on either side of $\tilde{E}_c$. For $\tilde{E}<\tilde{E}_c'$, the $(\sigma=+1)$-solutions correspond to the asymmetric ghost orbits, whereas for $\tilde{E}>\tilde{E}_c'$, we identify the two solutions for $\sigma=+1$ with the stable satellite orbit (marked by the $+$ in figure \[IEtaFig\]) and with the additional ghost ($-$ in figure \[IEtaFig\]). We now calculate the stationary values of the normal form, which correspond to action differences. They read $$\begin{aligned} \label{NFWirk}\eqalign{ \Phi_\pm = 4\left(\eta+\delta^2\right) \left(\delta\pm\sqrt{\eta+\delta^2}\right)+2\eta\delta \; , \\ \Phi_{-1} = -\frac{\varepsilon^2}{4(a-b)} \; . }\end{aligned}$$ Finally, we need the Hessian determinants of the normal form at the stationary points. We calculate them with respect to cartesian coordinates, because in polar coordinates the Hessian determinant at the central orbit $I=0$ is undefined, and obtain $$\begin{aligned} \label{NFHess}\eqalign{ {{\rm Hess}\,}_\pm = \left\{\varepsilon+2(a-3b)I_\pm-2cI_\pm^2\right\} \left\{\varepsilon+6(a-b)I_\pm+30cI_\pm^2\right\} \;, \\ {{\rm Hess}\,}_{-1}= \left\{\varepsilon+4aI_{-1}+4cI_{-1}^2\right\}^2 -4I_{-1}^2\left\{a-3b-2cI_{-1}\right\}^2 \;, \\ {{\rm Hess}\,}_0 = \varepsilon^2 \; . }\end{aligned}$$ We have now found a normal form which can serve as an analytical description of the complicated bifurcation scenario. Determination of normal form parameters {#ParmSec} ======================================= In order to use the normal form (\[NF\]) as an ansatz in the uniform approximation (\[unifApprox\]), we now have to determine the normal form parameters $\varepsilon, a, b,c$ so that the numerically observed action differences are quantitatively reproduced by the normal form results (\[NFWirk\]). Since, according to figure \[IEtaFig\], $\varepsilon$ measures the distance from the period quadrupling bifurcation, we choose $$\varepsilon = \tilde{E} - \tilde{E}_c \; ,$$ and then solve equations (\[NFWirk\]) for the parameters $a,b,c$. To achieve this, we introduce $$\begin{aligned} \label{hpmDef}\eqalign{ h_+ &=\frac{\Phi_++\Phi_-}{8} =\delta\left(\eta+\delta^2\right)+\frac{1}{2}\eta\delta \;, \\ h_- &=\frac{\Phi_+-\Phi_-}{8} =(\eta+\delta^2)^{3/2} \;. }\end{aligned}$$ The second equation gives $$\label{etaVonDelta} \eta = h_-^{2/3}-\delta^2 \;,$$ so that from the first equation we obtain $$\label{deltaGlg} \delta^3-3h_-^{2/3}\delta+2h_+ = 0 \;.$$ This is a cubic equation for $\delta$. Its discriminant reads $$D = h_+^2-h_-^2 = \frac{1}{16}\Phi_+\Phi_- \;,$$ and, using Cardani’s formula, we find its solutions $$\label{deltaLsg} \delta = \frac{\lambda}{2} {\sqrt[3]{-\left(\sqrt{\Phi_+}+\sqrt{\Phi_-}\right)^2}} +\frac{\lambda^\ast}{2} {\sqrt[3]{-\left(\sqrt{\Phi_+}-\sqrt{\Phi_-}\right)^2}} \;,$$ where $\lambda\in\left\{1,-\frac{1}{2}\pm \rmi\frac{\sqrt{3}}{2}\right\}$ is a cube root of unity. If $D>0$, $\lambda = 1$ yields the only real solution, whereas for $D<0$ all solutions are real. To proceed, we have to choose one of the three solutions. As can be seen from figure \[orbDat\] using the correspondence between stationary points and periodic orbits discussed above, we have $\Phi_+>0$, and there is an $\varepsilon_0<0$ such that $\Phi_->0$ for $\varepsilon>\varepsilon_0$ and $\Phi_-<0$ for $\varepsilon<\varepsilon_0$. Consequently, $D>0$ for $\varepsilon>\varepsilon_0$ and $D<0$ for $\varepsilon<\varepsilon_0$. To make $\delta$ real, we therefore have to choose $\lambda = 1$ if $\varepsilon<\varepsilon_0$. To determine $\lambda$ for $\varepsilon>\varepsilon_0$, we first observe that the parameters must depend on $\varepsilon$ continuously. Thus, $\lambda$ can only change at energies where equation (\[deltaGlg\]) has a double root, viz. $D=0$ or $\varepsilon \in \{0, \varepsilon_0\}$. Therefore, it suffices to determine $\lambda$ in a neighbourhood of $\varepsilon = 0$. From the plot of action differences in figure \[orbDat\] we find $$\begin{aligned} \Phi_+=\alpha^2\varepsilon^2+\Or\left(\varepsilon^3\right) \; , \\ \Phi_-=-\Gamma-\beta\varepsilon+\Or\left(\varepsilon^3\right) \; ,\end{aligned}$$ with positive constants $\alpha, \beta, \Gamma$. With the help of equations (\[deltaLsg\]) and (\[etaVonDelta\]) we can now expand $\eta$ in a Taylor series to first order in $\varepsilon$: $$\begin{aligned} \fl \eta =\left(\frac{1}{4}-({{\rm Re}\,}\lambda)^2\right)\Gamma^{2/3} \nonumber \\ +\left(\left(\frac{1}{4}-({{\rm Re}\,}\lambda)^2\right) \frac{2\beta}{3\Gamma^{2/3}} -{{\rm Re}\,}\lambda\,{{\rm Im}\,}\lambda\frac{4\alpha\sqrt{\Gamma}}{3\Gamma^{1/3}} {{\rm sign}\,}\varepsilon\right)\varepsilon +\Or\left(\varepsilon^2\right) \; .\end{aligned}$$ Requiring this result to reproduce the definition $$\eta = -\frac{1}{6c^{1/3}}\varepsilon\;,\qquad-\frac{1}{6c^{1/3}}>0\;,$$ we find the conditions $${{\rm Re}\,}\lambda=-\frac{1}{2}\;,\qquad {{\rm Im}\,}\lambda\frac{2\alpha\sqrt{\Gamma}}{3\Gamma^{1/3}} {{\rm sign}\,}\varepsilon>0\; ,$$ which lead to the correct choices of $\lambda$: $$\lambda = \cases{1 & : $\Phi_->0$\\ -\frac{1}{2}+\rmi\frac{\sqrt{3}}{2}{{\rm sign}\,}\varepsilon & : $\Phi_-<0$}\;.$$ Using this result, we can determine $\eta$ and $\delta$ from (\[etaVonDelta\]) and (\[deltaLsg\]). Equations (\[etaDeltaDef\]) and (\[NFWirk\]) then finally yield the desired parameter values $$\begin{aligned} \label{abc_param} \eqalign{ c=-\left(\frac{\varepsilon}{6\eta}\right)^3 \;,\\ a=3c^{2/3}\delta - \frac{\varepsilon}{8\Phi_{-1}}\;, \qquad b=3c^{2/3}\delta + \frac{\varepsilon}{8\Phi_{-1}} \; . }\end{aligned}$$ Note that from the parameters $a$, $b$, and $c$ are explicitly determined as functions of the energy $\varepsilon$ and the three action differences $\Phi_+$, $\Phi_-$, and $\Phi_{-1}$. In our case, we will determine the normal form coefficients from the scaled action differences shown in figure \[orbDat\]. To obtain the actual non-scaled coefficients for different values of the magnetic field strength $\gamma$, we need to derive scaling laws for the coefficients. As we shall display the semiclassical spectra as functions of scaled energy, we prefer not to scale the energy difference $\varepsilon=\tilde{E}-\tilde{E}_c$. Then, with the help of equations (\[NFWirk\]) and (\[NFHess\]), we can convince ourselves that the scaling prescriptions $$\tilde{a}=\gamma^{-1/3}a\; , \qquad \tilde{b}=\gamma^{-1/3}b\; , \qquad \tilde{c}=\gamma^{-2/3}c$$ fulfil the requirements of scaling actions according to $\tilde{S}=\gamma^{1/3}S$ while not scaling Hessian determinants. Evaluation of the uniform approximation {#EvalSec} ======================================= Now that the ansatz function $\Phi$ has been completely specified, a suitable approximation to the coefficient $X$ in (\[unifApprox\]) remains to be found. We assume $X$ to be independent of $\varphi$, and as the value of $X$ is known at the stationary points of $f$ at four different values of $I$ (including $I=0$), we approximate $X$ by the third order polynomial $p(I)$ interpolating between the four given points. This choice ensures that our approximation reproduces Gutzwiller’s isolated-orbits formula if, sufficiently far away from the bifurcations, we evaluate the integral in stationary-phase-approximation. Thus, the uniform approximation takes its final form $$\label{UnifSol} \fl d(E) = \frac{1}{2\pi^2m\hbar^2}\,{{\rm Re}\,}\exp\left\{\frac{\rmi}{\hbar}S_0(E) -\rmi\frac{\pi}{2}\hat{\nu}\right\} \int dY\,dP_Y'\,p(I) \exp\left\{\frac{\rmi}{\hbar}\Phi(Y,P_Y')\right\}\;,$$ which contains known functions only and can be evaluated numerically. It is important to point out, at this juncture, the progress encorporated in equation (\[UnifSol\]) as compared to previous literature results. Starting from the lowest-order normal form (\[NF2\]), Sieber and Schomerus [@Sie98] derived a similar formula for the contribution of an isolated generic period-quadrupling bifurcation to the density of states. However, as we have seen in section \[generic\_bif\], their normal form (\[NF2\]) describes the central periodic orbit and the stable and unstable satellites only, from which it is evident that the uniform approximation given by Sieber and Schomerus cannot take the presence of the additional ghost orbit and the occurence of the ghost orbit bifurcation into account. The effect of the latter is negligible far above the bifurcation energy $\tilde E_c$, where the asymptotic behaviour is determined by the three real orbits common to both (Sieber and Schomerus’s and our own) forms of the uniform approximation. The Sieber and Schomerus result, however, is not capable of describing the correct asymptotic behaviour below $\tilde E_c$, because one of the satellite orbits runs into a bifurcation unforeseen by the normal form (\[NF2\]) at $\tilde E_c'$, thus causing Sieber and Schomerus’s uniform approximation to diverge. Below $\tilde E_c'$, when the stable satellite orbit does not exist any more, their solution also ceases to exist because the required input data is no longer available. Thus, to obtain a smooth interpolation between the asymptotic Gutzwiller behaviour on either side of the bifurcations, we must make use of the extended uniform approximation (\[UnifSol\]), which takes the contribution of the ghost orbit bifurcation into account, as long as we do not go to the extreme semiclassical domain where Planck’s constant has become so small that below $\tilde E_c$ the asymptotic regime is reached before the ghost orbit bifurcation can produce a palpable effect. It is only in this limit that the impact of the ghost orbit bifurcation on the semiclassical spectrum vanishes. We calculated the uniform approximation (\[UnifSol\]) for three different values of the magnetic field strength $\gamma$. The results are shown in figure \[UnifFig\]. To suppress the highly oscillatory contributions originating from the factor $\exp\left\{\frac{\rmi}{\hbar}S_0(E)\right\}$, we plot the absolute value of (\[UnifSol\]) instead of the real part. As can be seen, the uniform approximation proposed is finite at the bifurcation energies, and, as the distance from the bifurcations increases, asymptotically goes over into the results of Gutzwiller’s trace formula. Even the complicated oscillatory structures in the density of states caused by interferences between the contributions from the different real orbits involved at $\tilde{E}>\tilde{E}_c=-0.342025$ are perfectly reproduced by our uniform approximation (see figures \[UnifFig\]b and \[UnifFig\]c). We also see that the higher the magnetic field strength, the farther away from the bifurcation the asymptotic (Gutzwiller) behaviour is acquired. In fact, for the largest field strength in figure \[UnifFig\]a ($\gamma = 10^{-10}$) the asymptotic region is not reached at all in the energy domain shown. The magnetic field dependence of the transition into the asymptotic regime can be traced back to the fact that, due to the scaling properties of our system, the scaling parameter $\gamma^{1/3}$ plays the rôle of an effective Planck’s constant, therefore the lower $\gamma$ becomes, the more accurate the semiclassical approximation will be. Conclusion ========== We have shown that in Hamiltonian systems with mixed regular-chaotic dynamics [*ghost orbit bifurcations*]{} can occur besides the bifurcations of real orbits. These are of special importance when they appear in the vicinity of bifurcations of real orbits, since they turn out to produce signatures in the semiclassical spectra much the same as those of the real orbits. Consequently, the traditional theory of uniform approximations for bifurcations of real orbits must be extended to also include the effects of bifurcating ghost orbits. We have illustrated the phenomenon of bifurcating ghost orbits in the neighbourhood of bifurcations of real orbits by way of example for the period-quadrupling of the balloon orbit in the diamagnetic Kepler problem, and have demonstrated how normal form theory can be extended for this case so as to allow the unified description of both real [*and*]{} complex bifurcations. We picked the example mainly for its simplicity, since (a) the real orbit considered is one of the shortest fundamental periodic orbits in the diamagnetic Kepler problem and (b) the period-quadrupling is the lowest period-$m$-tupling possible ($m=4$) that exhibits the island-chain-bifurcation typical of all higher $m$. Thus we expect ghost orbit bifurcations to appear also for longer-period orbits, and, in particular, in the vicinity of all higher period-$m$-tupling bifurcations of real orbits. In fact, a general discussion of the bifurcation scenarios described by the normal form (\[NF\]) and its more general variant given in [@Sad96] for different values of the parameters leads us to the conclusion that the appearance of ghost bifurcations in the vicinity of bifurcating real orbits is the rule, rather than the exception, in general systems with mixed regular-chaotic systems, and thus one of their generic features. It will be interesting and rewarding to study higher period-$m$-tuplings with respect to the appearance of ghost orbit bifurcations, and to extend ordinary normal form theory to also include the contributions of ghost orbit bifurcations for all higher $m$. Acknowledgments {#acknowledgments .unnumbered} =============== This work was supported in part by the Sonderforschungsbereich No. 237 of the Deutsche Forschungsgemeinschaft. J.M. is grateful to Deutsche Forschungsgemeinschaft for a Habilitandenstipendium (Grant No. Ma 1639/3). References {#references .unnumbered} ========== [88]{} Gutzwiller M C 1967 [*J. Math. Phys.*]{} [**8**]{} 1979 and 1971 [*J. Math. Phys.*]{} [**12**]{} 343 Gutzwiller M 1990 [*Chaos in Classical and Quantum Mechanics*]{} (New York: Springer) Ozorio de Almeida A M and Hannay J H 1987 [*J. Phys. A*]{} [**20**]{} 5873 Ozorio de Almeida A M 1988 [*Hamiltonian Systems: Chaos and Quantization*]{} (Cambridge University Press) Sieber M 1996 [*J. Phys. A*]{} [**29**]{} 4715 Schomerus H and Sieber M 1997 [*J. Phys. A*]{} [**30**]{} 4537 Sieber M and Schomerus H 1998 [*J. Phys. A*]{} [**31**]{} 165 Kuś M, Haake F, and Delande D 1993 [*Phys. Rev. Lett.*]{} [**71**]{} 2167 Main J and Wunner G 1997 [*Phys. Rev. A*]{} [**55**]{} 1743 Main J and Wunner G 1998 [*Phys. Rev. E*]{} [**57**]{} 7325 Schomerus H 1997 [*Europhys. Lett.*]{} [**38**]{} 423 Schomerus H and Haake F 1997 [*Phys. Rev. Lett.*]{} [**79**]{} 1022 Friedrich H and Wintgen D 1989 [*Phys. Rep.*]{} [**183**]{} 37 Hasegawa H, Robnik M, and Wunner G 1989 [*Prog. Theor. Phys. Suppl.*]{} [**98**]{} 198 Watanabe S 1993 in [*Review of Fundamental Processes and Applications of Atoms and Ions*]{}, ed. C. D. Lin (Singapore: World Scientific) Wintgen D 1987 [*J. Phys. B*]{} [**20**]{} L511 Mao J M and Delos J B 1992 [*Phys. Rev. A*]{} [**45**]{} 1746 Meyer K R 1970 [*Trans. AMS*]{} [**149**]{} 95 Birkhoff G D 1927, 1966 [*Dynamical Systems*]{}\ (Providence, RI: AMS Colloquium Publications vol. IX) Gustavson F 1966 [*Astron. J.*]{} [**71**]{} 670 Schomerus H 1998 [*J. Phys. A*]{} [**31**]{} 4167 Sadovskií D A and Delos J B 1996 [*Phys. Rev. E*]{} [**54**]{} 2033

--- abstract: 'We show that a configuration of four birefringent crystals and wave-plates can emulate almost any arbitrary unital channel for polarization qubits encoded in single photons, where the channel settings are controlled by the wave-plate angles. The scheme is applied to a single spatial mode and its operation is independent of the wavelength and the fine temporal properties of the input light. We implemented the scheme and demonstrated its operation by applying a dephasing environment to classical and quantum single-photon states with different temporal properties. The applied process was characterized by a quantum process tomography procedure, and a high fidelity to the theory was observed.' author: - 'A. Shaham' - 'T. Karni' - 'H. S. Eisenberg' title: Implementation of controllable universal unital optical channels --- Introduction ============ The wave-like nature of a quantum system is manifested by the existence of a well defined phase between its components. When the system is isolated, this phase is well defined. However, when the system interacts with its surrounding, the uncertainty of the phase may increase in a process called ’dephasing’ [@Nielsen]. When quantum information is encoded in the phase, dephasing processes are interpreted as an addition of noise to the stored information. As a result, quantum information protocols that rely on the certainty of the phase may slow down, and their success may be hindered (see for example [@Cirac_1999]). Dephasing processes belong to the family of unital channels, which is a special class of decohering processes. A general decohering processes reduces the information that is stored in a quantum state, where a unital channel also preserves the average of the input state. Thus, unital channels are connected with interactions that do not include an energy dissipation from the state to its surroundings. Dephasing channels are perhaps the most common type of the unital channel class since they occur naturally in atomic and solid state quantum systems (where they are commonly characterized using the $T_2$ decoherence time scale). In general, photonic implementations of quantum systems are more immune to decohering processes, including dephasing, since they only weakly interact with the environment. Thus, they serve as an appealing candidate in a variety of quantum information schemes [@Obrien_2009]. The noise robustness of single photons also makes them suitable for studying decohering processes: transmitting photons through a channel that successfully induces a certain type of decohering process will not be accompanied by other types of decoherence to the photons. Hence, photons are suitable for the demonstration of the effects of different types of decohering processes on quantum systems and protocols. In this work, we study a realization of unital channels that are applied to quantum bits (qubits) encoded in the polarization of single photons. Previously, different types of unital channels were implemented in several ways. Some works investigated the implementation of a general unital channel while other works implemented a specific channel type of the unital class, which mainly was a dephasing channel: general unital channels which were applied on single photons where implemented using scattering from different elements [@Puentes_2005]. Scrambling schemes that used Pockels cells [@Ricci_2004], liquid crystals [@Chiuri_2011], or mechanical stress on optical fibers [@Karpinski_2008] were also used to construct different types of unital channels. Concentrating in dephasing schemes, a channel that completely erases the polarization phase of the photons was implemented using a single birefringent crystal [@Kwiat_Dephasing]. Control over the dephasing level of such a channel was achieved by changing the birefringent crystal length [@Xu_2009; @Liu_2011]. Dephasing processes could also be emulated using scrambling schemes that use wave-plates [@Amselem_2009b] or liquid crystals [@Adamson_2007a]. Another method to implement a controllable dephasing channel is to couple between the polarization and two different spatial modes via a Sagnac interferometer [@Almeida_ESD] or a polarizing tunable beam displacer [@Urrego_2017]. Here, we study a controllable scheme that is composed of four birefringent crystals and wave-plates [@Shaham_iso_depo]. Previously, this scheme was only used to implement an isotropic depolarizing channel. Using a numeric search we now show that the four crystal scheme can emulate almost every arbitrary unital channel. We demonstrate the scheme operation by applying it as a dephasing channel on classical and quantum single-photon wave-packets. The measured processes are characterized using a quantum process tomography (QPT) procedure [@Chuang_1997] and present a high fidelity to dephasing processes. Unlike dephasing schemes that are composed of one or more birefringent crystals in a fixed orientation, the four crystal configuration has the advantage that its dephasing magnitude is known in advance and is not affected by the details of the temporal structure, such as the coherence time of the initial photonic wave-packet. This property was verified by applying the dephasing scheme to two photonic states that differ in their temporal properties. As expected, the two photonic states experienced the same dephasing magnitude and agreed well with the theoretical prediction. The structure of this article is as follows: in Sec. 2 we give a theoretical background on dephasing channels and their representations. The experimental setup for the generation and detection of single-photon states is presented in Sec. 3. Section 4 is dedicated to a theoretical study of the four birefringent crystal scheme, and for the demonstration of its operation as a dephasing channel for two photonic qubit wave-packets that differ in their coherence time. We summarize the results in Sec. 5. A complementary study of the temporal differences between the two photonic wave-packets that was performed using a Soleil-Babinet Compensator (SBC) is presented in the appendix. Theoretical background ====================== The state of a polarization qubit can be described either by the density matrix $\hat{\rho}$, or by a point in the Poincaré sphere representation. The Cartesian coordinates of this point are the Stokes parameters $\{S_1, S_2, S_3\}$, where $S_0\equiv1$. $S_1$ represents the linear horizontal and vertical polarizations $|h\rangle$ and $|v\rangle$, $S_2$ represents the linear diagonal polarizations $|p\rangle=(|h\rangle+|v\rangle)/\sqrt{2}$, and $|m\rangle=(-|h\rangle+|v\rangle)/\sqrt{2}$, and $S_3$ corresponds to the circular polarizations $|r\rangle=(|h\rangle+i|v\rangle)/\sqrt{2}$ and $|l\rangle=(|h\rangle-i|v\rangle)/\sqrt{2}$). Points inside the Poincaré sphere represent partially polarized states. Consider an arbitrary quantum channel $\mathcal{E}$ that acts on a single-qubit state $\hat{\rho}$. $\mathcal{E}$ is complete positive and linear. It can be represented as the mapping of the surface of the Poincaré sphere onto an ellipsoid which is contained within the sphere. The operation of $\mathcal{E}$ can also be described using the elements of the process matrix $\chi$ $$\label{process_definition} \mathcal{E}(\hat{\rho})=\sum_{m,n}\chi_{mn}\sigma_{m}\hat{\rho}\sigma_{n}^\dag,$$ which is presented here in the basis of the identity matrix $\sigma_0$, and the Pauli matrices $\{\sigma_1,\sigma_2,\sigma_3\}$. $\chi$ is positive, Hermitian, and satisfies $\textrm{Tr}(\chi)=1$ (i.e., the channel is lossless). Channels that obey $\mathcal{E}(\hat{I})=\hat{I}$ (mapping the maximally depolarized state at the origin) belong to the unital channel class. In the Poincaré sphere picture, unital channels are the case where the mapped ellipsoid and the sphere are concentric. Dephasing channels are a special case where the process can be written as $$\label{dephasing_channel} \mathcal{E}(\hat{\rho})=(1-P)\hat{\rho}+P\sigma_3\hat{\rho}\sigma_{3}.$$ They belong to the class of unital channels, and are characterized by the probability $P$ to obtain a change in the phase of the original state. A process matrix $\chi$ can describe a dephasing process if it has two eigenvalues that equal zero. Denote the highest eigenvalue of this $\chi$ by $\chi_0$, the probability $P$ of the corresponding dephasing process is: $$\label{dephasing_probability} P=1-\chi_0.$$ In order to implement a controllable dephasing channel that can vary from no dephasing to a complete dephasing it is sufficient to show that value of $P$ can have any value in the range of $\{0,0.5\}$. This is because dephasing channels with higher $P$ values can be represented as a combination of dephasing channels with $P$ that lies in the above mentioned range, and another deterministic bit-flip channel with a probability of $100\%$ that can easily be built. It is clear from Eqs. (\[dephasing\_channel\]) and (\[dephasing\_probability\]) that in the $\chi$ matrix representation it is sufficient to show that a fully controllable dephasing channel is obtained if the highest eigenvalue $\chi_0$ can have any value in the range of $\{0.5,1\}$, and the two lowest eigenvalues of $\chi$ remain zero for any $\chi_0$ settings. A useful representation of unital single-qubit quantum processes is the tetrahedron representation: The Choi-Jamio[ł]{}kowski isomorphism between complete-positive linear maps and quantum states connects between the process matrix of a single-qubit channel and the density operator of a two-qubit state [@Jamiolkowski_1972]. Thus, we can represent the 4-dimensional (4D) $\chi$ matrix of a single-qubit unital channel via the representation of the corresponding class of two-qubit states [@Horodecki_1996a] $$\label{Chi_decomposition} \chi=\frac{1}{4}\left(I\otimes{I}+\underset{m,n=1}{\overset{3}{\sum}}D_{mn}\sigma_{m}\otimes\sigma_{n}\right).$$ Here, $D$ is a $3\times 3$ real matrix, that epitomizes all the channel parameters. Ignoring rotations, the process of the channel can be geometrically represented by the coordinate vector $\vec{D}=\{D_1,D_2,D_3\}$, which is composed of the eigenvalues of $D$. The complete positivity of the process is equivalent to the requirements that $$\label{Legal_processes} |D_i\pm{D_j}|\leq|1\pm{D_k}|,$$ where $i\neq{j}\neq{k}$ [@King_2001]. Equation (\[Legal\_processes\]) dictates that all allowed $\vec{D}$ vectors, when drawn in a cartesian 3D coordinate systems, span the volume of a tetrahedron. The values of $\{D_1,D_2,D_3\}$ are related to the eigenvalues of the corresponding $\chi$ ($\chi_0,\chi_1,\chi_2,\chi_3$) by $$\label{Radii chi eigenvalues} D_i=\chi_0+\chi_i-\chi_j-\chi_k\,,$$ where $i\neq{j}\neq{k}\neq0$ [@King_2001]. It can be shown that the eigenvalues of $D$ are equal to the lengths of the three primary radii of the mapped ellipsoid in the Poincaré sphere representation of the process. In the tetrahedron representation, the identity process ($\mathcal{E}(\hat{\rho})=\hat{\rho}$)) is represented by the $\{1,1,1\}$ vertex of the tetrahedron, and any arbitrary dephasing channel with a corresponding probability $P$ is represented by a point that lies on the tetrahedron edge of $\vec{D}=\{1-2P,1-2P,1\}$. Experimental setup ==================  The experimental setup for the characterization of single-qubit channels. Photon pairs are generated in the BBO crystal, which is located after a lens (L1). The down-converted signal is filtered by passing through a dichroic mirror (DM), a single-mode fiber (SM) and an interference bandpass filter (IF). The photon pairs are split into two ports using a BS, where single-photon detectors (DET$_1$, DET$_2$) are located at the end of each port. The investigated channel ($\mathcal{E}$) is applied only on photons that emerge from one of the ports. The initial polarization state of these photons is determined using a polarizer (POL), a HWP ($\lambda/2$) and a QWP ($\lambda/4$), and their final polarization state is measured using a HWP, a QWP and a polarizer that are placed after the channel. (b) The four-crystal scheme, composed of four perpendicularly-oriented calcite crystals and three HWPs. The thickness of the two outer (inner) crystals is 1mm (2mm). (c) A Soleil-Babinet compensator, composed of two translatable quartz wedges and another rectangular fixed quartz crystal (see details in the appendix).](QPT_experimental_setup_scheme3.eps){width="100mm"} The experimental setup which was used to characterize the implementation of the dephasing channel is presented in Fig. \[Fig\_QPT\_Experimental\_setup\_scheme\](a). A pulsed 390nm pump laser is focused onto a 1mm thick type-I $\beta-\textrm{BaB}_{2}\textrm{O}_{4}$ (BBO) crystal. Photon pairs are collinearly generated via the process of spontaneous parametric down conversion. After the BBO crystal, the down converted signal is separated from the 390nm pump beam by passing through a dichroic mirror. Spatial and temporal filtering of the 780nm photon pairs is achieved by sending the photon pairs through a single-mode fiber, and a 5nm interference bandpass filter, respectively. Then, the two photons are split probabilistically using a beam splitter (BS), where one photon is sent directly to a detector, and the second photon is directed to the investigated single-qubit channel. The initial polarization state of the photon that is sent to the channel is set using a polarizer, a half- and a quarter-wave plates (HWP and QWP). After the channel, the output polarization state is measured using a standard quantum state tomography (QST) procedure [@James_2001]: the photon passes a QWP, a HWP and a polarizer before being coupled to the detector. We performed a complete QPT for every channel setting by characterizing the channel effect on $|h\rangle$, $|v\rangle$, $|p\rangle$, and $|r\rangle$ polarization states [@Chuang_1997]. When only the counts of photons that pass through the dephasing channel (counts from $\textrm{DET}_1$) are considered, the channel effect on a classical single-photon state (i.e., a weak signal photonic state that has the thermal statistics of the down conversion process) is measured. When the detection of the photon that experiences the dephasing noise is also conditioned in the detection of another photon that does not pass through the channel (i.e., coincidence counts from both $\textrm{DET}_1$ and $\textrm{DET}_2$ are counted), the channel effect on quantum single-photon states that have different temporal coherence properties is investigated. The classical and the quantum single-photon wave-packet states differ in their temporal properties, as the quantum wave-packet has a longer coherence time than classical wave-packet. In order to verify this difference, we studied the effect of a dephasing scheme which is composed of a SBC (see Fig. \[Fig\_QPT\_Experimental\_setup\_scheme\](c)) on the classical and the quantum wave-packets [@Branning_2000]. The dephasing results which clearly demonstrate the difference between the two wave-packet types are presented in the appendix. The typical count rate of single counts (photon hits in one detector disregarding the other one) was $\sim20,000$Hz, and the rate of the coincidence counts was $\sim1000$Hz. Single counts where considered and analyzed after the subtraction of a stray light signal noise (background counts were estimated to be in the order of $2000$Hz). As for the coincidence counts, no subtraction was required since the stray light signal noise was negligible. Errors were calculated using a maximal likelihood procedure and Monte Carlo simulations, assuming Poissonian noise for the photon counts [@Kwiat_Tomo; @Shaham_QPT]. Four-birefringent crystal scheme as a dephasing channel ======================================================= In order to apply decoherence to polarization qubits, one should entangle the polarization degree of freedom (DOF) with extra DOFs that are not going to be measured, effectively increasing the dimension of the state Hilbert space. Ignoring the extra DOFs, the measured density matrix of the polarization DOF is obtained after tracing out these extra DOFs. Such a coupling between the polarization and additional temporal DOFs of the wave-packet can be achieved using a depolarizer that is composed of birefringent crystals and wave-plates in between them [@Kwiat_Dephasing; @Shaham_2011]. Consider a polarized wave-packet of photons with a coherence time $\tau$ that passes through such a depolarizer. A crystal with a length $L$ induces a temporal delay $t=L\frac{\Delta{n}}{c}$ between the fast and the slow polarization modes, where $\Delta{n}$ is the birefringent index difference and $c$ is the speed of light. We require that all crystals are sufficiently long such that the temporal delay between different polarization modes of the output states is much larger than the coherence time of the photons $\tau$: $$\label{temporal delay} L\frac{\Delta{n}}{c}\gg\tau.$$ After the passage through the depolarizer, the wave-packet occupies a discrete number of temporal modes. These modes do not overlap in between them, and every different temporal mode is fully polarized by itself. The photon-detectors are insensitive to small temporal differences between the different modes and do not record them. As a result, the temporal DOFs of the photonic state are traced out and decohered mixed states are detected. The mixture level of the detected state depends only on the occupation of each temporal mode and is a function of the input polarization state, and the relative angle between the polarization state and the primary axes of the crystal [@Shaham_2011]. It does not depend on the temporal shape of the discrete modes which is inherited from the input temporal shape and properties. Thus, as long as the relation of Eq. (\[temporal delay\]) is kept, the decoherence process that is induced by such a depolarizing scheme is the same regardless of the exact temporal shape of the photonic wave-packet. Turning the wave-plates between the crystals affects the relative occupation of each different temporal mode and controls the decoherence properties that are to be measured. We study a decohering scheme that is composed of four fixed calcite crystals ($\textrm{C}_1,..,\textrm{C}_4$), and three tunable HWPs ($\theta_1,\theta_2,\theta_3$) in between them (see Fig. \[Fig\_QPT\_Experimental\_setup\_scheme\](b)) [@Shaham_iso_depo]. The length of $\textrm{C}_2$ and $\textrm{C}_3$ is 2mm, and the length of $\textrm{C}_1$ and $\textrm{C}_4$ is 1mm. The fast axes of $\textrm{C}_1$ and $\textrm{C}_3$, and the slow axes of $\textrm{C}_2$ and $\textrm{C}_4$ are parallel, and define the zero angle of the wave-plates. This scheme couples between the polarization DOF and seven possible temporal modes: recall that a 1mm crystal induces a time delay of $\Delta{t}$ between the fast and the slow polarization modes, the maximal time delay is obtained between a polarization mode that travels through the fast axis of every crystal in the configuration, and the polarization mode that travels through the slow axis of every crystal, and equals $(1+2+2+1)\Delta{t}=6\Delta{t}$. Polarization modes that travel through fast axis of some crystals and the slow axis of the other ones occupy all temporal modes which are the multiplications of $\Delta{t}$ that lie between $t=0$ and $t=6\Delta{t}$, so that seven different temporal modes are obtained (for a detailed mathematical description of the different temporal modes see also [@Shaham_2011]). Hence, a polarization qubit that passes through this scheme resides within a 14-dimensional time-polarization Hilbert space. For photons with a wavelength of 780nm, the temporal delay between two successive modes is $\sim570$fs. The coherence time $\tau$ of the down converted single-photon wave-packets before entering the scheme is $\sim180\,\textrm{fs}$ (it is mainly governed by the 5nm interference bandpass filter). Thus, the requirement of Eq. (\[temporal delay\]) is fulfilled and the seven different temporal modes can be regarded as discrete ones. Tuning the HWPs to different angles, different processes are induced by the scheme. The values of the different $D_i$ parameters (Eq. (\[Radii chi eigenvalues\])) that represent these processes as a function of the HWP angles are $$\begin{aligned} D_1 &=& -\sin(4\theta_1)\sin(4\theta_3)\cos^2(2\theta_2)+\cos(4\theta_1)\cos(4\theta_2)\cos(4\theta_3), \label{eq_four_crystal_D1}\\ D_2 &=& \sin(4\theta_2)\sin(2\theta_1+2\theta_3)\cos(2\theta_1)\cos(2\theta_3)-\cos^2(2\theta_2)\cos^2(2\theta_1+2\theta_3)-\nonumber\label{eq_four_crystal_R2}\\ & &-\frac{1}{2}\sin(4\theta_1)\sin(4\theta_3)\sin^2(2\theta_2), \label{eq_four_crystal_D2}\\ D_3 &=&-\sin(4\theta_2)\sin(2\theta_1+2\theta_3)\cos(2\theta_1)\cos(2\theta_3)-\cos^2(2\theta_2)\cos^2(2\theta_1+2\theta_3)-\nonumber\label{eq_four_crystal_R3}\\ & &-\frac{1}{2}\sin(4\theta_1)\sin(4\theta_3)\sin^2(2\theta_2).\label{eq_four_crystal_D3}\end{aligned}$$ We emphasize that these values are calculated up to rotations that flip the sign of two $D_i$ values. Cancelation of these rotations can be achieved by adding more fixed wave-plates before the crystals, or by tilting one of the crystals to change the induced birefringent phase. Using Eqs. (\[eq\_four\_crystal\_D1\])- (\[eq\_four\_crystal\_D3\]), we numerically investigated the spanned volume in the tetrahedron representation that represents all possible processes that can be emulated by this scheme. The spanned volume is shown in Fig. \[Fig\_Four\_crystal\_D\_phase\_diagraml\](a). The presented possible process range can be extended with the addition of polarization rotations that are described by two transformations: cyclic permutations between the three $D_i$ values, and sign-flips of two $D_i$ values. The extended process range is presented in Fig. \[Fig\_Four\_crystal\_D\_phase\_diagraml\](b). It can be seen from Fig. \[Fig\_Four\_crystal\_D\_phase\_diagraml\](b) that almost every complete positive unital qubit map can be implemented using the investigated four-crystal scheme. ![\[Fig\_Four\_crystal\_D\_phase\_diagraml\] Numerical investigation of the possible channels of the four-crystal scheme in the tetrahedron representation. (a) All possible channels using different orientations of the three HWPs of the four-crystal scheme. (b) All possible channels using the different orientations of the three HWPs of the four-crystal scheme, with the addition of all allowed polarization rotations. These rotations can be implemented using more wave-plates that are placed after the scheme. Points on the dashed yellow line along the edge of the tetrahedron in (a) correspond to a dephasing channel that preserves the length of $D_3$.](Four_crystal_radius_phase_diagram8.eps){width="90mm"} Observing Fig. \[Fig\_Four\_crystal\_D\_phase\_diagraml\](a), it can be inferred that a dephasing channel can be implemented using this scheme, since the spanned volume almost covers the edges of the tetrahedron. However, a careful examination reveals that only an approximation of the dephasing channel can be implemented. It can be proved that there is no solution to Eqs. (\[eq\_four\_crystal\_D1\])- (\[eq\_four\_crystal\_D3\]), that preserves the length of one of the $D_i$ values equal to 1, while the absolute values of the two other parameters have values in the range of (0,1). Thus, points on the edges which correspond to a dephasing process are not attainable. We examined the properties of the points that are in the vicinity of the edges, and found that an approximation to the dephasing channel is obtained for the following relation between the three HWPs angles of the four-crystal scheme: $$\label{eq_four_crystal_dephasing_condition} \theta_1=\theta_3=\frac{\theta_2}{2}.$$ Maintaining this relation, we calculated the corresponding process matrices $\chi$ for $0\leq\theta_1\leq45^\circ$. A plot of the eigenvalues of $\chi$ as a function of $\theta_1$ is presented in Fig. \[Fig\_dephasing\_channel\_four\_crystal\_results\](a). One of the eigenvalues remains zero for every angle setting, and a second one is approximately zero for small $\theta_1$ angles (see the gray area in Fig. \[Fig\_dephasing\_channel\_four\_crystal\_results\](a)). The two other eigenvalues participate in the process for every angle, and intersect when $\theta_1\simeq9^\circ$. Thus, an approximation to a dephasing channel is obtained when $\theta_1$ is in the range of $0\leq\theta_1\leq9^\circ$, and the other two HWP angles satisfy Eq. (\[eq\_four\_crystal\_dephasing\_condition\]). We denote the highest eigenvalue of $\chi$ by $\chi_m$. Using Eqs. (\[dephasing\_channel\]), (\[dephasing\_probability\]), (\[Radii chi eigenvalues\]), (\[eq\_four\_crystal\_D1\])- (\[eq\_four\_crystal\_D3\]) and (\[eq\_four\_crystal\_dephasing\_condition\]), the dephasing probability $P$ that is induced by the approximated dephasing channel can be written as a function of $\theta_1$: $$\label{eq_four_crystal_dephasing_probability} P=1-\chi_m=\frac{-3\cos^4(4\theta_1)+2\cos^2(4\theta_1)+1}{2}.$$ In a similar manner to the definition of the fidelity between two quantum states [@Jozsa_1995_fidelity], the fidelity $F$ between two different processes can be defined as $$\label{eq_process_fidelity_definition} F(\hat{\chi}_{1},\hat{\chi}_{2})=\left(\mathrm{\textrm{Tr}}\left(\sqrt{\sqrt{\hat{\chi}_{1}}\hat{\chi}_{2}\sqrt{\hat{\chi}_{1}}}\right)\right)^{2}.$$ Comparing between a theoretical ideal controlled dephasing channel and the theoretical prediction of the approximated dephasing processes in the range $0\leq\theta_1\leq9^\circ$ that have the same corresponding $P$, the minimal value of the calculated process fidelity is $99.6\%$. Actually, such a high value is better than the typical accuracy that is achieved in experimental realizations of such channels. It is important to mention that the induced processes are not accompanied by nontrivial rotations: defining the $S_1$ polarization directions as the directions of primary axes of the crystals, the channel maintains the $S_2$ value of the input light, and reduces the absolute values of $S_1$ and $S_3$. It also flips the signs of the $S_2$ and $S_3$ parameters - a sign flip that can be compensated for by adding a fixed HWP in a zero angle before or after the channel. ![\[Fig\_dephasing\_channel\_four\_crystal\_results\] Four-crystal scheme as a dephasing channel. (a) Calculated eigenvalues of the $\chi$ matrix as a function of the first HWP angle. (b) Measured eigenvalues of the $\chi$ matrix as a function of the first HWP angle for the dephasing range ($0\leq\theta_1\leq10^\circ$, gray area of (a)). Eigenvalues that were reconstructed from counts of a single detector (coincidence measurements) are presented as upright triangles (upside-down triangles). Solid lines are the theoretical prediction.](dephasing_channel_four_crystal_results3.eps){width="100mm"} Implementing the four-crystal scheme (Fig. \[Fig\_QPT\_Experimental\_setup\_scheme\](b)), we measured the processes that approximate the dephasing channel using the setup of Fig. \[Fig\_QPT\_Experimental\_setup\_scheme\](a). Figure \[Fig\_dephasing\_channel\_four\_crystal\_results\](b) presents the measured eigenvalues of the $\chi$ matrices that were reconstructed from the data of classical single-photon wave-packets, as well as those who were reconstructed from the data of quantum wave-packets (see details in the experimental setup section). The eigenvalues are presented as a function of $\theta_1$ along with the theoretical prediction. The measured results are in good agreement with theory, and the control over the dephasing level from no dephasing ($\theta_1=0$) to a complete dephasing ($\theta_1\simeq9^\circ$) is achieved. As expected, no significant differences between the dephasing processes of wave-packets with different temporal properties are observed. Neglecting channel rotations, the average fidelity between the measured processes and the theoretical dephasing processes (which were calculated using Eq. (\[eq\_four\_crystal\_dephasing\_probability\]) for the corresponding $\theta_1$ angle) is $97\pm2\%$, both for the processes which were applied to classical and to quantum single-photon wave-packets. For light of a broader spectral bandwidth we expect to obtain even better results: such a wave-packet has a shorter coherence time and the requirement of Eq. (\[temporal delay\]) is easily fulfilled since the overlap between different temporal modes after the passage through the crystal is more negligible. Thus, Eqs. (\[eq\_four\_crystal\_D1\])-(\[eq\_four\_crystal\_D3\]) more faithfully represent the decoherence of the polarization state, and a similar experiment will show a higher fidelity to the theoretical calculation when compared to the presented classical and quantum wave-packet cases. Conclusions and discussion ========================== To summarize, we investigated a controllable photonic dephasing channel of a single spatial mode that is composed of four birefringent crystals and wave-plates. This configuration can emulate almost every arbitrary unital channel. The operation of the channel as a dephasing channel was experimentally demonstrated, and was characterized using single-photon wave-packets with different coherence times. There is a high fidelity between the operation of the explored scheme and that of an ideal dephasing environment, where the dephasing rate is in agreement with the theoretical prediction. As expected, the noise probability of the four crystal scheme is known in advance and is not affected by changes in the temporal envelope of the photonic wave-packet. The presented scheme is not limited to single-photon states. It can be applied on a high-intensity classical light of a short coherence time (or a broader bandwidth), and serve as a depolarizer that emulates almost any unital process. In the case of a wave-packet that has a longer coherence time, the birefringent crystals can be replaced by polarization maintaining birefringent fibers which act as longer birefingent crystals. Appendix: SBC as a dephasing channel for classical and quantum single-photon wave-packets {#appendix-sbc-as-a-dephasing-channel-for-classical-and-quantum-single-photon-wave-packets .unnumbered} ========================================================================================= In order to show that the classical and quantum single-photon wave-packets differ in their fine temporal properties and that this difference affects the dephasing level that these wave-packets experience when transmitted through a common dephasing channel, we transmitted the same investigated states through a SBC dephasing scheme [@Branning_2000] (see Fig. \[Fig\_QPT\_Experimental\_setup\_scheme\](c) in the main text). The SBC dephasing scheme is composed of two birefringent crystal prisms with parallel optical axes, which may be followed by another birefringent rectangular crystal. The passage through the crystal prisms results in a temporal delay of $t=L\frac{\Delta{n}}{c}$ between the two polarization modes (here $L$ is the *total* optical path inside the birefringent medium). When one of the prisms is transversely translated, $L$ is changed, and a control over the time delay $t$ is attained. When $t$ is comparable or larger than the initial wave-packet coherence time $\tau$, the temporal overlap between the two polarization modes is reduced and the phase uncertainty between the two polarizations increases (i.e., the photons experience a dephasing process). If a rectangular crystal is added to the beam path (see Fig. \[Fig\_QPT\_Experimental\_setup\_scheme\](c)), such that its optical axes are perpendicularly oriented with respect to the two prisms, the polarization modes can be temporally overlapped again and a dephasing process with a small or a zero probability is emulated. The SBC dephasing channel was implemented using two wedge quartz crystals with a wedge angle of $15^{\circ}$, and another 9mm rectangular quartz crystal. The chosen wedge angles of the crystals enable a time delay up to $t\sim380\,\textrm{fs}$. Such a time delay is larger by almost 2 orders of magnitude than the maximal time delay that a typical Soleil-Babinet compensator can induce. As was mentioned before, the coherence time $\tau$ of the down converted photons is $\sim180\,\textrm{fs}$. As a result, we could continuously switch between different dephasing levels. For every translation setting, $t$ was calculated using the optical path inside the birefringent quartz medium. We define $t$ to be in the range of $0<t<\sim380\,\textrm{fs}$ when the primary axes of the rectangular crystal are perpendicular to those of the wedge crystals. The range of $t$ is extended up to 900fs using different settings of the third rectangular crystal: time range of $\sim270\,\textrm{fs}<t<\sim650\,\textrm{fs}$ is obtained when the rectangular crystal is omitted, and time range of $\sim530\,\textrm{fs}<t<\sim920\,\textrm{fs}$ is attained when the rectangular crystal primary axes are parallel to those of the wedge crystals. Performing a standard QPT procedure on the channel, we reconstructed the process matrix $\chi$ for different translation settings. The eigenvalues of the reconstructed $\chi$ matrices are presented in Fig. \[Fig\_Dephaser\_chi\_eigenvalues\_and\_s2\_oscillations\](a) as a function of the induced temporal delay $t$. As in the four crystal case, the processes were separately reconstructed from data of classical and quantum wave-packet states. It can be seen that both states experience a dephasing operation since for every $t$, two $\chi$ eigenvalues remain close to zero. When $t\simeq185\,\textrm{fs}$, the operation of the channel is very close to having no effect. The corresponding measured processes from both data sets for this temporal setting have fidelities higher than $95\pm2\%$ to the no-decohering process (disregarding polarization rotations). When compared to the quantum single-photon wave-packet, it is clear from Fig. \[Fig\_Dephaser\_chi\_eigenvalues\_and\_s2\_oscillations\](a) that the classical single-photon wave-packet experiences a faster dephasing process. This demonstrates the difference between the two wave-packet states, and shows the dependence of the dephasing probability of the SBC scheme in the temporal shape of the incoming photonic state, unlike the four crystal scheme (see Fig. \[Fig\_dephasing\_channel\_four\_crystal\_results\](b) in the main text). ![\[Fig\_Dephaser\_chi\_eigenvalues\_and\_s2\_oscillations\] (a) The eigenvalues of the $\chi$ matrix as a function of the time delay $t$ between the two polarization modes. Eigenvalues of process matrices that were reconstructed from counts of single detector (coincidence measurements) are presented as red upright triangles (blue upside-down triangles). Solid lines are Gaussian fits to lead the eye. (b) The oscillations of the Stokes parameter $S_2$ as a function of the time delay between the two polarization modes. Solid line represents a sinusoidal fit.](QPT_soleil_Babinet5.eps){width="100mm"} It is worth mentioning that although the wedge angle of the channel is larger with respect to that of a typical Soleil-Babinet compensator, the SBC dephasing scheme realization can also serve as a solid and stable polarization interferometer. Figure \[Fig\_Dephaser\_chi\_eigenvalues\_and\_s2\_oscillations\](b) presents the oscillations of the $S_2$ Stokes parameter as a function of the temporal delay. The oscillations were measured for the quantum single-photon wave-packet state, when a heavy dephasing process is applied to the photons ($P\sim0.45$). Analyzing the presented oscillations, the calculated wavelength of the photons is $775\pm15$nm, in agreement with the set value of 780nm. Funding {#funding .unnumbered} ======= We thank the Israel Science Foundation for supporting this work under Grants No. 546/10 and 793/13. We also thank the Israeli Ministry of Science and Technology for financial support through the Eshkol fellowship for A.S.. [99]{} M. A. Nielsen and I. L. Chuang, *Quantum Computation and Quantum Information* (Cambridge University, Cambridge, U.K., 2000). J. I. Cirac, A. K. Ekert, S. F. Huelga, and C. Macchiavello, “Distributed quantum computation over noisy channels,” Phys. Rev. A **59**(6), 4249-4254 (1999). J. l. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nature Photon. **3**, 687-695 (2009). G. Puentes, D. Voigt, A. Aiello, and J. P. Woerdman, “Experimental observation of universality in depolarized light scattering,” Opt. Lett. **30**, 3216-3218 (2005). M. Ricci, F. De Martini, N. J. Cerf, R. Filip, J. Fiuráš�ek, and C. Macchiavello, “Experimental purification of single qubits,” Phys. Rev. Lett. **93**, 170501 (2004). A. Chiuri, V. Rosati, G. Vallone, S. Pádua, H. Imai, S. Giacomini, C. Macchiavello, and P. Mataloni, “Experimental realization of optimal noise estimation for a general Pauli channel,” Phys. Rev. Lett. **107**, 253602 (2011). M. Karpiński, C. Radzewicz, and K. Banaszek, “Fiber-optic realization of anisotropic depolarizing quantum channels,” J. Opt. Soc. Am. B **25**(4), 668-673 (2008). P. G. Kwiat, A. J. Berglund, J. B. Altepeter, and A. G. White, “Experimental verification of decoherence-free subspaces,” Science **290**, 498-501 (2000). J.-S. Xu, C.-F. Li, X.-Y. Xu, C.-H. Shi, X.-B. Zou, and G.-C. Guo, “Experimental characterization of entanglement dynamics in noisy channels,” Phys. Rev. Lett. **103**, 240502 (2009). B.-H. Liu, L. Li, Y.-F. Huang, C.-F. Li, G.-Can Guo, E.-M. Laine, H.-P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems” Nature Physics **7**, 931�934 (2011). E. Amselem and M. Bourennane, “Experimental four-qubit bound entanglement,” Nature Phys. **5**, 748-752 (2009). R. B. A. Adamson, L. K. Shalm, and A. M. Steinberg, “Preparation of pure and mixed polarization qubits and the direct measurement of figures of merit,” Phys. Rev. A **75**, 012104 (2007). M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. S. Ribeiro, and L. Davidovich, “Environment-induced sudden death of entanglement,” Science **316**, 579-582 (2007). D. F. Urrego, J.-R. Álvarez, O. Calderón-Losada, J. Svozilík, M. Nuñez, and A. Valencia, ”Implementation and characterization of a controllable dephasing channel based on coupling polarization and spatial degrees of freedom of light,” Opt. Express **26**, 11940-11949 (2018). A. Shaham and H. S. Eisenberg, “Realizing a variable isotropic depolarizer,” Opt. Lett. **37**(13), 2643-2645 (2012). I. L. Chuang and M. A. Nielsen, “Prescription for experimental determination of the dynamics of a quantum black box,” J. Mod. Opt. **44**(11-12), 2455-2467 (1997). A. Jamio[ł]{}kowski, “Linear transformations which preserve trace and positive semidefiniteness of operators," Rep. Math. Phys. **3**(4), 275-278 (1972). M. Horodecki and R. Horodecki, “Information-theoretic aspects of inseparability of mixed states,” Phys. Rev. A **54**(3), 1838-1843 (1996). C. King and M. B. Ruskai, “Minimal entropy of states emerging from noisy quantum channels,” IEEE Trans. Inf. Theory **47**(1), 192-209 (2001). D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A **64**, 052312 (2001). D. Branning, A. L. Migdall, and A. V. Sergienko, “Simultaneous measurement of group and phase delay between two photons,” Phys. Rev. A **62**, 063808 (2000). J. B. Altepeter, E. R. Jeffrey, and P. G. Kwiat, “Photonic state tomography,” Adv. At. Mol. Opt. Phys. **52**, 105-159 (2005). A. Shaham and H. S. Eisenberg, “Quantum process tomography of single-photon quantum channels with controllable decoherence,” Phys. Scr. **T 147**, 014029 (2012). A. Shaham and H. S. Eisenberg, “Realizing controllable depolarization in photonic quantum-information channels,” Phys. Rev. A **83**, 022303 (2011). R. Jozsa, “Fidelity for mixed quantum states,” J. Mod. Opt. **41**, 2315-2323 (1994).

--- abstract: | In this paper we describe a Bayesian statistical method designed to infer the magnetic properties of stars observed using high-resolution circular spectropolarimetry in the context of large surveys. This approach is well suited for analysing stars for which the stellar rotation period is not known, and therefore the rotational phases of the observations are ambiguous. The model assumes that the magnetic observations correspond to a dipole oblique rotator, a situation commonly encountered in intermediate and high-mass stars. Using reasonable assumptions regarding the model parameter prior probability density distributions, the Bayesian algorithm determines the posterior probability densities corresponding to the surface magnetic field geometry and strength by performing a comparison between the observed and computed Stokes $V$ profiles. Based on the results of numerical simulations, we conclude that this method yields a useful estimate of the surface dipole field strength based on a small number (i.e. 1 or 2) of observations. On the other hand, the method provides only weak constraints on the dipole geometry. The odds ratio, a parameter computed by the algorithm that quantifies the relative appropriateness of the magnetic dipole model versus the non-magnetic model, provides a more sensitive diagnostic of the presence of weak magnetic signals embedded in noise than traditional techniques. To illustrate the application of the technique to real data, we analyse seven ESPaDOnS and Narval observations of the early B-type magnetic star LPOri. Insufficient information is available to determine the rotational period of the star and therefore the phase of the data; hence traditional modelling techniques fail to infer the dipole strength. In contrast, the Bayesian method allows a robust determination of the dipole polar strength, $B_d=911^{+138}_{-244}$G. author: - | V. Petit$^{1}$[^1], and G.A. Wade$^{2}$\ $^{1}$Dept. of Geology & Astronomy, West Chester University, West Chester, PA 19383\ $^{2}$Dept. of Physics, Royal Military College of Canada, PO Box 17000, Stn Forces, Kingston, Canada, K7K 4B4 bibliography: - 'bayes.bib' date: 'Accepted 2011 October 26. Received 2011 October 22; in original form 2011 August 17' title: 'Stellar magnetic field parameters from a Bayesian analysis of high-resolution spectropolarimetric observations[^2]' --- \[firstpage\] stars: magnetic fields– stars: early-type – techniques: polarimetric – methods: statistical. Introduction ============ The evolution of a massive star is strongly determined by its rotation, as well as the mass lost through its stellar wind, both of which can be strongly influenced by the presence of a magnetic field [e.g. @2010ApJ...714L.318T; @2011arXiv1106.3008W]. A magnetic field can furthermore couple different layers of a star’s interior, thereby modifying internal differential rotation and circulation currents . If a field has a large-scale component that extends outside the stellar surface, it can also channel the outflowing stellar wind, creating a structured wind - a magnetosphere - which will modify the rate and geometry of mass loss along with its observational properties . Furthermore, if the field couples the rotating surface of the star with the outflowing stellar wind, both effects will result in angular momentum loss (via the outflowing stellar wind), which differs strongly from that of a non-magnetic star [@2009MNRAS.392.1022U]. As angular momentum and mass loss are cornerstone inputs in stellar evolution calculations, it is crucial that the effect of magnetic fields in massive stars be understood properly. For example, evolutionary tracks and isochrones can be used to interpret large datasets associated with OB associations like the *VLT-FLAMES* surveys of massive stars . Over the last decade, our knowledge of the properties of massive star magnetic fields has significantly improved, in large part due to a new generation of powerful high-resolution spectropolarimetric instrumentation. Traditionally, stellar magnetic fields were modelled using measurements of the longitudinal magnetic field of the star, yielding a single quantity: the strength of the disc-integrated line-of-sight component of the magnetic field . In the absence of a field detection, the interpretation of such data was entirely dependent on the (unknown) stellar and magnetic geometry, and therefore highly ambiguous. However, modern high-resolution spectropolarimetry yields Doppler-broadened, velocity-resolved Stokes profiles measured across spectral lines, which encode additional information about the field geometry. These data therefore allow a more robust inference of the field characteristics. Remarkably detailed information about the strength and local/global structure of stellar magnetic fields can be extracted from high-resolution, phase-resolved spectropolarimetric datasets acquired in two or four Stokes parameters (i.e. Stokes $IV$ or Stokes $IVQU$) using techniques such as Magnetic Doppler Imaging . However, detailed modelling of this sort relies on extensive high-quality datasets that demand significant telescope time, and which can only be obtained for a small number of stars. The lower-quality data that can be obtained for stars with less suitable observational properties can still be approximately investigated using parametric models; nevertheless, even such a simple approach often requires approximately a dozen observations per star. Large observing programs, such as the Magnetism in Massive Stars (MiMeS) project, have dedicated significant resources to survey large samples of massive stars in search of magnetic fields [@2010arXiv1009.3563W]. Such surveys typically acquire a small number of Stokes $V$ observations (typically 1-3) per star, for a large number of stars. An outstanding problem concerns how to extract useful information about the magnetic field of a star from such a limited data set. In this paper we describe an approach to constrain the magnetic field strength of a star from small numbers of high-resolution Stokes $V$ observations, assuming the dipole oblique rotator paradigm, and expressed using the formalism of Bayesian statistics. In Sect. \[sec|line\_synth\] we describe briefly the model used to synthesise the emergent local circular polarisation produced by the Zeeman effect under the weak-field approximation. We present the disk-integrated emergent Stokes $V$ profiles obtained for a dipolar magnetic topology of a rotating star. In Sect. \[sec|bayes\] we introduce the Bayesian algorithm used to compare a set of high-resolution Stokes $V$ observations with a grid of synthetic line profiles. Sect. \[sec|test\] presents the application of the Bayesian algorithm to simulated data. We demonstrate that the Bayesian odds ratio can help detect a weak magnetic signal, by quantifying which target should be re-observed. It can also discriminate, with a few additional observations, whether an observation has a noise pattern that by chance looks like a magnetic signal versus a real magnetic signal. We also show that the Bayesian algorithm provides a meaningful upper limit on the magnetic strength in the case of a non-detection, and is able to reliably estimate the dipole field strength in the case of a detection. Some limited constraints can also be obtained for the geometry of the dipole. We compare the Bayesian diagnostics with the traditional global longitudinal field and signal detection probability diagnostics. Finally, in Sect. \[sec|lpori\], we present the Bayesian results for the magnetic B-type star LPOri [@2008MNRAS.387L..23P], obtained with observations of various signal-to-noise ratios. Line synthesis {#sec|line_synth} ============== Local profiles {#sec|model} -------------- Splitting of spectral lines due to the Zeeman effect corresponds to about 1kms$^{-1}$ per kG of surface field modulus. This implies that even relatively strong fields are difficult to detect reliably from Zeeman splitting in the spectrum of most stars. The situation is particularly challenging in the case of hot stars, where thermal broadening (of order a few kms$^{-1}$), turbulent broadening (of order a few tens of kms$^{-1}$) and rotational broadening (potentially a few hundreds of kms$^{-1}$) combine to fully obscure any modification of the line profile due to the Zeeman effect. The Zeeman effect provides us with a second useful tool, in the form of the polarisation of the Zeeman components. Zeeman components exist in two different types: $\pi$ components, spread symmetrically about the zero-field wavelength of the spectral line and $\sigma$ components, with symmetric wavelength offsets to the red and blue of the zero-field wavelength. If the external magnetic field is oriented parallel to the observer’s line of sight (a longitudinal field), the $\pi$ components vanish, while the two groups of $\sigma$ components have opposite circular polarisations. In a field aligned perpendicular to the line of sight (a transverse field), the $\pi$ components are linearly polarised perpendicular to the field direction, while the $\sigma$ components are linearly polarised parallel to the field direction. Therefore, spectropolarimetric observations are an extremely useful tool for detection and characterisation of both the strength and orientation of stellar magnetic fields. In order to correctly predict the polarised spectrum that will emerge from a stellar atmosphere, one must perform radiative transfer in an anisotropic medium, where the propagation properties depend on the propagation directions. In the weak-field regime, we can treat the anisotropic absorption and dispersion profiles as perturbations of the isotropic case [@2004ASSL..307.....L]. In that case, at first order, only the longitudinal component of the magnetic field contributes to the polarisation, and therefore only circular polarisation is predicted. The contribution of the transverse field, and the occurrence of linear polarisation, is a second-order effect. Therefore, circular polarisation (Stokes $V$ spectra) signatures are generally stronger, and used for large surveys as such observations provide the best detection threshold. Although there are some elaborate codes that are able to accurately synthesise the detailed circular polarisation profile of an arbitrary spectral transition , we will use the weak-field approximation, in order to draw a simple picture of the Stokes $V$ spectrum emerging from a star. The 1st order solution for the circular polarisation to the polarised radiative transfer equations is given by: $$V(v) = -\frac{e}{4\pi mc^2}cg_\mathrm{eff}\lambda_0B_{||}\frac{dI(v)}{dv}.\label{eq|stokesV}$$ The Stokes $V$ profile $V(v)$ emerging from a point at the surface of a star (referred to as a *local* profile) has the same shape as the derivative of the local intensity profile $I(v)$, scaled by the longitudinal component of the magnetic field $B_{||}$ at that point, by the wavelength $\lambda_0$ of the transition, and by the effective Landé factor $g_\mathrm{eff}$ of the transition. The effective Landé factor represents the separation of a triplet Zeeman pattern that approximates a more complex Zeeman pattern, and can be calculated under LS coupling, or taken from atomic experiments. Although we will be using the weak field approximation throughout this paper, the Bayesian algorithm can use synthetic profiles computed with any spectrum synthesis code. A multi-line approach will usually be applied – in order to increase the signal to noise ratio (s/n) – to the data for which a Bayesian approach will be useful. Although these multi-line techniques have been developed and refined for more than two decades now [e.g. @1981ApJ...247..569B; @1996SolarPhys...164...417S], the most widely-used approach is the Least Squares Deconvolution (LSD) method introduced by @1997MNRAS.291..658D. Under the weak field approximation, the LSD method assumes that all the spectral lines have the same shape, and that this shape is scaled by the line depth for the intensity line profiles (Stokes $I$) and by the product of the line depth, the wavelength and the effective Landé factor for the circularly polarised line profile (Stokes $V$). The LSD method therefore already approximates the Zeeman pattern of a line by a triplet whose separation is given by an effective Landé factor. For a complete discussion on the circumstances under which a LSD profile can be treated as a single spectral line, see . Disk integration {#sec|disk} ---------------- In order to predict the Stokes $V$ spectrum of a given stellar spectral line, we must take into account the contribution from every point on the visible hemisphere. In this paper, we consider a single spectral line of arbitrary depth and a width corresponding to the sum of the spectral resolution and an ad hoc local broadening width. The contribution of each point on the visible hemisphere is dictated by the effective surface area and a wavelength-independent limb-darkening of the form: $$\frac{I_c}{I_{c,0}} = 1-\epsilon + \epsilon\cos\theta,$$ where $I_{c,0}$ is the intensity at the stellar disk centre, $\theta$ is the angle between the normal to the surface and the line of sight, and $\epsilon$ is the limb-darkening factor [@1992oasp.book.....G]. We adopt $\epsilon=0.6$ for the rest of this paper. The surface velocity field of the stellar disk is defined by the projected equatorial velocity $v\sin i$ assuming rigid rotation. We consider a simple magnetic topology with a dipolar field of strength $B_p$, as the vast majority of intermediate mass and high mass stars’ magnetic fields appear to be predominantly dipolar . The orientation of the dipole with respect to the observer’s reference frame is described by the inclination $i$ of the rotational axis to the line of sight, the rotational phase $\varphi$ and the obliquity $\beta$ of the magnetic axis with respect to the rotational axis. Figure \[fig|model\_bz\] shows the radial field (top) and the longitudinal field (bottom) for a dipole field seen positive pole-on (left) and magnetic equator-on (right). The radial field is defined by its orientation with respect to the stellar surface normal. For example, when we are looking at the positive pole (left), the radial field is at its maximum (red) in the middle of the stellar disk. On the edge of the disk, the field is parallel to the surface, and the radial field is null (green). ![\[fig|model\_bz\] Radial (top) and longitudinal (bottom) surface magnetic field for a dipolar field seen pole-on (left) and equator-on (right). For the radial field, red, blue and green colours represent field oriented away from, into, and parallel to the star surface, respectively. For the longitudinal field, red, blue and green colours represent fields oriented toward, away from and perpendicular to the observer’s line of sight, respectively. The black grid represent the rotation reference frame, with an inclination $i=90^\circ$. The obliquity of the field is $\beta=90^\circ$ and the pole-on and equator-on configurations would correspond to rotational phases $\varphi=0^\circ$ and $\varphi=90^\circ$, respectively.](fig1.pdf){width="75mm"} The longitudinal field is the important quantity for the polarised radiative transfer. The longitudinal field represents the projection of the magnetic field vectors with respect to the observer’s line of sight (perpendicular to the paper plane). If we look at the magnetic field seen pole-on (bottom left), all of the magnetic vectors that are oriented toward the observer are colour-coded in red shades, green corresponds to a null longitudinal field component, and magnetic vectors oriented away from the observer are colour-coded in blue shades. As seen in Eq. (\[eq|stokesV\]), two longitudinal magnetic fields of same magnitude but opposite signs will produce inverse local Stokes $V$ profiles. However, a good fraction of the field components oriented away from the observer are located on the hidden hemisphere of the star. There is therefore a net positive longitudinal component on the visible hemisphere (i.e. the visible hemisphere is more red than blue). In terms of the total emergent Stokes $V$, positive local profiles (local profiles for which the longitudinal field is positive) will dominate the total line profile. This effect is even more pronounced when limb darkening is taken into account, as the edges of the disk contribute less to the total brightness. We now turn our attention to the case where we are looking at the magnetic equator (bottom right). In principle, there are as many magnetic vectors pointing toward the observer as away. The global longitudinal component is therefore null, as every magnetic vector on the left side of the stellar disk has its opposite on the right side of the disk. In terms of the emerging Stokes $V$ profile, each positive local Stokes $V$ profile will be cancelled out by the negative local Stokes $V$ profile of same amplitude coming from the opposite side of the visible stellar disk, even when considering limb-darkening. The resulting Stokes $V$ profile should therefore be null when the global longitudinal component is null. This is effectively the case, if the star is not rotating. If the star is rotating around a rotation axis oriented toward the top of the paper (as represented by the black grid), the line of sight component of that rotation will introduce some Doppler shifts to different points on the stellar surface. For example, if the star rotates from left to right, the point situated to the extreme left of the stellar disk will be shifted by $-v\sin i$. The point situated at the extreme right of the stellar disk, whose local Stokes $V$ profile would in principle cancel out the one produced by the leftmost point, will now be shifted by $+v\sin i$. Therefore, the rotation is able to separate in the velocity space the local Stokes $V$ line profiles that would otherwise cancel out, and a net circular polarisation profile can be seen even if the global longitudinal field is null. It has been shown that a $v\sin i$ as low as a few kms$^{-1}$ is enough for instruments able to resolve this effect in the line profile [@2000MNRAS.313..823W]. Figure \[fig|model\_nodetect1\_cont0\] shows an example of the shapes and relative amplitudes of Stokes $V$ profiles emerging from a star rotating at 50kms$^{-1}$ (top). The radial and longitudinal fields are shown (middle), as well as the global longitudinal field curve (bottom). The phases indicated by red dots on the curve indicate the rotational phases corresponding to each shown profile. We can see that for this dipole configuration ($i=90^\circ$, $\beta=90^\circ$), the emerging Stokes $V$ profile has an amplitude as strong when the global longitudinal field is null (2nd and 4th phases) than when it is at its maximum (1st, 3rd and last phases). We note the Stokes $V$ profiles at $B_l=0$G are symmetric around the centre of the line. If we were to observe such profiles with an instrument that does not resolve the line profiles, we would indeed obtain a null global longitudinal field, even if the profiles are as clearly detectable with high-resolution observations than when the longitudinal field is at its maximum. ![\[fig|model\_nodetect1\_cont0\] *Top:* Shape of the Stokes $V$ line profiles for five rotational phases of a star with a dipolar field with $i=\beta=90^\circ$. *Middle:* Corresponding radial field and longitudinal field components (colour schemes are as indicated in Figure \[fig|model\_bz\]). The rotation axis inclination $i=90^\circ$ is represented by the black grid and the magnetic field pole is perpendicular to the rotation axis ($\beta=90^\circ$). *Bottom:* Global longitudinal field curve, normalised to the dipole field strength. The dots show the phase of the profiles. Note how the Stokes $V$ profile do not disappear when the global longitudinal field goes to zero.](fig2.pdf "fig:"){width="84mm"}\ ![\[fig|model\_nodetect1\_cont2\] Same as Figure \[fig|model\_nodetect1\_cont0\] for a rotation axis inclined by $i=45^\circ$ and a magnetic pole of obliquity $\beta=45^\circ$. Note how the Stokes $V$ signal disappears when the global longitudinal field goes to zero because $i+\beta\sim90^\circ$. ](fig3.pdf){width="84mm"} Figure \[fig|model\_nodetect1\_cont2\] shows a second dipole configuration ($i=45^\circ$, $\beta=45^\circ$), where the rotation is not able to separate components that will cancel out. When the longitudinal field is null (middle profile, we are looking at the magnetic equator), we can see that as the star rotates from left to right, all the points situated on the left side of the stellar disk will be shifted to the blue. However, each half of the visible hemisphere contains as many field vectors oriented toward the observer as oriented away from the observer. Therefore, all the red-shaded points situated on the left side of the stellar disk will have a corresponding inverse profile that will be Doppler shifted by the same amount, and the cancellation will occur. This particular symmetry occurs when $\beta+i\sim90^\circ$, at a phase where we are looking at the magnetic equator. Due to the intrinsic symmetry of a dipole field, as well as the fact that circular polarisation is only sensitive to the longitudinal field components, some parameter degeneracy is encountered for the resulting Stokes $V$ profiles. For example, if a dipolar field of parameter $i$, $\beta$ and $\varphi$ produces a profile $f$, the following symmetries (or anti-symmetries) occur: $$\label{eq|model_sym} \begin{pmatrix} 90^\circ-i \\ 90^\circ-\beta \\ \varphi \\ f \end{pmatrix}, \begin{pmatrix} 90^\circ-i \\ \beta \\ 180^\circ+\varphi \\ -f \end{pmatrix}, \begin{pmatrix} i \\ 90^\circ-\beta \\ 180^\circ+\varphi \\ -f \end{pmatrix}.$$ The bayesian algorithm {#sec|bayes} ====================== Bayesian statistics have proven to be an useful tool to find plausible models to explain astronomical data . A probability density distribution $p(H_i|I)$ provides a quantitative encoding of the plausibility of a certain hypothesis ($H_i$), given our current state of knowledge ($I$). The main pathway to Bayesian statistics [@Jaynes2003] is the Bayes theorem: $$p(H_i|D,I) = \frac{p(H_i|I)p(D|H_i,I)}{p(D|I)}.$$ This theorem states that the posterior probability density $p(H_i|D,I)$ of an hypothesis, given a new dataset $D$ and the current state of knowledge, is equal to the product of prior knowledge of the plausibility of the hypothesis $p(H_i|I)$ and the likelihood $p(D|H_i,I)$ of obtaining the new dataset if the hypothesis is true. The global likelihood, $p(D|I)=\sum_i p(H_i|I)p(D|H_i,I)$, acts as a normalisation constant. The act of summing (or integrating in the case of a continuous hypothesis set) is called marginalisation. One difficulty of modelling the Stokes $V$ spectra of stars observed as part of a survey like MiMeS is that most of the time the rotational phase of the star at the moment of the observation is not known, because the rotational period is generally unknown. Our approach is therefore to use Bayesian probability nomenclature to find which set of phase-independent configurations $\mathcal{B} = (B_p, i, \beta)$ can reproduce the observations. In other words, while the geometry $\mathcal{B}$ of the field must stay the same for each observation $n$ of a given star, the phase $\varphi_n$ may in principle have any value. The hypothesis to be tested is the presence of an oblique dipolar magnetic field in a particular star. The predictions of this hypothesis are represented by the model $M_1$, parametrized with the parameters $\mathcal{B}$ (=\[$B_p$, $i$, $\beta$\]) and $\Phi$ (=\[$\varphi_1$..$\varphi_N$\]), the latter representing a set of phases associated with a set of Stokes $V$ observations $\mathcal{D}$ (=\[$D_1$..$D_N$\]). The Bayes theorem tells us that the joint posterior probability density for the parameters, assuming the veracity of the model $M_1$, is $$\label{eq|post_conf} p(\mathcal{B}, \Phi | \mathcal{D}, M_1) = \frac{p(\mathcal{B}, \Phi | M_1)p(\mathcal{D} | \mathcal{B}, \Phi, M_1)}{p(\mathcal{D} | M_1)}.$$ Our prior knowledge of the probability density for each model parameter, which can be as simple as its expected range, can be encoded in the prior term $p(\mathcal{B}, \Phi | M_1)$, whereas the information brought forward by the new observations is reflected in the likelihood term $p(\mathcal{D} | \mathcal{B}, \Phi, M_1)$. The global likelihood is the normalisation term $p(\mathcal{D} | M_1)$, which is equal to the total probability: $$\label{eq|globalLH} p(\mathcal{D} | M_1)=\int\int p(\mathcal{B}, \Phi | M_1)p(\mathcal{D} | \mathcal{B}, \Phi, M_1) \mathrm{d}\Phi \mathrm{d}\mathcal{B}.$$ We then treat the set of phases $\Phi$ as nuisance parameters by marginalising the joint posterior probability: $$\label{eq|post_conf_b} p(\mathcal{B} | \mathcal{D}, M_1) = \int p(\mathcal{B}, \Phi | \mathcal{D}, M_1) \mathrm{d}\Phi.$$ We therefore ensure that an adequate $\mathcal{B}$ possesses Stokes $V$ profiles at some phases that match all the observations. From this $\mathcal{B}$ posterior probability density, we can then explore the plausible region of the parameter space of an oblique dipole field given the data, assuming that the model is true. In the case of a limited number of spectropolarimetric observations, we will be generally interested in the field strength value, as only weak constraints can be placed on the geometrical parameters. This is particularly relevant in the case of a non-detection, where we wish to put an upper limit on the strength of an undetected dipole field. To obtain the posterior probability density for a given parameter, we need to marginalise the joint probability $p(\mathcal{B}|\mathcal{D}, M_1)$ over the other parameters. For example, the posterior probability density marginalised for the field strength is: $$p(B_p|\mathcal{D}, M_1) = \int p(\mathcal{B}|\mathcal{D}, M_1)\,\mathrm{d}i\,\mathrm{d}\beta .$$ Another powerful application of Bayesian statistics is the ability to quantitatively test the plausibility of one hypothesis versus another. We can therefore compare the plausibility of the oblique dipole model $M_1$, by computing the so-called odds ratio of this model compared to the model $M_0$ representing the absence of a magnetic field. To do so, we need to compute the posterior probability $p(M_i|D,I)$ of a given model which is, according to the Bayes theorem: $$p(M_i|D,I) = \frac{ p(M_i|I)p(D|M_i, I) }{ p(D|I) }.$$ The prior term $p(M_i|I)$ encodes the plausibility of the model, given our current knowledge, and the global likelihood $p(D|M_i, I)$ encodes the plausibility of the model, given the new observations. Therefore, the odds ratio of our two competing models, $M_0$ and $M_1$, can be written as: $$\mbox{odds ($M_0/M_1$)} = \frac{p(M_0|\mathcal{D},I)}{p(M_1|\mathcal{D},I)} = \frac{p(M_0|I)}{p(M_1|I)} \frac{p(\mathcal{D}|M_0, I)}{p(\mathcal{D}|M_1, I)}.$$ Typically, as no model is preferred prior to the acquisition of Stokes $V$ observations, the ratio of priors $p(M_0|I)/p(M_1|I)$ will be set to unity. The global likelihood of the dipole model is given by Eq. (\[eq|globalLH\]). As the model representing the absence of magnetic field has no parameters, its global likelihood is simply the product of the likelihoods that $V=0$ for each observation. According to @Jeffreys1998, the evidence in favour of a model is considered weak when the odds are $>10^{0.5}$ ($\sim$3:1), moderate when $>10^{1}$ ($\sim$10:1), strong when $>10^{1.5}$ ($\sim$30:1) and very strong when $>10^2$ ($\sim$100:1). Practical implementation ------------------------ Given the complexity of disk integration for a rotating magnetic star, it is not practical to solve the Bayesian problem analytically [for an example applied to the local Stokes profiles of the Sun, see @2011ApJ...731...27A]. A Markov-chain Monte-Carlo method works by choosing series of parameter sets, the parameter regions with a high likelihood being more likely to be picked. The sampling itself therefore reflects the posterior probability density. This method is not well suited to this problem because a large number of calculations are required to assess the probability of each $\mathcal{B}$ configuration, due to the marginalisation over all possible phases. We therefore chose a numerical, brute-force approach in order to explore the parameter space by sampling it with a regular grid. When evaluating Eqs. (\[eq|post\_conf\]) and (\[eq|post\_conf\_b\]), we can make use of the fact that the prior probability densities for each parameter are independent such that $p(\mathcal{B}, \Phi | M_1) = p(\mathcal{B}|M_1)p(\Phi|M_1)$. Furthermore, the prior probability for the phases are the same such that $p(\varphi_n)=p(\varphi)$. Finally, the likelihood $p(\mathcal{D}|\mathcal{B}, \Phi, M_1)$ can be factored into $\Pi_{n=1}^N p(D_n|\mathcal{B}, \varphi_n, M_1)$ given that the likelihood of one observation $D_n$ is only dependent on the phase $\varphi_n$. Therefore, Eq. \[eq|post\_conf\_b\] can be written as: $$\label{eq|post_conf_new} p(\mathcal{B}|D,M_1)=p(\mathcal{B}|M_1)\frac{ \prod_{n=1}^N \int p(\varphi|M_1)p(D_n|\mathcal{B},\varphi,M_1){\rm d}\varphi}{p(\mathcal{D}|M_1)}.$$ For each point in a \[$B_p$, $i$, $\beta$, $\varphi$\] parameter space, we can compute the likelihood between the synthetic profile produced by these parameters and each observation. In fact, given the Stokes $V$ profile symmetry properties described in Eq. (\[eq|model\_sym\]), only the synthetic profiles for a quarter of the parameter space need to be computed. Furthermore, synthetic profile interpolation is possible in the $B_p$ direction, as the local Stokes $V$ profile amplitude increases linearly although the shape remains the same. The maginalisation integrals are performed numerically using a five point Newton-Cotes algorithm. An adequate sampling of the $M_1$ parameter space was chosen to obtain accurate values of $p(M_1|\mathcal{D},I)$. Probability densities --------------------- For each parameter of the oblique dipole model, a prior probability density is needed to evaluate Eq. (\[eq|post\_conf\]). The dipole field strength $B_p$ is a parameter that can vary over several decades (from a few dozen G to a few kG). We therefore used a Jeffreys prior, which sets an equal probability per decade and is therefore scale invariant. We used a modified form [@2005ApJ...631.1198G] in order to eliminate the singularity at $B_p=0$: $$p(B_p|M_1) = \frac{1}{ (B_p+a) \ln\left(\frac{a+B_{p,\mathrm{max}}}{a}\right)}.$$ This prior function behaves as a flat prior when $B_p<a$, and as a Jeffreys prior when $B_p>a$. The maximum dipolar field strength $B_{p,\mathrm{max}}$ is adjusted according to the strength of the Stokes $V$ signal and by the quality of the data (for example, if the s/n is lower, a field of a higher strength can hide in the noise in the case of a non-detection). For this paper, the grid has 250 $B_p$ values. A finer sampling might be required when using a large number of observations, in order to avoid under sampling the joint posterior probability density. We tested the effect of the choice of the $a$ parameter, and we settled using a value about twice the $B_p$ grid step. The inclination of the rotation axis to the line of sight is a “position” parameter (invariant under a shift of zero position), for which a flat prior would be well suited. However, we know that if the orientations of the rotation axes of stars are generally randomly distributed, the probability that the inclination angle is between $i$ and $i+\mathrm{d}i$ is: $$\label{eq|prior_i} p(i|M_1) = \frac{1}{2}\sin i.$$ The inclination could in principle vary from $0^\circ$ to $180^\circ$. However, if a non-null $v\sin i$ is measured, it is possible to put a lower limit on the inclination by estimating the break-up velocity – typically around 700kms$^{-1}$ for a massive OB star. We used a grid of 37 $i$ values, giving a sampling at least every $\sim5^\circ$. The same reasoning could apply to the obliquity angle $\beta$ of the magnetic axis to the rotation axis. However, we ignore at this point if the magnetic axes are generally randomly distributed, or if a general relation exists for the position of the magnetic axis relative to the rotation axis. For example, have shown that the magnetic axis of long-period ApBp stars is generally aligned with their rotation axis. Therefore, instead of using a prior that assumes a randomness of the magnetic axis position, we kept a flat prior for the obliquity angle: $$p(\beta|M_1) = \frac{1}{\beta_\mathrm{max}-\beta_\mathrm{min}}.$$ The obliquity varies from $\beta_\mathrm{min}=0^\circ$ to $\beta_\mathrm{max}=180^\circ$, sampled every $5^\circ$. A flat prior probability is also used for the rotational phase: $$p(\varphi|M_1) = \frac{1}{\varphi_\mathrm{max}-\varphi_\mathrm{min}},$$ with $\varphi_\mathrm{min}=0^\circ$ and $\varphi_\mathrm{max}=360^\circ$, sampled every $5^\circ$. The likelihood function of a given data set $D_n$ corrupted with Gaussian noise is given by: $$\begin{aligned} p(D_n|\mathcal{B}, \varphi, M_1) = (2\pi)^{-\frac{M}{2}}\left[\prod_{k=1}^M\sigma_k^{-1}\right] \nonumber \\ \exp\left\{-\frac{1}{2}\sum_{k=1}^M\frac{(d_k-f_k)^2}{\sigma_k^2}\right\},\label{eq|LHodds}\end{aligned}$$ where $d_k$ represents one of the $M$ datapoints with its variance $\sigma_k$, and $f_k$ represents the model prediction. When performing a parameter estimation, assuming the veracity of a model, it is a good practice to treat the variance of our data as a model parameter itself, using our estimate of the variance (i.e. the error bars) as a starting point. Proceeding in this way, the final results will be less sensitive to our estimation of the variance, as noise propagation can be problematic in heavily processed data, as can be the case for LSD profiles. Furthermore, this approach treats any features that cannot be explained by this particular model (for example higher order polar field components or distortions of the profile due to abundance spots) as additional noise. According to the maximum entropy principle, a Gaussian distribution will be the most noncommittal about information we do not have, leading to the most conservative estimates of the parameters. If our estimate of the variance of each point is denoted by $s_k$, we introduce a “noise scaling” parameter $b$ [@2005blda.book.....G], in order to estimate the total noise variance $\sigma_k$: $$\label{eq|prior_b} \sigma^2_k=\frac{s^2_k}{b},$$ where $b$ varies around unity. The resulting expression for the likelihood is: $$\begin{aligned} p(D_n|\mathcal{B}, \varphi, b, M_1)=(2\pi)^{-M/2}(b)^{M/2}\left[\prod_{k=1}^Ms_k^{-1}\right] \nonumber \\ \exp\left\lbrace-\frac{b}{2}\sum_{k=1}^M\frac{(d_k-f_k)^2}{s_k^2}\right\rbrace.\label{eq|LHpar}\end{aligned}$$ The resulting posterior probability density will therefore be marginalised over $b$. This procedure is only used for the parameter estimation of model $M_1$. The noise scaling parameter is a scale parameter, and a Jeffreys prior will be used: $$p(b|M_1) = \frac{1}{b\ln \left( \frac{b_\mathrm{max}}{b_\mathrm{min}}\right) }.$$ The noise scaling parameter for the parameter estimation varies from $b_\mathrm{min}=0.1$ to $b_\mathrm{max}=2$, corresponding to an underestimation of the variance by a factor of 10 and an overestimation of the variance by a factor of 2, respectively. This conservative range can be verified a posteriori, by the probability density marginalised for $b$. Numerical tests {#sec|test} =============== In order to demonstrate our Bayesian method, we have simulated some realistic synthetic observations, and analysed them using our procedure. Ideal non detection ------------------- We start with a simple test representing an ideal non-detection. The local intensity line profiles are represented by Gaussians, with the width of the point spread function of an instrument with spectral resolution of 4.2kms$^{-1}$ (R=65000), typical of current high-resolution spectropolarimeters like ESPaDOnS, and an extra turbulent broadening of 5kms$^{-1}$, typical of the broadening encountered in hot stars. We used a shallow line with a depth of 0.17$I_\mathrm{c}$. The rotational velocity field is set to $v\sin i=54$kms$^{-1}$. The associated circular polarisation is set to zero (i.e. no magnetic field) and the error bars are set to represent a certain s/n. We used two simulated observations corresponding to a s/n of 12000 and 24000 respectively, values that can typically be achieved for the LSD profiles of hot stars. The effective Landé factor and the wavelength of the spectral line are set to 1.2 and 5000Å respectively. We performed the Bayesian analysis on a grid extending up to 1kG, with a maximum rotation velocity of 700kms$^{-1}$, leading to a minimum inclination of $8^\circ$. Figure \[fig|mar\_tri\_nodetect1\] shows the probability densities resulting from our Bayesian analysis, for the s/n of 12000. The three bottom panels show the posterior probability densities marginalised for each parameter – $B_p$, $i$ and $\beta$, from left to right respectively. The densities have been normalised by their maximum, in order to facilitate the display. The three top panels show the 2D posterior probability densities, marginalised for the $B_p$-$\beta$, $B_p$-$i$ and $i$-$\beta$ planes. The contours encircle the regions that contain 68.3%, 95.4%, 99.0% and 99.7% of the probability. ![\[fig|mar\_tri\_nodetect1\] Parameter estimation for the ideal non-detection (low s/n). *Bottom:* Posterior probability density functions (PDF) marginalised for the dipole field strength, the rotational axis inclination and the magnetic obliquity, from left to right respectively. The PDFs have been normalised by their maximum. *Top:* 2-D posterior probability density marginalised for the $B_p$-$i$, $B_p$-$\beta$ and $i$-$\beta$ planes. The 68.3%, 95.4%, 99.0% and 99.7% credible regions are shaded in dark to pale colours respectively. ](fig4.pdf){width="84mm"} As it can be seen from the $B_p$ probability density distribution, the bulk of the probability is concentrated at low $B_p$ values. The amplitude of the Stokes $V$ profiles produced for low $B_p$ values will remain under the noise level, for all possible dipole orientations. As $B_p$ increases, the amplitude of the Stokes $V$ profiles will progressively grow over the noise level, and only a restricted number of possible orientations will produce Stokes $V$ profiles with amplitudes below the noise level, as explained in section \[sec|disk\]. The probability density marginalised for $B_p$ therefore has a decreasing exponential-like shape, with a tail extending far away from the mode of the distribution. The shape of the 2D $B_p$-$i$ and $B_p$-$\beta$ planes gives us insight about the geometries that are more likely to produce a non-detection for a large field strength value. When we look at a star rotational equator-on ($i=90^\circ$), the obliquity required to obtain a low amplitude Stokes $V$ profile is around $0^\circ$ or $180^\circ$. Thus, as the field is nearly aligned with the rotational axis, the amplitude of the Stokes $V$ profiles will stay small for the whole phase range. For lower inclination angles, the Stokes $V$ signal will vanish only when the dipole field is seen nearly magnetic equator-on, therefore such an null observation is only possible on a restricted interval of rotation phases (as it was shown in Figure \[fig|model\_nodetect1\_cont2\]). Consequently, if we assume that a strong field is present but is not producing any Stokes $V$ signal, it is more likely that we are observing a dipole configuration for which the Stokes $V$ profiles always have a small amplitude, rather than a dipole configuration for which the Stokes $V$ signal vanishes for only a small fraction of the rotational period. This explains why the probability density is more extended toward higher $B_p$ values for $i=90^\circ$ and $\beta=0^\circ, 180^\circ$. The 2D $i$-$\beta$ plane shows that we have no constraint on the inclination and obliquity angles. The shape of the probability density marginalised for $i$ is in part due to the extra probability at higher $B_p$ for $i\sim90^\circ$, but is mainly driven by the prior probability that favours large inclinations (as given in Eq. (\[eq|prior\_i\])). Following the same reasoning, the aligned dipoles ($\beta\sim0^\circ$ or $180^\circ$) are slightly favoured. Table \[tab|nodetect1\] compiles the base ten logarithm of the odds ratio computed for each observation taken individually, and for the two observations combined together. In order to illustrate the effect of the priors, we also computed the odds ratios while using a flat prior for all the parameters. For a single low s/n observation, the model $M_0$ for the absence of a magnetic field is preferred by a factor of 2. When doubling the s/n, the odds in favour of $M_0$ only increase to a factor 2.8. When the two observations are combined together, the odds ratio goes up to a factor of 3. ------------ ----------------- ----------------- ----------------- $\log(M_0/M_1)$ $\log(M_0/M_1)$ $\log(M_0/M_1)$ low s/n high s/n joint obs. (2) (3) (4) Flat prior 0.976 1.267 1.355 Used prior 0.317 0.444 0.475 ------------ ----------------- ----------------- ----------------- : \[tab|nodetect1\]Base ten logarithm of the odds ratio $\log(M_0/M_1)$ for the ideal non-detections. If we had used a flat prior for all parameters, the odds ratios would have been in favour of the model $M_0$ by a factor of roughly 10 and 20, for the low and high s/n respectively. Jeffreys priors are used when we ignore the exact scale of a parameter. This serves to balance the high scales and the lower scales. In a non-detection case, the bulk of the probability is situated at low $B_p$ values. With a Jeffreys prior, the small scales have as much weight as the larger scales, hence the configurations at high $B_p$ values that produce bad fits dominate less the global likelihood. The flat prior effectively gives more weight to the larger scales, hence rejecting more strongly the $M_1$ model. We note here that the $M_0$ model is in fact a subset of the $M_1$ model, given that the field strength range extends down to 0G. This means that both models can reproduce this simulated dataset perfectly. However, the $M_1$ model is penalised by its complexity, which is mathematically encoded by the priors. Therefore, for a non-detection, the odds ratios will never be very strongly in favour of the $M_0$ model by a large number, as the discrimination comes mainly from “Occam’s razor”. Furthermore, the posterior probability density is sensitive to the exact choice of the prior in this case. Given the decreasing exponential behaviour of the probability density, we think that the more conservative Jeffreys prior is more suitable, as it is less eager to reject the $M_1$ model. Assuming that the dipole model is true, we can perform a parameter estimation to determine which dipole strengths would be admissible by our observations. This can be achieved using the marginalised probability density for $B_p$. Integrating this probability density between two $B_p$ values gives the probability that the true value of the field strength lies between these two values. The probability density itself can be used in many applications, such as building a statistical field strength distribution for a stellar population. In other applications, such as the study of magnetically confined wind shocks, it would be valuable to get an upper field strength limit. When a probability density has a Gaussian-like shape, it is customary to express our confidence intervals by regions enclosing a certain percentage of the total posterior probability, namely 68.3%, 95.4% and 99.7%. With a Gaussian distribution, the 95.4% region will be twice as extended as the 63.8% region, and the 99.7% will be 3 times as extended as the 68.3% region, analogous to the 1, 2, 3 $\sigma$ contours in frequentist statistics. However, as our probability distribution does not have a Gaussian shape, but is rather shaped like a decreasing exponential, these credible regions will not be regular (i.e. the 99.7% credible region will reach much farther than 3 times the 68.3% credible region, taking into account the extended tail of the distribution). We therefore added a credible region enclosing 99.0% of the probability. ----------------------- ------- ------- ------- ------- 68.3% 95.4% 99.0% 99.7% (G) (G) (G) (G) (2) (3) (4) (5) low s/n 30 112 258 456 high s/n 18 66 157 297 Joint 16 51 110 196 Joint, flat prior 29 138 398 734 Joint, no scale noise 20 67 142 251 ----------------------- ------- ------- ------- ------- : \[tab|nodetect1\_region\] Credible region upper limits for the ideal non-detection. Table \[tab|nodetect1\_region\] gives the upper limits of the credible regions (the lower limit being 0G in each case) for each percentage threshold of the probability. For example, the probability that the field strength is lower than 258G is 99.0% for the low s/n case. For the higher s/n case, all the credible regions are narrower. When we combine the two observations together, the 68.3% and 95.4% regions are similar to the high s/n case, but the probability density is more peaked (i.e. it has a less extended tail toward the high $B_p$ values), as can be seen from the narrower 99.0% and 99.7% credible regions (110G and 196G respectively). When combining two non-detection observations, the probability density of high $B_p$ values narrows around $i=90^\circ$ and $\beta=0^\circ$. The likelihood of the $i=90^\circ$, $\beta=0^\circ$ configurations stays the same, as none of the phases produce any Stokes $V$ signal. However, the likelihood of the other degenerate configurations decreases, as it becomes less likely that we have observed the star both times during the same narrow range of phases that do not produce a Stokes $V$ signal, if the observations were taken at random times. Therefore, the high-$B_p$ tail becomes less important as we combine multiple observations. Table \[tab|nodetect1\_region\] also gives the credible regions for the combined observations, while considering a flat prior for each parameter. In that case, the shape of the probability density is more weighted toward the high scales, and the credible regions are more extended toward the high $B_p$ values. We also give the credible regions when considering a fixed variance of our data. However, in this particular case, as the observations can be reproduced perfectly by both models (as we set Stokes $V$ strictly to zero), the inclusion of the noise scaling parameter does not change the probability density much, except for a slight tightening of the credible regions as we allowed for the possibility of variance overestimation. Ideal detections ---------------- In order to show the behaviour of the algorithm in the presence of a magnetic signal, we now add non-zero Stokes $V$ profiles to the simulated observation corresponding to a s/n of 12000. Once again, no random noise was added, to illustrate the ideal case. To draw a connection between the credible regions found in the preceding section, we chose field values corresponding roughly to the upper limits of these credible regions: 30G, 100G, 250G and 450G. The magnetic configuration was set to $i=90^\circ$ and $\beta=90^\circ$, at the rotational phase when we are looking straight at the magnetic pole ($\varphi=0^\circ$). This corresponds to a profile with a shape like the leftmost profile of Figure \[fig|model\_nodetect1\_cont0\]. Table \[tab|nodetect4\] gives the odds ratios for the used priors, as well as the ones computed with flat priors for comparison. We also give the credible regions containing 68.3%, 95.4%, 99.0% and 99.7% of the total probability. The lower limit is 0G, unless indicated otherwise. [D[(]{}[ (]{}[5,7]{} c c c c c c]{} & $\log(M_0/M_1)$ & $\log(M_0/M_1)$ & 68.3% & 95.4% & 99.0% & 99.7%\ & Used prior & Flat prior & (G) & (G) & (G) & (G)\ & (2) & (3) & (4) & (5) & (6) & (7)\ 30 (\~68.3%) & 0.303 & 0.939 & 34 & 128 & 290 & 498\ 100 (\~95.4%) & 0.0970 & 0.476 &123 & 338 & 625 & 825\ 250 (\~99.0%) & -2.98 & -3.22 &196 - 407 & 155 - 763 & 143 - 938 & 141 - 991\ 450 (\~99.7%) & -13.4 & -13.9 &410 - 658 & 384 - 928 & 379 - 995 & 363 - 1000\ The posterior probability density for the 30G observation is indistinguishable from the $V=0$ observation from the previous section. The odds ratio is still marginally in favour of the $M_0$ model, and the credible regions for $B_p$ are similar. At 100G, the odds ratio is still in favour of $M_0$, although both models are now nearly as likely ($\log(M_0/M_1=0.097)$). The odds ratio computed with a flat prior still rejects the $M_1$ model by a factor of three. The shape of the probability density, as shown in Figure \[fig|mar\_tri2\_nodetect4\], is also different than the non-detection case, with a secondary peak in the probability density marginalised for $B_p$, although the probability density still peaks at 0G. The credible regions for $B_p$ are therefore extended toward higher values, and the 68.3% credible region upper limit (123G) encompasses the real field value. As it was the case for the ideal non-detection, an aligned magnetic configuration is preferred, as it maximises the chances of observing this particular profile shape. The $i$-$\beta$ 2D plane is different from that of the ideal non-detection. The shape of the magnetic signal adds an extra constraint, by rejecting the configurations for which the positive pole is never located on the visible hemisphere. ![\[fig|mar\_tri2\_nodetect4\] Same as Figure \[fig|mar\_tri\_nodetect1\] for an ideal detection with a field strength of 100G.](fig5.pdf){width="84mm"} At 250G, the odds ratio has switched in favour of the dipole model, by three orders of magnitude ($\log(M_0/M_1)=-2.98$). However, our chosen prior is again more conservative than the flat prior, which favours $M_1$ more strongly ($\log(M_0/M_1)=-3.22$). As a flat prior overweights the larger scales compared to a Jeffreys prior, our used prior needs a more significant likelihood at large $B_p$ values in order to favour the magnetic model. The posterior probability density is now typical of a detectable magnetic field (Figure \[fig|mar\_tri3\_nodetect4\]). A sharp cut in the probability density marginalised for $B_p$ and the 2D planes shows a tight constraint on the lower field limit. The $B_p$ distribution peaks around 275G and decreases slowly toward higher $B_p$ values. Therefore, the credible regions for $B_p$ are not symmetric with respect to the mode of the distribution. The lowest field strength values correspond to configurations for which the positive magnetic pole is located at the stellar disk center. Larger field values are possible when we are looking at the positive pole from an angle. For example, if the field is aligned ($\beta=0^\circ$), the amplitude of the Stokes $V$ profiles will decrease as the inclination decreases, until we reach $i\sim0^\circ$ where the magnetic signal vanishes. Because there is only one observation, the aligned configurations are preferred, as shown by the probability density marginalised for $\beta$. Obviously, when the inclination is $90^\circ$, an aligned configuration would not provide any Stokes $V$ signal and cannot reproduce the observed profile. Some obliquity would be necessary in order to put the positive pole on the visible hemisphere, which explains the dip at $i=90^\circ$ in the probability density marginalised for $i$ and marginalised for the $B_p$-$i$ plane. When $i\sim0^\circ$ and the field is aligned, we are always looking directly at the positive pole, which makes this configuration quite likely. However, our prior knowledge tells us that a low inclination is less likely than a high inclination. For this reason, the posterior probability density marginalised for $i$ decreases toward $i\sim0^\circ$ and $180^\circ$. At the present $\mathcal{B}$ configuration ($i=90^\circ$ and $\beta=90^\circ$) the particular profile shape (anti-symmetrical) only occurs during a certain phase range (see Figure \[fig|model\_nodetect1\_cont0\]), which makes such a configuration less likely than the aligned configurations. Therefore, with only one observation, no meaningful constraints can be put on the angles. However, the field strength can be better constrained, and the probability density marginalised for $B_p$ can provide statistical insight into the more likely field strength values. At 450G, the odds ratio favours $M_1$ by more than 13 orders of magnitude. When the $M_0$ model is so strongly rejected, the difference in odds ratios between our chosen prior and the flat prior is less important than when the signal detection was more marginal. This is because the bulk of the likelihood is located at higher $B_p$ values, where the two priors are similar. The posterior probability density is quite similar in shape to the one with a 250G field, with the $B_p$ distributions shifted to higher values, as it can be seen from the credible regions in Table \[tab|nodetect4\]. ![\[fig|mar\_tri3\_nodetect4\] Same as Figure \[fig|mar\_tri\_nodetect1\] for an ideal detection with a field strength of 250G.](fig6.pdf){width="84mm"} Realistic case {#sec|real} -------------- So far, we have explored the behaviour of the algorithm in cases where the models can reproduce the data perfectly. Obviously, real observations will have some deviations from the predicted values, introduced by the noise. We will now explore how the noise corruption affects our detection capacity. The top spectrum of Figure \[fig|profiles\_nodetect7\] shows a simulated observation, consisting of only random noise generated from a normal distribution corresponding to a s/n of 12000. Figure \[fig|mar\_tri4\_nodetect7\] presents the posterior probability density for that specific observation. Notice how the $B_p$ distribution looks like the probability density for the 100G observation of the previous section (Figure \[fig|mar\_tri3\_nodetect4\]) [^3]. In fact, the odds ratio is even in favour of the dipole model ($\log(M_0/M_1)=-0.419$), even though the signal is pure noise. This is because the observation is better fitted by a non-null magnetic field. The absolute best fit to the data is given by the maximum of the likelihood. We illustrate this fit by the red profile overplotted on the data (Figure \[fig|profiles\_nodetect7\]). Therefore, the detection of a real signal embedded in the noise is ambiguous, as the noise could have – by chance – the shape of a magnetic profile. An observer would likely not feel confident to report a magnetic detection based on only one observation like this one. ![\[fig|profiles\_nodetect7\] Realistic simulated observations of pure Gaussian noise. The intensity line profile is shown at the bottom, and the five noisy Stokes $V$ profiles are shown at the top. The dotted lines display the range of the fits. The no magnetic field model $M_0$ ($V=0$) is shown by the black lines. The best fits of the oblique dipole model $M_1$ for each observation taken individually (represented by the maximum of the individual likelihoods) are shown in red. The best fit for all the observations taken together by a single $\mathcal{B}$ geometry (the maximum of the joint likelihood) is shown in green. The corresponding reduced $\chi^2$s are given on the right side, with matching colours.](fig7.pdf){width="84mm"} ![\[fig|mar\_tri4\_nodetect7\] Same as Figure \[fig|mar\_tri\_nodetect1\] for Obs. 1 of the realistic simulated observations of pure noise. ](fig8.pdf){width="84mm"} In order to verify if this ambiguity can be lifted by re-observing the star, we generated 4 additional simulated observations, again with pure normal noise (rest of Figure \[fig|profiles\_nodetect7\]). Each of these observations leads to an odds ratio in favour of $M_0$ by about a factor of two (Table \[tab|nodetect6\], column 2). The best fit achievable for each observation taken individually is again overplotted in red. However, nothing restricts the magnetic configuration $\mathcal{B}$ to be the same for each observation. The best fit produced by a single $\mathcal{B}$ configuration is given by the maximum of the joint likelihood, illustrated in green. Although the data can be reproduced by a non null magnetic field, the odds ratio of the combined observations is in favour of the $M_0$ model ($\log(M_0/M_1)=0.295$). The slight improvement of the fit produced by the oblique dipole model (see the reduced $\chi^2$ indicated in Figure \[fig|profiles\_nodetect7\]), is not enough to justify the use of a more complex model. ------------- ----------------- ------------- ----------------- $\log(M_0/M_1)$ $\varphi_n$ $\log(M_0/M_1)$ Noise only $B_p=125$G (2) (3) (4) Obs. 1 -0.419 0.76 -4.04 Obs. 2 0.306 0.43 -0.412 Obs. 3 0.295 0.40 -0.484 Obs. 4 0.178 0.20 -2.26 Obs. 5 0.281 0.60 -0.388 Combination 0.295 -9.05 ------------- ----------------- ------------- ----------------- : \[tab|nodetect6\] Odds ratios ($\log(M_0/M_1)$) for the realistic simulated observations for the pure noise case and the $B_p=125$G case. The rotational phases, chosen randomly, are also given. The posterior probability density (Figure \[fig|mar\_tri\_nodetect7\]) is similar in shape to the perfect non-detection that was shown in Figure \[fig|mar\_tri\_nodetect1\]. The high-$B_p$ tail is less extended that in the case of a single observation. This can by seen more easily by comparing the 2D probability densities for the $B_p$-$\beta$ and $B_p$-$i$ planes of Figures \[fig|mar\_tri\_nodetect7\] and \[fig|mar\_tri\_nodetect1\]. As explained in the previous sections, combining multiple observations has this effect because, in the case of non-detections, the high-$B_p$ inclined dipole configurations become less likely, as this would mean that all the observations were taken during a narrow range of phase. Obviously, the possibility that the observations were taken at the same rotational phase can generally be assessed by the time span of the observations and the possible range of rotational periods. In the case where it is known that the rotational period is quite long compared to the observation time span, more information is available than is assumed in this algorithm (that the phase of each observation can assume any value). In that case, it would be wiser to combine all the observations together and treat them as a single observation in order to get a more meaningful probability density. ![\[fig|mar\_tri\_nodetect7\] Same as Figure \[fig|mar\_tri\_nodetect1\] for the combined realistic simulated observations of pure noise.](fig9.pdf){width="84mm"} Table \[tab|nodetect7\] compiles upper limits to the 68.3%, 95.4%, 99.0% and 99.7% credible regions, extracted from the posterior probability density marginalised for $B_p$. Although we combined five observations, the 68.3% credible region for $B_p$ extends to 38G, which is higher than the value obtained for one perfect observation of low s/n (30G), because of the deviations introduced by the noise. However, as mentioned above, the $B_p$ distribution of the combined observations does not extend as far than as that from a single perfect observation, as illustrated by the 99.7% credible region that extends only to 370G, compared to 456G for the perfect observation. We also compiled the credible regions we would obtain without the use of the noise scaling parameter. As expected, the deviations from the model are consistent with our estimation of the variance (the error bars) and the credible regions are nearly the same in both cases. ----------------------- ------- ------- ------- ------- 68.3% 95.4% 99.0% 99.7% (G) (G) (G) (G) (2) (3) (4) (5) With noise scaling 38 108 214 370 Without noise scaling 38 107 213 368 ----------------------- ------- ------- ------- ------- : \[tab|nodetect7\] Credible region upper limits for the combined realistic simulated observations of pure noise. We have therefore demonstrated that a suspicious signal that has a shape similar to that of a magnetic signal can be verified by the acquisition of a small number of additional observations. We now demonstrate that the same is true for a real signal that is sufficiently embedded in the noise to render the odds ratio of a single observation ambiguous. Given that in the ideal detection case we were able to detect a field with a strength near the 95.4% upper limit of the ideal non-detection, we chose a field strength close to the upper limit of the 95.4% credible region of the previous example. The $\mathcal{B}$ configuration is given by $B_p=125$G, $i=90^\circ$ and $\beta=90^\circ$. Five phases were randomly generated ($\varphi= 0.76,\, 0.43,\, 0.40,\, 0.20,\, 0.60$), and the corresponding Stokes $V$ profiles were added to the previous simulated dataset of pure noise. The resulting simulated observations are shown in Figure \[fig|profiles\_nodetect6\]. The underlying real Stokes $V$ profile is shown as the black dashed curve for each observation. ![\[fig|profiles\_nodetect6\] Realistic simulated observations of noise with a Stokes $V$ signal corresponding to $B_p=125$G ($i=\beta=90^\circ$). The intensity line profile is shown at the bottom, and the five noisy Stokes $V$ profiles are shown at the top. The dotted lines display the range of the fits. The underlying real Stokes $V$ signals are shown with black dashed curves. The null magnetic field model $M_0$ ($V=0$) is shown by the black lines. The best fit by a single $\mathcal{B}$ geometry for all the observations taken together (the maximum of the joint likelihood) is shown in green. The fit produced by the MAP $\mathcal{B}$ parameters is shown in blue, and that produced by the modes of the posterior probability density marginalised for each parameter is shown in magenta. The corresponding reduced $\chi^2$s are given on the right side, with matching colours. ](fig10.pdf){width="84mm"} Table \[tab|nodetect6\] gives the odd ratio for each observation (column 4), all of which favour the dipole model $M_1$, although observations 2, 3 and 5 less strongly than observations 1 and 4. In fact, the odds ratios of the formers are similar to the odds ratio of the first observation of the pure noise test and on their own, each of these observations would result in an ambiguous signal detection. However, combining all the observations together, the odds ratio is now strongly in favour of the dipole model by 9 orders of magnitude ($\log(M_0/M_1)=-9.05$). Figure \[fig|mar\_tri\_nodetect6\] shows the posterior probability densities for the combined observations. There is a sharp lower limit on the possible magnetic strengths, illustrated by the probability density marginalised for $B_p$, and for the 2D $B_p$-$i$ and $B_p$-$\beta$ planes. Given that most of the likelihood is situated at non-null field strengths, the improvement of the fit to the data justifies the more complex model, as shown by the reduced $\chi^2$ on Figure \[fig|profiles\_nodetect6\] for the $M_0$ model fits (black lines) and the fit produced by maximum of the joint likelihood for $M_1$ (green curves). ![\[fig|mar\_tri\_nodetect6\] Same as Figure \[fig|mar\_tri\_nodetect1\] for the combined realistic simulated observations of noise plus a signal corresponding to $B_p=125$G.](fig11.pdf){width="84mm"} -------------------- ----- ------ -------- ----------- ----------- ----------- ----------- MAP Mode Median [68.3%]{} [95.4%]{} [99.0%]{} [99.7%]{} (2) (3) (4) (5) [(6)]{} [(7)]{} [(8)]{} $B_p$ (G) 210 195 220 155 - 261 118 - 508 109 - 793 102 - 892 $i$ ($^\circ$) 61 85 71 62 - 118 38 - 142 27 - 152 21 - 158 119 95 109 $\beta$ ($^\circ$) 70 90 66 54 - 125 23 - 157 14 - 165 12 - 168 110 113 -------------------- ----- ------ -------- ----------- ----------- ----------- ----------- Different choices are possible in order to express the derived values for the model parameters. One can choose for example the maximum of the joint posterior probability density (MAP), or the mode of the marginalised probability density for each parameter. Note however that if the probability distribution is complex, the parameters given by the MAP do not necessarily correspond to the mode of the marginalised probability densities. Usually, the MAP will produce the best fit to the data given that the a-priori information does not exclude any interesting parts of the parameter space, but does not necessarily represent the bulk of the probability. The mode of each parameter represents well the bulk of the probability, but does not necessarily give an excellent fit to the data. Using the median of the marginalised probability densities is usually a good compromise [@2005blda.book.....G]. In Table \[tab|nodetect6\_region\], we list the MAP, the mode and the median for each $\mathcal{B}$ parameter, which in this case are quite similar. The MAP, the modes and the medians all yield similarly good fits to the data (the MAP and the modes fits are shown in Figure \[fig|profiles\_nodetect6\] in blue and magenta respectively). The range of the credible regions of the probability density marginalised for each parameter are compiled in Table \[tab|nodetect6\_region\]. Using the median with the 68.3% credible region, we would infer a dipole strength $B_p=220\stackrel{+41}{_{-65}}$G, which is slightly higher than the input dipole field strength of 125G. We verified that this difference was due to the particular noise pattern used in the data, as additional noise simulations allowed us to recover the input value within the 68.3% region. The real values of the $i$ and $\beta$ angles are recovered by the 68.3% credible regions, although the constraints are poor, as expected. Comparison with traditional diagnostics --------------------------------------- With low-resolution instruments (such as FORS1 and FORS2 at the Very Large Telescope), one is generally only sensitive to the global longitudinal component of the magnetic field, as the rotationally-broadened spectral lines are not resolved. This global longitudinal field value is extracted from the spectrum using the “slope” method as described in . The presence of a magnetic field in a single observation is therefore diagnosed by the significance of the global longitudinal field measurement compared to its error bar. In the case of high-resolution spectropolarimetry (e.g. ESPaDOnS at Canada-France-Hawaii Telescope, Narval at Télescope Bernard-Lyot or HARPSPol at ESO La Silla 3.6m Telescope), the rotationally-broadened spectral lines are resolved and the field is generally diagnosed by the deviation of the circular polarisation profile with respect to $V=0$. The detection can be quantified by the probability that such a deviation from $V=0$ is produced by random noise (the false alarm probability or FAP). The detection probability $P$ can then be expressed as $P=1-FAP$. Following @1997MNRAS.291..658D, a field is generally considered detected if the FAP is less than $10^{-5}$ ($P>99.999\%$), and marginally detected when FAP is less than $10^{-3}$ ($P>99.9\%$). With high-resolution spectropolarimetry, it is also possible to integrate the signal over the line profile in order to recover a value equivalent to the global longitudinal field obtained from low-resolution instruments [@1997MNRAS.291..658D; @2000MNRAS.313..851W]. Although it is possible to detect a magnetic signal even when the global longitudinal field is null, this is a useful quantity for producing longitudinal field curves and for comparing with low-resolution data. [l D[\*]{}[3,3]{} c c ]{} & & $|B_l/\sigma|$ & $P(V)$\ & & & (%)\ & & (3) & (4)\ \  30G & 11\*33 & 0.34 & 0\ 100G & 37\*33 & 1.12 & 0\ 250G & 93\*33 & 2.76 & 52.0\ 450G & 168\*33 & 5.03 & 99.99997\ \ Obs. 1 & 72\*33 & 2.16 & 60.8\ Obs. 2 & 12\*33 & 0.38 & 64.1\ Obs. 3 & 2\*33 & 0.08 & 16.7\ Obs. 4 & -16\*33 & 0.48 & 87.9\ Obs. 5 & 10\*33 & 0.32 & 13.6\ \ Obs. 1 & 74\*33 & 2.22 & 99.5\ Obs. 2 & -28\*33 & 0.87 &91.9\ Obs. 3 & -35\*33 & 1.07 & 64.8\ Obs. 4 & 0\*33 & 0.01 & 99.8\ Obs. 5 & -27\*33 & 0.82 & 54.7\ We applied these traditional diagnostics to our simulated datasets and we report the results in Table \[tab\_bl\]. In the case of the ideal detections, both the longitudinal field (columns 2 and 3) and the detection probability (column 4) yield a detection when the field strength reaches 450G. However, the odds ratios are already in favour of the magnetic model at 250G. The detection probability only looks at the total deviation, whereas the odds ratio looks for a shape similar to that of the theoretical model. For the realistic case consisting only of noise, all five observations have longitudinal field measurements below $3\sigma$ significance (see column 3). Moreover, no signal is detected in the Stokes $V$ profiles, as shown by the detection probabilities. The same is also true for the realistic case consisting of noise plus a signal for $B_p=125$G, although the detection probability nearly reaches a marginal detection for Obs. 1 and 4. For these observations, the odds ratios were in favour of the magnetic model by 2 and 4 orders of magnitude respectively. For the remaining observations, the odds ratios were also in favour of the magnetic model, although at a lower significance. Therefore, if we had observed any of the simulated observations with $B_p=125\,G$, the traditional diagnostics would not have diagnosed the presence of a field, but the odds ratio would have indicated the possible magnetic signal. A few additional observations are enough to distinguish between real noise and a buried signal consistent with an oblique dipole field, as demonstrated by the example here. Therefore, Bayesian odds ratios provide a quantitative indication of stars worth re-observing in magnetic surveys. Application to real data {#sec|lpori} ======================== In order to present the application of this method to real data, we will use high-resolution spectropolarimetric observations of the magnetic B-type star LPOri (=Par1772, HD36982). LPOri is often considered a chemically peculiar star because it was first classified as a B1.5p star by @1952ApJ...116..251S. LPOri’s status as a He-strong or He-weak star is still uncertain. An inspection of our seven spectra (described below) does not reveal any significant variation in the He-line strength that would indicate a He-strong or He-weak star. Furthermore, comparing our spectra with the BSTAR grid of synthetic spectra from non-LTE <span style="font-variant:small-caps;">tlusty</span> models [@2007ApJS..169...83L], we find a reasonable agreement with a temperature of 20kK, a $\log g$ of 4.0 and a $v\sin i$ of 80kms$^{-1}$. LPOri is also a candidate Herbig Ae/Be star, because of its far-infrared excess, although no emission is present in the visible spectrum. @2002MNRAS.334..419M suggested that LPOri is an object transiting from the pre-main sequence to the main sequence. LPOri seems to be a single star. No radial velocity variation was found by @1991ApJ...367..155A nor was any speckle companion with a K-band ratio of less than 0.04 for a separation of 150mas or more [@1999NewA....4..531P]. The magnetic field of LPOri was first reported by @2008MNRAS.387L..23P. In that paper, they used three Stokes $V$ observations obtained with the ESPaDOnS and Narval spectropolarimeters to detect the magnetic field. ESPaDOnS and Narval are twin high-resolution spectropolarimetric instruments located at Canada-France-Hawaii Telescope and Télescope Bernard-Lyot respectively. A polarisation measurement consists of a set of 4 sub-exposures taken with different polarimeter configurations. From this measurement set, the circular polarisation Stokes $V$ spectrum is extracted, as well as a diagnostic null polarisation spectrum (labeled $N$) by combining the sub-exposures in such a way that the astronomical object’s polarisation should cancel out. ESPaDOnS frames were processed using the <span style="font-variant:small-caps;">Upena</span> pipeline provided by CFHT. The Narval frames were processed by the TBL archive pipeline. Both pipelines use the reduction package <span style="font-variant:small-caps;">Libre-ESPRIT</span> [@1997MNRAS.291..658D]. The spectral range of both instruments covers the 370nm to 1050nm wavelength band, with a resolution $R\sim65\,000$. @2008MNRAS.387L..23P used the Bayesian method described here to estimate the dipole strength of LPOri, obtaining $1150\stackrel{+320}{_{-200}}$G. For the present analysis, we have obtained one additional ESPaDOnS observation (within the context of the MiMeS CFHT Large Program) and an additional 3 Narval observations. The observation log, which includes the new observations as well as those analysed by Petit et al. (2008), is given in Table \[tab|lplog\][^4]. The sample of observations has a large range of s/n, and is therefore suitable for testing the behaviour of the Bayesian method on real data. [l l c c c D[\*]{}[4,4]{} c c D[\*]{}[4,4]{} c c ]{} & & HJD & t$_\mathrm{exp}$ & s/n & & $|B_l/\sigma|$ $V$& P $V$ & & $|B_l/\sigma|$ $N$ & P $N$\ & & (+2450000) & (s) & & & &(%) & & &(%)\ &&(3)&(4)&(5)&&(7)&(8)&&(9)&(11)\ 2007-11-08 & Nar. & 4413.54984 & 4000 & 73 & 300\*434 &0.69 & 60.0 & 441\*436 & 1.01 &31.9\ 2007-11-09 & Nar. & 4414.54299 & 6000 & 170 & 93\*174 &0.53 & 99.8 & -293\*173 & 1.69 &66.2\ 2007-11-10 & Nar. & 4415.50631 & 8800 & 232 & 199\*120 &1.66 & 87.1 & -3\*120 & 0.03 &96.7\ 2007-11-11 & Nar. & 4416.54958 & 6000 & 244 & 354\*113 &3.13 & **99.998** & -18\*112 & 0.17 &9.4\ 2006-01-11 & ESP. & 3747.79588 &9600 & 312 & 314\*91 & 3.45 & **100** & -244\*91 & 2.67 & 64.0\ 2007-03-06 & ESP. & 4166.76514 & 9600 & 510 & 98\*50 & 1.93 & **100** & -1\*51 & 0.02 &66.4\ 2010-02-23 & ESP. & 5250.77084 & 6400 & 426 & 347\*61 &5.62 & **100** & 97\*62 & 1.57 &97.7\ As is customary, we applied the LSD procedure to our observations. We used the i<span style="font-variant:small-caps;">LSD</span> code described by . We chose spectral lines from a Vienna Atomic Line Database [VALD; @2000BaltA...9..590K] list corresponding to an atmosphere model with $T=20$kK and $\log g=4.0$. From that list, we chose metallic lines and weak He lines that were unblended with strong Balmer lines and uncontaminated by telluric lines or strong nebular emission. The line depths have been slightly adjusted to match the spectra. The final line list is shown in Table \[tab|mask\]. The intensity ($d$) and polarisation ($dg\lambda$) weights of each line were normalised by 0.2 and 120 respectively. Therefore, the displayed $y$-axis scale of the resulting Stokes $V$ and diagnostic null LSD profiles (Figures \[fig|lpV\] and \[fig|lpN\]) corresponds to a line with a $d=0.2$ and $g\lambda=600$. ![\[fig|lpV\] *Bottom:* LSD line profiles of LPOri (black). The synthetic line profile used for the Bayesian analysis is shown in red. The dashed vertical lines display the range of the fits. *Top:* LSD Stokes $V$ profiles of LPOri. The grey curves correspond to the $M_0$ model ($V=0$). The red curves correspond to the best fit for each observation taken individually (represented by the maximum of the individual likelihood). The green curves represent the best fit for all the observations taken together by a single $\mathcal{B}$ geometry (the maximum of the joint likelihood). The blue curves correspond to the maximum of the posterior probability density (MAP). The corresponding reduced $\chi^2$s are given on the right side, with corresponding colours. ](fig12.pdf){width="84mm"} ![\[fig|lpN\] Same as Figure \[fig|lpV\] for LPOri null $N$ profiles. ](fig13.pdf){width="84mm"} The two sharp lines in the blue continuum of the Stokes $I$ LSD profiles are residuals from telluric lines adjacent to the He<span style="font-variant:small-caps;">I</span>$\lambda7281$ line. The emission bump present in the LSD profiles of the two first Narval observations are residuals from He emission, most likely of nebular origin because their centroid velocities and scaling match the nebular Balmer line emission. We computed the traditional diagnostics – the detection probability of the individual profiles and the global longitudinal field – by integrating $\pm115$kms$^{-1}$ around the line centre, for both the Stokes $V$ and $N$ profiles. The results are displayed in Table \[tab|lplog\]. Definite detections (column 8) are achieved for the three ESPaDOnS observations. Marginal detection is achieved only for one Narval observation. No signal is detected in the null profiles (column 11). The global longitudinal fields for the Stokes $V$ LSD profiles (column 6) show a positive trend, hinting that the positive magnetic pole is located somewhere on the visible hemisphere. Although the longitudinal field approaches zero, the magnetic signal is still detectable given a sufficient s/n (for example 2007-03-06). This hints that we are looking nearly at the magnetic equator at some phases. Assuming a reasonable range of possible rotation periods, defined by the $v\sin i$ and breakup velocity, the longitudinal field curve and the LSD profiles can be phased with various periods. Figure \[fig|blsin\] shows the longitudinal field measurements (black dots), along with sinusoidal curves for three possible periods. As no other variability has been observed by other means (spectral or photometric), this is a good example of a case where the rotational phases of the observations are not known. ![\[fig|blsin\] Longitudinal field curve for LPOri (black dots) to which we have superposed sinusoidal curves for three of the possible periods. ](fig14.pdf){width="84mm"} Given the range of global longitudinal field measurements, we extended the grid up to $B_p=3$kG. We set the $a$ parameter of the Jeffreys prior to 25G, corresponding to two times the parameter grid step. We performed the analysis on both the Stokes $V$ and null profiles. The fit of the Stokes $I$ profiles is shown in Figure \[fig|lpV\]. Vertical dotted lines represent the velocity range of the fits. The odds ratios for each individual observation are given in Table \[tab|lp\_odds\]. The odds ratios for the null profiles (column 3) all favour the absence of a magnetic field ($M_0$ model). The odds ratios for the Stokes $V$ observations all favour the oblique dipole model ($M_1$), but vary from more than 10 orders of magnitude for the high s/n observations to less than one order of magnitude for the low s/n observations. As mentioned for realistic numerical tests (Section \[sec|real\]), it is possible to obtain an odds ratio in favour of the dipole model when the noise happens to have a shape similar to a magnetic signal. When combining all the observations together, we get an odds ratio strongly in favour of the magnetic model, by 73 orders of magnitude, as expected given the definite signal detections in the ESPaDOnS observations. However, if only the low s/n observations were available (2007-11-08, -09 and -10), the situation would be more ambiguous. None of these observations lead to a traditional definite detection when considered on their own. We therefore combined and analysed these three observations in pairs, and then all together. The three possible pairs of observations led to odds ratios in favour of $M_1$ by more than one order of magnitude, and the combination of the three observations lead to an odds ratio $\log(M_0/M_1)=-3.3$, i.e. more than 3 orders of magnitudes. Therefore, the Bayesian algorithm is able to recover the magnetic signal that would have been buried in the noise and undetected by traditional diagnostics. ------------------ ----------------- ----------------- $\log(M_0/M_1)$ $\log(M_0/M_1)$ $V$ $N$ (2) (3) Nar. 2007-11-08 -0.32 0.15 Nar. 2007-11-09 -1.7 0.18 Nar. 2007-11-10 -0.86 0.15 Nar. 2007-11-11 -11.5 0.32 ESP. 2006-01-11 -13.5 0.26 ESP. 2007-03-06 -24.8 0.42 ESP. 2010-02-23 -16.9 0.43 Combination -73.3 0.52 2007-11 08+09 -2.3 0.20 2007-11 08+10 -1.3 0.17 2007-11 09+10 -2.7 0.16 2007-11 08+09+10 -3.3 0.17 ------------------ ----------------- ----------------- : \[tab|lp\_odds\] Stokes $V$ and null profiles odds ratio for LPOri observations taken individually, combined together, and for various combinations of the low s/n observations. In Figures \[fig|lpV\] and \[fig|lpN\], the grey dashed lines represent the $M_0$ model ($V=0$). The best fits achievable by a dipole model for each observation taken individually (maximum of the individual likelihoods) are shown in red. The best fit produced by a single $\mathcal{B}$ oblique dipole (maximum of the joint likelihood) is shown in green. The reduced $\chi^2$s are indicated with corresponding colours. For the Stokes $V$ observations where odds ratios strongly favour the $M_1$ model, the reduced $\chi^2$ is much improved with the addition of the more complex model compared to the $\chi^2$ of the $M_0$ model. Not only are the data reproduced by the dipolar profiles, they can all be simultaneously reproduced by a single $\mathcal{B}$ configuration, as shown by the similar $\chi^2$ for the individual fit (red) and the maximum of the joint likelihood (green). However, the reduced $\chi^2$ remains high in one case (2010-02-23; $\chi^2_\mathrm{red}$=1.95), meaning that there is some extra variance in the observation that neither models are able to reproduce (a 3$\sigma$ deviation would correspond to a reduced $\chi^2$ of 1.65). The reduced $\chi^2$s are low for all the null profile observations (Figure \[fig|lpN\]). This points toward a systematic deviation or a model-based effect for the Stokes $V$ observation of Feb. 2010 rather than an extra instrumental scatter or underestimated error bars. When performing parameter estimation, the extra scatter is addressed by the noise scaling term (Eq. \[eq|prior\_b\]). It is therefore possible to extract the probability density function marginalised for the noise scaling parameter, and determine if the model is able to reproduce the observations down to the noise level. Figure \[fig|scale\_noise\] shows that the Bayesian estimate of the variance is larger than the assumed variance (i.e. error bars) for both Stokes $V$ and the null profiles, but by less than a factor of two. ![\[fig|scale\_noise\] Posterior probability density marginalised for the noise scaling parameter for the Stokes $V$ (black) and null $N$ profiles (dotted red) of LPOri. ](fig15.pdf){width="84mm"} The posterior probability density for the Stokes $V$ observations is shown in Figure \[fig|lp\_postV\]. Dipole configurations with large inclination or obliquity are less favoured, as shown by the probability densities marginalised for $i$ and $\beta$, because the observations mainly show either the positive pole or the magnetic equator on the visible hemisphere. This is more likely to occur if the $i$ and $\beta$ angles are small and the negative pole spends little or no time on the visible hemisphere. The angle values are interrelated as shown by the $i$-$\beta$ 2D plane. If the inclination is small, the obliquity is more likely to be large, in order to display an equator-like magnetic signature. The probability density marginalised for the field strength shows a sharp lower limit. The high-$B_p$ tail of the distribution is attributable to the high-inclination (low obliquity) configurations, as shown in the 2D planes for $B_p$-$i$ and $B_p$-$\beta$. ![\[fig|lp\_postV\] Same as Figure \[fig|mar\_tri\_nodetect1\] for the combined Stokes $V$ observations of LPOri.](fig16.pdf){width="84mm"} Figure \[fig|lp\_postN\] shows the probability densities for the null profile observations, which are, as expected, similar to the realistic simulation of pure noise (as shown in Figure \[fig|mar\_tri\_nodetect7\]). We also show in Figure \[fig|lp\_postV345\] the probability densities we obtained when considering only the three low s/n observations. The constraints on the parameters, especially the angles, are worse than when we consider the full data set. Therefore, the constraints on the angles are mainly defined by the high s/n observations. ![\[fig|lp\_postN\] Same as Figure \[fig|mar\_tri\_nodetect1\] for the combined null $N$ observations of LPOri.](fig17.pdf){width="84mm"} ![\[fig|lp\_postV345\] Same as Figure \[fig|mar\_tri\_nodetect1\] for the combined low s/n Stokes $V$ observations of LPOri (2007-11-08, 2007-08-09 and 2007-11-10). ](fig18.pdf){width="84mm"} The credible regions of the probability density marginalised for $B_p$ are given in Table \[tab|lp\_cred\]. We also present the maximum of the joint posterior probability density (MAP), the mode of the probability density marginalised for $B_p$, as well as the median. For the Stokes $V$ observations, the MAP, mode and median values are all similar, therefore the fits to the data obtained for these values are nearly undistinguishable. The fit obtained with the MAP values is shown in Figure \[fig|lpV\] (blue curve). Using the 68.3% credible region and the median, the dipole field strength of LPOri is estimated to be $911\stackrel{+138}{_{-244}}$G. As was seen for the probability density in Figure \[fig|lp\_postV\], the lower limit on the field strength is sharp. The 68.3% credible region lower limit is 667G, and 557G for the 99.7% credible region. The distribution has a high-$B_p$ tail, and the 99.7% credible region extends up to 2.6kG. ------------------ ----- ------ -------- ------------ ------------ ------------ ------------ MAP Mode Median [68.3%]{} [95.4%]{} [99%]{} [99.7%]{} (G) (G) (G) (G) (G) (G) (G) (2) (3) (4) (5) [(6)]{} [(7)]{} [(8)]{} Combination 885 855 911 667 - 1049 590 - 1657 572 - 2348 557 - 2653 No noise scaling 930 930 967 762 - 1119 621 - 1659 660 - 2336 578 - 2672 Low s/n 555 585 633 386 - 825 0 - 1527 0 - 1141 0 - 2689 Combination 0 0 27 0 - 47 0 - 144 0 - 301 0 - 530 No noise scaling 0 0 27 0 - 46 0 - 142 0 - 296 0 - 524 Low s/n 0 0 86 0 - 163 0 - 502 0 - 1002 0 - 1624 ------------------ ----- ------ -------- ------------ ------------ ------------ ------------ The second row of Table \[tab|lp\_cred\] illustrates the effect of the noise scaling parameter on the inferred field values. When considering the variance as a model parameter, the MAP, mode and median are shifted to slightly lower $B_p$ values. The credible regions are somewhat larger, and also slightly shifted to lower values. The parameter estimation for the low s/n observations is less robust than that of the full dataset. The estimated field value from the median and the 68.3% credible region is $633\stackrel{+192}{_{-247}}$G. However, we do not have a good constraint on the lower limit, as the credible regions quickly go to zero. The 99.7% credible region extends to 2.7kG as well. For the null profile observations, the MAPs and modes are all 0G. Given the shape of the probability distribution, the median is non-null. However, given the probability density in favour of the $M_0$ model, it makes more sense to express the field strength estimation in terms of the upper limit of the credible regions. The null profiles are a good representation of what our data would look like in the absence of a stellar magnetic field. For the whole $N$ dataset, the upper limit of the 95.4% credible region is 144G, well below the inferred dipole strength ($\sim1$kG) from the Stokes $V$ observations. For the 99.0% credible region, we can say that the probability that an undetected field would have a dipole strength of more than 300G is only 1%. The noise scaling does not significantly change the credible regions. Interestingly, the dipole strength inferred from the low s/n Stokes $V$ observations is around 600G, and the odds ratio favours the magnetic model. From the low s/n null $N$ profile observation, we can see that a field with a dipole strength of the order of the 95.4% credible region upper limit (502G) can indeed be detected. Conclusion {#sec|conclu} ========== In this paper, we have described a method based on Bayesian statistics to infer the magnetic properties of stars observed spectropolarimetrically in the context of large surveys like the Magnetism in Massive Stars project. This approach is well-suited for stars for which the stellar rotation period, and therefore the rotational phases of the small number of observations, are not known. The model used to predict the expected Stokes $V$ profiles is that of an oblique dipolar magnetic field, parametrised by the field strength at the pole, the inclination of the rotational axis, the obliquity of the magnetic axis with respect to the rotational axis and the rotational phase (which is allowed to take any value for each individual observation). In the present case, the calculations are performed under the weak-field approximation, although any polarised spectral synthesis code can in principle be used with the Bayesian algorithm. The result of the analysis is a multidimensional posterior probability density that describes the relative likelihood of models spanning the parameter space of the dipolar field model. We have used synthetic observations to explore the behaviour of the Bayesian algorithm under ideal and realistic conditions. In the case of an ideal non-detection, the posterior probability density for the field strength has the form of a decreasing exponential, the extended tail of the field strength distribution being due to a specific family of dipole orientations. This tail becomes less extended with the addition of multiple observations, as some of these specific dipole orientations only present low-amplitude or null Stokes $V$ profiles over a restricted range of phases. However, the possibility that the dipole is aligned with the rotational axis and seen equator-on ($i=90^\circ$ and $\beta=0^\circ$) always remains as this configuration never produces any circular polarisation at any phase. When a detectable field is present, the probability distribution shows a sharp lower limit on the dipolar field strength, and a slow decrease towards higher field strengths, producing an asymmetrical distribution. With only one observation, not much can be inferred about the dipole orientation. The longitudinal field indicates which pole is located on the visible hemisphere and the shape of the Stokes V profile indicates when some obliquity is necessary because the pole is located at a non-null rotational radial velocity. A particularly useful quantity that can be computed from the posterior probabilities is the so-called “odds ratio”, which compares the relative compatibility between the observations and the magnetic dipole hypothesis versus the non-magnetic hypothesis. By adding a magnetic signal corresponding to the upper limit of the credible regions found for the ideal non-detection, we have explored the detection capability of the odds ratios. We find that fields corresponding to the 68.3% and 95.4% region are below detection, whereas those corresponding to the 99.0% and 99.7% regions are well detected with the odds ratios. In contrast, traditional diagnostics (detection probability and global longitudinal field significance) only detect fields corresponding to the 99.7% region. By using a set of five realistic simulated observations, we have also shown that in the case of noise emulating the shape of a weak magnetic signal, it is possible to use the odds ratios to distinguish between noise and real signal by obtaining a small number of additional observations. Combining all the observations together, it is therefore possible to detect a weak magnetic field (with a strength corresponding to roughly the upper limit of the 95.4% credible region of the noise-only case) under the detection capabilities of the traditional diagnostics. We have therefore shown that the odds ratio is an powerful quantitative indicator of which undetected stars in a survey should be re-observed in priority. In most applications where a field is indeed detected, the resulting probability density will generally be marginalised for the dipole strength, as only limited information can be obtained for the rotational inclination and the magnetic obliquity from a few observations (the magnetic geometry is recovered by the most probable inclination and obliquity, but the credible regions are quite extended). We have shown that the dipole strength probability distribution provides a reasonable estimate of the field strength. We have applied our method to real spectropolarimetric observations of the magnetic B-type star LPOri. The dataset consists of 3 ESPaDOnS and 4 Narval observations, of various signal-to-noise ratios. A magnetic signal is indeed detected by the odds ratios, even when only considering the low s/n observations where the traditional diagnostics do not detect the magnetic field. Using all the available spectra, we used the median of the marginalised posterior probability density, as well as the 68.3% credible region, to infer a dipolar field strength of $911\stackrel{+138}{_{-244}}$G. Although the probability density for the obliquity and inclination do not provide any tight constraints, geometries for which the negative magnetic pole spends little or no time on the visible hemisphere are preferred, since the dataset consists of positive or nearly null longitudinal field measurements. We also performed our analysis on the diagnostic null spectra, which resulted in a non-detection. The null profiles provide an useful verification of spurious signals and also provide an estimate of the field strengths that would be detectable with data of such quality. For example, when considering only the low s/n observations, the Stokes $V$ profile analysis yielded the detection of the $\sim600$G dipolar field and the diagnostic null profile analysis yielded an upper limit of $\sim500$G for the 95.4% credible region. Acknowledgments {#acknowledgments .unnumbered} =============== VP acknowledges support from Fonds québécois de la recherche sur la nature et les technologies. GAW acknowledges support from the Discovery Grants programme of the Natural Science and Engineering Research Council of Canada. We thank M. Gagné and R. P. Breton for useful discussions, and the anonymous referee whose helpful comments led to the improvement of this paper. ------------- ----------------------------------------------------- --------- ------------------ ------------- ----------------------------------------------------- --------- ------------------ ------------- ----------------------------------------------------- --------- ------------------ $\lambda_0$ Ion $d$ $g_\mathrm{eff}$ $\lambda_0$ Ion $d$ $g_\mathrm{eff}$ $\lambda_0$ Ion $d$ $g_\mathrm{eff}$ (nm) ($I_c$) (nm) ($I_c$) (nm) ($I_c$) (1) (2) (3) (4) (1) (2) (3) (4) (1) (2) (3) (4) 360.0943 Fe<span style="font-variant:small-caps;">iii</span> 0.159 1.075 437.1337 Fe<span style="font-variant:small-caps;">iii</span> 0.062 1.480 480.6021 Ar<span style="font-variant:small-caps;">ii</span> 0.048 1.600 360.1630 Al<span style="font-variant:small-caps;">iii</span> 0.214 1.100 437.2331 C<span style="font-variant:small-caps;">ii</span> 0.042 0.800 481.3333 Si<span style="font-variant:small-caps;">iii</span> 0.032 0.833 360.3890 Fe<span style="font-variant:small-caps;">iii</span> 0.156 0.670 437.2375 C<span style="font-variant:small-caps;">ii</span> 0.042 1.467 481.5552 S<span style="font-variant:small-caps;">ii</span> 0.100 1.300 386.7470 He<span style="font-variant:small-caps;">i</span> 0.408 1.250 437.2823 Fe<span style="font-variant:small-caps;">iii</span> 0.043 1.155 482.8951 Si<span style="font-variant:small-caps;">iii</span> 0.091 1.100 386.7482 He<span style="font-variant:small-caps;">i</span> 0.356 1.750 437.4281 C<span style="font-variant:small-caps;">ii</span> 0.058 1.214 494.2473 S<span style="font-variant:small-caps;">ii</span> 0.112 2.667 386.7630 He<span style="font-variant:small-caps;">i</span> 0.225 2.000 437.6582 C<span style="font-variant:small-caps;">ii</span> 0.042 1.100 500.1135 N<span style="font-variant:small-caps;">ii</span> 0.053 0.750 387.1791 He<span style="font-variant:small-caps;">i</span> 0.430 1.000 439.5755 Fe<span style="font-variant:small-caps;">iii</span> 0.181 1.515 500.1475 N<span style="font-variant:small-caps;">ii</span> 0.062 1.000 387.6038 C<span style="font-variant:small-caps;">ii</span> 0.054 0.700 440.9990 C<span style="font-variant:small-caps;">ii</span> 0.050 1.286 500.1959 Fe<span style="font-variant:small-caps;">ii</span> 0.073 1.150 387.6187 C<span style="font-variant:small-caps;">ii</span> 0.076 1.136 441.1152 C<span style="font-variant:small-caps;">ii</span> 0.052 0.900 501.4042 S<span style="font-variant:small-caps;">ii</span> 0.073 1.333 387.6393 C<span style="font-variant:small-caps;">ii</span> 0.069 1.056 441.1510 C<span style="font-variant:small-caps;">ii</span> 0.063 1.071 501.5678 He<span style="font-variant:small-caps;">i</span> 0.838 1.000 387.6653 C<span style="font-variant:small-caps;">ii</span> 0.061 0.929 441.4900 O<span style="font-variant:small-caps;">ii</span> 0.171 1.100 501.8440 Fe<span style="font-variant:small-caps;">ii</span> 0.035 1.935 387.8181 He<span style="font-variant:small-caps;">i</span> 0.045 1.000 441.6979 O<span style="font-variant:small-caps;">ii</span> 0.145 0.833 503.2126 C<span style="font-variant:small-caps;">ii</span> 0.057 1.100 391.1962 O<span style="font-variant:small-caps;">ii</span> 0.234 1.100 441.9596 Fe<span style="font-variant:small-caps;">iii</span> 0.166 1.665 503.2434 S<span style="font-variant:small-caps;">ii</span> 0.122 1.600 394.5034 O<span style="font-variant:small-caps;">ii</span> 0.336 1.500 443.7551 He<span style="font-variant:small-caps;">i</span> 0.471 1.000 503.5708 Fe<span style="font-variant:small-caps;">ii</span> 0.047 1.222 399.4997 N<span style="font-variant:small-caps;">ii</span> 0.346 1.000 444.7030 N<span style="font-variant:small-caps;">ii</span> 0.136 1.000 503.5946 C<span style="font-variant:small-caps;">ii</span> 0.040 0.833 399.7926 Si<span style="font-variant:small-caps;">ii</span> 0.106 0.929 451.2565 Al<span style="font-variant:small-caps;">iii</span> 0.132 0.833 504.1024 Si<span style="font-variant:small-caps;">ii</span> 0.130 0.833 403.9160 Fe<span style="font-variant:small-caps;">iii</span> 0.102 0.670 452.4675 S<span style="font-variant:small-caps;">ii</span> 0.044 1.067 504.5103 N<span style="font-variant:small-caps;">ii</span> 0.054 1.250 406.9621 O<span style="font-variant:small-caps;">ii</span> 0.097 0.500 452.4941 S<span style="font-variant:small-caps;">ii</span> 0.044 1.100 504.7738 He<span style="font-variant:small-caps;">i</span> 0.503 1.000 406.9881 O<span style="font-variant:small-caps;">ii</span> 0.113 0.900 452.8945 Al<span style="font-variant:small-caps;">iii</span> 0.060 1.067 507.3903 Fe<span style="font-variant:small-caps;">iii</span> 0.099 2.010 407.2005 Ar<span style="font-variant:small-caps;">ii</span> 0.063 1.200 452.9189 Al<span style="font-variant:small-caps;">iii</span> 0.155 1.100 512.2272 C<span style="font-variant:small-caps;">ii</span> 0.122 1.071 407.2150 O<span style="font-variant:small-caps;">ii</span> 0.126 1.071 454.9474 Fe<span style="font-variant:small-caps;">ii</span> 0.028 1.035 512.7387 Fe<span style="font-variant:small-caps;">iii</span> 0.068 1.180 407.4480 C<span style="font-variant:small-caps;">ii</span> 0.059 0.500 455.2410 S<span style="font-variant:small-caps;">ii</span> 0.125 0.833 512.7631 Fe<span style="font-variant:small-caps;">iii</span> 0.047 1.675 407.4543 C<span style="font-variant:small-caps;">ii</span> 0.078 0.900 455.2622 Si<span style="font-variant:small-caps;">iii</span> 0.259 1.250 513.2950 C<span style="font-variant:small-caps;">ii</span> 0.092 1.500 407.4841 C<span style="font-variant:small-caps;">ii</span> 0.096 1.071 456.7840 Si<span style="font-variant:small-caps;">iii</span> 0.313 1.750 513.3280 C<span style="font-variant:small-caps;">ii</span> 0.095 1.500 407.5852 C<span style="font-variant:small-caps;">ii</span> 0.114 1.167 457.4757 Si<span style="font-variant:small-caps;">iii</span> 0.232 2.000 514.3497 C<span style="font-variant:small-caps;">ii</span> 0.075 1.500 407.5859 O<span style="font-variant:small-caps;">ii</span> 0.138 1.167 459.0974 O<span style="font-variant:small-caps;">ii</span> 0.149 1.071 514.5167 C<span style="font-variant:small-caps;">ii</span> 0.144 1.600 407.8839 O<span style="font-variant:small-caps;">ii</span> 0.066 0.800 459.6172 O<span style="font-variant:small-caps;">ii</span> 0.131 0.900 515.1085 C<span style="font-variant:small-caps;">ii</span> 0.121 1.500 408.1007 Fe<span style="font-variant:small-caps;">iii</span> 0.071 1.375 460.1481 N<span style="font-variant:small-caps;">ii</span> 0.180 1.500 515.6111 Fe<span style="font-variant:small-caps;">iii</span> 0.117 1.245 412.8054 Si<span style="font-variant:small-caps;">ii</span> 0.118 0.900 460.7149 N<span style="font-variant:small-caps;">ii</span> 0.124 1.500 516.9033 Fe<span style="font-variant:small-caps;">ii</span> 0.060 1.325 413.0872 Si<span style="font-variant:small-caps;">ii</span> 0.048 1.029 460.9567 Ar<span style="font-variant:small-caps;">ii</span> 0.083 1.071 524.3306 Fe<span style="font-variant:small-caps;">iii</span> 0.085 1.300 413.0894 Si<span style="font-variant:small-caps;">ii</span> 0.110 1.071 461.3868 N<span style="font-variant:small-caps;">ii</span> 0.076 1.500 524.7952 Fe<span style="font-variant:small-caps;">ii</span> 0.030 0.735 413.2804 O<span style="font-variant:small-caps;">ii</span> 0.127 1.500 461.8559 C<span style="font-variant:small-caps;">ii</span> 0.103 0.929 545.3855 S<span style="font-variant:small-caps;">ii</span> 0.193 1.214 416.2665 S<span style="font-variant:small-caps;">ii</span> 0.194 1.167 461.9249 C<span style="font-variant:small-caps;">ii</span> 0.114 1.056 547.3614 S<span style="font-variant:small-caps;">ii</span> 0.094 1.333 416.4731 Fe<span style="font-variant:small-caps;">iii</span> 0.017 1.300 462.1396 N<span style="font-variant:small-caps;">ii</span> 0.092 1.500 563.9977 S<span style="font-variant:small-caps;">ii</span> 0.179 1.100 416.6840 Fe<span style="font-variant:small-caps;">iii</span> 0.010 1.700 462.5639 C<span style="font-variant:small-caps;">ii</span> 0.053 1.143 564.0346 S<span style="font-variant:small-caps;">ii</span> 0.048 1.071 417.4265 S<span style="font-variant:small-caps;">ii</span> 0.138 1.056 463.0543 N<span style="font-variant:small-caps;">ii</span> 0.243 1.500 564.5681 S<span style="font-variant:small-caps;">ii</span> 0.011 1.900 418.5440 O<span style="font-variant:small-caps;">ii</span> 0.152 0.929 463.8851 O<span style="font-variant:small-caps;">ii</span> 0.118 0.833 564.7020 S<span style="font-variant:small-caps;">ii</span> 0.153 0.833 418.9681 S<span style="font-variant:small-caps;">ii</span> 0.067 1.200 464.1810 O<span style="font-variant:small-caps;">ii</span> 0.160 1.100 569.6604 Al<span style="font-variant:small-caps;">iii</span> 0.158 1.167 418.9789 O<span style="font-variant:small-caps;">ii</span> 0.062 1.056 464.3090 N<span style="font-variant:small-caps;">ii</span> 0.129 1.500 572.2730 Al<span style="font-variant:small-caps;">iii</span> 0.139 1.333 419.0707 Si<span style="font-variant:small-caps;">ii</span> 0.064 1.100 464.9138 O<span style="font-variant:small-caps;">ii</span> 0.243 1.214 573.9734 Si<span style="font-variant:small-caps;">iii</span> 0.221 1.000 425.3589 S<span style="font-variant:small-caps;">iii</span> 0.192 1.167 465.0838 O<span style="font-variant:small-caps;">ii</span> 0.152 1.333 583.3938 Fe<span style="font-variant:small-caps;">iii</span> 0.073 1.375 426.6999 C<span style="font-variant:small-caps;">ii</span> 0.268 0.900 465.6757 S<span style="font-variant:small-caps;">ii</span> 0.034 1.833 587.5599 He<span style="font-variant:small-caps;">i</span> 0.177 2.000 426.7259 C<span style="font-variant:small-caps;">ii</span> 0.173 1.029 466.1635 O<span style="font-variant:small-caps;">ii</span> 0.135 1.467 587.5614 He<span style="font-variant:small-caps;">i</span> 0.248 1.333 426.7259 C<span style="font-variant:small-caps;">ii</span> 0.277 1.071 466.3046 Al<span style="font-variant:small-caps;">ii</span> 0.086 1.000 587.5615 He<span style="font-variant:small-caps;">i</span> 0.281 1.167 426.7762 S<span style="font-variant:small-caps;">ii</span> 0.111 1.100 467.6231 O<span style="font-variant:small-caps;">ii</span> 0.185 1.486 587.5625 He<span style="font-variant:small-caps;">i</span> 0.248 1.000 430.4767 Fe<span style="font-variant:small-caps;">iii</span> 0.142 1.150 469.9215 O<span style="font-variant:small-caps;">ii</span> 0.135 0.900 587.5640 He<span style="font-variant:small-caps;">i</span> 0.270 1.000 431.0355 Fe<span style="font-variant:small-caps;">iii</span> 0.084 1.220 470.5343 O<span style="font-variant:small-caps;">ii</span> 0.129 1.071 587.5966 He<span style="font-variant:small-caps;">i</span> 0.254 0.500 431.7136 O<span style="font-variant:small-caps;">ii</span> 0.081 1.500 471.3139 He<span style="font-variant:small-caps;">i</span> 0.310 1.250 637.1371 Si<span style="font-variant:small-caps;">ii</span> 0.176 1.333 431.7266 C<span style="font-variant:small-caps;">ii</span> 0.052 1.600 471.3156 He<span style="font-variant:small-caps;">i</span> 0.291 1.750 667.8154 He<span style="font-variant:small-caps;">i</span> 1.113 1.000 431.8643 S<span style="font-variant:small-caps;">ii</span> 0.061 1.486 471.3376 He<span style="font-variant:small-caps;">i</span> 0.238 2.000 728.1349 He<span style="font-variant:small-caps;">i</span> 0.524 1.000 431.9625 O<span style="font-variant:small-caps;">ii</span> 0.082 1.500 471.6271 S<span style="font-variant:small-caps;">ii</span> 0.025 1.867 436.6893 O<span style="font-variant:small-caps;">ii</span> 0.073 1.500 480.3288 N<span style="font-variant:small-caps;">ii</span> 0.113 1.333 ------------- ----------------------------------------------------- --------- ------------------ ------------- ----------------------------------------------------- --------- ------------------ ------------- ----------------------------------------------------- --------- ------------------ \[lastpage\] [^1]: E-mail: VPetit@wcupa.edu [^2]: Based on observations obtained at the Canada-France-Hawaii Telescope (CFHT) and at the Télescope Bernard Lyot (TBL). CFHT is operated by the National Research Concil of Canada, the Institut National des Sciences de l’Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii. TBL is operated by CNRS/INSU. [^3]: The $\beta$ density probability is different from the 100G ideal detection because the simulated noise pattern has a shape similar to a magnetic field whose pole is located at a non-null rotational radial velocity. Such a location requires a dipole that is not aligned. [^4]: During the analysis of the complete set of observations of LPOri for this paper, it was discovered that the sign of the Stokes $V$ profiles corresponding to two spectra employed by Petit et al. (2008) was inverted (the Jan. 2006 ESPaDOnS spectrum and the Narval spectrum). During the present analysis we have corrected the sign of these spectra and verified the sign of all others. We have verified that the results of Petit et al. (2008) are not substantially modified by this change.

=23.5truecm =6.25truein =6.25truein =1.5pt =[=1]{} =0 =0 =-1 =0 =0 =0 \#1\#2\#3 by 10==0 =-2 --00 -.8pt\#3by 1 \#1\#2\#3 [by 10==0 =-2 --00 -.8pt\#3by 1]{} \#1 \#2 \#1[Lemma]{}[*\#2*]{} \#1 \#2 \#1[Proposition]{}[*\#2*]{} \#1 \#2 \#1[Theorem]{}[*\#2*]{} \#1 \#2 \#1[Corollary]{}[*\#2*]{} \#1 \#2 \#1[Example]{}\#2 \#1 \#2 \#1[Exercise]{}\#2 \#1 \#2 \#1[Definition]{}[*\#2*]{} \#1 \#2 \#1[Theorem]{}[*\#2*]{} \#1 \#2 0==0 =-2 --00 -.8pt\#2by 1 \#1[\[\#1\]]{} \#1 \#1\#2\#3\#4\#5\#6\#7 [\[\#1\]  ]{}[\#2, ]{}[*\#3*]{}, [\#4]{} [**\#5**]{} [(\#6), \#7]{}. \#1\#2\#3\#4\#5\#6 [\[\#1\]  ]{}[\#2, ]{} [*\#3.*]{} [\#4]{}: [\#5]{}, [\#6]{}. \#1\#2\#3\#4\#5\#6\#7\#8 [\[\#1\]  ]{}[\#2, ]{}[*\#3*]{}. In: [\#4]{}, [\#5]{} [\#6]{} [\#7]{}, [\#8]{}. \#1\#2 0==0 =-2 --00 -.8pt\#2by 1 \#1[0==0 =-2 --00-.8pt]{} \#1 =msbm10 =msbm7 =msbm10 at 9truept =cmcsc10 =cmss10 \#1 \#1 =eufm10 \#1 \#1[$\underline{\hbox{#1}}$]{} \#1\#2 \#1 \#2 \#1\#2 \#1\#2[height \#1pt depth \#2pt width 0pt]{} \#1 \#1[\#1]{} \#1[\#1]{} \#1[\#1]{} =|| \#1[=0=0by 1]{} \#1[=by -\#1(.)]{} \#1\#2[\#1]{} **POLYSTABILITY AND THE HITCHIN-KOBAYASHI CORRESPONDENCE** Nicholas Buchdahl and Georg Schumacher =1 45pt 5.1in Using a quasi-linear version of Hodge theory, holomorphic vector bundles in a neighbourhood of a given polystable bundle $E_0$ on a compact Kähler manifold are shown to be (poly)stable if and only if their corresponding classes are (poly)stable in the sense of geometric invariant theory with respect to the action of the automorphism group of $E_0$ on the (finite-dimensional vector) space of infinitesimal deformations. In the first edition of his book [@MFK], Mumford introduced the notion of stability in algebraic geometry, giving a condition under which the set of orbits of an algebraic group action can be endowed with a “good" structure, a moduli space. A numerical characterisation of stability derived from the Hilbert criterion leads to the notion of stability for holomorphic vector bundles, with which Narasimhan and Seshadri [@NS] proved that stable holomorphic vector bundles on a compact Riemann surface correspond precisely to irreducible projective unitary representations of the fundamental group. Such representations in turn correspond to irreducible projectively flat unitary connections on a given hermitian vector bundle, and in this form Donaldson [@Do1] gave an analytical (as opposed to a representation-theoretic) proof of the theorem of Narasimhan and Seshadri. Donaldson [@Do3] subsequently generalised his results to the case of algebraic surfaces equipped with Hodge metrics by proving that a holomorphic vector bundle is stable if and only if it admits an irreducible Hermite-Einstein connection (see §1 for definitions). In the concluding section of his paper, following the seminal work of Atiyah and Bott [@AB], he explained how his result could be seen in a broader context, namely as an infinite-dimensional version of a theorem of Kempf and Ness [@KN] giving a metric characterisation of stability and relating categorical quotients of Mumford’s geometric invariant theory with the symplectic quotients of Marsden and Weinstein [@MW]. In addition to their remarkable consequences for the differential topology of smooth four-manifolds, Donaldson’s results have inspired a wide range of generalisations. In particular, Uhlenbeck and Yau [@UY] proved that a holomorphic vector bundle on a compact Kähler manifold is stable if and only if it admits an irreducible Hermite-Einstein connection, this (and other generalisations) known as the Hitchin-Kobayashi correspondence. In this paper, we demonstrate that the relationship between the two notions of stability—that in geometric invariant theory and that in Kähler geometry—is even more intimate than it might appear at first sight. For a holomorphic vector bundle $E$ on a compact complex manifold $X$ the space of infinitesimal deformations is the (finite-dimensional) cohomology group $H^1(X,End\,E)$, and Kuranishi theory describes the parameter space of a semi-universal deformation of $E$ as the zero set of a holomorphic function $\Psi$ defined in a neighbourhood ${\cal N}$ of zero in $H^1(X,End\,E)$ with values in $H^2(X,End\,E)$. If there are non-trivial automorphisms of $E$, different points in $\Psi^{-1}(0)$ may correspond to isomorphic bundles. The group $Aut(E)$ of holomorphic automorphisms of $E$ acts on the space of infinitesimal deformations, but in general it does not leave $\Psi^{-1}(0)$ invariant. Even if it does, it cannot be expected that points of the (set-theoretic/topological) quotient $\Psi^{-1}(0)/Aut(E)$ should represent isomorphism classes of holomorphic bundles near $E$ unless $E$ is simple, in which case the group action is trivial. For a simple holomorphic vector bundle $E$ on a compact complex manifold $X$, universal deformations exist and the base spaces of these can be glued together in a unique way to give a (generally non-Hausdorff) coarse moduli space. Using the methods of §6 of [@AHS] and the Hitchin-Kobayashi correspondence, coarse moduli spaces for stable holomorphic vector bundles in the category of reduced complex analytic spaces were constructed in [@Kim] and in [@FS] by a construction of local slices for the action of the complex gauge group, here viewing holomorphic structures as integrable semi-connections, with isomorphic such structures being (by definition) those in the same orbit of that group. If $X$ is equipped with a Kähler metric and if $E_0$ is a polystable vector bundle, several remarkable features of the local description of holomorphic bundles near $E_0$ emerge, these features otherwise failing to exist: Let $E_0$ be a polystable holomorphic vector bundle equipped with an Hermite-Einstein connection $d_0$ on a compact Kähler manifold $(X,\omega)$. Then the holomorphic function $\Psi\: {\cal N}\subsett H^1(X,End\,E_0)\to H^2(X,End\,E_0)$ is equivariant with respect to the action of $Aut\,E_0$ on $\db_0$-harmonic representatives. With the same objects as in Theorem 1 and restricting ${\cal N}$ if necessary, every holomorphic bundle $E$ corresponding to a point of $\Psi^{-1}(0)$ is semi-stable, and any destabilising subsheaf of $E$ is a subbundle. Restricting ${\cal N}$ further if necessary, the set of isomorphism classes of holomorphic vector bundles near to $E_0$ is in 1–1 correspondence with the set-theoretic quotient $\Psi^{-1}(0)/Aut(E_0)$; indeed, when each of these spaces is equipped with the quotient topology, the natural map from the latter into the space of isomorphism classes of holomorphic bundles near $E_0$ is a homeomorphism onto its image. This topological space is non-Hausdorff in general, but when attention is restricted to those bundles that are polystable—corresponding to an $Aut(E_0)$-invariant subset of $\Psi^{-1}(0)$—the quotient is Hausdorff. The group $Aut(E_0)$ is a complex reductive Lie group, acting linearly on the finite-dimensional vector spaces $H^q(X,End\,E_0)$ and extending the isometric actions of the unitary automorphisms when restricted to the $\db_0$-harmonic spaces equipped with the $L^2$ norm. At this point, the description of a neighbourhood of $E_0$ has, through the Hitchin-Kobayashi correspondence and its GIT analogue the Kempf-Ness theorem, an interpretation in terms of geometric invariant theory: A class $\alpha\in \Psi^{-1}(0)\subsett H^1(X,End\,E_0)$ is (poly)stable with respect to the action of $Aut(E_0)$ if and only if the corresponding bundle $E_{\alpha}$ is (poly)stable with respect to $\omega$. For those bundles that are semistable but not polystable, the following analogue of the standard GIT result holds: If $\bar\alpha$ is a polystable point in the closure of the orbit of $\alpha\in \Psi^{-1}(0)$ under the action of $Aut(E_0)$, the polystable bundle $E_{\bar\alpha}$ is isomorphic to the graded object $Gr(E)$ associated to a Seshadri filtration of $E$. In addition to these theorems, a number of other results of a quantitative but technical nature appear throughout the paper, many of these not restricted to integrable connections alone and being of considerable independent interest—indeed, the role played by integrability, apart from that of the connection defining $E_0$, turns out to be relatively minor throughout. The results of this paper shed further light on the close relationship between stability in the context of geometric invariant theory and stability for holomorphic vector bundles on compact Kähler manifolds, providing a framework for the further analysis of moduli spaces of stable bundles on such manifolds. Although some of these results may appear unsurprising at first sight, the fact that polystability lies at the borderline between stable and unstable renders inapplicable the standard techniques—notably, the implicit function theorem—so that new approaches lying at the interface between infinite and finite-dimensional analysis are required. The application of methods from geometric invariant theory to problems in geometric analysis is not new, and indeed some of the methods used here have appeared in the literature in the context of $K$-(poly)stability and the existence of constant scalar curvature Kähler metrics; see for example, [@S], [@CS], and also [@Th]. The (yet to be fully established) Yau-Tian-Donaldson conjecture is the analogue of the Hitchin-Kobayashi correspondence, postulating a one-to-one correspondence between objects satisfying an algebro-geometric condition and the zeros of an appropriate moment map, manifested as the solution of a particular differential equation. The results here are of a different nature, describing an actual correspondence between classical GIT and the theory of holomorphic vector bundles, as opposed to describing an analogue; the results are not subsumed by the (established) validity of the Hitchin-Kobayashi correspondence. Apart from providing potentially useful insight into an analogous theory, it is not the intention here to contribute to the developing theory of cscK metrics. The construction of slices for the action of the complex gauge group is given in  below, this in the general setting of a review of the Kuranishi theory of deformations of holomorphic vector bundles. Following a brief study of a neighbourhood of a semi-stable holomorphic bundle on a compact Kähler manifold in §3, the discussion is specialised in §4 to the case of a polystable bundle, with the bundle represented by an Hermite-Einstein connection. The equivariance of the function $\Psi$ described earlier is proved in , and the claim that the natural map mentioned above is a homeomorphism follows from the first statement of . In §5 there is a general discussion of some of the basic ideas associated with geometric invariant theory that will be well-known to experts, and the “if" part of the main theorem ( above) is established in . The more difficult “only if" part of the main theorem is proved in §§6-8, culminating in . Provided that polystable bundles considered are “sufficiently close" to $E_0$, a quantitative estimate of the distance between the corresponding Hermite-Einstein connections is provided by ; see also the second of the  following that result. The case of bundles close to $E_0$ that are not polystable is considered in §9, this corresponding  above. The paper concludes with some general remarks concerning the contents and possible extensions of the paper, and it commences in the next section, which is devoted to establishing terminology and notation and providing references. Let $X$ be a compact complex manifold. In this paper, as in [@Do3], holomorphic vector bundles on $X$ are viewed most often from the perspective of integrable $\db$-operators on fixed smooth bundles. Following Kobayashi [@Ko3], a [*semi-connection*]{} on a complex vector bundle $\rm E$ is a $\m C$-linear map $\db$ on differentiable local sections of $\et$ taking values in $\Lambda^{0,1}\otimes \et$ and satisfying the $\db$-Leibniz rule; here $\Lambda^{p,q}$ is the space of $(p,q)$-forms on $X$. Any such operator has natural prolongations $\Lambda^{0,q}\otimes{\rm E}\to \Lambda^{0,q+1}\otimes \rm E$, and the operator is integrable if $\db \circ \db=0$. As for connections, the set of semi-connections on $\rm E$ is an affine space with all the familiar properties of connections, including the ways in which they can be induced naturally on associated bundles. The semi-connections induced by integrable connections are again integrable. Let $\db_0$ be an integrable semi-connection on $\rm E$. Then every semi-connection $\db$ on the bundle can be written $\db=\db_0+a''$ for some unique $(0,1)$-form $a''$ with coefficients in $\rm End\,E$, and the integrability condition for $\db$ is $\db_0a'' + a''\w a''=0$. The notation, which will be used throughout, is inspired by the often-used convention to denote by $a'$ and $a''$ respectively the $(1,0)$- and $(0,1)$-components of a $1$-form $a$, where that $1$-form may take values in some vector bundle. The group ${\cal G}$ of complex automorphisms of $\rm E$ acts on the affine space of semi-connections as a “complex gauge group". This action, which preserves the integrability condition, is denoted by $g\cdot\db := g\circ \db \circ g^{-1}$. A holomorphic structure is defined by an integrable semi-connection, and two such structures are isomorphic if and only if they lie in the same orbit of ${\cal G}$. By virtue of the Newlander-Nirenberg theorem, this view of holomorphic structures is equivalent to the more usual one of holomorphic vector bundles being described by systems of holomorphic transition functions. Denote by $A^{p,q}(\rm E)$ the global smooth $(p,q)$-forms with coefficients in $\rm E$. For an integrable semi-connection $\db_0$ defining a holomorphic structure $E_0$, the Dolbeault cohomology groups $$H^q_{\overline\partial_0}(X,\rm E) ={\ker\db_0\: A^{0,q}(\rm E)\to A^{0,q+1}({\rm E})\over \im\db_0\: A^{0,q-1}({\rm E})\to A^{0,q}({\rm E})}$$ are denoted by $H^q(X,E_0)$, these being finite-dimensional spaces with $H^1(X,End\,E_0)$ being by definition the space of infinitesimal deformations of $E_0$. Analysis of the small deformations of $E_0$ is achieved with the introduction of metrics on both the bundle $\rm E$ and the manifold $X$. A hermitian metric ${\rm h}$ is fixed once and for all on the bundle $\rm E$, which is henceforth denoted $\et$. The group ${\cal G}$ is the complexification of the group ${\cal U}$ of unitary gauge transformations. The hermitian structure on $\et$ gives a one-to-one correspondence between semi-connections $\db$ and hermitian connections $d=\d+\db$ on $\et$, and if $d=d_0+a$ for some skew-adjoint $a\in A^1(\eet)$, then $\d=\d_0+a'$ and $\db=\db_0+a''$ where $a=a'+a''$ and $a'=-(a'')^*$. Henceforth, all connections are taken to be hermitian. The action of ${\cal G}$ on the space of semi-connections extends to an action on the space of connections via $g\cdot d := g^*{}^{-1}\circ \d \circ g^* + g \circ \db \circ g^{-1} = d+g^*{}^{-1}\d g^* -\db g\,g^{-1}$. The curvature $F(d)=d\circ d \in A^2(\eet)$ of a connection is a skew-adjoint $2$-form with coefficients in $\eet$, and the connection (i.e., the associated semi-connection) is integrable if and only if $F(d)$ is of type $(1,1)$. Since $F(d_0+a)=F(d_0)+d_0a+a\w a$, for $g\in {\cal G}$ it follows that $$\eqalignno{ F(g\cdot d) = g\,F^{0,2}(d)\,g^{-1} + g^*{}^{-1}\,F^{2,0}(d)\,g^* + \big( F^{1,1}&(d) + \db(g^*{}^{-1}\d g^*)-\d(\db g\,g^{-1})+ & \eqn \cr & (g^*{}^{-1}\d g^*)\w (\db g\,g^{-1}) + (\db g\,g^{-1})\w(g^*{}^{-1}\d g^*)\big)\;. }$$ Now fix a positive $(1,1)$-form $\omega$ on $X$. If $\dim X=n$, the associated volume form is $dV = \omega^n$, where the convention is adopted throughout that $\omega^q := (1/q!)\,\omega\w\cdots \w \omega$ ($q$ times). If $d$ is an integrable connection on $\et$, standard Hodge theory on compact manifolds gives a unique $\db$-harmonic representative in each Dolbeault cohomology class, where $\tau\in A^{0,q}(\et)$ is $\db$-harmonic if $\db\tau=0=\db^*\tau$ for $\db^*=-*\d*$, the formal adjoint of $\db$. So $\tau$ is $\db$-harmonic if and only if it lies in the kernel of the $\db$-Laplacian $\lap''=\db\db^* +\db^*\db$. In general, there is an $L^2$-orthogonal decomposition $$A^{0,q}(\et)=(\ker\db)^{\perp}\oplus (\ker\db^*)^{\perp}\oplus H^{0,q} = \overline{\im \db^*} \oplus \overline{\im\db}\oplus H^{0,q}$$ where $H^{0,q}=H^{0,q}(\db)$ is the space of $\db$-harmonic $(0,q)$-forms. Here, notation has been abused in that $A^{0,q}(\et)$ is no longer denoting the space of smooth sections of $\et$, but rather the space of global sections of $\Lambda^{0,1}\otimes \et$ that are square integrable, and the closures on the right are the closures in $L^2$ of the images under $\db$ and $\db^*$ of the spaces of smooth global sections. Standard elliptic regularity implies that the $\db$-harmonic sections are smooth, at least if the connections are. This abuse of notation will be employed throughout, so that $A^{0,q}(\et)$ will always denote a space of global sections of $\Lambda^{0,q}\otimes \et$ but with the degree of differentiability and/or integrability to be specified in the respective context. Sobolev spaces of functions are denoted by $L^p_k$, meaning all weak derivatives up to and including those of order $k$ lie in $L^p$. Having fixed a base connection on $\et$ once and for all, the spaces of $L^p_k$ elements of $A^{0,q}(\et)$ acquire norms that make them Banach spaces. Henceforth, a number $p>2n$ (for $n=\dim X$) will be fixed, so by the Sobolev embedding theorem there are compact embeddings $L^p_1\subset C^0$ and $L^p_2\subset C^1$. By standard elliptic theory on compact manifolds, for an integrable connection $d=\d+\db$ there is a constant $C>0$ (depending upon $d$) such that $$\norm{\tau}_{L^p_1}\le C\big(\norm{\db\tau}_{L^p}+ \norm{\db^*\tau}_{L^p}+\norm{\Pi^{0,q}\tau}_{L^2}\big)\quad{\rm for}\; \tau\in A^{0,q}(\et)\;,\myeq$$ where $\Pi^{0,q}\tau$ is the $L^2$-orthogonal projection of $\tau\in A^{0,q}(\et)$ in $H^{0,q}(\db)$. Connections on $\et$ will be permitted to have coefficients in $L^p_1$, and the fact that an integrable $L^p_1$ connection defines a holomorphic structure in the usual way follows from Lemma 8 of [@Bu2]. When not indicated by a subscript on the norm symbol, $\norm\tau$ will always mean $\norm{\tau}_{L^2}$. If $d$ is an integrable semi-connection and $g\in {\cal G}$, the Dolbeault cohomology groups defined by $\db$ and by $g\cdot \db$ are isomorphic, the isomorphism induced by mapping a $\db$-closed $(0,q)$-form $\tau\in A^{0,q}(\et)$ to the $(g\cdot\db)$-closed $(0,q)$-form $g\,\tau$. This isomorphism does not preserve harmonic representatives in general, unless $g\in {\cal U}$ in which case it also preserves $L^2$ norms. Subsequently in this paper it will be useful to consider connections that are not integrable, in which case the Dolbeault cohomology groups are not defined. However, one can still define the spaces $H^{0,q}(\db)$ of $\db$-harmonic $(0,q)$-forms as null spaces of the appropriate Laplacians, these still being finite-dimensional spaces consisting only of smooth forms (if $\db$ is itself smooth). If $d\omega=0$, the formal adjoints $\d^*$ and $\db^*$ have alternative expressions in terms of the Kähler identities: $$\d^*=i(\Lambda\db -\db\Lambda)\,,\quad \db^*=-i(\Lambda\d-\d\Lambda)\;,$$ where $\Lambda\: \Lambda^{p+1,q+1}\to \Lambda^{p,q}$ is the adjoint of $\omega$, so $\Lambda\omega=n$. For an integrable connection $d$ on $\et$ with curvature $F=F(d)$, the Bianchi and Kähler identities imply that the Yang-Mills equations $d^*F=0$ are equivalent to the equation $d\wh F=0$, where $\wh F := \Lambda F$, the central component of the curvature. Under these circumstances, the bundle and connection split into eigenspaces of the covariantly constant self-adjoint endomorphism $i\wh F$, and when restricted to any such eigenspace, the curvature of the restricted connection has central component that is a constant multiple of the identity. That is, it is an [*Hermite-Einstein*]{} connection, a term coined by Kobayashi in [@Ko1]. If $d_0$ is an Hermite-Einstein connection on $\et$ with $i\wh F(d_0)= \lambda\,1$, the constant $\lambda$ is a fixed positive multiple of the [*slope*]{} $\mu(\et)=deg(\et)/\rk(\et)$, where $deg(\et)=(n-1)!\,[\omega^{n-1}]\cdot c_1(\et)$. Kobayashi [@Ko2] proved that a holomorphic bundle $E$ defined by an Hermite-Einstein connection is semi-stable in the sense of Mumford-Takemoto [@Ta], meaning that any coherent subsheaf $A\subset E$ with non-zero torsion-free quotient should satisfy $\mu(A)\le \mu(B)$, and if the connection is irreducible (meaning that the unitary bundle-with-connection does not split), then the bundle is stable, meaning $\mu(A) <\mu(B)$. These results were proved independently by Lübke . Kobayashi had earlier mooted in [@Ko1] that there should exist a relationship between the algebro-geometric notion of stability and the existence of Hermite-Einstein connections, this also conjectured by Hitchin [@Hi] in the broader setting of arbitrary compact manifolds equipped with Gauduchon metrics [@G]. Following the work of Donaldson, the affirmation of that conjecture was established by Uhlenbeck and Yau [@UY] in the Kähler case and by Li and Yau [@LY] in the general case. Many other authors have made significant contributions to the theory, extending the correspondence in a variety of ways. Of relevance here is the semi-stable version of the correspondence due to Jacob [@J] and a version for torsion-free semi-stable sheaves due to Bando and Siu [@BS]. As in the previous section, let $X$ be a compact complex manifold and let $E_0$ be a holomorphic vector bundle on $X$. A vector bundle version of Kuranishi’s fundamental results [@Kur] on deformations of compact complex manifolds has been given by Forster and Knorr [@FK] using power series methods and by Miyajima [@Miy] using more analytical methods; (cf. also [@FS]). Either way, there is a holomorphic function $\Psi$ defined in a neighbourhood of $0\in H^1(X,End\,E_0)$ such that $\Psi^{-1}(0)$ is a complete family of small deformations of $E_0$. In this section, the construction of the function $\Psi$ will be presented in a manner to suit the purposes of the remainder of the paper. The entire discussion is essentially an $n$-dimensional version of the $2$-dimensional case presented in §6.4 of [@DK]. Fix a positive $(1,1)$-form $\omega$ on $X$, which at this stage is not assumed to be $d$-closed. Let $\et$ be the complex bundle underlying $E_0$ equipped with a fixed hermitian structure, and let $d_0$ be a connection on $\et$ inducing the holomorphic structure $E_0$. There is a number $\epsilon>0$ depending on $d_0$ with the property that for any integrable hermitian connection $d_a = d_0+a$ with $\norm{a}_{L^p_1} <\epsilon$ the $L^2$ orthogonal projection $ H^{0,q}(\db_a) \owns \tau \mapsto \Pi^{0,q}\tau\in H^{0,q}(\db_0) $ is injective. Write $a=a'+a''$ for $a''\in A^{0,1}(\eet)$. Suppose $\tau\in A^{0,q}(\et)$ satisfies $\db_0\tau+a''\w \tau=0=\db_0^*\tau-i\Lambda(a'\w \tau)$, so $\tau\in H^{0,q}(\db_a)$. If $\Pi^{0,q}\tau=0 \in H^{0,q}(d_0)$, then $\tau=\db_0\mu+\db_0^*\nu$ for some $\mu\in A^{0,q-1}(\et)$ and some $\nu\in A^{0,q+1}(\et)$, with $\mu$ and $\nu$ respectively orthogonal to the kernels of $\db_0$ and $\db_0^*$. Then $-a''\w\tau=\db_0\tau = \db_0^{}\db_0^*\nu$ and $i\,\Lambda(a'\w\tau) =\db_0^*\tau=\db_0^*\db_0^{}\mu$, so on taking inner products with $\nu$ and $\mu$ respectively it follows that $\norm{\db_0^*\nu}^2\le \norm{\nu}\,\norm{a''\w \tau}$ and $\norm{\db_0\mu}^2\le \norm{\mu}\,\norm{a'\w\tau}$, where $\norm{\cdot}$ is the $L^2$ norm. Since $p>2n$, the Sobolev embedding theorem gives $\sup|a| \le C\,\norm{a}_{L^p_1}$ for some constant $C$ independent of $a$ and $d_0$, so after adding the last two inequalities it follows that $\norm{\tau}^2 =\norm{\db_0\mu}^2+\norm{\db_0^*\nu}^2 \le C\,\norm{a}_{L^p_1}\,\norm{\tau}\big(\norm{\mu}+\norm{\nu}\big)$. Since $\mu$ and $\nu$ are orthogonal to $\ker\db_0$ and $\db_0^*$ respectively, ellipticity of $\lap_0''$ implies that $\norm{\mu}\le C_1\,\norm{\db_0\mu}$ and $\norm{\nu}\le C_2\,\norm{\db_0^*\nu}$ for some constants $C_1,\,C_2$, so $\norm{\tau}^2\le C_3\norm{a}_{L^p_1}\,\norm{\tau}^2$ for some new constant $C_3=C_3(d_0)$, giving the stated result. This proposition clearly implies the well-known and standard semi-continuity of cohomology. It also has the following useful consequence, resulting from equality of dimensions of cohomology groups: Under the hypotheses of , suppose in addition that $d_0$ and $d_a$ lie in the same ${\cal G}$-orbit. Then the map $H^{0,q}(\db_a)\to H^{0,q}(\db_0)$ induced by orthogonal projection is an isomorphism. Regardless of the integrability or otherwise of $\db_a=\db_0+a''$, if $\norm{a''}_{L^p_1}$ is sufficiently small, the Sobolev embedding theorem combined with  gives the following perturbed version of that estimate: There exists $\epsilon>0$ and $C>0$ depending on $d_0$ with the properties that if $a''\in A^{0,1}(\eet)$ satisfies $\norm{a''}_{L^p_1} < \epsilon$, then $$\norm{\tau}_{L^p_1} \le C\big(\norm{\db_0^*\tau}_{L^p} + \norm{\db_0\tau+a''\w\tau}_{L^p} + \norm{\Pi^{0,q}\tau}_{L^2}\big)\;, \qquad \tau\in A^{0,q}(\et)\;, \myeq$$ where $\Pi^{0,q}\:A^{0,q}(\eet)\to H^{0,q}(\db_0)$ is $L^2$ orthogonal projection. Replacing $\et$ by $\eet$, $d_0$ by the induced connection $d_0$ on $\eet$, taking $q=1$ and $\tau=a''$ gives There exists $\epsilon>0$ and $C>0$ with the properties that each $a''\in A^{0,1}(\eet)$ with $\norm{a''}_{L^p_1} < \epsilon$ satisfies $$\norm{a''}_{L^p_1} \le C\big(\norm{\db_0^*a''}_{L^p} + \norm{\db_0a''+a''\w a''}_{L^p} + \norm{\Pi^{0,1}a''}_{L^2}\big)\;, \myeq$$ where $\Pi^{0,1}\:A^{0,1}(\eet)\to H^{0,1}(\db_0)$ is $L^2$ orthogonal projection. If $a''\in A^{0,1}(\eet)$ and $g\in {\cal G}$, then $g\cdot(\db_0+a'') = \db_0-\db_0g\,g^{-1}+ga'' g^{-1}$. The map $${\cal G}\times A^{0,1}(\eet)\owns (g,a'')\mapsto \db_0^*(-\db_0g\,g^{-1}+ga'' g^{-1}) \in A^{0,0}(\eet)$$ maps into the subspace of $A^{0,0}(\eet)$ orthogonal to the kernel of $\db_0$, and its linearization at $(1,0)$ in the ${\cal G}$-direction is $A^{0,0}(\eet) \owns \gamma\mapsto - \lap_0''\,\gamma$. This is an isomorphism from the space of $L^p_2$ sections in $A^{0,0}(\eet)$ orthogonal to $\ker\db_0=H^{0,0}$ to the space of $L^p$ sections in $A^{0,0}(\eet)$ orthogonal to $\ker\db_0$. The implicit function theorem for Banach spaces now implies: There exist $\epsilon,\, C>0$ with the property that for each $a'' \in A^{0,1}(\eet)$ with $\norm{a''}_{L^p_1}<\epsilon$ there is a unique $\varphi\in A^{0,0}(\eet)\cap (\ker\db_0)^{\perp}$ with $\norm{\varphi}_{L^p_2} \le C\,\norm{\db_0^*a''}_{L^p}$ such that $\db_0^*(-\db_0g\,g^{-1}+g\,a''\,g^{-1})=0$, where $g = \exp(\varphi)$. This is a complex analogue of fixing a unitary gauge for hermitian connections near a given such connection, corresponding to the linear operation of projecting a $(0,1)$-form orthogonal to the range of $\db_0$. If $\db_0+a''$ is an integrable semi-connection with $\norm{a''}_{L^p_1}$ sufficiently small as dictated by this lemma, after applying an appropriate complex gauge transformation so that the new semi-connection $\db_0+\tilde a''$ satisfies $\db_0^*\tilde a''=0$,  gives an estimate of the form $\norm{\tilde a''}_{L^p_1} \le C\,\big\|\Pi^{0,1}\tilde a''\big\|_{L^2}$. Consequently, the well-known result that if $H^1(X,End\,E_0)=0$ then every small deformation of $E_0$ is isomorphic to $E_0$ follows immediately. A simple but pertinent example is given by a trivial bundle on $\m P_1$. It follows from  that for integrable semi-connections in the “good" complex gauge lying in a sufficiently small neighbourhood of $\db_0$ in $L^p_1$, the projection onto the $\db_0$-harmonic component is a homeomorphism onto a closed subset of an open neighbourhood of $0$ in $H^{0,1}$, where $H^{0,q}$ denotes $H^{0,q}(\db_0,\eet)$ in this section. This closed subset is the zero-set of the holomorphic function $\Psi$ mentioned in the introduction, as will now be discussed. If $a''\in A^{0,1}(\eet)$ is $\db_0$-closed, the semi-connection $\db_0+a''$ is not integrable in general, since $a''\w a''$ is not zero in general. But one might attempt to perturb $a''$ in such a way that the corresponding perturbed semi-connection is integrable. The derivative of the map $$\eqalignno{ \big(A^{0,1}\times A^{0,2}\big)&(\eet) \quad\longrightarrow \quad A^{0,2}(\eet)\cr &\upin \hskip1.4in \upin & \eqn \cr (a''&,\beta) \quad \mapsto \quad \db_0(a''+\db_0^*\beta)+(a''+\db_0^*\beta)\w (a''+\db_0^*\beta) }$$ in the $A^{0,2}$-direction at $(0,0)$ is $ A^{0,2}(\eet)\owns b\mapsto \db_0^{}\db_0^*\,b \in A^{0,2}(\eet) $, which is an isomorphism from the closed subspace of the $L^p_2$ sections in $A^{0,2}(\eet)$ that are orthogonal in $L^2$ to the kernel of $\db_0^*$ onto the closed subspace of the $L^p$ sections in $A^{0,2}(\eet)$ orthogonal in $L^2$ to the kernel of $\db_0^*$. The non-linear mapping 0 does not map into the latter closed subspace in general, but if $\Pi^{0,2}$ is the $L^2$ orthogonal projection of $A^{0,2}(\eet)$ onto the closed subspace $\ker\db_0^*$ and $\Pi^{0,2}_{\;\perp}= 1-\Pi^{0,2}$, the map 0 can be replaced by $$(a'',\beta)\mapsto \Pi^{0,2}_{\perp}\big[\,\db_0(a''+\db_0^*\beta)+ (a''+\db_0^*\beta)\w (a''+\db_0^*\beta)\,\big]$$ to obtain a map with the same linearization in the $A^{0,2}$-direction at $(0,0)$. Given $\db_0$-closed $a''\in A^{0,1}(\eet)$ in $L^p_1$ and $\beta\in A^{0,2}(\eet)$ in $L^p_2$, the form $\tau := \db_0(a''+\db_0^*\beta)+(a''+\db_0^*\beta) \w (a''+\db_0^*\beta)$ is an $L^p$ section in $A^{0,2}(\eet)$. The weak derivative $\db_0\tau$ of $\tau$ is $\db_0^{}\db_0^*\beta\w(a''+\db_0^*\beta) - (a''+\db_0^*\beta)\w \db_0^{}\db_0^*\beta$, which lies in $L^p$ (again using $p>2n$ and the Sobolev embedding theorem). This implies that $\Pi^{0,2}_{\perp}\tau$ lies $L^p$, and that the composition $(a'',\beta) \mapsto \Pi^{0,2}_{\perp}\,\tau$ is continuous with respect to the $L^p_2$ topology on the domain and the $L^p$ topology on the codomain. By ellipticity of $\lap_0''$ on $A^{0,2}(\eet)$, there is a constant $K$ such that $\sup_X|e|\le K$ for any $e\in H^{0,2}$ with $\norm{e}_{L^2}=1$. Since $\Pi^{0,2}(\db_0a''+a''\w a'')=\Pi^{0,2}(a''\w a'')$, it follows that $\norm{\Pi^{0,2}(\db_0a''+a''\w a'')}_{L^p} \le Const.\norm{a''}_{L^{4}}^2$, so another application of the implicit function theorem yields There exist $\epsilon>0$ and $C>0$ with the properties that for any $a''\in A^{0,1}(\eet)$ satisfying $\norm{a''}_{L^p_1} <\epsilon$ there is a unique $\beta\in (\ker\db_0^*)^{\perp}\subsett A^{0,2}(\eet)$ satisfying $\norm{\beta}_{L^p_2}\le C\,\norm{a''}_{L^4}^2$ such that $\db_0(a''+\db_0^*\beta)+(a''+\db_0^*\beta) \w (a''+\db_0^*\beta)$ is $\db_0^*$-closed. If $\beta$ is as in this lemma, then $\norm{a''+\db_0^*\beta}_{L^p_1}$ is uniformly bounded by a constant multiple of $\norm{a''}_{L^p_1}$. In fact, if $\tilde a'' := a''+\db_0^*\beta$, then $\tau := \db_0\tilde a''+\tilde a''\w \tilde a''$ satisfies $\db_0\tau + \tilde a''\w \tau-\tau\w \tilde a''=0=\db_0^*\tau$ weakly, so by elliptic regularity it follows that $\tau$ in fact lies in $L^p_1$. Hence by , if $\norm{a''}_{L^p_1}$ (and hence $\norm{\tilde a''}_{L^p_1}$) is sufficiently small, there is a constant $C = C(d_0)$ such that $\norm{\tau}_{L^p_1} \le C\,\norm{\Pi^{0,2}\tau}_{L^2}$.  remains valid even if $d_0$ is not integrable—all that is required is a uniform $C^0$ bound on $F^{0,2}(d_0)$. The proof as given only needs modification by noting that $\db_0\tau$ involves an extra term $F^{0,2}(d_0)\w a''-a''\w F^{0,2}(d_0)$, this lying in $L^p$ if $|F(d_0)|$ is bounded in $C^0$. If $\gamma$ is a complex automorphism of $\et$ satisfying $d_0\gamma=0$, then by the uniqueness statement of the lemma, $\beta(\gamma a''\gamma^{-1})=\gamma\beta(a'')\gamma^{-1}$, at least if $\norm{\gamma a''\gamma^{-1}}_{L^4}$ is sufficiently small. From  there is a constant $c>0$ depending only on $d_0$ such that any $\alpha\in A^{0,1}$ with $\db_0\alpha=0=\db_0^*\alpha$ satisfies $\norm{\alpha}_{L^p_1}\le c\,\norm{\alpha}$. Thus there is a number $\delta>0$ depending only on $d_0$ such that $\norm{\alpha}_{L^p_1} < \epsilon$ if $\norm{\alpha}<\delta$, and for such $\alpha$ there is a unique $\beta\in A^{0,2}(\eet)$ orthogonal to $\ker\db_0^*$ with $\norm{\beta}_{L^p_2}\le C'\norm{\alpha}^2$ for which the form $\tau = \db_0a''+a''\w a''$ is $\db_0^*$-closed, where $a'' := \alpha+\db_0^*\beta$. Moreover, $\norm{\tau}_{L^p_1}\le C\norm{\Pi^{0,2}\tau}$ for some constant $C=C(d_0)$. Define the function $\Psi$ on the set of $\db_0$-harmonic forms $\alpha\in A^{0,1}(\eet)$ with $\norm{\alpha}<\delta$ that takes values in $H^{0,2}(\db_0,\eet)$ by $$\Psi(\alpha) := \Pi^{0,2}(a''\w a'') \quad \hbox{for $a''=\alpha+\db_0^*\beta$ with $\beta=\beta(\alpha)$ as in \rAG.} \myeqn$$ Then the zero set of $\Psi$ parameterises precisely the integrable connections in this $L^p_1$ neighbourhood of $d_0$, in the sense of defining a semi-universal deformation. By fixing orthonormal bases for each of $H^{0,1}$ and $H^{0,2}$ and working in these bases, it is immediate that $\Psi$ is holomorphic. As will be shown later (), in the special case that $d_0$ is Hermite-Einstein, $\Psi$ is equivariant with respect to the action of $Aut(E_0)$ on these spaces, a consequence of the third remark above. With the same notation and conventions as in the previous section, we assume from now on that $(X,\omega)$ is Kähler. The following results on bounds of slopes of subsheaves may also be of independent interest. Let $E$ be a holomorphic bundle defined by an integrable semi-connection $\db$ on $\et$ and let $d=\db+\d$ be the associated hermitian connection. Then there is a constant $C$ depending only on $\omega$ such that $$\deg(A) \le C\,\norm{\widehat F(d)}_{L^1}\quad{\rm and}\quad \norm{c_1(A)} \le C\,\big(\norm{F(d)}_{L^1}+|\deg(A)|\big)$$ for any coherent analytic sheaf $A\subset E$ with torsion-free quotient. Here $\norm{c_1(A)}$ denotes the $L^2$ norm of the harmonic $(1,1)$-form representing the image of $c_1(A)$ in $H^2_{dR}(X)$. Suppose first that $A\subset E$ is a holomorphic sub- of $E$ and let $B := E/A$ be the quotient bundle. In a unitary frame for $\et = A_{\rm h}\oplus B_{\rm h}$, the connection $d=: d_E$ has the form $$d_E=\bmatrix{d_A& \beta\cr -\beta^*& d_B}$$ where $d_A$ and $d_B$ are the induced hermitian connections on $A$ and $B$, and $\beta\in A^{0,1}(Hom(B,A))$ is a $\db$-closed $(0,1)$-form representing the extension $0\to A\to E\to B\to 0$. The curvature $F_E=F(d_E)$ has the form $$F_E= \bmatrix{F_A-\beta\w\beta^* & d_{BA}\beta\cr -d_{AB}\beta^* & F_B-\beta^*\!\w \beta}\;,$$ where $d_{BA}$ here is the connection on $Hom(B,A)$ induced by $d_A$ and $d_B$. So if $\Pi_A$ is pointwise-orthogonal projection $E\to A$, it follows that $F_A = \Pi_A\,F_E\,\Pi_A +\beta\w\beta^*$. Since $\beta$ is a $(0,1)$-form, $i\,\tr\beta\w \beta^*$ is a [*non-positive*]{} $(1,1)$-form and therefore $i\,\tr F_A \le \tr (\Pi_A\,i F_E\,\Pi_A)$. Applying $\omega^{n-1}\w$ and integrating over $X$, it follows that $c_1(A)\cdot[\omega^{n-1}]$ is bounded above by a fixed multiple of $\norm{\widehat F_E}_{L^1}$. Now if $A$ is only a subsheaf of $E$ of rank $a>0$ with torsion-free quotient $B$, replace $E$ with $\Lambda^aE$, $A$ by the maximal normal extension of $\Lambda^aA$ in $\Lambda^aE$ (which is a line bundle) and then after blowing up the zero set of the induced section of $Hom(\det A,\Lambda^aE)$ and resolving singularities, there is again an upper bound on the degree of the desingularised subsheaf in terms of the $L^1$ norm of $F_E$ on the blowup. This upper bound depends on the metric used on the blowup, but as in §§2, 3 of [@Bu2], there is a family $\omega_{\epsilon}$ of such metrics converging to the pullback of $\omega$, and the resulting limit then gives the same bound: $\deg(A)$ is bounded above by a fixed multiple of $\norm{\widehat F(d)}_{L^1}$ for any subsheaf $A\subset E$ with torsion-free quotient. To obtain uniform bounds on $\norm{c_1(A)}$, suppose again initially that $A$ is a subbundle of $E$. For notational simplicity, let $f := i\,\tr F_A$ and $g := i\,\tr(\Pi_A F_E\Pi_A)$, so from the preceding arguments, $g-f\ge 0$ as hermitian forms on $TX$. The space $H^{1,1}_{\;\mm R}(X)$ of real harmonic $(1,1)$-forms is finite dimensional, so by picking an orthonormal basis, it is apparent that there is a constant $C>0$ depending on $\omega$ such that $-C\,\omega\le \varphi\le C\,\omega$ for any $\varphi\in H^{1,1}_{\;\mm R}(X)$ with $\norm{\varphi}=1$; equivalently, $C\omega\pm\varphi\ge 0$. Therefore $ (C\omega\pm\varphi)\w(g-f)\ge 0$, implying that $\pm\varphi\w f\le (C\omega\pm\varphi)\w g -C\omega\w f$ as real $(2,2)$-forms pointwise on $X$. Applying $\omega^{n-2}\w$ and integrating over $X$, it follows that $$\pm\int_X\omega^{n-2}\w\varphi\w f \le \int_X\omega^{n-2}\w (C\omega\pm \varphi)\w g-C\int_X\omega^{n-1}\w f\;,$$ and by allowing $\varphi$ to vary over the harmonic $(1,1)$-forms of norm $1$, it follows that $\norm{c_1(A)}_{L^2} \le C'\norm{F(d_E)}_{L^1} \le C'\big(\norm{F(d_E)}+|\deg(A)|\big)$ for some new constant $C'$ independent of $d_E$. This implies the second statement of the lemma when $A$ is a sub-bundle. In the case that $A$ is only a subsheaf rather than a subbundle, the same method as earlier can be used to reduce to the case of a line subbundle on a blowup of $X$. It need only be checked that for metrics of the kind $\omega_{\epsilon}$ mentioned earlier, $\lim_{\epsilon\to 0}\norm{c_1(L)}_{L^2(\omega_{\epsilon})} = 0$ for any line bundle $L$ on the blowup that is trivial off the exceptional divisor, which is straight-forward to verify.  implies the following result, showing that semi-stability is an open condition. Let $E_0$ be a semi-stable bundle on a compact Kähler manifold $(X,\omega)$, defined by an integrable connection $d_0$. Then there exists $\epsilon>0$ such that any integrable connection $d_0+a$ with $\norm{a}_{L^2_1}+\norm{a}_{L^4}<\epsilon$ defines a semi-stable holomorphic structure. If not, there is a sequence $(a_j)\in A^{1}(\eet)$ with $\norm{a_j}_{L^2_1}+\norm{a_j}_{L^4}\to 0$ and $d_j := d_0+a_j$ integrable such that the holomorphic bundle $E_j$ defined by $d_j$ is not semi-stable, so there is a subsheaf $A_j\subset E_j$ with torsion-free quotient that strictly destabilises $E_j$. Passing to a subsequence, it can be assumed that the ranks of the sheaves $A_j$ are constant, $a$ say. The hypotheses on $a_j$ imply that $\norm{F(d_j)}_{L^2}$ is uniformly bounded and therefore so too is $\norm{F(d_j)}_{L^1}$, and consequently  yields a uniform upper bound on $\deg(A_j)$. Since $\deg(A_j)$ is also uniformly bounded below, these bounds together with the bounds on $\norm{F(d_j)}_{L^2}$ then give uniform bounds on the $L^2$ norms of the harmonic representatives of the forms representing $c_1(A_j)$ in $H^2_{dR}(X)$, and hence there is a convergent subsequence. Since the image of $H^2(X,\m Z)$ in $H^2_{dR}(X)$ is discrete, this convergent subsequence must be eventually constant, so after passing to another subsequence, it can be assumed that $c_1(A_j)$ is constant, $c$ say. Since $X$ is Kähler, $Pic^c(X)$ is a compact torus, so after passing to another subsequence, it can be supposed that $\det A_j$ converges to a holomorphic line bundle $L$ on $X$. Since $(\det A_j)^*\otimes \Lambda^a E_j$ has a non-zero holomorphic section for each $j$, so too does $L^*\otimes \Lambda^a E_0$, and therefore $\Lambda^aE_0$ is strictly destabilised by $L$. But by Theorem 2 of [@J], semi-stability of $E_0$ implies that of $\Lambda^aE_0$, giving the desired contradiction. It is worth noting that this result is rather delicate in that the Kähler class must be fixed. For example, every non-split extension of the form $0\to {\cal O}(-1,1)\to E\to {\cal O}(1,0) \to 0$ on $\m P_1\times \m P_1$ is strictly stable with respect to $\omega_t := \pi_1^*\omega_0+t\,\pi_2^*\omega_0$ for $t>2$, is semi-stable but not polystable for $t=2$, and is strictly unstable if $0<t<2$. The result also fails in the non-Kählerian case, at least when the degree fails to be topological. For example, if $L$ is a non-trivial holomorphic line bundle on an Inoue surface with $L\otimes L$ trivial, the direct sum of $L$ with the trivial line bundle ${\bf 1}$ is polystable with respect to every Gauduchon metric, every small deformation is again a direct sum, and of these, the generic one is strictly unstable with respect to every Gauduchon metric. In this case, the automorphism group $\Gamma$ of $E_0=L\oplus {\bf 1}$ acts trivially on $H^1(X,End\,E_0)$, so each of its orbits is closed. A holomorphic bundle $E$ that is a non-split extension $0\to A\to E\to B\to 0$ by semistable bundles $A,B$ of the same slope is semistable but not polystable, and cannot be separated from the direct sum $A\oplus B$ in the quotient topology on $\{integrable~semiconnections\}/\{ complex~gauge\}$. Given this, it makes sense to focus on a neighbourhoods of polystable bundles, in which case a great deal more can be said than in the previous section. The same objects and definitions as in the last section are used here, but now $E_0$ is assumed to be a polystable holomorphic bundle. The following is a minor generalisation of a well-known result essentially due to Kobayashi [@Ko1]. Although its proof is elementary, it is presented here for the reason that in some respects, it is the pivotal result used in the paper. Let $d$ be a connection on a bundle $E$ with $\wh F(d)=0$. If $s\in A^{0,0}(\et)$ satisfies $\db s=0$, then $d s=0$. The equation $\db s=0$ implies $\d\<s,s\>=\<s,\d s\>$, using the convention that $\<\cdot,\cdot\>$ is conjugate-linear in the first variable. Therefore $\dbd\<s,s\>=\<\d s\,{\buildrel\w\over,}\,\d s\> + \<s,F^{1,1}(d)s\>$. Applying $i\Lambda$, it follows $\lap''|s|^2+|\d s|^2=0$, so integration over $X$ gives $\norm{\d s}^2=0$. Note that it is not assumed that $d$ should be integrable. For a connection $d_0$ on $\et$ with central curvature $\wh F(d_0)$ that is a constant multiple of the identity, the induced connection on $\eet$ has $\wh F$ identically zero, so  implies Suppose $d_0$ is a connection on $\et$ with $i\wh F(d_0)=\lambda\,1$ for some scalar $\lambda$. If $\sigma\in A^{0,0}(\eet)$ satisfies $\db_0\sigma=0$, then $d_0\sigma=0$. This corollary yields the equivariance property of the function $\Psi$ asserted at the end of §2: Under the hypotheses of , there is a constant $\epsilon=\epsilon(d_0)>0$ with the property that for any $\db_0$-harmonic $\alpha\in A^{0,1}(\eet)$ and $\db_0$-closed $\gamma\in {\cal G}$ satisfying $\norm{\alpha}_{L^2}+\norm{\gamma\alpha\gamma^{-1}}_{L^2} <\epsilon$, the form $\beta=\beta(\alpha)\in A^{0,2}(\eet)$ of  satisfies $\beta(\gamma\alpha\gamma^{-1})=\gamma\beta(\alpha)\gamma^{-1}$. Given $a''\in A^{0,1}(\eet)$ with $\norm{a''}_{L^p_1}$ sufficiently small,  guarantees the existence of a unique $\beta = \beta(a'')\in A^{0,2}(\eet)$ orthogonal to $\ker\db_0^*$ and with $\norm{\beta}_{L^p_2} \le C\norm{a''}_{L^4}^2$ such that $\tilde a'' := a''+\db_0^*\beta$ satisfies $\db_0^*\big(\db_0\tilde a''+\tilde a''\w \tilde a'')=0$. If $\gamma\in {\cal G}$ is $\db_0$-closed, then  implies that $\d_0\gamma=0$ and therefore conjugation by $\gamma$ commutes with both $\db_0$ and $\db_0^*$. Thus if $\beta$ is orthogonal to $\ker\db_0^*$ so too is $\gamma\beta\gamma^{-1}$ and also $\db_0^*\big(\db_0(\gamma\tilde a''\gamma^{-1}) +\gamma(\tilde a''\w \tilde a'')\gamma^{-1})=0$. Since $\gamma\tilde a''\gamma^{-1}=\gamma a''\gamma^{-1} +\db_0^*(\gamma\beta\gamma^{-1})$, it follows from the uniqueness statement of  that $\beta(\gamma a''\gamma^{-1}) =\gamma\beta(a'')\gamma^{-1}$ if both $a''$ and $\gamma a''\gamma^{-1}$ are sufficiently small in $L^p_1$. If $a''=\alpha$ is $\db_0$-harmonic, then so too is $\gamma\alpha\gamma^{-1}$, and the $L^p_1$ norms of these forms are uniformly bounded by a fixed multiple of their $L^2$ norms. The proof of  combined with  have the following useful consequence, which simplifies a number of subsequent arguments: Suppose $d_0$ is an integrable connection with $i\wh F(d_0)=\lambda\,1$ defining a polystable holomorphic structure $E_0$. There exists $\epsilon>0$ such that any integrable semi-connection $\db=\db_0+a''$ with $\norm{a''}_{L^p_1}<\epsilon$ defines a semi-stable holomorphic bundle $E$ with the property that any subsheaf $A\subset E$ with $\mu(A)=\mu(E)$ and with $E/A$ torsion-free is a sub- of $E$. If not, then from the proof of , there is a sequence of integrable semi-connections $\db_j=\db_0+a''_j$ with $\norm{a''_j}_{L^p_1} \to 0$ such that the corresponding holomorphic bundle $E_j$ has a subsheaf $A_j$ with $\rk A_j = a$ independent of $j$, $\mu(A_j)=\mu(E_j)=\mu(E_0)$ for all $j$, and with $E_j/A_j$ torsion-free but not locally free. Hence $\Lambda^a A_j$ is a rank $1$ subsheaf of $\Lambda^a E_j$, and the bundle $(\det A_j)^*\otimes \Lambda^aE_j$ has a holomorphic section that has a zero. After passing to a subsequence, the Hermite-Einstein connections on the line bundles $(\det A_j)^*$ can be assumed to converge to define a holomorphic line bundle $L$ on $X$, and compactness of the embedding of $L^p_1$ in $C^0$ implies that a subsequence of the integrable connections defining $(\det A_j)^*\otimes \Lambda^aE_j$ converges uniformly in $C^0$ to the Hermite-Einstein connection on $L\otimes E_0$. After scaling the sections of $(\det A_j)^*\otimes \Lambda^aE_j$ so as to have $L^2$ norm $1$, the convergence of the connections implies that a subsequence of the rescaled sections converges weakly in $L^2_1$ (say) and strongly in $L^2$ to a holomorphic section of $L\otimes \Lambda^a E_0$ of $L^2$ norm $1$. Every section in the sequence has a zero, and therefore so too does this limit: the Cauchy integral formula for a holomorphic function in a polydisk integrated over a poly-annulus shows that a sequence of holomorphic functions converging in $L^1$ on the polydisk converges uniformly on compact sub-polydisks. But the limiting section is a holomorphic section of a bundle of degree zero that admits a connection with $\wh F=0$, so it is either identically zero or nowhere zero. The conclusion of  can be strengthened to give a perturbed version; integrability of $d_0$ is again not required. Suppose $d_0$ is a connection on $\et$ with $\wh F(d_0)=0$. There is a constant $\epsilon=\epsilon(d_0)>0$ such that any connection $d_a=d_0+a$ with $a=a'+a''$, $\db_0^*a''=0$ and $\norm{a}_{L^p_1}<\epsilon$ has the property that any section $s\in \Gamma(X,\et)$ with $\db_0s+a''s=0$ must in fact satisfy $\db_0s=0=a''s$. Suppose $\db_0s+a''s=0$ for some section $s \in A^{0,0}(\et)$, and write $s=s_0+s_1$ where $\db_0s_0=0$ and $s_1$ is orthogonal in $L^2$ to the kernel of $\db_0$. By , $d_0s_0=0$. Then $\db_0s_1+a''s_1+a''s_0=0$, so after applying $\db_0^*= -i\,\Lambda\d_0$ it follows that $0=\lap_0'' s_1+i\,\Lambda (a''\w \d_0 s_1)= \lap_0'' s_1+ \db_0^*(a''s_1)$. Using the $L^2$ inner product, this implies that $\norm{\db_0s_1}^2=-\<\db_0s_1,a''s_1\> \le \sup|a''|\,\norm{\db_0s_1}\,\norm{s_1}$, and therefore $\norm{\db_0s_1} \le \sup|a''|\,\norm{s_1}$. Since $p>2n$, there is a constant $C$ such that $\sup|a''|\le C\norm{a''}_{L^p_1}$, and since $s_1$ is orthogonal to $\ker\db_0$, there is a constant $c=c(d_0)>0$ independent of $s$ such that $c\,\norm{s_1}\le \norm{\db_0s_1}$. So if $C\,\norm{a''}_{L^p_1} < c$, then $s_1$ must be $0$, giving $s=s_0\in \ker\db_0$, with $0=\db_0s+a''s=\db_0s_0+a''s_0=a''s_0$. This proposition leads to the verification of the assertion made in the introduction that if $d_0$ is an Hermite-Einstein connection on $\et$, then a neighbourhood of $[d_0]\in \Psi^{-1}(0)/Aut(E_0)$ is homeomorphic to a neighbourhood of $[d_0]$ in the set of isomorphism classes of integrable connections near $d_0$ equipped with the quotient topology: Let $d_0$ be an Hermite-Einstein connection defining a holomorphic structure $E_0$, and let $\Psi$ be the function of , defined in a neighbourhood of zero in $H^{0,1}=H^{0,1}(\et,\db_0)$ with values in $H^{0,2}$. Let ${\cal F}:= \{d_0+a\mid F^{0,2}(d_0+a)=0\}$, equipped with the $L^p_1$ topology. If $\Psi$ is restricted to a sufficiently small neighbourhood of zero, then with respect to the quotient topologies, the natural map $\Psi^{-1}(0)/Aut(E_0)\to {\cal F}/{\cal G}$ is a homeomorphism onto a neighbourhood of $[d_0]$. The continuity of the map is clear. If $\alpha_0\in H^{0,1}$, ellipticity of the $\db_0$-Laplacian implies that $\norm{\alpha_0}_{L^p_1}\le C\norm{\alpha_0}_{L^2}$ for some constant $C=C(d_0)$. Hence if $\norm{\alpha_0}_{L^2}$ is sufficiently small,  yields a unique $\beta_0\in A^{0,2}(\eet)$ orthogonal to $\ker\db_0^*$ with $\norm{\beta_0}_{L^p_2}\le C\norm{\alpha_0}_{L^2}^2$ such that $a_0'' := \alpha+\db_0^*\beta_0$ satisfies $\db_0^*(\db_0a_0''+a_0''\w a_0'')=0$. If $a=a'+a''$ is such that $\norm{a_0-a}_{L^p_1}$ is so small that  applies to $d_0+a$, then there is a unique $g\in {\cal G}$ of the form $g=\exp(\varphi)$ with $\varphi \in(\ker\db_0)^{\perp}$ and $\norm{\varphi}_{L^p_2}\le C\norm{\db_0^*a''}_{L^p}$ such that $d_0+a_1 := g\cdot (d_0+a)$ satisfies $\db_0^*a_1''=0$. Since $p>2n$, the Sobolev embedding theorem gives uniform $C^1$ estimates on $g$ and $g^{-1}$, implying that $\norm{a''-a_1''}_{L^p_1}\le C\norm{\db_0^*a''}_{L^p}=C\norm{\db_0^*(a''-a_0'')}_{L^p}$ for some new constant $C=C(d_0)$. So if $a$ is sufficiently close to $a_0$ in $L^p_1$,  now gives a uniquely determined $\alpha_1\in H^{0,1}$ and $\beta_1 \in (\ker\db_0^*)^{\perp}\subsett A^{0,2}(\eet)$ such that $a_1''=\alpha_1+\db_0^*\beta_1$, with $\db_0^*\big(\db_0a_1''+a_1''\w a_1'')=0$ and with $\norm{\beta_1}_{L^p_2} \le C\norm{\alpha_1}^2$. Thus for some new constant $C=C(d_0)$, $\norm{\alpha_1-\alpha_0}_{L^2} \le \norm{a_1''-a_0''}_{L^2} \le \norm{a_1''-a''}_{L^p_1}+\norm{a''-a_0''}_{L^p_1} \le C\norm{a''-a_0''}_{L^p_1}$. In summary, given $\alpha_0 \in H^{0,1}$ sufficiently close to $0$, for each $a\in A^{0,1}(\eet)$ sufficiently close to $a_0$ in $L^p_1$ there exists $g\in {\cal G}$ and $\alpha_1\in H^{0,1}$ such that $d_0+a$ is of the form $d_0+a = g^{-1}\cdot (d_0+a_1)$ for $a_1''=\alpha_1+\db_0^*\beta_1$ with $\norm{\alpha_1-\alpha_0}_{L^2}$ uniformly bounded by a multiple of $\norm{a-a_0}_{L^p_1}$. If ${\cal A}$ denotes $A^{0,1}(\eet)$ equipped with the $L^p_1$ norm and $\Gamma \subsett {\cal G}$ is the group of $\db_0$-closed (complex) automorphisms of $\et$, this implies that the natural map $[\alpha]\mapsto [\alpha+\db_0^*\beta]$ from a neighbourhood of $[0] \in H^{0,1}/\Gamma$ to ${\cal A}/{\cal G}$ is an open mapping, and therefore so too is the natural map $\Psi^{-1}(0)/\Gamma\to {\cal F}/{\cal G}$. To see that the mapping is 1–1, suppose that $d_0+a_0$, $d_0+a_1$ are (integrable) connections with $\db_0^*a_0''=0=\db_0^*a_1''$ that define isomorphic holomorphic structures; that is, there exists $g\in {\cal G}$ with $d_0+a_1=g\cdot(d_0+a_0)$. If $\norm{a_0}_{L^p_1}+\norm{a_1}_{L^p_2} < \epsilon$ with $\epsilon$ as in , apply that proposition to the connection on $\eet=\rm Hom(\et,\et)$ defined by $d_0+a_0$ on one side and $d_0+a_1$ on the other, with $s$ being the section $g\in\rm Hom(\et,\et)$. It follows from that result that $d_0g=0$ (so $g\in \Gamma$) and that $a_1g=ga_0$. Writing $a_j''=\alpha_j+\db_0^*\beta_j$ with $\beta_j$ determined by , orthogonality of the decomposition implies that $\alpha_1g=g\alpha_0$; that is, $\alpha_0,\,\alpha_1 \in H^{0,1}$ represent the same point in the quotient $H^{0,1}/\Gamma$. Note that again, the integrability of the connections $d_0+a_j$ is not critical; the essential ingredient is the “quasi-integrability" condition implicit in the statement of . The main application of will be to the connections on $\eet$ induced by connections $d_0$ on $\et$ with $\wh F(d_0)$ a scalar multiple of the identity, as in following theorem. Here $[x,y]=xy-yx$ is the usual Lie bracket on endomorphisms, and the reader is alerted to the fact that the hypotheses on $\sigma$ differ slightly in 1. and 2., although the conclusions are essentially the same. Let $d_0$ be a connection on $\et$ with $i\wh F(d_0)=\lambda\,1$, and let $d_0+a$ be a connection with $a=a'+a''$, where $a'=-(a'')^*$, $a''=\alpha+\db_0^*\beta$ for some $\alpha\in A^{0,1}(\eet)$ satisfying $\db_0\alpha =0=\db_0^*\alpha$ and $\beta\in A^{0,2}(\eet)$. There is a constant $\epsilon>0$ depending only on $d_0$ with the property that if $\norm{a}_{L^p_1}<\epsilon$ then the following hold for any endomorphism $\sigma\in A^{0,0}(\eet)$: =25 pt by 30pt by 10pt [1. ]{} If $\db_0\sigma+[a'',\sigma]=0$ then $d_0\sigma=0$ and $[\alpha,\sigma]=0=[\db_0^*\beta,\sigma]$. Furthermore, $[\beta,\sigma]=0$ if $\beta\in (\ker\db_0^*)^{\perp}$. [2. ]{} If $\db_0\sigma+[\alpha,\sigma]=0$ and $\beta\in (\ker\db_0^*)^{\perp}$ and $\db_0^*(\db_0^{}a''+a''\w a'')=0$ then $d_0\sigma=0$ and $[\alpha,\sigma]=0=[\beta,\sigma]$. [3. ]{} If $\db_0\sigma+[a'',\sigma]=0$ and $i\Lambda(\alpha\w\alpha^* +\alpha^*\w\alpha)$ is orthogonal to $\ker\db_0$ then $d_0\sigma=0$ and $[\alpha,\sigma]$, $[\db_0^*\beta,\sigma]$ and $[\alpha,\sigma^*]$ all vanish. Furthermore, $[\beta,\sigma]=0$ if $\beta\in (\ker\db_0^*)^{\perp}$. [4. ]{} If $\db_0\sigma+[a'',\sigma]=0$ and $\db_0^*(\db_0^{}a''+a''\w a'')=0$ and $i\Lambda(\alpha\w\alpha^*+\alpha^*\w\alpha)\in(\ker\db_0)^{\perp}$ and $\beta\in(\ker\db_0^*)^{\perp}$ then $d_0\sigma=0$ and $[\alpha,\sigma]$, $[\beta,\sigma]$, $[\alpha,\sigma^*]$ and $[\beta,\sigma^*]$ all vanish.   The connection induced on $\eet$ by $d_0$ has $\wh F=0$, so from  and , $ d_0\sigma=0=[a'',\sigma] =[\alpha,\sigma]+[\db_0^*\beta,\sigma] =[\alpha,\sigma]+\db_0^*[\beta,\sigma]\;. $ Applying $\db_0$ gives $\db_0^{}\db_0^*[\beta,\sigma]=0$, from which it follows that $\db_0^*[\beta,\sigma]=0=[\alpha,\sigma]$. If $\beta$ is orthogonal to $\ker \db_0^*$, so too is $[\beta,\sigma]$ by virtue of identity $\<\psi,[\beta,\sigma]\>=\<[\psi,\sigma^*],\beta\>$, and if $\psi\in \ker \db_0^*$ then so too is $[\psi,\sigma^*]$ since $d_0\sigma^*=0$. If $\tau := \db_0a''+a''\w a''$, then $\tau = \db_0^{}\db_0^*\beta+\alpha\w\db_0^*\beta +\db_0^*\beta\w\alpha+\alpha\w\alpha+\db_0^*\beta\w\db_0^*\beta$. Setting $\beta=0$ in 1. it follows from that case that $d_0\sigma=0=[\alpha,\sigma]$. Since $\sigma$ commutes with $\alpha$, it also commutes with $\alpha\w\alpha$, implying that $$[\sigma,\tau]= \db_0^{}\db_0^*[\sigma,\beta]+\alpha\w\db_0^*[\sigma,\beta] +\db_0^*[\sigma,\beta]\w\alpha + \db_0^*[\sigma,\beta]\w\db_0^*\beta +\db_0^*\beta\w\db_0^*[\sigma,\beta]\;. \myeqn$$ As in the proof of 1., because $\beta$ is orthogonal to $\ker\db_0^*$, so too is $[\sigma,\beta]$. On the other hand, since $\db_0^*\tau=0$, so too is $\db_0^*[\sigma,\tau]=0$. Thus $[\sigma,\beta]$ is orthogonal to $[\sigma,\tau]$. Taking the inner product on both sides of 0with $[\sigma,\beta]$, rearranging terms and estimating, it follows that $$\big\|\db_0^*[\sigma,\beta]\big\|^2 \le C\big(\sup|\alpha|+\sup|\db_0^*\beta|\big) \big\|\db_0^*[\sigma,\beta]\big\|\,\big\|[\sigma,\beta]\big\|\;.$$ Since $[\sigma,\beta]$ is orthogonal to $\ker\db_0^*$, there is a constant $c>0$ depending only on $d_0$ such that $\big\|{\db_0^*[\sigma,\beta]\big\|}^2 \ge c\nnorm{[\sigma,\beta]}^2$. Since $\alpha$ is $\db_0$-harmonic, $\sup|\alpha|$ is bounded by a constant multiple of $\norm{\alpha}$, and by  it can be assumed that $\sup|\beta|$ is bounded by a constant multiple of $\norm{\alpha}^2$. It follows that there is a new constant $\epsilon_1$ depending only on $d_0$ for which, if $\norm{\alpha}<\epsilon_1$, then necessarily $[\sigma,\beta]=0$. By 1., $d_0\sigma=0$ and $[\alpha,\sigma]=0=[\db_0^*\beta,\sigma]$. Since $\d_0\sigma=0$ it follows that $\db_0\sigma^*=0$. Using the fact that $\sigma$ commutes with $\alpha$, a short calculation gives $$\tr\big((\sigma^*\alpha-\alpha \sigma^*)^*\! \w(\sigma^*\alpha-\alpha \sigma^*)\big) =\tr\big((\sigma^*\sigma-\sigma\sigma^*) (\alpha^*\!\w\alpha+\alpha\w\alpha^*)\big)\;.$$ After applying $\omega^{n-1}\w$ to both sides and integrating over $X$, the fact that $(\sigma^*\sigma-\sigma^*\sigma)$ lies in $\ker\db_0$ together with the fact that $\Lambda(\alpha^*\!\w\alpha+\alpha\w\alpha^*)$ is orthogonal to $\ker\db_0$ imply that $\big\|\,[\sigma^*\!,\alpha]\,\big\|_{L^2}^2=0$. The vanishing of $d_0\sigma$, $[\alpha,\sigma]$ and $[\beta,\sigma]$ follows from 1. Then from 3. it follows that $[\alpha,\sigma^*]=0$, so applying 2. to $\sigma^*$ gives $[\beta,\sigma^*]=0$. Given $a''\in A^{0,1}(\eet)$ with $\norm{a''}_{L^p_1}$ small,  yields a uniquely determined $g\in {\cal G}$ near $1$ such that $\db_0^*(g\cdot a'')=0$, where $g\cdot a'' := ga''g^{-1}-\db_0gg^{-1}$. Also,  yields a uniquely determined $\beta\in A^{0,2}(\eet)$ for which $\tau := a''+\db_0^*\beta$ satisfies $\db_0\tau+\tau\w\tau \in \ker\db_0^*$. The two operations do not commute in general, but if $a_0''\in A^{0,1}(\eet)$ satisfies $\db_0^*a_0''=0$ and $\db_0^*\big(\db_0^{}a''+a''_0\w a_0''\big)=0$, then for $\beta\in A^{0,2}(\eet)$ and $g$ in ${\cal G}$ with both $\beta$ and $g-1$ sufficiently close to zero in $L^p_2$, the element $a_1''$ of $A^{0,1}(\eet)$ obtained by applying the operation of  to $g\cdot a_0''+\db_0^*\beta$ followed by the operation of  to the result of that has the form $a_1'' = \gamma a_0''\gamma^{-1}$ for some automorphism $\gamma\in \Gamma$ near $1$, where $\Gamma$ here and subsequently denotes the $\db_0$-closed elements of ${\cal G}$. All of this follows using the analysis of . The content of  implies and is implied by a corresponding result for connections on bundles of degree zero. For the purposes of transparency of the proof, the theorem was presented in terms of endomorphisms, but the alternative result is the following: Let $d_0$ be a connection on $\et$ with $\wh F(d_0)=0$, and let $d_0+a$ be a connection with $a=a'+a''$, $\db_0^*a''=0$, $a''=\alpha+\db_0^*\beta$ for some $\alpha\in A^{0,1}(\eet)$ satisfying $\db_0\alpha =0=\db_0^*\alpha$ and some $\beta\in A^{0,2}(\eet)$. There is a constant $\epsilon>0$ depending only on $d_0$ with the property that if $\norm{a}_{L^p_1}<\epsilon$ then the following hold for any section $s$ in $A^{0,0}(\et)$: =25 pt by 30pt by 10pt iIf $\db_0s+a''s=0$ then $d_0s=0$ and $\alpha s=0=(\db_0^*\beta) s$. Furthermore $\beta\,s=0$ if $\beta\in (\ker\db_0^*)^{\perp}$. iIf $\db_0s+\alpha s=0$ and $\beta\in(\ker\db_0^*)^{\perp}$ and $\db_0^*(\db_0^{}a''+a''\w a'')=0$ then $d_0s=0$ and $\alpha\,s=0=\beta\,s$; iIf $\db_0s+a''s=0$ and $i\Lambda(\alpha\w\alpha^* +\alpha^*\w\alpha) \in (\ker\db_0)^{\perp}$ then $d_0s=0$ and $\alpha s$, $(\db_0^*\beta) s$ and $\alpha^*s$ are all zero. Furthermore, $\beta\,s=0$ if $\beta\in (\ker\db_0^*)^{\perp}$. iIf $\db_0s+a''s=0$ and $\db_0^*(\db_0^{}a''+a''\w a'')=0$ and $i\Lambda(\alpha\w\alpha^*+\alpha^*\w\alpha) \in (\ker\db_0)^{\perp}$ and $\beta\in(\ker\db_0^*)^{\perp}$ then $d_0s=0$ and $\alpha s$, $\beta s$, $\alpha^*s$ and $\beta^*s$ are all zero. Hence $d_0s=0=d_as$. Let ${\bf 1}$ be the trivial line bundle equipped with its standard flat connection, and apply  to the endomorphism $\sigma = \bmatrix{0&s\cr 0&0}$ of $\et\oplus{\bf 1}$ using the direct sum connections on this bundle. When the connection $d_0+a$ of  is integrable, the second statement of the theorem implies that if the holomorphic structure $E$ defined by $d_0+a$ is not simple then the isotropy subgroup of $\Gamma$ at $\alpha$ has dimension greater than $1$; (since $\Gamma$ is acting by conjugation, the constant multiples of the identity are always in the isotropy subgroup). The converse of this follows from the third statement. As will be discussed in the next section, the condition that $i\Lambda(\alpha^*\!\w\alpha+\alpha\w\alpha^*)$ should be orthogonal to $\ker\db_0$ implies that the form $\alpha$ is an element of $H^{0,1}$ that is polystable with respect to the action of $\Gamma$ in the sense of geometric invariant theory, and in that context, the last statement of the theorem can also be interpreted as a holomorphic condition on $E$ determined by an algebraic condition on $\alpha$. This is a manifestation of one half of the “local" Hitchin-Kobayashi correspondence that is at the heart of this paper, made precise in  below. The other half, or “converse" of this is the focus of attention in §6, §7 and §8. This section commences with a summary of notions and ideas from geometric invariant theory, in particular of those related to stability, formulated to suit the purposes of this paper. The primary references for the definitions used here are [@MFK], [@Nes], [@Kir] and [@Th] in which the definitions are essentially consistent. It is a testament to the breadth of the applicability of the ideas that the literature is not always consistent in its definitions. Recall that when a reductive Lie group $G$ acts linearly on a finite-dimensional complex vector space $V$, a point $v\in V\-\{0\}$ is [*unstable*]{} for the action if $0\in \overline{G\cdot v}$, is [*semi-stable*]{} if $0\not\in \overline{G\cdot v}$, is [*polystable*]{} if $v$ is semi-stable and $G\cdot v$ is closed, and is [*stable*]{} if $v$ is polystable and the isotropy subgroup of $v$ is finite. Fixing a positive hermitian form on $V$, in the closure of each orbit there is a unique point of smallest norm. For each $v\in V$, the derivative at $1\in G$ of the function $G\owns g\mapsto \norm{g\cdot v}^2$ gives a function $\mu\: V \to \frak{g}^*$, the moment map, and its zeros on the set of semistable points are precisely the points of smallest norm in the closed orbits. If $G$ is the complexification of a real compact group $K$ acting isometrically on $V$ and if $W$ is a $G$-invariant subset contained in the open set of stable points, then $W/G$ is naturally identified with $(\mu^{-1}(0)\cap W)/K$, this a consequence of the fundamental results of Kempf and Ness [@KN]. The Hilbert-Mumford criterion states that a point $v\in V$ is stable if and only if it is stable for every $1$-parameter subgroup in $G$. A $1$-parameter subgroup is given by a homomorphism $\chi\: \m C^*\to G$, giving a representation of $\m C^*$ on $V$. The irreducible representations of $\m C^*$ are all $1$-dimensional, so $V$ splits as a direct sum of $1$-dimensional subspaces $V_j$ on each of which $\m C^*$ acts with a given weight $w_j\in \m Z$: $\m C^*\times V_j \owns (t,v) \mapsto \chi(t)\cdot v = t^{w_j}\,v$. If $v$ has $1$-dimensional orbit under $\chi$, it is clear that this orbit is closed if and only if $\m C^*\owns t\mapsto \chi(t)\cdot v\in V$ is proper, which in turn is equivalent to the condition that the maximum weight $w^{\sharp}$ and minimum weight $w^{\flat}$ of $\chi$ on the non-zero components of $v$ in its decomposition into irreducibles should differ in sign. Since $t\mapsto \chi(t^{-1})$ is another $1$-parameter subgroup of $G$ for which the maximum and minimum such weights are respectively $-w^{\flat}$ and $-w^{\sharp}$, the criterion reduces to the condition that $v$ is a stable point for the action of $G$ if and only if $w^{\flat}$ is negative for every $1$-parameter subgroup of $G$. In practice, this is the condition that $\limm_{t\to0}\big(\log\norm{\chi(t)\cdot v}/\log|t|\big) <0$ for every $1$-parameter subgroup $\chi$ of $G$, which is an analogue of the numerical condition in the definition of stability for a holomorphic vector bundle on a compact Kähler manifold. Consider now the situation discussed in the previous section: $(X,\omega)$ is a compact Kähler $n$-manifold and $\et$ is a complex vector bundle over $X$ equipped with a fixed hermitian structure. The group ${\cal G}$ of complex automorphisms of $\et$ acts on the affine space ${\cal A}$ of hermitian connections on $\et$, preserving the subspace of integrable such connections. The group ${\cal G}$ is the complexification of the group ${\cal U}$ of unitary gauge transformations. A connection $d_0\in {\cal A}$ that is integrable and has curvature $F(d_0)$ satisfying $i\wh F(d_0)=\lambda\,1$ is a minimum of the Yang-Mills functional. By , the group $\Gamma\subset {\cal G}$ of complex gauge transformations fixing $\db_0$ is the same group as the group that fixes $d_0$; these are the holomorphic automorphisms of the holomorphic structure $E_0$ defined by $d_0$. The group $\Gamma = Aut(E_0)$ acts on the space $H^1(X,End\,E_0)$ of infinitesimal deformations of $E_0$ by conjugation, and since each element of $\Gamma$ is covariantly constant with respect to $d_0$, this action preserves the harmonic subspaces $H^{0,q} = H^{0,q}(\db_0,\eet)$. From  the function $\Psi$ of  is equivariant with respect to the action of $\Gamma$ on $H^{0,1}$ (at least, near $0$), from which it follows that $\Psi^{-1}(0)\subsett H^{0,1}$ is invariant under $\Gamma$. The linearization of the action of $\Gamma$ on $H^{0,q}$ at $1\in \Gamma$ is given by the Lie bracket, $H^{0,0}\times H^{0,q}\owns (\varphi,\tau)\mapsto [\tau,\varphi]$. The group $\Gamma$ is the complexification of the subgroup $U(\Gamma)$ of $d_0$-closed unitary automorphisms of $\et$, so is a reductive Lie group. A form $\alpha\in H^{0,1}$ is of minimal norm in its orbit under $\Gamma$ if and only if $i\,\Lambda(\alpha\w\alpha^*+\alpha^*\w\alpha)\in A^{0,0}(\eet)$ is orthogonal to $\ker\db_0$. Given self-adjoint $\delta\in H^{0,0}$, let $\gamma_t := e^{t\delta}$ for $t\in \m R$. Then with $\alpha_t := \gamma_t\alpha \gamma_t^{-1}$, differentiation with respect to $t$ gives $\dot\alpha_t = [\delta,\alpha_t]=e^{t\delta}[\delta,\alpha]e^{-t\delta}$, using the fact that $\delta$ commutes with $e^{t\delta}$. Consequently $$\fract{d~}{dt}\norm{\alpha_t}^2 = 2\,\Re\big\<\alpha_t,[\delta,\alpha_t]\big\> =2\,\Re\big\<e^{2t\delta}\alpha e^{-2t\delta},[\delta,\alpha]\big\>$$ so $$\fract{d^2}{dt^2}\norm{\alpha_t}^2 = 4\,\Re\big\<[\delta,\alpha_t],[\delta,\alpha_t]\big\>\ge 0\;.$$ Thus any critical point of $t\mapsto \norm{\alpha_t}^2$ is a minimum, and since $\Re\<\alpha_t,[\delta,\alpha_t]\> = -\big\<\delta,i\Lambda(\alpha_t^{}\w\alpha_t^*+\alpha_t^*\w\alpha_t^{})\big>$ (using self-adjointness of $\delta$), the result follows. The assignment $$H^{0,1}\owns \alpha\mapsto m(\alpha) := \Pi^{0,0}\,i\,\Lambda\big(\alpha\w \alpha^*+\alpha^*\!\w \alpha\big) \in H^{0,0} \myeqn$$ maps elements of the hermitian vector space $H^{0,1}$ into the space of trace-free self-adjoint elements in $H^{0,0}$, the latter being $i$ times the Lie algebra of the group $SU(\Gamma)$ of $d_0$-closed unitary automorphisms of $\et$ with unit determinant, which is canonically identified with its real dual. In view of this lemma, it is natural to presume that $m$ is a moment map for the action of $SU(\Gamma)$ on $H^{0,1}$, where the latter is equipped with the symplectic form $$H^{0,1}\times H^{0,1}\owns (\alpha,\beta) \mapsto \underline{\omega}(\alpha,\beta) := \Int_X i\Lambda\,\tr\big(\alpha\w\beta^* -\beta\w\alpha^*\big)\dv=\<\alpha,\beta\>-\<\beta,\alpha\>\in \m R\;,$$ (with the complex structure $J(\alpha) = i\,\alpha$). Using the definition in [@Nes], to prove that this is the case, it must be shown that $m$ is equivariant with respect to the action of $SU(\Gamma)$ on $H^{0,1}$ and that $dm_{\delta}(\dot\alpha)=\underline{\omega}(X_{\delta},\dot\alpha)$ for each skew-adjoint $\delta\in H^{0,0}$ and each $\dot\alpha\in H^{0,1}$, where $X_{\delta}$ is the vector field on $H^{0,1}$ determined by $\delta$, this having the value $[\delta,\alpha]$ at $\alpha\in H^{0,1}$. Suppose that $\int_X\omega^n=1$. Then for any $u\in H^{0,0}$ and $v\in A^{0,0}(\eet)$ in $L^2$, $\tr(u^*\Pi^{0,0}v) = \int_X\tr(u^*v)\dV$. If $\varphi,\,\psi \in H^{0,0}$ are arbitrary, they are both covariantly constant and therefore so too is $\varphi^*\psi$. Hence the trace of this endomorphism is constant, which implies that $\tr(\varphi^*\psi) = \int_X\tr(\varphi^*\psi)\dV = \<\varphi,\psi\>$. Consequently, if $e_1,\dots, e_m$ is an $L^2$-orthonormal basis for $H^{0,0}$, the endomorphisms $e_1,\dots, e_m$ are pointwise orthonormal. Writing $u = \sum_j\<e_j,u\>\,e_j$ and $\Pi^{0,0}\,v = \sum_k\<e_k,v\>\,e_k$, it follows that $$\eqalign{ \tr (u^*\Pi^{0,0}v) = \sum_{j}\<u,e_j\>\,\<e_j,v\> &= \sum_{j}\<u,e_j\>\,\int_X\tr(e_j^*v)\dV \cr &= \int_X\sum_{j}\tr(u^*e_j)\,\tr(e_j^*v)\dV = \int_X\tr(u^*v)\dV\;. \qquad \qed }$$ It follows from this lemma that if $v\in A^{0,0}(\eet)$ is in $L^2$ and $u\in SU(\Gamma)$, then $\Pi^{0,0}(uvu^{-1})=u(\Pi^{0,0}v)u^{-1}$, which implies that the map $m$ of 0 is indeed equivariant with respect to the action of $SU(\Gamma)$. Furthermore, thinking of $\alpha$ as depending differentiably on a parameter $t\in \m R$ and differentiating at $t=0$, for $\dot\alpha_0 := d\alpha/dt|_{t=0}$ it follows $${d~\over dt}m(\alpha)\Big|_{t=0} = \Pi^{0,0}i\Lambda\big(\dot\alpha_0\w\alpha^*+\alpha\w\dot\alpha_0^* +\dot\alpha_0^*\w\alpha+\alpha^*\w\dot\alpha_0\big)\;.$$ Hence for $\delta\in H^{0,0}$, using  it follows that $$\eqalign{ {d~\over dt}\tr\big(m(\alpha)\delta\big)\Big|_{t=0} &= \int_X\tr i \big(\delta\,i\Lambda(\dot\alpha\w\alpha^* +\alpha\w\dot\alpha^* +\dot\alpha^*\w\alpha+\alpha^*\w\dot\alpha)\big) \cr &= \<[\alpha,\delta^*],\dot\alpha_0\>+\<\dot\alpha_0,[\alpha,\delta]\> \cr &=\underline{\omega}([\delta,\alpha],\dot\alpha_0) \quad\hbox{if $\delta=-\delta^*$.} }$$ Thus The assignment $H^{0,1}\owns \alpha\mapsto m(\alpha) =\Pi^{0,0}i\Lambda\big(\alpha\w\alpha^*+\alpha^*\w\alpha\big) \in H^{0,0}$ is the moment map for the action of $SU(\Gamma)$ on $H^{0,1}$. Combining  with the results of the previous sections gives some of the main results of this article: Let $d_0$ be an integrable connection on $\et$ with $i\wh F(d_0)=\lambda\,1$, and let $\Psi$ be the holomorphic function  defined in a neighbourhood of zero in $H^{0,1}$ with values in $H^{0,2}$. Let $\alpha\in\Psi^{-1}(0)$ be a non-zero form, and let $E_{\alpha}$ be the corresponding holomorphic structure. Then $\alpha$ is polystable with respect to the action of $\Gamma$ if $E_{\alpha}$ is polystable. Moreover, when $E_{\alpha}$ is polystable, $\alpha$ is stable with respect to the action of $\Gamma$ if and only if $E_{\alpha}$ is stable. The proof is by induction on the rank $r$ of $\et$, the case $r=1$ being elementary. Suppose first that $E_{\alpha}$ is stable but $\alpha$ is not $\Gamma$-polystable. Then the orbit of $\alpha$ under $\Gamma$ is not closed, and indeed, the infimum of $\norm{\gamma\cdot\alpha}^2$ over $\gamma\in \Gamma$ is not attained in that orbit. Let $\alpha_0\in H^{0,1}$ be a point of smallest norm in the closure of the orbit of $\alpha$ under $\Gamma$, unique up to conjugation by unitary elements of $\Gamma$. So there is a sequence $\gamma_j\in\Gamma$ with $\det\gamma_j=1$ for all $j$ such that $\norm{\gamma_j\cdot \alpha}^2$ is decreasing to $\norm{\alpha_0}^2$ but $\norm{\gamma_j}$ is not bounded. Let $\beta\in A^{0,2}(\eet)$ be the form orthogonal to $\ker\db_0^*$ determined by  such that $\db_0+\alpha+\db_0^*\beta$ is an integrable semi-connection defining the holomorphic structure $E_{\alpha}$ that is $L^p_1$-near to $E_0$, and let $a'' = \alpha+\db_0^*\beta$. If $\alpha_j := \gamma_j\cdot \alpha$ then $\norm{\alpha_j}$ is decreasing. From  it follows that $\norm{\beta_j}_{L^p_2}$ is uniformly bounded, where $\beta_j := \gamma_j\cdot\beta$. If $a_j'' := \alpha_j+\db_0^*\beta_j = \gamma_j\cdot a''$, it follows that $\norm{a_j''}_{L^p_1}$ is uniformly bounded, so after passing to a subsequence if necessary, the forms $a_j''$ converge weakly in $L^p_1$ and strongly in $C^0$ to a limit $a_{0}''$. The limiting connection $d_0+a_{0}$ is integrable, defining a holomorphic structure $E_{\alpha_0}$ on $\et$. After rescaling $\gamma_j$ to $\tilde\gamma_j$ with $\norm{\tilde\gamma_j} =1$, (a subsequence of) the automorphisms $\tilde\gamma_j$ converges to a non-zero limit $\tilde\gamma_0$ with $\norm{\tilde\gamma_0}=1$ and $\det\tilde\gamma_0=0$, this defining a holomorphic map from the holomorphic structure $E_{\alpha}$ defined by $d_0+a$ to the holomorphic structure defined by $d_0+a_{0}$. Since the latter can be assumed to be semistable (by ), there is a non-zero holomorphic map from $E_{\alpha}$ to a semistable bundle of the same degree and rank that is not an isomorphism, and this contradicts the assumption that $E_{\alpha}$ is stable. Therefore $\alpha$ is $\Gamma$-polystable. If $\alpha$ is polystable but not stable with respect to the action of $\Gamma$, the isotropy subgroup $\Gamma_{\alpha} \subset \Gamma$ of $\alpha$ has dimension greater than one. Then it follows from 2. of  that $E_{\alpha}$ is polystable but not stable, a contradiction. Suppose now that $E_{\alpha}$ is polystable but not stable. Then $E_{\alpha}$ splits into a direct sum of stable subbundles, each of the same slope. In terms of connections, the bundle-with-connection $(\et,d_0+a)$ has a unitary splitting into a direct sum of irreducible unitary bundles-with-connection. Orthogonal projection onto any of these subbundles (followed by inclusion) is a holomorphic endomorphism of $E_{\alpha}$, and by , such an endomorphism is in fact covariantly constant with respect to $d_0$ and also commutes with $\alpha$ and $\beta$. Thus the bundle-with-connection $(\et,d_0)$ has a unitary splitting into a direct sum of subbundles-with-connection, each of which defines a polystable subbundle of $E_0$ of the same slope. If $(\et,d_0)=\sum_j({\rm B}_{j},d_{0,j})$ is this last splitting, then $i\wh F(d_{0,j})=\lambda\,1$ and for some skew-adjoint $a_j\in A^1({\rm End\,}B_j)$, $(\et,d_0+a)=\sum_j({\rm B}_{j},d_{0,j} + a_j)$ corresponds to the splitting of $E_{\alpha}$ into stable components. The compatibility of the splittings implies that each $a_j$ is of the form $a_j=a_j'+a_j''$ with $a_j''=\alpha_j+\db_{0,j}^*\beta_j$ where $\alpha_j$ is a $\db_{0,j}$-harmonic $(0,1)$-form with coefficients in ${\rm End\, B_j}$. By the inductive hypothesis, each $\alpha_j$ is stable with respect to the action of $\Gamma_j$, the group of $\db_{0,j}$-closed automorphisms of $\rm B_j$. Hence there exists such an automorphism $\rho_j$ such that $\rho_j\cdot\alpha_j=: \hat\alpha_j$ is a zero of the moment map, which means that $\Pi^{0,0}_ji\Lambda(\hat\alpha_j^{}\w\hat\alpha_j^*+ \hat\alpha_j^*\w\hat\alpha_j^{}) =0$, where $\Pi^{0,0}_j$ is the orthogonal projection onto $\ker\db_{0,j}$. But this implies that there is an automorphism $\sigma\in \Gamma$ such that $\hat\alpha := \sigma\cdot\alpha$ satisfies $\Pi^{0,0}i\Lambda(\hat\alpha\w\hat\alpha^*+ \hat\alpha^*\w\hat\alpha)=0$. Therefore $\alpha$ is polystable. The last statement of the proposition follows immediately from . If $\alpha\in H^{0,1}$ satisfies $\Pi^{0,0}i\Lambda(\alpha\w\alpha^* +\alpha^*\w\alpha)=0$ and if $\Gamma_{\alpha}\subsett\Gamma$ is the isotropy subgroup of $\alpha$, then it follows from 2. of  that $\gamma$ commutes with $\beta(t\alpha)$ for $t$ sufficiently small, where $\beta(-)$ is the function defined implicitly in . Then by 3. of  it follows that $\gamma^*$ also commutes with $\alpha$, and indeed, it also commutes with $\beta(t\alpha)$. It then follows from Proposition 1.59 of [@K] that $\Gamma_{\alpha}$ is itself a complex reductive group. The group $\Gamma_{\alpha}$ acts fibrewise on $\et$, splitting each fibre into a direct sum of irreducible $\Gamma_{\alpha}$-invariant subspaces. These subspaces together form subbundles, namely the intersections of the eigen-bundles associated with the elements of $\Gamma_{\alpha}$. The fact that $\Gamma_{\alpha}$ is closed under adjoints implies that the splitting of $\et$ into $\Gamma_{\alpha}$-irreducible subbundles is a [*unitary*]{} splitting. For $|t|$ sufficiently small and $|s|\le1$, the connections $d_{t,s}=d_0+a_{t,s}$ given by $a_{t,s}''=t\alpha+s\db_0^*(t\alpha)$ preserve these splittings, and restrict to irreducible unitary connections on each of the $\Gamma_{\alpha}$-irreducible subbundles. As mentioned at the end of §4,  is one half of a local version of the Hitchin-Kobayashi correspondence: $\omega$-(poly)stability of $E_{\alpha}$ implies $\Gamma$-(poly)stability of $\alpha$, where the latter term means polystable with respect to the action of $\Gamma$ in the sense of geometric invariant theory. The more difficult task is to establish the other half; that is, the converse, and this effectively involves solving differential equations. This will be the subject of the next three sections. As in previous sections, let $(X,\omega)$ be a compact Kähler $n$-manifold and let $\et$ be a fixed complex $r$-bundle equipped with a fixed hermitian metric; all conventions and notations from previous sections are also retained. In this section, the study of §2 into an $L^p_1$ neighbourhood of a given (hermitian) connection will be continued but the focus is now is on the central component $\wh F=\Lambda F$ of the curvature $F$ rather than the $(0,2)$-component. As previously, ${\cal G}$ is the group of complex automorphisms of $\et$, with ${\cal U}\subset {\cal G}$ the subgroup preserving the given hermitian metric. Let ${\cal A}$ denote the space of hermitian connections $d=\d+\db$ on $\et$, so the action of ${\cal G}$ on ${\cal A}$ is given by $${\cal G}\times {\cal A}\owns (g,d)\;\mapsto\; g\cdot d := g^*{}^{-1}\circ \d\circ g^*+g\circ \db\circ g^{-1} \;=\; d+g^*{}^{-1}\d g^*-\db gg^{-1}\;\in\; {\cal A}\;. \myeqn$$ Unless otherwise stated, elements of ${\cal G}$ are assumed to lie in $L^p_2$ and elements of ${\cal A}$ to lie in $L^p_1$. Projection to the central component of the curvature defines a function $\Phi$ on ${\cal G}\times {\cal A}$ with values in the space of self-adjoint endomorphisms of $\et$ lying in $L^p$ given by $$\Phi \: {\cal G}\times {\cal A}\to A^{0,0}(\eet)\;, \quad \Phi(g,d) := i\Lambda F(g\cdot d)\;, \myeqn$$ and it is the properties of this function and its derivatives with respect to each of its arguments on which the analysis concentrates in this section. Let $d_0=\d_0+\db_0$ be a connection on $\et$, and let $a=a'+a''$ be an element of $A^1(\eet)$ with $a'=-(a'')^*$. If $d_a= d_0+a$, the curvature of this connection is $$F(d_a)=F(d_0)+d_0a+a\w a\;. \myeqn$$ It follows that if $a=a(t)$ depends differentiably on the real parameter $t$, then $${d~\over dt}\big[F(d_a)\big]= d_0\dot a+\dot a\w a+a\w \dot a = d_a\dot a\;, \myeqn$$ where $\cdot$ denotes differentiation with respect to $t$. The action of ${\cal G}$ on ${\cal A}$ has the explicit form $$g\cdot d_a = d_0 + \big(g^*{}^{-1}a'g^*+g^*{}^{-1}\d_0g^*\big) +\big(ga''g^{-1}-\db_0gg^{-1}\big)\;, \quad g\in {\cal G}\;. \myeqn$$ If $g=g(t)$ also depends differentiably on $t$, then by direct calculation from 0, it follows that $${d~\over dt}\big[g\cdot d_a\big] = \big(g^*{}^{-1}\dot a'g^*+\d_{g\cdot d_a}(g^*{}^{-1}\dot g^*)\big) + \big(g\dot a''g^{-1}-\db_{g\cdot d_a}(\dot gg^{-1})\big)\;. \myeqn$$ Hence, from 2 and 0, it follows that $$\fract{d~}{dt}\big[F(g\cdot d_a)\big] = d_{g\cdot d_a}\big(g^*{}^{-1}\dot a' g^*+ g\dot a''g^{-1}+\d_{g\cdot d_a}(g^*{}^{-1}\dot g^*) -\db_{g\cdot d_a}(\dot g g^{-1})\big)\;. \myeqn$$ Applying $i\Lambda$ to both sides and recalling that $\d^*=i\Lambda\db$ and $\db^*=-i\Lambda\d$ on $1$-forms, $$\eqalignno{ \fract{d~}{dt}\big[i\wh F(g\cdot d_a)\big] &= i\Lambda\db_{g\cdot d_a}(g^*{}^{-1}\dot a' g^*) +i\Lambda\d_{g\cdot d_a}(g\dot a''g^{-1}) +i\Lambda\db_{g\cdot d_a}\!\d_{g\cdot d_a}(g^*{}^{-1}\dot g^*) -i\Lambda\d_{g\cdot d_a}\!\db_{g\cdot d_a}(\dot g g^{-1}) \cr &= \d^*_{g\cdot d_a}(g^*{}^{-1}\dot a' g^*) -\db^*_{g\cdot d_a}(g\dot a''g^{-1}) +\d^*_{g\cdot d_a}\!\d^{}_{g\cdot d_a}(g^*{}^{-1}\dot g^*) +\db^*_{g\cdot d_a}\!\db^{}_{g\cdot d_a}(\dot g g^{-1}) &\eqn }$$ Since $\Lambda(\db\d+\d\db)=\wh F$, writing $\sigma := \dot gg^{-1}$ and decomposing into self-adjoint and skew-adjoint components gives the following conclusion, which will be used frequently in this section: Let $a=a(t)\in A^1(\eet)$ be a differentiable $1$-real parameter family of skew-adjoint forms, and let $g=g(t)$ be a differentiable $1$-real parameter family of complex automorphisms of $\et$. If $d_0=\d_0+\db_0$ is a connection on $\et$ and $d_a=d_0+a$, then $$\fract{d~}{dt}\big[i\wh F(g\cdot d_a)\big]= \d^*_{g\cdot d_a}(g^*{}^{-1}\dot a' g^*) -\db^*_{g\cdot d_a}(g\dot a''g^{-1}) +\lap_{g\cdot d_a}\sigma_+ -[i\wh F(d_{g\cdot d_a}),\sigma_-]\;, \myeqn$$ where $\sigma := \dot gg^{-1}$ and $\sigma_{\pm} := {1\over 2}(\sigma\pm\sigma^*)$. Taking $a\in A^1(\eet)$ to be independent of $t$, it follows from 0 that the linearization of the map ${\cal G}\owns g\mapsto i\wh F(g\cdot d_a)= \Phi(g,d_a)$ at a connection $g_0\cdot d$ is $$(D_1\Phi)_{(g_0,d_a)}(\sigma) = \lap_{g_0\cdot d_a}\sigma_+-[i\wh F(g_0\cdot d_a),\sigma_-],\, \quad \sigma\in A^{0,0}(\eet)\;.$$ In particular, if $g_0=1$ and $a=0$, the linearization at $(1,d_0)\in {\cal G}\times {\cal A}$ of the action of ${\cal G}$ on the space of $L^p_1$ (hermitian) connections is an isomorphism from the space of $L^p_2$ self-adjoint sections of $\eet$ that are orthogonal to the $d_0$-closed sections to the space of self-adjoint $L^p$ sections of $\eet$ that are again orthogonal to the kernel of $d_0$. From now on, let $d_0$ be a connection with $i\wh F(d_0)=\lambda\,1$, so by , $\ker d_0=\ker\db_0$. In general, the map ${\cal G}\times A^1(\eet)\owns (g,a)\mapsto \Phi(g,d_a)= i\,\widehat F(g\cdot d_a)$ takes values in the self-adjoint endomorphisms of $\et$, but it does not map into the space orthogonal to $\ker \db_0$. However, if $\Pi^{0,0}$ is the $L^2$ projection of $A^{0,0}(\eet)$ onto $\ker \db_0$ and $\Pi_{\;\perp}^{0,0} = 1-\Pi^{0,0}$ is the projection onto the orthogonal complement, then for any skew-adjoint $L^p_1$ section $a\in A^1(\eet)$, the composition $$A^{0,0}(\eet) \owns \varphi \mapsto \Pi_{\;\perp}^{0,0}\,i\,\widehat F\big(\exp(\varphi)\cdot(d_0+a)\big) \in A^0(\eet)$$ maps the Banach space of self-adjoint $L^p_2$ sections in $A^{0,0}(\eet)$ orthogonal to $\ker\db_0$ into the Banach space of such sections lying in $L^p$, and when $a=0$, this map has the same linearization at $0$ as the earlier map. The implicit function theorem for Banach spaces then yields: -100 There exists $\epsilon>0$ depending only on $d_0$ with the property that for each skew-adjoint section $a\in A^1(\eet)$ satisfying $\norm{a}_{L^p_1} < \epsilon$ there is a unique self-adjoint $\varphi \in (\ker \db_0)^{\perp}\subset A^{0,0}(\eet)$ for which $\Pi^{0,0}_{\perp}\,i\wh F\big(\exp(\varphi)\cdot(d_0+a)\big)=0$. Furthermore, there is a constant $C$ depending only on $d_0$ such that $\varphi$ satisfies $\norm{\varphi}_{L^p_2}\le C\,\nnorm{\Pi_{\;\perp}^{0,0}\,\Lambda(d_0a+a\w a)}_{L^p}$. The power series expansion in $\varphi$ at $\varphi=0$ of $i\,\wh F(\exp(\varphi)\cdot d_a)$ is, to first order, $$\eqalignno{ i\wh F\big(&\exp(\varphi)\cdot(d_0+a)\big) = \;i\wh F(d_0)+i\Lambda(d_0a+a\w a) -[i\wh F(d_0)+i\Lambda(d_0a+a\w a),\varphi_-] \cr &\; + \lap_0\varphi_++2i\Lambda\big(a''\w \d_0\varphi_++\d_0\varphi_+\w a''- a'\w\db_0\varphi_+-\db_0\varphi_+\w a'\big)+[d_0^*a,\varphi_+] \cr &\qquad+i\Lambda\big([a',\varphi_+]\w a'' + a''\w[a',\varphi_+] -a'\w[a'',\varphi_+]-[a'',\varphi_+]\w a'\big) +R_2(\varphi)\;,&\eqn }$$ where the term $R_2(\varphi)$ involves products of $\varphi$ and its first and second derivatives with respect to $d_a$ with at least two such factors, but where the second-order derivatives appear linearly and the first-order derivatives appear at most quadratically. Consequently, if $a$ satisfies the hypotheses of  and if $\varphi\in A^{0,0}(\eet)$ satisfies the inequality in the statement of that proposition, then $\norm{R_2(\varphi)}_{L^p} \le C\,\norm{\Lambda(d_0a+a\w a)}_{L^p}^2$ for some constant $C = C(d_0)$. If $\varphi$ is self-adjoint and if $\db_0^*a''=0$, the formula 0 simplifies somewhat. In this case, it is useful to project both sides into $\ker\db_0$ and $(\ker\db_0)^{\perp}$, respectively giving $$\eqalignno{ \Pi^{0,0}i\wh F\big(\exp(\varphi)\cdot d_a\big) &= \lambda\,1 + i\,\Pi^{0,0}\,\Lambda\big(a'\w a''+a''\w a'\big) &\eqn{\rm (a)}\cr &\qquad +i\,\Pi^{0,0}\Lambda\big([a',\varphi]\w a''+a''\w[a',\varphi] -[a'',\varphi]\w a'-a'\w [a'',\varphi]\big)+\Pi^{0,0} R_2(\varphi)\;,\cr \strutt{13}0 \global\advance\eqcount by -1 \Pi^{0,0}_{\;\perp}i\wh F\big(\exp(\varphi)\cdot d_a\big) &= \Pi^{0,0}_{\;\perp}\,i\,\Lambda\big(a'\w a''+a''\w a'\big) &\eqn{\rm (b)}\cr &\;\;\quad+\lap_0\varphi+2i\Lambda\big(a''\w \d_0\varphi+\d_0\varphi\w a'' -a'\w\db_0\varphi-\db_0\varphi\w a'\big) \cr &\quad\quad +\Pi^{0,0}_{\;\perp}i\Lambda\big([a',\varphi]\w a''+a''\w[a',\varphi] -[a'',\varphi]\w a'-a'\w [a'',\varphi]\big) +\Pi^{0,0}_{\;\perp}R_2(\varphi)\;. }$$ If $\varphi=\varphi(a)$ is the endomorphism of , the left-hand side of 0(b) vanishes, giving the equation that effectively determines $\varphi(a)$. Since $\Lambda d_0a=0$ now, the uniform estimate on $\varphi$ provided by that proposition then implies $\norm{\varphi(a)}_{L^p_2}\le C\norm{a}_{L^{2p}}^2$, so the remainder term $R_2(\varphi)$ appearing in 0 is uniformly bounded by a constant multiple of $\norm{a}_{L^{2p}}^4$, something that is also true of the other terms on the second line of 0(a). If $a''=\alpha+\db_0^*\beta$ for $\db_0$-harmonic $\alpha\in A^{0,1}(\eet)$ and $\beta\in(\db_0^*)^{\perp} \subsett A^{0,2}(\eet)$, the (negative of the) terms involving $a'\w a''+a''\w a'$ in 0 expand to $$\eqalignno{ i\,\Lambda(\alpha^*\!\w\alpha+\alpha\w\alpha^*) +i\,\Lambda(&\d_0^*\beta^*\!\w\db_0^*\beta+\db_0^*\beta\w\d_0^*\beta^*) \cr &+i\,\Lambda(\alpha^*\!\w\db_0^*\beta+\d_0^*\beta^*\!\w\alpha +\alpha\w\d_0^*\beta^*+\db_0^*\beta\w\alpha^*)\;. &\eqn }$$ If $\sigma\in A^{0,0}(\eet)$ is $\db_0$-closed, then $\d_0\sigma=0$ by , and $$\big\<\sigma,i\,\Lambda(\alpha^*\!\w\db_0^*\beta+\d_0^*\beta^*\!\w\alpha +\alpha\w\d_0^*\beta^*+\db_0^*\beta\w\alpha^*)\big\> = \<[\alpha,\sigma],\db_0^*\beta\>+\<[\db_0^*\beta,\sigma]),\alpha\>\;. \myeqn$$ The first term on the right vanishes because $\db_0[\alpha,\sigma]= 0$, and the second term vanishes because $[\db^*_0\beta,\sigma]= \db^*_0[\beta,\sigma]$; consequently the projection of the term in the second line of 1 on $\ker\db_0$ is zero. To summarize the discussion so far: Suppose that $a=a'+a''$ satisfies the hypotheses of and $a''=\alpha+\db_0^*\beta$ where $\alpha$ is $\db_0$-harmonic. If $\varphi=\varphi(a)$ is the endomorphism of that proposition, then $$\eqalignno{ i\wh F\big(\exp(\varphi)\cdot&(d_0+a)\big) -\lambda\,1 = -i\,\Pi^{0,0}\,\Lambda(\alpha^*\w\alpha+\alpha\w\alpha^*) -i\,\Pi^{0,0}\, \Lambda(\d_0^*\beta^*\w\db_0^*\beta+\db_0^*\beta\w\d_0^*\beta^*) \cr +\;i\,&\Pi^{0,0}\Lambda\big([a',\varphi]\w a''+a''\w[a',\varphi] -[a'',\varphi]\w a'-a'\w [a'',\varphi]\big)\;+R(a) &\eqn }$$ where $\norm{R(a)}_{L^p} \le C\norm{a}_{L^{2p}}^4$ for some constant $C=C(d_0)$. The term $\Pi^{0,0}\Lambda(\alpha^* \w \alpha+\alpha\w \alpha^*)$ is $O(\norm{\alpha}^2)$ in general, whereas if $\beta$ is as in , all the other terms on the right of 0 are $O(\norm{\alpha}^4)$. But if $\Pi^{0,0}\Lambda(\alpha^* \w \alpha+\alpha\w \alpha^*)$ vanishes (as considered in §5) then the whole of the right-hand side of 0 is $O(\norm{\alpha}^4)$ and the connection $\exp(\varphi(a))\cdot (d_0+a)$ is very close to having central curvature equal to $-i\,\lambda\,1$. Given that $\varphi(a)$ is orthogonal to $\ker\db_0$, it can be hoped that a small perturbation by an element of $\ker\db_0$ will yield a connection with $i\wh F \equiv\lambda\,1$. For $\gamma\in\Gamma$, $\gamma\cdot (d_0+a)=d_0+\gamma\cdot a := d_0+\gamma^*{}^{-1}a'\gamma^*+\gamma a''\gamma^{-1}$, so since $\exp(\varphi+\delta)$ is close to $\exp(\varphi)\exp(\delta)$ for small $\varphi\in (\ker\db_0)^{\perp}$ and small $\delta \in \ker\db_0$, an alternative is to perturb $\alpha$ by conjugation with an element of $\Gamma$ close to $1$. Such an argument will involve an application of the inverse function theorem in finite dimensions, for which purpose the variation in $\wh F$ as $\alpha$ is varied in this way must be determined. With ‘$sk$’ denoting skew-adjoint and ‘$sa$’ denoting self-adjoint, consider first the function $G$ defined on $A^{0,0}(\eet)\times A^1_{sk}(\eet)$ with values in $A^{0,0}_{sa}(\eet)$ defined by $$G(\psi,a) := i\wh F(\exp(\psi)\cdot(d_0+a)) \qquad \hbox{($=\Phi(\exp(\psi),d_0+a)$ for $\Phi$ as in \eFtwo).}$$ If $G_0 := \Pi^{0,0}G$ and $G_1 := \Pi^{0,0}_{\;\perp}G$, the conclusion of  is that for $a\in A^1_{sk}(\eet)$ with $\norm{a}_{L^p_1}<\epsilon$, $G_1(\varphi(a),a)\equiv 0$, where $a\mapsto \varphi(a)$ is the function specified in that proposition, the existence of which is guaranteed by the implicit function theorem. If $a$ moves in a differentiable $1$-parameter family $a(t)$, then it follows that $$0\equiv {d~\over dt}\Big[G_1(\varphi(a(t)),a(t))\Big] = \big((D_1G_1)_{(\varphi(a),a)}\circ (D\varphi)_{a}\big)(\dot a) + (D_2G_1)_{(\varphi(a),a)}(\dot a)\;,$$ where $D_1G_1$, $D_2G_1$ are respectively the partial derivatives of $G_1$ with respect to its first and second arguments and $\dot a$ denotes the derivative with respect to $t$ as before. The implicit function theorem implies that if $(\psi,a)$ is sufficiently close to $(0,0)$, then $(D_1G_1)_{(\psi,a)}$ is an isomorphism from the space of self-adjoint elements of $A^{0,0}(\eet)$ orthogonal to $\ker\db_0$ lying in $L^p_2$ to the space of self-adjoint elements of $A^{0,0}(\eet)$ orthogonal to $\ker\db_0$ lying in $L^p$, so $$(D\varphi)_{a}(\dot a) = -\big((D_1G_1)_{(\varphi(a),a)}^{-1}\circ (D_2G_1)^{}_{(\varphi(a),a)}\big)(\dot a)\;.$$ The variation in $G(\varphi(a),a)$ at $a$ is therefore given by $$\eqalignno{ {d~\over dt}\Big[G(\varphi(a),a)\Big] &= \hphantom{-}\big((D_1G)_{(\varphi(a),a)}\circ (D\varphi)_{a}\big)(\dot a) + (D_2G)_{(\varphi(a),a)}(\dot a) &\eqn \cr &=-\big((D_1G_0)_{(\varphi(a),a)}\circ (D_1G_1)_{(\varphi(a),a)}^{-1} \circ (D_2G_1)_{(\varphi(a),a)}\big)(\dot a) + (D_2G_0)_{(\varphi(a),a)}(\dot a)\;. }$$ The partial derivatives appearing here are determined by the formula : setting $d_b := \exp(\varphi(a))\cdot (d_0+a)$ and substituting $g=e^{\varphi(a)}$ into  gives $$\eqalignno{ (D_2G)_{(\varphi(a),a)}(\dot a) &= \d_b^*(e^{-\varphi(a)}\dot a' e^{\varphi(a)}) -\db_b^*(e^{\varphi(a)}\dot a'' e^{-\varphi(a)}) \cr &= e^{\varphi(a)}\d_a^*\big(e^{-2\varphi(a)}\dot a' e^{2\varphi(a)} \big)e^{-\varphi(a)} -e^{-\varphi(a)}\db_a^*\big(e^{2\varphi(a)}\dot a'' e^{-2\varphi(a)}\big) e^{\varphi(a)}\,, &\eqn }$$ where $\dot a=\dot a' +\dot a''$, with the derivatives $D_2G_0$ and $D_2G_1$ obtained by projecting into $\ker\db_0$ and its orthogonal complement respectively. Similarly, the partial derivative of $G$ with respect to its first variable $\psi$ at $(\varphi(a),a)$ is obtained from by substituting $\dot a=0$ and $\sigma=({de^{\psi}}\!/{dt})\,e^{-\psi}\big|_{\psi=\varphi(a)}$ for a $1$-parameter family $\psi(t)$ into that formula, so $$(D_1G)_{(\varphi,a)}(\sigma)= \db_b^*\db_b^{}\sigma+\d_b^*\d_b^{}\sigma^* =\lap_b\,\sigma_+-[i\wh F(d_b),\sigma_-]\,, \myeqn$$ again with the derivatives of $G_0$ and $G_1$ obtained by taking $L^2$ projection into $\ker\db_0$ and its orthogonal complement. Note that $\sigma$ is not self-adjoint in general, but satisfies $\sigma^* = e^{-\varphi(a)}\sigma e^{\varphi(a)}$. Thus the second term on the right of 0 is not zero in general, unless $i\wh F(d_b)=\lambda\,1$. Under the hypotheses of , suppose in addition that $\db_0^*a''=0$. Then there is a constant $C=C(d_0)$ such that, for skew-adjoint $\dot a\in A^1(\eet)$, $$\nnorm{\big((D_1G_0)_{(\varphi(a),a)}\circ (D_1G_1)_{(\varphi(a),a)}^{-1} \circ (D_2G_1)^{}_{(\varphi(a),a)}\big)(\dot a)}_{L^2} \le C\,\norm{a}_{L^p_1}^3\,\big(\norm{\db_0^*\dot a}_{L^p} +\norm{\dot a}^{}_{L^p}\big)\;. \myeqn$$ The assumption that $\db_0^*a''=0$ implies that $\Lambda d_0a=0$, so the bound of  implies that $\norm{\varphi}_{L^p_2}\le C\norm{a}_{L^{2p}}^2$ for some uniform constant $C$. By the Sobolev embedding theorem, there is a similar such bound on the $C^1$ norm of $\varphi$, so (given that $\norm{a}_{L^p_1}$ is sufficiently small), an arbitrary endomorphism $\psi\in A^{0,0}(\eet)$ will satisfy a pointwise bound of the form $\big|\psi-e^{\varphi}\psi e^{-\varphi}\big| \le C\norm{a}_{L^{2p}}^2\,|\psi|$, which can be seen by orthogonally diagonalizing $\varphi$ at the point in question. Since $(\db_0e^{\varphi})e^{-\varphi}$ is uniformly bounded in $C^0$ by a multiple of $\norm{a}_{L^{2p}}^2$, it follows that $b''=e^{\varphi}a''e^{-\varphi}- (\db_0e^{\varphi})e^{-\varphi}$ also satisfies a pointwise bound of the form $|b''-a''|\le C\norm{a}_{L^{2p}}^2$, and therefore $(\db_{a}e^{\varphi})e^{-\varphi} =a''-b''$ satisfies this bound; similarly, $e^{-\varphi}\db_ae^{\varphi}$ also satisfies such a bound. Consequently, for any $\chi\in A^{0,1}(\eet)$, there is a uniform pointwise bound of the form $$\big|\,e^{-\varphi}\db_a^*\big(e^{2\varphi} \chi e^{-2\varphi}\big) e^{\varphi} - \db_a^*\chi\,\big| \le C\norm{a}_{L^{2p}}^2\big(|\db_a^*\chi|+|\chi|\big)\;,$$ and since $\db_a^*\chi =\db_0^*\chi -i\Lambda(a'\w\chi+\chi\w a')$, $$\big|\,e^{-\varphi}\db_a^*\big(e^{2\varphi} \chi e^{-2\varphi}\big) e^{\varphi} - \db_a^*\chi\,\big| \le C\norm{a}_{L^{p}_1}^2\big(|\db_0^*\chi|+|\chi|\big) \myeqn$$ for some new uniform constant $C$. Taking $\chi=\dot a$ in , it follows that $$\big|(D_2G)_{(\varphi,a)}(\dot a)\big| \le C\norm{a}^2_{L^{p}_1}\big(|\db_0^*\dot a|+|\dot a|\big) \quad {\rm pointwise.} \myeqn$$ It follows from this that the $L^2$ norm of $(D_2G)_{(\varphi,a)}(\dot a)$ is uniformly bounded above by a constant multiple of $\norm{a}^2_{L^p_1}\big(\norm{\db_0^*\dot a}_{L^2}+\norm{\dot a}_{L^2}\big)$, which implies the same such bound for its orthogonal projection onto $\ker\db_0$. Since $\ker\db_0$ is finite dimensional, the $L^2$ norm on this space is equivalent to any other, so it follows from 0 that there is uniform bound of the form $$\nnorm{(D_2G_1)_{(\varphi,a)}(\dot a)}_{L^p} \le C\norm{a}^2_{L^p_1} \big(\norm{\db_0^*\dot a}_{L^p}+\norm{\dot a}_{L^p}\big)\;. \myeqn$$ From the proof of  using the implicit function theorem, the operator $(D_1G_1)_{(\varphi,a)}$ is an isomorphism from the space of self-adjoint $L^p_2$ sections of $A^{0,0}(\eet)$ orthogonal to $\ker\db_0$ to the same such space of $L^p$ sections, so $$\nnorm{(D_1G_1)_{(\varphi,a)}^{-1} \big((D_2G_1)^{}_{(\varphi,a)}\big)(\dot a)\big)}_{L^p_2} \le C\nnorm{(D_2G_1)_{(\varphi,a)}(\dot a)}_{L^p} \myeqn$$ for some new constant $C=C(d_0)$. For a section $\sigma\in A^{0,0}(\eet)$,  gives $(D_1G_0)_{(\varphi,a)}(\sigma)= \Pi^{0,0}\big(\db_b^*\db_b^{}\sigma+\d_b^*\d_b^{}\sigma^*\big)$, where $d_b$ is the connection $d_b= e^{\varphi(a)}\cdot d_a$. Since $\db_b^*\db_b^{}\sigma = \db_0^*\db_b^{}\sigma-i\Lambda(b'\w\db_b\sigma +\db_b\sigma\w b')$ for which the first term is annihilated by $\Pi^{0,0}$, estimation of the second term gives $\nnorm{\Pi^{0,0}\db_b^*\db_b^{}\sigma}_{L^2} \le C\big(\norm{b}_{L^2}\norm{\sigma}_{L^2_1} + \norm{b}_{L^4}^2\norm{\sigma}_{L^2}\big)$. The a priori $L^p_1$ bound on $a$ and the estimate on $\varphi(a)$ from  implies that $\norm{b}_{L^{2p}}$ is uniformly bounded above by a multiple of $\norm{a}_{L^{2p}}$, so $\nnorm{\Pi^{0,0}\db_b^*\db_b^{}\sigma}_{L^2} \le C\norm{a}_{L^{2p}}\norm{\sigma}_{L^2_1}$. Taking adjoints, this same estimate applies to $\Pi^{0,0}\d_b^*\d_b^{}\sigma^*$ to give $\nnorm{(D_1G_0)_{(\varphi(a),a)}(\sigma)}_{L^2} \le C\norm{a}_{L^{2p}}\norm{\sigma}_{L^2_1}$. Combining this with the estimates of 0 and 1 gives . In accordance with the strategy outlined following the statement of ,  provides sufficient information to analyse the variation of $i\wh F(e^{\varphi(a)}\cdot d_a)$ as $a''=\alpha+\db_0^*\beta(\alpha)$ is varied according to $H^{0,1}\owns \alpha\mapsto \gamma\alpha\gamma^{-1}$ for $\gamma\in \Gamma$, at least when the connections $d_a$ are sufficiently near to $d_0$; here the notion of “near" here depends on the class $\alpha\in H^{0,1}$, which is the explanation for the reference to [*near*]{} in the title of the next section. Retaining all of the notation and definitions of the previous section, suppose now that $\gamma_t\in\Gamma$ is a family depending differentiably on the real variable $t$, with $\gamma_0=1$ and $\dot\gamma_t|_{t=0}=\delta\in H^{0,0}$. Suppose that $a=a'+a''$ satisfies the hypotheses of  as well as $\db_0^*a''=0$, and let $a_t := \gamma_t\cdot a = \gamma_t^{} a''\gamma_t^{-1} +\gamma_t^*{}^{-1}a'\gamma_t^*$. Then at $t=0$, $\dot a''_0 := \dot a''_t|_{t=0}=-[a'',\delta]$, so $\db_0^*\dot a_0''=0$ and  gives an estimate of the contribution of the first term on the right of  to the variation in $i\wh F$ in terms of $\nnorm{[a'',\delta]}_{L^p}$. But since $\delta$ is $d_0$-closed, its $C^0$ norm is bounded by a uniform multiple of its $L^2$ norm so $\norm{[a'',\delta]}_{L^p} \le C\norm{a}_{L^p} \norm{\delta}$ for $C=C(d_0)$, and therefore that contribution is uniformly bounded above by a constant multiple of $\norm{a}_{L^p_1}^4\norm{\delta}$, (the fourth power coming from ). It follows that if $\norm{a}_{L^p_1}$ is sufficiently small, the dominant term in the variation of $i\wh F$ is that coming from $(D_2G_0)_{(\varphi_0,a)}\big([a'',\delta]_+\big)$, given by the projection of  onto $\ker\db_0$, provided that this is appropriately non-degenerate as a function of $\delta$. As in the proof of ,  gives a bound $$\nnorm{\, e^{-\varphi}\db_a^*\big(e^{2\varphi}[a'',\delta] e^{-2\varphi}\big) e^{\varphi}-\db_a^*[a'',\delta]\,}_{L^2} \le C\,\norm{a}_{L^{p}_1}^2\,\nnorm{[a'',\delta]}\;.$$ Noting that $\dot a''=-[a'',\delta]$ and $\db_a\delta=[a'',\delta]$, it follows that $$\eqalign{ \big\<\delta,e^{-\varphi}\db_a^*\big(e^{2\varphi}[a'',\delta] e^{-2\varphi}\big) e^{\varphi}\big\> &\ge \big\<\delta,\db_a^*[a'',\delta]\big\> - C\norm{a}^2_{L^{p}_1}\,\nnorm{[a'',\delta]}\,\nnorm{\delta} \cr &= \nnorm{[a'',\delta]}^2 - C\norm{a}^2_{L^{p}_1}\,\nnorm{[a'',\delta]}\,\nnorm{\delta}\cr &\ge {1\over 2}\nnorm{[a'',\delta]}^2-C'\nnorm{a}_{L^p_1}^4\,\nnorm{\delta}^2\;. }$$ By taking adjoints and using $\dot a' = +[a',\delta^*]$, the same estimates apply to the other term in  with $\delta$ replaced by $\delta^*$. Then if $\delta$ is taken to be self-adjoint, by combining these estimates with those of  the following conclusion is reached: Under the hypotheses of , assume in addition that $\db_0^*a''=0$. Let $\wh \Phi$ be the function defined in a neighbourhood of $1\in \Gamma$ with values in $H^{0,0}$ given by $\wh \Phi(\gamma) := i\wh F(\varphi(\gamma\cdot a)\cdot\gamma\cdot (d_0+a))$, where $\varphi(-)$ is the function of . Then there is a constant $C=C(d_0)>0$ such that $$\nnorm{\,[a'',\delta]\,}^2 \le 2\,\big\<\delta,(D\wh \Phi)(\delta)\big\> +C\norm{a}_{L^p_1}^4\norm{\delta}^2$$ for any self-adjoint $\delta\in H^{0,0}$. With $a''$ and $\delta$ as in this proposition, write $a''=\alpha+\db_0^*\beta$ where $\alpha\in H^{0,1}$ and $\beta\in A^{0,2}(\eet)$. Given that $d_0\delta=0$, $[a'',\delta]= [\alpha,\delta]+[\db_0^*\beta,\delta]= [\alpha,\delta]+\db_0^*[\beta,\delta]$, and since this is an orthogonal decomposition, it follows that $\nnorm{[a'',\delta]}^2=\nnorm{[\alpha,\delta]}^2 +\nnorm{\db_0^*[\beta,\delta]}^2$. Assuming that $\beta$ satisfies the uniform bound of , from  it follows that there is a bound of the form $\norm{a}_{L^p_1} \le C\,\norm{\alpha}$ for some constant $C=C(d_0)$. Hence for some new constant $C=C(d_0)$, the bound of  implies a bound of the form $$\nnorm{[\alpha,\delta]}^2 +\nnorm{\db_0^*[\beta,\delta]}^2 \le 2\,\big\<\delta,(D\wh\Phi)(\delta)\big\> +C\norm{\alpha}^4\norm{\delta}^2 \quad\hbox{for $\delta=\delta^*\in H^{0,0}$.} \myeqn$$ This estimate has been derived under the assumption that $\beta$ satisfies the uniform bound given in . In particular, it applies if $\beta=\beta_0(\alpha)$ where $\beta_0(\alpha)$ is the unique element of $A^{0,2}(\eet)$ specified in that result, but more generally, it also applies if $\beta=s\,\beta_0(\alpha)$ where $s\in [0,1]$. This observation facilitates some homotopy arguments to follow. The leading term on the right of  is the negative of $m(\alpha) := \Pi^{0,0}i\Lambda(\alpha\w\alpha^*+\alpha^*\w\alpha)$. Fixing $\alpha$ temporarily, this gives a map $\Gamma \to H^{0,0}$ give by $\Gamma\owns \gamma\mapsto m(\gamma\alpha\gamma^{-1})$, and the derivative of this map at $\gamma=1$ is given by $$\big(D_{\gamma}\big[m(\gamma\alpha\gamma^{-1})\big]\big) \big|_{\gamma=1}(\delta)= \Pi^{0,0}i\Lambda\big([\delta,\alpha]\w\alpha^* +\alpha\w[\alpha^*,\delta^*]+ [\alpha^*,\delta^*]\w\alpha+\alpha^*\w[\delta,\alpha]\big)\;, ~~ \delta\in H^{0,0}\;. \myeqn$$ If $m_{\alpha}\: H^{0,0}\to H^{0,0}$ is the $\m R$-linear function on the right of 0 and $L_{\alpha}\: H^{0,0}\to H^{0,1}$ is the $\m C$-linear function $L_{\alpha}(\delta) := [\alpha,\delta]$, it is clear that the kernel of $L_{\alpha}$ is contained in that of $m_{\alpha}$. In fact, by direct calculation, for $\delta\in H^{0,0}$ with $\delta=\delta_++\delta_-$ and $(\delta_{\pm})^*=\pm\delta_{\pm}$, $$\Re\<\delta,m_{\alpha}(\delta)\> =\<\delta_+,m_{\alpha}(\delta)\> = -2\nnorm{[\alpha,\delta_+]}^2+\big\<[\delta_-,\delta_+],m(\alpha)\big>\;. \myeqn$$ Thus when acting on the self-adjoint elements of $H^{0,0}$, $m_{\alpha}$ and $L_{\alpha}$ have the same kernel. Since $m_{\alpha}$ is itself self-adjoint as an $\m R$-linear map $H^{0,0}\to H^{0,0}$, it follows that $m_{\alpha}$ maps the space of self-adjoint elements of $H^{0,0}$ that are orthogonal to $\ker L_{\alpha}$ isomorphically into the same space. At this point, arguments are simplified if it is assumed that $\alpha$ is stable with respect to the action of $\Gamma$, not just polystable. Given that $\alpha$ is $\Gamma$-polystable, the assumption of stability is equivalent to the condition that $\ker L_{\alpha}=span\,\{1\}$. For $g\in {\cal G}$,  implies $\tr i\wh F(g\cdot (d_0+a)) = \tr i\wh F(d_0+a)+i\,\dbd\log\det(g^*g) = r\lambda+i\Lambda d\,\tr a+i\,\Lambda\dbd\log\det(g^*\!g)$, so if $\db_0^*a''=0$ then $\tr i\wh F(g\cdot (d_0+a)) = r\lambda+\lap'\log\det(g^*\!g)$. If $g=\exp(\varphi)\gamma$ for self-adjoint $\varphi\in A^{0,0}(\eet)$ orthogonal to $\ker\db_0$ and $\gamma\in \Gamma$, $\log\det(g^*\!g)=2\,\tr\varphi+\log\det(\gamma\gamma^*)$, so $\tr\big(i\wh F(g\cdot (d_0+a))-\lambda\,1\big)=\lap\tr\varphi$. Hence for $\db_0^*$-closed $a''$ and $g=\exp(\varphi)\gamma$, by taking the trace-free part of $\varphi$ and rescaling $\gamma$ to have determinant $1$, it can be assumed without loss of generality that $\tr i\wh F(d_0+a) = r\lambda=\tr i\wh F(g\cdot(d_0+a))$, with $\det g\equiv 1 =\det\gamma$. If $V$ is a hermitian vector space, the real vector space $End^{\,sa}(V)$ of self-adjoint endomorphisms has a canonically induced orientation. This can be seen by induction on $\dim V$, given that the space of self-adjoint endomorphisms of $V\oplus \m C$ is canonically isomorphic to $End^{\,sa}(V)\oplus V\oplus \m R$. So the space of self-adjoint automorphisms of $V$, which is a real closed submanifold of $Aut(V)$, is orientable. Similarly, the space of trace-free self-adjoint endomorphisms of $V\oplus\m C$ is canonically isomorphic to $End^{\,sa}(V)\oplus V$, so the space of self-adjoint automorphisms of determinant $1$, which is a real closed submanifold of $Aut(V)$, is orientable. The function $\Gamma\owns\gamma\mapsto m(\gamma\alpha\gamma^{-1})$ restricted to the space $\Gamma^{sa}_0$ of self-adjoint elements of $\Gamma$ of determinant $1$ maps $\Gamma^{sa}_0$ into the space $(H^{0,0})^{sa}_0$ of trace-free self-adjoint elements of $H^{0,0}$; that is, into its tangent space at $1\in \Gamma^{sa}_{0}$, and (given that $\alpha$ is $\Gamma$-stable) its derivative at $1$ is an isomorphism between $T_{1}\Gamma^{sa}_0$ and $(H^{0,0})^{sa}_0$. Replacing $\alpha$ in with $t\alpha$ for $t>0$ sufficiently small (depending on $\alpha$), it follows from that estimate that once $t$ is sufficiently small, the kernel of the $\m R$-linear function $(D\wh\Phi)_1$ acting on the trace-free self-adjoint elements of $H^{0,0}$ is zero, and this holds for any such $t$ and any $s\beta(t\alpha)$ with $|s|\le 1$ where $\beta(\alpha)\in A^{0,2}(\eet)$ is the section specified by . Now view $\wh\Phi$ in  as a function of $\gamma\in \Gamma$ and $\db_0^*$-closed $a''\in A^{0,1}(\eet)$, taking values in the trace-free self-adjoint elements of $H^{0,0}$. When restricted to those $a''\in A^{0,1}(\eet)$ of the form $a''=t\alpha+s\,\db_0^*\beta_0^{}(t\alpha)$ for $0<t\le 1$ and $0\le s\le 1$ and self-adjoint $\gamma$ near $1$ of determinant $1$, there is an induced map $\Gamma^{\rm sa}_0 \to (H^{0,0})^{sa}_0$, for which, from , the derivative with respect to its first variable $\gamma$ at $1$ is injective once $t$ is sufficiently small, where “sufficiently" depends upon $\alpha$; more specifically, on the first non-zero eigenvalue of $L^*_{\alpha}L^{}_{\alpha}$. Suppose $\alpha\in H^{0,1}$ satisfies $\Pi^{0,0}\Lambda(\alpha\w\alpha^*+\alpha^*\w\alpha) =0$. Then for $t>0$ sufficiently small (depending on $\alpha$) and $s\in [0,1]$ there exists self-adjoint $\delta\in \ker\db_0$ and self-adjoint $\varphi\in(\ker\db_0)^{\perp}$ (both depending on $s$ and $t$) such that $i\wh F\big(\exp(\varphi)\cdot \exp(\delta)\cdot (d_0+a_{s,t})\big)=\lambda\,1$, where $a_{s,t}^{}=a'_{s,t}+a''_{s,t}$, $a''_{s,t}=t\alpha+ s\db_0^*\beta(t\alpha)$, and $\beta(-)\in A^{0,2}(\eet)$ is determined by . Furthermore, there is a constant $C=C(d_0,\alpha)$ such that $\norm{\delta}\le Ct^2$ and $\norm{\varphi}_{L^p_2} \le Ct^2$ for $t>0$ sufficiently small (depending on $\alpha$). The assumption on $\alpha$ implies that it is polystable with respect to the action of $\Gamma$. Assume initially that $\alpha$ is stable with respect to this action. If $t>0$ is sufficiently small and $s\in [0,1]$, $(D_1\wh\Phi)_{(1,a''_{s,t})}$ gives an isomorphism between the tangent space to $\Gamma^{\rm sa}_0$ at $1$ and $(H^{0,0})^{sa}_0$. The manifold $\Gamma^{\rm sa}_0$ is orientable, and the degree of $\wh\Phi(-,a''_{s,t})$ at $\lambda\,1$ is independent of $s$ and sufficiently small $t$, and is therefore equal to the degree at $\lambda\,1$ for such $t$ and $s=0$. If $\varphi_t := \varphi(a_{0,t})$, then by ,   and the remarks following that lemma, for $\delta\in H^{0,0}$ near $0$, there is a function $R_4=R_4(t,\delta)$ with $\norm{R_4(t,\delta)}_{L^p} \le Ct^4$ such that, for $\gamma_{\delta} := \exp(\delta)$, $$\eqalignno{ \wh\Phi(\gamma_{\delta},a''_{0,t}) &=\Pi^{0,0}i\wh F\big(\exp(\varphi(\gamma_{\delta}\cdot a_{0,t})) \cdot (d_0+\gamma_{\delta}\cdot a_{0,t})\big) \cr &= \lambda\,1+\lap_0\varphi(a_{0,t})-i\,t^2\Pi^{0,0}\Lambda(\alpha\w\alpha^* +\alpha^*\w\alpha)+t^2\,m_{\alpha}(\delta)+R_4(t,\delta)\cr &=\lambda\,1+t^2\,m_{\alpha}(\delta)+R_4(t,\delta)\;. &\eqn }$$ Since both $\wh\Phi-\lambda\,1$ and $m_{\alpha}$ take their values in the space of trace-free self-adjoint elements of $H^{0,0}\!$, so too does $R_4$. Since $m_{\alpha}$ is an isomorphism on this space and $R_4(t,\delta)/t^4$ is uniformly bounded as $t\to 0$, it follows that $\lambda\,1$ is in the range of $\wh\Phi(-,a''_{0,t})$ for $t$ sufficiently small, and indeed that the degree of $\wh\Phi(-,a''_{0,t})$ at $\lambda\,1$ is precisely $1$. Therefore, for $s\in [0,1]$ and $t>0$ sufficiently small (depending on $\alpha$), there exists $\gamma\in\Gamma$ such that $i\wh F\big(\exp(\varphi(\gamma\cdot a_{s,t}))\cdot\gamma\cdot (d_0+a_{s,t})\big)=\lambda\,1$. The invertibility of $m_{\alpha}$ on the space $(H^{0,0})^{sa}_0$ of trace-free self-adjoint elements of $H^{0,0}$ implies that there is a solution $\gamma= \exp(\delta(t))$ to $i\wh F\big(\exp(\gamma\cdot a_{s,t}))\cdot \gamma \cdot (d_0+a_{s,t})\big)=\lambda\,1$ depending continuously on $t$, and from 0, the section $\delta(t)$ satisfies a bound of the form $\norm{\delta(t)}\le Ct^2/c_{\alpha}$ where $c_{\alpha}$ is the lowest eigenvalue of $L_{\alpha}^*L_{\alpha}^{}$ acting on $(H^{0,0})^{sa}_0$ and $C=C(d_0,\alpha)$. Then uniform bounds on $\exp(\delta(t))$ give estimates on $\exp(\delta(t))\cdot\alpha$, and together with the estimates of , a uniform $L^p_2$ bound on $\varphi$ of the form $Ct^2/c_{\alpha}$ follows for some new constant $C$ depending on $d_0$ and $\alpha$. If now $\alpha$ is assumed to be polystable but not stable, then the isotropy subgroup $\Gamma_{\alpha}$ has dimension greater than one. Hence there are non-zero trace-free elements of $H^{0,0}$ commuting with $\alpha$, and by 2. of , so too do their adjoints, as they all do with $\beta(t\alpha)$ for any (small) $t>0$. Hence there are non-zero trace-free self-adjoint elements of $H^{0,0}$ commuting with $\alpha$ and $\beta(t\alpha)$ for any such $t$, and therefore there are non-zero trace-free self-adjoint endomorphisms of $\et$ that are covariantly constant with respect to $d_0+a_{s,t}$ for all $s,t$. Any such endomorphism gives a unitary splitting of the bundle and connections, and these splittings are all compatible with one another, including the splitting of $(\et,d_0)$. With respect to such a splitting, the form $\alpha$ splits into a collection of endomorphism-valued $(0,1)$-forms on $X$, each of which is $\db$-harmonic with respect to the induced connection. The “off-diagonal" components of $\alpha$ with respect to such a splitting are zero, since $\alpha$ commutes with the trace-free endomorphisms determining the splitting. Since $\alpha$ is of minimal norm in its orbit under $\gamma$, each of the “diagonal" components of $\alpha$ must be of minimal norm under the action of the subgroup of $\Gamma$ that is the automorphism group of the corresponding component of $(\et,d_0)$. Hence each of these components defines an element of the corresponding space that is polystable with respect to the action of the corresponding automorphism group, so it follows by induction on the rank $r$ that for $t$ sufficiently small, for each of these new bundles with connection, once $t>0$ is sufficiently small there is a complex gauge transformation that gives a new connection with $i\wh F$ a scalar multiple of $1$, (the case $r=1$ being elementary). Since the splitting of $(\et,d_0)$ is unitary and $i\wh F(d_0)=\lambda\,1$, the relevant scalar in all cases is $\lambda$. For each summand, the estimates on the corresponding endomorphisms $\delta$ and $\varphi$ imply an estimate of the required form on the direct sum connection, verifying the last statement of the theorem. A direct proof of  that does not use induction on rank appears to be possible, but raises a number of interesting representation-theoretic questions. Under the hypotheses of , if $t>0$ is sufficiently small and $d_0+a_{s,t}$ is integrable, then the corresponding holomorphic bundle is polystable, and is stable only if $\alpha$ is stable with respect to the action of $\Gamma$. By  and the results of Kobayashi and Lübke, the holomorphic bundle defined by $d_0+a_{s,t}$ is polystable. If it is not stable, then there exists a non-zero trace-free holomorphic endomorphism of this bundle. By , this endomorphism is covariantly constant with respect to $d_0$ and commutes with $\alpha$, these facts contradicting the assumption that $\alpha$ is stable with respect to the action of $\Gamma$. by 3pt From the viewpoint of deformation theory, an unobstructed infinitesimal deformation is (poly)stable with respect to the action of $\Gamma$ if and only if there is a $1$-complex parameter family of (poly)stable holomorphic structures whose tangent at $E_0$ is the given infinitesimal deformation. Of course, in the case that the latter is polystable but not stable, there may be families of bundles that are semi-stable but not polystable with that tangent vector, which will often be the case if $H^2(X,End\,E_0)$ vanishes. (Direct sums of non-isomorphic line bundles of degree zero on a torus provide an example when this is not the case.) A relatively straightforward application of the continuity method applied to the assignment $t\mapsto g_t\in {\cal G}$ solving $i\wh F(g_t\cdot(d_0+a_{s,t})=\lambda\,1$ gives a more quantitative version of , namely that the dependence of $t$ on $\alpha$ stated in the theorem can be made explicit if the constant $C$ there is replaced by by $C_0\norm{\alpha}^2/c_{\alpha}$ where $C_0$ is a constant depending only on $d_0$ and where $c_{\alpha}$ is the first non-zero eigenvalue of $L_{\alpha}^*L_{\alpha}^{}\: H^{0,0}\to H^{0,0}$. In the interests of brevity, an explicit proof will not be given. The following proposition is the companion uniqueness result to (existence). The proof of the first statement is based on the proof of Corollary 9 in [@Do3]: Suppose $\alpha\in H^{0,1}$, $a''=\alpha+\db_0^*\beta$, $a=-(a''{})^*+a''$, and $\norm{a}_{L^p_1} <\epsilon$ where $\epsilon>0$ is as in . If $i\wh F(g_1\cdot (d_0+a))=\lambda\,1=i\wh F(g_2\cdot(d_0+a))$, then $g_2=u\,g_1\gamma$ for some $u\in {\cal U}$ and $\gamma\in \Gamma_{\alpha}$. Furthermore, there exists $g_0\in {\cal G}$ with $i\wh F(g_0\cdot(d_0+a))=\lambda\,1$ satisfying the conditions that $g_0^{}=g^*_0$ is positive, $\det g_0\equiv 1$, and $\Pi^{0,0}(g^*_0g_0^{})\in H^{0,0}$ is orthogonal to the space of trace-free elements of $\ker L_{\alpha}$, with these conditions determining $g_0$ uniquely up to conjugation by unitary elements of $\Gamma_{\alpha}$. Let $d_a := d_0+a$ and set $d_b := g_1\cdot d_a$, so for $g := g_2^{}g_1^{-1}$ it follows that $g_2\cdot d_a=g\cdot d_b$ with $i\wh F(d_b)=\lambda\,1 =i\wh F(g\cdot d_b)$. After a unitary change of gauge applied to $g_2\cdot d_a$, it can be supposed that $g$ is positive self-adjoint, with $g=\exp(v)$ for some self-adjoint $v$. If $y\in \m R$ and with $d_y := \exp(y\,v)\cdot d_b$, by the function $\m R\owns y\mapsto \<i\wh F(\exp(y\,v)\cdot d_b)-\lambda\,1,v\>\in\m R$ has derivative $\<\lap_yv,v\>=\norm{d_yv}^2\ge 0$ and is therefore a non-decreasing function on $\m R$. Since it attains the value $0$ at both $y=0$ and $y=1$, it must be constant on $[0,1]$ with derivative identically $0$. Hence $d_bv=0$, which implies that $\db_a(g_1^{-1}g^{}_2)=0$. By 1. of , $\gamma := g_1^{-1}g_2^{}$ is $d_0$-covariantly constant and commutes with $\alpha$. Thus $ug_2=g_1\gamma$ for some $u\in {\cal U}$ and $\gamma\in \Gamma_{\alpha}$. To prove the second statement, note that $\Gamma$ acts freely on ${\cal G}$ by right multiplication, as does the closed subgroup $\Sigma_{\alpha}\subset \Gamma$ of elements in $\Gamma_{\alpha}$ of unit determinant. Given a fixed $g\in {\cal G}$ there is a constant $c=c(g)$ such that $c\norm{\gamma}^2 \le \norm{g\gamma}^2\le c^{-1}\norm{\gamma}^2$, so there exists $\gamma_0\in \Sigma_{\alpha}$ minimising $\norm{g\gamma}^2$ over all $\gamma\in\Sigma_{\alpha}$. The Euler-Lagrange equation for this functional on $\Sigma_{\alpha}$ is $\Pi(g_0^*g_0^{})=0$, where $\Pi$ is $L^2$-orthogonal projection onto the space of trace-free elements in $\ker L_{\alpha}$, the Lie algebra of $\Sigma_{\alpha}$. Suppose now that $g_1, \, g_2\in {\cal G}$ are as in the statement of the proposition, with $g_2=ug_1\gamma$ for some $u\in{\cal U}$ and some $\gamma\in \Gamma_{\alpha}$. Suppose in addition that both $g_1$ and $g_2$ have unit determinant, are both positive and self-adjoint, and that $\Pi^{0,0}(g_j^*g_j^{})$ is orthogonal to the trace-free elements of $\ker L_{\alpha}$ for $j=1,2$. Then for any trace-free $\phi\in\ker L_{\alpha}$, and using the fact that the trace of a covariantly constant endomorphism is constant, $$0=\<g_2^*g_2^{},\phi\> =\<\gamma^*g_1^*g_1^{}\gamma,\phi\> =\<g_1^*g_1^{},\gamma\phi\gamma^*\> =\<g_1^*g_1^{},{\tr\gamma\phi\gamma^*\over r}\,1\> =\norm{g_1}_{L^2}^2\,{\tr\gamma\phi\gamma^*\over r} = \norm{g_1}^2_{L^2}{\<\gamma^*\gamma,\phi\>\over r}\;.$$ Therefore $\gamma^*\gamma$ is a multiple of $1$, and this multiple must be $1$ since $1=\det u\,\det\gamma$. Thus $\gamma\in U(\Gamma_{\alpha})$, the group of unitary elements in $\Gamma$ commuting with $\alpha$. Then since $g_1$ and $g_2$ are both self-adjoint, $g_2^2=g_2^*g_2^{}=\gamma^*g_1^*g_1^{}\gamma= (\gamma^{-1}g_1\gamma)^2$, and positivity implies $g_2=\gamma^{-1} g_1 \gamma$. From $g_2=ug_1\gamma$, it then follows that $u=\gamma^{-1}$. In general, if $g\in {\cal G}$ satisfies $i\wh F(g\cdot d_a)=\lambda\,1$, the particular form of $a =a'+a''$ for $a''=\alpha+\db_0^*\beta$ imposes certain conditions on $h := g^*g$, at least if $\alpha$ is sufficiently small. Specifically, if $s\in A^{0,0}(\eet)$ is $\db_a$-closed, then by  so too is $s^*$ and both commute with $\alpha$ and $\beta$. Then the fact that $g^{-1}sg$ is $\db_b$-closed for $d_b := g\cdot d_a$ together with the condition that $i\wh F(d_b)=\lambda\,1$ implies that $h^{-1}s^*h$ is $\db_a$-closed and indeed, $d_a$-closed. The same applies to $s^*$, so $H^{0,0}\owns s\mapsto h^{-1}sh\in H^{0,0}$ is an automorphism of $H^{0,0}$, one that is self-adjoint and positive. Any eigenvector with eigenvalue that is not $1$ must be a $d_0$-closed nilpotent endomorphism of $\et$ that commutes with $\alpha$ and $\beta$ and their adjoints. The bundle-with-connection $(\et,d_a)$ has a unitary splitting into bundles-with-connection that are irreducible with respect to the action of $\Gamma_{\alpha}$, these splittings being compatible with splittings of $(\et,d_0)$ into a direct sum of polystable components. If $\Pi$ is orthogonal projection onto the direct sum of all components isomorphic to a given component, then $h^{-1}\Pi h$ must map that sum to itself, but is not necessarily self-adjoint. In sum, this structure raises a number of interesting questions associated with the representation theory of $\Gamma_{\alpha}$.  gives a condition under which a connection near $d_0$ has a connection with central component of the curvature equal to a scalar multiple of the identity in its orbit under ${\cal G}$, but the deficiency of the result is that how near to $d_0$ the connection must be depends on the connection itself, dictated by the relative sizes of the eigenvalues of $L_{\alpha}^*L_{\alpha}^{}$. This issue is addressed in the next section. As stated at the end of the previous section, the objective of this section is to remove the dependency of  on $\alpha$ other than through $\norm{\alpha}$. That is, retaining all of the notion of that section, the objective is to prove the following result: Let $d_0$ be a connection on $\et$ with $i\wh F(d_0)=\lambda\,1$. Then there is a constant $\epsilon=\epsilon(d_0)$ with the following property: If $\alpha\in H^{0,1}$ is polystable with respect to the action of $\Gamma$ and $\norm{\alpha}<\epsilon$, and if $\beta\in A^{0,2}(\eet)$ is as in , then there exists $g\in {\cal G}$ with $i\wh F(g\cdot (d_0+a))=\lambda\,1$, where $a=a'+a''$ for $a''=\alpha+\db_0^*\beta$. The approach to proving this result is to ensure that the analysis is performed in a sufficiently small neighbourhood of $d_0$ where the connections are well-approximated by their linearizations, which has the effect of reducing the problem to a finite-dimensional question that is naturally attacked using the methods of classical geometric invariant theory. Before commencing the proof of the theorem, there are several remarks and observations that simplify matters considerably. First, consider the case in which the rank $r$ of the bundle $\et$ is $1$. Then $\alpha$ is a harmonic $(0,1)$-form on $X$, $\beta$ must be zero since $\alpha\w\alpha=0$, and the connection $d_a=d_0+(\alpha-\alpha^*)$ has curvature $F(d_0)$, which already satisfies the condition $i\wh F=\lambda$. Thus $g\equiv 1$ solves the equation. In the general case, if $i\wh F(g\cdot(d_0+a)) =\lambda\,1$, then on taking the trace of both sides it follows that $i\Lambda\big(\tr(F(d_0)+d_0a+a\w a)+\ddb\log\det(g^*\!g))=r\,\lambda$, which implies that $i\Lambda\,\ddb\log\det(g^*\!g)\equiv 0$ and hence that $\det(g^*\!g)$ is constant. After rescaling $g$ by a constant, it can therefore be assumed that $|\det g|\equiv 1$. Second, given that $\alpha$ is polystable with respect to the action of $\Gamma$, it may be assumed without loss of generality that $\alpha$ is of minimal norm in its orbit under $\Gamma$, and therefore $i\Lambda(\alpha\w\alpha^*+\alpha^*\w\alpha)$ is orthogonal to $\ker\db_0$, by . Third, as it was for the proof of , if $\alpha$ is polystable but not stable, precisely the same argument using induction on $r$ that was employed at the end of the proof of  reduces the problem to the case when $\alpha$ is stable with respect to the action of $\Gamma$. Given this, the uniqueness result  implies that the only freedom in choice of $g$ is that of replacing $g$ by $ug$ for $u\in {\cal U}$. Hitherto, little use has been made of unitary gauge freedom ${\cal U}\owns u\mapsto u\cdot d$ for a connection $d$, as this is subsumed into the complex gauge freedom ${\cal G}\owns g\mapsto g\cdot d$. But since the equation $i\wh F(d)=\lambda\,1$ is invariant under unitary gauge transformations of the connection $d$, it is helpful to make use of the opportunity to place connections in good (unitary) gauges: There are constants $\epsilon>0$ and $C$ depending only on $d_0$ with the property that if $d_0+b$ is a connection with $\norm{b}_{L^p_1}<\epsilon$ then there is a unique skew-adjoint section $\psi\in A^{0,0}(\eet)$ orthogonal to $\ker d_0$ for which $d_0^*\big(e^{\psi}\cdot (d_0+b)-d_0\big)=0$, with $\norm{\psi}_{L^p_2} \le C\norm{d_0^*b}_{L^p}$. The linearization of the function ${\cal U}\owns u\mapsto d_0^*\big(ubu^{-1}-d_0u\,u^{-1}\big)$ at $u=1$ and $b=0$ is $A^{0,0}(\eet)\owns \sigma \mapsto \lap_0\sigma$, which is an isomorphism from the space of skew-adjoint elements of $A^{0,0}(\eet)$ orthogonal to $\ker d_0$ lying in $L^p_2$ to the same such space of elements lying in $L^p$. Since the original function takes values in the latter space, an application of the implicit function theorem implies that there is a number $\epsilon>0$ such that the equation $d_0^*\big(e^{\psi}be^{-\psi}-(d_0e^{\psi})e^{-\psi}\big) =0$ has a unique skew-adjoint solution $\psi\in (\ker d_0)^{\perp} \subsett A^{0,0}(\eet)$ if $\norm{b}_{L^p_1}<\epsilon$, and moreover $\norm{\psi}_{L^p_2} \le C\norm{d_0^*b}_{L^p}$ for some $C=C(d_0)$. Turning now to the proof of  and retaining most of the notation of the previous section, suppose that $\alpha\in H^{0,1}$ is stable with respect to the action of $\Gamma$ and is of minimal norm in its orbit under this action, with $\norm{\alpha}=1$. For $t >0$ sufficiently small that  is valid, let $\beta_t := \beta(t\alpha)$ and let $a_t=a_t'+a_t''$ for $a_t'':= t\alpha+\db_0^*\beta_t$, with $d_t := d_0+a_t$. Fix a number $\epsilon_0\in (0,1]$, the precise value of which will be fixed later, but for the moment should satisfy the condition that for any $t\in (0,\epsilon_0]$, $a_t$ satisfies the hypotheses of , , ,  and . Now let $$\eqalign{ S:= \Big\{t_0\in (0,\epsilon_0]\;\;\vrule height 10pt depth 5pt width .2pt\;\; &\hbox{\sl for every $t\in (0,t_0]$ there exists $g\in {\cal G}$ with $\;\nnorm{(\db_{t}g)g^{-1}}_{L^p_1}<t$} \cr % $\varphi=\varphi^*\in (\ker\db_0)^{\perp}$, and $\gamma\in \Gamma$}\cr \noalign{\vskip-6pt} % &\hbox{\sl for which $g:= e^{\psi}e^{\varphi}\gamma$ satisfies &\hbox{\sl \quad for which $g\cdot d_t =: d_0+b_t$ satisfies $d_0^*b_t^{}=0\;$ and $\;i\wh F(d_0+b_t)=\lambda\,1$.}\Big\} }$$By , for $t>0$ sufficiently small (depending on $\alpha$) there exist trace-free self-adjoint $\delta\in \ker\db_0$ and $\varphi\in (\ker\db_0)^{\perp}$ with $\norm{\delta}^2+\norm{\varphi}_{L^p_2}\le C_{\alpha}t^2$ such that $i\wh F(g\cdot d_t)=\lambda\,1$ for $g=\exp(\varphi) \exp(\delta)$. Then $(\db_tg)g^{-1} =(\db_0g)g^{-1}+a_t''-ga_t''g^{-1} =(\db_0e^{\varphi})e^{-\varphi}+[a_t'',g]g^{-1}$. The first term is bounded in $L^p_1$ by $C_{\alpha}t^2$, and since $g^{-1}$ is uniformly bounded in $C^0$ whilst $\norm{g-1}_{L^p_1} \le C_{\alpha}t$, the bound $\norm{a''_t} \le Ct$ implies that $\norm{(\db_tg)g^{-1}}_{L^p_1}\le C_{\alpha}t^2$ for some new constant $C_{\alpha}$. Since $\db_0^*a''_t=0$, it follows easily that $ \nnorm{\db_0^*\big(-(\db_0g)g^{-1}+ga''_tg^{-1}\big)}_{L^p} \le C_{\alpha}t^2 $, so by , after a unitary gauge transformation $u\cdot g\cdot d_t = d_0+b_t$ so that $d_0^*b_t=0$, the complex automorphism $\tilde g =ue^{\varphi}e^{\delta}\in{\cal G}$ satisfies the requirements for $t$ to lie in $S$ once $t>0$ is sufficiently small. Thus $S$ is not empty. The fact that $S$ is open (if $\epsilon_0$ is sufficiently small) will be shown shortly, this being a straightforward consequence of the implicit function theorem. The proof that $S$ is closed is more involved, involving a priori estimates on solutions. To see that $S$ is open, suppose that $t_0\in (0,\epsilon_0)\cap S$, and let $g_0\in{\cal G}$ satisfy $\nnorm{(\db_{t_0}g_0^{})g_0^{-1}}_{L^p_1}<t_0$, $d_0^*b_0^{}=0$ for $d_0+b_0 := g_0\cdot d_{t_0}$ and $i\wh F(d_{t_0})=\lambda\,1$. The linearization of the function ${\cal G}\owns g\mapsto i\wh F(g\cdot d_{t_0})$ at $g_0\in {\cal G}$ is $A^{0,0}(\eet)\owns \sigma \mapsto d_{b_0}^*d_{b_0}^{}\sigma_+ \in A^{0,0}(\eet)$. If $\sigma$ is in the kernel of this map, then $\db_{b_0}\sigma_{{\!}_+\,}=0$, so $(\db_0+ a_{t_0}'')(g_0^{-1}\sigma_{{\!}_+\,}g_0^{})=0$. Given that $\norm{a_{t_0}}_{L^p_1} \le Ct_0$ and $t_0$ is sufficiently small, it follows from  that the endomorphism $g_0^{-1}\sigma_{{\!}_+\,}g_0^{}$ is covariantly constant with respect to $d_0$ and commutes with $\alpha$ as well as with $\beta_{t_0}$. Since $\alpha$ is $\Gamma$-stable, this implies that $g_0^{-1}\sigma_{{\!}_+\,}g_0^{}$ is a scalar multiple of the identity, and therefore so too is $\sigma_+$. Hence the kernel of $\lap_{b_0}=d_{b_0}^*d_{b_0}^{}$ acting on the trace-free self-adjoint sections of $A^{0,0}(\eet)$ is zero, and so an application of the implicit function theorem implies that there is a small neighbourhood of $t_0$ in $(0,\epsilon_0)$ that lies in $S$, proving that $S$ is open. It remains to show that $S$ is also closed. Suppose now that $(0,t_0) \subset S$, and for each $t\in (0,t_0)$, let $g_t\in {\cal G}$ satisfy $\det g_t\equiv 1$, $i\wh F(g_t\cdot d_t)=\lambda\,1$ and $d_0^*b_t^{}=0$ for $g_t\cdot d_t=:d_0+b_t$, with $\norm{(\db_t^{}g_t^{})g_t^{-1}}_{L^p_1} < t$. Since $b_t''=-(\db_0^{}g_t^{})g_t^{-1}+ g_t^{}a_t''g_t^{-1} = -(\db_t^{}g_t^{})g_t^{-1}+a_t''$, it follows that $\norm{b_t}_{L^p_1} \le Ct$ for some constant $C=C(d_0)$, and therefore the preparatory results , , , and  apply to the connection $d_0+b_t$ if $\epsilon_0$ is sufficiently small. The equation $i\wh F(d_0+b_t)=\lambda\,1=i\wh F(d_0)$ implies that $i\Lambda(d_0b_t+b_t\w b_t)=0$, which can be re-written as $\d_0^*b_t'-\db_0^*b_t''=-i\Lambda(b_t'\w b_t''+b_t''\w b_t')$. Since $0=d_0^*b_t=\d_0^*b_t'+\db_0^*b_t''$, it follows that $2\db_0^*b_t''=i\Lambda(b_t'\w b_t''+b_t''\w b_t')$, implying that $\norm{\db_0^*b_t''}_{L^p} \le Ct^2$ for some $C=C(d_0)$. Note that since $\db_0^*a''_t=0$, this is a bound on the $L^p$ norm of $\db_0^*\big((\db_tg_t)g_t^{-1}\big)$, and because of the uniform bounds on $a_t$, this can also be seen as a uniform bound on the $L^p$ norm of $\db_t^*\big((\db_t^{}g_t^{})g_t^{-1}\big)$. By , there is a unique self-adjoint $\varphi_t \in A^{0,0}(\eet)$ orthogonal to $\ker\db_0$ such that $$\db_0^*\big(e^{-\varphi_t}\db_0e^{\varphi_t}+e^{-\varphi_t}b_t''e^{\varphi_t}\big)=0 \quad {\rm with} \quad \norm{\varphi_t}_{L^p_2}\le C\norm{\db_0^*b_t''}_{L^p}\le Ct^2\;. \myeqn$$ If $d_0+c_t := \exp(-\varphi_t)\cdot (d_0+b_t) = (\exp(-\varphi_t)g_t)\cdot(d_0+a_t)$, then $\db_0^*c_t''=0 =\db_0^*a_t''$, so by the first statement of (using the connection on $\eet=\rm Hom(\et,\et)$ induced by $d_0+a_t$ and $d_0+c_t$ respectively), $e^{-\varphi_t}g_t =: \gamma_t$ is $d_0$-covariantly constant and $c_t''\gamma_t=\gamma_ta_t''$. Note that since $\varphi_0$ must be trace-free (being orthogonal to $1$) and $\det g_t =1$, it follows that $\det\gamma_t=1$. Since $\gamma_t$ is invertible, $c_t'' =t\,\gamma_t^{} \alpha\gamma_t^{-1}+\gamma_t^{}\db_0^*\beta_t\gamma_t^{-1}$, but since $t\le\epsilon_0$ is sufficiently small, $\gamma_t^{}\db_0^*\beta_t\gamma_t^{-1} =\db_0^*\beta(t\gamma_t^{}\alpha\gamma_t^{-1})$. Since $\db_0+c_t''=\exp(-\varphi_t)\cdot(\db_0+b_t'')$ and $\norm{\varphi_t}_{L^p_2}\le Ct^2$ whilst $\norm{b_t''}_{L^p_1}\le Ct$, it follows that $\norm{c_t''}_{L^p_1} \le Ct$ and in particular, $\norm{c_t''}_{L^2}\le Ct$. But since $$\norm{c_t''}^2_{L^2} =t^2\nnorm{\gamma_t^{}\alpha\gamma_t^{-1}}_{L^2}^2 +\nnorm{\db_0^*\beta(t\gamma_t^{}\alpha\gamma_t^{-1})}_{L^2}^2\;, \myeqn$$ it follows that $\alpha_t := \gamma_t^{}\alpha\gamma_t^{-1}$ is uniformly bounded in $L^2$ independently of $t$ and $\alpha$, and since $\alpha$ is stable with respect to the action of $\Gamma$, the assignment $\Gamma\owns \gamma\mapsto \norm{\gamma\alpha\gamma^{-1}}$ is proper on the elements $\gamma \in \Gamma$ that have unit determinant, and therefore $\gamma_t$ converges to some $\gamma_0\in\Gamma$ as $t\to t_0$. Since $t_0$ is sufficiently small that  applies, $\gamma_t^{}\beta_t^{}\gamma_t^{-1} =\beta(t\gamma_t^{}\alpha\gamma_t^{-1})$ converges in $L^p_2$ to $\beta(t_0\gamma_0^{}\alpha\gamma_0^{-1})$ as $t\to t_0$. The uniform $L^p_2$ bounds on $\varphi_t$ imply that these converge weakly in $L^p_2$ and by the Sobolev embedding theorem, strongly in $C^1$ to $\varphi_0\in (\ker\db_0)^{\perp}$, so the corresponding automorphisms $g_t=\exp(\varphi_t)\gamma_t$ have the same convergence. By ellipticity of the $\db_0$-Laplacians on $A^{0,1}(\eet)$ and $A^{0,2}(\eet)$, the function $\beta$ of  depends smoothly on its argument, and therefore $\beta(t\gamma_t^{}\alpha\gamma_t^{-1})$ converges smoothly to $\beta(t_0\gamma_0^{}\alpha\gamma_0^{-1})$ as $t\to t_0$. Ellipticity of the equations $\d_t^*(h_t^{-1}\d_t^{}h_t^{}) = \lambda\,1-h_t^{-1}F(d_t^{})h_t^{}$ for $h_t := g_t^*g_t^{}$ and smooth dependence of $a_t$ on $t$ then imply that the family $(h_t)$ depends smoothly on $t$, and the uniqueness of the unitary gauge stated in then gives smooth dependence of $(g_t)$ on $t$, so $g_t$ converges smoothly to some $g_0\in {\cal G}$ as $t\to t_0$, with $d_0+b_0 := g_0\cdot(d_0+a_{t_0})$ satisfying $i\wh F(d_0+b_0)=\lambda\,1$, $d_0^*b_0^{}=0$, and $\nnorm{(\db_{t_0}g_0^{})g_0^{-1}}_{L^p_1} \le t_0$. The proof that $S$ is closed will be complete if it can be shown that this is a strict inequality, but ensuring this is the critical issue. As above, $\alpha_t := \gamma_t\cdot \alpha = \gamma_t^{}\alpha\gamma_t^{-1}$ is uniformly bounded in $L^2$ and hence in $L^p_2$ independent of $t\in (0,\epsilon_0]$ and of $\alpha$. Since $c_t''=t\alpha_t +\db_0^*\beta(t\alpha_t)$, $\norm{c_t''-t\alpha_t}_{L^2} \le C\norm{\beta(t\alpha_t)}_{L^p_1} \le C\norm{t\alpha_t}_{L^2}^2\le Ct^2$ for some uniform constant $C$, using here . Again using the $L^p_2$ bounds on $\varphi_t$, it then follows that $$t^2\nnorm{i\Lambda\big(\alpha_t^{}\w\alpha_t^*+\alpha_t^*\w\alpha_t\big)} \le \nnorm{i\Lambda\big(b''_t\w b'_t+b'_t\w b''_t\big)} + Ct^3 = Ct^3\;. \myeqn$$ Thus, if $ m(\sigma) := \Pi^{0,0}i\Lambda\big(\sigma\w\sigma^*+\sigma^*\w\sigma\big)$ for $\sigma\in H^{0,1}$ is the moment map for the action of $\Gamma$ as in §5, then $\norm{m(\alpha_t)} \le Ct$ for some constant $C$ that is independent of $\alpha$ and $t$. In fact, it follows from (a) that there is a better estimate, namely $\norm{m(\alpha_t)}\le Ct^2$ for some new constant $C$ independent of $\alpha$ and $t$, provided that $\epsilon_0$ is sufficiently small. The relationship between $\norm{\sigma}$ and $\norm{m(\sigma)}$ for $\sigma\in H^{0,1}$ (or more precisely, the behaviour of the projectively-invariant function $\norm{m(\sigma)}/\norm{\sigma}^2$) has been studied by a number of experts in the field, lying at the heart of the interaction between geometric invariant theory and symplectic geometry. It plays a critical role in determining the topology of quotients; see, for example, [@Kir], [@Nee]. For every $\epsilon>0$ there is a $\delta>0$ such that $\norm{\gamma\alpha\gamma^{-1}}^2-1 < \epsilon$ for every polystable $\alpha\in H^{0,1}$ with $m(\alpha)=0$ and $\norm{\alpha}=1$ for which $\norm{m(\gamma\alpha\gamma^{-1})} < \delta$. Assuming for the moment that this “uniform continuity" result holds, the proof of  is easily completed. As before, $$\db_tg_t^{}g_t^{-1} =\db_0g_t^{}g_t^{-1}+a''_t-g_t^{}a_t''g_t^{-1} = (\db_0e^{\varphi_t})e^{-\varphi_t}+a_t''- e^{\varphi_t}\gamma_t^{}a_t''\gamma_t^{-1}e^{-\varphi_t}\;,$$ so with the earlier estimates on $\norm{\varphi_t}_{L^p_2}$ and on $\norm{\beta}_{L^p_2}$, $$\norm{\db_tg_t^{}g_t^{-1}}_{L^p_1} \le t\norm{\alpha_t-\alpha}_{L^p_1} + Ct^2 \le C't\norm{\alpha_t-\alpha}_{L^2} + Ct^2 \myeqn$$ for some uniform constants $C$ and $C'$. The unitary gauge transformations provided by  have been applied to the connections with central curvature $-i\lambda\,1$, but as yet, none has been applied to the connections $d_t$. This is now done by writing $\gamma_t = p_tu_t$ for some uniquely determined positive self-adjoint $p_t\in \Gamma$ and some unitary $u_t\in \Gamma$. The convergence of $\gamma_t$ to $\gamma_0$ implies convergence of $p_t$ and $u_t$ to some positive $p_0$ and $u_0$ in $\Gamma$ respectively. Note that all the estimates above apply equally with no change of constants when $\alpha$ is replaced by $\alpha_t$ since they depended on $\alpha$ only through $\norm{\alpha}$. Since $\alpha_t' := u_t^{}\alpha u_t^{-1}$ is of minimal norm in its orbit under $\Gamma$ (i.e., that of $\alpha$), it follows that $$\nnorm{\alpha_t-\alpha_t'}^2= \norm{\alpha_t}^2-2\,\Re\big<\alpha_t,\alpha_t'\big\>+\norm{\alpha_t'}^2 =\norm{\alpha_t}^2-2\,\big\<p_t^{1/2}\cdot\alpha_t'\,,\,p_t^{1/2}\cdot\alpha_t'\big\> +\norm{\alpha_t'}^2 \le \norm{\alpha_t}^2-1\;.$$ From 0, $\norm{\db_tg_t^{}g_t^{-1}}_{L^p_1}$ will be less than $t/2$ if both $t\le t_0$ and $t_0$ is sufficiently small (depending only on $d_0$) and $\norm{\alpha_t-\alpha_t'}$ is also sufficiently small. The latter condition will hold if $\norm{\alpha_t}^2$ is sufficiently close to $1$, and by , this in turn will hold if $\norm{m(\alpha_t)}$ is sufficiently small. From , that last condition will be satisfied provided that $t$ is sufficiently small, where “sufficiently small" is a condition that depends only on $d_0$, and not on $\alpha$. Consequently, provided that $\epsilon_0$ is chosen to be sufficiently small, the set $S$ will be closed as well as open, and hence be equal to $(0,\epsilon_0]$, completing the proof of . It remains to prove , which will be a consequence of the following: Suppose $\alpha\in H^{0,1}\-\{0\}$ satisfies $m(\alpha)=0$. Then for any $\gamma\in\Gamma$, $\nnorm{\gamma\alpha\gamma^{-1}}^2-\norm{\alpha}^2 \le C\nnorm{m(\gamma\alpha\gamma^{-1})}$ for some constant $C$ depending only on $d_0$. The polystable holomorphic bundle $E_0$ splits as a direct sum $\bigoplus_{i=1}^m E_i$ of stable bundles all of the same slope. With respect to this splitting of $E_0$, a form $\tau\in H^{0,1}$ corresponds to an $m\times m$ matrix $[\tau_{j}^{\;i}]$ of $(0,1)$-forms, with $\tau_j^{\;i}$ being $\db$-harmonic with respect to the induced Hermite-Einstein connection on $Hom(E_j,E_i)$. Then $$\norm{\tau}^2 = \sum_{i=1}^m\sum_{j=1}^m\norm{\tau_j^{\;i}}^2\;.$$ Moreover, $m(\tau)=\Pi^{0,0}i\Lambda(\tau\w\tau^*+\tau^*\w\tau)$ corresponds to an $m\times m$ matrix for which the $i$-th diagonal entry is $$m(\tau)_i^{\;i}=\sqrt{-1}\sum_{j=1}^m\Pi^{0,0}_{\;i} \Lambda\big((\tau_j^{\;i})^*\w\tau_j^{\;i} +\tau_i^{\;j}\w (\tau_i^{\;j})^*\big)\;,$$ where $\Pi^{0,0}_{\;i}$ is $L^2$-orthogonal projection onto the $(i,i)$-component (of $\ker\db_0$). Since $E_i$ is stable for each $i$, $Aut(E_i)=\m C^*$ and so the projection $\Pi^{0,0}_{\;i}$ here is simply given by integrating the trace over $X$. Thus $$m(\tau)_i^{\;i}=\sum^m_{j=1}\big(\norm{\tau_j^{\;i}}^2 -\norm{\tau_i^{\;j}}^2\big)\;.$$ (More precisely, $m(\tau)_{i}^{\;i}$ is the number on the right multiplied by the identity endomorphism of $E_i$, but this fact only changes estimates by combinatorial factors bounded by a combinatorial function of $r$.) Suppose now that $\alpha\in H^{0,1}$ satisfies $m(\alpha)=0$ and $\gamma\in\Gamma$. Using a Cartan decomposition of $\Gamma$ into $\Gamma= U\,T\,U$ where $T$ is a maximal complex torus and $U=U(\Gamma)$, it follows from the left and right unitary invariance of the norm and the unitary equivariance of $m$ that $\gamma$ may be assumed to lie in $T$; that is, $\gamma=diag(t_1,\dots, t_m)$ for some $t_j\in \m C^*$. Instead of working directly with $\alpha$, it is more convenient to work with $\tau := \alpha\gamma^{-1}$, so $m(\tau\gamma)=0$ and it must be shown that $\norm{\gamma\tau}^2-\norm{\tau\gamma}^2\le C\norm{m(\gamma\tau)}$. This will follow if it can be shown that $$\sum_{j=1}^m\big(|t_j|^2\norm{\tau_j^{\;i}}^2 -|t_i|^2\norm{\tau_i^{\;j}}^2) = 0 \quad \hbox{\sl for $i=1,\dots,m$ implies that} \myeqn$$ $$\sum_{i=1}^m\sum_{j=1}^m \big(|t_i|^2\norm{\tau_j^{\;i}}^2 -|t_j|^2\norm{\tau_j^{\;i}}^2\big) \le C\,\sum_{i=1}^m\bigg|\sum_{j=1}^m\big(|t_i|^2\norm{\tau_j^{\;i}}^2 -|t_j|^2\norm{\tau_i^{\;j}}^2\big)\bigg| \myeqn$$ [*for some constant $C$*]{}, using here the fact that the $\ell_1$ and $\ell_2$ norms on $H^{0,0}$ are equivalent in this representation. Observe that $$\big(|t_i|^2-|t_j|^2\big)\big(\norm{\tau_{j}^{\;i}}^2- \norm{\tau_i^{\;j}}^2\big) = \big(|t_i|^2\norm{\tau_{j}^{\;i}}^2- |t_j|^2\norm{\tau_j^{\;i}}^2\big) +\big(|t_i|^2\norm{\tau_i^{\;j}}^2 -|t_j|^2\norm{\tau_j^{\;i}}^2\big)\;,$$ and by 1, for each fixed $i$ the second term on the right sums to zero on application of $\sum_j$. Similarly, $$\big(|t_i|^2-|t_j|^2\big)\big(\norm{\tau_{j}^{\;i}}^2+ \norm{\tau_i^{\;j}}^2\big) = \big(|t_i|^2\norm{\tau_j^{\;i}}^2-|t_j|^2\norm{\tau_i^{\;j}}^2\big) +\big(|t_i|^2\norm{\tau_{i}^{\;j}}^2-|t_j|^2\norm{\tau_j^{\;i}}^2\big)\;,$$ and again 1 implies that the second term on the right sums to zero on application of $\sum_j$. So 0 is equivalent to $$\sum_{i=1}^r\sum_{j=1}^r \big(|t_i|^2-|t_j|^2\big)\big(\norm{\tau_{j}^{\;i}}^2- \norm{\tau_i^{\;j}}^2\big) \le C\,\sum_{i=1}^r\bigg|\sum_{j=1}^r \big(|t_i|^2-|t_j|^2\big)\big(\norm{\tau_{j}^{\;i}}^2+ \norm{\tau_i^{\;j}}^2\big)\bigg|\;.$$ After renumbering, it can be assumed that $|t_i|\ge |t_j|$ if $i<j$. The summand on the left is symmetric under interchange of $i$ and $j$, so it can be written as $2\sum_{i=1}^r\sum_{j=i+1}^r \big(|t_i|^2-|t_j|^2\big)\big(\norm{\tau_{j}^{\;i}}^2- \norm{\tau_i^{\;j}}^2\big)$. Then the desired inequality will certainly follow if it can be shown that $$\sum_{i=1}^r\sum_{j=i+1}^r \big||t_i|^2-|t_j|^2\big|\big(\norm{\tau_{j}^{\;i}}^2+ \norm{\tau_i^{\;j}}^2\big) \le C\,\sum_{i=1}^r\bigg|\sum_{j=1}^r \big(|t_i|^2-|t_j|^2\big)\big(\norm{\tau_{j}^{\;i}}^2+ \norm{\tau_i^{\;j}}^2\big)\bigg|\;.$$ That this is true is a consequence of the following: Let $S = (s_{ij})$ be a skew-symmetric $m\times m$ matrix with $s_{ij}\ge 0$ if $i<j$. Then $$\sum_{i=1}^m \sum_{j=i+1}^ms_{ij} \le 2^{m-1}\sum_{i=1}^m\bigg|\sum_{j=1}^m s_{ij}\bigg|\;.$$ It will be shown inductively that for $k\in \{1,\dots, m\}$, $$\sum_{i=1}^k\sum_{j=i+1}^ms_{ij} \le 2^{k-1}\sum_{i=1}^k\bigg|\sum_{j=1}^m s_{ij}\bigg|\;. \myeqn$$ For $k=1$, the inequality clearly holds since $s_{1j}\ge 0$ for every $j$. Suppose that the inequality has been shown to hold for $k=1,\dots,\ell-1$. Then for $k=\ell$, the new term on the left is $L_{\ell} := \sum_{j=\ell+1}^m s_{\ell j}$, and on the right the new term is $R_{\ell} := \big|\sum_{j=1}^m s_{\ell j}\big|$. Let $A_{\ell} := -\sum_{j=1}^{\ell-1} s_{\ell j}\ge 0$ and $B_{\ell} := \sum_{j=\ell+1}^m s_{\ell j}\ge 0$, so $R_{\ell} = \big|B_{\ell}-A_{\ell}\big|$ and by inspection, $B_{\ell}=L_{\ell}$. By skew-symmetry of $S$, $A_{\ell}=\sum_{j=1}^{\ell-1}s_{j\ell}=\sum_{i=1}^{\ell-1}s_{i\ell} \le \sum_{i=1}^{\ell-1}\sum_{j=\ell}^ms_{ij} \le 2^{\ell-2}\sum_{i=1}^{\ell-1}\big|\sum_{j=1}^m s_{ij}\big|$, using the inductive hypotheses. Now if $B_{\ell}-A_{\ell}\le 0$, then $ L_{\ell} = B_{\ell}\le A_{\ell} \le 2^{\ell-2}\sum_{i=1}^{\ell-1}\big|\sum_{j=1}^m s_{ij}\big|\;, $ so $$\sum_{i=1}^{\ell}\sum_{j=i+1}^ms_{ij} = B_{\ell}+\sum_{i=1}^{\ell-1}\sum_{j=i+1}^m s_{ij} \le A_{\ell}+\sum_{i=1}^{\ell-1}\sum_{j=i+1}^m s_{ij} \le 2^{\ell-1}\sum_{i=1}^{\ell-1}\bigg|\sum_{j=1}^m s_{ij}\bigg| \le 2^{\ell-1}\sum_{i=1}^{\ell}\bigg|\sum_{j=1}^m s_{ij}\bigg|\;.$$ On the other hand, if $B_{\ell}-A_{\ell}\ge 0$, then $$B_{\ell}=(B_{\ell}-A_{\ell})+A_{\ell} = \big|B_{\ell}-A_{\ell}\big|+A_{\ell} \le \big|B_{\ell}-A_{\ell}\big|+2^{\ell-2} \sum_{i=1}^{\ell-1}\bigg|\sum_{j=1}^m s_{ij}\bigg|\;,$$ so by the inductive hypothesis again, $$\sum_{i=1}^{\ell}\sum_{j=i+1}^ms_{ij} = B_{\ell}+\sum_{i=1}^{\ell-1}\sum_{j=i+1}^m s_{ij} \le \big|B_{\ell}-A_{\ell}\big|+2^{\ell-1} \sum_{i=1}^{\ell-1}\bigg|\sum_{j=1}^m s_{ij}\bigg| \le 2^{\ell-1}\sum_{i=1}^{\ell}\bigg|\sum_{j=1}^m s_{ij}\bigg|\;.$$ This completes the proof of the lemma, and with it, the proof of  and hence of . There is a alternative proof of  that is less direct but which ties in well with several deeper results in the context of geometric invariant theory. Namely, one can study the downwards gradient flow for the function $H^{0,0}\owns \alpha \mapsto \norm{\alpha}^2$, which is a flow that preserves the orbits of $\Gamma$, a fact easily checked by solving $\dot\gamma_t^{}\gamma_t^{-1} =[m(\alpha_t),\alpha_t]$ for $\alpha_t=\gamma_t^{}\alpha\gamma_t^{-1}$. Modulo reparameterisation, this flow turns out to be the same as the downwards gradient flow for $H^{0,0} \owns \alpha \mapsto \norm{m(\alpha)}^2$. The latter covers the downwards gradient flow for $\m P(H^{0,0})\owns [\alpha] \mapsto \norm{m(\alpha)}\big/\norm{\alpha}^2$, $m(\alpha)/\norm{\alpha}^2$ being the moment map for the action of $\Gamma$ on $\m P(H^{0,0})$ ([@Nes]). An unpublished theorem of Duistermaat using the Łojasiewicz inequality presented in [@Le] shows that this flow defines a (strong) deformation retract of the set of $\Gamma$-polystable points onto the zero set of the moment map (analogous to the result of Neeman [@Nee] in the algebraic setting), and  follows quite easily from this. The fact that it is possible to obtain the very precise estimate given in  is perhaps a reflection of the quasi-linearity of the Yang-Mills equations, and does not ordinarily hold in the fully non-linear setting; cf. [@CS]. -1000 is a version of the Hitchin-Kobayashi correspondence for bundles in an $L^p_1$ neighbourhood of a polystable bundle, but it does not provide much detail in the case of connections and/or classes that are not polystable. Whereas non-zero elements of $H^{0,1}$ may be unstable with respect to the action of $\Gamma$—that is, zero is in the closures of their orbits,  states that there are no strictly unstable bundles near $E_0$, so the correspondence between the two different notions of stability is not perfect. However, it is nevertheless true that the interrelation between the two notions goes further than just that described by , as will be seen in this section. All notation from earlier sections continues to be retained. In general, if ${\cal E}$ is an arbitrary torsion-free semi-stable sheaf that is not stable, there is a non-zero subsheaf ${\cal S} \subset {\cal E}$ with $\mu({\cal S})=\mu(\cal E)$ and with torsion-free quotient ${\cal Q}={\cal E}/{\cal S}$ for which $\mu({\cal Q})=\mu({\cal E})$. Both ${\cal S}$ and ${\cal Q}$ are necessarily semi-stable, and if ${\cal S}$ is of maximal rank, then ${\cal Q}$ is stable. Iterating this process yields a filtration of ${\cal E}$, $0={\cal S}_0\subset {\cal S}_1 \subset {\cal S}_2\subset\cdots \subset {\cal S}_k = {\cal E}$ such that the successive quotients are all torsion-free and stable. Any such filtration is known as a [*Seshadri filtration*]{} or sometimes a Jordan-Hölder filtration, and although it is not unique, the graded object $Gr({\cal E})= \bigoplus_{j=1}^k({\cal S}_j/ {\cal S}_{j-1})$ is unique after passing to the double-dual. In the current setting of holomorphic structures $E$ near to $E_0$,  states that $E$ is semistable whilst states that any destabilising subsheaf of $E$ is a subbundle. In this case therefore, there is a Seshadri filtration of $E$ defined by subbundles, so the graded object $Gr(E)$ associated to $E$ is a polystable holomorphic structure on $\et$ close to $d_0$ in $L^p_1$. Recall from the proof of  that if $A$ is a holomorphic subbundle of $E$ with quotient $B$, then in a unitary frame for $A$ and $B$, a hermitian connection $d_E$ on $E$ and its curvature $F_E$ have the form $$d_E=\bmatrix{d_A& \beta\cr -\beta^*& d_B}\,, \qquad F_E= \bmatrix{F_A-\beta\w\beta^* & d_{{}_{BA}}\beta\cr -d_{{}_{AB}}\beta^* & F_B-\beta^*\!\w \beta}\;,$$ where now $\beta\in A^{0,1}(Hom(B,A))$ is a $\db$-closed form representing the extension $0\to A\to E\to B\to 0$ and where $d_A$ and $d_B$ are the connections on $A$ and $B$ induced by the hermitian structure and $d_E$. If $\rk A=a$ and $\rk B=b$, and if $t>0$, let $ h_t=\bmatrix{t^b&0\cr 0 & t^{-a}} $ so $h_t$ is covariantly constant with respect to the direct sum connection $d_{A\oplus B}$ on $A\oplus B$, $\det h_t=1$, and $h_t\cdot d_E$ has the same form as $d_E$ with $\beta$ replaced by $t^{a+b}\beta$. So as $t\to 0$, $h_t\cdot d_E \to d_{A\oplus B}$. Proceeding inductively, it follows easily that there exist $g_t\in {\cal G}$ such that $g_t\cdot (d_0+a)$ converges in $C^{\infty}$ to the Hermite-Einstein connection on $Gr(E)$. Note that by , every holomorphic endomorphism of $Gr(E)$ is also $\db_0$-closed, and is therefore covariantly constant with respect to $d_0$, by . Thus the automorphisms $g_t\in {\cal G}$ can even be taken to lie in $\Gamma$. The following result gives something of a converse to this observation, albeit in a rather special case. In its hypotheses, how small is “sufficiently small" is determined by the connection $d_0$, so that  is applicable. Let $d_0+a$ be an integrable connection on $\et$ with $\norm{a}_{L^p_1}$ sufficiently small, and suppose that $a''=\alpha+\db_0^*\beta$ for some $\alpha\in H^{0,1}$ and $\beta\in A^{0,2}(\eet)$. Then the following are equivalent: by 35pt ** [1. ]{} For any $\epsilon>0$ there exists $\gamma\in\Gamma$ such that $\norm{\gamma\alpha\gamma^{-1}}<\epsilon$; [2. ]{} For any $\epsilon>0$ there exists $g\in {\cal G}$ such that $\norm{g\cdot(d_0+a)-d_0}_{L^p_1} <\epsilon$. The implication 1. $\Rightarrow$ 2. follows immediately from . For the converse, assume $\epsilon>0$ is smaller than the number specified in  and let $g\in {\cal G}$ be an automorphism such that $\norm{g\cdot(d_0+a)-d_0}_{L^p_1}<\epsilon$. Applying  to the semi-connection $g\cdot(\db_0+a'')$ yields a unique $\varphi\in A^{0,0}(\eet)$ orthogonal to $\ker\db_0$ such that $d_0+\tilde a:= \exp(\varphi)\cdot g\cdot (d_0+a)$ satisfies $\db_0^*\tilde a''=0$, with $\norm{\tilde a''}$ bounded by a fixed multiple of $\epsilon$. Applying  to the connection on $\rm Hom(\et,\et)$ induced by $d_0+\tilde a$ and $d_0+a$ and the section $\exp(\varphi)g$ of this bundle, it follows that if $\epsilon$ is sufficiently small then $\exp(\varphi)g =: \gamma$ must be $\db_0$-closed with $a''\gamma=\gamma\tilde a''$. Then if $\tilde a'' = \tilde\alpha +\db_0^*\tilde\beta$, orthogonality of the decompositions gives $\gamma^{-1}\alpha\gamma = \tilde\alpha$, and $\norm{\tilde\alpha}$ is bounded by a fixed multiple of $\epsilon$ since $\norm{\tilde a''}_{L^p_1}$ is. Consider now an integrable connection $d_0+a$, with $\norm{a}_{L^p_1}$ assumed to be appropriately small and with $a''=\alpha+\db_0^*\beta$ for some $\alpha\in H^{0,1}$ and some $\beta\in A^{0,2}(\eet)$ orthogonal to the kernel of $\db_0^*$. Under the action of $\Gamma$ on $H^{0,1}$, there is a point $\bar\alpha\in H^{0,1}$ of smallest norm in the closure of the orbit of $\alpha$ unique up to conjugation by unitary elements in $\Gamma$, and this is a $\Gamma$-polystable point (if not zero). Since $\bar\alpha$ is in the closure of the orbit of $\alpha$ and each element near $0$ in this orbit lies in the analytic set $\Psi^{-1}(0)$, there is a unique section $\bar\beta\in A^{0,2}(\et)$ such that $\db_0+\bar a'' := \db_0+\bar\alpha+\db_0^*\bar\beta$ is integrable, so by  the corresponding holomorphic bundle $\bar E$ is polystable. The following is the main result of this section: With the preceding definitions, let $E$ be the holomorphic structure defined by $d_0+a$. Then $\bar E \simeq Gr(E)$. The proof, which is principally by induction on the rank $r$ of $\et$ (with the initial case $r=1$ being self-evident) proceeds in several stages, corresponding to three cases: 1. [*That $\alpha=0$*]{};   2. [*That $\alpha$ is not zero and is not $\Gamma$-semistable*]{}; and   3. [*That $\alpha$ is $\Gamma$-semistable*]{}. The first is the totally degenerate case for which $\alpha=0$: Let $d_0+a$ be an integrable connection on $\et$ with $\db_0^*a''=0$, and let $E$ be the corresponding holomorphic structure. If $\norm{a}_{L^p_1}$ is sufficiently small then $\Pi^{0,1}a''=0$ if and only if $E\simeq E_0$. If $\Pi^{0,1}a''=0$, then it follows from  that $a''=0$, and therefore $a=0$. Conversely, if $E\simeq E_0$, then there exists $g\in {\cal G}$ such that $g\cdot d_0=d_0+a$, or equivalently, $\db_0g+a''g=0$. Applying  to the connection on $Hom(\et,\et)$ induced by $d_0$ and $d_0+a$, it follows that $d_0g=0=a''g$, so $a''=0$. Let $d_0+a$ be as above with $a''=\alpha+\db_0^*\beta$. Choose a sequence $(\gamma_j)$ in $\Gamma$ with $\det\gamma_j=1$ for every $j$ such that $\norm{\gamma_j^{}\alpha\gamma_j^{-1}}^2 \to \inf\limits_{\gamma\in\Gamma} \norm{\gamma\alpha\gamma^{-1}}^2$ as $j\to\infty$, so after passing to a subsequence if necessary, it can be assumed that $\alpha_j := \gamma_j^{}\alpha\gamma_j^{-1}$ converges to $\bar\alpha\in H^{0,1}$. If $\beta_j := \gamma_j^{}\,\beta\,\gamma_j^{-1}$ and $a_j'' := \alpha_j+\db_0^*\beta_j$, then $d_0+a_j =\gamma_j\cdot(d_0+a)$ is an integrable connection defining a holomorphic structure isomorphic to $E$, with $\db_0^*a_j''=0$. By , $\norm{a''_j}_{L^p_1}$ is uniformly bounded independent of $j$, so after passing to another subsequence if necessary, the connections $d_0+a_j$ can be assumed to converge weakly in $L^p_1$ and strongly in $C^0$ (say) to a limiting connection $d_0+\bar a$, with $a\in L^p_1$. Elliptic regularity combined with integrability of the connection together with the equation $\db_0^*\bar a''=0$ imply that $\bar a$ is in fact smooth. Indeed, using the analysis of §1, the forms $\beta_j$ can be assumed to be converging in $L^p_2$ to a limit in $A^{0,2}(\eet)$ that is orthogonal to $\ker\db_0^*$, and by the uniqueness statement of , this limit must be the form $\bar\beta$ mentioned earlier, with $\bar a'' =\bar\alpha+\db_0^*\bar\beta$. Since $\det\gamma_j=1$ for every $j$, it follows that if $\norm{\gamma_j}$ is uniformly bounded then a subsequence can be found converging to some $\gamma_0\in \Gamma$, and then $d_0+\bar a=\gamma_0\cdot (d_0+a)$. This is the case considered in the previous section when $\alpha\in H^{0,1}$ is a $\Gamma$-polystable point. So it may be supposed that $\norm{\gamma_j}$ is not uniformly bounded, and after rescaling $\gamma_j$ to $\eta_j := \gamma_j/\norm{\gamma_j}$, these may be assumed to converge to some $\gamma_0\in H^{0,0}$ with $\norm{\gamma_0}=1$ and $\det\gamma_0=0$. It may also be assumed without loss of generality that $\gamma_j$ is self-adjoint and positive for each $j$, so $\gamma_0$ is also self-adjoint and non-negative. The equation $\gamma_j\cdot (d_0+a)=d_0+a_j$ is equivalent to $\db_j\gamma_j=0$ where $d_j$ is the connection on $Hom(\et,\et)$ induced by $d_0+a$ and $d_0+a_j$, these connections converging to the connection on this bundle induced by $d_0+a$ and $d_0+\bar a$. So $\gamma_0$ defines a non-zero holomorphic map from $E$ to $\bar E$, this map having determinant $0$. Since $\gamma_0$ must be of constant rank on $X$, its kernel $K$ is a holomorphic subbundle of $E$, necessarily of the same slope as that of $\et$. Thus $E$ may be expressed as an extension by holomorphic semi-stable bundles $0\to K\to E\to Q\to 0$, where $Q := E/K$. Consider now Case 2. of , namely when $\alpha$ is non-zero and is not $\Gamma$-semistable. By definition, zero is in the closure of the orbit of $\alpha$ under $\Gamma$, so $\bar\alpha=0$ and therefore $\bar E=E_0$ by . For notational convenience, set $E_1 := K = \ker\gamma_0$ and $E_2 := Q = E/K$. Since $\gamma_0$ is self-adjoint, $E_2$ can be identified with $E_1^{\perp} \subset \et$ as a unitary bundle. The holomorphic structures on $E_1$ and $E_2$ are those induced from $E$ as holomorphic sub- and quotient bundles. But since $E_1=\ker\gamma_0$ and $\gamma_0$ is a $d_0$-closed self-adjoint endomorphism of the holomorphic bundle $E_0$, both $E_1$ and $E_2$ have hermitian connections induced from $d_0$, with respect to which the connections are Hermite-Einstein with the same Einstein constant as $E_0$. These connections will be denoted by $d_{0,1}$, $d_{0,2}$ respectively, so $d_0=d_{0,1}\oplus d_{0,2}$ using self-evident notation. The limit $\gamma_0$ of the (rescaled) automorphisms $\gamma_j$ is $d_0$-closed and satisfies $\gamma_0\alpha=0=\gamma_0\beta$ and also $\gamma_0a''=0$, so in terms of the splitting $\et=E_1\oplus E_2$, $$\gamma_0=\bmatrix{0&0\cr0&\bar\gamma}\,,\quad \alpha=\bmatrix{\alpha_{11}&\alpha_{12}\cr 0 & 0}\,,\quad \beta=\bmatrix{\beta_{11}&\beta_{12}\cr 0 & 0}\,, \quad a''=\bmatrix{a_{11}'' & a_{12}''\cr 0 &0}\,, \myeqn$$ where $\bar\gamma=\bar\gamma^*$ has non-zero determinant. Since $\gamma_0$ is $d_0$-closed, the connection on $E_2$ induced by the connection $d_0+a$ (i.e., as a quotient of $E$) is the same as the connection on this bundle induced by $d_0$ (i.e., as a subbundle of $E_0$), so the holomorphic bundle $E_2$ is isomorphic to a direct sum of stable summands of $E_0$. The connection on $E_1$ induced by $d_0+a$ is identified with $d_{0,1}+a_{11}$, with $a_{11}=\alpha_{11}+\db_{0,1}^*\beta_{11}$. By the inductive hypothesis (of ), under the action of $\Gamma_1 = Aut(E_1(d_{0,1}))$, there are connections in the orbit of $d_{0,1}+a_{11}$ that are arbitrarily close in $L^p_1$ to the Hermite-Einstein connection $d_G$ on $Gr(E_1)$. Then using automorphisms of the form $h_t$ as described earlier, the off-diagonal term $a''_{12}$ in can be made arbitrarily small whilst leaving the diagonal terms fixed, from which it follows that there exist $g_t\in {\cal G}$ for $t>0$ such that $g_t\cdot(d_0+a)\to (d_{G}\oplus d_{0,2})$ as $t\to 0$. Thus the two Hermite-Einstein connections $d_0$ and $d_{G\oplus E_2}$ lie in the closure of the orbit of $d_0+a$ under ${\cal G}$, but up to unitary isomorphism, there is at most one such connection since the space of Yang-Mills connections modulo unitary gauge is Hausdorff, by the reasoning of §6 of [@AHS]. So $Gr(E_1)\oplus E_2\simeq E_0$, which implies that $Gr(E) \simeq E_0$. (Note that this argument has proved that $\alpha_{11}\in H^{0,1}(E_{0,1})$ is not $\Gamma_1$-semi-stable, which is not self-evident.) It remains to complete the proof of  in Case 3., this being when $\alpha\in H^{0,1}$ is $\Gamma$-semistable but not $\Gamma$-polystable. With the same objects as earlier, this is the case that $\bar\alpha\not=0$, so $\bar E$ is not isomorphic to $E_0$ (by ), but $\bar E$ is polystable, by . By , every endomorphism of $\et$ that is holomorphic with respect to $\db_0+\bar a''$ is in fact covariantly constant with respect to $d_0$ and commutes with $\bar a$, so $\bar\Gamma := Aut(\bar E)$ is the subgroup of $\Gamma=Aut(E_0)$ commuting with $\bar a$. The connection $d_0+\bar a$ defines an $\omega$-polystable point, and $\gamma_j\cdot (d_0+a)\to d_0+\bar a$. From , once $d_0+a$ has been placed in the good complex gauge $d_0+\tilde a$ of  with respect to the Hermite-Einstein connection $\bar d$ inducing $\bar E$, there are automorphisms $\bar\gamma_j$ that are $\bar d$-closed such that $\bar\gamma_j\cdot(d_0+\tilde a)\to \bar d$. But now Case 2. applies with $\bar E$ replacing $E_0$, for which the conclusion is that the bundle $Gr(E)$ is isomorphic to $\bar E$, as desired. Consequently, the proof of is complete. We end this paper with several concluding remarks. The results presented here appear to be of some significance even in the case of compact Riemann surfaces. When the degree and the rank of $\et$ are coprime, the moduli space of stable holomorphic structures on $\et$ is a smooth compact manifold, of considerable interest in its own right, not least because this space carries a natural hyper-Kähler metric of Weil-Petersson type. When the rank and degree of $\et$ are not coprime, the stable bundles are naturally compactified by adding the polystable bundles, and the results here provide a description of neighbourhoods of boundary points. The degeneration of the metric at the boundary of the stable moduli space can be interpreted in terms of quotient singularities arising from the Kähler quotient of a smooth metric on a framed moduli space. The case of compact Riemann surfaces is also helpful for obtaining a better understanding of several of the results presented here. There are no integrability conditions to be considered, and the only singularities occurring in moduli spaces result from quotient singularities which can be viewed in the light of isotropy for the action of $\Gamma$ on classes in $H^{0,1}$. The cases of genus $0$, $1$ and $2$ for $E_0$ being the trivial bundle of rank $2$, or the direct sum of a non-trivial line bundle of degree $0$ with the trivial line bundle all provide considerable insight. In the case $n=2$, relatively explicit examples of moduli spaces can be computed, particularly when $X$ is a ruled surface and even more particularly when $X=\m P_1\times\m P_1$. Using monads, moduli spaces of $2$-bundles have been computed explicitly in [@Bu1], including an explicit description of the space of deformations of the bundle ${\cal O}(1,-1)\oplus {\cal O}(-1,1)$, which again illustrates many of the results here; cf. the  following . Monads also feature in Donaldson’s paper [@Do2], which presents another illustration of the interrelation between the notions of stability in Kähler geometry and geometric invariant theory, one that is not independent of the results in this paper. Because the results here focus on a [*neighbourhood*]{} of a fixed polystable bundle, it is reasonable to expect that they will hold mutatis mutandis on arbitrary compact complex manifolds equipped with Gauduchon metrics. However, in light of the  following the proof of , there may be some unforeseen subtleties. For the sake of brevity and simplicity, we have considered only the Kähler case. It is evident from the analysis that the assumption of integrability for connections is not nearly as important as might be expected, as the $(0,2)$ and $(2,0)$ components of the curvature are well-controlled by , given that the calculations are local to $d_0$. This highlights the interesting class of solutions $d$ of the Yang-Mills equations on a compact Kähler manifold for which $\d F^{0,2}(d)=0=\db\wh F(d)$ (which includes some [*self*]{}-dual solutions on compact surfaces), these bearing some formal similarities to solutions of the Seiberg-Witten equations. At its heart, the proof of  is a manifestation of a very coarse compactness property of stable bundles, a desirable property used to great effect in gauge theory. Moduli spaces of stable holomorphic bundles on a Kähler surface can fail to be compact in two ways, one reflecting the degeneration from stable to polystable and the other in terms of the concentration of curvature of Hermite-Einstein connections. The former is the subject of this paper, whereas the latter is considered in [@Bu3]. Although the failure of moduli spaces of stable bundles on compact Kähler surfaces can be controlled to some extent as described in that reference, in higher dimensions there is less control on the degeneration and one is forced to consider compactifications in terms of sheaves ([@BS]). The Bogomolov inequality $(c_2-(r-1)c_1^2/2r) \cdot \omega^{n-1} \ge 0$ for semi-stable sheaves and bundles does not provide sufficient control on subbundles in dimensions greater than $2$. As alluded to in the introduction, there are profound relationships between the theory of stable holomorphic vector bundles on compact Kähler manifolds and the theory of constant scalar curvature Kähler metrics, these relationships mediated by geometric invariant theory. In that the former theory is a quasi-linear analogue of the latter (in the sense of partial differential equations), it can be hoped that the results here may provide useful directions for the further investigation of moduli of compact complex manifolds and their geometries. To conclude on an even more speculative note, in view of the critical importance of Yang-Mills theory and of representation theory in contemporary physics, it might also be hoped that the our results may provide deeper insight into the nature of elementary particles and their interactions. **References** [308]{}[1983]{}[523–615]{} [Self-duality in four-dimensional Riemannian geometry]{}[Proc. Roy. Soc. London Ser. A]{}[362]{}[1978]{}[425–461]{} [Math. Ann.]{}[280]{}[1988]{}[625–648]{} [407–454]{} [Comm. Math. Phys.]{}[93]{}[1984]{}[453–460]{} [Proc. London Math. Soc.]{}[50]{}[1985]{}[1–26]{} [The geometry of four-manifolds]{}[Oxford University Press]{}[New York]{}[1990]{} [209]{}[1974]{}[291–346]{} [The moduli space of Hermite-Einstein bundles on a compact Khler manifold]{}[Proc. Japan Acad. Ser. A Math. Sci.]{}[63]{}[1987]{}[69–72]{} [Le théorème de l’excentricité nulle]{}[C. R. Acad. Sci. Paris Sér. A-B]{}[285]{}[1977]{}[A387–A390]{} [\[Hi\]  ]{} N. J. Hitchin, Review of “[*La topologie d’une surface hermitienne d’Einstein*]{}" by P. Gauduchon, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), A509–A512. In: Mathematical Reviews MR571563 (81e:53052), 1981. [18]{}[2014]{}[859–883]{} [Springer]{}[Berlin-Heidelberg-New York]{}[1978]{}[233–244]{} [Math. Z.]{}[195]{}[1987]{}[143–150]{} [Proc. Japan Acad. Ser. A Math. Sci.]{}[58]{}[1982]{}[158–162]{} [Princeton University Press]{}[Princeton, NJ]{}[1987]{} [Enseign. Math.]{}[51]{}[2005]{}[117–127]{} [Manuscripta Math.]{}[42]{}[1983]{}[245–257]{} [Reduction of symplectic manifolds with symmetry]{}[Rep. Mathematical Phys.]{}[5]{}[1974]{}[121–130]{} [1989]{}[301–320]{} [Springer]{}[Berlin]{}[1994]{} [Stable and unitary vector bundles on a compact Riemann surface]{}[Ann. of Math.]{}[82]{}[1965]{}[540–567]{} [Am. J. Math.]{}[132]{}[2010]{}[1077–1090]{} [Nagoya Math. J.]{}[47]{}[1972]{}[29–48]{} [On the existence of Hermitian-Yang-Mills connections in stable vector bundles]{}[Comm. Pure Appl. Math.]{}[39]{}[1986]{}[S257–S293]{} =.5in School of Mathematical Sciences University of Adelaide Adelaide, Australia 5005 -.25in[*E-mail address*]{}: [nicholas.buchdahl@adelaide.edu.au]{} .15in Fachbereich Mathematik und Informatik Philipps-Universität Marburg Lahnberge, Hans-Meerwein-Strasse D-35032 Marburg, Germany -.25in[*E-mail address*]{}: [schumac@mathematik.uni-marburg.de]{}

--- abstract: 'We report on the deterministic coupling between single semiconducting nanowire quantum dots emitting in the visible and plasmonic Au nanoantennas. Both systems are separately carefully characterized through microphotoluminescence and cathodoluminescence. A two-step realignment process using cathodoluminescence allows for electron beam lithography of Au antennas near individual nanowire quantum dots with a precision of . A complete set of optical properties are measured before and after antenna fabrication. They evidence both an increase of the NW absorption, and an improvement of the quantum dot emission rate up to a factor two in presence of the antenna.' address: - '$^1$ [Univ. Grenoble Alpes, F-38000 Grenoble, France]{}' - '$^2$ [CNRS, Institut Néel, “Nanophysique et semiconducteurs” group, F-38000 Grenoble, France]{}' - '$^1$ [CEA, INAC-SP2M, “Nanophysique et semiconducteurs” group, F-38000 Grenoble, France]{}' author: - 'Mathieu Jeannin$^{1,2}$, Pamela Rueda-Fonseca$^{1,3}$, Edith Bellet-Amalric$^{1,3}$, Kuntheak Kheng$^{1,3}$ and Gilles Nogues$^{1,2}$' title: Deterministic radiative coupling between plasmonic nanoantennas and semiconducting nanowire quantum dots --- [*Keywords*]{}: semiconductor, nanowire, plasmonics, nanoantenna, nanofabrication, cathodoluminescence Introduction ============ Recent progress in semiconductor growth research now allows for fabrication of nanowires (NWs) structures which are of great interest for their high crystalline quality, strain free character and practical geometry. A key step towards realization of nano-optical circuits and applications relies on the coupling of single NWs with other structures, like photonic crystal cavities [@Birowosuto2014] or plasmonic nanoantennas [@Ozel2013; @Casadei2014; @Casadei2015; @Ramezani2015] (NAs). To better use the advantages and the versatility of NWs, it is now necessary to adapt previous studies on self-assembled or colloidal quantum dots (QDs) to their NW counterparts. While numerous results have been obtained in the coupling of self-assembled [@Curto2010; @Pfeiffer2010; @Nogues2013; @Belacel2013; @Kukushkin2014] and colloidal [@Curto2013; @Hoang2016] QDs with plasmonic nanostructures, all the existing studies on control of the optical properties of nanowire quantum dots (NWQDs) rely on a photonic approach, using the nanowire itself as an antenna [@Claudon2009; @Munsch2012; @Cremel2014]. Following previous work in our group [@Nogues2013] on droplet epitaxy QDs, we show how combining cathodoluminescence (CL) with standard e-beam lithography technique allows to fabricate at will plasmonic NAs in the vicinity of II-VI NWQDs emitting around . Both systems are initially fully characterized using CL for the antennas and micro-photoluminescence (PL) and time-resolved spectroscopy for the emitters. Previous studies have made use of morphological criteria to detect nanoemitters, using scanning electron microscopy (SEM) or atomic force microscopy [@Pfeiffer2010]. In contrast, our method has the advantage of allowing localization of embedded structures undetectable with the previous methods. Nanowire quantum dot properties {#sec:NWproperties} =============================== Our nanoemitters are single (Cd,Mn)Te QDs (Mn fraction $\approx 5\%$) inserted inside ZnTe/(Zn,Mg)Te core/shell nanowires \[figure \[fig:NW\_presentation\] (a)\]. The NWs are grown by molecular beam epitaxy on a (111)B GaAs substrate covered by a thick buffer layer of ZnTe [@Artioli2013; @Rueda-Fonseca2014]. Dewetted gold droplets are used as catalysts, and temperature variations allow to favor different growth mechanisms resulting in the final heterostructure. The core growth is both longitudinal and lateral, leading to the conical aspect of the nanowire, with a core diameter of at the top to $\approx$ at the base, and a thick shell. The QD emission is measured by confocal PL spectroscopy. It features a main emission peak around , blue-shifted by compared to the exciton transition in bulk CdTe [@Horodysky2005] due to the confinement and the strain induced by the surrounding shell [@Ferrand2014]. It is significantly broadened (FWHM ) by the presence of the magnetic Mn atoms creating a fluctuating magnetic field inside the QD, randomly shifting the exciton line in time by Zeeman effect [@stepanov:tel-00994939] \[figure \[fig:NW\_presentation\] (b)\]. Amongst all the studied NWs, the central emission wavelength ranges from 605 to . NWs are detached from their growth substrate and dispersed onto a host substrate for optical studies. The host is a Si substrate pre-patterned by optical lithography and dry etching to fabricate coarse localization marks. It is covered by a thick Au layer deposited by e-gun evaporation and a thick Al$_2$O$_3$ spacing layer by atomic layer deposition. The mirror and the spacer thickness are designed to give constructive interferences between the light directly emitted from the QD and the light reflected by the mirror in order to maximize luminescence collection. NWs are finally dispersed by mechanical contact between the host and growth substrates. A first selection of NWs is performed using CL at [@Nogues2013; @Artioli2013]. The luminescence from the QD is spectrally filtered and detected by an avalanche photodiode (APD). CL imaging of the emission profile at the QD luminescence wavelength is obtained for a set of NWs. To avoid spurious emission from exciton trapped in defects at the base of the NWs, we conserve only structures presenting a well localized emission in the top half of the conical NW and having the typical spectral structure of figure \[fig:NW\_presentation\] (b). The selected QDs are then fully characterized by PL spectroscopy at . The NWs are excited at by a frequency doubled, ps pulsed Ti:Sapphire. Time-resolved measurements are performed at very low pumping power, well below saturation of the QD, with a spectral integration bandwidth of to ensure that we only detect the excitonic transition, resulting in a monoexponential time trace. They show decay times ranging from 0.2 to \[figure \[fig:NW\_presentation\] (c)\]. The average lifetime ($\tau=$ ) is about five times longer than for self-assembled CdTe/ZnTe QDs [@man2015]. In addition, the emission from the QD is linearly polarized in a direction parallel to the NW axis in $70\%$ of the cases, and orthogonal in all the others. The average degree of linear polarization is $0.7\pm 0.2$. These optical properties can be explained by the very small energy difference between the valence bands of CdTe and ZnTe [@stepanov:tel-00994939]. The resulting nature of the ground hole state strongly depends on additional energy shifts induced by strain or QD aspect ratio [@Zielinski2013; @Ferrand2014]. In our case, the orbital hole wavefunction is probably poorly confined inside the QD, resulting in low electron-hole wavefunction overlap and thus long radiative decay time. We note that the measured decay rate $1/\tau = \gamma_r + \gamma_{nr}$, where $\gamma_r$ (resp. $\gamma_{nr}$) is the radiative (resp. non-radiative) decay rate, might be limited by non-radiative processes. Furthermore, a large variability from dot to dot strongly mixes the light- and heavy-hole bands, explaining the polarization results. This variability is not detrimental to our study as we compare a complete set of optical properties *on the one and same NW before and after antenna fabrication*. Nanoantenna characterization {#sec:NACarac} ============================ Au nanoantennas are fabricated on the host substrate, in a region empty of NWs. They consist in single rectangles of fixed width, as sketched in figure \[fig:Spectral\_map\] (a). Their length L varies from 85 to . They are fabricated by electron-beam lithography on a thick Poly(methyl methacrylate) (PMMA) bi-layer resist, using PMMA with molecular weights of and to create a mechanical mask with the resist. of Au is deposited on the sample by e-gun evaporation. N-methyl-2-pyrrolidone (NMP) lift-off with 80$^{\circ}$C heating is finally performed to remove the resist. The antennas are characterized using CL at room temperature [@GarciadeAbajo2010; @Vesseur2007]. The CL spectrum of each antenna is obtained by raster-scanning the electron beam over its surface. It presents a large peak at a fixed wavelength around and a second resonance at lower energy. CL spectra for all the antenna are presented in figure \[fig:Spectral\_map\] (b), revealing that the second resonance red-shifts with increasing antenna length. We note that the optimization of the Al$_2$O$_3$ spacer thickness in order to increase the collected signal at results in destructive interferences for the collected light in the 700 to wavelength range, decreasing the detection contrast in this spectral region. Further information is obtained by imaging the plasmon local density of states [@GarciadeAbajo2010] (LDOS). It is reconstructed by slowly scanning the electron beam over the antenna and collecting the CL emission as a function of the beam position filtered in a spectral window around the resonance energy on an APD. This spectral integration bandwidth is chosen to maximize the collected signal from the antenna. For $L\geq$, it is smaller than the energy separation between two consecutive modes so that only a single resonance contributes to the LDOS image. The LDOS of the red-shifting resonance \[figure \[fig:Spectral\_map\] (c)\] clearly displays two lobes characteristic of a longitudinal dipolar mode. In this regime the antenna sustains a single radiating mode whose dispersion relation can be extracted from figure \[fig:Spectral\_map\] (b)(blue squares). On the contrary, the LDOS at \[figure \[fig:Spectral\_map\] (d)\] shows no precise spatial structure at any length. We have fabricated nanoantennas of different shapes, sizes and spacer thicknesses and observed that it is always present. The energy of its maximum only depends on the Al$_2$O$_3$ spacer thickness (see Supplementary Material [@Supplementary]). We attribute this peak to the scattering by the NA of the continuum of surface plasmon polariton modes sustained by the Au/Al$_2$O$_3$/air multilayer system. LDOS evaluation based on numerical calculations of the surface plasmon polariton dispersion relation[@Davis2009] show that at this energy a lot of lossy modes contribute to the signal. We note that we use here a thick spacer for which no significant coupling occurs between the antenna and the metal film, which simply acts as a mirror to reflect emitted and scattered light towards the collection objective. To determine the relevant dimensions for enhancing emission processes at the QD wavelength, we plot a cut of figure \[fig:Spectral\_map\] (b) at $\lambda_{QD}=$ \[green solid line, figure \[fig:Spectral\_map\] (e)\]. It predicts an increase in scattering efficiency for an antenna length around due to the contribution of the dipolar mode. It also predicts an enhancement for lengths above . This enhancement comes from the onset of a higher order mode theoretically predicted to appear at this length [@Davis2009; @Filter2012]. We note however that for the target coupled NW-NA system, the presence of the very high refractive index nanowire (n$_{\mathrm{ZnTe}}\approx$ 3) in the near-field of the antenna significantly modifies its plasmonic properties. Hence a red-shift in the plasmon dispersion relation curve is expected, and the enhancement peaks observed in figure \[fig:Spectral\_map\] (e) will occur for smaller antenna lengths. Hybrid NW-NA structures ======================= Fabrication ----------- Two examples of the target structure are shown in figures \[fig:repos\] (d)-(e). Fine alignment marks are first fabricated around the chosen NWs using electron-beam lithography, metal evaporation and lift-off \[figure \[fig:repos\] (a)\]. We then record a SEM image \[figure \[fig:repos\] (b)\] together with the CL intensity image at $\lambda_{QD}=$ \[figure \[fig:repos\] (c)\]. Both images are acquired after aligning the microscope beam using the fine alignment marks. Acquisition times are limited to a few seconds to limit mechanical and electrostatic drifts. We fit the QD emission peak with a bi-dimensional Gaussian profile and superimpose it to the corresponding SEM image. It allows to precisely localize the QD inside the NW in the frame defined by the fine alignment marks. NAs are then fabricated with the same process as the previously characterized antennas. Before insulation of the resist, a final alignment step is performed onto the fine marks under the resist. The antennas have a fixed width $w=$ and a variable length L from $50$ to . This range is based on the previous CL results and takes into account the shift due to the presence of the NW. The antenna sides of length L are parallel to the measured polarization direction of the QD emission. We aim at having an antenna to NW gap equal to zero. SEM images reveal an average gap of $\pm$, thus some antennas are on top of the NW. The final error of has different sources. Alignment of the lithography setup has a typical error of but is degraded in our case due to poor contrast of the fine alignment marks image under the resist. We evaluate the thermal, mechanical and electrostatic drifts error during CL at low temperature to . We note that the NW core and shell are already thick ($\approx$ at the QD position). The resulting QD to antenna distance ranges from 50 to . This distance is always large enough to prevent luminescence quenching, and the variations due to the positioning error only have a moderate effect on the NA to QD coupling [@Anger2006]. Photoluminescence observations ------------------------------ The coupled QD-NAs systems are characterized using PL spectroscopy with a pulsed excitation laser at a wavelength of . Figure \[fig:PL\_results\] (a) compares the integrated QD emitted intensity as a function of the exciting laser power $P_{exc}$ before and after antenna fabrication for $L=$ . The PL intensity first increases with the excitation power and then saturates because of the complete occupation of the discrete excitonic state in the QD. For a same excitation power one clearly sees a much higher collected signal in presence of the NA. Similar measurements on other QDs present on the substrate which have experienced the same process except the final antenna fabrication show no change in their properties. Previous studies revealed modification of the luminescence collection by redirection of light by the antenna [@Ramezani2015; @Curto2010; @Nogues2013]. We have carried out Fourier plane microscopy by imaging the back focal plane of the microscope objective in the PL setup (see supplementary material [@Supplementary]) to observe the radiation diagram of the NWQDs [@Grzela2012], and we did not observed significant change in the radiation pattern after antenna fabrication. Clearly, the higher PL intensity we measure is not due to the redirection of light by the antennas. Additional information is provided by measurements of the exciton lifetime before ($\tau_0$) and after ($\tau_{NA}$) antenna fabrication under the same excitation conditions. We observe no significant change, with an average ratio $\tau_{NA}/\tau_{0} = 1.1 \pm 0.2$ (see Supplementary Materials [@Supplementary]). To better analyse the effect of the antennas, we evaluate the saturation intensity $I_s$ at high excitation power, and the slope at the origin $g$ of the power-dependent PL curve. The ratio of $I_s$ before and after antenna fabrication is plotted as a function of antenna length in figure \[fig:PL\_results\] (b), showing a net increase after NA fabrication. As shown in figure \[fig:PL\_results\] (c), we also observe a greater increase in the slope $g$ after NA fabrication. Discussion ---------- Assuming that we only detect the excitonic transition and considering that the nanowire excitation is done with a pulsed laser, the saturation intensity $I_s$ is equal to $f\times \gamma_{r}/(\gamma_{r}+\gamma_{nr})\times\alpha_{coll}$, where $f$ is the laser repetition rate, $Y=\gamma_{r}/(\gamma_{r}+\gamma_{nr})$ is the quantum yield and $\alpha_{coll}$ is the fraction of the emitted power collected by the microscope objective. As we measure no change in the radiation diagram after NA fabrication, it is reasonable to assume that $\alpha_{coll}$ is unperturbed. As a consequence the enhancement observed in figure \[fig:PL\_results\] (b) is directly related to an increase in the quantum yield $Y$. This enhancement is found to be moderate for antennas with a length greater than but stronger for the smallest antenna, up to a factor $2.5$. As expected from section \[sec:NACarac\], it occurs for shorter antennas than the determined resonant length from figure \[fig:Spectral\_map\] (e). Moreover since the QD response at saturation is independent of the excitation power, the change of $I_s$ after antenna fabrication is not due to an increase of the absorption inside the NW. Nevertheless, CL spectra of the NAs at $\lambda_{exc} = $ show that there is still a measurable LDOS at this energy that can modify the absorption of the excitation laser inside the NW (see Supplementary Material [@Supplementary]). The main contribution to this LDOS comes from the fixed resonance observed in Figure \[fig:Spectral\_map\] and does not depend on the antenna length. The change of the absorption can be retrieved by analyzing the slope at the origin $g$. In the limit of low excitation regime, $g$ is expected to be proportional to $Y\times \alpha_{abs} \times \alpha_{coll}$, where $\alpha_{abs}$ is the fraction of power absorbed in the NW. As $\alpha_{coll}$ remains unchanged after NA fabrication, the ratio $(g/g_0)/(I_s/I_{s0})$ directly gives the change in $\alpha_{abs}$. The comparison of figures \[fig:PL\_results\] (b) and (c) shows that the absorption in the NW is enhanced by a factor 2.2$\pm$1.1 over the whole antenna length range. Further illustration of the respective changes in $I_s$ and $g$ are presented in the Supplementary Materials [@Supplementary] with two other power-dependent PL curves. Our main conclusion is that, for the shorter antenna lengths, enhanced NW absorption and improvement of the quantum yield contribute equally at $\sim$50% to the four-fold increase of $g$ that we observe. Finally, lifetime measurements show that there is no change of the total decay rate $\gamma_{r}+\gamma_{nr}$. This is an indication that the total decay rate is dominated by the non-radiative term $\gamma_{nr}$ and is moderately affected by a change in $\gamma_r$. The increase in $Y$ that we observe is therefore essentially due to an increase of the radiative rate $\gamma_r$ of the QD because of the coupling to the NA. Conclusion ========== In conclusion, we demonstrated deterministic coupling between single semiconducting nanowire quantum dots and plasmonic nanoantennas using CL and electron-beam lithography with a precision of . Our method has the advantage of relying only on the luminescence of the emitters for their precise localization. It also grants full characterization of individual nanoemitters and antennas. CL spectroscopy and LDOS imaging of individual NAs is demonstrated as a powerful technique to experimentally determine antennas parameters for the fabrication of coupled plasmonic-semiconductor emitters. Furthermore, we demonstrate two effects of the NA on the QD: an absorption enhancement of a factor 2, and a light emission enhancement due to radiative coupling to the antenna up to a factor 2.5 in a region of high plasmonic losses, extending the control of light emission from semiconducting nanostructures towards the visible spectral region. The effect could be greatly increased for a smaller QD-antenna distance, i.e. using a thinner NW shell. Implementation of the method is a crucial step towards fabricating more complex and versatile coupled structures. It can be applied to all kind of nanoemitters, aiming at controlling their optical properties like their polarization response [@Kukushkin2014; @Casadei2015] or emission diagram [@Curto2010; @Belacel2013]. We acknowledge the help of Institut Néel technical support teams Nanofab (clean room) and optical engineering (CL, Fabrice Donatini). This work was supported by the French National Research Agency (project Magwires, ANR-11-BS10-013 and Labex LANEF du Programme d’Investissements d’Avenir ANR-10-LABX-51-01) References {#references .unnumbered} ========== [10]{} url \#1[[\#1]{}]{} urlprefix \[2\]\[\][[\#2](#2)]{} Birowosuto M D, Yokoo A, Zhang G, Tateno K, Kuramochi E, Taniyama H, Takiguchi M and Notomi M 2014 [*Nat Mater*]{} [**13**]{} 279–285 ISSN 1476-4660 <http://dx.doi.org/10.1038/nmat3873> Ozel T, Bourret G R, Schmucker A L, Brown K A and Mirkin C A 2013 [*Advanced Materials*]{} [**25**]{} 4515–4520 ISSN 0935-9648 <http://dx.doi.org/10.1002/adma.201301367> Casadei A, Pecora E F, Trevino J, Forestiere C, Rüffer D, Russo-Averchi E, Matteini F, Tutuncuoglu G, Heiss M, Fontcuberta i Morral A and Dal Negro L 2014 [*Nano Lett.*]{} [**14**]{} 2271–2278 Casadei A, Llado E A, Amaduzzi F, Russo-Averchi E, Rüffer D, Heiss M, Negro L D and Morral A F i 2015 [*Scientific Reports*]{} [**5**]{} 7651 ISSN 2045-2322 <http://dx.doi.org/10.1038/srep07651> Ramezani M, Casadei A, Grzela G, Matteini F, Tütüncüoglu G, Rüffer D, Fontcuberta i Morral A and Gómez Rivas J 2015 [*Nano Lett.*]{} [**15**]{} 4889–4895 ISSN 1530-6992 <http://dx.doi.org/10.1021/acs.nanolett.5b00565> Curto A G, Volpe G, Taminiau T H, Kreuzer M P, Quidant R and van Hulst N F 2010 [*Science*]{} [**329**]{} 930–933 ISSN 1095-9203 <http://dx.doi.org/10.1126/science.1191922> Pfeiffer M, Lindfors K, Wolpert C, Atkinson P, Benyoucef M, Rastelli A, Schmidt O G, Giessen H and Lippitz M 2010 [*Nano Lett.*]{} [**10**]{} 4555–4558 ISSN 1530-6992 <http://dx.doi.org/10.1021/nl102548t> Nogues G, Merotto Q, Bachelier G, Hye Lee E and Dong Song J 2013 [*Applied Physics Letters*]{} [**102**]{} 231112 ISSN 0003-6951 <http://dx.doi.org/10.1063/1.4809831> Belacel C, Habert B, Bigourdan F, Marquier F, Hugonin J P, Michaelis de Vasconcellos S, Lafosse X, Coolen L, Schwob C, Javaux C and et al 2013 [ *Nano Lett.*]{} [**13**]{} 1516–1521 ISSN 1530-6992 <http://dx.doi.org/10.1021/nl3046602> Kukushkin V I, Mukhametzhanov I M, Kukushkin I V, Kulakovskii V D, Sedova I V, Sorokin S V, Toropov A A, Ivanov S V and Sobolev A S 2014 [*Physical Review B*]{} [**90**]{} ISSN 1550-235X <http://dx.doi.org/10.1103/PhysRevB.90.235313> Curto A G, Taminiau T H, Volpe G, Kreuzer M P, Quidant R and van Hulst N F 2013 [*Nat Comms*]{} [**4**]{} 1750 ISSN 2041-1723 <http://dx.doi.org/10.1038/ncomms2769> Hoang T B, Akselrod G M and Mikkelsen M H 2016 [*Nano Lett.*]{} [**16**]{} 270–275 ISSN 1530-6992 <http://dx.doi.org/10.1021/acs.nanolett.5b03724> Claudon J, Bleuse J, Malik N S, Bazin M, Jaffrennou P, Gregersen N, Sauvan C, Lalanne P and Gérard J M 2009 [*Nature Photon*]{} [**3**]{} 116–116 ISSN 1749-4893 <http://dx.doi.org/10.1038/nphoton.2009.287> Munsch M, Claudon J, Bleuse J, Malik N S, Dupuy E, Gérard J M, Chen Y, Gregersen N and Mørk J 2012 [*Physical Review Letters*]{} [**108**]{} ISSN 1079-7114 <http://dx.doi.org/10.1103/PhysRevLett.108.077405> Cremel T, Elouneg-Jamroz M, Bellet-Amalric E, Cagnon L, Tatarenko S and Kheng K 2014 [*Phys. Status Solidi C*]{} [**11**]{} 1263–1266 ISSN 1862-6351 <http://dx.doi.org/10.1002/pssc.201300737> Artioli A A, Rueda-Fonseca P P, Stepanov P P, Bellet-Amalric E, Den Hertog M, Bougerol C, Genuist Y, Donatini F, André R R, Nogues G and et al 2013 [ *Applied Physics Letters*]{} [**103**]{} 222106 ISSN 0003-6951 <http://dx.doi.org/10.1063/1.4832055> Rueda-Fonseca P, Bellet-Amalric E, Vigliaturo R, den Hertog M, Genuist Y, André R, Robin E, Artioli A, Stepanov P, Ferrand D and et al 2014 [*Nano Lett.*]{} [**14**]{} 1877–1883 ISSN 1530-6992 <http://dx.doi.org/10.1021/nl4046476> Horodyský P and Hlídek P 2005 [*physica status solidi (b)*]{} [**243**]{} 494–501 ISSN 0370-1972 <http://dx.doi.org/10.1002/pssb.200541402> Ferrand D and Cibert J 2014 [*Eur. Phys. J. Appl. Phys.*]{} [**67**]{} 30403 ISSN 1286-0050 <http://dx.doi.org/10.1051/epjap/2014140156> Stepanov P S 2013 [*Magneto-optical spectroscopy of magnetic semiconductor nanostructures*]{} https://tel.archives-ouvertes.fr/tel-00994939 Universit[é]{} de Grenoble <https://tel.archives-ouvertes.fr/tel-00994939> Man M T and Lee H S 2015 [*Scientific Reports*]{} [**5**]{} 8267 ISSN 2045-2322 <http://dx.doi.org/10.1038/srep08267> Zieliński M 2013 [*Phys. Rev. B*]{} [**88**]{} ISSN 1550-235X <http://dx.doi.org/10.1103/PhysRevB.88.115424> García de Abajo F J 2010 [*Rev. Mod. Phys.*]{} [**82**]{} 209–275 ISSN 1539-0756 <http://dx.doi.org/10.1103/RevModPhys.82.209> Vesseur E J R, de Waele R, Kuttge M and Polman A 2007 [*Nano Lett.*]{} [ **7**]{} 2843–2846 ISSN 1530-6992 <http://dx.doi.org/10.1021/nl071480w> Davis T 2009 [*Optics Communications*]{} [**282**]{} 135–140 ISSN 0030-4018 <http://dx.doi.org/10.1016/j.optcom.2008.09.043> Filter R, Qi J, Rockstuhl C and Lederer F 2012 [*Physical Review B*]{} [ **85**]{} ISSN 1550-235X <http://dx.doi.org/10.1103/PhysRevB.85.125429> Anger P, Bharadwaj P and Novotny L 2006 [*Physical Review Letters*]{} [ **96**]{} ISSN 1079-7114 <http://dx.doi.org/10.1103/PhysRevLett.96.113002> Grzela G, Paniagua-Domínguez R, Barten T, Fontana Y, Sánchez-Gil J A and Gómez Rivas J 2012 [*Nano Lett.*]{} [**12**]{} 5481–5486 ISSN 1530-6992 <http://dx.doi.org/10.1021/nl301907f>

--- abstract: 'Algorithmic fairness has attracted significant attention in the past years. Surprisingly, there is little work on fairness in networks. In this work, we consider fairness for link analysis algorithms and in particular for the celebrated PageRank algorithm. We provide definitions for fairness, and propose two approaches for achieving fairness. The first modifies the jump vector of the Pagerank algorithm to enfonce fairness, and the second imposes a fair behavior per node. We also consider the problem of achieving fairness while minimizing the utility loss with respect to the original algorithm. We present experiments with real and synthetic graphs that examine the fairness of Pagerank and demonstrate qualitatively and quantitatively the properties of our algorithms.' author: - Sotiris Tsioutsiouliklis - Evaggelia Pitoura - Panayiotis Tsaparas - Ilias Kleftakis - Nikos Mamoulis bibliography: - 'FairLAR.bib' title: 'Fairness-Aware Link Analysis' --- =1

--- abstract: 'Requirements are informal and semi-formal descriptions of the expected behavior of a system. They are usually expressed in the form of natural language sentences and checked for errors manually, *e.g.*, by peer reviews. Manual checks are error-prone, time-consuming and not scalable. With the increasing complexity of cyber-physical systems and the need of operating in safety- and security-critical environments, it became essential to automatize the consistency check of requirements and build artifacts to help system engineers in the design process.' author: - Simone Vuotto bibliography: - 'reference.bib' title: Consistency Checking of Functional Requirements --- Introduction ============ The assessment of requirements is an important yet costly and complex task, still largely carried manually. The Requirements Engineering (RE)[@nuseibeh2000requirements] research field aims at developing tools and techniques to analyze and handle requirements in a more efficient and automatic way. One of the main challenges is to evaluate requirements *consistency*: informally, it means detecting errors, missing information and deficiencies that can compromise the interpretation and implementation of the intended system behavior. At a syntactic level, this may involve the check for compliance with standards and guidelines, such as the use of a restricted grammar and vocabulary. We call this task *Compliance Checking*. However, most of the inconsistencies reside at a semantic level, *i.e.* in their intended meaning. This call for an interpretation and reasoning of requirements semantics. The formalization and translation of requirements into a formal representation is an interesting and open research question. A recurrent solution in the literature is the use of Property Specification Patterns (PSPs), first introduced by [@dwyer1999]. PSPs provide a direct mapping from English-like structured natural languages to one or more logics. A survey of all available patterns and their translation has been made by [@autili2015aligning]. Other approaches, like [@ghosh2016arsenal], employ Natural Language Processing techniques to extract the representation directly from fully natural language requirements. Given the set of requirements represented in a formal logic, the main research question is what kind of reasoning we can employ and how to do that. We formally define this task *Consistency Checking* analysis [@heitmeyer1996automated]. Consistency Checking can range from simple variables type and domain checks to more complex activities, like the evaluation of the intended system behavior over time. In particular, we are interested in checking if the set of requirements together “make sense”, namely answering the question: *Given the set of requirements, does a system exist that can satisfy them all at the same time?* The choice of which logic to use is a key research question and it largely affects the reasoning power and the kind of requirements that can be formalized: qualitative, real-time and/or probabilistic. We decided to use Linear Temporal Logic (LTL)[@pnueli1977temporal] because it is widely used in the literature and it has a good balance between expressiveness and complexity. In particular, answering the aforementioned question can be easily translated in a LTL satisfiability check, largely studied and with many efficient tools available [@rozier2010ltl]. The satisfiability check in turn brings other two research questions: - *Vacuity Check*: if the formula is satisfiable, is it satisfiable in a meaningful way? For example, the linear temporal logic (LTL) specification $\Box (msg \rightarrow \Diamond rcv)$ (“every message is eventually received”) is satisfied vacuously in a model with no messages, probably not the expected behavior. - *Inconsistent Requirements Explanation*: if the requirements are inconsistent, which is the minimum set of them that create the inconsistency? The number of requirements may be really large, but only few of them making the system unfeasible. Finally, the formalization of requirements and the consistency checking are enablers for other tasks we would like to tackle in this Ph.D. project, namely the automatic generation of test suites and runtime monitors. The full overview of the tool that we are designing is depicted in Figure \[fig:main\_framework\]. We are now focusing on the NL2FL and Consistency Checker modules. Consistency of Property Specification Patterns ============================================== Our first contribution [@narizzano2018consistency], developed in the context of the H2020 CERBERO European Project [@cerbero], presented a tool for the consistency checking of qualitative requirements expressed in form of PSPs with constrained numerical signals. An example of requirement that we can handle is: *Globally, it is always the case that if $proximity\_sensor < 20$ holds, then $arm\_idle$ eventually holds.* We first translate every requirement $r_i \in R$ in LTL$(\mathcal{D}_C)$, an extension of LTL over a constraint system $D_C$ = ($\mathbb{R}$,$<$,$=$), with atomic constraints of the form $x < c$ and $x = c$ (where $c \in \mathbb{R}$ is a constant real number and ‘$<$’’ and “$=$” have the usual interpretation). We then show how the new problem can be reduced to LTL satisfiability. Let $X(\phi)$ be the set of numerical variables and $C(\phi)$ be the set of constants that occur in $\phi$. We compute: - the LTL$(\mathcal{D}_C)$ formula $\phi_i$ for every requirement $r_i \in R$; - the conjunctive formula $\phi = \phi_1 \wedge ... \wedge \phi_n$; - a set $M_x(\phi)$ of boolean propositions representing possible values of $x \in X(\phi)$; - the formula $Q_M$ encoding the constraints over $M_x(\phi)$ $\forall x \in X(\phi)$; - the formula $\phi'$ that substitute all $x \in X(\phi)$ in $\phi$ with a set of boolean propositions from $M_x(\phi)$; Given the LTL($\mathcal{D}_C$) formula $\phi$ over the set of Boolean atoms $Prop$ and the terms $C(\phi) \cup X(\phi)$ we have that $\phi$ is satisfiable if and only if the LTL formula $\phi_M \rightarrow \phi'$ is satisfiable. This result is important because it shows that LTL($\mathcal{D}_C$) is decidable and that we can exploit state-of-the-art LTL model checkers. In the second part of the paper we translate our encoding in different formats for of-the-shelf model checkers and we compare their performance. We conclude with the scalability analysis and the methodology application to a robotic arm use-case. Future work =========== In order to reduce the number of errors in the specification, we have partially implemented an algorithm to check the relationship among requirements. This is a first step to prevent vacuous results, but more work is needed. #### Connected Requirements Check Given a set of requirements $R = \{r_1, ..., r_n\}$, we want to check if one or more of them are completely unrelated from the others, meaning that they describe some behaviors that don’t interact with the main bulk of the system. This may happen in an underspecified requirements set or for some spelling errors (*e.g.* $armidle$ is written in place of $arm\_idle$ in the previous example). To find these faulty requirements we first build the undirected graph $G = (V, E)$ representing the connections in $R$, such that: - $v_i \in V$ $\forall r_i \in R$; - $(i, j) \in E$ if $X(r_i) \cap X(r_j) \neq \emptyset$ $\forall r_i, r_j \in R, i \neq j$ where $X(r_i)$ is set of variables, boolean or numerical, that appear in $r_i$. We then compute all the connected components in $G$. If the number of components in greater then one, we find the smallest one (*i.e.* the component with the lowest number of vertex) and report it to the user.\ Currently we are also focusing our attention on the Inconsistent Requirements Explanation problem. We implemented a simple algorithm that iterate over all $r_i \in R$ and perform the consistency check on the set $R \setminus r_i$. We keep $r_i$ in $R$ only if the new set is shown consistent, and we discard it otherwise. The algorithm terminates when all the requirements in the original set are checked. This algorithm effectively find a solution, but it is quite inefficient. Therefore, we are seeking for a better algorithm which exploits the structure of the problem. Finally, for future works we would also like to both extend the natural language interface with less restrictive constraints and adopt a more expressive logic. In particular, we are interested in probabilistic logics such PCTL, but the consistency checking problem is difficult to define in this case and more research is needed. #### Acknowledgments The research of Simone Vuotto is funded by the EU Commission H2020 Programme under grant agreement N.732105 (CERBERO project).

--- abstract: 'In this work we address the reheating issue in the context of $F(R)$ gravity, for theories that the inflationary era does not obey the slow-roll condition but the constant-roll condition is assumed. As it is known, the reheating era takes place after the end of the inflationary era, so we investigate the implications of a constant-roll inflation era on the reheating era. We quantify our considerations by calculating the reheating temperature for the constant-roll $R^2$ model and we compare to the standard reheating temperature in the context of $F(R)$ gravity. As we demonstrate, the new reheating temperature may differ from the standard one, and in addition we show how the reheating era may restrict the constant-roll era by constraining the constant-roll parameter.' author: - 'V.K. Oikonomou,$^{1,2}$[^1]' title: 'Reheating in Constant-roll $F(R)$ Gravity' --- Introduction ============ One of the challenges in modern cosmology is to describe the era which connects the end of the inflationary period with the subsequent radiation and matter domination eras, an era known as reheating era. The Universe after the inflationary era is very cold, due to the abrupt nearly de Sitter expansion, and the matter contained in the Universe needs somehow to be thermalized. It is conceivable that such a task is quite complicated and there exist various proposals in the literature, see for example the recent review [@Amin:2014eta]. The reheating era invokes many procedures that are involved in what is now known as Big Bang Nucleosynthesis, and during this era the energy density that drove the quasi de Sitter expansion during the inflationary era, will thermalize the matter content of the Universe. In most well-known approaches in reheating, the inflaton field transfers its energy to the Standard Model particles via direct couplings to these fields [@Kofman:1997yn; @Greene:1997fu; @Kofman:1994rk]. However, in this context there are some drawbacks of having to fine-tune the couplings significantly, in order to avoid large couplings during the inflationary era. An alternative approach for describing the reheating era is offered by modified gravity, and particularly from $F(R)$ gravity [@Mijic:1986iv], in which case the reheating effects take place once the inflationary era ends. In the modified gravity description, gravity has an effect of the effective equation of state of the matter fields, and in effect, these produce a non-trivial effect on the field equations, which in turn thermalize the matter content of the Universe, see Ref. [@Mijic:1986iv] for the $R^2$ model description of the reheating era. In most approaches on the description of the reheating era, a slow-roll era is assumed for the preceding inflationary era. Recently however, another interesting research stream described an alternative evolutionary possibility for the inflationary period, know as constant-roll inflation [@Inoue:2001zt; @Tsamis:2003px; @Kinney:2005vj; @Tzirakis:2007bf; @Namjoo:2012aa; @Martin:2012pe; @Motohashi:2014ppa; @Cai:2016ngx; @Motohashi:2017aob; @Hirano:2016gmv; @Anguelova:2015dgt; @Cook:2015hma; @Kumar:2015mfa; @Odintsov:2017yud; @Odintsov:2017qpp; @Gao:2017owg], see also [@Lin:2015fqa; @Gao:2017uja] for an alternative perspective on this issue, and also see Ref. [@Nojiri:2017qvx] for the $F(R)$ gravity generalization. The constant-roll inflationary models have the interesting property of predicting non-Gaussianities [@Chen:2010xka], even in the context of the single scalar field models [@Inoue:2001zt; @Tsamis:2003px; @Kinney:2005vj; @Tzirakis:2007bf; @Namjoo:2012aa; @Martin:2012pe; @Motohashi:2014ppa; @Cai:2016ngx; @Motohashi:2017aob], so this makes these models conceptually appealing and robust towards future observations of non-Gaussianities in the power spectrum of primordial curvature perturbations. In a recent work we provided a generalization of the constant-roll inflationary era in the context of $F(R)$ gravity [@Nojiri:2017qvx], see also [@Motohashi:2017vdc] for an alternative approach. The focus in this letter is to investigate what are the effects of a constant-roll inflationary era on the reheating process, in the context of $F(R)$ gravity. We shall use the $R^2$ model [@Starobinsky:1980te; @Barrow:1988xh] and we shall calculate the reheating temperature for the case that the preceding inflationary era was a constant-roll one, and we will compare the resulting reheating temperature to the one corresponding to the case that the inflationary era was a usual slow-roll one. As we demonstrate, the ratio of the two reheating temperatures can be quite large, depending strongly on the $F(R)$ gravity model, and also we show that the reheating era can constraint the constant-roll era, since the parameters that quantify the constant-roll era must be constrained for consistency. This paper is organized as follows: in section II we provide some essential information for the $F(R)$ constant-roll inflationary era. We focus on the $R^2$ inflation case, and as we show, the constant-roll condition affects the rate of the quasi-de Sitter expansion. In section III we investigate the effects of the constant-roll quasi-de Sitter expansion on the reheating era, and we calculate the ratio of the reheating temperatures corresponding to the constant-roll and slow-roll preceding inflationary eras. As we show, the ratio depends on the constant-roll parameter that quantifies the constant-roll era. Finally, the conclusions follow at the end of the article. The Constant-roll Inflation Condition with $R^2$ Gravity ======================================================== The constant-roll $F(R)$ gravity inflationary era was introduced in [@Nojiri:2017qvx] (see also [@Motohashi:2017vdc] for an alternative viewpoint), and the main assumption was that the constant-roll condition becomes as follows, $$\label{constantrollcondition} \frac{\ddot{H}}{2H\dot{H}}\simeq \beta\, ,$$ with $\beta$ being a real parameter. The condition (\[constantrollcondition\]) is a natural generalization of the constant-roll condition for a canonical scalar field, since the expression in (\[constantrollcondition\]) is the second slow-roll index in the $F(R)$ gravity frame. For the purposes of this paper, we shall consider a vacuum $F(R)$ gravity (for reviews see [@reviews1; @reviews2; @reviews3]), with the gravitational action being of the following form, $$\label{JGRG7} S_{F(R)}= \int d^4 x \sqrt{-g} \left( \frac{F(R)}{2\kappa^2} \right)\, ,$$ with $g$ being the trace of the background metric, which we shall assume to be a flat Friedmann-Robertson-Walker (FRW) metric with line element, $$\label{metricfrw} ds^2 = - dt^2 + a(t)^2 \sum_{i=1,2,3} \left(dx^i\right)^2\, ,$$ where $a(t)$ is the scale factor. Upon variation of the action (\[JGRG7\]) with respect to the metric tensor, the gravitational equations of motion are, $$\begin{aligned} \label{eqnmotion1} 3F_RH^2=& \frac{F_RR-F}{2}-3H\dot{F}_R \, , \\ \label{eqnmotion2} -2F_R\dot{H}=& \ddot{F}-H\dot{F} \, ,\end{aligned}$$ where the expression $F_R$ stands for $F_R=\frac{\partial F}{\partial R}$ and the “dot” indicates differentiation of the corresponding quantity with respect to the cosmic time. In Ref. [@Nojiri:2017qvx], we explored the inflationary dynamics of constant-roll inflation in the context of $F(R)$ gravity, and as we showed it is possible to obtain observational indices of inflation compatible with the Planck observational data. We used two well-known $F(R)$ gravity models in order to exemplify our results, and particularly the $R^2$ model and a power-law $F(R)$ gravity model, and in this paper we shall use the $R^2$ inflation model in order to investigate how the reheating era is modified if a constant-roll inflationary era precedes the reheating era. The $F(R)$ gravity function in the case of the $R^2$ model [@Starobinsky:1980te] has the following form, $$\label{r2inflation} F(R)=R+\frac{1}{36H_i}R^2\, ,$$ with $H_i$ being a phenomenological parameter with dimensions of mass$^2$, and we assume that $H_i\gg 1$. During the inflationary era we shall assume that the first slow-roll index $\epsilon_1=-\frac{\dot{H}}{H^2}$ satisfies $\epsilon_1\ll 1$, and also that the constant-roll condition (\[constantrollcondition\]) holds true. Hence, during the era for which the conditions $\dot{H}\ll H^2$ and also $\ddot{H}\sim 2\beta H\dot{H}$ hold true, the gravitational equations (\[eqnmotion1\]) and (\[eqnmotion2\]) can be written as follows, $$\label{frweqnsr2} \ddot{H}-\frac{\dot{H}^2}{2H}+3H_iH=-3H\dot{H}\, ,\quad \ddot{R}+3HR+6H_iR=0\, .$$ In view of the constant-roll condition $\ddot{H}\sim 2\beta H\dot{H}$, the first differential equation appearing in Eq. (\[frweqnsr2\]), can be written as follows $$\label{rsquarebasic} \dot{H}H \left( 2\beta+\frac{\epsilon_1}{2}+3 \right)\dot{H}=-3H_i\, ,$$ and due to the fact that $\epsilon_1\ll 1$, by eliminating the $\epsilon_1$-dependent term in Eq. (\[rsquarebasic\]), and by solving the resulting differential equation, we obtain the following solution at leading order, $$\label{hubblersquare} H(t)=H_0-H_I(t-t_k)\, ,$$ with the parameter $H_0$ being arithmetically of the order $\mathcal{O}(H_i)$. Also, the parameter $H_I$ appearing in Eq. (\[hubblersquare\]) is equal to, $$\label{hI} H_I=\frac{3H_i}{2\beta+3}\, ,$$ and in addition, the time instance $t=t_k$ corresponds to the horizon crossing time instance. Clearly, the cosmological evolution (\[hubblersquare\]) is a quasi-de Sitter evolution, just as in the ordinary slow-roll $R^2$ inflation model, with the difference being that in the ordinary $R^2$ model case, $\beta=0$ and hence $H_I\to H_i$. As it was shown in [@Nojiri:2017qvx], the constant-roll inflationary era for the $R^2$ model comes to an end, due to the production of curvature fluctuation, which ends the inflationary era. Then, when the constant-roll era ends, the constant-roll condition does not hold true and due to the presence of the term $\ddot{H}$ in the gravitational equations, the oscillating reheating era commences, as we evince in the next section. Reheating in Constant-roll $R^2$ Gravity ======================================== After the graceful exit from the inflationary era, the Universe enters an intermediate era, which should make a connection between the inflationary era and the radiation and matter domination eras. During the reheating era, the Standard Model particles are thermalized by the Universe, and this is an important feature of any viable cosmological, since after the inflationary era the Universe is cold due to the abrupt nearly exponential expansion that the Universe underwent during the inflationary era. As we mentioned, we shall investigate how the reheating era is affected due to a constant-roll $F(R)$ gravity era, and we directly compare the resulting picture with the ordinary slow-roll $F(R)$ gravity model. We shall focus on the $R^2$ model of Eq. (\[r2inflation\]), although it is expected that similar results can be obtained for any $F(R)$ gravity. In the following we adopt the approach and notation of Ref. [@Mijic:1986iv]. As we show, the effects of a constant-roll quasi-de Sitter evolution can be found directly on the reheating temperature, so we shall make a comparison of the ordinary slow-roll case and the constant-roll case. The reheating era brings new cosmological features into play since the term $\ddot{R}$ in the corresponding differential equation in Eq. (\[frweqnsr2\]), cannot be omitted. In this case, the scalar curvature evolves as a damped oscillation, with a restoring force being of the form $\sim 3H_i$. During the reheating era, the Hubble rate can be found by solving the following differential equation, $$\label{maindiffhubble} \ddot{H}-\frac{\dot{H}^2}{2H}+3H_iH=-3H\dot{H}\, .$$ During the reheating era the terms $\sim \ddot{H}$ and $\sim \frac{\dot{H}^2}{2H}$ start to dominate the evolution, however the term $\sim \dot{H}H$ is comparably negligible. Consider that $t=t_r$ is the time instance that the reheating era commences, so for $t<t_r$, the Hubble rate is given in Eq. (\[hubblersquare\]), and for $t>t_r$, by solving the differential equation (\[maindiffhubble\]), we get the following solution, $$\label{htsolutionreheating} H(t)\simeq \frac{\cos^2\omega (t-t_r)}{\frac{3}{\omega}+\frac{3}{4}(t-t_r)+\frac{3}{8\omega}\sin 2\omega (t-t_r)}\, .$$ The corresponding scale factor during the reheating is, $$\label{reheatingscalefactor} a(t)=a_r\left(1+\frac{\omega (t-t_r)}{4} \right)^{2/3}\, ,$$ with $a_r$ being equal to $a_r=a_0 {\mathrm{e}}^{\frac{H_0^2}{2H_I}-\frac{1}{12}}$, and $a_0$ is the scale factor at the beginning of inflation. In Eq. (\[htsolutionreheating\]), the parameter $\omega$ is affected by the transition from constant-roll to reheating at $t=t_r$, and can be determined by using the following condition, $$\label{condition1equal} \left|\frac{\dot{H}^2}{2H}\right| = \left| 3H\dot{H} \right|\, .$$ The condition (\[condition1equal\]) combined with the quasi-de Sitter evolution (\[hubblersquare\]) and with the reheating evolution (\[htsolutionreheating\]), yields the following form of $\omega$ and $t_r$, $$\label{omegareheating} \omega=\sqrt{\frac{3H_I}{2}}\, \quad t_r\simeq H_I H_0\, .$$ An approximate form of the scalar curvature can also be determined, since during the reheating era, the Ricci scalar is approximately equal to $R\simeq 6\dot{H}$, thus we approximately have, $$\label{approxscalarcurvaturereheating} R(t)\simeq -\frac{6\omega \sin 2\omega (t-t_r)}{\left(\frac{3}{\omega}+\frac{3}{4}(t-t_r)+\frac{3}{8\omega}\sin 2\omega (t-t_r) \right)}\, .$$ At this point recall that the parameter $H_I$ contains a hidden $\beta$-dependence, as it can be seen in Eq. (\[hI\]), and recall that the parameter $\beta$ is the constant-roll condition parameter of Eq. (\[constantrollcondition\]). Essentially the parameter $\beta$ determines the shape and the size of the reheating phase. Now let us quantify the effects of the constant-roll on the $R^2$ gravity reheating by comparing the reheating temperature for the constant-roll and slow-roll $R^2$ model. In order to calculate the reheating temperature, it is assumed that the matter content consists of a scalar field $\phi$ with gravitational equation $g^{\mu \nu}\phi_{;\mu \nu}=0$. As it was shown in [@Mijic:1986iv], the effect of the matter content is connected to the square average of the scalar curvature $R$, and the $(t,t)$ component of the field equations yields, $$\label{eqnmotionreheatingmain} H^2+\frac{1}{18H_I}\left(HR-\frac{R^2}{12}RH^2 \right)=\frac{8\pi}{3}G\rho_c\, ,$$ where energy-density term $\rho_c$ is defined to be, $$\label{rhomatter} \rho_c=\frac{N}{a^4}\int_{t_r}^t\frac{\omega}{1152\pi} \bar{R}^2a^4 d t\, ,$$ and it refers to the constant-roll case. By assuming that during the reheating era, the energy density $\rho_c$ is totally converted to radiation energy $\rho_r=\frac{N\pi^2}{30}T_{rc}^4$, where $N$ is the total number of relativistic particles, while $T_{rc}$ is the reheating temperature corresponding to the case that a constant-roll case precedes the reheating epoch. By denoting $\rho_s$ and $T_{rs}$ the energy density and the reheating temperature corresponding to the case that a slow-roll inflationary era precedes the reheating epoch, by using Eq. (\[rhomatter\]) and the corresponding formula for the slow-roll pre-reheating era, and also by assuming that the final time instance in the integral appearing in Eq. (\[rhomatter\]), is $t\simeq t_r+10\omega $, we obtain the following relations, $$\label{reheatingtemperature1} \frac{\rho_c}{\rho_s}=\frac{\omega^4}{\omega_s^4},\,\,\, \frac{T_{rc}}{T_{rs}}=\frac{\omega}{\omega_s}\, ,$$ where the parameter $\omega_s$ for the slow-roll era quasi-de Sitter evolution reads, $$\label{omegas} \omega_s=\sqrt{\frac{3H_i}{2}}\, .$$ By combining Eqs. (\[hI\]), (\[omegareheating\]) and (\[omegas\]), we finally obtain the ratio of the constant-roll to slow-roll reheating temperatures, $$\label{finalrelation} \frac{T_{rc}}{T_{rs}}=\sqrt{\frac{3}{2 \beta +3}}\, ,$$ in which it can clearly be seen that the parameter $\beta$ affects the reheating temperature, so the constant-roll era preceding the reheating era, can affect the reheating era. ![Behavior of the ratio $\frac{T_{rc}}{T_{rs}}$ as a function of the constant-roll parameter $\beta$.[]{data-label="plot1"}](plot1.eps "fig:"){width="18pc"} ![Behavior of the ratio $\frac{T_{rc}}{T_{rs}}$ as a function of the constant-roll parameter $\beta$.[]{data-label="plot1"}](plot2.eps "fig:"){width="18pc"} In order to have a quantitative picture of how the constant-roll parameter $\beta$ affects the reheating temperature, we shall plot the ratio $\frac{T_{rc}}{T_{rs}}$, as a function of $\beta$. In Fig. \[plot1\], we plot the behavior of the ratio $\frac{T_{rc}}{T_{rs}}$ as a function of the constant-roll parameter $\beta$. As it can be seen in the left plot, the ratio $\frac{T_{rc}}{T_{rs}}$ takes large values as $\beta\to -3/2$, since the ratio is singular at $\beta=-3/2$. This means that for $\beta\to -3/2$, the constant-roll reheating temperature may differ significantly from the corresponding slow-roll reheating temperature. However, as $\beta$ takes larger values, the ratio $\frac{T_{rc}}{T_{rs}}$ tends to zero, which means that the reheating temperature in the two cases is almost the same. Moreover, the equation (\[finalrelation\]) can be considered as a constraint on the parameter $\beta$ for the constant-roll $R^2$ model, since $\beta$ must be $\beta>-3/2$, so the reheating era imposes some constraints on the constant-roll inflationary era. Conclusions =========== In this paper we discussed the implications of a constant-roll inflationary era on the dynamics of reheating, in the context of $F(R)$ gravity. After discussing how the constant-roll condition modifies the dynamics of inflation in $F(R)$ gravity, we focused on the $R^2$ model and we demonstrated that the constant-roll condition predicts a quasi-de Sitter evolution which governs the inflationary era. As we demonstrated this quasi-de Sitter evolution affects the reheating era, and it modifies directly the reheating temperature. We compared the reheating temperature corresponding to a constant-roll inflationary era preceding the reheating era, to the corresponding slow-roll case for the $R^2$ model, and we found that when the constant-roll parameter approaches $\beta\to -3/2$, the two reheating temperatures may differ significantly. Finally, we demonstrated that the reheating era may restrict the constant-roll parameter and in effect the whole constant-roll inflationary era, but this feature is strictly model dependent. For example, in the case of the $R^2$ model, the constant-roll parameter must be $\beta>-3/2$ for physical consistency reasons. In principle the results we found in this work may apply to other $F(R)$ gravities as well, and it would be interesting to investigate this issue in the context of string inspired modified gravities, such Gauss-Bonnet scalar theories, a task we hope to address in a future work. Acknowledgments {#acknowledgments .unnumbered} =============== This work is supported by the Russian Ministry of Education and Science, Project No. 3.1386.2017 (V.K.O). [99]{} M. A. Amin, M. P. Hertzberg, D. I. Kaiser and J. Karouby, Int. J. Mod. Phys. D [**24**]{} (2014) 1530003 \[arXiv:1410.3808 \[hep-ph\]\]. L. Kofman, A. D. Linde and A. A. Starobinsky, Phys. Rev. D [**56**]{} (1997) 3258 \[hep-ph/9704452\]. P. B. Greene, L. Kofman, A. D. Linde and A. A. Starobinsky, Phys. Rev. D [**56**]{} (1997) 6175 \[hep-ph/9705347\]. L. Kofman, A. D. Linde and A. A. Starobinsky, Phys. Rev. Lett.  [**73**]{} (1994) 3195 \[hep-th/9405187\]. M. B. Mijic, M. S. Morris and W. M. Suen, Phys. Rev. D [**34**]{} (1986) 2934. S. Inoue and J. Yokoyama, Phys. Lett. B [**524**]{} (2002) 15 doi:10.1016/S0370-2693(01)01369-7 \[hep-ph/0104083\]. N. C. Tsamis and R. P. Woodard, Phys. Rev. D [**69**]{} (2004) 084005 doi:10.1103/PhysRevD.69.084005 \[astro-ph/0307463\]. W. H. Kinney, Phys. Rev. D [**72**]{} (2005) 023515 doi:10.1103/PhysRevD.72.023515 \[gr-qc/0503017\]. K. Tzirakis and W. H. Kinney, Phys. Rev. D [**75**]{} (2007) 123510 doi:10.1103/PhysRevD.75.123510 \[astro-ph/0701432\]. M. H. Namjoo, H. Firouzjahi and M. Sasaki, Europhys. Lett.  [**101**]{} (2013) 39001 doi:10.1209/0295-5075/101/39001 \[arXiv:1210.3692 \[astro-ph.CO\]\]. J. Martin, H. Motohashi and T. Suyama, Phys. Rev. D [**87**]{} (2013) no.2, 023514 doi:10.1103/PhysRevD.87.023514 \[arXiv:1211.0083 \[astro-ph.CO\]\]. H. Motohashi, A. A. Starobinsky and J. Yokoyama, JCAP [**1509**]{} (2015) no.09, 018 doi:10.1088/1475-7516/2015/09/018 \[arXiv:1411.5021 \[astro-ph.CO\]\]. Y. F. Cai, J. O. Gong, D. G. Wang and Z. Wang, JCAP [**1610**]{} (2016) no.10, 017 doi:10.1088/1475-7516/2016/10/017 \[arXiv:1607.07872 \[astro-ph.CO\]\]. H. Motohashi and A. A. Starobinsky, arXiv:1702.05847 \[astro-ph.CO\]. S. Hirano, T. Kobayashi and S. Yokoyama, Phys. Rev. D [**94**]{} (2016) no.10, 103515 doi:10.1103/PhysRevD.94.103515 \[arXiv:1604.00141 \[astro-ph.CO\]\]. L. Anguelova, Nucl. Phys. B [**911**]{} (2016) 480 doi:10.1016/j.nuclphysb.2016.08.020 \[arXiv:1512.08556 \[hep-th\]\]. J. L. Cook and L. M. Krauss, JCAP [**1603**]{} (2016) no.03, 028 doi:10.1088/1475-7516/2016/03/028 \[arXiv:1508.03647 \[astro-ph.CO\]\]. K. S. Kumar, J. Marto, P. Vargas Moniz and S. Das, JCAP [**1604**]{} (2016) no.04, 005 doi:10.1088/1475-7516/2016/04/005 \[arXiv:1506.05366 \[gr-qc\]\]. S. D. Odintsov and V. K. Oikonomou, arXiv:1703.02853 \[gr-qc\]. S. D. Odintsov and V. K. Oikonomou, arXiv:1704.02931 \[gr-qc\]. Q. Gao, arXiv:1704.08559 \[astro-ph.CO\]. J. Lin, Q. Gao and Y. Gong, Mon. Not. Roy. Astron. Soc.  [**459**]{} (2016) no.4, 4029 doi:10.1093/mnras/stw915 \[arXiv:1508.07145 \[gr-qc\]\]. Q. Gao and Y. Gong, arXiv:1703.02220 \[gr-qc\]. S. Nojiri, S. D. Odintsov and V. K. Oikonomou, arXiv:1704.05945 \[gr-qc\]. X. Chen, Adv. Astron.  [**2010**]{} (2010) 638979 doi:10.1155/2010/638979 \[arXiv:1002.1416 \[astro-ph.CO\]\]. H. Motohashi and A. A. Starobinsky, arXiv:1704.08188 \[astro-ph.CO\]. A. A. Starobinsky, Phys. Lett.  [**91B**]{} (1980) 99. doi:10.1016/0370-2693(80)90670-X J. D. Barrow and S. Cotsakis, Phys. Lett. B [**214**]{} (1988) 515. doi:10.1016/0370-2693(88)90110-4 S. Nojiri and S. D. Odintsov, Phys. Rept.  [**505**]{} (2011) 59 doi:10.1016/j.physrep.2011.04.001 \[arXiv:1011.0544 \[gr-qc\]\] S. Nojiri and S. D. Odintsov, eConf C [**0602061**]{} (2006) 06 \[Int. J. Geom. Meth. Mod. Phys.  [**4**]{} (2007) 115\] doi:10.1142/S0219887807001928 \[hep-th/0601213\]. S. Nojiri, S. D. Odintsov and V. K. Oikonomou, arXiv:1705.11098 \[gr-qc\]. [^1]: v.k.oikonomou1979@gmail.com

--- abstract: 'Discriminative localization is essential for fine-grained image classification task, which devotes to recognizing hundreds of subcategories in the same basic-level category. Reflecting on discriminative regions of objects, key differences among different subcategories are subtle and local. Existing methods generally adopt a two-stage learning framework: *The first stage* is to localize the discriminative regions of objects, and *the second* is to encode the discriminative features for training classifiers. However, these methods generally have two limitations: (1) *Separation* of the two-stage learning is *time-consuming*. (2) *Dependence* on object and parts annotations for discriminative localization learning leads to heavily *labor-consuming* labeling. It is highly challenging to address these two important limitations *simultaneously*. Existing methods only focus on one of them. Therefore, this paper proposes *the discriminative localization approach via saliency-guided Faster R-CNN* to address the above two limitations at the same time, and our main novelties and advantages are: (1) *End-to-end network* based on Faster R-CNN is designed to *simultaneously* localize discriminative regions and encode discriminative features, which accelerates classification speed. (2) *Saliency-guided localization learning* is proposed to localize the discriminative region automatically, avoiding labor-consuming labeling. Both are jointly employed to simultaneously accelerate classification speed and eliminate dependence on object and parts annotations. Comparing with the state-of-the-art methods on the widely-used CUB-200-2011 dataset, our approach achieves both the best classification accuracy and efficiency.' author: - 'Xiangteng He, Yuxin Peng\* and Junjie Zhao' bibliography: - 'sigproc.bib' title: | Fine-grained Discriminative Localization\ via Saliency-guided Faster R-CNN --- Introduction ============ Fine-grained image classification is a highly challenging task due to large variance in the same subcategory and small variance among different subcategories, as shown in Figure \[interintra\], which is to recognize hundreds of subcategories belonging to the same basic-level category. These subcategories look similar in global appearances, but have subtle differences at discriminative regions of objects, which are crucial for classification. Therefore, most researchers focus on localizing discriminative regions of objects to promote the performance of fine-grained image classification. ![Examples of CUB-200-2011 dataset [@wah2011caltech]. First row shows large variance in the same subcategory, and second row shows small variance among different subcategories.[]{data-label="interintra"}](interintra.pdf){width="1\linewidth"} Most existing methods [@zhang2014part; @zhang2014fused; @krause2015fine; @xiao2015application; @zhang2016picking; @simon2015neural; @he2017spatial] generally follow a two-stage learning framework: The first learning stage is to localize discriminative regions of objects, and the second is to encode the discriminative features for training classifiers. Girshick et al. [@girshick2014rich] propose a simple and scalable detection algorithm, called R-CNN. It generates thousands of region proposals for each image via bottom-up process [@uijlings2013selective] first. And then extracts features of objects via convolutional neural network (CNN) to train an object detector for each class, which is used to discriminate the probabilities of the region proposals being objects. This framework is widely used in fine-grained classification. Zhang et al. [@zhang2014part] utilize R-CNN with geometric constraints to detect object and its parts first, and then extract features for the object and its parts, finally train a one-versus-all linear SVM for classification. It needs both object and parts annotations. Krause et al. [@krause2015fine] adopt the box constraint of Part-based R-CNN [@zhang2014part] to train part detectors with only object annotation. These methods generally have two limitations: (1) Separation of the two-stage learning is time-consuming. (2) Dependence on object and parts annotations for discriminative localization learning leads to heavily labor-consuming labeling. It is highly challenging to address these two limitations simultaneously. Existing works only focus on one of them. {width="1\linewidth"} For addressing the first limitation, researchers focus on the end-to-end network. Zhang et al. [@zhang2016spda] propose a Part-Stacked CNN architecture, which first utilizes a fully convolutional network to localize parts of object, and then adopts a two-stream classification network to encode object-level and part-level features simultaneously. It is over two order of magnitude faster than Part-based R-CNN [@zhang2014part], but relies heavily on object and parts annotations that are *labor consuming*. For addressing the second limitation, researchers focus on how to localize the discriminative regions under the weakly supervised setting, which means neither object nor parts annotations are used in training or testing phase. Xiao et al. [@xiao2015application] propose a two-level attention model: object-level attention is to select relevant region proposals to a certain object, and part-level attention is to localize discriminative parts of object. It is the first work to classify fine-grained images without using object or parts annotations in both training and testing phase, but still achieves promising results [@zhang2016weakly]. Simon and Rodner [@simon2015neural] propose a constellation model to localize parts of object, leveraging CNN to find the constellations of neural activation patterns. A part model is estimated by selecting part detectors via constellation model. And then the part model is used to extract features for classification. Zhang et al. [@zhang2016picking] incorporate deep convolutional filters for both parts selection and description. He and Peng [@he2017spatial] integrate two spatial constraints for improving the performance of parts selection. These methods rarely depend on object or parts annotations, but their classification speeds are *time consuming* due to the separation of localization and encoding. Different from them, this paper proposes a discriminative localization approach via saliency-guided Faster R-CNN, which is the first attempt based on discriminative localization to simultaneously accelerate classification speed and eliminate dependence on object and parts annotations. Its main novelties and contributions can be summarized as follows: - ***End-to-end network.***  Most existing discriminative localization based methods [@xiao2015application; @simon2015neural; @zhang2016picking] generally localize discriminative regions first, and then encode discriminative features. The separated processes cause highly *time-consuming* classification. For addressing this important problem, we propose an *end-to-end network* based on Faster R-CNN to *accelerate* the classification speed by simultaneously localizing discriminative regions and encoding discriminative features. Localization exploits discriminative regions with subtle but distinguishing features from other subcategories, and encoding generates representative descriptions. They have synergistic effect with each other, which further improves the classification performance. - ***Saliency-guided localization learning.***  Existing methods as [@huang2016part] combine localization and encoding to accelerate classification speed. However, localization learning relies heavily on object or parts annotations, which is *labor consuming*. For addressing this important problem, we propose a *saliency-guided localization learning* approach, which *eliminates the heavy dependence on object and parts annotations* by localizing the discriminative regions automatically. We adopt a neural network with global average pooling (GAP) layer, which is called saliency extraction network (SEN), to extract the saliency information for each image. And then share convolutional weights between SEN and Faster R-CNN to transfer knowledge of discriminative features. This takes the advantages of both SEN and Faster R-CNN to boost the discriminative localization and avoid the labor-consuming labeling simultaneously. The rest of this paper is organized as follows: Section 2 presents our approach in detail, and Section 3 introduces the experiments as well as the results analyses. Finally Section 4 concludes this paper. Saliency-guided Faster R-CNN ============================ We propose a discriminative localization approach via saliency-guided Faster R-CNN without using object or parts annotations. Saliency-guided Faster R-CNN is an end-to-end network to localize discriminative regions and encode discriminative features simultaneously, which not only achieves a notable classification performance but also accelerates classification speed. It consists of two components: saliency extraction network (SEN) and Faster R-CNN. SEN extracts saliency information of each image for generating the bounding box which is used to guide the discriminative localization learning of Faster R-CNN. They are two localization learning stages, and their jointly learning further achieves better performance. An overview of our approach is shown as Figure \[framework\]. Weakly supervised Faster R-CNN ------------------------------ We propose a weakly supervised Faster R-CNN to accelerate classification speed and achieve promising results simultaneously without using object or parts annotations. A saliency extraction network (SEN) is proposed to generate bounding box information for Faster R-CNN first. It takes a resized image as an input and outputs a saliency map for generating the bounding box of discriminative region. We follow the work of Zhou et al. [@zhou2016cvpr] to model this process by utilizing global average pooling (GAP) to produce the saliency map. We sum the feature maps of last convolutional layer with weights to generate the saliency map for each image. Figure \[saliencymap\] shows some examples of saliency maps obtained by our approach. Finally we perform binarization operation on the saliency map with a adaptive threshold, which is obtained via OTSU algorithm [@otsu1979threshold], and take the bounding box that covers the largest connected area as the discriminative region of object. For a given image $I$, the value of spatial location $(x,y)$ in saliency map for subcategory $c$ is defined as follows: $$\begin{gathered} M_c(x,y) = \sum \limits_u w_u^c f_u(x,y)\end{gathered}$$ where $M_c(x,y)$ directly indicates the importance of activation at spatial location $(x,y)$ leading to the classification of an image to subcategory $c$, $f_u(x,y)$ denotes the activation of neuron $u$ in the last convolutional layer at spatial location $(x,y)$, and $w_u^c$ denotes the weight that corresponding to subcategory $c$ for neuron $u$. Instead of using the image-level subcategory labels, we use the predicted result as the subcategory $c$. Faster R-CNN [@ren2015faster] is proposed to accelerate the process of detection as well as achieve promising detection performance. However, the training phase needs ground truth bounding boxes of objects for supervised learning, which is heavily labor consuming. In this paper, we propose weakly supervised Faster R-CNN to localize the discriminative region, which is guided by the saliency information extracted by SEN. Faster R-CNN is composed by region proposal network (PRN) and Fast R-CNN [@Girshick_2015_ICCV], both of them share convolutional layers for better performance. ![Some examples of saliency maps extracted by SEN in our Saliency-guided Faster R-CNN approach.[]{data-label="saliencymap"}](saliencymap.pdf){width="1\linewidth"} Instead of using time-consuming bottom-up process such as selective search [@uijlings2013selective], RPN is adopted to quickly generate region proposals of images by sliding a small network over the feature map of last shared conolutional layer. At each sliding-window location, $k$ region proposals are simultaneously predicted, and they are parameterized relative to $k$ anchors. We apply $9$ anchors with $3$ scales and $3$ aspect ratios as Faster R-CNN. For training RPN, a binary class label of being an object or not is assigned to each anchor, which depends on the Intersection-over-Union (IoU) [@everingham2015pascal] overlap with a ground truth bounding box of object. But in our approach, we compute the IoU overlap with the bounding box of discriminative region generated by SEN rather than the ground truth bounding box of object, which avoids using the labor-consuming object and parts annotations. And the loss function for an image is defined as: $$\begin{gathered} L(\{p_i\},\{t_i\})=\frac{1}{N_{cls}} \sum_i L_{cls}(p_i,p_i^*)\nonumber \\ + \lambda \frac{1}{N_{reg}} \sum_i {p_i^*L_{reg}(t_i,t_i^*)}\end{gathered}$$ where $i$ denotes the index of an anchor in a mini-batch, $p_i$ denotes the predicted probability of anchor $i$ being a discriminative region, $p_i^*$ denotes the label of being a discriminative region of object or not depending on the bounding box $t_i^*$ generated by SEN , $t_i$ is the predicted bounding box of discriminative region, $L_{cls}$ is the classification loss defined by log loss, and $L_{reg}$ is the regression loss defined by the robust loss function (smooth $L_1$) [@Girshick_2015_ICCV]. For the localization network, Fast R-CNN [@Girshick_2015_ICCV] is adopted. In Fast R-CNN, a region of interest (RoI) pooling layer is employed to extract a fixed-length feature vector from feature map for each region proposal generated by RPN. And each feature vector passes forward for two outputs: one is predicted subcategory and the other is predicted bounding box of discriminative region. Through Faster R-CNN, we obtain the discriminative region and subcategory of each image simultaneously, accelerating classification speed. Saliency-guided localization learning ------------------------------------- The saliency-guided localization learning schedules the training process of SEN and Faster R-CNN to make full use of their advantages: (1) SEN learns the saliency information of image to tell which region is important and discriminative for classification, and saliency information guides the training of Faster R-CNN, and (2) RPN in Faster R-CNN generates region proposals that relevant to the discriminative regions of images, which accelerates the process of region proposal rather than using bottom-up process as selective search [@uijlings2013selective]. Considering that training RPN needs bounding boxes of discriminative regions provided by SEN, and Fast R-CNN utilizes the proposals generated by RPN, we adopt the strategy of sharing convolutional weights between SEN and Faster R-CNN to promote the localization learning. First, we train the SEN. This network is first pre-trained on the ImageNet 1K dataset [@imagenet_cvpr09], and then fine-tuned on the fine-grained image classification dataset, such as CUB-200-2011 [@wah2011caltech] in our experiment. And then, we train the PRN. Its initial weights of convolutional layers are cloned from SEN. Instead of fixing the shared convolutional layers, all layers are fine-tuned in the training phase. Besides, we train RPN and Fast R-CNN follows the strategy in Ren et al. [@ren2015faster]. Experiments =========== Dataset and evaluation metrics ------------------------------ We conduct experiments on the widely-used CUB-200-2011 [@wah2011caltech] dataset in fine-grained image classification. Our proposed Saliency-guided Faster R-CNN approach is compared with 18 state-of-the-art methods to verify its effectiveness. **CUB-200-2011** [@wah2011caltech] is the most widely-used dataset in fine-grained image classification task, which contains 11788 images of 200 subcategories belonging to bird, 5994 images for training and 5794 images for testing. And each image has detailed annotations: a image-level subcategory label, a bounding box of object, and 15 part locations. In our experiments, only image-level subcategory label is used to train the networks. **Accuracy** is adopted to comprehensively evaluate the classification performances of our Saliency-guided Faster R-CNN approach and compared methods, which is widely used in fine-grained image classification [@zhang2016weakly; @zhang2016picking; @zhang2014part], and its definition is as follow: $$\begin{gathered} Accuracy = \frac{R_a}{R}\end{gathered}$$ where $R$ denotes the number of images in testing set, and $R_a$ denotes the number of images that are correctly classified. **Intersection-over-Union (IoU)** [@everingham2015pascal] is adopted to evaluate whether the predicted bounding box of discriminative region is a correct localization, and its formula is defined as: $$\begin{gathered} IoU = \frac{area(B_p \cap B_{gt})}{area(B_p \cup B_{gt})}\end{gathered}$$ where $B_p$ denotes the predicted bounding box of discriminative region, $B_{gt}$ denotes the ground truth bounding box of object, $B_P \cap B_{gt}$ denotes the intersection of the predicted and ground truth bounding boxes, and $B_p \cup B_{gt}$ denotes their union. We consider the predicted bounding box of discriminative region is correctly localized, if the IoU exceeds 0.5. Details of the networks ----------------------- Our Saliency-guided Faster R-CNN approach consists of three networks: saliency extraction network (SEN), region proposal network (RPN) and Fast R-CNN. They are all based on 16-layer VGGNet [@simonyan2014very], which is widely used in image classification task. The basic CNN can be replaced with the other CNN. SEN extracts the saliency information of images to provide bounding boxes needed by Faster R-CNN. For VGGNet in SEN, we remove the layers after conv$5\_3$ and add a convolutional layer of size $3 \times 3$, stride $1$, pad $1$ with $1024$ neurons, which is followed by a global average pooling layer and a softmax layer [@zhou2016cvpr]. We adopt the object-level attention of Xiao et al. [@xiao2015application] to select relevant image patches for data extension. And then we utilize the extended data to fine-tune SEN for learning discriminative features. The number of neurons in softmax layer is set as the number of subcategories in the dataset. Faster R-CNN shares the weights of layers before conv$5\_3$ with SEN for better discriminative localization as well as classification performance. The architecture of Fast R-CNN is the same with VGGNet except that pool5 layer is replaced by a RoI pooling layer, and has two outputs: one is predicted subcategory and the other is predicted bounding box of discriminative region. At training phase, for SEN, we initialize the weights with the network pre-trained on the ImageNet 1K dataset, and then use SGD with a minibatch size of 20. We use a weight decay of 0.0005 with a momentum of 0.9 and set the initial learning rate to 0.001. The learning rate is divided by 10 every 5K iterations. We terminate training at 35K iterations. For Faster RCNN, we initialize the weights with the SEN, and then start SGD with a minibatch size of 128. We use a weight decay of 0.0005 with a momentum of 0.9 and set the initial learning rate to 0.001. We divide the learning rate by 10 at 30K iterations, and terminate training at 50K iterations. Comparisons with state-of-the-art methods ----------------------------------------- This subsection presents the experimental results and analyses of our Saliency-guided Faster R-CNN approach as well as the state-of-the-art methods on the widely-used CUB-200-2011 [@wah2011caltech] dataset. We verify the effectiveness of our approach from accuracy and efficiency of classification. ### Accuracy of classification Table \[cub\] shows the comparison results on CUB-200-2011 dataset at the aspect of classification accuracy. Object, parts annotations and CNN used in these methods are listed for fair comparison. Traditional methods as [@berg2013poof] choose SIFT [@lowe2004distinctive] as features, even using both object and parts annotations its performance is limited and much lower than our approach. Our approach achieves the highest classification accuracy among all methods under the same weakly supervised setting that neither object nor parts annotations are used in training or testing phase, and obtains 0.45% higher accuracy than the best result of TSC [@he2017spatial] (85.14% vs. 84.69%), which jointly considers two spatial constraints in parts selection. Despite achieving better classification accuracy, our approach is over two order of magnitude faster than TSC, due to the end-to-end network, as shown in Table \[cubtime\]. The efficiency analysis will be described in Section 3.3.2. And our approach performs better than the method of Bilinear-CNN [@lin2015bilinear], which combines two different CNNs: VGGNet [@simonyan2014very] and VGG-M [@chatfield2014return]. Its classification accuracy is 84.10%, which is lower than our approach by 1.04%. Furthermore, our approach even outperforms these methods using object annotation in both training and testing phase by at least 2.24%, such as Coarse-to-Fine [@yao2016coarse], PG Alignment [@krause2015fine] and VGG-BGLm [@zhou2016fine]. Moreover, our approach outperforms these methods that use both object and parts annotations [@zhang2014part; @xu2016webly]. Neither object nor parts annotations are used in our Saliency-guided Faster R-CNN approach, which leads fine-grained image classification to practical application. Besides, end-to-end network in our approach simultaneously localizes discriminative region and encodes discriminative feature for each image, and discriminative localization promotes the classification performance. [|p[7.4cm]{}&lt;|p[4cm]{}&lt;|p[4cm]{}&lt;|]{} Methods & Testing Speed (fps) & Net\ **Our Saliency-guided Faster R-CNN Approach** & **10.07** & VGGNet\ Bilinear-CNN [@lin2015bilinear]& 4.52 & VGGNet&VGG-M\ TSC [@he2017spatial] & 0.34 & VGGNet\ TL Atten [@xiao2015application] & 0.25 &VGGNet\ NAC [@simon2015neural] & 0.10 &VGGNet\ **Our Saliency-guided Faster R-CNN Approach** & **17.09** & AlexNet\ PS-CNN [@huang2016part] & 14.30 & AlexNet\ [|p[7.4cm]{}&lt;|p[4cm]{}&lt;|p[4cm]{}&lt;|]{} Methods & Classification Accuracy(%) & Localization Accuracy(%)\ **Our Saliency-guided Faster R-CNN Approach** & **85.14** & **46.05**\ SEN & 77.50 & 44.93\ {width="0.95\linewidth"} ### Efficiency of classification Experimental results at the aspect of efficiency on CUB-200-2011 dataset is presented in Table \[cubtime\]. We get the testing speed on the computer with NVIDIA TITAN X @1417MHZ and Intel Core i7-6900K @3.2GHZ, and use frames per second (fps) to evaluate the efficiency. Comparing with state-of-the-art methods, our Saliency-guided Faster R-CNN approach achieves the best performance on not only the classification accuracy but also the efficiency. We split state-of-the-art methods into two groups by the basic CNNs used in their methods: VGGNet [@simonyan2014very] and AlexNet [@krizhevsky2012imagenet]. Results of these methods in first group are obtained by their authors’ source codes. Comparing with these methods, our approach improves about 123% than Bilinear-CNN at the aspect of classification speed (10.07 fps vs. 4.52 fps). Besides, our classification accuracy is also 1.04% higher than Bilinear-CNN. Even more, our approach is over two orders of magnitude faster than these methods with two separated stages of localization and encoding. When utilizing AlexNet as the basic network, our approach is still faster than PS-CNN [@huang2016part] and improves about 19.51%, which also utilizes AlexNet. And when applying AlexNet as basic CNN in our approach, the classification accuracy is 73.58%. It is noted that neither object nor parts annotations are used in our approach, while all used in PS-CNN. The classification speed of PS-CNN [@huang2016part] is reported as 20 fps in their paper. They provide a reference that a single CaffeNet [@jia2014caffe] runs at 50 fps under their experimental setting (NVIDIA Tesla K80). In our experiments, a single CaffeNet runs at 35.75 fps, so we calculate the speed of PS-CNN in the same experimental setting with ours as 20\*35.75/50=14.30 fps. Our approach avoids the time-consuming classification process by the design of end-to-end network, and achieves the best classification performance by the mutual promotion between localization and classification. This leads the fine-grained image classification to practical application. Effectiveness of discriminative localization -------------------------------------------- Saliency-guided localization learning is proposed to train SEN and Faster R-CNN for improving the localization and classification performance simultaneously. Since we devote to localizing the discriminative region which is generally located at the object, we adopt the IoU overlap between discriminative region and ground truth bounding box of object to evaluate the correctness of localization. We consider a bounding box of discriminative region to be correctly predicted if IoU with ground truth bounding box of object is larger than 0.5. The accuracy of localization is shown in Table \[component\]. Our Saliency-guided Faster R-CNN approach achieves the accuracy of 46.05%. Considering that neither object nor parts annotations are used, it is a promising result. And comparing with “SEN” which means directly using SEN to generate bounding box, our approach achieves improvements both in classification and localization, which verifies the effectiveness of our saliency-guided localization learning approach. [|p[1.47cm]{}&lt;|p[1.47cm]{}&lt;|p[1.47cm]{}&lt;|p[1.47cm]{}&lt;|p[1.47cm]{}&lt;|p[1.47cm]{}&lt;|p[1.47cm]{}&lt;|p[1.47cm]{}&lt;|p[1.47cm]{}&lt;|]{} Parts & back & beak & belly & breast & crown & forehead & left eye & left leg\ PCL (%) & 96.33 & 96.49 & 94.00 & 95.29 & 97.38 & 97.07 & 97.49 & 89.92\ Parts & left wing & nape & right eye & right leg & right wing & tail & throat & **average**\ PCL (%)& 92.60 & 96.60 & 96.79 & 91.85 & 97.00 & 85.03 & 96.38 & **94.68**\ ![IoU histogram. Abscissa denotes the IoU overlap between predicted bounding box of discriminative region and ground truth bounding box of object. And ordinate means the number of testing images that have the IoU overlap at the range. []{data-label="ioucurve"}](ioucurve.pdf){width="0.9\linewidth"} We show some samples of predicted bounding boxes of discriminative regions and ground truth bounding boxes of objects at different ranges of IoU (e.g. 0$\sim$0.2, 0.2$\sim$0.4, 0.4$\sim$0.6, 0.6$\sim$0.8, 0.8$\sim$1) on CUB-200-2011 dataset, as Figure \[boundingbox\]. We have some predicted bounding boxes whose IoUs with ground truth bounding boxes of objects are lower than 0.5. But these predicted bounding boxes contain discriminative regions of objects, such as heads and bodies. It verifies the effectiveness of our approach in localizing discriminative region of object for achieving better classification performance. Figure \[ioucurve\] shows the histogram of IoU. We can observe that most testing images lie in the range of 0.4$\sim$1. To further verify the effectiveness of discriminative localization in our approach, results are given in terms of the Percentage of Correctly Localization (PCL) in Table \[parts\], estimating whether the predicted bounding box contains the parts of object or not. CUB-200-2011 dataset provides 15 part locations, which denote the pixel locations of centers of parts. We consider our predicted bounding box contain a part if the part location lies in the area of the predicted bounding box. Table \[parts\] shows that about average 94.68% of the parts located in our predicted bounding boxes. It shows that our discriminative localization can detect the distinguishing information of objects to promote classification performance. Analysis of misclassification ----------------------------- Figure \[confusion\] shows the classification confusion matrix for our approach, where coordinate axes denote subcategories and different colors denote different probabilities of misclassification. The yellow rectangles show the sets of subcategories with the higher probability of misclassification. We can observe that these sets of subcategories locate near the diagonal of the confusion matrix, which means that these misclassification subcategories generally belong to the same genus with small variance. The small variance is not easy to measure from the image, which leads the high challenge of fine-grained image classification. Figure \[confusionexamples\] shows some examples of the most probably confused subcategory pairs. One subcategory is most confidently classified as the other in the same row. The subcategories in the same row look almost the same, and belong to the same genus. For example, “Brandt Cormorant” and “Pelagic Cormorant” look the same in the appearance, both of them have the same attributes of black feather and long neck, and belong to the genus of “Phalacrocorax”. It is extremely difficult for us to distinguish between them. ![Classification confusion matrix on CUB-200-2011 dataset with 200 subcategories. The yellow rectangles show the sets of subcategories with the higher probability of misclassification. []{data-label="confusion"}](confusion.pdf){width="0.95\linewidth"} ![Examples of the most confused subcategory pairs. One subcategory is mostly confidently classified as the other in the same row when in the testing phase.[]{data-label="confusionexamples"}](confusion_examples.pdf){width="1\linewidth"} Comparison with baselines ------------------------- Our Saliency-guided Faster R-CNN approach is based on Faster-RCNN [@ren2015faster], and adopts VGGNet [@simonyan2014very] as the basic model. To verify the effectiveness of our approach, we present the results of our approach as well as the baselines in Table \[faster\]. “VGGNet” denotes the result of directly using fine-tuned VGGNet, and “Faster R-CNN (gt)” denotes the result of directly adopting Faster R-CNN with ground truth bounding box to guide training phase. Our approach achieves the best performance even without using object or parts annotations. We adopt VGGNet as the basic model in our approach, but its classification accuracy is only 70.42%, which is much lower than ours. It shows that the discriminative localization has promoting effect to classification. With discriminative localization, we find the most important regions of images for classification, which contains the key variance from other subcategories. Comparing with “Faster R-CNN (gt)”, our approach also achieves better performance. It is an interesting phenomenon that worth thinking about. From the last row in Figure \[boundingbox\], we observe that not all the areas in the ground truth bounding boxes are helpful for classification. Some ground truth bounding boxes contain large area of background noise that has less useful information and even leads to misclassification. So discriminative localization is significantly helpful for achieving better classification performance. And comparison with “Ours (without shared conv layers)” verifies the effectiveness of our saliency-guided localization learning represented in Section 2.2, which promotes not only discriminative localization but also classification. [|c|c|]{} Methods & Accuracy (%)\ & **85.14**\ Ours (without shared conv layers) & 83.95\ Faster R-CNN (gt) & 82.46\ VGGNet & 70.42\ Conclusion ========== In this paper, discriminative localization approach via saliency-guided Faster R-CNN has been proposed for weakly supervised fine-grained image classification. We first propose saliency-guided localization learning approach to localize discriminative region automatically for each image, which uses neither object nor parts annotations to avoid using labor-consuming annotations. And then an end-to-end network based on Faster R-CNN with guide of saliency information is proposed to simultaneously localize discriminative region and encode discriminative features, which not only achieves a notable classification performance but also accelerates classification speed. And combining them, we simultaneously accelerate classification speed and eliminate dependence on object and parts annotations. Comprehensive experimental results show our Saliency-guided Faster R-CNN approach is more effective and efficient compared with state-of-the-art methods on the widely-used CUB-200-2011 dataset. The future works lie in two aspects: First, we will focus on learning better discriminative localization via exploiting the effectiveness of fully convolutional networks. Second, we will also attempt to localize several discriminative regions with different semantic meanings simultaneously, such as the head or body of bird, to improve fine-grained image classification performance. Acknowledgments ================ This work was supported by National Natural Science Foundation of China under Grants 61371128 and 61532005.

--- abstract: 'Acyclic overlays used for broker–based publish/subscribe systems provide unique paths for content–based routing from a publisher to interested subscribers. Cyclic overlays may provide multiple paths, however, the *subscription broadcast process* generates one content-based routing path per subscription. This poses serious challenges in offering *dynamic routing* of notifications when congestion is detected because instantaneous updates in routing tables are required to generate alternative routing paths. This paper introduces the first subscription–based publish/subscribe system, `OctopiS`, which offers inter–cluster dynamic routing when congestion in the output queues is detected. `OctopiS` is based on a formally defined *Structured Cyclic Overlay Topology (SCOT)*. SCOT is divided into *homogeneous clusters* where each cluster has equal number of brokers and connects to other clusters through multiple *inter–cluster* overlay links. These links are used to provide parallel routing paths between publishers and subscribers connected to brokers in different clusters. While aiming at deployment at data center networks, `OctopiS` generates subscription–trees of shortest lengths used by *Static Notification Routing (SNR)* algorithm. *Dynamic Notification Routing (DNR)* algorithm uses a bit–vector mechanism to exploit the *structuredness* of a clustered SCOT to offer inter–cluster dynamic routing without making updates in routing tables and minimizing load on overwhelmed brokers and congested links. Experiments on a cluster testbed with real world data show that `OctopiS` is scalable and reduces the number of inter–broker messages in subscription delivery by 89%, subscription delay by 77%, end–to–end notification delay in static and dynamic routing by 47% and 58% respectively, and the lengths of output queues of brokers in dynamic routing paths by 59%.' author: - Muhammad Shafique bibliography: - 'refs\_p1.bib' title: 'Content–based Dynamic Routing in Structured Overlay Networks' --- Introduction ============ = \[draw, -latex’\] [0.24]{} = \[thick, scale=1\] (1) at (+,+) [$1$]{}; (2) at (+2,+) [$2$]{}; (3) at (+3,+) [$3$]{}; (4) at (+,) [$4$]{}; (5) at (+2,) [$5$]{}; (6) at (+3,) [$6$]{}; (S1) at (+0.1,0) [$S1$]{}; (P) at (+4-0.3, 0) [$P$]{}; (S2) at (+4-0.1,+) [$S2$]{}; (5) – (4) (6) – (5) (3) – (6) (2) – (3) (1) – (2) (4) – (1) (5) – (2); (S1) – (4) (P) – (6) (S2) – (3); (S1) to \[out=30,in=150\] (4); (4) to \[out=30,in=155\] (5); (4) to \[out=110,in=255\] (1); (5) to \[out=30,in=155\] (6); (1) to \[out=30,in=155\] (2); (2) to \[out=30,in=155\] (3); (S2) to \[out=150,in=30\] (3); (3) to \[out=210,in=330\] (2); (2) to \[out=210,in=330\] (1); (6) to \[out=210,in=330\] (5); (5) to \[out=210,in=330\] (4); (3) to \[out=300,in=70\] (6); (+1.5,-0.3) – (+1.5,1.8); (+2.5,-0.3) – (+2.5,1.8); at (+2.4, 1.5) [$R_1$]{}; at (+3.4,1.5) [$R_2$]{}; at (+4.4,1.5) [$R_3$]{};   [0.24]{} = \[thick, scale=1\] (1) at (+,+) [$1$]{}; (2) at (+2,+) [$2$]{}; (3) at (+3,+) [$3$]{}; (4) at (+,) [$4$]{}; (5) at (+2,) [$5$]{}; (6) at (+3,) [$6$]{}; (S1) at (+0.1,0) [$S1$]{}; (P) at (+4-0.3,0) [$P$]{}; (S2) at (+4-0.1,+) [$S2$]{}; (5) – (4) (6) – (5) (2) – (3) (1) – (2) (4) – (1); (S1) – (4) (P) – (6) (S2) – (3); (5) – (2) (3) – (6); (S1) to \[out=30,in=150\] (4); (4) to \[out=30,in=155\] (5); (4) to \[out=110,in=255\] (1); (5) to \[out=30,in=155\] (6); (1) to \[out=30,in=155\] (2); (2) to \[out=30,in=155\] (3); (S2) to \[out=150,in=30\] (3); (3) to \[out=210,in=330\] (2); (2) to \[out=210,in=330\] (1); (5) to \[out=330,in=210\] (6); (4) to \[out=330,in=210\] (5); (1) to \[out=300,in=70\] (4); (+1.5,-0.3) – (+1.5,1.8); (+2.5,-0.3) – (+2.5,1.8); at (+2.4, 1.5) [$R_1$]{}; at (+3.4,1.5) [$R_2$]{}; at (+4.4,1.5) [$R_3$]{};   [0.24]{} = \[thick, scale=1\] (1) at (+,+) [$1$]{}; (2) at (+2,+) [$2$]{}; (3) at (+3,+) [$3$]{}; (4) at (+,) [$4$]{}; (5) at (+2,) [$5$]{}; (6) at (+3,) [$6$]{}; (S1) at (+0.1,0) [$S1$]{}; (P) at (+4-0.3,0) [$P$]{}; (S2) at (+4-0.1,+) [$S2$]{}; (5) – (4) (6) – (5) (2) – (3) (1) – (2) (4) – (1); (S1) – (4) (P) – (6) (S2) – (3); (5) – (2) (3) – (6); (S1) to \[out=30,in=150\] (4); (4) to \[out=30,in=155\] (5); (4) to \[out=110,in=255\] (1); (5) to \[out=30,in=155\] (6); (1) to \[out=30,in=155\] (2); (2) to \[out=30,in=155\] (3); (S2) to \[out=150,in=30\] (3); (3) to \[out=210,in=330\] (2); (2) to \[out=210,in=330\] (1); (6) to \[out=210,in=330\] (5); (5) to \[out=210,in=330\] (4); (3) to \[out=300,in=70\] (6); (+1.5,-0.3) – (+1.5,1.8); (+2.5,-0.3) – (+2.5,1.8); at (+2.4, 1.5) [$R_1$]{}; at (+3.4,1.5) [$R_2$]{}; at (+4.4,1.5) [$R_3$]{};   [0.28]{} = \[thick, scale=1\] (1) at (+-0.5,+) [$1$]{}; (2) at (+2,+) [$2$]{}; (3) at (+3-0.5,+) [$3$]{}; (4) at (+-0.5,) [$4$]{}; (5) at (+2,+) [$5$]{}; (6) at (+3-0.5,+) [$6$]{}; (S1) at (+0.1,) [$S1$]{}; (P) at (+4-1.1,) [$P$]{}; (S2) at (+4-1.1,+) [$S2$]{}; (S3) at (+1.5,+0.6) [$S3$]{}; (5) – (4) (6) – (5) (2) – (3) (1) – (2) (4) – (1); (S1) – (4) (P) – (6) (S2) – (3) (S3)–(5); (5) – (2) (3) – (6); (S1) to \[out=30,in=150\] (4); (4) to \[out=15,in=165\] (5); (4) to \[out=110,in=255\] (1); (5) to \[out=30,in=155\] (6); (1) to \[out=15,in=165\] (2); (2) to \[out=15,in=165\] (3); (S2) to \[out=150,in=30\] (3); (3) to \[out=200,in=340\] (2); (2) to \[out=195,in=350\] (1); (5) to \[out=330,in=200\] (6); (4) to \[out=345,in=195\] (5); (3) to \[out=250,in=110\] (6); (S3) to \[out=350,in=130\] (5); (5) to \[out=60,in=290\] (2); (6) to \[out=60,in=290\] (3); (4) to \[out=60,in=290\] (1); (5) to \[out=50,in=140\] (6); (5) to \[out=150,in=30\] (4); (+1.9,-0.3) – (+1.9,1.8); (+3.5,-0.3) – (+3.5,1.8); at (+2.4,1.5) [$R_1$]{}; at (+4.4,1.5) [$R_2$]{}; at (+5.4,1.5) [$R_3$]{}; *Content–based Publish/Subscribe (CPS)* systems are used for many–to–many communication among loosely coupled distributed entities. A *publisher* publishes data in form of *notifications*, while a *subscriber* registers its interest in form of a *subscription* (set of filters or predicates) to receive notifications of interest [@MANY_FACES; @carz_thesis; @PADRESBookChapte; @Tarkoma]. In *broker–based* CPS systems, a dedicated *overlay network*, formed by a set of inter–connected brokers, is used to connect publishers and subscribers while keeping them anonymous from each other [@MANY_FACES; @carz_thesis]. Publish/subscribe is an active area of research due to its increasing popularity and gradual adoption in different application domains [@Tarkoma]. It is the communication substratum in social networking systems [@KaiwenZhang_1; @WormHole], business process monitoring [@muthy_thesis], software defined networking [@TariqPLEROMA], massive multi–player online games [@MMOG_Canas], and many commercial applications [@PNUTS; @G_CLOUD_PS; @MS_PULSE; @KAFKA; @WormHole]. A subscription is broadcast in the overlay network and saved in *routing table* of each broker in order to form a subscription–tree. Upon receiving a notification, a broker calculates next destination–paths by matching contents in the notification with filters in saved subscriptions. This technique is called the *content–based* or *filter–based* routing using *Reverse Path Forwarding* [@RPF; @carz_thesis]. A CPS system in this paper refers to a subscription–based publish/subscribe system that uses a dedicated broker–based overlay network for content–based routing [@SIENA_WIDE_AREA]. Most CPS systems use acyclic overlay topologies, which provide single routing path and offers limited flexibility to deal with network conditions like load imbalance, and link congestion. To stabilize a CPS system, subscribers are shifted from overloaded to less loaded brokers in a network area. Unfortunately, finding less loaded brokers requires extra in-broker processing and generates additional network traffic, which exerts more load on the system. This not only keeps a system unstable for quite sometime until the load shift process is complete but also causes loss of messages [@dynamic_LoadBalancing]. Cyclic overlay networks are expected to improve performance and throughput by offering multiple paths among publishers and subscribers. When a link congestion or load imbalance is detected, the best available path can be selected to offer *dynamic routing*. Although multiple paths may be available, at most one routing path, activated by a *subscription–tree*, can be used to route notifications [@SIENA_WIDE_AREA]. If a link is congested, no alternative routing paths are available for content–based routing. The available multiple links can be exploited to generate new content-based routing paths and avoid shifting subscribers, however, this requires an intelligent algorithm to search for alternative routing paths and then make a large number of updates in routing tables, which is costly and not scalable for large network settings (cf. Sec. 2). This indicates that for dynamic routing, acyclic and cyclic overlays suffer from almost the same limitations. Ideally, for high throughput and scalability, dynamic routing should be achieved without requiring updates in routing tables. Notification routing in cyclic overlay networks has received a little attention and, to the best of our knowledge, no CPS system offers dynamic routing. In addition to the limitation of one subscription–tree, the traditional CPS systems have more fundamental issues. For example, each subscription should be uniquely identified to avoid loops. Notifications should carry identifications of matching subscriptions to identify routing paths. Extra inter-broker messages, and larger lengths of routing paths with no support for dynamic routing (cf. Sec. 2). This paper introduces the first CPS system, `OctopiS`, which offers *inter–cluster* dynamic routing. The system is based on a purpose-built topology called *Structured Cyclic Overlay Topology (SCOT)* generated from Cartesian product of graphs (cf. Sec. 3). We use a novel *clustering* technique to divide SCOT into groups of brokers (i.e., *clusters*). Classifications of clusters, brokers, and links are introduced to define of a SCOT (cf. Sec. 4). `OctopiS` exploits the to generate subscription–trees of shortest lengths and do not require unique identifications to detect loops. The system offers *inter-cluster* dynamic routing without making updates in routing tables (cf. Secs. 4 & 5). The proposed *Static Notification Routing (SNR)* algorithm uses subscription–trees to send notifications to interested subscribers, while *Dynamic Notification Routing (DNR)* algorithm reduces delivery delay by offering inter–cluster dynamic routing when congestion is detected. In summary, the contributions of this paper are as follows. (i) Sec. 2 identifies issues surrounding the CPS systems that use cyclic overlays. (ii) Sec. 3 introduces how Cartesian product of graphs can be used to formally describe SCOT. Additional constraints are introduced to optimize SCOT for content–based routing. (iii) Sec. 4 describes a clustering approach with additional classifications for topology elements to prevent loops in SCOT. This section also introduces a lightweight bit–vector mechanism to identify *target clusters* for inter–cluster dynamic routing. (iv) A subscription broadcast algorithm that generates subscription–trees of shortest lengths in a clustered SCOT is described in Sec. 5. (v) Details on static routing (by *SNR* algorithm) and dynamic routing (by *DNR* algorithm) are provided in Sec. 6. (vi) Comparison with state–of–the–art *identification–based* routing is discussed in Sec. 7. We describe related work in Sec. 8, and conclude in Sec. 9. Background Issues ================= In this section, we use Fig. 1 to discuss different issues related to content–based routing in cyclic overlays. In particular, we discuss the issue of: (i) adding a *Unique Identification* to each subscription to avoid cycles in subscription broadcast, (ii) *Extra Inter–broker Messages (IMs)* to detect and discard duplicate subscriptions, (iii) larger *Lengths of Subscription–trees*, and (iv) *Path Identification* for notification routing. In this paper, an overlay link is represented as $l\langle source, destination\rangle$, where the *source* and *destination* are message sending and receiving brokers respectively.\ **I1 Unique Identification**: Content–based routing generates loops in cyclic overlays. Loops route a message indefinitely if it is not detected and discarded. The *Subscription Broadcast Process (SBP)* broadcasts subscriptions to form subscription–trees for notifications routing. Since multiple paths can be available, a broker may receive duplicate subscriptions (or duplicates). To solve this issue, each broker adds its unique identification, called *Broker Identification (or BID)*, in a subscription of the *local subscriber*. The subscription and BID form a network-wide unique identification, which is saved in routing tables to detect and discard duplicates.\ **I2 Extra Inter–broker Messages (IMs)**: Extra IMs have to be generated to detect and discard duplicates. In Fig. 1(a), broker 4 receives the subscription of S2 from broker 5. The link $l \langle 5,4\rangle$ is added in the subscription–tree of S2, assuming that broker 4 discards a second copy of the subscription received from broker 1. Similarly, broker 5 discards a second copy of the subscription of S1 received from broker 2, assuming that the first copy was already received from broker 4 (there is, therefore, no subscription–tree link from broker 2 to broker 5). This indicates that, despite using BIDs, extra IMs are generated to detect and discard duplicates.\ **I3 Subscription–tree Length**: In an acyclic overlay, SBP generates a unique subscription–tree even if a subscription is issued multiple times (after calling unsubscribe from the same broker). However, this is not the case in cyclic overlays, where multiple subscribe calls issued from the same broker may generate multiple subscription–trees with different lengths. SBP in cyclic overlays is an *uncontrolled* process and selects the first available link (or broker) as the next destination. This may generate subscription–trees of larger lengths when load on the links and brokers is uneven. For example, subscription–trees of S1 and S2 in Fig. 1(a) have shortest lengths (number of hops), however, the subscription–tree of S2 in Fig. 1(b) has larger length. Presumably, the links $l \langle 2,5\rangle$ and $l \langle 3,6\rangle$ had heavy network traffic when S2 issued the subscription. Broker 6 received the subscription of S2 from broker 5 and discarded the duplicate received from broker 3. Although P and S2 are hosted by the brokers in the same region (i.e., $R_{3}$), S2 receives notifications from P after they are processed by brokers in $R_{1}$ and $R_{2}$. Subscription–trees of longer lengths increase in–broker computation, generate extra IMs, waste network bandwidth, and cause high latency in notification delivery. Ideally, a subscription–tree should always has the shortest length, even if some links have high loads when the subscription is issued (e.g., the subscription–trees in Fig. 1(c)). To the best of our knowledge, no CPS system generates subscription–trees of shortest lengths.\ **I4 Path Identification**: The issue of loops is also relevant for delivering notifications. For example, the matching process executed at broker 6 in Fig. 1(a) indicates that a notification $n$ from P should be forwarded onto the links $l \langle6,3\rangle$ and $l \langle6,5\rangle$. Broker 6 creates two copies of $n$, $n_{1}$ and $n_{2}$, to forward to brokers 5 and 3, respectively. S1 receives $n_{1}$ from broker 4 and S2 receives $n_{2}$ from broker 3. Unfortunately, the matching process at broker 3 indicates that the subscription of S1 matches $n_{2}$, and a copy of $n_{2}$, say $n_{2}^{'}$, should be forwarded to broker 2. $n_{2}^{'}$ ultimately reaches S1 after being forwarded by broker 4. Again, the matching process at broker 4 identifies that S2 should also receive a copy of $n_{2}^{'}$, and this process continues indefinitely until identified and stopped. Similarly, $n_{1}$, forwarded by broker 6 onto $l \langle6,5\rangle$, is received more than once. To prevent receiving duplicates, host broker of a publisher adds BIDs, assigned to matching subscriptions, to a notification. Reverse path forwarding technique is used to route the notification along the paths identified by the BIDs and each interested broker removes its BID from the notification before forwarding it to the next brokers. Routing is stopped when no BID is left in the notification. This approach is further explained by using Fig. 1(d). For any notification $m$ from P, broker 6 creates two copies, $m_{1}$ and $m_{2}$, where $m_{1}$ with the BID of broker 3 is forwarded onto link $l \langle6,3\rangle$, and $m_{2}$ with the BIDs of brokers 4 and 5 is forwarded onto link $l \langle6,5\rangle$. Because of carrying an additional BID, payload of $m_{2}$ is greater than payload of $m_{1}$. After receiving $m_{2}$, broker 5 removes its own BID and forwards one copy of $m_{2}$ to S3, and another copy to broker 4. After receiving $m_{2}$, broker 4 removes its own BID and forwards a copy of $m_{2}$ to S1. Since no BID is left, broker 4 does not forward $m_{2}$ any further. In a large overlay network, a notification may be received by a large number of subscribers hosted by many brokers and many BIDs may have to be added in a notification [@MS_PULSE]. In scenarios where scalability is a major requirement, BID–based routing is a bottleneck.\ **I5 Single Routing Path**: As SBP generates one subscription–tree per subscription, updates in routing tables have to made to offer dynamic routing. Fig. 1(a) shows that there are three paths from broker 6 to broker 3 and only the path (with link $l \langle3, 6\rangle$), being part of the subscription–tree of S2, is used as a content-based routing path. If the link $l \langle6,3\rangle$ is congested, S2 has to be moved to some other broker in the less loaded network area, which requires *unsubscribe* and *subscribe* calls generating more network traffic and requiring updates in a number of routing tables. If the link $l \langle6,3\rangle$ is broken for some reason, a new subscription–tree has to be generated to send notifications to S2. This requires an intelligent algorithm that makes updates in routing tables of brokers 2, 3, 5, and 6 to remove the link $l \langle3, 6\rangle$ and add the link $l \langle2, 5\rangle$. Looking at the decoupled nature of CPS systems, such algorithms are difficult to design and not scalable for large networks. Structured Cyclic Topology ========================== In this section we describe our approach of designing a structured cyclic topology for loop free content-based routing. Structured cyclic topologies provide parallel links that can be used as alternative routing paths when congestion is detected. Our goal is to use inter-cluster parallel links as alternative content-based routing paths when congestion is detected. This requires no updates in routing tables to offer inter-cluster dynamic routing. We use the Cartesian Product of Undirected Graphs (CPUG) to design large, structured overlay cyclic networks based on small graph patterns [@theGeneralizedPUG]. Preliminaries ------------- A graph is an ordered pair $G = (V_{G}, E_{G})$, where $V_{G}$ is a finite set of vertices and $E_{G}$ is a set of edges or links that connect two vertices in $G$. The number of vertices of $G$ (called *order*) is $\mid G\mid$ (or $|V_{G}|$). Similarly, the number of edges in $G$ is $\parallel G \parallel$ (or $|E_{G}|$). A graph in which each pair of vertices are connected by an edge is called a *complete graph*. The diameter of a graph G, represented as *diam(G)*, is the shortest path between the two most distant nodes in $G$. A graph product is a binary operation that takes two small graph operands—for example $G(V_{G}, E_{G})$ and $H(V_{H}, E_{H})$—to produces a large graph whose vertex set is given by $V_{G} \mathsf{X} V_{H}$. Many types of graph products exist, but we find the Cartesian product most suitable for content-based routing. Other products, for example, the Direct product and the Strong product can be used but their rule-based interconnection of vertices increases node degree and makes routing complex. The $CPUG$ of two graphs $G(V_{G}, E_{G})$ and $H(V_{H}, E_{H})$ is denoted by $G \square H$, with vertex set $V_{G \Box H}$ and set of edges $E_{G \Box H}$. Two vertices $(g, h) \in V_{G \Box H}$ and $(g', h') \in V_{G \Box H}$ are adjacent if $g=g'$ and $hh' \in E_{G \Box H}$ or $gg' \in E_{G \Box H}$ and $h=h'$. Formally, the sets of vertices and edges of a CPUG are given as [@CPUG_Book]. $$V_{G \square H} = \{ (g, h) | g \in V_{G} \wedge h \in V_{H}\}$$ $$\left.\begin{aligned} E_{G \square H} = \{ \langle (g, h)(g', h') \rangle | (g=g', hh' \in E_{H}) \\ \vee (gg' \in E_{G}, h=h')\} \end{aligned} \right\}$$ The operand graphs $G$ and $H$ are called factors of $G \square H$. CPUG is commutative—that is, $G \Box H = H \square G$. Although CPUG of $n$ number of graphs is possible, we are concerned with CPUG of only two graphs. Structured Cyclic Overlay Topology ---------------------------------- The *Structured Cyclic Overlay Topology (SCOT)* is a $CPUG$ of two graphs. One graph, represented by $G_{af}$, is called *SCOT acyclic factor*, while the second graph operand, represented by $G_{cf}$, is called *SCOT connectivity factor*. A SCOT has two important properties: (i) *Acyclic Property* emphasizes that the $G_{af}$ must be an acyclic graph, and (ii) *Connectivity Property* requires that $G_{cf}$ must be a complete graph. These properties augment a SCOT with essential characteristics that are used for generating subscription–trees of shortest lengths. $V_{af}$ and $V_{cf}$ are the sets of vertices of $G_{af}$ and $G_{cf}$, while $E_{af}$ and $E_{cf}$ are the sets of edges of $G_{af}$ and $G_{cf}$, respectively. For a *singleton graph* of vertex set $\{h\} \subset V_{cf}$, the graph $G_{af}^h$ generated by $G_{af} \Box \{h\}$ is called a $G^h_{af}-fiber$ with *index h*. Similarly, for a singleton graph of vertex set $\{m\} \subset G_{af}$, the graph $G^m_{cf}$ generated by $\{m\} \square G_{cf}$ is called a $G^m_{cf}-fiber$ with *index m*. We describe the importance of using indexes in SCOT fibers in Section 4. The definitions of the fibers indicate that, for each vertex of $G_{cf}$, $CPUG$ generates one replica of $G_{af}$, and for each vertex of $G_{af}$, *CPUG* generates one replica of $G_{cf}$. The number of distinct fibers of $G_{af}$ and $G_{cf}$ is equal to $|V_{cf}|$ and $|V_{af}|$ respectively. = \[minimum size=7mm\] (a) at (,) [$a$]{}; (b) at (+,) [$b$]{}; (c) at (+2,) [$c$]{}; (d) at (+3,) [$d$]{}; (e) at (+4,) [$e$]{}; (f) at (+5,) [$f$]{}; (b) to \[out=30,in=150\] (e); (a) – (b) (b) – (c) (d) – (e) (e) – (f); (op) at (+6,) ; (va) at (+7,) [$1$]{}; (vb) at (+8,) [$2$]{}; (vc) at (+7+0.5,+0.6) [$0$]{}; (va) – (vb) (vb) – (vc) (vc) – (va); \[fig:Operands\] In addition to acyclic and connectivity properties, a SCOT has two more properties: (i) *Index property*, which emphasizes that the labels of nodes of $G_{cf}$ must be a sequence of unique integers starting from zero, and (ii) *Label Order property*, which requires that the first operand (from left to right) of a $CPUG$ node should be from node of $G_{af}$. The index property implies that the index of each fiber of $G_{af}$ is always an integer. The label order property indicates that the first part of the label of a SCOT node comes from the corresponding vertex of $V_{af}$, and the second part is the label of the corresponding vertex of $V_{cf}$, as indicated by Eq. 1. Reversing the order of operands does not generate extra links or nodes, since CPUG is commutative. These two properties are used for clustering and routing purposes (cf. Secs. 4–6). In Fig. 2, the left operand of $\square$ operator, an acyclic $H-graph$, is the $G_{af}$, while the second operand $G_{cf}$ is a triangle, which is a complete graph. More details on SCOT are available in the technical report [@OctopiA_TR]. \(1) at (,6) [$a,0$]{}; (2) at (+,6) [$b,0$]{}; (3) at (+2,6) [$c,0$]{}; (4) at (+3,6) [$d,0$]{}; (5) at (+4,6) [$e,0$]{}; (6) at (+5,6) [$f,0$]{}; (2) to \[out=15,in=165\] (5); (11) at (,5.1) [$a,1$]{}; (12) at (+,5.1) [$b,1$]{}; (13) at (+2,5.1) [$c,1$]{}; (14) at (+3,5.1) [$d,1$]{}; (15) at (+4,5.1) [$e,1$]{}; (16) at (+5,5.1) [$f,1$]{}; (12) to \[out=15,in=165\] (15); (21) at (,4.2) [$a,2$]{}; (22) at (+,4.2) [$b,2$]{}; (23) at (+2,4.2) [$c,2$]{}; (24) at (+3,4.2) [$d,2$]{}; (25) at (+4,4.2) [$e,2$]{}; (26) at (+5,4.2) [$f,2$]{}; (22) to \[out=15,in=165\] (25); (1) – (2) (2) – (3) (4) – (5) (5) – (6) (11) – (12) (12) – (13) (14) – (15) (15) – (16) (21) – (22) (22) – (23) (24) – (25) (25) – (26); (1) – (11) (11) – (21) (2) – (12) (12) – (22) (3) – (13) (13) – (23) (4) – (14) (14) – (24) (5) – (15) (15) – (25) (6) – (16) (16) – (26); (1) to \[out=240,in=120\] (21) (2) to \[out=240,in=120\] (22) (3) to \[out=240,in=120\] (23) (4) to \[out=240,in=120\] (24) (5) to \[out=240,in=120\] (25) (6) to \[out=240,in=120\] (26); (-0.5,6.45) – (8,6.45) – (8,5.55) – (-0.5,5.55) – (-0.5,6.45); (-0.5,5.55) – (8,5.55) – (8,4.65) – (-0.5,4.65) – (-0.5,5.55); (-0.5,4.65) – (8,4.65) – (8,3.9) – (-0.5,3.9) – (-0.5,4.6); at (+9,6.2) [$C_0$]{}; at (+9,5.2) [$C_1$]{}; at (+9,4.3) [$C_2$]{}; (-0.5,3.5) – (-0.5,6.6); (+0.7,3.5) – (+0.7,6.6); (+2,3.5) – (+2,6.6); (+3.3,3.5) – (+3.3,6.6); (+4.9,3.5) – (+4.9,6.6); (+6.3,3.5) – (+6.3,6.6); (+8,3.5) – (+8,6.6); at (+0.9,3.6) [$R_a$]{}; at (+2.3,3.6) [$R_b$]{}; at (+3.7,3.6) [$R_c$]{}; at (+5.1,3.6) [$R_d$]{}; at (+6.5,3.6) [$R_e$]{}; at (+7.9,3.6) [$R_f$]{}; (-0.5, 6.7) – (0.1, 6.7); at (2.1, 6.7) [*Intra-cluster overlay link (aCOL)*]{}; (4, 6.7) – (4.6, 6.7); at (6.6, 6.7) [*Intet-cluster overlay link (iCOL)*]{}; \[fig:SCOT1\] Clustering for Structuredness ============================= Parallel paths or links provided by a SCOT are not enough to handle issues presented in Sec. 2. This section describes a set of classifications for brokers and links to build a *structuredness* in SCOT. The *structuredness* divides a SCOT into uniquely identifiable group of brokers called *clusters*. Another pattern of grouping divides a SCOT into multiple *regions*. Types for clusters, brokers, and links are used to generate subscription–trees of shortest lengths, avoid use of unique identifications to detect loops, and support inter-cluster dynamic routing (cf. Secs. 5 & 6). More details are provided in the following. Cluster and Region ------------------ Each $G^i_{af}-fiber$ in a SCOT is a separate group of brokers called a *SCOT Cluster* (or simply a cluster) and represented by $C_{i}$, where $i \in V_{cf}$ is known as *Cluster Index*. A cluster index is the label of a vertex of $V_{cf}$ that generates the cluster (or $G^i_{af}-fiber$) when a CPUG is calculated. Similarly, each $G^j_{cf}-fiber$ is called a *Region* and is represented as $R_{j}$, where $j \in V_{af}$ is a *Region Index*. A region index is the label of a vertex of $V_{cf}$ that generates the region (or $G^j_{cf}-fiber$) when a CPUG is calculated. There are $|V_{cf}|$ and $|V_{af}|$ number of clusters and regions in a SCOT, respectively. The SCOT in Fig. 3 contains three clusters (horizontal layers) each identified by $C_{i}$, where $i \in \{0, 1, 2\}$, and six regions (vertical layers) each identified by a unique $R_{j}$, where $j \in {\{a,b,c,d,e,f\}}$. Overlay Links and Messaging --------------------------- A SCOT has two types of links: (i) an *intra-cluster overlay link (aCOL)*, and (ii) an *inter-cluster overlay link (iCOL)*. aCOLs connect brokers in the same cluster, while iCOLs connect brokers in the same region. Messaging along aCOLs and iCOLs is referred to as *intra-* and *inter-cluster messaging*, respectively. The set of all aCOLs in a cluster $C_{i}$ is $\{l \langle (x, i,), (y, i) \rangle | x, y \in V_{af} \wedge xy \in E_{af}\}$, while the set of all aCOLs in a SCOT is $\{l \langle (x, z,), (y, z) \rangle | x, y \in V_{af} \wedge xy \in E_{af} \wedge z \in V_{cf} \}$. Similarly, the set of all iCOLs in a region $R_{j}$ is $\{ l \langle (j, x'), (j, y') \rangle | x',y' \in V_{cf} \wedge x'y' \in E_{cf} \}$, while the set of all iCOLs in a SCOT is $\{ l \langle (z', x'), (z', y') \rangle | x',y' \in V_{cf} \wedge x'y' \in E_{cf} \wedge z' \in V_{af} \}$. There are $|E_{af}|.|V_{cf}|$ number of aCOLs, and $|V_{af}|.|E_{cf}|$ iCOLs in a SCOT. A *target link* refers to an overlay link that is part of a notification routing path. Classification of Clusters and Brokers -------------------------------------- Classification or types of SCOT brokers and clusters is used for cluster-level routing of subscriptions and notifications (cf. Secs. 5 & 6). The cluster that contains the host broker of a client is the *Primary or Host Cluster*, while all other are the *Secondary Clusters* of the client. The *Primary neighbours* of a broker belong to the same cluster, while the *Secondary neighbours* are those in the same region. Primary and secondary neighbours are also called direct neighbours. This arrangement of secondary brokers requires only one iCOL to forward messages from one cluster to any other cluster. The host (or secondary) cluster of a publisher is its *Target Cluster* (or *Target Secondary Cluster (TSC)*) if the cluster hosts at least one interested subscriber. An *edge broker* has at most one primary neighbour, while an *inner broker* has at least two primary neighbours. All brokers in a region are the same type (i.e., are either inner or edge). A SCOT broker is represented by $B(x, y)$, where $x \in V_{af}$ and $y \in V_{cf}$ (from Eq. 1), and is aware of its own type (i.e., edge or inner), the types of its primary and secondary neighbours, and the types of its links (i.e., aCOLs and iCOLs). in [-0.5, 0, 0.5, 1, 1.5, 2, 2.5, 3]{} (B1) at (, 1) ; at (-0.5, 1.5) [7]{}; at (0, 1.5) [6]{}; at (0.5, 1.5) [5]{}; at (1, 1.5) [4]{}; at (1.5, 1.5) [3]{}; at (2, 1.5) [2]{}; at (2.5, 1.5) [1]{}; at (3, 1.5) [0]{}; in [-0.5, 0, 0.5, 1, 1.5, 2.5, 3]{} at (, 1) [0]{}; at (2, 1) [1]{}; (-0.5, 1.7) – (3, 1.7) node \[black,midway,yshift=0.4cm\] [*Bit Indexes*]{}; (-0.7, 0.85) – node\[below=10pt\][*Bit Values*]{} (3.2, 0.85) ; in [-0.5, 0, 0.5, 1, 1.5, 2, 2.5, 3]{} (B2) at (+4.2, 1) ; at (-0.5+4.2, 1.5) [7]{}; at (0+4.2, 1.5) [6]{}; at (0.5+4.2, 1.5) [5]{}; at (1+4.2, 1.5) [4]{}; at (1.5+4.2, 1.5) [3]{}; at (2+4.2, 1.5) [2]{}; at (2.5+4.2, 1.5) [1]{}; at (3+4.2, 1.5) [0]{}; in [-0.5, 0, 0.5, 1, 1.5, 2.5]{} at (+4.2, 1) [0]{}; at (2+4.2, 1) [1]{}; at (3+4.2, 1) [1]{}; (-0.5 + 4.2, 1.7) – (3 + 4.2, 1.7) node \[black,midway,yshift=0.4cm\] [*Bit Indexes*]{}; (-0.7+4.2, 0.85) – node\[below=10pt\][*Bit Values*]{}(3.2+4.2, 0.85); at (1.2, -0.2) [(a) $CBV_{s}$ for a subscription.]{}; at (5.5, -0.2) [(b) $CBV_{p}$ for a notfication.]{}; \[fig:SRA2\] Cluster Bit Vector ------------------ Cluster Bit Vector (CBV) is a row (vector) of bits used to identify the host cluster of a subscriber and the TSCs of a publisher. It has two contexts: (i) *the subscription context* $CBV_{s}$ identifies host cluster of the subscriber, and (ii) *the publication context* $CBV_{p}$ identifies TSCs in inter-cluster dynamic routing (cf. Sec. 6). Bits in CBV are indexed from right to left, with the index of the right most bit being zero. Each bit of CBV is reserved for a SCOT cluster where index of the bit is same as index of the cluster it represents. Since each subscriber has at most one host cluster, there is only one *meaningful* bit in each $CBV_{s}$. The $CBV_{s}[2]$ in Fig. 4(a) indicates that $C_{2}$ is the host cluster of the subscriber (index of the bit and cluster is 2). Each broker is aware of index bit of its cluster and set it to 1 before a local subscription is broadcast. The $CBV_{p}$ of a notification in Fig. 4(b) indicates that $C_{0}$ and $C_{2}$ are the TSCs of the publisher and should receive the notification. The number of significant bits in a CBV is equal to the number of clusters in a SCOT. The SCOT in Fig. 3 has three clusters and requires only three bits in CBV to identify all possible TSCs. `OctopiS` saves $CBV_{s}$ in routing tables while $CBV_{p}$ is carried with a notification in inter-cluster dynamic routing. Example ------- We use Fig. 2 and Fig. 3 to explain structure of a clustered SCOT. Fig. 2 shows that $|G_{af}| = 6$ (i.e., $||G_{af}|| = 5$), while $|G_{cf}| = 3$ and $||G_{cf}|| = 3$. There are 3 clusters and 6 regions where each cluster is identified by $C_{i}$, where $i \in \{0,1,2\}$. Each region is identified by $R_{j}$, where $j \in \{a, b, c, d, e, f\}$. The set of all aCOLs in $C_{0}$ (or $G^0_{af}-fiber$) is given as. $\{l \langle (a, 0,), (b, 0) \rangle | a, b \in V_{af} \wedge ab \in E_{af}\}$.\ Similarly, the set of iCOLs in $R_{a}$ (or $G^a_{cf}-fiber$), is given by. $\{l \langle (a, x',), (a, y') \rangle | x', y' \in V_{cf} \wedge x'y' \in E_{cf} \}$.\ All brokers in regions $R_{a}, R_{c}, R_{d},$ and $R_{f}$ are edge, while in $R_{b}$, and $R_{e}$ are inner brokers. *B(a,0)*, and *B(c,0)* are the primary, while *B(b,1)* and *B(b, 2)* are the secondary neighbours of *B(b,0)*. $\big(|E_{af}||V_{cf}|$ + $|V_{af}||E_{cf}|\big)$ is 33. Subscription Broadcast ====================== The *Subscription Broadcast Process (SBP)* in traditional CPS systems is a one–step process in which a subscription reaches every broker of a cyclic overlay. However, `OctopiS` performs SBP in two–steps. We use this approach to exploit structuredness of SCOT for a *controlled* SBP to generate subscription–trees of shortest–lengths, avoid using unique identification for each subscription, and prevent loops and extra IMs (eliminating **I1**, **I2**, and **I3**). Each broker of a subscriber’s host cluster performs the two–steps, while the host broker of the subscriber sets its cluster index bit in $CBV_{s}$ to 1 before forwarding a subscription. $CBV_{s}[i]$ for S1, S2, and S3 in Fig. 5 are $CBV_{s}[0]$, $CBV_{s}[0]$, and $CBV_{s}[1]$, respectively. Each subscription in clustered SCOT has two states: (i) *primary state*, and (ii) *secondary state*. In the first step, a subscription is forwarded to brokers in a subscriber’s host cluster and state of the subscription is primary. No loops occur, as the host cluster is a replica of an acyclic factor $G_{af}$ (recall the acyclic property of SCOT graph operand) and the subscription broadcast is similar to in an acyclic overlay, which generates subscription–tree of the minimum length. No unique identification is needed, and no duplicates appear. aCOLs of the host cluster are added in the subscription–tree, which has the maximum length $\le diam(G_{af})$. In the second step, each broker in the host cluster of the subscriber changes state of the subscription to secondary and forwards it to secondary neighbours. The secondary neighbours do not forward a secondary subscription to any other broker. In this step, all iCOLs are added to the subscription–tree and the maximum length is $\le \big(diam(G_{af}) + 1 \big)$. A subscription with its $CBV_{s}$ is saved as *{subscription, last hop, $CBV_{s}$}* tuple in routing tables. A broker can find state of the subscription by examining $CBV_{s}$ saved with the subscription. For a primary subscription, the index of the bit with value 1 should be same as the cluster (or broker) index. As shown in Fig. 5, in the first step of SBP for S1, *B(a,0)* forwards the subscription to the primary broker *B(b,0)*, and in the second step, to the secondary brokers *B(a,1)* and *B(a, 2)*. Each broker of $C_{0}$ repeats the first and second steps to generate the subscription–tree of S1. Similarly, the subscription–trees of S2 and S3 are generated. Contrary to traditional CPS systems, SBP in `OctopiS` is `controlled`, which always generates a unique subscription–tree for a subscription issued from a broker. Each broker is aware of its type (i.e, primary or secondary), and each cluster is treated as an exclusive acyclic overlay. SBP uses this information to forward a subscription onto specific links to brokers (e.g., in Fig. 5). Uneven load in brokers or links does not effect structure or length of a subscription–tree. This pattern of subscription broadcast does not generate duplicates. Algorithm 1 provides more details about the two-steps of SBP. The *state* attribute of a subscription has two values: *PRIMARY* and *SECONDARY*. The host broker of a subscriber sets the host cluster index bit $CBV_{s}[i]$ to 1 (lines 4-7), and forwards the subscription to direct neighbours. The *isPrimary(n)* method checks the type of the next broker and the state attribute of the subscription is set accordingly (lines 10-11). The subscription message is saved on the primary and secondary brokers (line 15). = \[draw, -latex’\] = \[draw, latex-’\] (1) at (,) [$a,0$]{}; (2) at (+ 1,) [$b,0$]{}; (3) at (+2,) [$c,0$]{}; (4) at (+3,) [$d,0$]{}; (5) at (+4,) [$e,0$]{}; (6) at (+5,) [$f,0$]{}; (11) at (,-) [$a,1$]{}; (12) at (+ 1,-) [$b,1$]{}; (13) at (+ 2,-) [$c,1$]{}; (14) at (+ 3,-) [$d,1$]{}; (15) at (+ 4,-) [$e,1$]{}; (16) at (+ 5,-) [$f,1$]{}; (21) at (,-2) [$a,2$]{}; (22) at (+ 1,-2) [$b,2$]{}; (23) at (+ 2,-2) [$c,2$]{}; (24) at (+ 3,-2) [$d,2$]{}; (25) at (+ 4,-2) [$e,2$]{}; (26) at (+ 5,-2) [$f,2$]{}; (S1) at (-1,) [$S1$]{}; (S2) at (+ 6,) [$S2$]{}; (S3) at (+ 6, -) [$S3$]{}; (1) to \[out=30,in=150\] (2); (6) to \[out=210,in=330\] (5); (2) to \[out=30,in=150\] (3); (2) to \[out=210,in=330\] (1); (5) to \[out=30,in=150\] (6); (5) to \[out=210,in=330\] (4); (5) – (4); (2) – (3); (2) to \[out=25,in=155\] (5); (5) to \[out=210,in=335\] (2); (S2) to \[out=150,in=30\] (6); (S1) to \[out=30,in=150\] (1); \(1) to \[out=260,in=90\] (11); (1) to \[out=245,in=115\] (21); (1) to \[out=245,in=110\] (11); (1) to \[out=295,in=65\] (21); \(2) to \[out=260,in=90\] (12); (2) to \[out=245,in=115\] (22); (2) to \[out=245,in=110\] (12); (2) to \[out=295,in=65\] (22); \(3) to \[out=260,in=90\] (13); (3) to \[out=245,in=115\] (23); (3) to \[out=245,in=110\] (13); (3) to \[out=295,in=65\] (23); \(4) to \[out=260,in=90\] (14); (4) to \[out=245,in=115\] (24); (4) to \[out=245,in=110\] (14); (4) to \[out=295,in=65\] (24); \(5) to \[out=260,in=90\] (15); (5) to \[out=245,in=115\] (25); (5) to \[out=245,in=110\] (15); (5) to \[out=295,in=65\] (25); \(6) to \[out=260,in=90\] (16); (6) to \[out=245,in=115\] (26); (6) to \[out=245,in=110\] (16); (6) to \[out=295,in=65\] (26); (S1) – (1) (S2) – (6); (S3) – (16); (11) to \[out=70,in=275\] (1); (12) to \[out=70,in=275\] (2); (13) to \[out=70,in=275\] (3); (14) to \[out=70,in=275\] (4); (15) to \[out=70,in=275\] (5); (16) to \[out=70,in=275\] (6); (11) to \[out=290,in=85\] (21); (12) to \[out=290,in=85\] (22); (13) to \[out=290,in=85\] (23); (14) to \[out=290,in=85\] (24); (15) to \[out=290,in=85\] (25); (16) to \[out=290,in=85\] (26); (16) to \[out=210,in=330\] (15); (15) to \[out=210,in=330\] (14); (12) to \[out=330,in=210\] (13); (12) to \[out=210,in=330\] (11); (S3) to \[out=160,in=20\] (16); (15) to \[out=155,in=30\] (12); \[fig:theSFP\] */\* PRIMARY state indicates host cluster of the subscriber \*/* */\* subscription is in SECONDARY state, not forwarded to any broker \*/* $RT.insert(s)$ \[algo:A1\] The subscription–trees in clustered SCOT require no updates in secondary brokers when a subscriber relocates to some other broker in the same cluster as the second step of SBP does not change. This provides robust fault tolerance when a broker fails for some reasons (fault tolerance in `OctopiS` is not within scope of this paper). Notification Routing ==================== This section describes our static and inter–cluster dynamic routing approaches in clustered SCOT. `OctopiS` uses cluster–based static routing to deliver notifications using subscription–trees of shortest lengths, and switches to dynamic routing when congestion is detected. We also outline state–of–the–art BID-based routing in unclustered SCOT. BID–based Static Routing ------------------------ An unclustered SCOT has no types or groups of brokers and links, which is prerequisite to use BID–based routing algorithm. To identify routing paths in an unclustered SCOT, BID–based routing algorithm requires a notification to carry BIDs assigned to matching subscriptions. SBP in unclustered SCOT is uncontrolled and lengths of routing paths satisfy the relation: $max\big(d\langle(x_{1},y_{1}), (x_{2}, y_{2})\rangle \big) \leq \big (|G_{cf}| (diam(G_{af}) + 1)-1 \big)$ where $(x_{1},y_{1})$ and $(x_{2}, y_{2})$ are any brokers. Increase in payload due to adding BIDs is an important concern because normally a BID is formed by a combination of IP address and a broker level unique identifier [@PADRESBookChapte]. For a large network, where a large number of brokers may host interested subscribers, BID–based routing is not scalable because of additional payload and impeding in-broker processing in matching process [@carz_thesis; @Li_ADAP].\ As Algorithm 2 shows, the host broker of a publisher adds BIDs of the matching subscriptions in *bidList* attribute of a notification $n$ (lines 3-7). *splitBIDs* method creates hash map, which uses a next destination path as hash key and the list of BIDs of brokers down to the next destination path as object of the hash key. The presence of an object *localBID* in the hash map indicates that each local subscriber with a matching subscriptions should receive a copy of $n$ (lines 11-16). Afterwards, a copy of $n$ with the corresponding list of the remaining BIDs are forwarded onto the next destinations (lines 18-21). */\* host broker of publisher adds BIDs\*/* */\* next destination based split list of BIDs \*/* $nextBIDs \gets splitBIDs(n.bidList)$ */\* send notifications to local interested subscribers\*/* */\* send notifications to next brokers in routing paths \*/* Cluster–based Static Routing ---------------------------- *Static Notification Routing (SNR)* algorithm uses subscription–trees of shortest lengths for routing notifications in a clustered SCOT. The algorithm deals with two scenarios. (i) The host cluster of the publisher is the only target cluster, and therefore lengths of the routing paths satisfy the relation: $max\big(d\langle(u_{1},v_{1}), (u_{2}, v_{2})\rangle \big) \leq diam(G_{af})$ where $(u_{1},v_{1})$ and $(u_{2}, v_{2})$ are any brokers in the same cluster. (ii) At least one interested subscriber is hosted by a secondary cluster and therefore lengths of the routing paths satisfy the relation: $max\big(d\langle(x_{1},y_{1}), (x_{2}, y_{2})\rangle \big) \leq \big( diam(G_{af}) + 1 \big)$ where $(x_{1},y_{1})$ and $(x_{2}, y_{2})$ are any brokers in a clustered SCOT. The host broker of a publisher forwards a notification onto target aCOLs and iCOLs (recall that a target link is part of a routing path to next destination). The notification is routed to interested subscribers in the host and TSCs of the publisher using reverse path forwarding technique. No loops appear and no path identifications are required because each cluster is an acyclic overlay (eliminating **I4**). In Fig. 6, a notification from *P1* propagates onto the routing path in $C_{2}$ to reach *S4*. *B(f,2)* creates two additional copies of the notification to forward to *B(f,1)* and *B(f,0)* in the TSCs. For *S1* and *S2* only one copy of the notification is forwarded to *B(f,0)* to avoid duplicates. Similarly, notifications from *P2* and *P3* are forwarded. Note that *P2* has no primary target cluster, while *P3* has no TSC.\ In Algorithm 3, *getDistinctSubs* uses matching subscriptions to return a list of distinct next *destinations* (line 5). The method *getDistinctSubs* assures that only one copy of notification *n* is forwarded onto a next destination link. The *else* part (lines 7-9) handles intra–cluster messaging when *n* is received by a broker other than the host broker of the publisher. $nextDistinct\_aCOLs$ provides a list of the next distinct destinations to forward *n* in the same cluster. = \[draw, -latex’\] (1) at (,) [$a,0$]{}; (2) at (+ 1,) [$b,0$]{}; (3) at (+2,) [$c,0$]{}; (4) at (+3,) [$d,0$]{}; (5) at (+4,) [$e,0$]{}; (6) at (+5,) [$f,0$]{}; (11) at (,-) [$a,1$]{}; (12) at (+ 1,-) [$b,1$]{}; (13) at (+ 2,-) [$c,1$]{}; (14) at (+ 3,-) [$d,1$]{}; (15) at (+ 4,-) [$e,1$]{}; (16) at (+ 5,-) [$f,1$]{}; (21) at (,-2) [$a,2$]{}; (22) at (+ 1,-2) [$b,2$]{}; (23) at (+ 2,-2) [$c,2$]{}; (24) at (+ 3,-2) [$d,2$]{}; (25) at (+ 4,-2) [$e,2$]{}; (26) at (+ 5,-2) [$f,2$]{}; (1) – (11) (11) – (21) (2) – (12) (12) – (22) (3) – (13) (13) – (23) (4) – (14) (14) – (24) (5) – (15) (15) – (25) (6) – (16) (16) – (26); (2) to \[out=20,in=160\] (5); (12) to \[out=20,in=160\] (15); (22) to \[out=20,in=160\] (25); (22) – (23) (22) – (21) (25) – (26) (25) – (24) (1) – (2) (2) – (3) (4) – (5) (5) – (6) (11) – (12) (13) – (12) (14) – (15) (15) – (16); (1) to \[out=240,in=120\] (21) (2) to \[out=240,in=120\] (22) (3) to \[out=240,in=120\] (23) (4) to \[out=240,in=120\] (24) (5) to \[out=240,in=120\] (25) (6) to \[out=240,in=120\] (26); (S1) at (-1.2,) [$S1$]{}; (S2) at (+6,) [$S2$]{}; (S3) at (+ 6, -) [$S3$]{}; (S4) at (-1.2,-2) [$S4$]{}; (P2) at (+6,-1.8) [$P2$]{}; (P1) at (+ 6,-2) [$P1$]{}; (P3) at (-1.2,-) [$P3$]{}; (S1) – (1) (S2) – (6) (S3) – (16) (S4) – (21) (P2) – (16) (P1) – (26) (P3) – (11); (P2) to \[out=180,in=320\] (16); (16) to \[out=60,in=300\] (6); (16) to \[out=235,in=110\] (26); (6) to \[out=210,in=330\] (5); (5) to \[out=205,in=335\] (2); (2) to \[out=210,in=330\] (1); (1) to \[out=210,in=330\] (S1); (6) to \[out=330,in=210\] (S2); (26) to \[out=210,in=330\] (25); (25) to \[out=145,in=35\] (22); (22) to \[out=210,in=330\] (21); (21) to \[out=210,in=330\] (S4); (P1) to \[out=150,in=30\] (26); (16) to \[out=335,in=205\] (S3); (26) to \[out=150,in=30\] (25); (25) to \[out=155,in=25\] (22); (22) to \[out=150,in=30\] (21); (21) to \[out=150,in=30\] (S4); (26) to \[out=60,in=300\] (16); (26) to \[out=130,in=230\] (6); (6) to \[out=30,in=150\] (S2); (6) to \[out=150,in=30\] (5); (5) to \[out=155,in=25\] (2); (2) to \[out=150,in=30\] (1); (1) to \[out=150,in=30\] (S1); (P3) to \[out=30,in=150\] (11); (11) to \[out=30,in=150\] (12); (12) to \[out=25,in=155\] (15); (15) to \[out=30,in=150\] (16); (16) to \[out=30,in=150\] (S3); \[fig:SRA\] */\* forward n onto all target links \*/* = \[draw, -latex’\] [0.23]{} = \[minimum size=7mm\] (a) at (,) [$a$]{}; (b) at (+,) [$b$]{}; (c) at (+2,) [$c$]{}; (a) – (b) (b) – (c); (op) at (+3-0.43,) ; (va) at (+3,) [$1$]{}; (vb) at (+4,) [$2$]{}; (vc) at (+3+0.5,+0.6) [$0$]{}; (va) – (vb) (vb) – (vc) (vc) – (va); (-1.2, 3.6) – (-0.7, 3.6); at (1.2, 3.6) [*Intra-cluster notification routing*]{}; (-1.2, 3.4) – (-0.7, 3.4); at (1.2, 3.4) [*Inter-cluster notification routing*]{}; (-1.2, 3.1) – (-0.7, 3.1); at (1.4, 3.1) [*Intra-cluster $CBV_{p}-N$ routing*]{}; (-1.2, 2.85) – (-0.7, 2.85); at (1.4, 2.85) [*Inter-cluster $CBV_{p}-N$ routing*]{}; (-1.2, 2.6) – (-0.7, 2.6); at (1.2, 2.6) [*Overloaded aLink*]{}; (-1.2, 2.4) – (-0.7, 2.4); at (1.2, 2.4) [*Overloaded iLink*]{}; (lbroker) at (-1,2.15) ; at (0, 2.15) [*Broker*]{}; (lpub) at (-1, 1.8) ; at (0.7, 1.8) [*Publisher or Subscriber*]{};   [0.23]{} = \[thick\] (1) at (,+2) [$a,0$]{}; (2) at (+,+2) [$b,0$]{}; (3) at (+2,+2) [$c,0$]{}; (4) at (,+) [$a,1$]{}; (5) at (+,+) [$b,1$]{}; (6) at (+2,+) [$c,1$]{}; (7) at (,) [$a,2$]{}; (8) at (+,) [$b,2$]{}; (9) at (+2,) [$c,2$]{}; (S1) at (+2+1,+2) [$S1$]{}; (P) at (+,-0.7) [$P$]{}; (S2) at (+2+1,+) [$S2$]{}; (S3) at (,-0.7) [$S3$]{}; (S4) at (+2,-0.7) [$S4$]{}; (1) – (2) (2) – (3) (4) – (5) (5) – (6) (7) – (8) (8) – (9) ; (1) – (4) (4) – (7) (8) – (5) (6) – (9) (5) – (2) (3) – (6); (P) – (8) (S1) – (3) (S2) – (6) (S3) – (7) (S4) – (9); (1) to \[out=300,in=60\] (7); (2) to \[out=240,in=120\] (8); (3) to \[out=240,in=120\] (9); (P) to \[out=60,in=290\] (8); (5) to \[out=60,in=300\] (2); (8) to \[out=30,in=150\] (9); (5) to \[out=30,in=150\] (6); (2) to \[out=30,in=150\] (3); (8) to \[out=60,in=300\] (5); (6) to \[out=30,in=150\] (S2); (8) to \[out=150,in=30\] (7); (3) to \[out=30,in=150\] (S1); (9) to \[out=250,in=110\] (S4); (7) to \[out=250,in=110\] (S3);   [0.23]{} = \[thick\] (1) at (,+2) [$a,0$]{}; (2) at (+,+2) [$b,0$]{}; (3) at (+2,+2) [$c,0$]{}; (4) at (,+) [$a,1$]{}; (5) at (+,+) [$b,1$]{}; (6) at (+2,+) [$c,1$]{}; (7) at (,) [$a,2$]{}; (8) at (+,) [$b,2$]{}; (9) at (+2,) [$c,2$]{}; (S1) at (+2+1,+2) [$S1$]{}; (P) at (+,-0.7) [$P$]{}; (S2) at (+2+1,+) [$S2$]{}; (S3) at (,-0.7) [$S3$]{}; (S4) at (+2,-0.7) [$S4$]{}; (1) – (2) (2) – (3) (4) – (5) (5) – (6) (7) – (8) (8) – (9) ; (1) – (4) (4) – (7) (2) – (5) (3) – (6) (6) – (9); (P) – (8) (S1) – (3) (S2) – (6) (S3) – (7) (S4) – (9); (1) to \[out=300,in=60\] (7); (2) to \[out=240,in=120\] (8); (3) to \[out=240,in=120\] (9); (8) – (5); (P) to \[out=60,in=290\] (8); (9) to \[out=110,in=250\] (6); (9) to \[out=60,in=300\] (3); (8) to \[out=30,in=150\] (9); (6) to \[out=30,in=150\] (S2); (8) to \[out=150,in=30\] (7); (3) to \[out=30,in=150\] (S1); (9) to \[out=250,in=110\] (S4); (7) to \[out=250,in=110\] (S3);   [0.25]{} = \[thick\] (1) at (,+2) [$a,0$]{}; (2) at (+,+2) [$b,0$]{}; (3) at (+2,+2) [$c,0$]{}; (4) at (,+) [$a,1$]{}; (5) at (+,+) [$b,1$]{}; (6) at (+2,+) [$c,1$]{}; (7) at (,) [$a,2$]{}; (8) at (+,) [$b,2$]{}; (9) at (+2,) [$c,2$]{}; (S1) at (-+0.2,+2) [$S1$]{}; (P) at (+,-0.7) [$P$]{}; (S2) at (+2+1,+) [$S2$]{}; (S3) at (+2,-0.7) [$S3$]{}; (S4) at (+2+0.7,-0.7) [$S4$]{}; (1) – (2) (2) – (3) (4) – (5) (5) – (6) (7) – (8); (1) – (4) (4) – (7) (6) – (9); (8) – (9); (P) – (8) (S1) – (1) (S2) – (6) (S3) – (9) (S4) – (9); (1) to \[out=300,in=60\] (7); (2) to \[out=240,in=120\] (8); (3) to \[out=240,in=120\] (9); (8) – (5) (2) – (5) (3) – (6); (P) to \[out=60,in=290\] (8); (8) to \[out=60,in=300\] (5); (6) to \[out=60,in=300\] (3); (8) to \[out=30,in=150\] (9); (5) to \[out=30,in=150\] (6); (2) to \[out=150,in=30\] (1); (3) to \[out=150,in=30\] (2); (6) to \[out=30,in=150\] (S2); (1) to \[out=150,in=30\] (S1); (9) to \[out=0,in=100\] (S4); (9) to \[out=250,in=110\] (S3); Inter–cluster Dynamic Routing ----------------------------- *Dynamic routing* refers to the capability of a network system to alter a routing path in response to overloaded or failed links and/or routers. Multiple techniques have been proposed to offer dynamic routing in address–based networks where routing paths are calculated from a global view of a network topology graph that is saved on every network router [@routing_book]. IP addresses may be used to form clusters in a network area where the same network mask requires a single routing entry. However, these techniques are not applicable in broker–based CPS systems because brokers in these systems are aware of only their direct neighbours and destination addressing is based on contents (i.e, subscriptions). Therefore dynamic routing decisions are decentralized and have to be made without having a global view of an overlay. This section describes inter–cluster dynamic routing when one or more target inter–cluster overlay links (i.e., iCOLs) are overloaded and notifications start queuing up in the output queues. Overloading and subsequent queuing can happen in two cases: (i) when a broker is not able to process high volume of outgoing notifications and become overwhelmed, and (ii) when bandwidth is limited. As SBP generates only one subscription–tree per subscription, dynamic routing is difficult in broker-based CPS systems and never supported before. We introduce a unique approach, which deviates from traditional reverse path forwarding technique and uses structuredness of the proposed topology (i.e., SCOT) along with subscription–trees of the matching subscriptions to offer inter-cluster dynamic routing. Our approach is scalable because it does not require updates in routing tables and can reduce delivery delays when a large number of notifications start accumulating in the output queues. SNR algorithm adds exclusive copies of a notification in the output queues of the target links. If a publisher generates $\gamma$ number of notifications in $t_{w}$ time window, and there are $\alpha$ number of target aCOLs and $\beta$ target iCOLs, the host broker of the publisher enqueue $(\alpha + \beta).\gamma$ number of copies of notifications. A High Rate Publisher (HRP) with a high value of $\gamma$ can overwhelm brokers in a routing paths when SNR algorithm is used. Our inter-cluster *Dynamic Notification Routing (DNR)* algorithm alleviates overwhelmed brokers and selects an unoverloaded iCOL to forward a notification to a TSC. The algorithm dynamically selects an unoverloaded iCOL, which may not be part of a subscription–tree to the TSC. As parallel iCOLs are available in a clustered SCOT, one can be selected without making updates in routing tables (partially eliminating **I5**). DNR adds at most *one copy* of a notification in the congested output queues of a target link at a broker. The algorithm sets the cluster index bits in $CBV_{p}$ to 1 to identify those TSCs that are not forwarded the notification due to overloaded target iCOLs. $CBV_{p}$ is added in the header of the notification, called *the $CBV_{p}-Notification$ (or $CBV_{p}-N$)*. $CBV_{p}-N$ is an indication for a broker that the routing is dynamic, and a copy of the notification should be forwarded to TSCs if unoverloaded target iCOLs are available. Using this technique, DNR algorithm keeps the length of the congested output queues of an overwhelmed broker to a minimum, and the load of forwarding the notification is shifted to other brokers using the heuristic that unoverloaded target iCOLs are available down the routing path. `OctopiS` uses Eq. 3 to find whether an output queue is congested, and DNR algorithm should be activated. $$(Q_{\ell}) . (1+Q_{in}, 1+Q_{out})_{t_{w}} > \tau$$ $Q_{in}$ and $Q_{out}$ are the number of notifications that enter into or leave the output queue in the time window $t_{w}$, respectively. The term $(1+Q_{in}, 1+Q_{out})_{t_{w}}$ is the ratio of $(1+Q_{in})$ to $(1+Q_{out})$, and is known as the *Congestion Element (CE)*. $CE > 1$ indicates that the congestion is increased, while $CE < 1$ shows that congestion is decreased in the last $t_{w}$ interval. An output queue is congestion–free when $CE$ is 1 and the queue length $Q_{\ell}$ is 0. `OctopiS` saves the values of $Q_{in}$ and $Q_{out}$ in a *Link Status Table (LST)* on each broker, and the values are updated after each $t_{w}$ interval. Inter-cluster dynamic routing by DNR algorithm is further explained in the following with help of three cases.\ **Case I – Unoverloaded Target iCOL**: When the output queue of at least one target iCOL is uncongested (i.e., the corresponding link is unoverloaded), DNR algorithm does not enqueue a notification in the congested output queues of the target iCOLs. Instead, only *one copy* of the notification is enqueued in the uncongested output queue of the unoverloaded target iCOL. If more than one unoverloaded target iCOL is available at a broker, $CBV_{p}-N$ is forwarded onto the target iCOL, which has the least value of $Q_{\ell}$. The number of notifications added to the output queues of the target links in $t_{w}$ interval is $(\alpha + \theta ).\gamma$, where $\theta$ is the number of unoverloaded target iCOLs and $\theta < \beta$. Fig. 7(b) indicates that the iCOL $l \langle(b,2),(b,0)\rangle$), which forwards notifications from *B(b,2)* to $C_{0}$, is overloaded. After finding the overloaded iCOL from the LST, *B(b, 2)* sets the index bit of $C_{0}$ to 1 and adds $CBV_{p}-N$ (with *$CBV_{p}$ 001*) in the output queue of the unoverloaded target iCOL $l \langle(b,2),(b,1)\rangle$) to forward $CBV_{p}-N$ to *B(b, 1)*. Since the output queue of the iCOL $l \langle(b,1),(b,0)\rangle$ is uncongested, *B(b, 1)* sets the index bit of $C_{0}$ to 0, removes the $CBV_{p}$ from the notification as all index bits in $CBV_{p}$ are zero, and forwards the notification to *B(b, 0)* and *B(c, 1)*. The notification is forwarded to S2, S3, and S4 using their subscription–trees. However, to avoid overloaded link $l \langle(b,2),(b,0)\rangle$, DNR algorithm deviates from the subscription–tree of S1 and dynamically selects the iCOL $l \langle(b,1),(b,0)\rangle$ for routing without making updates in routing tables. To forward one notification from P to interested subscribers, each of SNR and DNR algorithms generate 6 IMs, although DNR algorithm excluded one overloaded link, the dynamic routing path for S1 contains 4 brokers, which is one more than the number of brokers in path adopted by SNR algorithm.\ **Case II – All Target iCOLs Overloaded**: When the output queues of all target iCOLs are congested and at least one output queue of a target aCOL is uncongested, DNR algorithm uses the target aCOLs to find unoverloaded target iCOLs. $CBV_{p}-N$ is added in the uncongested output queue of the target aCOL. The load of forwarding the notification to TSCs is shifted to the next primary broker. The number of notifications added to the output queues of the target links in $t_{w}$ interval is $\alpha . \gamma$ (here $\beta$ is zero as *no copy* of the notification is added into the congested output queues of the target iCOLs). Fig. 7(c) indicates that the two target iCOLs, $l \langle(b,2), (b,1) \rangle$ and $l \langle(b,2), (b,0) \rangle$, are overloaded, and notifications are sent only to *B(a, 2)* and *B(a, 0)* in the host cluster of P. Since two unoverloaded target aCOLs are available, the target aCOL with the least $Q_{\ell}$ is selected to forward $CBV_{p}-N$ with $CBV_{p}$ 011 (presumably, aCOL $l \langle(b,2), (c,2) \rangle$ has least value of $Q_{\ell}$). As the target iCOLs are not overloaded at *B(c,2)*, the notification is routed to the TSCs after removing $CBV_{p}$. To forward one notification in this case, DNR algorithm generates 4 IMs, while SNR algorithm generates 6 IMs. Furthermore, two overloaded iCOLs are excluded from the dynamically generated routing paths. */\* list of interested subscribers \*/* $IS \gets getInterestedSubs(n)$ */\* local hosted interested subscribers \*/* $local \gets getHostedSubscribers(IS)$ */\* send n to local hosted subscribers \*/* */\* next unique destinations \*/* $\mu \gets nextUniqueDestinations(IS-local)$ */\* get overloaded iCOLs from $\mu$ unique destinations \*/* $\eta1 \gets getOverloaded\_iCOLs(\mu)$ $CBV_{p} \gets 0$ */\* Set index bits of the overloaded iCOLs \*/* $\eta2 \gets getLeastLoaded\_iCOL(\mu)$ $\ell \gets getLeastLoaded\_aCOL(\mu)$ */\* Send n to unoverloaded target links (aCOLs and iCOLs) \*/* \ **Case III – All Target Links Overloaded**: If all target links are overloaded, $CBV_{p}-N$ is forwarded onto the least loaded target iCOL. Because of the possibility of having more overloaded aCOLs down the routing path, $CBV_{p}-N$ is not added into the output queue of a target aCOL even if it is the least overloaded link. The number of notifications added to the output queues of the target links in $t_{w}$ interval is $(\alpha + 1).\gamma$. Fig. 7(d) shows that $CBV_{p}-N$ is forwarded onto (presumably) the least overloaded link $l \langle(b,2),(b,1)\rangle$ with $CBV_{p}$ is 001. As the target iCOL $l \langle(b,1),(b,0)\rangle$ is also overloaded, *B(b, 1)* forwards $CBV_{p}-N$ onto the target aCOL $l \langle(b,1),(c,0)\rangle$. The overloaded link $l \langle(c,1),(c,0)\rangle$ is the only target iCOL available to forward the notification to TSC $C_{0}$, DNR algorithm is unable to find unoverloaded target iCOL and the notification has to be forwarded onto the overloaded iCOL $l \langle(c,1),(c,0)\rangle$. The number of IMs generated by SNR and DNR algorithms in this case are 6 and 5 respectively. Although DNR algorithm successfully avoided the overloaded iCOL $l \langle(b,2),(b,0)\rangle$, the notification has to be forwarded onto another overloaded target iCOL $l \langle(c,1),(c,0)\rangle$. Additionally, the generated dynamic routing path for S1 has 3 additional brokers.\ Algorithm 4 provides further details about the inter-cluster dynamic notification routing in `OctopiS`. Upon receiving a notification $n$, the overwhelmed broker prepares a list of local subscribers interested in $n$ (line 5). A copy of $n$ for each subscriber is added in the next destinations list (lines 7-9). Next, $\eta1$, a list of overloaded target iCOLs is prepared to set index bits in $CBV_{p}$ (lines 13-18). Note that the index bit of the current notification routing cluster is not set (line 17) because this information does not effect inter-cluster dynamic routing. A copy of $n$ for each unoverloaded target link is added in the next destination list *DL* (lines 22-24). In the end, the algorithm shows how three cases are handled. The first *if–statement* handles the Case II when all target iCOLs are overloaded and an unoverloaded target aCOL is available (lines 26-29). The second condition (in the *else–if* block) is valid when an unoverloaded target iCOL is available to carry $CBV_{p}$ (lines 30-33). The *else* block handles the case when all target links are overloaded and a copy of *n* is added in *DL* to sent onto least overloaded target iCOL $\eta2$. DNR is a best–effort algorithm and depends on the subscription–trees laid–on by interested subscribers in a target cluster. The algorithm does not guarantee finding an unoverloaded target iCOL, even if one exists. For example, in Fig. 7(d), DNR algorithm does not use the unoverloaded link $l\langle(b,2),(a,2)\rangle$ for inter-cluster dynamic routing because the link is not a target aCOL. Forwarding the $CBV_{p}-N$ on such links can generate loops among different clusters [@OctopiA_TR]. DNR algorithm currently does not support *intra–cluster dynamic routing*, and this is the reason that `OctopiS` eliminates **I5** only partially. *intra–cluster dynamic routing* is part of the future work. Evaluation ========== We implemented SNR, and DNR algorithms in `OctopiS`, developed on top of a publish/subscribe tool PADRES [@PADRESBookChapte]. For a comparison with state–of–the–art, we also implemented BID–based routing in PADRES. Importantly, we created a subscription–based publish/subscribe version of PADRES, as the tool supports advertisement–based semantics [@Li_ADAP]. SNR and DNR algorithms exploit structuredness of clustered SCOT, while BID–based routing algorithm uses unclustered SCOT. Setup ----- Fig. 8 shows factors of the SCOT topology $\mathbb{S}_{e}$, which we used for evaluation and comparison of SNR, DNR, and BID–based routing algorithms. $G_{af}$ factor of $\mathbb{S}_{e}$ is an acyclic topology of 15 brokers (5 inner brokers (*vi, vii, viii, ix* and *x*) and 10 edge brokers), while $G_{cf}$ factor has 5 brokers, which generates $|V_{cf}|-1$ number of secondary neighbours for each broker in $G_{af}$. This results in 25 inner brokers and 50 edge brokers (for a total of 75 brokers), forming 5 clusters and 15 regions in $\mathbb{S}_{e}$. `OctopiS` was deployed on a cluster of 35 physical computing nodes, where each node had one quad core processor of 2.4 GHz with 4 GB RAM, and running 64-bit JDK on Linux OS. Each broker was loaded in a separate instance of JVM with 1 GB initial memory. One dedicated high throughput Gigabit Ethernet switch was used for connectivity. $\mathbb{S}_{e}$ was deployed in such a way that the primary and secondary neighbours of each broker were always on different computing nodes. §[0.4]{} \(1) at (+ 0,) [$i$]{}; (2) at (+ 1,) [$ii$]{}; (3) at (+ 2,) [$iii$]{}; (4) at (+ 3,) [$iv$]{}; (5) at (+ 4,) [$v$]{}; (11) at (+ 0,-) [$vi$]{}; (12) at (+ 1,-) [$vii$]{}; (13) at (+ 2,-) [$viii$]{}; (14) at (+ 3,-) [$ix$]{}; (15) at (+ 4,-) [$x$]{}; (21) at (+ 0,-2) [$xi$]{}; (22) at (+ 1,-2) [$xii$]{}; (23) at (+ 2,-2) [$xiii$]{}; (24) at (+ 3,-2) [$xiv$]{}; (25) at (+ 4,-2) [$xv$]{}; (1) – (11) (11) – (12) (12) – (13) (14) – (15) (12) – (22) (11) – (21) (13) – (23) (13) – (14) (14) – (24) (15) – (25) (2) – (12) (3) – (13) (4) – (14) (5) – (15) ; (op) at (4.65,-) ; (c0) at (6.4,-0.1) [$0$]{}; (c1) at (5.3, -) [$1$]{}; (c2) at (7.4,-) [$4$]{}; (c3) at (5.6,-2) [$2$]{}; (c4) at (7,-2) [$3$]{}; (c0) – (c1) (c0) – (c2) (c0) – (c3) (c0) – (c4) (c1) – (c3) (c1) – (c2) (c1) – (c4) (c3) – (c4) (c3) – (c2) (c4) – (c2); \[fig:evalTop\] Stock datasets are commonly used to generate workloads for evaluations of CPS systems [@agg15]. We used a dataset of 500 stock symbols from Yahoo Finance!, where each stock notification had 10 distinct attributes. This high dimension data require high computation for filtering information during in–broker processing. Subscriptions were generated synthetically. We randomly distributed publishers and subscribers, where each subscriber registered one subscription with 2% selectivity. Metrics ------- Through experiments with real world data, we evaluated `OctopiS` using primitive metrics, such as subscription, notification, and matching delays.\ ***Subscription delay***: The subscription (forwarding) delay is the maximum time elapsed as a subscription reaches brokers in an overlay network. A subscription is expected to take less time to reach brokers in a close proximity to the host broker of a subscriber than to brokers in a far–off region. Since the SBP in a clustered SCOT generates subscription–trees of shortest lengths, it is important to measure the difference between the average subscription delays in unclustered and clustered SCOT.\ ***Notification delay***: The notification delay measures end–to–end latency from the time a notification is generated to the time it is received by a subscriber. As the average length of subscription–trees for BID–based routing is higher than a clustered SCOT for SNR and DNR algorithms, knowing the difference in latencies in these two cases is a worthwhile area of inquiry.\ ***Matching delay***: The matching delay is the time taken to find subscriptions that match with the notification contents. BID–based routing is expected to have less matching delay as the matching is done at only the brokers which host publishers and interested subscribers.\ ***Inter–broker Messages (IMs)***: The number of IMs depends on the lengths of subscription–trees, as well as the relative distance between publishers and subscribers. As the average length of subscription–trees in clustered SCOT is less than in unclustered SCOT, the number of IMs generated by SNR and DNR algorithms is expected to be less than BID–based routing algorithm. Aggregation techniques like *covering* are developed for acyclic overlays to reduce size of routing tables. These techniques can be used with clustered SCOT as each cluster is an acyclic overlay. However, for a comparison with state–of–the–art BID–based routing, covering in not considered in evaluation because no covering technique for cyclic overlays is reported in literature. [0.24]{}   [0.24]{}   [0.24]{}   [0.24]{} [0.24]{}   [0.24]{}   [0.24]{}   [0.24]{} [0.3]{}   [0.3]{}   [0.3]{} Results ------- The results presented in this section cover three important aspects of the evaluation: (i) *Subscriber Scalability*, which studies the behaviour of the routing algorithms when the number of subscribers increases, while the number of publishers remains constant; (ii) *Publisher Scalability* is about study of the algorithms with a varying number of publishers and constant number of subscribers; (iii) *Burst Scenario*, in which an HRP starts sending notifications at a high rate and causes congestion in the output queues.\ ***Subscriber Scalability:*** We gradually increased the number of subscribers from 500 to 10000, and used 100 publishers, each sending 1000 notifications at the rate of 60 notifications per minute (npm). All publishers start sending the notifications in the first 5 seconds after all the subscriptions register. We adopted this pattern of generating controlled workload to count IMs and notifications received by the subscribers. Fig. 9(a) shows that the number of IMs generated by SBP in clustered $\mathbb{S}_{e}$ (legends SNR, and DNR) are 89% less than in unclustered $\mathbb{S}_{e}$ (legend BID). There are two reasons for this significant difference: (i) larger average lengths of subscription–trees, and (ii) extra IMs generated to discard duplicate subscriptions in unclustered $\mathbb{S}_{e}$. The average length of the subscription–trees generated by `OctopiS` is 14% less than the PADRES. Furthermore, almost 80% of the generated IMs are used to detect and discard duplicates in unclustered $\mathbb{S}_{e}$ [@Li_ADAP]. `OctopiS` uses a pattern of subscription broadcast that does not generate duplicates and extra IMs. Fig. 9(b) shows that the average subscription delay in clustered $\mathbb{S}_{e}$ is 77% less than the unclustered $\mathbb{S}_{e}$. The difference is due to the larger lengths of subscription–trees and extra IMs generated in unclustered $\mathbb{S}_{e}$ to detect duplicate subscriptions. Importantly, there is no difference in subscription delay when SNR and DNR algorithms are used, since SBP does not use inter-cluster dynamic routing even if some output queues are congested. This approach generates subscription–trees of shortest lengths even if some links are overloaded. The average subscription delay in unclustered and clustered $\mathbb{S}_{e}$ is nearly constant. Fig. 9(c) shows that the end–to–end notification delay in SNR and DNR is less than BID–based routing approach. More specifically, SNR and DNR algorithms reduce the notification delay by 47% when compared with BID–based routing algorithm. There are three reasons for this difference: (i) the larger length of subscription–trees in BID–based routing, (ii) high payload due to carrying BIDs of the matching subscriptions, and (iii) extra processing by brokers to prepare and split lists of BIDs down the routing paths to find the next destinations. In large networks with thousands of brokers in multiple data centres (e.g., in [@WSP_BING]), a notification may have to carry thousands of BIDs to identify routing paths. `OctopiS` does not require carrying BIDs with notifications (a lightweight bit vector $CBV_{p}$ is added when the notification routing is dynamic). Delay in static and dynamic routing is almost the same as this experiment does not generate dynamic routing paths for small workloads. Fig. 9(d) shows that the number of IMs generated by the same number of notifications (1000 per publisher) in SNR and DNR algorithms is 12% less than BID–based approach. Again, the difference in the number of IMs is due to larger lengths of subscription–trees generated by the BID-based subscription routing. The difference between the number of generated IMs decreases with the increase in the number of subscribers because of the possibility of having less *forwarder–only* brokers in a routing path. Fig. 10(a) shows that the average matching delay in BID–based routing is 25% less than SNR and DNR algorithms, and increase almost linearly. In BID–based notification routing, only the host brokers of publishers and interested subscribers execute the matching process, while no matching occurs at the intermediate brokers [@Li_ADAP], while SNR and DNR algorithms performs matching at each broker, which results in larger matching delays. The difference between matching delays in the three algorithms decrease with increase in the number of subscribers, which decreases forwarder–only brokers.\ ***Publisher Scalability:*** We increase the number of publishers from 100 to 500 (each sending 500 notifications at a fixed rate of 100 npm) with 3000 subscribers. Fig. 10(b) shows that the average notification delay in SNR and DNR algorithms is 42% less than BID–based routing algorithm. The difference is due to the larger length of subscription–trees and higher payload due to carrying BIDs with notifications. Because of the difference in lengths of subscription–trees, the number of IMs generated by BID–based routing is 36.4% higher than SNR and DNR algorithms (Fig. 10(c)). The average matching delay in SNR and DNR algorithms is 265% higher than BID–based algorithm (Fig. 10(d)). BID–based routing matches a notification only at the host brokers of publishers and interested subscribers, while SNR and DNR algorithms match a notification at each broker of a routing path. As the number of subscribers is constant, the matching delays introduced by three algorithms is linear.\ ***Stability Analysis:*** The stability analysis tells how quickly a CPS system converges to a normal state after an HRP finishes sending notifications. To study this behaviour, we set the value of $\tau$ to 10 and $t_{w}$ to 50 milliseconds. We used 5000 subscribers, and 100 publishers each issued 2000 notifications at the rate of 60 npm. 0.2% of the subscribers subscribe to receive notifications from the HRP. We execute three simulations with the HRP sending 100K notifications at rates of 100K, 80K, and 60K npm. All clusters of $\mathbb{S}_{e}$ are target of the HRP. Furthermore, the HRP and interested subscribers are hosted by different brokers exerting more load on iCOLs and aCOLs. The burst continued for 60 to 100 seconds depending on the rate of the HRP. Each point in the graphs in Fig. 11 is a maximum delivery delay of 1000 notifications received in a sequence. This approach helps in analysing the stability without graphing too many points. Each simulation is run until all notifications are received. Fig. 11 shows that DNR algorithm stabilized `OctopiS` before SNR algorithm, while BID–based routing algorithm is not able to stabilize PADRES for the same workload. On average, in the first 18 minutes and 30 seconds, the maximum delay of a notification (out of 1000) is the same in the three routing algorithms and no tendency toward stability is observed. This indicates that, due to the high rates of notifications, the state of the system (`OctopiS`) does not converge to normal until the condition $CE < 1$ is maintained for some time (on average, 16 minutes and 40 seconds for the three simulations). DNR starts stabilizing `OctopiS` before the other two algorithms. The average value of $Q_{\ell}$ of target links at the brokers in routing paths of notifications from the HRP when DNR is used is 48% less than SNR and 59% less than BID–based routing algorithms. There are 5 target clusters of the HRP and DNR tends to add the minimum number of copies of a notification when the output queues of the target links are congested. This decreased the length of $Q_{\ell}$ when compared with SNR and BID–based algorithms. The average notification delay in DNR algorithm when the notification rate is 100K is 13%, and 58% less than SNR, and BID–based routing algorithms, respectively. Similar improvements, when the rates are 80K, 60K npm, are 12.1% and 53%, and 11% and 49.2%, respectively. On average, for the three simulations, DNR algorithm stabilizes the system 239 seconds before SNR algorithm and generates only 0.32% IMs more than SNR and 17.2% less than BID–based routing algorithms. This shows that our approach of adding at most one copy of a notification when the output queues are congested significantly reduces delivery delay and queue length. Analysis of the collected data indicate that the number of notifications that have delivery delays less than 1 second in DNR, SNR and BID–based routing algorithms are 44.2%, 39% and 28%, respectively. We also conducted several experiments with HRPs in each cluster and sending notifications with different rates. The performance difference between DNR and SNR algorithms decreases with an increase in the number of HRPs that start sending notifications simultaneously. This indicates that the improvement due to inter-cluster dynamic routing in `OctopiS` diminishes when more congested output queues and overloaded iCOLs appear. Related Work ============ Content-based routing in distributed broker-based CPS systems has been focus of many research efforts. SIENA [@SIENA_WIDE_AREA], JEDI [@JEDI], Rebeca [@Rebeca], PADRES [@PADRESBookChapte], Kyra [@Kyra], and MEDYN [@MEDYN] are just few examples. Notifications routing in cyclic overlays has got a little attention from the research community. SIENA introduces a notifications routing scheme for cyclic overlays and detects duplicates using BIDs. Latency-based distance-vector algorithms generate routing paths for advertisement- and subscription-based CPS systems [@carz_thesis]. However, the algorithms do not generate subscription–trees of shortest lengths and, despite multi-path overlay networks, dynamic routing is not supported. Subscription–trees generated for CPS systems in [@carz_thesis] can be improved by periodically sending subscription messages to find links with the minimum latency; however, this refinement generates extra traffic in the network and is infeasible for a large network settings. Li et al [@Li_ADAP] offers dynamic routing without making updates in routing tables in advertisement-based publish/subscribe systems. A large number of IMs are generated in the advertisement broadcast process to detect duplicates. Unique path identifications are added in notifications for routing to interested subscribers. Dynamic routing relies on brokers with intersecting routing paths generated by multiple advertisements matching one subscription. Therefore dynamic routing for a subscription (or subscriber) matching with one advertisement is not possible. Furthermore, intersecting routing paths are not always possible even if a subscription matches with multiple advertisements [@Li_ADAP]. The probability of having brokers that publish intersecting routing paths also depends upon the number of multiple advertisements matching a subscription. Finally, a subscription has to be delivered multiple times to the broker that publishes more than one advertisement intersecting that subscription [@OctopiA_TR]. Shuping et al [@MERC] propose an approach which divides an overlay into interconnected clusters to apply content-based and destination based intra- and inter-cluster routing. Algorithms for routing using shortest paths are developed, however, the approach requires embedding routing path identifications, which increases payload and consume network bandwidth inefficiently. A broker has to be aware of all other brokers in its host cluster, which increases topology maintenance cost. Dynamic routing requires updates in routing tables to generate alternative routing paths, which increases network traffic and delivery delays.\ `OctopiS` offers inter-cluster dynamic routing without requiring updates in routing tables. Thanks to the structuredness of clustered SCOT overlays, which provide multiple inter-cluster routing paths. `OctopiS` neither generates redundant IMs or duplicates in SBP nor requires BIDs. Notifications are delivered using subscription–trees of shortest lengths and without requiring unique identifications to identify routing paths to prevent loops. Brokers in SCOT overlay are decoupled and aware of their direct neighbours only, which requires a very low topology maintenance cost. All these properties make `OctopiS` scalable and suitable for large content-based routing networks. Conclusion ========== In this paper, we present the design and evaluation of the first (subscription–based) CPS system, `OctopiS`, that offers inter–cluster dynamic routing of notifications without requiring updates in routing tables. The system uses a novel structured cyclic topology SCOT, which is constructed applying optimizations on Cartesian product of two graphs. A homogeneous clustering technique is introduced, which divides a SCOT into similar identifiable blocks of brokers. We further exploit the structuredness of clustered SCOT to generate subscription–trees of shortest lengths, without generating extra IMs to detect and discard duplicates. A static routing algorithm SNR uses subscription–trees of shortest lengths to send notifications to interested subscribers, while DNR algorithm offers inter–cluster dynamic routing using a lightweight bit vector mechanism. Both algorithms do not require a global knowledge of an overlay topology and dynamic routing is activated when congestion is detected in the output queues of the target links. The evaluation of SNR and DNR algorithms, and comparison with BID–based routing algorithm indicates that `OctopiS` scales better than state–of–the–art and suitable for large network settings.\

--- abstract: 'In this note we derive a type of a three critical point theorem which we further apply to investigate the multiplicity of solutions to discrete anisotropic problems with two parameters.' author: - 'Marek Galewski, Piotr Kowalski' title: Three solutions to discrete anisotropic problems with two parameters --- Introduction ============ The main aim of this note is to develop further a type of the three critical point theorem by providing some general version which would be applicable for various types of nonlinear problems depending on numerical parameters. The main result of this note says that a coercive functional acting on a reflexive strictly convex Banach space under some geometric conditions concerning local behaviour around $0$ has at least three critical points. The research connected with the existence of at least three critical points to action functionals, both smooth and nonsmooth, connected with boundary value problems has received some considerable attention lately. It begun with the celebrated results of Ricerri [@ricceri.2; @ricceri.3] and was further developed in many subsequent papers, see for example [@ricceri.4; @ricceri.5]. The three critical theorem was later generalized, simplified and next extensively applied, see for example [@molica.bisci.bonanno; @bonnano.chinne] and references in [@ricceri.1]. Recently another type of a three critical point theorem was developed in [@cabada.iannizzotto.tersian] and further generalized in [@cabada.iannizzotto] to the case of $p-$laplacians and in [@galewski.wieteska] to the case of anisotropic problems. In this note, we base ourselves on results in [@cabada.iannizzotto; @galewski.wieteska] in order to provide yet another type of a three critical point theorem, which would hold for problems to which the results mentioned cannot be applied. Moreover, our main result generalizes main theorems in [@cabada.iannizzotto; @galewski.wieteska]. As a model problem to which our general multiplicity results could be applied is the following discrete boundary value problem: $$\left\{ \begin{array}{l} \begin{array}{l} -{\Delta {}\left({{\left| {{\Delta {x}\left({k-1}\right)}} \right|_{} }^{p(k-1)-2} {\Delta {x}\left({k-1}\right)}}\right)} +\\ +\gamma {{g} \left({k,x(k)}\right)} +\lambda {{f} \left({k,x(k)}\right)}=0 \end{array} ,\: k\in \left[ 1,T\right] \\ x(0)=x(T+1)=0, \end{array} \right. \label{zad}$$ where $\gamma ,\lambda >0$ are numerical parameters, $f,g:[1,T]\times {\mathbb{R}}\rightarrow {\mathbb{R}}$ are continuous functions subject to some assumptions, $[1,T]$ is a discrete interval $\{1,2,...,T\}$, ${\Delta {x}\left({k-1}\right)}=x(k)-x(k-1)$ is the forward difference operator, $p: \left[ 0,T+1\right] \rightarrow {\mathbb{R}}_{+}$, $p^{-}={\operatornamewithlimits{min}\limits}_{k\in \left[ 0,T+1\right]} p(k) >1,$ $p^{+}={\operatornamewithlimits{max}\limits}_{k\in \left[ 0,T+1\right] }p(k) $. Solutions to will be investigated in a space $$X={\left\lbrace {x:[0,T+1]\rightarrow {\mathbb{R}}:x(0)=x(T+1)=0}\right\rbrace}$$ which considered with a norm ${\left\| {x} \right\|_{}}={{} \left({ {\sum\limits^{T+1}_{k=1} }{\left| {{\Delta {x}\left({k-1}\right)}} \right|_{} }^2}\right)}^\frac{1}{2}$ becomes a Hilbert space. The research concerning the discrete anisotropic problems of type have only been started, [@kone.ouaro], [@mihailescu.radulescu.tersian], where known tools from the critical point theory are applied in order to get the existence of solutions. In [@bereanu.jebelean.serban] the authors undertake the existence of periodic or Neumann solutions for the discrete $p(k)-$Laplacian. The so called ground state solutions are considered in [@bereanu.jebelean.serban.2]. Continuous version of problems like are known to be mathematical models of various phenomena arising in the study of elastic mechanics, [@zhikov], electrorheological fluids, [@ruzicka], or image restoration, [@chen.levine.rao]. Variational continuous anisotropic problems have been started by Fan and Zhang in [@fan.zhang] and later considered by many methods and authors, [@harjulehto.hasto.le.nuortio], for an extensive survey of such boundary value problems. For some related papers let us also mention, far from being exhaustive, the following [@agarwal.perera.oregan; @cai.yu; @liu.su; @yang.zhang; @cheng.zhang; @zhang]. These papers employ in the discrete setting the variational techniques already known for continuous problems, of course with necessary modifications. The tools employed cover the Morse theory, the mountain pass methodology and linking arguments. Paper is organized as follows. Firstly we provide a variational framework and assumptions for problem in Section 2. Next, in Section 3 we comment on three critical point theorems which we apply. In Section 4 we give a general multiplicity result which we apply for problem in Section 5. Variational framework ===================== In this section we provide a variational framework for problem . We connect solutions to with critical points to the following action functional $$E_{\gamma ,\lambda }(x)={\sum\limits^{T+1}_{k=1} } \frac{1}{p(k-1)}{\left| {{\Delta {x}\left({k-1}\right)}} \right|_{} }^{p(k-1)} +\lambda {\sum\limits^{T}_{k=1} } F(k,x(k))+\gamma {\sum\limits^{T}_{k=1} }G(k,x(k)),$$where $F(k,s)={\int\limits_{{0}}^{{s}}}f(k,t)dt$, $G(k,s)={\int\limits_{{0}}^{{s}}}g(k,t)dt $. With any fixed $\gamma ,\lambda >0$ functional $E_{\gamma ,\lambda }$ is differentiable in the sense of Gâteaux. Its Gâteaux derivative reads $$\begin{array}{l} {\left\langle {E_{\gamma ,\lambda }^{^{\prime }}(x)},{v} \right\rangle_{}}={\sum\limits^{T+1}_{k=1} }{\left| {{\Delta {x}\left({k-1}\right)}} \right|_{} }^{p(k-1)-2} {\Delta {x}\left({k-1}\right)} {\Delta {v}\left({k-1}\right)} +\\ +\lambda {\sum\limits^{T}_{k=1} } f(k,x(k)) v(k) +\gamma {\sum\limits^{T}_{k=1} }g(k,x(k))v(k). \end{array}$$ A critical point to $E_{\gamma ,\lambda }$ is a point $x\in X$ such that ${\left\langle {E_{\gamma ,\lambda }^{^{\prime }}(x)},{v} \right\rangle_{}}=0$ for all $v\in X$ and is a weak solution to . Summing by parts we see that any weak solution to is in fact a strong one. Hence in order to solve we need to find critical points to $E_{\gamma ,\lambda }$ and further investigate their multiplicity. We will need the following assumptions. We note that the assumptions on $f$ are similar to those considered in [@galewski.wieteska] but this problem cannot be easily tackled by method from [@galewski.wieteska] since we have another term $g$ which also depends on a numerical parameter. That is why we must provide another three critical point theorem in order to investigate the multiplicity of solutions. Now we provide example of nonlinear terms which satisfy assumptions \[A.1\]-\[A.5\]. Let $T$ be a positive integer, $T\geq 2.$ Let us consider a continuous function $f:\left[ 1,T\right] \times \mathbb{R}\rightarrow \mathbb{R}$ given by the formula $$f(k,x)=\left\{ \begin{array}{l l} \alpha(k) \cdot \frac{1}{2} x&,|x|<2\\ \alpha(k) \cdot {{} \left({-x+3 {\operatorname{sgn}}(x)}\right)}&,2\leq |x|<4\\ \alpha(k) \cdot {{} \left({-{\operatorname{sgn}}(x)}\right)}&,4\leq |x|<6\\ \alpha(k) \cdot {{} \left({x-7 {\operatorname{sgn}}(x)}\right)}&,6\leq |x|<8\\ \alpha(k) \cdot {\operatorname{sgn}}(x) e^{-|x|+8}&, |x|\geq 8 \end{array} \right.$$ where $\alpha [1,T]\rightarrow (0,+\infty)$ is an arbitrary function. Let us consider another function $g:\left[ 1,T\right] \times {\mathbb{R}}\rightarrow {\mathbb{R}}$ given by the formula $$g(k,x) = \beta(k) \cdot {{} \left({0.5- e^{-x^2}}\right)}$$ where $\beta [1,T]\rightarrow (0,+\infty)$ is an arbitrary function. Then \[A.1\] is satisfied since ${\operatornamewithlimits{lim}\limits}_{x->\infty} F(k,x) = {\int\limits_{{0}}^{{\infty}}} f(k,x) = 0$ for every $k \in [1,T]$. \[A.2\] and \[A.5\] are also satisfied with $m=2$ and $s_2=s_1 = 6$. Then for any $x \in [-2,2] \setminus {\left\lbrace {0}\right\rbrace}$ and $k \in [1,T]$ $$F(k,x) = \alpha(k) \frac{1}{4} x^2 > 0$$ and $$\begin{array}{c} F(k,6) = {\int\limits_{{0}}^{{6}}} f(k,x) = \\\\ ={\int\limits_{{0}}^{{2}}} \alpha(k) \cdot \frac{1}{2} x + {\int\limits_{{2}}^{{4}}} \alpha(k) \cdot {{} \left({-x+3 {\operatorname{sgn}}(x)}\right)} + {\int\limits_{{4}}^{{6}}} \alpha(k) \cdot {{} \left({-{\operatorname{sgn}}(x)}\right)} =\\\\ = \alpha(k) + 0 - 2\alpha(k) < 0 \end{array}$$ Since function $g$ is negative in neighbourhood of $0$ for every $k\in [1,T]$ thus $G$ defined as $G(k,x) := {\int\limits_{{0}}^{{x}}} g(k,s) ds$ is nonpositive in this neighbourhood. On the other hand, for sufficiently large ${\left| {t} \right|_{} }$: $$G(k,t)\geq \beta(k) \frac{1}{4}\cdot {\left| {t} \right|_{} }.$$ Thus $${\operatornamewithlimits{liminf}\limits}_{{\left| {t} \right|_{} } \to +\infty} \frac{G(k,t)}{{\left| {t} \right|_{} }} \geq \beta(k) \frac{1}{4 } > 0$$ which implies that \[A.4\] holds. Remarks on a three critical point theorems ========================================== In this section we comment on some recently obtained results pertaining to the existence of three critical points to action functionals. \[theo\_from\_Cabada\] Let $(X,{\left\| {\cdot} \right\|_{}} )$ be a uniformly convex Banach space with strictly convex dual space, $J\in {\operatorname{C}^{{1}} \left( {X} \right)}$ be a functional with compact derivative, $x_{0},$ $x_{1}\in X$, $p,r \in {\mathbb{R}}$ be such that $p>1$ and $r>0$. Let the following conditions be satisfied: 1. ${\operatornamewithlimits{liminf}\limits}_{{\left\| {x} \right\|_{}}\to+\infty}\frac{J(x)}{{\left\| {x} \right\|_{}}^{p}}\geq 0$ 2. ${\operatornamewithlimits{inf}\limits}_{x \in X} J(x)< {\operatornamewithlimits{inf}\limits}_{{\left\| {x-x_0} \right\|_{}}\leq r } J(x) $ 3. ${\left\| {x_1-x_0} \right\|_{}}<r$ and $J(x_1) < {\operatornamewithlimits{inf}\limits}_{{\left\| {x-x_0} \right\|_{}}=r} J(x)$. Then there exists a nonempty open set $A\subseteq (0,+\infty )$ such that for all $\lambda \in A$ the functional $x\rightarrow \frac{{\left\| {x-x_{0}} \right\|_{}}^{p}}{p}+\lambda J(x)$ has at least three critical points in $X$. The above theorem initiated some later research as concerning its applicability to anisotropic problems, see [@galewski.wieteska], where the term $ {\left\| {x} \right\|_{}}^{p}$ is replaced by some convex coercive functional. Namely, the result from [@galewski.wieteska] reads: \[Galewski.Wieteska\] Let $(X,{\left\| {\cdot} \right\|_{}})$ be a uniformly convex Banach space with strictly convex dual space, $J\in {\operatorname{C}^{{1}} \left( {X,{\mathbb{R}}} \right)}$ be a functional with compact derivative, $\mu \in {\operatorname{C}^{{1}} \left( {X,{\mathbb{R}}_+} \right)}$ be a convex coercive functional such that its derivative is an operator $\mu^{\prime}:X\rightarrow {{X}^{\ast}}$ admitting a continuous inverse, let $\widetilde{x}\in X$ and $r>0$ be fixed. Assume that the following conditions are satisfied: 1. ${\operatornamewithlimits{liminf}\limits}_{{\left\| {x} \right\|_{}}\to\infty} \frac{J(x)}{\mu(x)} \geq 0$ \[C.1\] 2. ${\operatornamewithlimits{inf}\limits}_{x \in X} J(x)< {\operatornamewithlimits{inf}\limits}_{\mu{x}\leq r} J(x) $ \[C.2\] 3. ${{\mu} \left({\widetilde{x}}\right)}<r$ and $J(\widetilde{x}) < {\operatornamewithlimits{inf}\limits}_{\mu(x)=r} J(x)$. \[C.3\] Then there exists a nonempty open set $A\subseteq (0,+\infty )$ such that for all $\lambda \in A$ the functional $\mu +\lambda J$ has at least three critical points in $X$. Note that when $\mu(x) ={\left\| {x} \right\|_{}}^{p}$ then Theorem \[theo\_from\_Cabada\] follows from Theorem \[Galewski.Wieteska\]. Some further question can be asked when examining assumptions and proof of Theorem \[Galewski.Wieteska\]. Namely whether this is possible to weaken assumptions \[C.1\]-\[C.3\]. We try to answer these questions in this note providing some related multiplicity result. In our proof we will base on Theorem \[Galewski.Wieteska\] and also on the following lemma, which can be easily derived from [@ricceri.1 Proposition 2.2] and [@bonnano Theorem 1] \[Key-lemma\] Let $(X,{\left\| {\cdot} \right\|_{}})$ be a reflexive Banach space, $I\subseteq {\mathbb{R}}_+$ be an interval, $\Phi \in {\operatorname{C}^{{1}} \left( {X} \right)}$ be a sequentially weakly l.s.c. functional whose derivative admits a continuous inverse, $J\in {\operatorname{C}^{{1}} \left( {X} \right)}$ be a functional with compact derivative. Moreover, assume that there exist $x_{1},x_{2}\in X$ and $\sigma \in {\mathbb{R}}$ such that: 1. $\Phi (x_{1})<\sigma <\Phi (x_{2})$ \[D.1\] 2. ${\operatornamewithlimits{inf}\limits}_{\Phi (x)\leq \sigma} J(x)>\frac{\left( \Phi \left( x_{2}\right) -\sigma \right) J(x_{1})+\left( \sigma -\Phi (x_{1})\right) J(x_{2})}{\Phi (x_{2})-\Phi (x_{1})}$ \[D.2\] 3. ${\operatornamewithlimits{lim}\limits}_{{\left\| {x} \right\|_{}}\to \infty} \left[ \Phi (x)+\lambda J(x)\right] =+\infty $ for all $\lambda \in I$. \[D.3\] Then there exists a nonempty open set $A\subseteq I$ such that for all $\lambda \in A$ the functional $\Phi +\lambda J$ has at least three critical points in X. We will provide our main results in terms of a kind of comparison theorems. In this section we provide the following simple observation: Let $(X,{\left\| {\cdot} \right\|_{}})$ be a uniformly convex Banach space with strictly convex dual space, $J\in {\operatorname{C}^{{1}} \left( {X,{\mathbb{R}}} \right)}$ be a functional with compact derivative. $\mu _{1}\in {\operatorname{C}^{{1}} \left( {X,{\mathbb{R}}} \right)}$ and $\mu_{2}\in {\operatorname{C}^{{1}} \left( {X,{\mathbb{R}}_+} \right)}$ be a convex coercive functional such that its derivative is an operator $\mu_{2}^{\prime }:X\rightarrow {{X}^{\ast}}$ admitting a continuous inverse, let $y\in X$ and $r>0$ be fixed. Assume the following conditions are satisfied: 1. ${\operatornamewithlimits{liminf}\limits}_{{\left\| {x} \right\|_{}}\to\infty} \frac{J(x)}{\mu_2(x)} \geq 0$ \[E.1\] 2. ${\operatornamewithlimits{inf}\limits}_{x \in X} J(x)< {\operatornamewithlimits{inf}\limits}_{{{\mu_1} \left({x}\right)}\leq r} J(x) $ \[E.2\] 3. ${{\mu_2} \left({\widetilde{x}}\right)}<r$ and $J(\widetilde{x}) < {\operatornamewithlimits{inf}\limits}_{{{\mu_2} \left({x}\right)}=r} J(x)$. \[E.3\] 4. For all $x \in X$ if $\mu_2(x) \leq r$ then $\mu_1(x) \leq \mu_2(x)$. Then there exists a non empty open set $A\subset (0,+\infty )$ such that for all $\lambda \in A$ the functional $x\rightarrow \mu_{2}(x)+\lambda J(x)$ has at least three critical points in $X$. If $z\in {\left\lbrace { x:\mu _{2}(x) \leq r}\right\rbrace} $ then $\mu_{1}(z)\leq r$. Thus $${\operatornamewithlimits{inf}\limits}_{\mu_{1}(x)\leq r}J(x) \leq {\operatornamewithlimits{inf}\limits}_{\mu_{2}\leq r} J(x)$$We apply Theorem \[Galewski.Wieteska\] with $\mu :=\mu _{2}$. A general multiplicity result ============================= In this section we provide our main result. \[3CP.Kowalski.Galewski\]\[Main Theorem\] Let $(X,{\left\| {\cdot} \right\|_{}})$ be a uniformly convex Banach space with strictly convex dual space, $J\in {\operatorname{C}^{{1}} \left( {X,{\mathbb{R}}} \right)}$ be a functional with compact derivative. Assume that $\mu_{1}\in {\operatorname{C}^{{1}} \left( {X,{\mathbb{R}}} \right)}$ is sequentially w.l.s.c and coercive. Let $\mu_{2}\in {\operatorname{C}^{{1}} \left( {X,{\mathbb{R}}_+} \right)}$ be a convex coercive functional. Assume that derivative of $\mu_{1}$ is an operator $\mu_{1}^{\prime }:X\rightarrow {{X}^{\ast}}$ admitting a continuous inverse. Let $y\in X$ and $r>0$ be fixed. Assume the following conditions are satisfied: 1. ${\operatornamewithlimits{liminf}\limits}_{{\left\| {x} \right\|_{}}\rightarrow \infty }\frac{J(x)}{\mu_{2}\left( x\right) }\geq 0$ \[F.1\] 2. ${\operatornamewithlimits{inf}\limits}_{x\in X}J(x)<{\operatornamewithlimits{inf}\limits}_{\mu _{1}\left( x\right) \leq r}J(x)$ \[F.2\] 3. $\mu_{2}\left( y\right) <r$ and $J(y)<{\operatornamewithlimits{inf}\limits}_{\mu_{2}\left( x\right) =r}J(x)$ \[F.3\] 4. ${\operatornamewithlimits{\forall}\limits}_{x\in X}\mu_{2}\left( x\right) \leq r {\Rightarrow}\mu _{1}\left( x\right) \leq \mu _{2}\left( x\right) $ and $\mu_{1}\left( x\right) \geq \mu _{2}\left( x\right) $ for ${\left\| {x} \right\|_{}} \geq M$, where $M>0$ is some constant. \[F.4\] 5. $J$ is convex on the convex hull of $B:={\left\lbrace {x\in X:\mu _{1}(x)\leq r}\right\rbrace}$ \[F.5\] Then there exists a non empty open set $A\subset (0,+\infty )$ such that for all $\lambda \in A$ the functional $x \to \mu _{1}\left( x\right) +\lambda J(x)$ has at least three critical points. We will use Lemma \[Key-lemma\]. Set $I=(0,+\infty )$ and observe that for any $\lambda \in I$ we have for sufficiently large ${\left\| {x} \right\|_{}}$ by \[F.1\] and \[F.4\] that $\frac{J(x)}{\mu _{2}(x)}>-\frac{1}{2\lambda}$. Thus $$\mu _{1}(x)+\lambda J(x)>\mu _{2}(x)-\lambda \frac{1}{2\lambda }\mu _{2}(x)= \frac{1}{2}\mu _{2}(x)\rightarrow +\infty$$ as ${\left\| {x} \right\|_{}}\to +\infty $. So we have condition\[D.3\] of Lemma \[Key-lemma\] satisfied. We define $C:={\left\lbrace { x\in X:\mu _{2}(x)\leq r}\right\rbrace} $. We claim there exists $x_{1}$ such that $\mu_{1}(x_1)<r$ and $J(x_{1})={\operatornamewithlimits{inf}\limits}_{x\in B}J(x)$. Note that $C\subset B$. Since $\mu _{2}$ is continuous and convex, the set $ C $ is weakly closed. Since $\mu _{2}$ is coercive, it follows that $C$ is weakly compact. Since $J$ has a compact derivative, so it is s.w.l.s.c. and therefore its restriction to $C$ attains its infimum. We shall refer to its minimizer as $z$. Take $y$ as in \[F.3\]. We can distinguish the three following cases 1. $y$ minimizes also $J$ over $B$. \[main.proof.case.1\] 2. $y$ does not minimize $J$ over $B$ but $z$ does. \[main.proof.case.2\] 3. neither $y$ and nor $z$ minimize $J$ over $B$. \[main.proof.case.3\] In \[main.proof.case.1\] we put $y=x_{1}$ since $r>\mu _{2}(y)\geq \mu _{1}(y)$. Which proves the case. In \[main.proof.case.2\] we take $z=x_{1}$ since $$J(z)={\operatornamewithlimits{inf}\limits}_{x\in C}J(x)={\operatornamewithlimits{inf}\limits}_{x\in B}J(x)$$ Suppose $z\in {\partial {C}}$, then $$J(z)={\operatornamewithlimits{inf}\limits}_{x\in {\partial {C}}}J(x)>J(y)>J(z)$$ contradiction. Thus $r>\mu _{2}(z)\geq \mu _{1}(z)$. In \[main.proof.case.3\] if neither $y$ and nor $z$ minimize $J$ in $B$, there would exist such $s\in B\setminus C$ such that $J(s)<J(z)\leq J(y)$. $C$ is convex and closed thus there would exists such $\alpha \in (0,1)$ that $t:=\alpha s+(1-\alpha )z\in {\partial {C}}$. Then by \[F.5\] we see that $$J(t)\geq {\operatornamewithlimits{inf}\limits}_{x\in {\partial {C}}}J(x)=J(z)>J(s)$$ Since $J$ in convex $$J(t)\leq \alpha J(s)+(1-\alpha )J(z)<J(z)$$ We see that it is impossible. Thus we have $x_{1}$ such that $\mu _{1}(x_{1})<r$ and $J(x_{1})={\operatornamewithlimits{inf}\limits}_{x\in B}J(x)$. By \[F.2\] there exist $x_{2}$ such that $\mu _{1}(x_{2})>r$ and $J(x_{2})<{\operatornamewithlimits{inf}\limits}_{x\in B}J(x)=J(x_{1})$. Putting $\phi =\mu _{1}$, $\delta =r$ we see that condition \[D.1\] of Lemma \[Key-lemma\] is satisfied. Finally $$\begin{array}{l} {\operatornamewithlimits{inf}\limits}_{x\in B}J(x)=J(x_{1})=\frac{J(x_{1})(\mu _{1}(x_{2})-\mu _{1}(x_{1}))}{ (\mu _{1}(x_{2})-\mu _{1}(x_{1}))}= \\ \\ =\frac{(\mu _{1}(x_{2})-r)J(x_{1})+(r-\mu _{1}(x_{1}))J(x_{1})}{(\mu _{1}(x_{2})-\mu _{1}(x_{1}))}> \\ \\ >\frac{(\mu _{1}(x_{2})-r)J(x_{1})+(r-\mu _{1}(x_{1}))J(x_{2})}{(\mu _{1}(x_{2})-\mu _{1}(x_{1}))}. \end{array}$$ Thus condition \[D.2\] of Lemma \[Key-lemma\] holds. Since we aim at applications for finite dimensional systems and we will work in a finite dimensional Hilbert space, the assumptions of Theorem \[3CP.Kowalski.Galewski\] can be relaxed. Other types of discrete BVPs can also be considered with this approach. In the finite dimensional context we obtain the following: \[3CP.HilbertSpace\]Let $(X,{\left\| {\cdot} \right\|_{}})$ be a finite dimensional Hilbert space and let $J\in {\operatorname{C}^{{1}} \left( {X,{\mathbb{R}}} \right)}$ be a functional with compact derivative. Assume that $\mu_{1}\in {\operatorname{C}^{{1}} \left( {X,{\mathbb{R}}} \right)}$ is coercive, and $\mu_{2}\in {\operatorname{C}^{{1}} \left( {X,{\mathbb{R}}_+} \right)}$ be a convex coercive functional. Assume that there derivative of $\mu_{1}$ is an operator $\mu_{1}^{\prime }:X\rightarrow {{X}^{\ast}}$ admitting a continuous inverse, let $y\in X$ and $r>0$ be fixed. Assume the following conditions are satisfied: 1. ${\operatornamewithlimits{liminf}\limits}_{{\left\| {x} \right\|_{}}\rightarrow \infty }\frac{J(x)}{\mu_{2}\left( x\right) }\geq 0$ \[G.1\] 2. ${\operatornamewithlimits{inf}\limits}_{x\in X}J(x)<{\operatornamewithlimits{inf}\limits}_{\mu _{1}\left( x\right) \leq r}J(x)$ \[G.2\] 3. $\mu_{2}\left( y\right) <r$ and $J(y)<{\operatornamewithlimits{inf}\limits}_{\mu_{2}\left( x\right) =r}J(x)$ \[G.3\] 4. ${\operatornamewithlimits{\forall}\limits}_{x\in X}\mu_{2}\left( x\right) \leq r {\Rightarrow}\mu _{1}\left( x\right) \leq \mu _{2}\left( x\right) $ and $\mu_{1}\left( x\right) \geq \mu _{2}\left( x\right) $ for ${\left\| {x} \right\|_{}} \geq M$, where $M>0$ is some constant. \[G.4\] 5. $J$ is convex on the convex hull of $B:={\left\lbrace {x\in X:\mu _{1}(x)\leq r}\right\rbrace}$ \[G.5\] Then there exists a non empty open set $A\subset (0,+\infty )$ such that for all $\lambda \in A$ the functional $x \to \mu _{1}\left( x\right) +\lambda J(x)$ has at least three critical points. We see that $X$ is a Banach space with a strictly convex dual. Since $X$ is finite dimensional weak convergence is equivalent to the strong one, so $\mu $ and $J$ are weakly continuous. Thus we may apply Theorem \[3CP.Kowalski.Galewski\]. Existence and multiplicity results for problem =============================================== \[norm.equivalence\] For any $u \in X$ following inequality holds $$\frac{2}{\sqrt{T+1}}{\left\| {u} \right\|_{C}} \leq {\left\| {u} \right\|_{}} \leq 2 \sqrt{T} {\left\| {u} \right\|_{C}}$$ First it is obvious that $$2 {\left\| {u} \right\|_{C}} \leq {\sum\limits^{T}_{k=1} } {\left| {{\Delta {u}\left({k}\right)}} \right|_{} }$$ By Hölders inequality we know that $${\sum\limits^{T}_{k=1} } {\left| {{\Delta {u}\left({k}\right)}} \right|_{} } \leq {{} \left({T+1}\right)} \sqrt{{\sum\limits^{T+1}_{k=1} } {{} \left({{\Delta {u}\left({k}\right)}}\right)}^2} = {{} \left({T+1}\right)} {\left\| {u} \right\|_{}}$$ Which proves the first inequality. The other one is proven as follows: $$\begin{array}{c} {\left\| {u} \right\|_{}} = \sqrt{{\sum\limits^{T+1}_{k=1} } {{} \left({{\Delta {u}\left({k}\right)}}\right)}^2} = \sqrt{{\sum\limits^{T+1}_{k=1} } {{} \left({{{u} \left({k}\right)}-{{u} \left({k-1}\right)}}\right)}^2} \leq \\\\ \sqrt{{\sum\limits^{T}_{k=1} } 2 \cdot {\left\| {u} \right\|_{C}}^2 + 2 \cdot {\left\| {u} \right\|_{C}} \cdot {\left\| {u} \right\|_{C}} } = 2\sqrt{T} {\left\| {u} \right\|_{C}}. \end{array}$$ \[Theorem.Coerciveness\]Let $p^{-}\geq 2$. Assume that conditions \[A.1\]-\[A.5\] hold. Then for all $\lambda >0$, $\gamma >0$ problem has at least one solution. Let us define $$\mu_{1}(x)={\sum\limits^{T+1}_{k=1} }{{} \left({ \frac{1}{p(k-1)} {\left| {{\Delta {x}\left({k-1}\right)}} \right|_{} }^{p(k-1)}+\gamma G(k,x(k)) }\right)},$$ $$\mu _{2}(x)={\sum\limits^{T+1}_{k=1} }{{} \left({ \frac{1}{p(k-1)} {\left| {{\Delta {x}\left({k-1}\right)}} \right|_{} }^{p(k-1)}}\right)}.$$ Let $$J(x)={\sum\limits^{T}_{k=1} } F(k,x(k)).$$ Then $E_{\gamma ,\lambda }(x)=\mu _{1}(x)+\lambda J(x)$. Since in [@galewski.wieteska] it was shown that $\mu_{2}(x)\rightarrow \infty $ as ${\left\| {x} \right\|_{}} \to \infty $, so $\mu _{1}(x)\rightarrow \infty $ as ${\left\| {x} \right\|_{}} \to \infty $. Next we see that by \[A.1\] it follows that $E_{\gamma ,\lambda }(x)\to \infty $ as ${\left\| {x} \right\|_{}} \to \infty $. Since $E_{\gamma ,\lambda }$ is differentiable, continuous, coercive and $X$ is a finite dimensional space, it has at least one critical point which is a weak and thus a strong solution to (\[zad\]). As an application of Theorem \[3CP.HilbertSpace\] to problem we have the following: \[Main.Theorem\] Let $p^{-}\geq 2$. Assume that conditions \[A.1\]-\[A.5\] hold. Then there exists $\gamma_{max} >0 $ such that for every $\gamma \in (0,\gamma_{max})$ there exists $A_\gamma \subseteq (0,+\infty )$ such that for all $\lambda \in A$ problem has at least two nontrivial solutions. We will show step by step that the assumptions of Lemma \[3CP.HilbertSpace\] hold. We will start by proving \[G.1\]. Let ${\left( {x}_{n} \right)_{{n} \in {{\mathbb{N}}}} }$ such that ${\left\| {x_n} \right\|_{}}\to \infty$. Let $\epsilon >0$. By norm equivalence there exists such $c>0$ that: $${\left\| {x} \right\|_{}}^{p^{-}} \geq c {\sum\limits^{T}_{k=1} } {\left| {x(k)} \right|_{} }^{p^{-}}.$$ We set $c_1 = \frac{c T^{\frac{2-p^{-}}{2}}}{2p^+}$. By \[A.1\] there exists such $K_1 \in {\mathbb{N}}$ that $${\operatornamewithlimits{\forall}\limits}_{k \in [1,T]} {\operatornamewithlimits{\forall}\limits}_{{\left| {t} \right|_{} }>K_1} \frac{-F(k,t)}{{\left| {t} \right|_{} }^{p^{-}}} < \epsilon \frac{c_1}{T}$$ Let $K_2 \in {\mathbb{N}}$ be such that for all $n>K_2$, ${\left\| {x_n} \right\|_{}}>1$. It is easy to see that $$\mu(x_n) \geq \frac{T^\frac{2-p^{-}}{2}}{p^{+}} {\left\| {x_n} \right\|_{}}^{p^{-}} - \frac{T+1}{p^{+}}$$ Then there exists $K_3 \geq {\operatornamewithlimits{max}\limits}{\left\lbrace {K_2,K_1}\right\rbrace}$ such that for all $n \geq K_3$ $$\mu_2(x_n) \geq \frac{T^\frac{2-p^{-}}{2}}{2p^{+}} {\left\| {x_n} \right\|_{}}^{p^{-}} \geq c_1 {\sum\limits^{T}_{k=1} } {\left| {x_n(k)} \right|_{} }^{p^{-}}$$ Let denote as $M= {\operatornamewithlimits{max}\limits}{\left\lbrace {{\left| {F(k,t)} \right|_{} } : k\in [1,T], {\left| {t} \right|_{} }\leq K_1}\right\rbrace}$. By coerciveness of $\mu_2$ there exists such $K_4$ that for all $n\geq K_4$ $$\mu_2(x_n) \geq \frac{MT}{\epsilon}$$ Let $k = {\operatornamewithlimits{max}\limits}{\left\lbrace {K_3,K_4}\right\rbrace}$. Let $n \geq k$, then: $$\begin{array}{c} -\frac{J(x_n)}{\mu_2(x_n)} = \frac{- {\sum\limits^{T}_{k=1} } F(k,x_n(k))}{\mu_2(x_n)} \leq \\ \leq \frac{ {\sum\limits^{T}_{k=1, {\left| {x(k)} \right|_{} }\leq K_1} } {\left| {F(k,x_n(k))} \right|_{} }}{\mu_2(x_n)} + \frac{ {\sum\limits^{T}_{k=1, {\left| {x(k)} \right|_{} }> K_1} } {\operatornamewithlimits{max}\limits}\{- F(k,x_n(k)) , 0\} }{\mu_2(x_n)} \leq \\ \leq \frac{ {\sum\limits^{T}_{k=1, {\left| {x(k)} \right|_{} }\leq K_1} } {\left| {F(k,x_n(k))} \right|_{} }}{\mu_2(x_n)} + \frac{ {\sum\limits^{T}_{k=1, {\left| {x(k)} \right|_{} }> K_1} } {\operatornamewithlimits{max}\limits}\{- F(k,x_n(k)) , 0\}}{c_1 {\sum\limits^{T}_{k=1} } {\left| {x_n(k)} \right|_{} }^{p^{-}}} \leq \\ \leq {\sum\limits^{T}_{k=1, {\left| {x(k)} \right|_{} }\leq K_1} } \frac{ M \epsilon }{M T} + {\sum\limits^{T}_{k=1, {\left| {x(k)} \right|_{} }> K_1} } \frac{\epsilon}{T} = \epsilon \end{array}$$ Thus $${\operatornamewithlimits{\forall}\limits}_{\epsilon >0} {\operatornamewithlimits{\exists}\limits}_{k \in {\mathbb{N}}} {\operatornamewithlimits{\forall}\limits}_{n \geq k} \quad \frac{J(x_n)}{\mu_2(x_n) } \geq -\epsilon.$$ Which proves \[G.1\]: $${\operatornamewithlimits{liminf}\limits}_{{\left\| {x} \right\|_{}}\to + \infty} \frac{J(x)}{\mu_2(x)} \geq 0$$ Now we will prove \[G.4\]. By coerciveness and ${{\mu_2} \left({0}\right)}=0$ there exists $r^\ast >0$ such that $$\forall x \in X, \quad \mu_2(x) \leq r^\ast {\Rightarrow}{\left\| {x} \right\|_{}} \leq 1.$$ Let $0<r < r_1 = {\operatornamewithlimits{min}\limits}{\left\lbrace {{{} \left({\frac{2 M_1}{\sqrt{T+1}}}\right)}^{p^+} \frac{T^{\frac{p^{+}-2}{2}}}{p^+},r^\ast}\right\rbrace}$, and let $x\in X$ be such that $\mu_2(x) \leq r$. Then $$\mu_2(x) \leq {{} \left({\frac{2 M_1}{\sqrt{T+1}}}\right)}^{p^+} \frac{T^{\frac{p^{+}-2}{2}}}{p^+}.$$ We know that $$\mu_2(x) = {\sum\limits^{T+1}_{k=1} } \frac{1}{p(k-1)} {\left| {{\Delta {x}\left({k-1}\right)}} \right|_{} }^{p(k-1)} \geq \frac{1}{p^{+}} {\sum\limits^{T+1}_{k=1} } {\left| {{\Delta {x}\left({k-1}\right)}} \right|_{} }^{p(k-1)}.$$ When ${\left\| {x} \right\|_{}}\leq 1$ it follows that $${\sum\limits^{T+1}_{k=1} } {\left| {{\Delta {x}\left({k-1}\right)}} \right|_{} }^{p(k-1)} \geq T^{\frac{p^{+}-2}{2}} {\left\| {x} \right\|_{}}^{p^{+}}.$$ So we have the following: $${\left\| {x} \right\|_{C}}\leq \frac{\sqrt{T+1}}{2} {\left\| {x} \right\|_{}}.$$ Thus $${\left\| {x} \right\|_{}}\leq \frac{2 M_1}{\sqrt{T+1}} \text{ and } {\left\| {x} \right\|_{C}}\leq M_1.$$ By \[A.4\] for all $k \in [1,T]$ we see that $G(k,x(k)) \leq 0$ and so ${\sum\limits^{T}_{k=1} }G(k,x(k)) \leq 0$. For any $\gamma >0$ this implies that $\mu_1(x) \leq \mu_2(x)$. We will prove the second part of condition \[G.4\]. From \[A.4\] we have that $${\operatornamewithlimits{\exists}\limits}_{d>0} {\operatornamewithlimits{\forall}\limits}_{k \in [1,T]} \quad {\operatornamewithlimits{liminf}\limits}_{{\left| {t} \right|_{} }\to \infty} \frac{G(k,t)}{{\left| {t} \right|_{} }} > d.$$ Let $t_k >0 $ be such real number that $${\operatornamewithlimits{\forall}\limits}_{k \in [1,T]} {\operatornamewithlimits{\forall}\limits}_{{\left| {t} \right|_{} }>t_k} \quad \frac{G(k,t)}{{\left| {t} \right|_{} }} > \frac{d}{2}.$$ By $G$ continuity it is obvious that there exists such $l<0$ that $${\operatornamewithlimits{\forall}\limits}_{k \in [1,T]} {\operatornamewithlimits{\forall}\limits}_{t \in{\mathbb{R}}} \quad G(k,t) \geq l.$$ Let $x \in X$ such that $${\left\| {x} \right\|_{}} \geq M := {\operatornamewithlimits{min}\limits}{\left\lbrace {2 \sqrt{T} t_k, \frac{- 4 l T\sqrt{T}}{d}}\right\rbrace} >0$$ By $${\left\| {x} \right\|_{}}\leq 2 \sqrt{T} {\left\| {x} \right\|_{C}}$$ we conclude that ${\left\| {x} \right\|_{C}} \geq {\operatornamewithlimits{max}\limits}{\left\lbrace {t_k, \frac{-2lT}{d}}\right\rbrace}$. Let $q \in [1,T]$ be such index that ${\left\| {x} \right\|_{C}} = {\left| {x(q)} \right|_{} }$. Then $${\sum\limits^{T+1}_{k=1} } G(k,x(k)) = {\sum\limits^{T+1}_{q\neq k=1} } G(k,x(k)) + G(q,x(q)) \geq l\cdot T + G(q,x(q)).$$ Since ${\left| {x(q)} \right|_{} }={\left\| {x} \right\|_{C}}>t_k$ then $G(q,x(q)) > \frac{d}{2} {\left| {x(q)} \right|_{} }$. Moreover, since ${\left| {x(q)} \right|_{} }={\left\| {x} \right\|_{C}}> \frac{-2lT}{d}$ we obtain that $$l \cdot T + G(q,x(q)) \geq l \cdot T + \frac{d}{2} {\left| {x(q)} \right|_{} } \geq l \cdot T + \frac{d}{2} \cdot \frac{-2lT}{d} = 0.$$ then for every $\gamma >0$ we conclude that $\mu_1(x) \geq \mu_2(x)$ which proves the case \[G.4\]. We will now prove \[G.2\]. Let $$r < r_2 = {\operatornamewithlimits{min}\limits}{\left\lbrace {{{} \left({\frac{2 M_1}{\sqrt{T+1}}}\right)}^{p^+} \frac{T^{\frac{p^{+}-2}{2}}}{p^+},{{} \left({\frac{2 m}{\sqrt{T+1}}}\right)}^{p^+} \frac{T^{\frac{p^{+}-2}{2}}}{p^+},r^\ast}\right\rbrace}$$ For such $r<r_2\leq r_1$ the proof of \[G.4\] holds. Lets define $\gamma_{max} = \frac{r_2 - r}{-(T+1) l}$. Let $x \in X$ such that $\mu_1(x) \leq r$, and $\gamma \in (0,\gamma_{max})$. We observe that $$\mu_1(x) = \mu_2(x) + \gamma {\sum\limits^{T+1}_{k=1} } G(k,x(k)) \leq r.$$ We have the following chain of estimations: $$\begin{array}{c} \mu_2(x) \leq r - \gamma {\sum\limits^{T+1}_{k=1} } G(k,x(k))\leq r - \gamma l (T+1) \leq r - \gamma_{max} l (T+1)\\\\ \leq r - \frac{r_2-r}{-(T+1) l } l (T+1) \leq r_2 \end{array}$$ Since $r_2 \leq {{} \left({\frac{2 m}{\sqrt{T+1}}}\right)}^{p^+} \frac{T^{\frac{p^{+}-2}{2}}}{p^+}$ we obtain, in similar way as before $$\mu_2(x) \leq {{} \left({\frac{2 m}{\sqrt{T+1}}}\right)}^{p^+} \frac{T^{\frac{p^{+}-2}{2}}}{p^+}.$$ Since $${\sum\limits^{T+1}_{k=1} } {\left| {{\Delta {x}\left({k-1}\right)}} \right|_{} }^{p(k-1)} \leq {{} \left({\frac{2 m}{\sqrt{T+1}}}\right)}^{p^+} T^{\frac{p^{+}-2}{2}}$$ we see that: $${\left\| {x} \right\|_{}} \leq \frac{2m}{\sqrt{T+1}} \text{ and }{\left\| {x} \right\|_{C}}\leq m$$ Which proves that $J(x) \geq 0$. Since $x$ was taken arbitrary we have that ${\operatornamewithlimits{inf}\limits}_{\mu_1(x)\leq r} J(x) \geq 0$. On the other hand, if we choose $$x(k) = \left \lbrace \begin{array}{c l} \frac{s_1+s_2}{2} &, k \in [1,T] \\ 0 &, k=0 \lor k=T+1 \end{array} \right.$$ then $J(x) = {\sum\limits^{T}_{k=2} } F(k,\frac{s_1+s_2}{2}) < 0$. Which proves \[G.2\] $${\operatornamewithlimits{inf}\limits}_{x \in X} J(x) < 0 \leq {\operatornamewithlimits{inf}\limits}_{\mu_1(x) \leq r} J(x).$$ By \[A.5\] $J$ is convex on ${\left\lbrace {x: {\left\| {x} \right\|_{C}}\leq m}\right\rbrace}$. Thus it is convex on convex hull of ${\left\lbrace {x: \mu_1(x) \leq r}\right\rbrace}$ since it is the smallest convex set that contains ${\left\lbrace {x: \mu_1(x) \leq r}\right\rbrace}$. Then \[G.5\] holds. Finally we prove \[G.3\]. For the same $r < r_2$, let $x \in X$ such that $\mu_2(x) \leq r$. Then $\mu_2(x) \leq r_2$. Since $r_2 \leq {{} \left({\frac{2 m}{\sqrt{T+1}}}\right)}^{p^+} \frac{T^{\frac{p^{+}-2}{2}}}{p^+}$ we obtain, in similar way as before $$\mu_2(x) \leq {{} \left({\frac{2 m}{\sqrt{T+1}}}\right)}^{p^+} \frac{T^{\frac{p^{+}-2}{2}}}{p^+}.$$ Since $${\sum\limits^{T+1}_{k=1} } {\left| {{\Delta {x}\left({k-1}\right)}} \right|_{} }^{p(k-1)} \leq {{} \left({\frac{2 m}{\sqrt{T+1}}}\right)}^{p^+} T^{\frac{p^{+}-2}{2}}$$ we see that: $${\left\| {x} \right\|_{}} \leq \frac{2m}{\sqrt{T+1}} \text{ and }{\left\| {x} \right\|_{C}}\leq m$$ Thus $J(x) \geq 0$. Let $y=0$. Then off course $J(y)=0$. Let $z \in X$ such that $\mu_2(z)=r$. We know that $J(z) \geq 0$ and we will prove in fact $J(z) >0$. Indeed, it is obvious that $$\mu_2(x)=0 \iff x=0$$ Since $0<r=\mu_2(z) {\Rightarrow}z \neq 0$. Since $z \neq 0$ then there exists such $q \in [1,T]$ that $0<{\left\| {z} \right\|_{C}}={\left| {z(q)} \right|_{} }$. Then $$J(z) \geq {{F} \left({q,z(q)}\right)} > 0.$$ Finally $${\operatornamewithlimits{inf}\limits}_{\mu_2(x) = r} J(x) = {\operatornamewithlimits{min}\limits}_{\mu_2(x) = r} J(x) >0 = J(0)=J(y)$$ which completes the proves by proving \[G.2\]. [99]{} R.P. Agarwal, K. Perera, D. O’Regan, Multiple positive solutions of singular discrete p-Laplacian problems via variational methods, Adv. Difference Equ. 2005 (2) (2005) 93–99. C. Bereanu, P. Jebelean, C. Şerban, Periodic and Neumann problems for discrete $p(\cdot )-$Laplacian. J. Math. Anal. Appl. 399 (2013), no. 1, 75–87. C. Bereanu, P. Jebelean, C. Şerban, Ground state and mountain pass solutions for discrete $p(\cdot )-$Laplacian. Bound. Value Probl. 2012, 2012:104. G. Molica Bisci, G. Bonanno, Three weak solutions for elliptic Dirichlet problems. J. Math. Anal. Appl. 382 (2011), no. 1, 1–8. G. Molica Bisci, D. Repovs, Nonlinear algebraic systems with discontinuous terms, J. Math. Anal. (2012), doi:10.1016/j.jmaa.2012.09.046. G. Bonanno, A minimax inequality and its applications to ordinary differential equations, J. Math. Anal. Appl. 270 (2002) 210–229. G. Bonanno, A. Chinně, Existence of three solutions for a perturbed two-point boundary value problem. Appl. Math. Lett. 23 (2010), no. 7, 807–811. A. Cabada, A. Iannizzotto, A note on a question of Ricceri, Appl. Math. Lett. 25 (2012), 215-219. A. Cabada, A. Iannizzotto, S. Tersian, Multiple solutions for discrete boundary value problems. J. Math. Anal. Appl. 356 (2009), no. 2, 418–428. X. Cai, J. Yu, Existence theorems of periodic solutions for second-order nonlinear difference equations, Adv. Difference Equ. 2008 (2008) Article ID 247071. Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image processing, SIAM J. Appl. Math. 66 (2006), No. 4, 1383-1406. X.L. Fan , H. Zhang, Existence of Solutions for $p(x)-$Lapacian Dirichlet Problem, Nonlinear Anal., Theory Methods Appl. 52, No. 8, A, 1843-1852 (2003). M. Galewski, R. Wieteska, A note on the multiplicity of solutions to anisotropic discrete BVP’s, Appl. Math. Lett, DOI: 10.1016/j.aml.2012.11.002, (2012). P. Harjulehto, P. Hästö, U. V. Le, M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal. 72 (2010), 4551-4574. B. Kone, S. Ouaro, Weak solutions for anisotropic discrete boundary value problems, to appear J. Difference Equ. Appl. A. Kristály, M. Mihǎilescu, V. Rǎdulescu, S. Tersian, Spectral estimates for a nonhomogeneous difference problem, Commun. Contemp. Math. 12 (2010), 1015-1029. A. Kristály, W. Marzantowicz, C. Varga, A non-smooth three critical points theorem with applications in differential inclusions, J. Glob. Optim. 46 (2010), 49-62. J.Q. Liu, J.B. Su, Remarks on multiple nontrivial solutions for quasi-linear resonant problemes, J. Math. Anal. Appl. 258 (2001) 209–222. M. Mihǎilescu, V. Rǎdulescu, S. Tersian, Eigenvalue problems for anisotropic discrete boundary value problems. J. Difference Equ. Appl. 15 (2009), no. 6, 557–567. B. Ricceri, A three critical points theorem revisited, Nonlinear Anal. 70 (2009), 3084–3089. B. Ricceri, On a three critical points theorem, Arch. Math. (Basel) 75 (2000), 220–226. B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000), 401–410. B. Ricceri, A further refinement of a three critical points theorem. Nonlinear Anal. 74 (2011), no. 18, 7446–7454. B. Ricceri, A further three critical points theorem. Nonlinear Anal. 71 (2009), no. 9, 4151–4157. M. Ržička, Electrorheological fluids: Modelling and Mathematical Theory, in: Lecture Notes in Mathematics, vol. 1748, Springer-Verlag, Berlin, 2000 Y. Yang, J. Zhang, Existence of solution for some discrete value problems with a parameter, Appl. Math. Comput. 211 (2009), 293–302. G. Zhang, S.S. Cheng, Existence of solutions for a nonlinear system with a parameter, J. Math. Anal. Appl. 314 (1) (2006) 311–319. G. Zhang, Existence of non-zero solutions for a nonlinear system with a parameter, Nonlinear Anal. 66 (6) (2007) 1400–1416. V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR Izv. 29 (1987), 33-66. ------------------------------------------------------ Marek Galewski, Piotr Kowalski Institute of Mathematics, Technical University of Lodz, Wolczanska 215, 90-924 Lodz, Poland, marek.galewski@p.lodz.pl, piotr.kowalski.1@p.lodz.pl ------------------------------------------------------

--- abstract: 'We present the evolutionary models of metal-free stars in the mass range from 0.8 to $1.2 {\ensuremath{M_\odot}}$ with up-to-date input physics. The evolution is followed to the onset of hydrogen mixing into a convection, driven by the helium flash at red giant or asymptotic giant branch phase. The models of mass $M \ge 0.9 {\ensuremath{M_\odot}}$ undergo the central hydrogen flash, triggered by the carbon production due to the [3$\alpha$ reaction]{}. We find that the border of the off-center and central ignition of helium core flash falls between 1.1 and 1.2 ${\ensuremath{M_\odot}}$; the models of mass $M \leq 1.1 {\ensuremath{M_\odot}}$ experience the hydrogen mixing at the tip of red giant branch while the models of $M = 1.2 {\ensuremath{M_\odot}}$ during the helium shell flashes on the asymptotic giant branch. The equation of state for the Coulomb liquid region, where electron conduction and radiation compete, is shown to be important since it affects the thermal state in the helium core and influences the red giant branch evolution. It is also found that the non-resonant term of [3$\alpha$ reaction]{} plays an important role, although it has negligible effect in the evolution of stars of younger populations. We compare our models with the computations by several other sets of authors, to confirm the good agreement except for one study which finds the helium ignition much closer to the center with consequences important for subsequent evolution.' author: - 'Takuma Suda, Masayuki Y. Fujimoto' - Naoki Itoh title: 'Evolution of Low-Mass Population III Stars' --- Introduction ============ Although the initial mass function of metal-free stars formed out of primordial matter has not yet been determined, there is evidence that metal-free stars of low and intermediate mass were formed. For example, the fact that the frequency of stars with strong carbon enhancements is much larger than in the case of population I and II stars is consistent with theoretical predictions. One of the most prominent characteristics of models of low-mass and intermediate-mass extremely metal-poor stars (EMPSs) is that they become carbon stars at an earlier stage of evolution than the stars of younger populations [@fuj00 hereinafter FII00]. Recently, it is argued that EMP stars survived to date were formed as a secondary component in the binary systems of massive companions [@Komiya07]. As another example, the peculiar abundance characteristics of the most metal poor stars yet discovered [@chr02] can be understood in terms of the evolutionary properties of $Z = 0$ or extremely iron-poor models in an intermediate-mass interacting binary [@sud04]. Thus, the evolutionary characteristics of $Z = 0$ models have direct relevance to discussions of star formation in the early universe. EMP stars may alter their initial surface abundances by bringing to the surface products of internal nuclear transformations which occur in unique ways. For initial CNO abundances $\textrm{Z}_{\rm CNO} \lesssim 10^{-7}$, the outer edge of the convective zone, generated by the first off-center helium flash in the hydrogen-exhausted core extends into the hydrogen-rich layers, eventually leading to the enrichment of carbon and nitrogen in the surface (Fujimoto et al. 1990, hereinafter FIH90; Hollowell et al. 1990). This “He-flash driven deep mixing (He-FDDM)” mechanism is distinguished from the third dredge-up in asymptotic giant branch (AGB) stars of Populations I and II which enriches surface material in $^{12}$C [@ibe75]. Even if we take into account the possible surface pollution by the accretion of metal-rich gas after birth, the interior evolution of metal-poor stellar models is not altered as long as the original CNO abundance satisfies $\textrm{Z}_{\rm CNO} \leq 10^{-8}$ [@fuj95]. Many calculations of Population III (hereinafter [Pop.III]{}) star evolution have been published [see FIH90; @sud04 and references therein]. Based on a top heavy initial mass function for the primordial cloud @eze71 first computed sets of massive metal-free models. Computations of the evolution of low mass metal-deficient stars were first carried out by [@wag74] in order to provide a set of stellar lifetimes for use in calculations of galactic chemical evolution. Computations of low-mass, zero-metallicity star evolution were performed by @cas75 up to the exhaustion of hydrogen at the center. [@dan82] pursued the evolution of metal-free models of mass 1 ${\ensuremath{M_\odot}}$ to the red giant branch (RGB) phase, but because of a large time step in her computation, did not find any abnormal behavior. [@gue83] suggested that a convective instability occurs during the central hydrogen-burning phase and FIH90 found this to be the case. The “peculiar” evolution of $Z=0$ model stars and its interpretation is revealed by the calculations of FIH90 and [@hol90], who followed the evolution of a $Z=0$ star of mass $1 {\ensuremath{M_\odot}}$ and gave explanations for the core He-H flash, a shell He-H flash, and the He-FDDM phenomenon. By computing in detail the progress of the hydrogen-mixing event and the subsequent evolution, @hol90 first demonstrated that metal-free stars become CN rich carbon stars at the red giant stage. @cas93 and @cas96 found the core He-H flash in models of $Z=10^{-10}$, but, since their computations did not continue beyond the initiation of the helium core flash, they found no evidence for the additional events just described. [@wei00] (hereinafter W00) also made computations of low mass metal-free stars, but terminated their computations just after the ignition of the helium core flash while the flash-driven convective zone was still growing in mass. Computations by @sch01 [hereinafter S01] support the reality of the He-FDDM phenomenon for low-mass zero metallicity stars, whereas computations by [@sie02] (hereinafter SLL02) do not. While they found the mixing of hydrogen into the helium-flash driven convective zone, the failure of SLL02 to reproduce the He-FDDM phenomenon may be ascribed to their assumption of instantaneous mixing of elements in the helium flash convective zone; in consequence, nuclear energy released from the mixed hydrogen is distributed throughout the entire convective zone and the entropy of the zone is not built up sufficiently for the edge of the convective zone to reach deeply into the hydrogen profile. Because mixing and burning of hydrogen occur on a very short timescale in the middle of the helium convective zone, following the He-FDDM event properly requires a treatment of time-dependent mixing [@hol90]. @hol90 first derived the surface composition caused by the He-FDDM phenomenon for a $Z = 0$ model of mass $M = 1 {\ensuremath{M_\odot}}$. @sch02 also derived the surface composition for a model of mass $M = 0.82 {\ensuremath{M_\odot}}$ and compared the results with the observations of EMPSs. After the discovery of HE0107-5240 [@chr02], which held the distinction of being the most metal-poor giant known before the discovery of HE1327-2326 [@fre05], models of EMPSs were computed in an effort to determine the evolutionary history of the observed star. Both @pic04 [ hereinafter P04] and @wei04 found the He-FDDM phenomenon for $Z=0$ models of mass $M = 0.8 {\ensuremath{M_\odot}}$ and $M = 0.82 {\ensuremath{M_\odot}}$, respectively. All but one of the works just cited find the core He-H flash; thus far, only Fujimoto and his coworkers have found the shell He-H flash driven by a violent CN-cycle burning at the base of the hydrogen-burning shell. Why various groups find different evolutionary characteristics for similar masses and compositions has not yet been fully understood. One reason for this is that the computation of $Z=0$ models requires not only a careful treatment of the input physics but also a careful numerical solution. In particular, the treatment of convective regions requires consideration of many complicated factors including uncertain parameters. S01 made comparisons among models by changing parameters related to abundances, diffusion, stellar mass, and the mixing length algorithm for convection, and suggestions have been made for the causes of differences such as radiative and/or conductive opacities and neutrino energy-loss rates (W00; S01; SLL02). However, we are not satisfied that the basic causes for differences have been pinpointed satisfactorily. In this paper, we examine the differences in the evolutionary characteristics of low-mass $Z = 0$ models by taking into account the choice not only of the radiative and conductive opacity and of neutrino energy-loss rates, but also of nuclear reaction rates. Thanks to the many efforts until today, useful sets of input physics are available and they have been more and more accurate for numerical computations. But we think it is worth trying to check the dependence on the input physics and to elucidate the structural characteristics in detail because these efforts are still going on and because the evolution of low-mass Popualtion III stars has not been established yet. In particular, as for the three alpha reaction rates, the determination of resonance states in [${}^{12}$]{}C is still controversial [@fyn05]; the simulations of three body interactions are challenging, and yet, become feasible task in near future thanks to the improved computer resources [see e.g., @kur05 and the references there]. In the next section, we elaborate the input physics adopted in the stellar evolution program. In §3, we present the result of computations of evolution of low mass $Z=0$ model stars with updated input physics, and discuss the model characteristics and their dependences on the input physics including the resonant and non-resonant reaction rates of 3 $\alpha$ reactions. In section 4, comparisons with other works with the different input physics taken into account, and, in §5, we summarize conclusions. The Computational Program ========================= The original program to compute stellar evolution was constructed by @ibe65 and has been modified periodically [e.g., see @ibe75; @ibe92]. The size of each mass shell and the time step are regulated, respectively, by the gradients with respect to space and time of structure and composition variables. Typically, 200-300 mesh points are required for main sequence models and 700-800 for core helium-burning models with hydrogen-burning shells. The number of equilibrium models required to follow evolution from the zero-age main sequence to the end of RGB or AGB phase varies from 5000 and 30000, with the exact number depending on initial mass. The stellar structure equations are typically satisfied to better than one part in $10^{5}$. In the program, the radiative opacity is obtained by interpolation in OPAL tables [@igl96] and in tables by @ale94 [ hereinafter, AF94 tables] and the conductive opacity is from @ito83. The equation of state involves fits by [@ibe92] to work by [@abe59], [@bow60], [@sla80], [@sla82], [@iye82], [@han73], [@han78], [@coh55], and [@car61]. Neutrino energy-loss rates are from @ito96 [ in the following I96] for plasma, photo, pair and bremsstrahlung processes. In order to compare with other works, we make use of two sets of nuclear reaction rates: those given by @cau88 [ in the following CF88] and those given in the latest NACRE compilation [@ang99]. Nuclear screening factors for weak and strong screening are also taken into consideration using standard prescriptions [see, e.g., @bohm58], but only weak screening is dominant in the actual computational range in this work. Nine nuclear species are considered: [${}^{1}$]{}H, [${}^{3}$]{}He, [${}^{4}$]{}He, [${}^{12}$]{}C, [${}^{14}$]{}N, [${}^{16}$]{}O, [${}^{18}$]{}O, [${}^{22}$]{}Ne, and [${}^{25}$]{}Mg. Abundances of these isotopes are determined by 16 nuclear reactions which include proton, electron, and alpha captures. In regions of high temperature and high density which are not covered by OPAL tables, we use the analytical radiative opacities from the fits by [@ibe75] to [@cox70a; @cox70b] ones. At table boundaries, no interpolation is made between table values and analytical values. Hence, jumps in the radiative opacity occur at table boundaries, but jumps are normally smaller than a factor of 2 and do not seriously affect model convergence, primarily because, in regions not covered by the tables, the overall opacity is dominated by electron conductivity. At low temperature and low density, interpolation between OPAL and AF94 tables is accomplished by setting $\kappa=\kappa_{\rm OPAL}\ (1-\Theta)+\kappa_{\rm AF}\ \Theta$, where $\Theta=\sin^2{(\pi/2)(T-T_1)/(T_2-T_1)}$ for the temperature $T$ between $T_1=8000$ K and $T_2=10000$ K, and $\kappa_{\rm OPAL}$ and $\kappa_{\rm AF}$ are OPAL and AF94 opacities, respectively. In surface layers of all models discussed here, AF94 tables provide opacities for all densities and temperatures encountered. In regions of overlap, OPAL and AF94 opacities agree rather well, so switching from one table to the other introduces little uncertainty. Although the quantum correction to the conductivity at low temperatures has been calculated [@mit84], it is now believed that the approximation employed is not appropriate at the temperatures considered. Therefore, we use the analytic fits devised by I83 for the conductive opacity at low temperature; the liquid metal phase for various elemental compositions as well as strong degeneracy are taken into consideration. Strong degeneracy prevails if $T \ll T_{\rm F}$, where $T_{\rm F}(K) = 5.930 \times 10^{9}\ ((1+1.018(\rho_{6}/\mu_e)^{2/3} )^{1/2} -1 )$, $\rho_{6}$ is the density in units of $10^{6}$ g cm$^{-3}$, and $\mu_e$ is the electron molecular weight. For the actual computations, since their approximations are good for $T \lesssim T_{\rm F}$, we adopt the analytical approximations of @ito83 for $T \leq 0.5~T_{\rm F}$. For $T \geq 2~T_{\rm F}$, we adopt the conductive opacity used in the fits by @ibe75 to calculations by [@hub69] for non-relativistic electrons and to [@can70] for relativistic electrons. For intermediate temperatures, interpolation between the two approximations is achieved by “sine square” weighting, analogous to that used in interpolating between OPAL and AF tabular opacities. Finally, the conductive opacity is subjected to a limit described in the Appendix. Interpolated conductive opacities are given in Figure \[fig:cop\], which covers the overall range of application for stellar evolution. For neutrino energy-loss rates, we adopt the fitting formulae presented by I96, who include pair, photo, plasma, and bremsstrahlung with weak degeneracy, the liquid-metal phase with low-temperature quantum corrections and the crystalline lattice phase, and the recombination neutrino processes. Among these latter processes, we do not adopt the low-temperature correction to the liquid-metal phase as calculated by [@ito84] since the corrections are generally small, as stated in I96. For $T \leq 10^{7}$ K, we assume that neutrino energy-losses of all kinds can be neglected. The treatment of a multicomponent gas, which is important for bremsstrahlung, is considered in applying the neutrino energy-loss rates of I96, as described in the Appendix. Mass loss is neglected in our work because we are interested primarily in the stellar interior and the evolution of the helium core is not affected by modest surface mass loss. To determine the temperature gradient in convective regions, we use the standard mixing length recipe by [@bohm58] [see, e.g., @cox68] and the mixing length is taken to be 1.5 times the local pressure scale height. Neither overshooting nor semi-convection are considered. In our calculations we focus on the evolutionary trajectory up to the He-FDDM event and on the thermal structure of a star at stages of interest along its path to this event; our computations are terminated at the onset of hydrogen mixing into the convective zone driven by a helium flash. This event can occur during an off-center helium flash, and/or at the beginning of the thermally pulsing AGB (TPAGB) phase, depending on the initial mass and input physics. Subsequent evolution after the mixing event is beyond the scope of thispaper and will be discussed in detail in a separate paper (Suda, Fujimoto, & Iben, in preparation). Evolution of Low-Mass [Pop.III]{} Stars ======================================= The evolution of $Z = 0$ stars has been computed from the zero-age main sequence (ZAMS) through the RGB and/or through the TPAGB phase for model masses of $0.8 - {1.2} {\ensuremath{M_\odot}}$. For all models, the initial chemical composition is $X = 0.767$, $Y = 0.233$, and $Z=0$, and the initial abundance by mass of [${}^{3}$]{}He is $2 \times 10^{-5}$. These abundances are based on models of big bang nucleosynthesis and are the same as those chosen by FII00. Table \[tab:model\] lists the models computed with all the input physics updated for this paper. The first two columns give, respectively, the model identifier and the initial model mass; the labels “nac” and “cf” mean that the model has been computed with the choice of nuclear reaction rates of NACRE and CF88, respectively. The third to the ninth columns give the effective temperature and surface luminosity at the turn-off point and at the tip of the RGB, the times to reach these two stages, respectively. Figures \[fig:hrd\] and \[fig:rhot\] show, respectively, evolutionary tracks in the H-R diagram and in the central-density and central-temperature plane for ”cf” and ”nac” models of mass in the range $0.8 -1.2 {\ensuremath{M_\odot}}$. All models are terminated at the onset of hydrogen mixing episodes. We found that $M \leq 1.1 {\ensuremath{M_\odot}}$ models ignite the helium burning in the off-center shell and the engulfment of hydrogen by the flash-driven convection occurs just after the peak of the major core helium-flash at the tip of RGB. On the other hand, $1.2 {\ensuremath{M_\odot}}$ models undergo the helium core flash at the center without the hydrogen mixing, and encounter the hydrogen mixing in the helium-flash convection during the helium shell flash at the beginning of thermally pulsating AGB (TP-AGB) phase. The two models display almost the same trajectory irrespective of the choice of nuclear reactions, with the small differences stemming from the difference in the [3$\alpha$ reaction]{} rates, less than a factor of 2 in the temperature range $\log T = 7.8 - 7.9$. The evolutionary characteristics of the core helium flash and the hydrogen mixing event are summarized in Table \[tab:he-flash\]. Each column gives the model identifier defined in Table \[tab:model\], the helium core mass $M_{1}{\ensuremath^{{\textrm{max}}}}$ and the helium burning rate $L{\ensuremath_{{\textrm{He}}}}{\ensuremath^{{\textrm{max}}}}$ when the helium burning rate reaches maximum, the mass coordinate $M{\ensuremath_{{\textrm{BCS}}}}$ and the maximum temperature $T{\ensuremath_{{\textrm{BCS}}}}{\ensuremath^{{\textrm{max}}}}$ at the base of the convective shell, driven by the core helium flash, the helium burning rate $L{\ensuremath_{{\textrm{He}}}}{\ensuremath^{{\textrm{mix}}}}$ at the onset of hydrogen mixing at the RGB, and the time intervals, $\Delta t^\prime$ and $\Delta t{\ensuremath_{{\textrm{mix}}}}$, to it from the appearance of helium flash-driven convection and from the stage of maximum helium burning, respectively. The mass $M_1$ of helium core is defined as the mass coordinate where the abundance of hydrogen is half of the surface abundance of hydrogen. In this section, we discuss in detail the evolutionary behavior of [Pop.III]{} models with respect to the differences in initial mass and input physics. Hydrogen Burning Phase ---------------------- Variations in several quantities characterizing model stars are summarized in Figure \[fig:phys\] as a function of the hydrogen abundance at the center. In a [Pop.III]{} model, due to the absence of CNO catalysts, p-p chain reactions are initially the only mode of energy generation by hydrogen burning. Because of the weak temperature dependence of the energy-generation rate, the central temperature keeps rising as the hydrogen abundance decreases (top panel of Fig. \[fig:phys\]). As the central temperature increases, the [3$\alpha$ reaction]{} gradually becomes active and CNO-cycle reactions begin to occur. Eventually, CNO-cycle reactions dominate the p-p chains with regard to total energy production. When this first occurs, the central abundance by mass $X {\ensuremath_{{\textrm{CNO}}}}$ of catalysts is $10^{-11} - 10^{-9}$ at $\log T_{c} \simeq 7.8 - 7.9$ (top and middle panels of Fig. \[fig:phys\]). Because of lower central temperatures, the less massive the star, the later is the evolutionary stage (and the smaller is the central hydrogen abundance) at which the transition from burning dominated by the p-p chains to burning dominated by the CNO-cycle reactions takes place. For $M \leq 0.8 {\ensuremath{M_\odot}}$, hydrogen is depleted in the center before a transition can take place. After the CNO cycle takes over as the main source of energy generation, because of the strong temperature dependence of CNO-cycle reactions, the central temperature remains nearly constant. At the same time, the core expands because of the central concentration of energy generation and the rate of production of catalysts slows down; the abundances of catalysts saturate at $X {\ensuremath_{{\textrm{CNO}}}} \simeq 10^{-10} - 10^{-8}$. The transition to the CNO-cycle dominated phase is accompanied by the formation of a convective zone which develops outward from the center as described in the lower panel of Fig. \[fig:phys\]. The growth of the convective core is also evident in the middle panel of Fig. \[fig:phys\], which shows that the hydrogen abundance at the center, $X_{c}$, stops decreasing monotonically and increases for a time as evolution progresses. Central convection is caused by a thermonuclear runaway and persists after the transition; due to the larger temperature dependence of the CNO-cycle energy-generation rate, energy generation is highly concentrated toward the center. Both the inward mixing of hydrogen from outer hydrogen-rich layers and the outward mixing of CNO elements generated near the center during the thermonuclear runaway amplify the average hydrogen-burning rate over the region encompassed by convection relative to the average rate in the absence of convection. For stars of mass $0.9 \leq M / {\ensuremath{M_\odot}}\leq 1.2$, the transition from the p-p chain dominated phase to the CNO-cycle dominated phase is delayed until the core has already begun to contract rapidly and electrons have begun to become degenerate at the center; the maximum energy generation has already shifted away from the center and core contraction has initiated the expansion of the envelope. A thermonuclear runaway called the core helium-hydrogen (He-H) flash takes place [@fuj90]. In Fig. \[fig:rhot\], a first increase in the central temperature with decreasing central density indicates that the electron degeneracy is lifted; then, the temperature turns to decrease with the density so as to settle in the thermal equilibrium state where the nuclear energy generation balances the energy loss from the core. In this runaway, helium burning plays a key role through the production of CNO-cycle catalysts. Because of the strong temperature dependence of CNO-cycle reaction rates, the flash starts even when the contribution of the CNO cycle to the total energy-generation rate is smaller than that of p-p chain reactions in central regions. Smaller mass stars experience stronger flashes since the central entropy is smaller and the electron degeneracy is stronger at the onset of the thermonuclear runaway. Thus, the temperature reaches larger, leading to a greater production of CNO catalysts and a greater extension of convection. For stars of mass $M \leq 0.8 {\ensuremath{M_\odot}}$, the central temperature does not become large enough for the production of sufficient carbon to activate the CNO cycle, irrespective of the choice of reaction rates. We see a small hump along the trajectories of $M = 0.8 {\ensuremath{M_\odot}}$ models in Fig. \[fig:rhot\], which marks where the exhaustion of hydrogen at the center quenches a thermonuclear runaway. These evolutionary characteristics during core hydrogen burning are common to all of the models, regardless of the adopted nuclear reaction rates since the evolutionary tracks prior to the transition are dominated by the p-p chain reactions with relatively small temperature dependences. All the models of masses in the range $0.9 \leq M / {\ensuremath{M_\odot}}\leq 1.2$ experience the core He-H flash. We may notice that the differences in the ignition temperatures are rather small because of the strong temperature dependence of the [3$\alpha$ reaction]{} rate, the flash grows weaker for models of larger mass. Core convection lasts until hydrogen is almost exhausted at the center ($X_{c} \lesssim 10^{-4}$). After the model settles in the thermal equilibrium, the central density increases again as the hydrogen abundance decreases. The central temperature at first rises, and then, decreases as the nuclear energy-generation rate decreases at the center (when $X_{c} < 2 - 3 \times 10^{-4}$). Note that since the nuclear energy generation is dominated by contribution from off-center burning, the model at this phase tends to have an isothermal core with the central temperature directly reflecting the temperature in the H-burning shell. In Fig. \[fig:rhot\], we can see the rise in central temperature with the growth of core, a sudden jump in temperature occurring when the hydrogen burning shell passes the shells occupied by the flash convection, and hence, have larger CNO abundances. The resultant loop, as seen in Fig. \[fig:rhot\], is characteristics of flash, implying that He-H core flash eventually exerts work through the expansion and contraction of core, during which the envelope first contacts and then expands again, as seen in Fig. \[fig:hrd\]. In the subsequent evolution, our models do not encounter the phenomena, so called “shell He-H flash” at the base of hydrogen burning phase, as FIH90 discovered. FIH90 discuss the behavior of hydrogen burning shell in the $Z=0$ environment and found the convective instability by the production of carbon and of nuclear energy by CN cycles. This is the counterpart of core He-H flash, i.e., electrons are degenerate at the base of hydrogen burning shell, which causes He-H flash. However, no other groups than Fujimoto and his collaborators find such a event. The different consequence is due to the consideration of non-resonant effect in [3$\alpha$ reaction]{} rate as we will discuss in §\[sec:tal\]. Helium Burning Phase {#sec:heburn} -------------------- Characteristic properties of the helium core flash are given in Table \[tab:he-flash\]. The thermal structure of the core when the helium core flash is ignited are determined by (1) neutrino energy losses which produce cooling, (2) the radiative and conductive opacity which controls heat flow, and (3) the temperature and energy generation in the hydrogen-burning shell which affects the heating due to gravitational compression. If neutrino cooling in the central region is effective enough to produce a strong positive temperature gradient, the helium core flash is ignited off center. Model mass must be larger than $M \geq 1.2 {\ensuremath{M_\odot}}$ for central helium ignition to occur (Fig. \[fig:rhot\]). The mass $M_{1}$ of the helium core when the helium core flash occurs differs greatly between the off-center and central ignition cases, as seen from the third column of Table \[tab:he-flash\]. Central ignition occurs before neutrino cooling becomes appreciable, and, hence, core masses at ignition are small: $M_{1} \lesssim 0.4 {\ensuremath{M_\odot}}$. Once neutrino cooling becomes effective in the central region, off-center ignition is delayed until the core mass is much larger, namely, $M_1 \gtrsim 0.5 {\ensuremath{M_\odot}}$. Figure \[fig:str\] shows the interior temperature as a function of density for models of mass 0.8 ${\ensuremath{M_\odot}}$ and core mass $M_{1} = 0.49 {\ensuremath{M_\odot}}$. A constant temperature “plateau” spreads inward from the base of the hydrogen-burning shell. This implies that the thermal structure of the helium core results mainly from the temperature in the hydrogen burning shell and neutrino energy losses in central regions [@fuj84]. Along with a decrease in the initial abundance of CNO elements, the hydrogen-burning rate decreases, and, in order to compensate for this, the temperature in the hydrogen-burning shell increases, which makes the contribution of compressional heating smaller. Accordingly, an isothermal plateau develops in the region where the radiative heat transport dominates over electron conduction for the metallicity of $[{\rm Fe}/{\rm H}] \lesssim -5$ [see @fuj95]. In our models, the compressional heating plays a small part to raise the maximum temperature in the helium zone slightly higher than the temperature in the hydrogen burning shell, as seen from the lowest mass model; note that for the massive models, the helium burning already contributes appreciably to increase the temperatures in the right shoulder of structure lines in this figure. This behavior contrasts with that of model stars of younger population in which the compressional heating play the dominant part in determining the maximum temperature in the helium core and makes it much larger than the temperature in the hydrogen-burning shell. Because of high temperature in the hydrogen burning shell, therefore, $Z=0$ model stars experience central ignition at smaller initial masses than do model stars of younger populations for which the minimum initial mass for central helium ignition is $\sim 2.5 {\ensuremath{M_\odot}}$. While the density and temperature of the hydrogen burning shell are in local maximum just before the helium ignition, hot CNO-cycle is not still effective. At this stage, the nuclear timescale of ${{}^{13}{\rm N}}$ against proton capture reaction ($\sim 10000$ sec) is much larger than that against $\beta$-decay reaction (863 sec) and is negligible in the outcome of neither the nucleosynthesis nor the nuclear energy output. If neutrino energy losses are sufficiently effective before helium is ignited, the central region cools, and helium is ignited off-center as is the case in low-mass stars of younger populations. In the central region of the model, plasma neutrinos are the dominant neutrino energy-loss mechanism and produce a steep gradient in the rate of released energy. On the other hand, the conduction, which is important in the electron-degenerate core, transport the energy towards the center where the neutrino loss works. Consequently, both conductivity and neutrino energy-loss rates promote cooling of the core and, thus, delay the off-center ignition of helium until a larger core mass gives rise to a larger hydrogen-shell burning rate and a larger temperature in the hydrogen-burning shell, as seen from Table \[tab:he-flash\]. In all of the models which experience off-center ignition, convection driven by helium burning extends into the upper hydrogen-rich layers during the decay phase of the core helium flash, because of smaller entropy in the hydrogen burning shell, as shown by [@fuj90; @fuj95]. The ingestion of hydrogen into the helium convective zone begins a sequence of events that leads to the enrichment of the surface with carbon and nitrogen [@hol90; @fuj00; @sch02; @pic04; @wei04]. Characteristics of hydrogen mixing are also given in Table \[tab:he-flash\]. The ingestion of hydrogen occurs within a matter of days after the helium-burning luminosity reaches its peak. In the models undergoing central helium burning, we do not find a hydrogen-mixing event; The core helium flash is rather weak (see $L_{\rm He}^{\rm max}$ in Table \[tab:he-flash\]), which makes it difficult for the outer edge of the convective region driven by helium burning to reach the hydrogen-containing layer [@fuj77]. For the models of $M=1.2 {\ensuremath{M_\odot}}$, we follow the evolution through the thermally pulsing AGB phase to find that the He-FDDM is triggered during the helium shell flashes. Discussion and Comparison with Other Works ========================================== In this section, we compare the models in the literature with our models adopting similar input physics to see which cause the dominant effect on the differences in the evolution and to confirm the correctness of numerical computations. In addition, we compare the models without non-resonant effect for $\alpha$-capture reactions to see the influence of uncertainty in [3$\alpha$ reaction]{} rates on the stellar structure at $Z=0$. Comparison with $0.8 {\ensuremath{M_\odot}}$ Models --------------------------------------------------- A model comparable with our $0.8 {\ensuremath{M_\odot}}$ model is Model 1 of @pic04 [P04], of composition $Y = 0.23$ and $Z = 0$. In their computations, release 4.98 of FRANEC was used and time-dependent convective mixing was calculated; neutrino energy-loss rates by plasma-neutrino emission were modified, with consequences being reported as minimal. Reaction rates and conductive opacities are, respectively, common with our models. The neutrino energy-loss rates are common with our model for photo- and pair-neutrino processes, but they use energy-loss rate of @dic76 for bremsstrahlung and of @bea67 for recombination processes. For plasma neutrino energy losses, they adopt an energy-loss rate [@esp03] which differs only slightly from I96 in the temperature and density ranges relevant to the ignition of the helium flash. Although there are some differences in adopted input physics, the evolution of the helium core flash of the P04 model is similar to that of our 08cf model. The helium core mass at the onset of the core helium flash is the same in both models, namely, $M_{1} = 0.52 {\ensuremath{M_\odot}}$. When CNO-cycle reactions are the dominant contributors to the hydrogen-burning luminosity, so that the hydrogen profile is very steep, the quantity $M_{\rm{He}}$ defined by P04 as the mass of the helium core when the maximum hydrogen-burning luminosity is reached is nearly the same as the quantity $M_{1}$ we have defined as the mass of the helium core when the core helium flash begins. The maximum helium-burning luminosities differ by less than a factor of two, being $L_{\rm He} = {\ensuremath{{1.2} \times 10^{10}}} L_{\sun}$ in the P04 model and $L_{\rm He} = {\ensuremath{{7.6} \times 10^{9}}} L_{\sun}$ in our model 08cf. The mass at the outer edge of the convective shell at the onset of hydrogen mixing is the same in both cases, namely, $0.506 {\ensuremath{M_\odot}}$. The values of $\Delta t_{\rm{mix}}$ (see Table \[tab:he-flash\]) and $X_{\rm C}$, the mass fraction of carbon in the helium convective shell, are also comparable: $\Delta t_{\rm{mix}} = {\ensuremath{{2.1} \times 10^{5}}}$ sec in the P04 model versus ${\ensuremath{{1.87} \times 10^{5}}}$ sec in model 08cf, and $X_{\rm C} = {\ensuremath{{4.15} \times 10^{-2}}}$ in the P04 model versus $X_{\rm C} = {\ensuremath{{4.28} \times 10^{-2}}}$ in model 08cf. There is a large difference in model characteristics when the helium convective shell first appears; in the P04 model, the convective shell driven by helium burning appears at a mass shell $M_{\rm BCS} = 0.348 {\ensuremath{M_\odot}}$ when $L_{\rm{He}} = 0.658 L_{\sun}$, while, in our model 08cf, $M_{\rm BCS} = 0.3825 {\ensuremath{M_\odot}}$ when $L_{\rm He} = {\ensuremath{{1.57} \times 10^{2}}} L_{\sun}$. The factor of 200 difference in the helium-burning luminosity when shell convection begins is probably simply a typographical error in P04, an interpretation reinforced by the fact that the time for the helium-burning luminosity to reach its maximum value is almost the same in the P04 model (723 yr) and in ours (715 yr). It takes more than $\sim 3\times 10^4$ yr for the helium-burning rate to increase by a factor of 200 in this range. The model of $0.8 {\ensuremath{M_\odot}}$ and $Z=0$ in FII00 is based on those of @fuj95 (hereafter F95) and the comparable results are given in their Table 1. In F95, the values of $M_{1} {\ensuremath^{{\textrm{max}}}}$, $M_{\rm BCS}$, and ${\ensuremath{\log L_{\textrm{He}}}}{\ensuremath^{{\textrm{max}}}} (L_\odot) $ are $0.5116 {\ensuremath{M_\odot}}$, $0.3705 {\ensuremath{M_\odot}}$, and 9.983, respectively. This core mass $M_1$ coincides with our 08nac very closely despite the differences in the input physics; F95 took into account only the resonant [3$\alpha$ reaction]{} reactions [@aus71], which is smaller by a factor of $\sim 2.4$ than the NACRE rate at the relevant temperature range ($\log T \simeq 7.94$). The smaller helium burning rate tends to delay the ignition of helium core flash. On the other hand, the I83 formulae adopted here give larger conductivity than the Iben’s fitting formulae used by F95 in the region of coulomb-liquid regime where the maximum temperature in the helium zone occurs (see fig. \[fig:cop\]), which works to delay the helium ignition due to the enhanced cooling of helium zone through the inward heat conduction in our model. These two effects compensate for each other, while the effect of larger conduction is manifest in inner ignition (or in smaller $M_{\rm BCS}$) in the 08nac model. In actuality, the 08cf model with the same input physics as the 08nac model except for the nuclear reaction rates results in a larger core mass than the 08nac model. This is because the cf88 [3$\alpha$ reaction]{} rate is smaller by $20 \%$ than the NACRE [3$\alpha$ reaction]{} rate around the temperature relevant here. In any case, the dependence of $M_1$ on the nuclear reaction rate is very small because of strong temperature dependence of [3$\alpha$ reaction]{} rate. It is also worth noting that the non-resonant reaction [@nom85] has little to do with the ignition of helium core flash because of rather high temperatures in the helium zone $\log T > 7.9$, although it takes a major part in the later phase of core hydrogen burning and the omission delays the depletion of hydrogen until higher temperature is reached in the center ($\Delta T_{\rm c} \simeq 0.04$ and $\Delta \rho_{\rm c} \simeq 0.25$). Since our calculation does not follow the burning of mixed-in hydrogen, we cannot assess the results of FII00 and P04 with regard to the occurrence of a hydrogen-burning flash in the middle of helium convective zone, the splitting into two convective shells, and the merging of the upper convective shell with the surface convective zone. This remains for further investigation and will be discussed in a separate paper (Suda, Fujimoto, & Iben, in preparation). Comparison with $1 {\ensuremath{M_\odot}}$ Models ------------------------------------------------- In this section, we compare our results with those of FIH90, W00, S01, and SLL02 for the evolution to the beginning of the core helium flash of models of mass $1 {\ensuremath{M_\odot}}$ and initial composition $Z=0$. In all of the cited calculations, the core He-H flash occurs, although the size of the blue loop differs among the different works. We first compare our results with the model of FIH90. The distinctive feature of FIH90 model is the He-H shell flash during the hydrogen shell-burning, as stated in the introduction. We will show in the following subsection that the instability of the hydrogen shell burning and the resultant shell flash are solely attributable to the exclusion of non-resonant rate of [3$\alpha$ reaction]{} reactions [@nom85][^1]. Furthermore, the FIH90 model ended in a significantly larger core mass at the ignition of helium core flash ($M_{1} = 0.528 {\ensuremath{M_\odot}}$). It is even larger as compared with the $0.8 {\ensuremath{M_\odot}}$ model of FII00 with the same input physics except for the equation of state (EOS), despite the general tendency of decrease for larger stellar masses as seen in Table.  \[tab:he-flash\]; Since these two computations differ only in the equation of state (EOS) for a Coulomb liquid and solid among the input physics [see @ibe92 for the adopted the EOS], the main reason for the larger core mass may be the larger radius of the helium core in the FIH90 model (see their Table 1 in FIH90), resulting from the difference in the adopted Coulomb corrections in the EOS. A larger core radius implies a smaller gravity of the core, and hence, a smaller temperature in the hydrogen-burning shell for a given core mass, to defer the ignition of helium core flash until a larger core mass is achieved. In actuality, in FIH90 model, a He-H shell flash is postponed until the core mass grows as large as $M_1 = 0.505 {\ensuremath{M_\odot}}$ and ignited at a low density in the bottom of hydrogen burning shell under a flat configuration ($V \gg 4$) but under non electron-degenerate conditions [see their Fig. 2 in @fuj82]. Because of a large core mass, the He-H shell flash has little effect on the thermal state of the inner core and FIH90 find off-center ignition of the helium core flash. Next, we compare with the work by @wei00 [ W00] who use a code which differs from the one used by @sch01 [ S01] with regard to the EOS [@str88], reaction rates [@thi87 which include the non-resonant term in the [3$\alpha$ reaction]{} rate], and the radiative opacity [old version of OPAL, @rog92; @igl92]. The conductive opacity and the neutrino energy-loss rates are not described in their paper. Since they do not follow the progress of the helium core flash, they do not find the He-FDDM event while helium is ignited off center. Their value of $t {\ensuremath_{{\textrm{TO}}}} = 6.31$ Gyr is close to that of our model 10cf (6.54 Gyr); the small difference may be due to the use of different versions of OPAL opacities. As for the core He-H flash, their values for the location of the outer edge of the central convective zone $M_{ECS} = 0.11 {\ensuremath{M_\odot}}$ and for the helium-burning luminosity $L {\ensuremath_{{\textrm{He}}}} {\ensuremath^{{\textrm{max}}}} = 2.57 \times 10^{-7} L_{\sun}$ are similar to our values of $M_{ECS} = 0.115 {\ensuremath{M_\odot}}$ and $L {\ensuremath_{{\textrm{He}}}} {\ensuremath^{{\textrm{max}}}} = 3.27 \times 10^{-7} L_{\sun}$ for our model 10cf. Element abundances at maximum nuclear burning luminosity are also comparable; W00 find $X_{12} = {\ensuremath{{6.50} \times 10^{-12}}}$, $X_{14} = {\ensuremath{{2.19} \times 10^{-10}}}$, and $X_{16} = {\ensuremath{{2.77} \times 10^{-12}}}$, and we find $X_{12} = {\ensuremath{{7.90} \times 10^{-12}}}$, $X_{14} = {\ensuremath{{1.20} \times 10^{-10}}}$, and $X_{16} = {\ensuremath{{8.44} \times 10^{-13}}}$. These abundances are influenced slightly by the choice of time step. At the tip of the RGB, we can compare with their “canonical” model that element diffusion is not considered in their work; their $M_{1} {\ensuremath^{{\textrm{tip}}}} = 0.497 {\ensuremath{M_\odot}}$ and $\log (L_s^{\rm tip} /L_{\sun}) = 2.357$ differ slightly from our 10cf model: $M_1^{\rm tip} = 0.5054 {\ensuremath{M_\odot}}$ and $\log (L_s^{\rm tip}/L_{\sun}) = 2.413$ (See Tables \[tab:model\] and \[tab:he-flash\]). In the S01 calculations, evolution is followed from the main sequence to the TPAGB phase. The S01 input physics differs from ours with regard to the EOS in regions of electron degeneracy [@kip90], the energy-loss rates for photo, pair, and plasma neutrinos [@mun85], and weak screening for nuclear reactions (they use the Salpeter formula). For the EOS in core regions, they adopt a simplified equation of state for a degenerate electron gas [@kip90] but they provide no explicit statement as to their treatment of the ion gas. Their nuclear reaction rates and conductive opacities are the same as those which we have used. S01 give numerical results only for the helium core flash phase. Their value of $\log (L/L_{\sun}) = 2.314$ at the start of the core helium flash is slightly smaller than our model 10cf value and their value of $M_{1} {\ensuremath^{{\textrm{max}}}} = 0.482 {\ensuremath{M_\odot}}$ is also smaller than ours. However, they find $M {\ensuremath_{{\textrm{BCS}}}} = 0.151 {\ensuremath{M_\odot}}$, while our model 10cf gives $M {\ensuremath_{{\textrm{BCS}}}} = 0.3390 {\ensuremath{M_\odot}}$ and none of our other models ignite a helium core flash with $M_{\rm BCS}$ smaller than $0.3 {\ensuremath{M_\odot}}$, except in the case of central ignition (Table \[tab:he-flash\]). Because of the much larger mass between the base of the convective shell and the location of the base of the hydrogen-rich layer, the time required for the outer edge of the convective shell to reach hydrogen-rich material is much larger in the S01 model than in our models: $\Delta t {\ensuremath_{{\textrm{mix}}}} = 10$ yr for S01 versus ${\Delta t}_{\rm mix} = 10^{-3}$ - $10^{-2}$ yr for all our models, irrespective of the input physics. We note that it takes more than 1000 yr for convection generated in the center to reach the maximum extension in mass and ${\Delta t}_{\rm mix}$ for S01 falls in the middle of our two cases. In an effort to reproduce the S01 results, we constructed a $1.0 {\ensuremath{M_\odot}}$ model 10cf$^\prime$, using the fitting formula for neutrino energy-loss rates given by @mun85 which does not include neutrino bremsstrahlung. The contribution of neutrino loss is rather small as compared with the gravitational energy release in the core, and yet, affects the internal structure of helium core significantly. In Fig. \[fig:str\], we compare the structure line at core mass $M_1 = 0.49 {\ensuremath{M_\odot}}$ in which the neutrino energy loss rate ($L_\nu = 0.70 L_\odot$) is smaller than the rate of gravitational energy release ($L_g = 2.3 L_\odot$) and the helium burning rate is still small ($L_{\rm He} = 0.35 L_\odot$). We see that the maximum temperature shifts to the inner shell as much as $\Delta \log \rho \simeq 0.13$. The reason for this inward shift is that, near the stellar center, the energy-loss rate due to neutrino bremsstrahlung is comparable to the energy-loss rate due to other neutrino processes; neglect of the neutrino bremstrahlung contribution means that the cooling rate in central regions is reduced from what it would otherwise have been. Accordingly, the initiation of a helium-burning thermonuclear runaway occurs for a smaller core mass than would otherwise be the case. We find that, when $M_{1} = 0.4968 {\ensuremath{M_\odot}}$, helium is ignited at a mass point $M {\ensuremath_{{\textrm{BCS}}}} = 0.2933 {\ensuremath{M_\odot}}$, which is about 10% smaller than we find for our 10cf model when neutrino bremstrahlung is included. The 10% reduction we have found is, however, far too small to account for the S01 result but we suspect that differences in neutrino energy-loss rates are in part responsible for the small value of $M {\ensuremath_{{\textrm{BCS}}}}$ found by S01. Since radiative opacities, conductive opacities, and nuclear reaction rates are presumably also not responsible, differences in the EOS may be another source of the discrepancy. In dense stellar matter, Coulomb corrections reduce the pressure, leading to an increase in the density and to a reduction of core radius. The increase in the gravity entails the higher temperature in the hydrogen burning shell and heats up the core. To explore this point more quantitatively, we examine conditions in the 10cf$^\prime$ model for the same core mass as the S01 model ignites helium ($M_1 = 0.482 {\ensuremath{M_\odot}}$). In Fig. \[fig:str\], we locate by filled circles the density and temperature of the hydrogen-burning shell and the density and temperature at the mass point $M_{r} \approx 0.151 {\ensuremath{M_\odot}}$ when $M_1 = 0.482 {\ensuremath{M_\odot}}$. We conclude that the S01 structure curve must be fairly different from ours in the sense that their internal core might be kept much hotter despite the neutrino energy loss, presumably due to larger Coulomb corrections. Finally, we compare the $1 {\ensuremath{M_\odot}}$ model of SLL02 with our model 10nac. SLL02 provide many evolutionary tracks of zero metallicity models covering a large mass range. They do not encounter the He-FDDM phenomenon and their evolutionary calculations for low mass stars extend to the AGB phase. Much of the input physics they adopt is the same as we have used to construct model “nac”. The conductive opacities, radiative opacities, nuclear reaction rates, and the mixing length parameter ($\alpha = 1.5$) are presumably the same. However, they use a different EOS and a different treatment of the nuclear screening factor [@gra73]. They do not comment on the choice of neutrino energy-loss rates. With respect to the core He-H flash, the minimum carbon abundance by mass for the appearance of convection at the center in their models is $\log X_{12C} \simeq -11.5$ for various model masses and this agrees well with our results for lower mass models. The main sequence lifetime, measured by the age at the turn-off point, is $t {\ensuremath_{{\textrm{TO}}}} = 6.56$ Gyr in excellent agreement with our result of $6.53$ Gyr (see Table \[tab:model\]); this agreement is to be expected since both opacities and nuclear reaction rates are the same in both cases. For the same reason, the CN-cycle takes over as the main energy-production mechanism at essentially the same abundance of carbon at the center: $X_{c} = {\ensuremath{{5.8} \times 10^{-4}}}$ in the SLL02 model and ${\ensuremath{{5.71} \times 10^{-4}}}$ in ours. The maximum mass of the convective core and the maximum helium-burning luminosity are very similar in the two cases: $M {\ensuremath_{{\textrm{ECS}}}} \simeq 0.095 {\ensuremath{M_\odot}}$ and $L {\ensuremath_{{\textrm{He}}}} {\ensuremath^{{\textrm{max}}}} \simeq 10^{-7} L_{\sun}$ in the SLL02 model, compared with $M {\ensuremath_{{\textrm{ECS}}}} = 0.104 {\ensuremath{M_\odot}}$ and $L {\ensuremath_{{\textrm{He}}}} {\ensuremath^{{\textrm{max}}}} \simeq = {\ensuremath{{2.51} \times 10^{-7}}} L_{\sun}$ in ours. At the maximum helium-burning luminosity, element abundances are $X_{12} = {\ensuremath{{6.50} \times 10^{-12}}}$, $X_{14} = {\ensuremath{{2.19} \times 10^{-10}}}$, and $X_{16} = {\ensuremath{{2.77} \times 10^{-12}}}$ in the SLL02 model, compared with $X_{12} = {\ensuremath{{6.64} \times 10^{-12}}}$, $X_{14} = {\ensuremath{{1.19} \times 10^{-10}}}$, and $X_{16} = {\ensuremath{{8.97} \times 10^{-13}}}$ in our model. SLL02 do not find a shell He-H flash, consistent with our results. Very similar results are obtained for stellar luminosity and the mass of the helium core at the RGB tip; they find $\log L {\ensuremath^{{\textrm{tip}}}} = 2.357$ and $M_{1} {\ensuremath^{{\textrm{tip}}}} = 0.497 {\ensuremath{M_\odot}}$, compared with $\log L {\ensuremath^{{\textrm{tip}}}} = 2.372$ and $M_{1} {\ensuremath^{{\textrm{tip}}}} = 0.4922 {\ensuremath{M_\odot}}$ for our model. Core helium burning begins when $M {\ensuremath_{{\textrm{BCS}}}} = 0.31 {\ensuremath{M_\odot}}$ for SLL02 and $M {\ensuremath_{{\textrm{BCS}}}} = 0.3320 {\ensuremath{M_\odot}}$ in model 10nac (see Table \[tab:he-flash\]). SLL02 actually find that hydrogen mixes into the convective zone driven by core helium flash, but, because the hydrogen abundance that appears in the zone is so small ($X < 10^{-8}$), the luminosity due to hydrogen burning is relatively small ($L {\ensuremath_{{\textrm{H}}}} \lesssim 10^{4} L_{\sun}$), so they neglect the effects. As mentioned earlier, their result may be an artifact occasioned by their assumption that convective mixing is instantaneous. For completeness, we discuss additional properties of our $1 M_\odot$ model 10nac and, when possible, compare with properties of the other models. At the turn-off point, our model has $M_{1} = 0.2032 {\ensuremath{M_\odot}}$ and the abundance of hydrogen at the center is finite with the value $X_{c} = {\ensuremath{{1.57} \times 10^{-2}}}$. After 553 Myr of evolution beyond the turnoff point, a convective core is formed, driven by the He-H core flash. Still 3.06 Myr later, the hydrogen-burning luminosity reaches a maximum of $L {\ensuremath_{{\textrm{H}}}} {\ensuremath^{{\textrm{max}}}}=24.6 L_{\sun}$, with $L_{\rm pp}=18.3 L_{\sun}$ and $L_{\rm CN} = 6.29 L_{\sun}$ being, respectively, the contributions of the pp-chain reactions and of the CN-cycle reactions. The blue loop in the H-R diagram is characterized by $3.795 \leq \log {\ensuremath{T_{\textrm{eff}}}}$ (K) $\leq$ 3.863, and 1.27 $\leq \log L_{s} \leq 1.33$. During the flash, the convective core grows to a maximum mass of $M {\ensuremath_{{\textrm{ECS}}}} = 0.104 {\ensuremath{M_\odot}}$, which is between the masses found by W00 and SLL02. The maximum carbon abundance is achieved at nearly the same time that the convective shell achieves its maximum mass and has the value $X_{12} {\ensuremath^{{\textrm{max}}}} = {\ensuremath{{7.146} \times 10^{-12}}}$. As evident from Fig. \[fig:rhot\], the inner CN-burning shell passes through the site of this central convective zone when, at the center, $\log \rho_{c} = 5.017$ and $\log T_{c} = 7.801$. A temperature plateau develops in the outer core, as is characteristic of all zero metallicity models, and as the core mass continues to increase, a temperature inversion is formed [@fuj84]. Thereafter, energy transport into the center plays a crucial role in determining the maximum temperature in the core. Our model center evolves into a region of strong degeneracy, reaching the maximum central density $\log \rho_{c} {\ensuremath^{{\textrm{max}}}} =6.074$ at $\log T_{c} = 7.840$. When the helium flash ignites off-center, the core mass is $M_{1} {\ensuremath^{{\textrm{max}}}} = 0.5028 {\ensuremath{M_\odot}}$ (see Table \[tab:model\]) and the luminosity is $L {\ensuremath^{{\textrm{max}}}} = 251.6 L_\odot$. At maximum helium luminosity, the central abundances of CNO elements are $X_{12}={\ensuremath{{1.18} \times 10^{-5}}}$, $X_{14}={\ensuremath{{5.35} \times 10^{-10}}}$, and $X_{16}={\ensuremath{{9.21} \times 10^{-8}}}$. In our model, $T_{c}={\ensuremath{{5.622} \times 10^{7}}}$ K and $\rho_{c}={\ensuremath{{8.498} \times 10^{5}}}$ g cm$^{-3}$ at this time. Influence of [3$\alpha$ reaction]{} rates ----------------------------------------- \[sec:tal\] In this subsection, we discuss the relevance of characteristics of [3$\alpha$ reaction]{} rates to the evolution of zero-metallicity stars. Recently, the properties of resonances other than the so-called Hoyle resonance in [${}^{12}$]{}C and their effects on [3$\alpha$ reaction]{} rates have been discussed experimentally and theoretically since the compilation of NACRE [see for example, @ito04b; @fyn05; @kur05]. On the other hand, the non-resonant term of [3$\alpha$ reaction]{} rate may also be subject to uncertainty in the cross section because the nuclear theory hardly determines the behavior of three body interactions (K. Kat$\bar{\rm o}$ 2007, private communication). Considering these improvement in the field of nuclear physics, it is important to examine the effects of resonant and non-resonant terms of [3$\alpha$ reaction]{}s, separately. Indeed, for $Z=0$ models, the non-resonant term of [3$\alpha$ reaction]{} rate can change the evolutionary behavior drastically at the hydrogen shell burning phase. As a test for exploring the contribution of non-resonant term, we compute the models of 1.0 and $1.1 {\ensuremath{M_\odot}}$ by adopting the nuclear reaction rates of @fow75 (hereafter FCZ75) with the same other input physics as the models in this work. The main difference of FCZ75 rates from the NACRE rates is the consideration of non-resonant term in [3$\alpha$ reaction]{}. Since FCZ75 have not yet taken into account the effect, the [3$\alpha$ reaction]{} rate drops by far more rapidly than the NACRE rates below $\log T \lesssim 7.89$. Other cross sections may differ by within factor of 2 or 3 [@sud03] and do not affect the qualitative results. Figures \[fig:shellHeH\] shows the evolutionary tracks and interior structures in the density-temperature plane for these models with the FCZ75 rates (refereed as models 10fcz and 11fcz, respectively, in the following) and compare them with those of our models with the NACRE rates. The effect of different nuclear reaction rates is apparent before the depletion of hydrogen in the center. Both models with and without the non-resonant term experience the core He-H flash, the FCZ models postpone it until higher central temperature than the NACRE models; in the latter models, it is ignited at the low temperatures where the non-resonant term is effective ($\log T < 7.89$). Because of stronger electron degeneracy, the FCZ75 models undergo much stronger flashes ($L_{\rm He} = 1.06 \times 10^4$ and $78 L_\odot$ for models 10fcz and 11fcz, respectively), as seen from larger loops in this figure, than our models 10nac and 11nuc ($L_{\rm He} = 24$ and $29 L_\odot$, respectively), which entails larger extension of flash convection; for models 10fcz and 11fcz, the flash convection reaches to the shells of $M_{\rm conv} = 0.181$ and $0.149 {\ensuremath{M_\odot}}$ at the maximum extension, respectively, about twice as large as compared to our models 10nac and 11nac, with the CN abundance of $X_{\rm CN} = 6.0 \times 10^{-10}$ and $ 2.6 \times 10^{-10}$, respectively. There is also a remarkable difference in the evolutionary tracks during the hydrogen-shell burning between the models with and without the non-resonant terms. In the hydrogen burning-shell, the CN-burning with carbon produced by [3$\alpha$ reaction]{} contributes considerably to the total energy generation though the contribution to the total energy is still smaller than that of p-p chain reactions for the stars of mass $1.2 > M/ {\ensuremath{M_\odot}}> 0.9 $, because of lower entropy in the hydrogen burning shell as compared with the stars of younger populations. In the models without the non-resonant term, in particular, there form two local maxima in the energy generation rates, a narrow one associated with CN burning and localized at the base of small hydrogen abundance ($X < 0.01$), and a broad one associated with pp-chain reactions in the middle of large hydrogen abundance (nearly half of the surface hydrogen abundance). When the hydrogen burning shell passes across the inner sphere, occupied by the convection during the core He-H flash, a thermonuclear runaway is triggered at the base of hydrogen-burning shell under electron degeneracy, which is a similar situation as the ignition of a core He-H flash in the center; in addition, at the base of hydrogen burning shell, the pressure scale height is much smaller than the radial distance to the center, i.e., $V = r / \vert d r / d \ln P \vert = G M_r \rho / r P \gg 1$, and the flat configuration also contributes to the instability [@sug78]. It occurs in the shell of mass $M_r = 0.212 {\ensuremath{M_\odot}}$ with $M_1 = 0.340 {\ensuremath{M_\odot}}$ for mode 10fcz and $M_r = 0.170 {\ensuremath{M_\odot}}$ with $M_1 = 0.352 {\ensuremath{M_\odot}}$ for the first shell flash of model 11fcz. On the other hand, the models with the non-resonant term stably burn hydrogen, although electrons are degenerate ($\Psi \simeq 8$) at the base of the hydrogen burning shell. This different behavior stems mainly from the difference in the temperature dependence of the resonant and non-resonant terms, rather than from the difference in the burning rate itself, as can be seen from the analysis of thin shell burning by @sch65 [see also Fujimoto 1982]. At temperatures of $\log T < 7.9$, the [3$\alpha$ reaction]{} rates by FCZ75 and NACRE give a large difference in their temperature dependences when compared at the same burning rates. For less massive FCZ75 models of $M \le 0.9 {\ensuremath{M_\odot}}$, the entropy is smaller and hydrogen is burnt before carbon production becomes appreciable. For FCZ75 models of mass $M \ge 1.2 {\ensuremath{M_\odot}}$, higher temperature as well as weaker electron degeneracy tend to stabilize hydrogen shell burning; in addition, as the contribution of CN burning with products of [3$\alpha$ reaction]{} reaction as catalysis overweighs the p-p chain reactions, the pressure scaleheight grows comparable to the radial distance to the center, whicl also stabilize the shell burning by making the heat capacity negative with hydrostatic readjustment [e.g., see @fuj82]. It is noted that the temperature dependence of resonant reaction rate decreases with increase in the temperature, which also stabilizes the hydrogen shell-burning in combination with the reduction in electron degeneracy when the core grows more massive than $M_1 \gtrsim 0.4 {\ensuremath{M_\odot}}$, as is the case for 10fcz model. During the shell flash, the CN-cycle burning shell expands, and the hydrogen exhausted core also expands due to the reduction in the weight of overlying layers; the central temperature and density decrease almost adiabatically (Label “A” in Fig. \[fig:shellHeH\]). During the decay phase of the first shell flash, after the flash-driven convective zone disappears, the core is heated by the flow of energy from the burning shell. For model 11fcz, the second shell flash is ignited at the shell of mass $M_r = 0.287 {\ensuremath{M_\odot}}$ with $M_1 = 0.374 {\ensuremath{M_\odot}}$, when the hydrogen shell-burning passes across the shells, incorporated into the convection and enriched in CN elements during the first shell flash. This flash grows so strong as to drive the convection deep into the hydrogen-rich envelope up to the shell of $M_1 = 0.404 {\ensuremath{M_\odot}}$ (Label “B”), which expedites the growth of helium core to cause the increase in central density. Since the matter in the convective zone driven by the shell flash is enriched in CNO elements, the CN-cycle reactions dominate the energy generation rate while the burning shell traverse in the site of convective zone. During this stable burning phase, the growth rate of core is large because of small hydrogen abundance therein. Accordingly, concomitant rapid compression of helium core increases the central temperature. This enhanced growth of helium core after the shell flashes leads directly to the ignition at the center of a helium core flash (Label “C”) with a small core masses. This is the case if the convective zone is sufficiently large, as in the case for the second shell flashes of 11fcz model, while it is barely missed in 10fcz model. The most important point by this test is that the contribution of the non-resonant term is discernible in the circumstance of $Z=0$, although it may be difficult to detect differences by the observations. There is a difference in the mass range that the off-center helium core flashes occur. If the non-resonant term is included in the [3$\alpha$ reaction]{} rate, the central helium burning occurs at $\geq 1.2 {\ensuremath{M_\odot}}$. Otherwise, it occurs at $\geq 1.0 {\ensuremath{M_\odot}}$ if we use the smaller conductivity by Iben’s approximates to @hub69 and by @can70 and the neutrino loss rates by @bea67 as in FII00, both slightly smaller than ours. Since the lifetime of $1.0 {\ensuremath{M_\odot}}$ model is $\sim$ 7 Gyr (Table \[tab:model\]), such stars cannot be seen in the present halo if they were born. Only clue to those objects will be binary mass transfer between giants of mass $\ge1.0 {\ensuremath{M_\odot}}$ and dwarfs of mass $\le 0.8 {\ensuremath{M_\odot}}$. Since FII00 predict the abundance ratio of C/N $\sim 1$ for He-FDDM at RGB, while C/N $\gtrsim 5$ for He-FDDM at AGB, it will be crucial for the constraint on the estimate of nuclear reaction rate to determine the abundance ratio of C/N and the mass of the primary for $Z=0$. Conclusions =========== We have explored the evolution of low-mass, zero-metallicity stars with the most recent input physics. The mass range is $0.8 {\ensuremath{M_\odot}}\leq M \leq 1.2 {\ensuremath{M_\odot}}$ in step of $0.1 {\ensuremath{M_\odot}}$ and the initial composition is $X = 0.767$, $Y=0.233$, and $Z=0$. Calculations extend from the zero-age main sequence to the beginning of hydrogen mixing into the helium convective region on the RGB or at the start of the TPAGB phase, depending on the characteristics of the helium core flash. 1. The emergence of CN-cycle reactions as important contributors to nuclear energy production occurs during the core hydrogen-burning phase in models of mass $M \geq 0.9 {\ensuremath{M_\odot}}$ in consequence of the formation of carbon by the 3$\alpha$ reaction. This phenomenon is independent of the adopted physics and its importance is a function only of the initial mass and metallicity. 2. The models of $M \leq 1.1 {\ensuremath{M_\odot}}$ undergo an off-center helium flash and hydrogen-mixing into helium flash-convection, leading to helium-flash driven deep mixing at the tip of red giant branch. On the other hand, the models of mass $M \ge 1.2{\ensuremath{M_\odot}}$ ignite the core helium flash at the center and postpone the He-FDDM until the helium shell flashes occur during the early phase of thermal pulsation at the asymptotic giant branch. For models of mass $0.8 {\ensuremath{M_\odot}}$ and $0.9 {\ensuremath{M_\odot}}$, our results coincide qualitatively with those first found by [@fuj90] and @fuj00. For $1.0 {\ensuremath{M_\odot}}\leq M \leq 1.1 {\ensuremath{M_\odot}}$, whether or not a helium core flash is ignited off the center and hydrogen is mixed inward into the convective zone driven by it depend on the adopted nuclear reaction rates. We have also compared our results with those of other investigations. Our models made with the most up-to-date input physics agree well with the models of @pic04, @wei00 and [@sie02] during evolution on the main sequence and the RGB. In particular, we obtain nearly the same results as do [@sie02], although we do not follow evolution after the mixing of hydrogen into the helium flash driven convective zone at the beginning of the helium core flash. To check the behavior of the He-FDDM event, we need to treat mixing with a time-dependent algorithm. On the other hand, even after adopting the same radiative opacities, conductive opacities, nuclear reaction rates, and neutrino energy-loss rates as [@sch01], we are not able to obtain the inner ignition at the onset of core helium burning which they find; we suspect that the discrepancy may be due to differences in the EOS for the Coulomb corrections in the liquid and solid states and due to the neglect in their work of neutrino energy losses associated with neutrino bremsstrahlung. By treating the resonant and non-resonant rates of [3$\alpha$ reaction]{}s separately, we demonstrate that the non-resonant term plays a critical role in the low-mass, zero-metal stars. The neglect of non-resonant term causes the lower border in mass of central helium burning, i.e., $\geq 1.0 {\ensuremath{M_\odot}}$. This explains the discrepancy of the results between ours and @fuj00, which stems mainly from the difference in the temperature dependence rather than in the energy generation rates themselves. We first point out the possibility of discerning the effect of non-resonant term of [3$\alpha$ reaction]{} from the evolution of stars other than at low temperature regime in the accreting degenerate stars [@nom85]. It is important to precisely determine the abundances and the properties of extremely metal-poor stars to constrain the nuclear reaction rates. We are grateful to I. Iben Jr. for improving and revising our manuscript. We wish to thank A. Ohnishi and K. Kat$\bar{\rm o}$ for valuable comments on uncertainties on nuclear reaction rates. This work is part of a PhD. thesis constructed at Hokkaido University and is in part supported by a Grant-in-Aid for Science Research from the Japanese Society for the Promotion of Science (15204010, 18104003). Conductive Opacities ==================== The most up-to-date conductive opacities are those of @ito83 [hereinafter I83]. We consider an ion mixture consisting of $n$ species of nuclei. The conductive opacity $\kappa_{c}$ for temperature $T_{8}$ (units of $10^{8}$ K) and density $\rho_{6}$ (units of $10^{6}$ g cm$^{-3}$) is taken from eq. (7) in I83, $$\kappa_{c} = {\ensuremath{{1.280} \times 10^{-3}}} \left( {\sum_{i=1}^{n}}X_{i} A_{i} \left\langle S \right\rangle_{i} \right) \left[ 1 + 1.018 \left( {\sum_{i=1}^{n}}\frac{Z_{i}}{A_{i}} X_{i} \right)^{2/3} \rho_{6}^{2/3} \right] \left( \frac{T_{8}}{\rho_{6}} \right)^{2} \left[ \textrm{cm}^{2} \textrm{g}^{-1} \right],$$ where $\langle \rangle$ denotes the average over the nuclear species and $Z_{i}$, $X_{i}$, and $A_{i}$ are the atomic number, the mass fraction, and the atomic mass number of $i^{\rm th}$ nucleus. The average value is taken as $$\begin{aligned} \left\langle S \right\rangle_{i} &=& {\left\langle {S_{-1}} \right\rangle}_{i} - \frac{1.018 \left( \sum Z_{i} X_{i} / A_{i} \right)^{2/3} \rho_{6}^{2/3}}{1 + 1.018 \left( \sum Z_{i} X_{i} / A_{i} \right)^{2/3} \rho_{6}^{2/3}} {\left\langle {S_{+1}} \right\rangle}_{i}. \end{aligned}$$ The quantities ${\left\langle {S_{-1}} \right\rangle}_{i}$ and ${\left\langle {S_{+1}} \right\rangle}_{i}$ are calculated with eqs. (8) and (9) in I83 and with the parameters according to eq.(19) of @ito04: $$\begin{aligned} {\ensuremath{\Gamma_{\! i}}}&=& \displaystyle \frac{Z_{i}^{5/3} e^{2}}{a_{e} k_{B} T} = 0.2275 \frac{Z_{i}^{5/3}}{T_{8}} \left( \rho_{6} {\sum_{j=1}^{n}}\frac{X_{j} Z_{j}}{A_{j}} \right)^{1/3} \\ x_{i} &=& 0.45641 \ln \Gamma_{i} - 1.31636 \\ r_{s} &=& {\ensuremath{{1.388} \times 10^{-2}}} \left( {\sum_{j=1}^{n}}\frac{Z_{j}}{A_{j}} X_{j} \rho_{6} \right)^{-1/3}, \end{aligned}$$ where [$\Gamma_{\! i}$]{} is the Coulomb coupling constant for the $i^{\rm th}$ nucleus, $a_{e}$ the electron-sphere radius defined as $a_{e} = (3 / 4 \pi n_{e})^{1/3}$ with the electron number density $n_{e}$, and $r_{s}$ the electron density parameter. The expressions for various mixtures are to be be considered as first approximations, compared with the formulae in I83 which are for pure compositions and are accurate solutions. In particular, the approximation for a mixture of elements with very different $Z$’s (e.g., ${{}^{1}{\rm H}}$ and ${{}^{56}{\rm Fe}}$) is not very accurate; however, the exact solution for such mixtures is not presently attainable. For mixtures of elements of comparable $Z$’s (e.g., ${{}^{12}{\rm C}}$ and ${{}^{16}{\rm O}}$), the approximation is fairly accurate. The I83 results are strictly applicable only for $T \leq T_{F}$, where $T_{F}$ is the Fermi temperature defined by eq. (1) in I83. In this work, $\kappa_{c}$ as defined by I83 and $\kappa_{c}$ as defined by @ibe75 are interpolated with one other over the range $0.5 \leq T / T_{F} \leq 2.0$ using a sin squared algorithm. Another algorithm is used to extrapolate I83 results for values of $\Gamma$ outside the region defined by $2 \leq \Gamma \leq 160$. Both ${\left\langle {S_{-1}} \right\rangle}$ and ${\left\langle {S_{+1}} \right\rangle}$ are constrained by ${\left\langle {S_{-1}} \right\rangle} \leq {\left\langle {S_{-1}} \right\rangle}_{\rm lim}$ and ${\left\langle {S_{+1}} \right\rangle} \leq {\left\langle {S_{+1}} \right\rangle}_{\rm lim}$, where the upper limits are calculated by demanding that, in eq. (6) in I83, $$\begin{aligned} S \left( \frac{k}{2 k_{F}} \right) &=& 1, \\ \epsilon \left( \frac{k}{2 k_{F}} , 0 \right) &=& \displaystyle \frac{k^{2} + k_{TF}^{2}}{k^{2}}. \end{aligned}$$ where $k_{\rm TF}$ is the Thomas-Fermi wavenumber. The first condition assumes that interactions between ions can be neglected and the second condition expresses the Thomas-Fermi approximation for electron screening. These conditions give for the upper limits: $$\begin{aligned} {\left\langle {S_{-1}} \right\rangle}_{lim} &=& \frac{1}{2} \left[ \ln \left( 1 + \frac{4 k_{F}^{2}}{k_{TF}^{2}} \right) -\left(1+{\frac{k_{TF}^{2}}{4 k_{F}^{2}}}\right)^{-1} \right], \label{eq:sm1} \\ {\left\langle {S_{+1}} \right\rangle}_{lim} &=& \frac{1}{2} - {\frac{k_{TF}^{2}}{4 k_{F}^{2}}}\left[ \ln \left(1+ \frac{4 k_{F}^{2}}{k_{TF}^{2}} \right) + \frac{1}{2} - \left(1 + {\frac{k_{TF}^{2}}{4 k_{F}^{2}}}\right)^{-1} \right] + \frac{1}{2} \left( {\frac{k_{TF}^{2}}{4 k_{F}^{2}}}\right)^{2} \left(1 + {\frac{k_{TF}^{2}}{4 k_{F}^{2}}}\right)^{-1} \label{eq:sp1} \end{aligned}$$ The factor $k_{TF}^{2} / (4 k_{F}^{2})$ is given by $$\begin{aligned} {\frac{k_{TF}^{2}}{4 k_{F}^{2}}}&=& \frac{\alpha}{\pi} \left( 1 + \frac{1}{b^{2}} \right)^{1/2}, \\ \alpha &=& \frac{1}{137}, \\ b &=& \frac{\hbar k_{F}}{m c} = \frac{1}{137} \left( \frac{9 \pi}{4} \right)^{1/3} r_{s}^{-1} \end{aligned}$$ where $\alpha$ is the fine structure constant and $b$ is taken from I83. This holds both for relativistic degeneracy ($\rho_{6} \geq 1$) and for non-relativistic degeneracy ($\rho_{6} \leq 1$). The interpolation between I83 and @hub69 as approximated by @ibe75 is shown in Fig. \[fig:cop\]. In the actual computations, the upper limits \[eq:sm1\] and \[eq:sp1\] are infrequently invoked. Neutrino Losses By Bremsstrahlung ================================= We describe here the estimate of the neutrino energy-loss rate due to bremsstrahlung for the case of an ion mixture as discussed by @ito96 [hereinafter I96]. For any nuclear species, gas, liquid, and solid states for ions are taken into consideration. The total energy-loss rate is given by $${\ensuremath{Q_{\textrm{brems}}}} = {\sum_{i}} X_{i} \left( {\ensuremath{Q_{\textrm{gas}}}}^{i} + {\ensuremath{Q_{\textrm{liq}}}}^{i} + {\ensuremath{Q_{\textrm{sol}}}}^{i} \right) ,$$ where the $X_{i}$ are nuclear abundances and the summation is taken over the species ${{}^{4}{\rm He}}$, ${{}^{12}{\rm C}}$, and ${{}^{16}{\rm O}}$. The gas state corresponds to the scheme for weakly degenerate electrons and ${\ensuremath{Q_{\textrm{gas}}}}^{i}$ is based on eq. $(5.1)$ in I96: $${\ensuremath{Q_{\textrm{gas}}}}^{i} = 0.5738 \, \frac{Z_{i}^{2}}{A_{i}} T_{8}^{6} \rho \left( C_{+} {\ensuremath{F_{\textrm{gas}}}} - C_{-} {\ensuremath{G_{\textrm{gas}}}} \right),$$ where $$\begin{aligned} C_{+} = \frac{1}{2} \left\{ \left( C^{2}_{V} + C^{2}_{A} \right) + n \left( C^{\prime 2}_{V} + C^{\prime 2}_{A} \right) \right\}, \\ C_{-} = \frac{1}{2} \left\{ \left( C^{2}_{V} - C^{2}_{A} \right) + n \left( C^{\prime 2}_{V} - C^{\prime 2}_{A} \right) \right\}. \end{aligned}$$ All values of the right hand side of these equations are defined in I96. ${\ensuremath{F_{\textrm{gas}}}}$ and ${\ensuremath{G_{\textrm{gas}}}}$ are independent on the nuclear species other than the number of electrons. ${\ensuremath{F_{\textrm{gas}}}}$ is given by eq. $(5.2)$ in I96, but the parameter $\eta$ is given by $$\eta = {\sum_{i} \frac{X_{i} Z_{i}}{A_{i}} \rho}\left( {\ensuremath{{7.05} \times 10^{6}}} \, T_{8}^{1.5} + {\ensuremath{{5.12} \times 10^{4}}} \, T_{8}^{3} \right)^{-1}.$$ ${\ensuremath{G_{\textrm{gas}}}}$ is given as follows according to eq. $(5.9)$ in I96: $$\begin{aligned} {\ensuremath{G_{\textrm{gas}}}} = \left[\left( 1 + 10^{-9} {\sum \frac{X_{i} Z_{i}}{A_{i}} \rho}\right) \left( a_{3} + a_{4} T_{8}^{-2} + a_{5} T_{8}^{-5} \right) \right]^{-1} \nonumber \\ + \left[b_{3} \left( {\sum \frac{X_{i} Z_{i}}{A_{i}} \rho}\right)^{-1} + b_{4} + b_{5} \left( {\sum \frac{X_{i} Z_{i}}{A_{i}} \rho}\right)^{0.656} \right]^{-1}\end{aligned}$$ According to I96, the applicable range of ${\ensuremath{Q_{\textrm{gas}}}}^{i}$ is determined by $T > 0.01 \, T_{F}$ for ${{}^{4}{\rm He}}$ and $T > 0.3 \, T_{F}$ for ${{}^{12}{\rm C}}$ and ${{}^{16}{\rm O}}$, where $T_{F}$ is the Fermi temperature: $$T_{F} = {\ensuremath{{5.9302} \times 10^{9}}} \, \left\{ \sqrt{1 + 1.018 \left( \sum \frac{X_{i} Z_{i}}{A_{i}} \rho_{6} \right)^{2/3} } - 1 \right\} \quad \left[ \textrm{K} \right].$$ ${\ensuremath{Q_{\textrm{liq}}}}^{i}$ is calculated according to §5.2 in I96. For $T < 0.3 \, T_{F}$ (or $T < 0.01 \, T_{F}$ for ${{}^{4}{\rm He}}$), the formula for the bremsstrahlung process in the liquid state is adopted if the parameter, which considers ion-ion correlation effect according to @ito04, $${\ensuremath{\Gamma_{\! i}}}= 0.2275 \, \frac{Z_{i}^{5/3}}{T_{8}} \left( {\sum_{j=1}^{n}}\frac{X_{j} Z_{j}}{A_{j}} \rho_{6} \right)^{1/3}$$ satisfies ${\ensuremath{\Gamma_{\! i}}}< 180$. Here, ${\ensuremath{Q_{\textrm{liq}}}}^{i}$ is given in cgs units as $${\ensuremath{Q_{\textrm{liq}}}}^{i} = 0.5738 \, \frac{Z_{i}^{2}}{A_{i}} T_{8}^{6} \rho \left( C_{+} {\ensuremath{F_{\textrm{liq}}}} - C_{-} {\ensuremath{G_{\textrm{liq}}}} \right),$$ where [$F_{\textrm{liq}}$]{} and [$G_{\textrm{liq}}$]{} are provided by eqs. $(5.19)$ and $(5.20)$, respectively, in I96. These quantities are functions of $u$ and [$\Gamma_{\! i}$]{} via $$\begin{aligned} v_{i} &=& {\displaystyle \sum_{m=0}^{3}}\alpha_{m} {\ensuremath{\Gamma_{\! i}}}^{- \frac{m}{3}}, \\ w_{i} &=& {\displaystyle \sum_{m=0}^{3}}\beta_{m} {\ensuremath{\Gamma_{\! i}}}^{- \frac{m}{3}}.\end{aligned}$$ For ${\ensuremath{\Gamma_{\! i}}}\geq 180$, we assume for the bremsstrahlung process that $${\ensuremath{Q_{\textrm{sol}}}}^{i} = {\ensuremath{Q_{\textrm{lattice}}}}^{i} + {\ensuremath{Q_{\textrm{phonon}}}}^{i}$$ where, according to eq. $(5.29)$ in I96, $$\begin{aligned} {\ensuremath{Q_{\textrm{lattice}}}}^{i} &=& 0.5738 \, \frac{Z_{i}^{2}}{A_{i}} T_{8}^{6} \rho \left( C_{+} {\ensuremath{F_{\textrm{lattice}}}} - C_{-} {\ensuremath{G_{\textrm{lattice}}}} \right) \label{eq:b13}, \\ {\ensuremath{Q_{\textrm{phonon}}}}^{i} &=& 0.5738 \, \frac{Z_{i}^{2}}{A_{i}} T_{8}^{6} \rho \left( C_{+} {\ensuremath{F_{\textrm{phonon}}}} - C_{-} {\ensuremath{G_{\textrm{phonon}}}} \right) \label{eq:b14}.\end{aligned}$$ Note that we adopt $f_{\textrm{band}} = 1$ in ${\ensuremath{Q_{\textrm{lattice}}}}^{i}$. Each factor such as $F$ and $G$ in the right-hand side of \[eq:b13\] and \[eq:b14\] depends on the nuclei via the parameters: $$\begin{aligned} v_{i} &= {\displaystyle \sum_{m=0}^{3}}\alpha_{m} {{\ensuremath{\Gamma_{\! i}}}^{- \frac{m}{3}}}\hspace{5mm} w_{i} &= {\displaystyle \sum_{m=0}^{3}}\beta_{m} {{\ensuremath{\Gamma_{\! i}}}^{- \frac{m}{3}}}\\ v_{i}^{\prime} &= {\displaystyle \sum_{m=0}^{3}}\alpha_{m}^{\prime} {{\ensuremath{\Gamma_{\! i}}}^{- \frac{m}{3}}}\hspace{5mm} w_{i}^{\prime} &= {\displaystyle \sum_{m=0}^{3}}\beta_{m}^{\prime} {{\ensuremath{\Gamma_{\! i}}}^{- \frac{m}{3}}}.\end{aligned}$$ Abe, R. 1959, Prog. Theor. Phys., 22, 213 Alexander, D. R. & Ferguson, J. W. 1994, , 437, 879 Angulo et al. 1999, Nucl. Phys. A, 656, 3 Austin, S.M, Trentelman, G.F. & Kashy, E. 1971, , 163,L79 Beaudet, G., Petrosian, V., & Salpeter, E. E. 1967, , 150, 979 Böhm-Vitense, E., 1958, Zs. f. Ap., 46, 108 Bowers, D. L. & Salpeter, E. E. 1960, Physical Review Series, 119, 1180 Caughlan, G. R., & Fowler, A. W. 1988, At. Data Nucl. Data Tables, 40, 283 Cameron, A. G. W. 1959, , 130, 916 Canuto, V. 1970, , 159, 641 Carr, W. J. 1971, Physical Review Series, 122, 1437 Cassisi, S. & Castellani, V. 1993, , 88, 509 Cassisi, S., Castellani, V., & Tornambè, A. 1996, , 459, 298 Castellani, V. & Paolicchi, P. 1975, , 35, 185 Christlieb, N. et al. 2002, , 419, 904 Christy, R. F. 1966, , 144, 108 Clayton, D. D. 1968, in Principles of Stellar Evolution and Nucleosynthesis, (USA: McGRAW-HILL book company), 357 Cohen, M. H., & Keffer, F. 1955, Physical Review Series, 99, 1128 Cox, A. N., & Giuli, P. T. 1968, in Principles of Stellar Evolution, Vol.1 (New York: Gordon and Breach), 281 Cox, A. N., & Stewart, J. N. 1970, , 19, 243 Cox, A. N., & Stewart, J. N. 1970, , 19, 261 D’Antona, F. 1982, , 115, L1 Dicus, D. A., Kolb, E. D., Schramm, D. N., & Tubbs, D. L. 1976, , 210, 481 Esposito, S., Mangano, G., Miele, G., Picardi, I., & Pisanti, O. 2003, Nuclear Physics B, 658, 217 Ezer, D. & Cameron, A. G. W. 1971, , 14, 399 Festa, G. G., & Ruderman, M. A. 1969, Physical Review Series, 180, 1227 Fowler, W. A., Caughlan, G. R., & Zimmerman, B. A. 1975, , 13, 69 Frebel, A., Aoki, W., Christlieb, N. et al. 2005, Nature, in press. Fujimoto, M. Y. 1977, , 29, 331 Fujimoto, M. Y. 1982, , 257, 767 Fujimoto, M. Y., Hanawa, T., Iben, I. Jr., & Richardson, M. B. 1984, , 278, 813 Fujimoto, M. Y., Iben, I. Jr., & Hollowell, D. 1990, , 349, 298 Fujimoto, M. Y., Sugiyama, K., Hollowell, D., & Iben, I. Jr. 1995, , 444, 175 Fujimoto, M. Y., Ikeda, Y., & Iben, I. Jr. 2000, , 529, L25 Fynbo, H. O. et al. 2005, , 433, 136 Graboske, H. C., DeWitt, H. E., Grossman, A. S., & Cooper, M. S. 1973, , 181, 457 Guenther, D. B. & Demarque, P. 1983, , 118, 262 Hansen, J. P. 1973, , 8, 3096 Hansen, J. P. & Mazighi, R. 1978, , 18, 1282 Hubbard, W. B., & Lampe, M. 1969, , 18, 297 Hollowell, D., Iben, I. Jr., & Fujimoto, M. Y. 1990, , 351, 245 Iben, I. Jr. 1965, , 141, 993 Iben, I. Jr. 1975, , 196, 525 Iben, I. Jr., Tutukov, A., V. 1984, , 282, 615 Iben, I. Jr., Fujimoto, M., Y., & MacDonald, J. 1992, , 388, 521 Iglesias, C. A., Rogers, F. J., & Wilson, B. G. 1992, , 397, 717 Iglesias, C. A. & Rogers, F. J. 1996, , 464, 943 Itoh, N., Mitake, S., Iyetomi, H., & Ichimaru, S. 1983, , 273, 774 Itoh, N., Kohyama, Y., Matsumoto, N., & Seki, M. 1984, , 280, 787 Itoh, N., Hayashi, H., Nishikawa, A., & Kohyama, Y. 1996, , 102, 411 Itoh, N., Asahara, R., Tomizawa, N., Wanajo, S., & Nozawa, S. 2004, , 611, 1041 Itoh, M. et al. 2004, Nuclear Physics A, 738, 268 Iyetomi, H. & Ichimaru, S. 1982, , 25, 2434 Kippenhahn, R. & Weigert, A. 1990, Stellar Structure and Evolution (Berlin : Springer) Komiya, Y., Suda, T., Minaguchi, H. Shigeyama, T., Aoki, W., & Fujimoto, M. Y., 2007, , 658, 367 Kurokawa, C. & Kat$\bar{o}$, K. 2005, , 71. 021301 Lucatello, S., Tsangarides, S., Beers, T. C., Carretta, E., Gratton, R. G., & Ryan, S. G. 2005 , 625, 825 Mitake, S., Ichimaru, S., & Itoh, N. 1984, , 277, 375 Munakata, H., Kohyama, Y., & Itoh, N. 1985, , 296, 197; erratum 304, 580 (1986) Nomoto, K., Thielemann, F. K., & Miyaji, S. 1985, , 149, 239 Picardi, I., Chieffi, A., Limongi, M., Pisanti, O., Miele, G., Mangano, G., & Imbriani, G. 2004 , 609, 1035 Rogers, F. J. & Iglesias, C. A. 1992, , 79, 507 Rossi, S., Beers, T.C., & Sneden, C. in ASP Conf. Ser. 165, The third Stromlo Symposium: The Galactic Halo, eds. B. K. Gibson, T. S. Axelrod, & . E. Putman (San Francisco), 264. Schlattl, H., Cassisi, S., Salaris, M., & Weiss, A. 2001, , 559, 1082 Schlattl, H., Salaris, M., Cassisi, S., & Weiss, A. 2002, , 395, 77 Schwarzschild, M. & Härm, R. 1965, , 142, 855 Siess, L., Livio, M., & Lattanzio, J. 2002, , 570, 329 Slattery, W. L., Doolen, G. D., & DeWitt, H. E. 1980, , 14, 840 Slattery, W. L., Doolen, G. D., & DeWitt, H. E. 1982, , 26, 2255 Straniero, O. 1988, , 76, 157 Suda, T. 2003, PhD. Thesis Suda, T., Aikawa, M., Machida, M. N., Fujimoto, M. Y., & Iben, I. Jr. 2004, , 611, 476 Sugimoto, D., & Fujimoto, M. Y. 1978, , 30, 467 Thielemann, F. K., Arnould, M., & Truran, J. W. 1987, in Advances in Nuclear Astrophysics, ed. E. Vangioni-Flam, J. Audouze, M. Casse, J.-P. Chieze, & J. Tran Thanh Van(Gif-sur-Yvette: Ed. Frontières), 525 Wagner, R. L., 1974, , 191, 173 Weiss, A., Cassisi, S., Schlattl, H., & Salaris, M. 2000, , 553, 413 Weiss, A., Schlattl, H., Salaris, M., & Cassisi, S. 2004, , 422, 217 ------- ---------------------------- --------------------------------------------------------------------- -------------------------------------------- ---------------------------------------------------------------------- --------------------------------------------- ----------------------------------- ------------------------------------ Model $M$ $\log {\ensuremath{T_{\textrm{eff}}}}{\ensuremath^{{\textrm{TO}}}}$ $\log L_{s} {\ensuremath^{{\textrm{TO}}}}$ $\log {\ensuremath{T_{\textrm{eff}}}}{\ensuremath^{{\textrm{tip}}}}$ $\log L_{s} {\ensuremath^{{\textrm{tip}}}}$ $t {\ensuremath_{{\textrm{TO}}}}$ $t {\ensuremath_{{\textrm{tip}}}}$ (${\ensuremath{M_\odot}}$) (K) ($M_{\odot}$) (K) ($L_{\odot}$) (Gyr) (Gyr) 08nac 0.8 3.833 0.4221 3.697 2.425 14.03 15.84 09nac 0.9 3.866 0.6559 3.698 2.411 9.31 10.45 10nac 1.0 3.909 0.8589 3.700 2.401 6.53 7.28 11nac 1.1 3.952 1.0403 3.703 2.369 4.77 5.28 12nac 1.2 3.987 1.1991 3.858 1.870 3.76 4.17 08cf 0.8 3.831 0.4126 3.696 2.446 14.03 15.88 09cf 0.9 3.864 0.6589 3.697 2.438 9.35 10.47 10cf 1.0 3.908 0.8613 3.700 2.413 6.54 7.28 11cf 1.1 3.949 1.0398 3.703 2.386 4.77 5.28 12cf 1.2 3.986 1.2086 3.852 1.872 3.61 4.00 ------- ---------------------------- --------------------------------------------------------------------- -------------------------------------------- ---------------------------------------------------------------------- --------------------------------------------- ----------------------------------- ------------------------------------ : Evolutionary Characteristics of Low-Mass, Population III Models with Up-to-Date Input Physics[]{data-label="tab:model"} ------- ---------------------------------------- ------------------------------------ --------------------------------------------------------------------- ------------------------------------------------------------------- --------------------------------------------------------------------- --------------------- ------------------------------------------- Model $M_{1} {\ensuremath^{{\textrm{max}}}}$ $M {\ensuremath_{{\textrm{BCS}}}}$ ${\ensuremath{\log L_{\textrm{He}}}}{\ensuremath^{{\textrm{max}}}}$ $T {\ensuremath_{{\textrm{BCS}}}} {\ensuremath^{{\textrm{max}}}}$ ${\ensuremath{\log L_{\textrm{He}}}}{\ensuremath^{{\textrm{mix}}}}$ $\Delta t^{\prime}$ $\Delta t {\ensuremath_{{\textrm{mix}}}}$ (${\ensuremath{M_\odot}}$) (${\ensuremath{M_\odot}}$) ($L_{\odot}$) (10$^{6}$K) ($L_{\odot}$) (yr) (yr) 08nac 0.5113 0.3579 9.913 244.5 9.263 750 [${6.07} \times 10^{-3}$]{} 09nac 0.5078 0.3480 9.876 241.1 9.166 813 [${7.51} \times 10^{-3}$]{} 10nac 0.5028 0.3320 9.834 237.1 9.038 846 [${9.81} \times 10^{-3}$]{} 11nac 0.4929 0.2986 9.722 226.9 8.744 953 [${1.82} \times 10^{-2}$]{} 12nac 0.3840 0.0 3.829 126.8 - - - 08cf 0.5151 0.3717 9.882 248.6 9.264 715 [${5.91} \times 10^{-3}$]{} 09cf 0.5121 0.3629 9.858 245.3 9.231 749 [${6.49} \times 10^{-3}$]{} 10cf 0.5054 0.3390 9.831 239.4 9.026 858 [${9.99} \times 10^{-3}$]{} 11cf 0.4966 0.3115 9.728 231.1 8.776 941 [${1.69} \times 10^{-2}$]{} 12cf 0.3829 0.0 3.393 124.2 - - - ------- ---------------------------------------- ------------------------------------ --------------------------------------------------------------------- ------------------------------------------------------------------- --------------------------------------------------------------------- --------------------- ------------------------------------------- : Characteristics of Helium Core Flash of Model Stars with Up-to-Date Input Physics[]{data-label="tab:he-flash"} [^1]: The computation of FIH90 was done at the University of Tokyo Observatory in 1984 when one of authors (I.I., jr.) visited it as JSPS fellow.

--- abstract: 'Correlations between the QCD coupling $\alpha_s$, the gluon condensate $\la \alpha_s G^2\ra$, and the $c,b$-quark running masses $\overline{m}_{c,b}$ in the $\overline{MS}$-scheme are explicitly studied (for the first time) from a global analysis of the (axial-)vector and (pseudo)scalar charmonium and bottomium spectra using optimized ratios of Laplace sum rules (LSR) evaluated at the $\mu$-subtraction stability point where PT @N2LO, N3LO, $\la \alpha_s G^2\ra$ @NLO and LO $D=6-8$-dimensions non-perturbative condensates corrections are included. Our results clarify the (apparent) discrepancies between different estimates of $\la \alpha_s G^2\ra$ from $J/\psi$ sum rule but also shows the sensitivity of the sum rules on the choice of the $\mu$-subtraction scale which does not permit a high-precision estimate of $\overline{m}_{c,b}$. We obtain from the (axial-)vector \[resp. (pseudo)scalar\] channels: $\la\alpha_s G^2\ra=(8.5\pm3.0)$ \[resp. $(6.34\pm0.39)]\times 10^{-2}$ GeV$^4$ , $\overline{m}_c(\overline{m}_c)= 1256(30)$ \[resp. 1266(16)\] MeV and $\overline{m}_b(\overline{m}_b)=4192(15)$ MeV. Combined with our recent determinations from vector channel, one can deduce the average: $\overline{m}_c(\overline{m}_c)\vert_{\rm average}= 1263(14)$ MeV and $\overline{m}_b(\overline{m}_b)\vert_{\rm average}=4184(11)$ MeV. Adding the two above values of the gluon condensate to previous estimates in Table\[tab:g2\], one obtains the [*new sum rule average:*]{} $\la\alpha_s G^2\ra\vert_{\rm average}=(6.35\pm 0.35)\times 10^{-2}$ GeV$^4$. The mass-splittings $M_{\chi_{0c(0b)}}-M_{\eta_{c(b)}}$ give @N2LO: $\alpha_s(M_Z)=0.1183(19)(3)$ in good agreement with the world average (see more detailed discussions in the section: addendum).' address: | Laboratoire Univers et Particules , CNRS-IN2P3,\ Case 070, Place Eugène Bataillon, 34095 - Montpellier Cedex 05, France.\ Email address: snarison@gmail.com author: - Stephan Narison title: 'QCD Parameters Correlations from Heavy Quarkonia[^1]' --- \ Introduction ============ ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Sources $\la \alpha_s G^2\ra\times 10^{2}$ \[GeV$^4$\] References ------------------------------------- ---------------------------------------------------------------------------- --------------------------------------------------------------------------- -- -- -- -- $q^2=0$-moments $4\pm 2$ SVZ 79 [@SVZa; @SVZb] (guessed error) $q^2\not=0$-moments $5.3\pm 1.2$ RRY 81-85 [@RRY] – $9.2\pm 3.4$ Miller-Olssson 82 [@OLSSON] – $\approx 6.6^*$ Broadhurst et al. 94 [@BAIKOV] – $2.8\pm 2.2 $ Ioffe-Zyablyuk 07[@IOFFEa; @IOFFEb] – $7.0\pm 1.3$ Narison 12a[@SNcb2] Exponential $12\pm 2$ Bell-Bertlmann 82 [@BELLa; @BELLb; @BERTa; @BERTb; @BERTc; @BERTd; @NEUF] – $17.5\pm 4.5$ Marrow et al. 87[@SHAW] – $7.5\pm 2.0$ Narison 12b[@SNcb3] Exponential $M_\psi-M_{\eta_c}$ $10\pm 4$ Narison 96[@SNHeavy; @SNHeavy2] Exponential $M_{\chi_b}-M_\Upsilon$ $6.5\pm 2.5$ Narison 96[@SNHeavy; @SNHeavy2] Non-rel. moments $5.5\pm 3$ Yndurain 99[@YND] Exponential $0.9\sim 6.6^*$ Eidelman et al. 79[@EID] Ratio of Exponential $4\pm1$ Launer et al. 84[@LNT] FESR $13\pm 6$ Bertlmann et al. 88[@PEROTTETa; @PEROTTETb] Infinite norm $1 \sim 30^*$ Causse-Mennessier[@MENES] $\tau$-like decay $7\pm 1$ Narison 95[@SNIa; @SNIb] Axial spectral function $6.9\pm 2.6$ Dominguez-Sola 88[@SOLA] [****]{} **& [**Prior 2017**]{}\ &\ ALEPH collaboration &$6.3\pm 1.2$ & Duflot 95[@DUFLOT]\ CLEO II collaboration& $2.4\pm1.0$& Duflot 95[@DUFLOT]\ OPAL collaboration &$-0.9\sim +4$& Ackerstaff et al. 99[@OPAL]\ ALEPH collaboration& $-5\sim +6$& Schael et al. 05[@ALEPH]\ ALEPH collaboration&$-12\sim -0.6$& Davier et al. 14[@DAVIER]\ &\ [O]{}($\alpha_s^{12}$) & $\approx 13$& Rakow 05[@RAKOW; @BURGIO; @HORLEY]\ [O]{}($\alpha_s^{35}$) & $\approx 27$& Bali-Pineda 15[@BALIa; @BALIb]\ Average plaquette &$\approx 44$&Lee 14[@LEE]\ ** ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- \[tab:g2\] Gluon condensates introduced by SVZ[@SVZa; @SVZb; @ZAKA] play important rôle in gluodynamics and in the QCD spectral sum rules analysis where they enter as high-dimension operators in the OPE of the hadronic correlators. In particular, this is the case for the heavy quark systems and the pure Yang-Mills gluonia/glueball channels[@NSVZ; @VENEZIA; @SNG] where the light quark loops and condensates are absent to leading order. The heavy quark condensate contribution can be absorbed into the gluon one through the relation [@SVZa; @SVZb]: |QQ=-[\_s G\^2/ (12M\_Q)]{}+.... where a similar relation holds for the mixed heavy quark-gluon condensate $\la \bar QGQ\ra$ . $G$ is the short hand notation for the gluon field strength $G^a_{\mu\nu}$ and $M_Q$ is the pole mass. The SVZ orignal value[@SVZa; @SVZb]: \_s G\^20.04  [GeV]{}\^4 , \[eq:standard\] extracted (for the first time) from charmonium sum rules [@SVZa; @SVZb] has been challenged by different authors (for reviews, see e.g [@SNB1; @SNB2; @SNB3; @SNB4] and Table\[tab:g2\]). One can see in Table \[tab:g2\] that the results from standard SVZ and FESR sum rules for heavy and light quark systems vary in a large range but all of them are positive numbers, while the ones from analysis of the modified $\tau$-decays moments allow negative values. However, one should notice from the original QCD expression of the $\tau$-decay rate[@BNPa; @BNPb] that the $\la\alpha_s G^2\ra$ gluon condensate contribution is absent to leading order indicating that it is a bad place for extracting a such quantity[@SNTAU]. The presence of $\la\alpha_s G^2\ra$ in the analysis of [@DUFLOT; @OPAL; @ALEPH; @DAVIER] is only an aritfact of the high-moments where the systematic errors needs to be better controlled. Earlier lattice calculations indicate a non-zero positive value of $\la\alpha_s G^2\ra$[@GIACOa; @GIACOb; @GIACOc; @GIACOd] while recent estimates in Table\[tab:g2\] give positive values about 2-7 times higher than the phenomenological estimates. However, the subtraction of the perturbative contribution in the lattice analysis which is scheme dependent is not yet well-understood[@LEE] and does not permit a direct comparison of the lattice results obtained at large orders of PT series with the ones from the truncated PT series used in the phenomenological analysis. These previous results indicate that $\la\alpha_s G^2\ra$ is not yet well determined and motivate a reconsideration of its estimate. A first step for the improvement of the estimate of the gluon condensate was the recent direct determination of the ratio of the dimension-six gluon condensate $\la g^3f_{abc} G^3\ra$ over $\la\alpha_s G^2\ra$ from the heavy quark systems with the value[@SNcb1; @SNcb2; @SNcb3]: g\^3f\_[abc]{} G\^3/ \_s G\^2=(8.21.0) [GeV]{}\^2, \[eq:rcond\] which differs significantly from the instanton model estimate[@NIKOL2; @SHURYAK; @IOFFE2] and may question the validity of this approximation. Earlier lattice results in pureYang-Mills found: $\rho\approx 1.2$ GeV$^2$[@GIACOa; @GIACOb; @GIACOc; @GIACOd] such that it is important to have new lattice results for this quantity. Note however, that the value given in Eq.\[eq:rcond\] might also be an effective value of the unknown high-dimension condensates not taken into account in the analysis of [@SNcb1; @SNcb2; @SNcb3] when requiring the fit of the data by the truncated OPE at that order. We shall see that the effect of this term is a small correction at the stability region where the optimal results are extracted. In this paper, we pursue a such program by reconsidering the extraction of the lowest dimension QCD parameters from the (axial-)vector and (pseudo)scalar charmonium and bottomium spectra taking into account the correlations between $\alpha_s$, the gluon condensate $\la \alpha_s G^2\ra$, and the $c,b$-quark running masses. We shall use these parameters for predicting the known masses of the (pseudo)scalar heavy quarkonia ground states and also re-extract $\alpha_s$ and $\la \alpha_s G^2\ra$ from the mass-splittings $M_{\chi_{0c(0b)}}-M_{\eta_{c(b)}}$. In so doing, we shall work with the example of the QCD Laplace sum rules (LSR) where the corresponding Operator Product Expansion(OPE) in terms of condensates is more convergent than the moments evaluated at small momentum. The QCD Laplace sum rules ========================= Form of the sum rule {#form-of-the-sum-rule .unnumbered} -------------------- We shall work with the Finite Energy version of the QCD Laplace sum rules (LSR) and their ratios: \^c\_n=\_[4m\_Q\^2]{}\^[t\_c]{} dt  t\^n e\^[-t]{}[Im]{}\_[V(A)]{}(t) ,\^c\_n()= , where $\tau$ is the LSR variable, $t_c$ is the threshold of the “QCD continuum” which parametrizes, from the discontinuity of the Feynman diagrams, the spectral function ${\rm Im}\Pi_{V(A)}(t,m_Q^2,\mu)$ associated to the transverse part $\Pi_{V(A)}(q^2,m_Q^2,\mu)$ of the two-point correlator: \^\_[V(A)]{}(q\^2)&& i d\^4x  e\^[-iqx]{}0 J\^\_[V(A)]{}(x)J\^\_[V(A)]{}(0)\^0\ &=& -g\^q\^2-q\^q\^\_[V(A)]{}(q\^2)+q\^q\^\^[(0)]{}\_[V(A)]{} (q\^2), \[eq:2-point\] where : $J_{V(A)}^\mu(x)=\bar Q \gamma^\mu(\gamma_5) Q(x)$ is the heavy quark local vector (axial-vector) current. In the (pseudo)scalar channel associated to the local current $J_{S(P)}=\bar Q i(\gamma_5) Q(x)$, we work with the correlator: \_[S(P)]{}(q\^2)=i d\^4x  e\^[-iqx]{}0 J\_[S(P)]{}(x)J\_[S(P)]{}(0)\^0, \[eq:2-pseudo\] which is related to the longitudinal part $\Pi^{(0)}_{V(A)}(q^2)$ of the (axial-)vector one through the Ward identity[@SNB1; @SNB2; @BECCHI]: q\^2 \^[(0)]{}\_[A(V)]{}(q\^2)=\_[P(S)]{}(q\^2)-\_[P(S)]{}(0) . Working with $\Psi_{P(S)}(q^2)$ is safe as $\Psi_{P(S)}(0)$ should affect the $Q^2$-moments and the exponential sum rules derived from $\Pi^{(0)}_{A(V)}(q^2)$ which is not accounted for in e.g [@RRY; @BERTb; @SHAW] . Originally named Borel sum rules by SVZ because of the appearance of a factorial suppression factor in the non-perturbative condensate contributions into the OPE, it has been shown by [@SNR] that the PT radiative corrections satisfy instead the properties of an inverse Laplace sum rule though the present given name here. Parametrisation of the spectral function {#parametrisation-of-the-spectral-function .unnumbered} ---------------------------------------- ${\rm Im}\Pi_V(t)$ is related to the ratio $R_{e^+e^-}$ of the total cross-section of $\sigma(e^+e^-\to$ hadrons) over $\sigma(e^+e^-\to \mu^+\mu^-)$ through the optical theorem. Expressed in terms of the leptonic widths and meson masses, it reads in a narrow width approximation (NWA): R\_[e\^+e\^-]{}12 \_V(t) =\_M\_[V]{}\_[Ve\^+e\^-]{}(t-M\_V\^2, where $M_{V}$ and $\Gamma_{V\to e^+e^-}$ are the mass and leptonic width of the $J/\psi$ or $\Upsilon$ mesons; $Q_V=2/3 (-1/3)$ is the charm (bottom) electric charge in units of $e$; $\alpha=1/133$ is the running electromagnetic coupling evaluated at $M^2_{V}$. We shall use the experimental values of the $J/\psi$ and $\Upsilon$ parameters compiled by PDG[@PDG]. We include the contributions of the $\psi(3097)$ to $\psi(4415)$ and $\Upsilon(9460)$ to $\Upsilon(11020)$ within NWA. The high-energy part of the spectral function is parametrized by the “QCD continuum" from a threshold $t_c$ (we use $\sqrt{t_c^c}=4.6$ GeV and $\sqrt{t_c^b}$= 11.098 GeV just above the last resonance). In the case of the axial-vector and (pseudo)scalar channels where there are no complete data, we use the duality ansatz: {(t);(t)}f\_H\^2 M\_H\^[{0;2}]{} (t-M\_H\^2) +(t-t\_c) “QCD [continuum]{}"; where $M_H$ and $f_H$ are the lowest ground state mass and coupling analogue to $f_\rho$ and $f_\pi$. This implies : \^[c]{}\_nM\_H\^2 , indicating that the ratio of moments appears to be a useful tool for extracting the masses of hadrons [@SNB1; @SNB2; @SNB3; @SNB4]. We shall work with the lowest ratio of moments $ {\cal R}^{c}_0$. Exponential sum rules have been used successfully by SVZ for light quark systems [@SVZa; @SVZb; @SNB1; @SNB2; @SNB3; @SNB4] and extensively by Bell and Bertlmann for heavy quarkonia in their relativistic and non-relativistic versions [@BELLa; @BELLb; @BERTa; @BERTb; @BERTc; @BERTd; @NEUF; @SHAW; @SNHeavy; @SNHeavy2]. QCD Perturbative expressions @N2LO {#qcd-perturbative-expressions-n2lo .unnumbered} ---------------------------------- The perturbative QCD expression of the vector channel is deduced from the well-known spectral function to order $\alpha_s$ within the on-shell renormalization scheme [@KALLEN; @SCHWINGER]. The one of the axial-vector current has been obtained in[@SCHILCHER; @BROAD; @GENER; @RRY]. To order $\alpha_s^2$ (N2LO), the spectral functions are usually parametrized as: R\^[(2)]{}C\_F\^2R\^[(2)]{}\_A+C\_AC\_FR\^[(2)]{}\_[NA]{}+C\_FT\_Qn\_lR\^[(2)]{}\_l +C\_FT\_Q(R\^[(2)]{}\_F+ R\^[(2)]{}\_S+R\^[(2)]{}\_G) , which are respectively the abelian (A), non-abelian (NA), massless (l) and heavy (F) internal quark loops, singlet (S) and double bubble gluon (G) contributions. $C_F=4/3,$ $ C_A=3, T_Q=1/2$ are usual SU(3) group factors and $n_l$ is the number of light quarks. We use the (approximate) but complete result in the on-shell scheme given by[@CHET0] for the abelian and non-abelian contributions. The one from light quarks comes from[@TEUBNER; @HOANGa; @CHET1]. The one from heavy fermion internal loop comes from[@CHET2] for the vector current while the one from the axial current is (to our knowledge) not available. The singlet one due to double triangle loop comes from[@CHET3]. The one from the gluonic double-bubble reconstructed from massless fermions comes from[@TEUBNER; @HOANGa; @CHET2]. The previous on-shell expressions are transformed into the $\overline{MS}$-scheme through the relation between the on-shell $M_Q$ and running $ \overline{m}_Q(\mu)$ quark masses[@SNB1; @SNB2; @SNB3; @SNB4; @TAR; @COQUEa; @COQUEb; @SNPOLEa; @SNPOLEb; @BROAD2a; @AVDEEV; @BROAD2b; @CHET2a; @CHET2b] @N2LO: M\_Q &=& \_Q() 1+ a\_s+ (16.2163 -1.0414 n\_l)a\_s\^2\ &&+[ln]{} a\_s+(8.8472 -0.3611 n\_l) a\_s\^2+[ln]{}\^21.7917 -0.0833 n\_la\_s\^2+, \[eq:pole\] for $n_l$ light flavours where $\mu$ is the arbitrary subtraction point and $a_s\equiv \alpha_s/\pi$, ln$\equiv \ln{\ga{\mu}/{ M_Q}\dr^2}$. QCD Non-Perturbative expressions @LO {#qcd-non-perturbative-expressions-lo .unnumbered} ------------------------------------ Using the OPE à la SVZ, the non-perturbative contributions to the two-point correlator can be parametrized by the sum of higher dimension condensates: (t)=C\_[2n]{}(t,m\^2,\^2))O\_[2n]{} : n=1,2,... where $C_{2n}$ are Wilson coefficients calculable perturbatively and $ \la O_{2n}\ra$ are non-perturbative condensates. In the exponential sum rules, the order parameter is the sum rule variable $\tau$ while for the heavy quark systems the relevant condensate contributions at leading order in $\alpha_s$ are the gluon condensate $\la \alpha_s G^2\ra$ of dimension-four[@SVZa; @SVZb], the dimension-six gluon $ \la g^3f_{abc} G^3\ra $ and light four-quark $\alpha_s\la\bar uu\ra^2 $ condensates [@NIKOLa; @NIKOLb]. The condensates of dimension-8 entering in the sum rules are of seven types[@NIKOL2]. They can be expressed in different basis depending on how each condensate is estimated (vacuum saturation[@NIKOL2] or modified vacuum saturation[@BAGAN]). Our estimate of these D=8 condensates is the same as in[@SNcb2]. For the vector channel, we use the analytic expressions of the different condensate contributions given by Bertlmann[@BERTb]. We shall not include the eventual $D=2$ coperator induced by a tachyonic gluon mass[@ZAK1; @CNZ1] as it is dual to the contribution of large order terms[@SNZ], which we estimate using a geometric growth of the PT series. In various examples, its contribution is numerically negligible[@SND2]. Initial QCD input parameters {#initial-qcd-input-parameters .unnumbered} ----------------------------- In the first iteration, we shall use the following QCD input parameters: \_s(M\_)&=&0.325\^[+0.008]{}\_[-0.016]{}  ,        \_s G\^2= (0.070.04)  [GeV]{}\^4.\ \_c(\_c)&=& (126117)  [MeV]{}  , \_b(\_b)= (417711)  [MeV]{}  , \[eq:param\] \[eq:mcmom\] The central value of $\alpha_s$ comes from $\tau$-decay[@SNTAU; @PICHTAUa; @PICHTAUb; @BETHKEa; @BETHKEb; @BETHKEc]. The range covers the one allowed by PDG[@PDG; @BETHKEa; @BETHKEb; @BETHKEc] (lowest value) and the one from our determination from $\tau$-decay (highest value). The values of $\overline{m}_{c,b}(\overline{m}_{c,b})$ are the average from our recent determinations from charmonium and bottomium sum rules [@SNcb1; @SNcb2]. The value of $\la\alpha_s G^2\ra$ almost covers the range from different determinations mentioned in Table\[tab:g2\] and reviewed in[@SNB1; @SNB2; @SNHeavy; @SNHeavy2]. We shall use the ratio of condensates given in Eq.\[eq:rcond\]. For the light four-quark condensate, we shall use the value: \_s|uu\^2=(5.81.8)10\^[-4]{} [GeV]{}\^6 , obtained from the original $\tau$-decay rate[@SNTAU] where the gluon condensate does not contribute to LO[@BNPa; @BNPb] and by some other authors from the light quark systems[@SNB1; @SNB2; @LNT; @JAMI2a; @JAMI2b; @JAMI2c] where a violation by a factor about 3–4 of the vacuum saturation assumption has been found. Charmonium Ratio of Moments $ {\cal R}_{J/\psi(\chi_{c1})}$ ============================================================ Convergence of the PT series {#convergence-of-the-pt-series .unnumbered} ---------------------------- \ ![Behaviour of the ratio of moments ${\cal R}$ versus $\tau$ in GeV$^{-2}$ at different orders of perturbation theory. The input and the meaning of each curve are given in the legends: [**a)**]{} $J/\psi$ and [**b)**]{} $\chi_{c1}$.[]{data-label="fig:pertc"}](psi-pert.pdf "fig:"){width="14cm"}\ \ ![Behaviour of the ratio of moments ${\cal R}$ versus $\tau$ in GeV$^{-2}$ at different orders of perturbation theory. The input and the meaning of each curve are given in the legends: [**a)**]{} $J/\psi$ and [**b)**]{} $\chi_{c1}$.[]{data-label="fig:pertc"}](chi1c-pert.pdf "fig:"){width="14cm"} In so doing, we shall work with the renormalized (but non-resummed renormalization group) perturbative (PT) expression where the subtraction point $\mu$ appears explicitly. We include the known N2LO terms. The $D=(6)8$ condensates contributions are included for the (axial-)vector current. The value of $\sqrt{t_c}=4.6$ GeV is chosen just above the $\psi(4040)$ mass for the vector current where the sum of all lower mass $\psi$ state contributions are included in the spectral function. For the axial current, we use (as mentioned) the duality ansatz and leave $t_c$ as a free parameter which we shall fix after an optimisation of the sum rule. We evaluate the ratio of moments at $\mu=2.8$ GeV and for a given value of $t_c=20$ GeV$^2$ for the $\chi_{c1}$ around which they will stabilize (as we shall show later on). The analysis is illustrated in Fig.\[fig:pertc\]. On can notice the importance of the N2LO contribution which is dominated by the abelian and non-abelian contributions. The N2LO effects go towards the good direction of the values of the experimental masses. LSR variable $\tau$-stability and Convergence of the OPE {#lsr-variable-tau-stability-and-convergence-of-the-ope .unnumbered} --------------------------------------------------------- The OPE is done in terms of the exponential sum rule variable $\tau$. We show in Fig.\[fig:npertc\] the effects of the condensates of different dimensions. One ca notice that the presence of condensates are vital for having $\tau$-stabilities which are not there for the PT-terms alone. The $\tau$-stability is reached for $\tau\simeq 0.6$ GeV$^{-2}$. At a given order of the PT series, the contributions of the $D=8$ condensates are negligible at the $\tau$-stability region while the $D=6$ contribution goes again to the right track compared with the data. \ ![The same as in Fig.\[fig:pertc\] but for different truncation of the OPE: [**a)**]{} $J/\psi$ and [**b)**]{} $\chi_{c1}$.[]{data-label="fig:npertc"}](psi-npert.pdf "fig:"){width="14cm"}\ \ ![The same as in Fig.\[fig:pertc\] but for different truncation of the OPE: [**a)**]{} $J/\psi$ and [**b)**]{} $\chi_{c1}$.[]{data-label="fig:npertc"}](chi1c-npert.pdf "fig:"){width="14.5cm"} ![Behaviour of the ratio of moments ${\cal R}_{\chi_{c1}}$ versus $\tau$ in GeV$^{-2}$. The input and the meaning of each curve are given in the legend.[]{data-label="fig:chi1c-tau"}](chi1c-tau.pdf){width="14cm"} Continuum threshold $t_c$-stability for $ {\cal R}_{\chi_{c1}}$ {#continuum-threshold-t_c-stability-for-cal-r_chi_c1 .unnumbered} ---------------------------------------------------------------- We show the analysis in Fig.\[fig:chi1c-tau\] where the curves correspond to different $t_c$-values. We find nice $t_c$-stabilities where we take the value : t\_c(17\~22)  [GeV]{}\^2 , \[eq:tc-chi\] where the lowest value corresponds to the phenomenological estimate $M_{\chi_{c1}}(2P)-M_{\chi_{c1}}(1P)\approx M_{\psi}(2S)-M_{\psi}(1S)$ while the higher one corresponds to the beginning of $t_c$-stability. This range of $t_c$-values induces an error of about 8 MeV in the meson mass determination. ![Behaviour of the ratio of moments ${\cal R}_{J/\psi}$ and ${\cal R}_{\chi_{c1}}$ versus $\mu$ for $t_c=20$ GeV$^2$. The inputs and the meaning of each curve are given in the legends.[]{data-label="fig:muc"}](muc.pdf){width="8cm"} Subtraction point $\mu$-stability {#subtraction-point-mu-stability .unnumbered} ---------------------------------- The subtraction point $\mu$ is an arbitrary parameter. It is popularly taken between 1/2 and 2 times an “ad hoc" choice of scale. However, the physical observables should be not quite sensitive to $\mu$ even for a truncated PT series. In the following, like in the previous case of external (unphysical) variable, we shall fix its value by looking for a $\mu$-stability point if it exists at which the observable will be evaluated. This procedure has been used recently for improving the LSR predictions on molecules and four-quark charmonium and bottoming states[@SNSU3; @SNCHI; @SNCHI2; @SNX; @X3A]. Taking here the example of the ratios of moments, we show in Fig.\[fig:muc\] their $\mu$- dependence. We notice that $ {\cal R}_{\psi}$ is a smooth decreasing function of $\mu$ while $ {\cal R}_{\chi_{c1}}$ presents a slight stability at : =(2.8\~2.9) [GeV]{}, \[eq:muc\] at which we shall evaluate the two ratios of moments. On can notice that at a such higher scale, one has a better convergence of the $\alpha_s(\mu)$ PT series. ![Correlation between $\la\alpha_s G^2\ra$ and $\overline{m}_c(\overline{m}_c)$ for the range of $\alpha_s$ values given in Eq.\[eq:param\] and for $\mu$ given in Eq.\[eq:mub\].[]{data-label="fig:mc-g2"}](mc-g2.pdf){width="12cm"} Correlations of the QCD parameters {#correlations-of-the-qcd-parameters .unnumbered} ---------------------------------- Once fixed these preliminaries, we are now ready to study the correlation between $\alpha_s$, the gluon condensate $\la \alpha_s G^2\ra$, and the $c$-quark running masses $\overline{m}_{c}(\overline{m}_{c})$. In so doing we request that the $ \sqrt{\cal R}_{J/\psi}$ sum rule reproduces within (2-3) MeV accuracy the experimental measurement, while the $\chi_{c1}$ mass is reproduced within (8–10) MeV which is the error induced by the choice of $t_c$ in Eq.\[eq:tc-chi\]. The results of the analysis are obtained at the $\tau$-stability points which are about 1.1 (resp. 0.6) GeV$^{-2}$ for the ${J/\psi}$ (resp. $\chi_{c1}$) channels. They are shown in Fig.\[fig:mc-g2\] for the two values of $\mu$ given in Eq.\[eq:muc\]. One can notice that, $\la \alpha_s G^2\ra$ decreases smoother from the $\chi_{c1}$ (grey region) than from the $J/\psi$ sum rule when $\overline{m}_{c}$ increases. In the $J/\psi$ sum rule, it moves from 0.15 to 0.02 GeV$^4$ for $\overline{m}_{c}(\overline{m}_{c})$ varying from 1221 to 1301 MeV. This feature may explain the apparent discrepancy of the results reviewed in the introduction from this channel. One should notice that the results from the $J/\psi$ sum rules are quite sensitive to the choice of the subtraction point (no $\mu$-stability) which then does not permit accurate determinations of $\la \alpha_s G^2\ra$ and $\overline{m}_{c}(\overline{m}_{c})$. Some accurate results reported in the literature for an “ad hoc " choice of $\mu$ may be largely affected by the $\mu$ variation. One can also see from Fig.\[fig:mc-g2\] that within the alone $J/\psi$ sum rule the values of $\la \alpha_s G^2\ra$ and $\overline{m}_{c}(\overline{m}_{c})$ cannot be strongly constrained[^2]. Once the constraint from the $\chi_{c1}$ sum rule is introduced, one obtains a much better selection. Taking as a conservative result the range covered by the change of $\mu$ in Eq.\[eq:muc\], one deduces: \_s G\^2= (8.53.0)10\^[-2]{} [GeV\^4]{},        \_[c]{}(\_[c]{})=(125630) [MeV]{}. \[eq:respsi\] We improve this determination by including the N3LO PT[@N3LO] corrections and NLO $\la \alpha_s G^2\ra$ gluon condensate (using the parametrization in [@IOFFEa; @IOFFEb]) contributions [@BAIKOV]. The effects of these quantities on $\sqrt{{\cal R}_{J/\psi}}$ and $\sqrt{{\cal R}_{\chi_{c1}}}$ is about $(1\sim 2)$ MeV at the optimization scales which induces a negligible change such that the results quoted in Eq.\[eq:respsi\] remain the same @N3LO PT and @NLO gluon condensate approximations. This value of $\la \alpha_s G^2\ra$ is in good agreement with the one $ (7.5\pm 2.0)\times 10^{-2}~{\rm GeV^4}$ from our previous analysis of the charmonium Laplace su rules using resummed PT series[@SNcb3] indicating the self-consistency of the results. However, these results do not favor lower ones quoted in Table\[tab:g2\]. Taking the weighted average of different sum rule determinations given in Table\[tab:g2\] with the new result in Eq.\[eq:respsi\], we obtain the [*sum rule average*]{}: \_s G\^2\_[average]{} = (6.300.45)10\^[-2]{} [GeV\^4]{}, \[eq:g2average\] where the error may be optimistic but comparable with the one of the most precise predictions given in Table\[tab:g2\]. These results agree within the errors within our recent estimates of $\la \alpha_s G^2\ra$ and $\overline{m}_{c}(\overline{m}_{c})$[@SNcb1; @SNcb2; @SNcb3] obtained from the moments and their ratios subtracted at finite $Q^2=n\times 4m_c^2$ with $n=0,1,2.$ and from the heavy quark mass-splittings[@SNHeavy; @SNHeavy2]. Hereafter, we shall use the value of $\la \alpha_s G^2\ra$ in Eq.\[eq:g2average\]. ![Behaviour of the ratio of moments ${\cal R}_{\Upsilon}$ versus $\tau$ in GeV$^{-2}$ for different truncation of the OPE. The input and the meaning of each curve are given in the legend.[]{data-label="fig:upsilon-tau"}](upsilon-tau.pdf){width="13cm"} ![Behaviour of the ratio of moments ${\cal R}_{\chi_{b1}}$ versus $\tau$ for different values of $t_c$. The input and the meaning of each curve are given in the legend.[]{data-label="fig:chi1b-tc"}](chi1b-tc.pdf){width="13.7cm"} Bottomium Ratios of Moments $ {\cal R}_{\Upsilon(\chi_{b1})}$ ============================================================== $\tau$ and $t_c$-stabilities and test of convergences {#tau-and-t_c-stabilities-and-test-of-convergences .unnumbered} ----------------------------------------------------- The analysis is very similar to the previous $J/\psi$ sum rule. The relative perturbative and non-perturbative contributions are very similar to the curves in Figs.\[fig:pertc\] to \[fig:npertc\]. We use the value: $ \mu=9.5$ GeV which we shall justify later on. However, it is informative to show in Fig.\[fig:upsilon-tau\] the $\tau$-behaviour of ${\cal R}_{\Upsilon}$ for different truncation of the OPE where $\tau$-stability is obtained at $\tau\simeq 0.22$ GeV$^{-2}$. In Fig.\[fig:chi1b-tc\], we show the $\tau$-behaviour of ${\cal R}_{\chi_{b1}}$ for different values of $t_c$ from which we deduce a stability at $\tau\simeq 0.28$ GeV$^{-2}$ and $t_c$-stability which we shall take to be $\sqrt{t_c}\simeq 11$ GeV. A much better convergence of the $\alpha_s$ series is observed as the sum rule is evaluated at a higher scale $\mu$. The OPE converges also faster as $\tau$ is smaller here. $\mu$-stability {#mu-stability .unnumbered} --------------- The two sum rules are smooth decreasing functions of $\mu$ but does not show $\mu$-stability. Instead, their difference presents $\mu$-stability at: (9\~10) [GeV]{}, \[eq:mub\] as shown in Fig.\[fig:delta\] at which we choose to evaluate the two sum rules. ![Behaviour of $M_{\chi_{b1}}-\sqrt{{\cal R}_{\Upsilon}}$ versus $\mu$.[]{data-label="fig:delta"}](delta.pdf){width="12.cm"} ![Behaviour of $\Delta\alpha_s(M_\tau)$ versus $\overline{m}_{b}(\overline{m}_{b})$from the ratio of moments ${\cal R}_{\Upsilon}$ The horizontal band corresponds to the range of $\alpha_s$ value given in Eq.\[eq:param\]. The input and the meaning of each curve are given in the legend.[]{data-label="fig:mb-alfas"}](mb-alfas.pdf){width="12.3"} Mass of $\chi_{b1}(1^{++})$ from $ {\cal R}_{\chi_{b1}}$ {#mass-of-chi_b11-from-cal-r_chi_b1 .unnumbered} -------------------------------------------------------- Using the previous value of the QCD parameters, we predict from the ratio of $\chi_{b1}$ moments: M\_[\_[b1]{}]{}9677(26)\_[t\_c]{}(8)\_[\_s]{}(11)\_[G\^2]{}(9)\_[m\_b]{}(99)\_ [MeV]{} , which is (within the error) about 100 MeV lower than the experimental mass $M_{\chi_{b1}}^{exp}=9893$ MeV. The agreement between theory and experiment may be improved when more data for higher states are available or/and by including Coulombic corrections shown to be small for the vector current (see e.g[@SNcb1]) and not considered here. Correlation between $\alpha_s(\mu)$ and $\overline{m}_{b}(\overline{m}_{b})$ from $ {\cal R}_{\Upsilon }$ {#correlation-between-alpha_smu-and-overlinem_boverlinem_b-from-cal-r_upsilon .unnumbered} --------------------------------------------------------------------------------------------------------- From the previous analysis, one can notice that the $\chi_{b1}$ channel cannot help from a precise study of the correlation between $\alpha_s$ and $\overline{m}_{b}(\overline{m}_{b})$. We show in Fig.\[fig:mb-alfas\] the result of the analysis from the $\Upsilon$ channel by requiring that the experimental value of $ \sqrt{\cal R}_{\Upsilon }$ is reproduced within $(1\sim 2)$ MeV accuracy. First, one can notice that the error due to the gluon condensate with the value given in Eq.\[eq:respsi\] is negligible. Given the range of $\alpha_s$ quoted in Eq.\[eq:param\], one can deduce the prediction: \_[b]{}(\_[b]{})=4192(15)(8)\_[coul]{} [MeV]{} , \[eq:resmb\] where we have added in Eq.\[eq:resmb\] an error of about 8 MeV from Coulombic corrections as estimated in [@SNcb3]. The previous result in Eq.\[eq:resmb\] corresponds to: \_s(M\_)=0.321(12) \_s(M\_Z)=0.1186(15)(3) \[eq:alfas-mb\] given by the range in Eq.\[eq:param\]. The running from $M_\tau$ to $M_Z$ due to the choice of the thresholds induces the last error (3). This result is consistent with the ones from moments sum rules quoted in Eq.\[eq:param\] and with [@SNcb3]: \_[b]{}(\_[b]{})=4212(32) [MeV]{} , \[eq:mblsr\] from LSR with RG resummed PT expressions. Taking the average of our three determinations, we obtain the final estimate: \_[b]{}(\_[b]{})\_[average]{}=(418411) [MeV]{} , \[eq:mbaverage\] where the errors come from the most precise determination. Due to the large errors induced by the subtraction scale as shown in Fig\[fig:mb-alfas\], one cannot accurately extract the value of $\alpha_s$ given the present value of $\overline{m}_{b}$. ![Behaviour of $M_{\eta_c}$ versus $\tau$ for different values of $t_c$.[]{data-label="fig:etac-tau"}](etac-tau.pdf){width="10.5cm"} ![Behaviour of $M_{\chi_{c0}}$ versus $\tau$ for different values of $t_c$.[]{data-label="fig:chic-tau"}](chic-tau.pdf){width="10.2cm"} (Pseudo)scalar charmonium ========================= In these channels, we shall work with the ratio of sum rules associated to the two-point correlator $\Psi_{P(S)}(q^2)$ defined in Eq.\[eq:2-pseudo\] which is not affected by $\Psi_{P(S)}(0)$. We shall use the PT expression known @N2LO[@TEUBNER; @HOANGa; @CHET1; @CHET2; @CHET3], the contribution of the gluon condensates of dimension 4 and 6 to LO[@NIKOLa; @NIKOLb]. $\eta_c$ and $\chi_{c0}$ masses {#eta_c-and-chi_c0-masses .unnumbered} ------------------------------- The $\eta_c$ sum rule shows a smooth decreasing function of $\mu$ but does not present a $\mu$-stabiity. Then, we choose the value of $\mu$ given in Eq.\[eq:muc\] for evaluating it. We show in Fig.\[fig:etac-tau\] the $\tau$-behaviour of the $\eta_c$-mass for different values of $t_c$ which we take from 10 GeV$^2$ \[around the mass squared of the $\eta_c(2P)$ and $\eta_c(3P)$\] until 13 GeV$^2$ ($t_c$-stability) . Similar analysis is done for the $\chi_{c0}$ associated to the scalar current $\bar Q(i)Q$ which is shown in Fig.\[fig:chic-tau\], where we take $t_c\simeq (16\sim 24)$ GeV$^2$. Using the averaged values of $\la \alpha_s G^2\ra$ and $\overline{m}_{c}(\overline{m}_{c})$ in Eqs.\[eq:g2average\] and \[eq:mcaverage\], we deduce the optimal result in units of MeV: M\_[\_c]{}&=&2979(5)\_(11)\_[t\_c]{}(11)\_[\_s]{}(30)\_[m\_c]{}(10)\_[G\^2]{} ,\ M\_[\_[c0]{}]{}&=&3411(1)\_(17)\_[t\_c]{}(26)\_[\_s]{}(30)\_[m\_c]{}(20)\_[G\^2]{} , \[eq:chic\] \[eq:etac\] in good agreement within the errors with the experimental masses: $M_{\eta_c}=2984$ MeV and $M_{\chi_{c0}}$=3415 MeV but not enough accurate for extracting with precision the QCD parameters. Correlation between $\overline{m}_{c}(\overline{m}_{c})$ and $\la \alpha_s G^2\ra$ {#correlation-between-overlinem_coverlinem_c-and-la-alpha_s-g2ra .unnumbered} ---------------------------------------------------------------------------------- We study the correlation between $\overline{m}_{c}(\overline{m}_{c})$ and $\la \alpha_s G^2\ra$ by requiring that the sum rules reproduce the masses of the ${\eta_c}$ and ${\chi_{c0}}$ within the error induced by the choice of $t_c$ repsectively 11 and 17 MeV. We show the result of the analysis in Fig.\[fig:mc-g2-etac\] keeping only the strongest constraint from $M_{\eta_c}$. We deduce: \_[c]{}(\_[c]{})=1266(16)  [MeV]{} , \[eq:mcetac\] ![Behaviour of $\Delta \overline{m}_{c}(\overline{m}_{c})$ versus $\Delta\la \alpha_s G^2\ra$ from $M_{\eta_c}$. The dashed region corresponds to $\Delta \alpha_s=0$ and for different values of $t_c\simeq 10\sim 13$ GeV$^2$. The two extremal lines correspond to $\Delta\alpha_s\pm 12$. We use the range of $\alpha_s$ in Eq.\[eq:param\] and of $\la \alpha_s G^2\ra$ in Eq.\[eq:g2average\]. []{data-label="fig:mc-g2-etac"}](mc-g2-etac.pdf){width="12cm"} in good agreement with the one in [@ZYAB] from pseudoscalar moments. We combine our determinations in Eqs.\[eq:respsi\] and \[eq:mcetac\] with the two determinations[@SNcb1; @SNcb2] from vector moments sum rules quoted in Eq.\[eq:param\]. As a final result , we quote the average from exponential and moment sum rules from a global fit of the quarkonia spectra: \_[c]{}(\_[c]{})\_[average]{}=(126314) [MeV]{}, \[eq:mcaverage\] where we have retained the error from the most precise prediction rather than from the weighted average. It is remarkable that this value agrees with the original SVZ estimate[@SVZa; @SVZb] of the euclidian mass. (Pseudo)scalar bottomium ======================== $\eta_b$ and $\chi_{b0}$ masses {#eta_b-and-chi_b0-masses .unnumbered} ------------------------------- ![Behaviour of $M_{\eta_b}$ versus $\tau$ for different values of $t_c$.[]{data-label="fig:etab"}](etab-tau.pdf){width="8.cm"} ![Behaviour of $M_{\chi_{b0}}$ versus $\tau$ for different values of $t_c$.[]{data-label="fig:chib"}](chib-tau.pdf){width="8.5cm"} The masses of the $\eta_b(0^{-+})$ and $\chi_{b0}(0^{++})$ are extracted in a similar way using the value of $\mu$ in Eq.\[eq:mub\] and the parameters in Eqs.\[eq:g2average\] and \[eq:mbaverage\]. We take the range $\sqrt{t_c}=(9.5\sim 12)$ \[resp. $(10.5 \sim 13)$\] GeV for the $\eta_b$ \[resp. $\chi_{b0}$\] channels, as shown in Figs\[fig:etab\] and \[fig:chib\] from which we deduce in units of MeV: M\_[\_b]{}&=&9394(16)\_(30)\_[t\_c]{}(7)\_[\_s]{}(16)\_[m\_b]{}(8)\_[G\^2]{} ,\ M\_[\_[b0]{}]{}&=&9844(7)\_(35)\_[t\_c]{}(6)\_[\_s]{}(17)\_[m\_b]{}(29)\_[G\^2]{} , \[eq:mchib\] \[eq:metab\] in good agreement with the data $M_{\eta_b}=9399$ MeV and $M_{\chi_{b0}}=9859$ MeV. Correlation between $\overline{m}_{b}(\overline{m}_{b})$ and $\la \alpha_s G^2\ra$ {#correlation-between-overlinem_boverlinem_b-and-la-alpha_s-g2ra .unnumbered} ---------------------------------------------------------------------------------- The analysis done for charmonium is repeated here where we request that the sum rule reproduces the $\eta_b$ and $\chi_{b0}$ masses with the error induced by the choice of $t_c$. Unfortunately, this constraint is too weak and leads to $\overline{m}_{b}(\overline{m}_{b})$ with an accuracy of about 40 MeV which is less interesting than the estimate from the vector channel in Eq.\[eq:resmb\]. $\alpha_s$ and $\la \alpha_s G^2\ra$ from $M_{\chi_{0c(0b)}}-M_{\eta_{c(b)}}$ ============================================================================= \[sec:7\] As the sum rules reproduce quite well the absolute masses of the (pseudo)scalar states, we can confidently use their mass-spliitngs for extracting $\alpha_s$ and $\la \alpha_s G^2\ra$. We shall not work with the Double Ratio of LSR[@DRSR; @SNFORM1; @SNGh3; @SNGh1; @SNGh5; @SNmassa; @SNmassb; @SNhl; @HBARYON1; @HBARYON2; @NAVARRA; @SNB1; @SNB2] as each sum rule does not optimize at the same points. We check that, in the mass-difference , the effect of the choice of the continuum threshold is reduced and induces an error from 6 to 14 MeV instead of 11 to 35 MeV in the absolute value of the masses. The effect due to $\overline{m}_{c,b}$ in Eqs.\[eq:mcaverage\] and \[eq:mbaverage\] and to $\mu$ in Eqs.\[eq:muc\] and \[eq:mub\] induce respectively an error of about (1–2) MeV and 8 MeV. The largest effects are due to the changes of $\alpha_s$ and $\la \alpha_s G^2\ra$. We show their correlations in Fig\[fig:alfas-g2\] where we have runned the value of $\alpha_s$ from $\mu=2.85$ GeV to $M_\tau$ in the charm channel and from $\mu=9.5$ GeV to $M_\tau$ in the bottom one where the values of $\mu$ correspond to the scales at which the sum rules have been evaluated. We have requested that the method reproduces within the errors the experimental mass-splittings by about 2-3 MeV. With the central values given in Eqs.\[eq:g2average\] and \[eq:alfas-mb\], the allowed region leads to our final predictions: \_s(M\_)&=&0.318(15) \_s(M\_Z)=0.1183(19)(3) ,\ \_s G\^2&=&(6.340.39)10\^[-2]{} [GeV]{}\^4 . \[eq:alfas\] \[eq:glue1\] Adding into the analysis the range of input $\alpha_s$ values given in Eq.\[eq:param\] (light grey horizontal band in Fig.\[fig:alfas-g2\]), one can deduce stronger constraints on the value of $\la\alpha_s G^2\ra$: \_s G\^2=(6.390.35)10\^[-2]{} [GeV]{}\^4. \[eq:glue2\] Combining the previous values in Eqs.\[eq:respsi\], \[eq:glue1\] and \[eq:glue2\] with the ones in Table\[tab:g2\], one obtains the [*new sum rule average*]{}: \_s G\^2\_[average]{}=(6.350.35)10\^[-2]{} [GeV]{}\^4, \[eq:glue2mean\] where we have retained the error from the most precise determination in Eq.\[eq:glue2\] instead of the weighted error of 0.23. This result definitely rules out some eventual lower and negative values quoted in Table\[tab:g2\]. ![Correlation between $\alpha_s$ and $\la \alpha_s G^2\ra$ by requiring that the sum rules reproduce the (pseudo)scalar mass-splittings.[]{data-label="fig:alfas-g2"}](alfas-g2.pdf){width="12.cm"} Summary and Conclusions ======================= We have explicitly studied (for the first time) the correlations between $\alpha_s,~\la \alpha_s G^2\ra$ and $\overline{m}_{c,b}$ using ratios of Laplace sum rules @N3LO of PT QCD and including the gluon condensate $\la \alpha_s G^2\ra$ of dimension 4 @NLO and the ones of dimension 6-8 @LO in the (axial-)vector charmonium and bottomium channels. We have used the criterion of $\mu$-stability in addition to the usual sum rules stability ones (sum rule variable $\tau$ and continuum threshold $t_c$) for extracting our optimal results. Our final result from the $J/\psi$ channel in Eq.\[eq:glue2\] and the [*sum rule average*]{} in Eq.\[eq:glue2mean\] including this new value confirm and improve our previous estimates of $\la \alpha_s G^2\ra$ from moments within $n$ (number of moments) stability criterion [@SNcb2] and from Laplace sum rules within $\tau$ stability criterion [@SNcb3] in the vector channels quoted in Table\[tab:g2\] and from the heavy quark mass-spittings obtained in[@SNHeavy; @SNHeavy2]. The correponding values of $\overline{m}_{c,b}$ from vector moments and Laplace sum ruels quoted in Eqs\[eq:param\] and \[eq:mblsr\] are also confirmed by the present determinations given in Eqs.\[eq:respsi\] to \[eq:mcaverage\] and in Eqs.\[eq:resmb\] to \[eq:mbaverage\]. We have extended the analysis to the (pseudo)scalar channels where the experimental masses of the lowest ground states are reproduced quite well. The $\eta_c$ sum rule also leads to an alternative prediction of $\overline{m}_{c}$ in Eq.\[eq:mcetac\]. The $\chi_{c0(b0)}-\eta_{c(b)}$ mass-splittings lead to improved values of the gluon condensate $\la \alpha_s G^2\ra$ in Eqs.\[eq:glue1\] and \[eq:glue2\] which give the [*new sum rule average*]{} in Eq.\[eq:glue2mean\]. The $\chi_{c0(b0)}-\eta_{c(b)}$ mass-splittings also provide a new prediction of $\alpha_s$ in Eq.\[eq:alfas\] in good agreement with the world average[@PDG; @BETHKEa; @BETHKEb; @BETHKEc]. Addendum : $\alpha_s(\mu)$ from $M_{\chi_{0c(0b)}}-M_{\eta_{c(b)}}$@N2LO ======================================================================== In this complementary note, we present a more detailed discussion of the $\alpha_s-$results obtained previously @ N2LO in Section\[sec:7\] for two different subtraction scales $\mu$ from the (pseudo)scalar heavy quarkonia mass-spliitings $M_{\chi_{0c(0b)}}-M_{\eta_{c(b)}}$.This complementary discussion is useful for a much better understanding of these results. Optimized subtraction scales {#optimized-subtraction-scales .unnumbered} ---------------------------- Besides the usual sum rules optimization procedure (sum rule variables and QCD continuum threshold) studied in details in previous sections, we deduce from Figs.4 and 8 that the ratios of charmonium and bottomium moments are optimized respectively at the values of the subtraction scales: \_c=(2.8\~2.9)  [GeV     and   ]{} \_b=(9\~10) [GeV]{}. $\alpha_s$ and $\la \alpha_s G^2\ra$ correlation {#alpha_s-and-la-alpha_s-g2ra-correlation .unnumbered} ------------------------------------------------ We study, in Fig.\[fig:alfas-g2\], the correlation between $\alpha_s$ and $\la \alpha_s G^2\ra$ where the charmonium (resp. bottomium) sum rules have been evaluated at $\mu_c$ (resp. $\mu_b$) but runned to the scale $M_\tau$ for a global comparison of the results. For the range of $\la \alpha_s G^2\ra$ values allowed by different analysis ($x$-axis) and requiring that the sum rule reproduces the experimental mass-splittings $M_{\chi_{0c}}-M_{\eta_{c}}$ by about $(2\sim 3)$ MeV, one obtains the grey band limited by the two green (continuous) curves in Fig.\[fig:alfas-g2\] which lead to: && \_s(2.85)=0.262(9) \_s(M\_)=0.318(15)  \_s(M\_Z)=0.1183(19)(3)  . \[eq:muc\] In the same way, the $M_{\chi_{0b}}-M_{\eta_{b}}$ bottomium sum rule evaluated at the optimization scale $\mu_b=$9 GeV gives (sand colour band limited by two dotted red curves): && \_s(9.50)=0.180(8) \_s(M\_)=0.312(27) \_s(M\_Z)=0.1175(32)(3)  . ![Comparison with the running of the world average $\alpha_s(M_Z)=0.1181(11)$[@BETHKEa; @PDG] (grey band limited by the two green curves) of our predictions at three different scales: $\mu_\tau=M_\tau$ for the original $\tau$-decay width[@SNTAU] (open circle), $\mu_c$=2.85 GeV for $M_{\chi_{c0}}-M_{\eta_c}$ (full triangle) and $\mu_b=$9.5 GeV for $M_{\chi_{b0}}-M_{\eta_b}$ (full square)[@SN18].](run-alfas.pdf){width="12.cm"} \[fig:alfas\] Comparison with the world average {#comparison-with-the-world-average .unnumbered} --------------------------------- These values of $\alpha_s(\mu)$ estimated at different $\mu$-scales are shown in Fig.\[fig:alfas\] where they are compared with the running of the world average $\alpha_s(M_Z)=0.1181(11)$[@BETHKEa; @PDG]. We have added, in the figure, your previous estimate of $\alpha_s(M_\tau)$[@SNTAU] obtained from the original $\tau$-decay rate (lowest moment)[@BNPa; @BNPb]: \_s(M\_)=0.325(8) , where one should note that non-perturbative corrections beyond the standard OPE (tachyonic gluon mass and duality violations) do not affect sensibly the above value of $ \alpha_s(M_\tau)$ as indicated by the coïncidence of the central value with the recent one from high-moments[@PICHTAUb]. Our most precise prediction for $\alpha_s$ from the heavy-quarkonia mass-splittings comes from the (pseudo)scalar charmonium one in Eq.\[eq:muc\] which corresponds to: \_s(M\_Z)=0.1183(19)(3)  , which agrees with the world average: $\alpha_s(M_Z)=0.1181(11)$[@BETHKEa; @PDG]. [999]{} M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, [*Nucl. Phys.*]{} [**B147**]{} (1979) 385. M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, [*Nucl. Phys.*]{} [**B147**]{} (1979) 448. For a review, see e.g.: V.I. Zakharov, talk given at the Sakurai’s Price, [*Int. J. Mod .Phys.*]{} [**A14**]{}, (1999) 4865. V.A Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, [*Nucl. Phys.*]{} [**B191**]{} (1981) 301. S. Narison and G. Veneziano, [*Int. J. Mod. Phys.*]{} [**A4**]{} (1989) no. 11, 2751. S. Narison, [*Nucl. Phys.*]{} [**B509**]{} (1998) 312. For a review, see e.g.: S. Narison, [*Cambridge Monogr. Part. Phys. Nucl. Phys. Cosmol.*]{} [**17**]{} (2004) 1-778 \[hep-ph/0205006\]. For a review, see e.g.: S. Narison, [*World Sci. Lect. Notes Phys.*]{} [**26**]{} (1989) 1. For a review, see e.g.: S. Narison, [*Phys. Rept.*]{} [**84**]{} (1982) 263. S. Narison, [*Acta Phys. Pol.*]{} [**B 26**]{}(1995) 687. S.N. Nikolaev and A.V. Radyushkin, [*Nucl. Phys.*]{} [**B213**]{} (1983) 285. S.N. Nikolaev and A.V. Radyushkin, [*Phys. Lett.*]{} [**B110**]{} (1983) 476. L. J. Reinders, H. Rubinstein and S. Yazaki, [*Phys. Rept.* ]{} [**127**]{} (1985) 1 and references therein. K.J. Miller and M.G Olsson, [*Phys. Rev.*]{} [**D25**]{} (1982) 1247. D. J. Broadhurst, P. A. Baikov, V. A. Ilyin, J. Fleischer, O. V. Tarasov and V. A. Smirnov, [*Phys. Lett.*]{} [**B329**]{} (1994) 103. B.L. Ioffe and K.N. Zyablyuk, [*Eur. Phys. J.*]{} [**C 27**]{} (2003) 229. B.L. Ioffe, [*Prog. Part. Nucl. Phys.*]{} [**56**]{} (2006) 232. S. Narison, [*Phys. Lett.*]{} [**B706**]{} (2012) 412. J.S. Bell and R.A. Bertlmann, [*Nucl. Phys.*]{} [**B177**]{}, (1981) 218. J.S. Bell and R.A. Bertlmann, [*Nucl. Phys.*]{} [**B187**]{}, (1981) 285. R.A. Bertlmann, [*Acta Phys. Austriaca*]{} [**53**]{}, (1981) 305. R.A. Bertlmann,[*Nucl. Phys.*]{} [**B204**]{}, (1982) 387. R.A. Bertlmann,[*Non-pertubative Methods,*]{} ed. Narison, World Scientific (1985) R.A. Bertlmann,Nucl. Phys. (Proc. Suppl.) [**B23**]{} (1991) 307. R. A. Bertlmann and H. Neufeld, [*Z. Phys.*]{} [**C27**]{} (1985) 437. J. Marrow, J. Parker and G. Shaw, [*Z. Phys.*]{} [**C37**]{} (1987) 103. S. Narison, [*Phys. Lett.*]{} [**B707**]{} (2012) 259. S. Narison, [*Phys. Lett.*]{} [**B387**]{} (1996) 162. S. Narison, [*Nucl. Phys. (Proc. Suppl)*]{} [**A54**]{} (1997) 238. F.J. Yndurain, [*Phys. Rept.*]{} [**320**]{} (1999) 287 \[arXiv hep-ph/9903457\]. S.I Eidelman, L.M Kurdadze, A.I Vainshtein, [*Phys. Lett.*]{} [B82]{} ( 1979) 278. G. Launer, S. Narison and R. Tarrach, [*Z. Phys.*]{} [**C26**]{} (1984) 433. R.A. Bertlmann, G. Launer and E. de Rafael, [*Nucl. Phys.*]{} [**B250**]{}, (1985) 61. R.A. Bertlmann et al., [*Z. Phys.*]{} [**C39**]{}, (1988) 231. M.B. Causse and G. Mennessier, [*Z. Phys.*]{} [**C47**]{} (1990) 611. S. Narison, [*Phys. Lett.*]{} [**B300**]{} (1993) 293. S. Narison, [*Phys. Lett.*]{} [**B361**]{} (1995) 121. C.A Dominguez and J. Sola, [*Z. Phys.*]{} [**C40**]{} (1988)63. L. Duflot, [*Nucl. Phys. (Proc. Suppl)*]{} [**B40**]{} (1995) 37. The OPAL collaboration, K. Ackerstaff et al., [*Eur. Phys. J*]{} [**C8**]{}, 183 (1999). The ALEPH collaboration, S. Schael et al., [*Phys. Rept.*]{} [**421**]{}, 191 (2005). M. Davier et al., [*Eur. Phys. J.*]{} [**C 74**]{}, (2014)n$^0$3 2803. P.E. Rakow, [*arXiv:hep-lat/0510046* ]{}. G. Burgio, F. Di Renzo, G. Marchesini and E. Onofri, [*Phys. Lett.*]{} [**B422**]{} (1998) 219. R. Horley, P.E.L. Rakow and G. Schierholz, [*Nucl. Phys. (Proc. Sup.)*]{} [**B 106**]{} (2002) 870. G. S. Bali, C. Bauer and A. Pineda, [*Phys. Rev. Lett.*]{} [**113**]{} (2014) 092001. G. S. Bali, C. Bauer and A. Pineda, [*AIP Conf. Proc.*]{} [**1701**]{} (2016) 030010. T. Lee, [*Nucl. Part. Phys. Proc.*]{} [**258-259**]{} (2015)181. E. Braaten, S. Narison and A. Pich, [*Nucl. Phys.*]{} [**B373**]{} (1992) 581. S. Narison and A. Pich, [*Phys. Lett.*]{} [**B211**]{} (1988) 183. S. Narison, [*Phys. Lett.*]{} [**B673**]{} (2009) 30. A. Di Giacomo, Non-pertubative Methods, ed. Narison, World Scientific (1985). M. Campostrini, A. Di Giacomo, Y. Gunduc [*Phys. Lett.*]{} [**B225**]{} (1989) 393. A. Di Giacomo and G.C. Rossi, [*Phys. Lett.*]{} [**B100**]{} (1981) 481. M. D’Elia, A. Di Giacomo and E. Meggiolaro, [*Phys. Lett.*]{} [**B408**]{} (1997) 315. S. Narison, [*Phys. Lett.*]{} [**B693**]{} (2010) 559, erratum [*ibid*]{} [**705**]{} (2011) 544. S.N. Nikolaev and A.V. Radyushkin, [*Phys. Lett.*]{} [**B 124**]{} (1983) 243. T. Schafer and E.V. Shuryak, [*Rev. Mod. Phys.*]{} [**70**]{} (1998) 323. B.L. Ioffe and A.V. Samsonov, [*Phys. At. Nucl.*]{} [**63**]{} (2000) 1448. C. Becchi, S. Narison, E. de Rafael and F.J. Yndurain, [*Z. Phys.*]{} [**C8**]{} (1981) 335. S. Narison and E. de Rafael, [*Phys. Lett.*]{} [**B 522**]{}, (2001) 266. PDG, C. Patrignani et al. (Particle Data Group), [*Chin. Phys.*]{} [**C40**]{}, 100001 (2016) and 2017 update. A.O.G. Kallen and A. Sabry, [*Kong. Dan. Vid.Sel. Mat. Fys. Med.*]{} [**29N17**]{} (1955) 1. J. Schwinger, [*Particles sources and fields, ed. Addison-Wesley Publ.*]{}, [**Vol 2**]{} (1973). K. Schilcher, M.D Tran and N.F Nasrallah, [*Nucl. Phys.*]{} [**B181**]{} (1981) 104. D.J. Broadhurst and S.C Generalis, Open University preprint OUT-4102-12. S.C Generalis, Open University, PhD thesis, preprint OUT-4102-13 (unpublished). K. Chetyrkin, J.H. Kuhn and M. Steinhauser, [*Nucl. Phys.*]{} [**B505**]{} (1997) 40. A. Hoang, and T. Teubner, [*Nucl. Phys.*]{} [**B 519**]{} (1998) 285. K. Chetyrkin et al., [*Phys.Lett.* ]{}[**B384**]{} (1996) 233. K. Chetyrkin, J.H. Kuhn and M. Steinhauser, [*Nucl. Phys.*]{} [**B482**]{} (1996) 213. K. Chetyrkin, R. Harlander and M. Steinhauser, [*Phys.Rev.*]{} [**D58**]{} (1998) 014012. K. Chetyrkin, R. Harlander and M. Steinhauser, [*Nucl. Phys.*]{} [**B503**]{} (1997) 339. R. Tarrach, [*Nucl. Phys.*]{} [**B183**]{} (1981) 384. R. Coquereaux, [*Annals of Physics*]{} [**125**]{} (1980) 401. P. Binetruy and T. Sücker, [*Nucl. Phys.*]{} [**B178**]{} (1981) 293. S. Narison, [*Phys. Lett.*]{} [**B197**]{} (1987) 405. S. Narison, [*Phys. Lett.*]{} [**B216**]{} (1989) 191. N. Gray, D.J. Broadhurst, W. Grafe, and K. Schilcher, [*Z. Phys.*]{} [**C48**]{} (1990) 673. L.V. Avdeev and M.Yu. Kalmykov, [*Nuc. Phy.*]{} [**B 502**]{} (1997) 419. J. Fleischer, F. Jegerlehner, O.V. Tarasov, and O.L. Veretin, [*Nucl. Phys.*]{} [**B539**]{} (1999) 671. K.G. Chetyrkin and M. Steinhauser, [*Nucl. Phys.*]{} [**B573**]{} (2000) 617. K. Melnikov and T. van Ritbergen, [*Phys. Lett.*]{} [**B482**]{} (2000) 99. E. Bagan, J.I Latorre, P. Pascual and R. Tarrach, [*Nucl. Phys.*]{} [**B254**]{} (1985) 555. V.I. Zakharov, [*Nucl. Phys. Proc. Suppl.*]{} [**164**]{} (2007) 240. K. Chetyrkin, S. Narison and V.I. Zakharov, [*Nucl. Phys.*]{} [**B550**]{} (1999) 353. S. Narison and V.I. Zakharov, [*Phys. Lett.*]{} [**B522**]{} (2001) 266. S. Narison, [*Nucl. Phys. Proc. Suppl.*]{} [**164**]{} (2007) 225. Y. Chung et al., [*Z. Phys.*]{} [**C25**]{} (1984) 151. H.G. Dosch, [*Non-Perturbative Methods (Montpellier 1985)*]{} ed. S. Narison, World Scientific (Singapore). H.G Dosch, M. Jamin and S. Narison, [*Phys. Lett.*]{} [**B220**]{} (1989) 251. A. Pich and A. Rodriguez-Sanchez, [*Mod. Phys. Lett.*]{} [**A31**]{}(2016) no.30, 1630032. A. Pich and A. Rodriguez-Sanchez, [*Phys.Rev.*]{} [**D94**]{} (2016) no.3, 034027 For a review, see e.g: S. Bethke, [*Nucl. Part. Phys. Proc.*]{} [**282-284**]{} (2017)149. For a review, see e.g: A. Pich, arXiv:1303.2262, \[PoSConfin.X,022(2012)\]. For a review, see e.g: G. Salam, arXiv:1712.05165 \[hep-ph\]. R. Albuquerque, S. Narison, D. Rabetiarivony, G. Randriamanatrika , arXiv:1709.09023 (2017). R. Albuquerque et al., [*Int. J. Mod. Phys.*]{} [**A31**]{} (2016) no.36, 1650196. R. Albuquerque et al., [*Nucl. Part. Phys. Proc.*]{} [**282-284**]{} (2017) 83. R. Albuquerque, S. Narison, A. Rabemananjara and D. Rabetiarivony, [*Int. J. Mod. Phys.*]{} [**A31**]{} (2016) no. 17, 1650093. R.M. Albuquerque, F. Fanomezana, S. Narison and A. Rabemananjara, [*Phys. Lett.*]{} [**B715**]{} (2012) 129-141. B. Dehnadi, A.H. Hoang,V. Mateu, [*Nucl. Part. Phys.Proc.*]{} [**270-272**]{} (2016) 113. B. Dehnadi, A.H. Hoang,V. Mateu and S.M Zebarjad, [*JHEP*]{} [**1309**]{} (2013) 103. K. Chetyrkin, R. Harlander, and J. H. Kuhn, [*Nucl. Phys.*]{} [**B586**]{} (2000) 56. K.N. Zyablyuk, [*JHEP*]{} [**0301**]{} (2003) 081 S. Narison, [*Phys. Lett.*]{} [**B210**]{} (1988) 238. S. Narison, [*Phys. Lett.*]{} [**B337**]{} (1994) 166; S. Narison, [*Phys. Lett.*]{} [**B322**]{} (1994) 327; S. Narison, [*Phys. Lett.*]{} [**B387**]{} (1996) 162; S. Narison, [*Phys. Lett.*]{} [**B358**]{} (1995) 113; S. Narison, [*Phys.Rev.*]{} [**D74**]{} (2006) 034013; S. Narison, [*Phys.Lett.*]{} [**B466**]{} (1999) 345; S. Narison, [*Phys. Lett.*]{} [**B605**]{} (2005) 319; R.M. Albuquerque and S. Narison, [*Phys. Lett.*]{} [**B694**]{} (2010) 217; R.M. Albuquerque, S. Narison and M. Nielsen, [*Phys. Lett.*]{} [**B684**]{} (2010) 236; S. Narison, F. Navarra and M. Nielsen, [*Phys. Rev.*]{} [**D83**]{} (2011) 016004. [^1]: Some preliminary versions of this work have been presented @ QCD17, Montpellier - FR and @ HEPMAD16 and 17, Antananarivo-MG. [^2]: Similar relations from vector moments have been obtained[@IOFFEa; @IOFFEb] while the ones between $\alpha_s$ and $\overline{m}_{c}$ have been studied in[@DEHNADIa; @DEHNADIb].

--- author: - ZEUS Collaboration date: June 2010 title: | Measurement of $\mathbf{D^{+}}$ and $\mathbf{\Lambda_{c}^{+}}$ production\ in deep inelastic scattering at HERA ---

--- abstract: 'We present calculations of the conductivity tensor $\sigma$ of a $2D$–system with the Rashba spin–orbit interaction (SOI) in an orthogonal magnetic field, with allowance for electron elastic scattering by a Gaussian $\delta$–correlated random potential in the self–consistent Born approximation. The calculations are performed proceeding from the Kubo formula using a new exact representation of the one–particle Green function of the $2D$–system with SOI in an arbitrary magnetic field. We have obtained the analytical expressions for the density of states and $\sigma$ which have a simple interpretation in terms of the two–subband model and hold good in a wide range from the classical magnetic fields $(\omega_{c}^{}\tau\ll 1)$ up to the quantizing ones $(\omega_{c}^{}\tau\gtrsim 1)$. The numerical analysis of the Shubnikov — de Haas oscillations of the kinetic coeffitients and of their behavior in the classical fields region is performed.' author: - | S.G. Novokshonov$_{}^{1}$, and A.G. Groshev$_{}^{2}$\ $_{}^{1}$Institute of Metal Physics, Ural Division of RAS, Ekaterinburg, Russia\ $_{}^{2}$Physical-Technical Institute, Ural Division of RAS, Izhevsk, Russia title: Diffusive magnetotransport in a $2D$ Rashba system --- Introduction ============ The growing interest in studying the spin–orbit interaction (SOI) in semiconductor $2D$–structures is mostly due to its potential application to the spin–based electronic devices [@zutic_etal_2004]. There are two main types of SOI in the quantum well based on zinc–blende–lattice semiconductors. First, the Dresselhaus interaction [@dressel_1955] that originates from the bulk inversion asymmetry (BIA); second, the Rashba interaction [@rashba] induced by structural inversion asymmetry (SIA) of the confined field of a quantum well. Both these interactions lead to the momentum-dependent spin splitting of the electron energy spectrum and to the formation of quantum states with the hard linked spatial and spin degrees of freedom of the electrons. They are responsible for many interesting effects in the transport phenomena like beatings in the Shubnikov — de Haas (SdH) oscillations [@rashba; @luo_etal]; weak antilocalization [@iord_etal_1994; @pikus_pikus_1995; @knap_etal_1996; @golub_2005]; current–induced non–equilibrium spin polarization [@levit_etal_1985; @edelst_1990]; spin Hall effect [@dyak_etal_1971; @mura_etal_2003], and so on. At present there are some sufficiently well developed theories of the kinetic and spin phenomena in $2D$–systems with SOI in zero or classical weak $(\omega_{c}^{}\tau\ll 1)$ orthogonal magnetic fields. Here $\omega_{c}^{}= |e|B/mc$ is the cyclotron frequency, and $\tau$ is the electron scattering time. As for theoretical studies of the considered systems in strong, and especially in quantizing $(\omega_{c}^{}\tau\gtrsim 1)$ magnetic fields, there is still no satisfactory analytical description of the kinetic phenomena even in the usual diffusive regime (without quantum corrections). The complex form of the eigenspinors and energy spectrum of an electron in the presence of SOI and a strong magnetic field [@rashba] is the main cause of such a situation. Direct employment of this basis forces one to proceed almost right from the start to the numerical analysis of very cumbersome expressions [@wang_etal_2003; @lange_etal_2004; @averk_etal_2005; @wang_etal_2005]. The strong magnetic field is however one of the most efficient tools for investigation of SOI [@luo_etal] and manipulation of the spin degrees of freedom in the semiconductor $2D$–structures. Thus, a rather simple theoretical description of the kinetic phenomena in the $2D$–systems with SOI in a strong orthogonal magnetic field becomes a necessity. In the present work we consider the problem of calculation of the longitudinal and Hall resistances of a disordered Rashba system in the self–consistent Born approximation (SCBA). We have found the [*exact*]{} relation between the one–particle Green function (GF) of the Rashba $2D$–electron in an arbitrary orthogonal magnetic field and the well known GF of an “ideal” electron, that is an electon with the ideal value of the Zeeman coupling $(g_{0}^{}=2)$ and without SOI. This allows one to obtain the analytical expressions for the density of states (DOS) and the conductivity tensor $\sigma_{ij}^{}$ in the SCBA. These expressions hold good in a wide range, from the classically weak magnetic fields $(\omega_{c}^{}\tau\ll 1)$ up to the quantizing ones $(\omega_{c}^{}\tau \gtrsim 1)$. They have a simple physical interpretation in the framework of the two–subband conductor model. On the basis of these results, we perform the numerical analysis of the beatings of the SdH oscillations of the considered kinetic coefficients, as well as of their behavior in the classical magnetic fields region. Model ===== Let us consider a two–dimensional $(|| OXY)$ degenerate gas of electrons with effective mass $m$, and effective Zeeman coupling $g$ that moves in a Gaussian $\delta$–correlated random field $U({\bf r})$ in the presence of an external orthogonal $({\bf B}|| OZ)$ magnetic field ${\bf B}=\nabla\times {\bf A}$. We assume the Rashba interaction to be the dominant mechanism of the energy spin splitting in the absence of a magnetic field. This situation occurs, for example, in the narrow–gap semiconductor heterostructures, such as ${\rm In\,As}/{\rm Ga\,Sb}$ [@luo_etal], ${\rm In\,Ga\,As}/{\rm In\,Al\,As}$ [@nitta_etal]. The one–particle Hamiltonian of the considered system has the form $$\label{eq:model_h} {\cal H}+U=\frac{{\mbox{\boldmath $\pi$}}_{}^{2}}{2m}+\alpha({\mbox{\boldmath $\sigma$}}\times{\mbox{\boldmath $\pi$}})\cdot{\bf n} +\frac{1}{4}g\omega_{c}^{}\sigma_{z}^{}+U({\bf r})$$ ($\hbar=1$). Here ${\mbox{\boldmath $\pi$}}={\bf p}-e{\bf A}/c=m{\bf v}$ is the operator of the kinematic electron momentum; ${\mbox{\boldmath $\sigma$}}=(\sigma_{x}^{},\,\sigma_{y}^{},\, \sigma_{z}^{})$ is the vector formed by the Pauli spin matrices; $\alpha$ is the Rashba spin–orbit coupling; $g$ is the effective Zeeman coupling ($g$–factor). In the gauge ${\bf A}=(0,Bx,0)$, the components of the eigenspinors of the Hamiltonian ${\cal H}$ (\[eq:model\_h\]) of a free ($U({\bf r})=0$) Rashba electron are expressed through the Landau wave functions $\psi_{n,X}^{} ({\bf r})$ depending on the Landau level number $n=0,1,2,\ldots$ and the $X$–coordinate of the cyclotron orbit centre $X=-k_{y}^{}/m\omega_{c}^{}$ [@rashba] \[eq:r\_basis\] $$\label{eq:r_spinor} {\displaystyle}\widehat\Psi_{\alpha}^{}({\bf r})=\frac{1} {\sqrt{1+C_{s,n}^{2}}}\left[\begin{array}{c} {\displaystyle}C_{s,n}^{}\psi_{n-1,X}({\bf r})\\[4pt] {\displaystyle}\psi_{n,X}^{}({\bf r})\end{array}\right]\,,\quad\alpha=(s,n,X)\,.$$ The corresponding energy levels have the following form $$\label{eq:r_spectrum} {\cal E}_{s,n}^{}=\left\{\begin{array}{ll} -\omega_{c}\delta\,, & n=0,~~s=+1\,,\\ \omega_{c}^{}\big[n+s\sqrt{\delta_{}^{2}+2\gamma_{}^{2}n}\,\big]\,, & n>0,~~s=\pm 1\,.\end{array}\right.$$ Here $C_{s,n}^{}=\gamma\sqrt{2n}/\big[s\sqrt{\delta_{}^{2}+2\gamma_{}^{2}n} -\delta\big]$ is a normalizing coefficient; $\delta=(g-2)/4$ is the relative deviation of the effective Zeeman coupling from its ideal value $g_{0}^{}=2$ (for definiteness, it is assumed that $\delta<0$ in these equations, but all the following results are valid for any sign of $\delta$); and, finally, $\gamma =\alpha\sqrt{m/\omega_{c}^{}}$ is the dimensionless Rashba spin–orbit coupling. The quantum number $s=\pm 1$ describes the [*helicity*]{} of the Rashba electron eigenstate as in the absence of a magnetic field [@edelst_1990]. Indeed, it can be verified immediately that $s=\pm 1$ is the eigenvalue of the operator $$\label{eq:helic_oper} \nu=\frac{\big[\alpha{\mbox{\boldmath $\sigma$}}\times{\mbox{\boldmath $\pi$}}+\omega_{c}^{}\delta{\mbox{\boldmath $\sigma$}}\big]\cdot {\bf n}}{\sqrt{2m\alpha_{}^{2}{\cal H}_{0}^{}+ \omega_{c}^{2}\delta_{}^{2}}}\,,$$ that is diagonal in the basis (\[eq:r\_spinor\]) and approaches the helicity operator $({\mbox{\boldmath $\sigma$}}\times{\bf p})\cdot{\bf n}/|{\bf p}|$ as $B\to 0$. Here ${\bf n}$ is the unit normal vector to the considered $2D$–system; $$\label{eq:ideal_h} {\cal H}_{0}^{}=\frac{{\mbox{\boldmath $\pi$}}_{}^{2}}{2m}+\frac{1}{2}\omega_{c}^{}\sigma_{z}^{}$$ is the Hamiltonian of the “ideal” ($g_{0}^{}=2$) electron in a magnetic field, which commutes with ${\mbox{\boldmath $\sigma$}}\cdot{\bf n}$, $({\mbox{\boldmath $\sigma$}}\times{\mbox{\boldmath $\pi$}})\cdot{\bf n}$, and with ${\cal H}$ (\[eq:model\_h\]). In spite of this analogy with the $B=0$ case, we cannot say that the Rashba electron has in the states (\[eq:r\_spinor\]) the spin projection $\pm 1/2$ onto the direction $\alpha{\mbox{\boldmath $\pi$}}\times{\bf n}+\omega_{c}^{}\delta {\bf n}$, because the components of the kinematic momentum operator ${\mbox{\boldmath $\pi$}}$ are not commuting motion integrals. Nevertheless, this interpretation makes sense in the quasiclassical limit, when one can speak about the electron path in a magnetic field. Namely, the quantum number $s=\pm 1$ determines the value of the spin projection on the instant direction of $\alpha{\mbox{\boldmath $\pi$}}\times{\bf n}+\omega_{c}^{}\delta{\bf n}$ that changes along the quasiclassical electron path. Thus, the spin configurations of the Rashba electron states form [*vortices*]{} in the $XY$–plane with center at the origin. The conductivity tensor $\hat{\sigma}$ of the considered system has just one independent circularly polarized component $\sigma=\sigma_{xx}^{}+ i\sigma_{yx}^{}$. In the one–electron approximation, it has the form [@gerhar_1975] $$\label{eq:kubo} \sigma=\sigma_{}^{I}+\sigma_{}^{II}= \frac{e_{}^{2}}{8\pi}{\mathop{\rm Tr}\nolimits}\,V_{+}^{}\!\left[\left.\big[2\Phi_{EE}^{RA}- \Phi_{EE}^{RR}-\Phi_{EE}^{AA}\big]\right|_{E=E_{F}^{}}^{} +\!\left.\int_{-\infty}^{E_{F}^{}}\!\big(\partial_{E}-\partial_{E_{}'}\big) \!\big[\Phi_{EE_{}'}^{AA}-\Phi_{EE_{}'}^{RR}\big]\right|_{E_{}'=E}^{}\! {\rm d}E\right]\!.$$ Here, $\Phi_{EE_{}'}^{XY}=\big\langle\hat{G}_{}^{X}(E)V_{-}^{}\hat{G}_{}^{Y} (E_{}')\big\rangle$ is the current vertex operator; $V_{\pm}^{}=V_{x}^{}\pm iV_{y}^{}=v_{\pm}^{}\pm 2i\alpha\sigma_{\pm}^{}$ are circularly polarized components of the full velocity operator \[the corresponding components ${\mbox{\boldmath $\sigma$}}$ are defined as $\sigma_{\pm}^{}=(\sigma_{x}^{}\pm i\sigma_{y}^{})/2$\], where the last term occurs due to SOI (\[eq:model\_h\]). $\hat{G}_{}^{R(A)}(E) =1/(E-{\cal H}-U\pm i0)$ is the resolvent (retarded $(R)$ or advanced $(A)$) of the Hamiltonian (\[eq:model\_h\]), and angular brackets $\langle\ldots\rangle$ denote the averaging over the random field $U$ configurations. One–electron Green function =========================== By definition, the one–particle GF is the averaged resolvent of the Hamiltonian (\[eq:model\_h\]) $\langle\hat{G}_{}^{R(A)}(E)\rangle=\langle 1/(E-{\cal H}-U\pm i0)\rangle$. It is connected with the electron self–energy operator $\hat{\Sigma}_{}^{R(A)}(E)$ by the relation $(X=R,A)$ $$\label{eq:gf_def} \langle\hat{G}_{}^{X}(E)\rangle=\left[\begin{array}{cc} \langle G_{{\uparrow\!\uparrow}}^{X}(E)\rangle & \langle G_{{\uparrow\!\downarrow}}^{X}(E)\rangle\\ \langle G_{{\downarrow\!\uparrow}}^{X}(E)\rangle & \langle G_{{\downarrow\!\downarrow}}^{X}(E)\rangle \end{array}\right]=\frac{1}{E-{\cal H}- \hat{\Sigma}_{}^{X}(E)}\,.$$ The direct employment of the eigenspinors (\[eq:r\_basis\]) for calculation of (\[eq:gf\_def\]), or kinetic and thermodynamic properties of the Rashba system in a strong magnetic field leads to very complicated expressions. One is forced almost from the first steps either to turn to numerical calculations [@wang_etal_2003; @averk_etal_2005; @wang_etal_2005], or to make simplifying approximations like the momentum–independent spin–splitting energy [@lange_etal_2004]. This makes more difficult the interpretation of the results obtained in such a way, as well as the understanding of the whole physical picture. But it turns out that the GF of the free $(U=0)$ Rashba system is expressed [*exactly*]{} through the GF of the “ideal” electron in a magnetic field. This opens up new possibilities for analytical studies of the considered system. Indeed, it is easy to check that the Hamiltonian of the free Rashba systems can be presented in the following form $$\label{eq:h_rd_connect} {\cal H}={\cal H}_{0}^{}+\nu\sqrt{2m\alpha_{}^{2}{\cal H}_{0}^{} +\omega_{c}^{2}\delta_{}^{2}}\,.$$ Here $\nu$ is the helicity operator defined in Eq. (\[eq:helic\_oper\]), ${\cal H}_{0}^{}$ is the Hamiltonian of the “ideal” electron (\[eq:ideal\_h\]). The substitution of the Hamiltonian (\[eq:h\_rd\_connect\]) into the resolvent $\hat{G}(E)=(E-{\cal H})_{}^{-1}$ gives, after some simple algebra, the following result (here and below, we drop superscripts $R(A)$, if this does not lead to misunderstundings. Sometimes, for brevity of notations, we shall not write explicitly the energy arguments of the resolvents or GF’s.) $$\label{eq:step_one} \hat{G}(E)=\frac{E-{\cal H}_{0}^{}+\big[\alpha({\mbox{\boldmath $\pi$}}\times{\bf n})+ \omega_{c}^{}\delta{\bf n}\big]\cdot{{\mbox{\boldmath $\sigma$}}}}{(E+m\alpha_{}^{2}- {\cal H}_{0}^{})_{}^{2}-{\displaystyle}\frac{1}{4}\Omega_{B}^{2}}\,,$$ where $$\label{eq:spin_preces} \Omega_{B}^{}=2\sqrt{2m\alpha_{}^{2}E+m_{}^{2}\alpha_{}^{4}+\omega_{c}^{2} \delta_{}^{2}}=\sqrt{\Omega_{}^{2}+4\omega_{c}^{2}\delta_{}^{2}}\,.$$ The quantity $\Omega_{B}^{}$ is equal to the magnetic field–dependent frequency of the spin precession of the electron with energy $E$ that is responsible for the Dyakonov — Perel spin relaxation mechanism [@dyak_etal_1971a]; $\Omega$ is the same frequency in the absence of a magnetic field. It should be noted that the same representation of the one–electron GF can be also obtained for a system with the momentum–linear Dresselhaus SOI. For example, in the case of a $[001]$–grown quantum well based on the ${\rm A}_{\rm III}^{}{\rm B}_{\rm V}^{}$ semiconductors, it suffices to replace ${\mbox{\boldmath $\pi$}}\to\tilde{{\mbox{\boldmath $\pi$}}}=(\pi_{y}^{},\pi_{x}^{})$ in the definition of the helicity operator (\[eq:helic\_oper\]), change the sign before the Zeeman term ($g_{0}^{}=-2$!) in the Hamiltonian of the “ideal”electron (\[eq:ideal\_h\]), and, finally, to redefine the parameter $\delta\to\delta_{D}^{}=(g+2)/4$. The denominator of the right–hand side of Eq. (\[eq:step\_one\]) depends on the “ideal” electron Hamiltonian alone. Expanding this expression into the partial fractions, we obtain the desired representation of the one–electron GF of the free Rashba system $$\begin{aligned} \label{eq:gf_repres} &\!\!\!\hat{G}&\!\!\!\!(E)=\nonumber\\ &\!\!\!=&\!\!\!\frac{1}{2\Omega_{B}^{}}\sum_{s=\pm 1/2}\frac {\Omega_{B}^{}+4s\big[m\alpha_{}^{2}-\omega_{c}^{}\delta_{}^{}\sigma_{z}^{} -\alpha_{}^{}({\mbox{\boldmath $\pi$}}\times{\bf n})\cdot{\mbox{\boldmath $\sigma$}}\big]}{E+m\alpha_{}^{2} +s\Omega_{B}^{}-{\cal H}_{0}^{}}\nonumber\\ &\!\!\!=&\!\!\!\sum_{s=\pm 1/2}\left[\Phi_{s}^{}-2s\frac{\alpha_{}^{}({\mbox{\boldmath $\pi$}}\times{\bf n})\cdot{\mbox{\boldmath $\sigma$}}}{\Omega_{B}^{}}\right]\hat{G}(E+m\alpha_{}^{2} +s\Omega_{B}^{})\,.\nonumber\\\end{aligned}$$ We use here the same notation ($\hat{G}$) for the GF of the Rashba electron and for the GF of the “ideal” electron. However, this does not lead to confusion since the latter depends always on the energy arguments like $E+m\alpha_{}^{2}+s\Omega_{B}^{}$ etc. It is important that the same representation can be obtained for the averaged resolvent of the Rashba system in the SCBA. We restrict ourselves here to an approximation in which the electron self–energy operator is diagonal in the spin space. Then, the SCBA equation for $\Sigma_{}^{X}(E)$ has the following form $$\label{eq:scba_def} \hat{\Sigma}(E)=W\langle{\mathop{\rm Sp}\nolimits}\hat{G}(E)\rangle= \left[\begin{array}{cc} \Sigma_{{\uparrow\!\uparrow}}^{}(E) & 0\\ 0 & \Sigma_{{\downarrow\!\downarrow}}^{}(E)\end{array}\right],$$ where ${\mathop{\rm Sp}\nolimits}$ denotes the trace only over the spatial degrees of freedom; $W$ is the mean–square fluctuation of the Gaussian random field ($\langle U({\bf r}) U({\bf r}_{}')\rangle=W\delta({\bf r}-{\bf r}_{}')$). Therefore, it suffices to make everywhere in Eq. (\[eq:gf\_repres\]) the following substitutions $$\label{eq:substitution} E\,\to\,E-\Sigma_{e}^{}(E)\,\qquad g\omega_{c}^{}\,\to\,g\omega_{c}^{}+ 4\Sigma_{o}^{}(E)$$ to obtain the desired representations for the averaged GF’s in the SCBA. Here $\Sigma_{e(o)}^{}(E)=\big[\Sigma_{{\uparrow\!\uparrow}}^{}(E)\pm\Sigma_{{\downarrow\!\downarrow}}^{}(E)\big]/2$ are the even and odd parts of the electron self–energy. The first ($\Sigma_{e}^{}=\Delta_{e}^{}\pm i/2\tau_{e}^{}$) describes the perturbation (shift $\Delta_{e}^{}$ and broadening $1/\tau_{e}^{}$) of the one–electron energy levels by a random field. The real part of $\Sigma_{o}^{}= \Delta_{o}^{}\pm i/2\tau_{o}^{}$ defines the renormalization of the Zeeman coupling (\[eq:substitution\]), while its imaginary part $\propto 1/ \tau_{o}^{}$ makes a contribution to the overall broadening of the one–electron energy levels. As a result, we obtain a expression like Eq. (\[eq:gf\_repres\]) for the averaged GF, where $$\label{eq:ideal_gf} \hat{G}_{}^{R(A)}(E+m\alpha_{}^{2}+s\Omega_{B}^{})=\frac{1}{E+m\alpha_{}^{2} +s\Omega_{B}^{}-{\cal H}_{0}^{}\pm{\displaystyle}\frac{i}{2\tau_{s}^{}}}$$ is the averaged retarded (advanced) GF of the “ideal” electron, and $$\begin{aligned} \label{eq:renorm} \Omega_{B}^{}&\!\!\!=&\!\!\!\frac{1}{2}\big(\Omega_{B}^{R}+\Omega_{B}^{A}\big) \,,\nonumber\\ \frac{1}{\tau_{s}^{}}&\!\!\!=&\!\!\!\frac{1}{{\tau}_{e}^{}}-is\big( \Omega_{B}^{R}-\Omega_{B}^{A}\big)=\left(1+s\frac{4m\alpha_{}^{2}} {\Omega_{B}^{}}\right)\frac{1}{\tau_{e}^{}}+s\frac{4\omega_{c}^{}\delta} {\Omega_{B}^{}}\frac{1}{\tau_{o}^{}}\nonumber\\\end{aligned}$$ are the disorder–modified frequency of the spin precession (\[eq:spin\_preces\]) and the inverse life time of an electron in the $s$–th spin–split subband. As usual, we do not take explicitly into consideration in (\[eq:ideal\_gf\]) the one–electron energy levels shift $\Delta_{e}^{}$ that is absorbed by the normalization condition, but we mean here that the odd shift $\Delta_{o}^{}$ is included in the definition of the effective $g$–factor in accordance with (\[eq:substitution\]). The explicit allowance for the Zeeman coupling renormalization is particularly important in the SdH oscillations regime. Density of states and self–energy ================================= We first consider the calculation of the DOS $n(E)={\mathop{\rm Im}\nolimits}\langle{\mathop{\rm Tr}\nolimits}\hat {G}_{}^{A}(E)\rangle/\pi$ using the above–obtained expression for the one–particle GF (\[eq:gf\_repres\]). Here, the symbol ${\mathop{\rm Tr}\nolimits}$ denotes the trace over the spatial and spin degrees of freedom. For the sake of simplicity, we shall deal with the case of large filling numbers $(E\gg\omega_{c}^{})$. Calculating the trace of resolvent (\[eq:gf\_repres\]) over the spatial and spin degrees of freedom, we obtain the following expression for the DOS $$\begin{aligned} \label{eq:dos_scba} n(E)&\!\!\!=&\!\!\!\sum_{s=\pm 1/2}\frac{m_{s}^{}}{m}n_{}^{(0)}\big[E+ m\alpha_{}^{2}+s(\Omega_{B}^{}\pm\omega_{c}^{})\big]\nonumber\\ &\!\!\!=&\!\!\!\sum_{s=\pm 1/2}\frac{m_{s}^{}}{m}n_{s}^{(0)}(E)\,.\end{aligned}$$ Here, we take into account that the DOS of a spinless electron in an orthogonal magnetic field $n_{}^{(0)}(E)$ satisfies $n_{}^{(0)}(E)=n_{}^{(0)} (E\pm\omega_{c}^{})$ at large filling factors $(E\gg\omega_{c}^{})$. The sign before $\omega_{c}$ is chosen in such a way as to ensure the right–hand limit $s(\Omega_{B}^{}\pm\omega_{c}^{})\to\pm sg\omega_{c}^{}/2$, as the spin–orbit coupling approaches zero. The effective mass $m_{s}^{}$ in the $s$–th subband is defined as $$\label{eq:eff_mass} m_{s}^{}=m\left(1+s\frac{4m\alpha_{}^{2}}{\Omega_{B}^{}}\right)= m\partial_{E}^{}(E+s\Omega_{B}^{})\,.$$ In the considered case this expression coincides with the usual definition of the transport and cyclotron effective masses in the isotropic nonparabolic band [@tsidil_1978]. The symbol $\partial_{E}^{}$ denotes the derivative with respect to energy $E$. In full accordance with the two–subband model, the DOS in Eq. (\[eq:dos\_scba\]) is presented as a sum of partial contributions. Using this expression for the DOS, we can obtain the analytical form of the equation for the electron concentration $n=\int_{}^{E_{F}^{}}n(E){\rm d}E$ that is the normalization condition for the Fermi level determination. For example, at $B=0$ we have $$\label{eq:zero_norm} n=\frac{m}{\pi}\big(E_{F}^{}+m\alpha_{}^{2}\big)=\frac{m}{\pi}E_{0}^{}\,.$$ Thus, the energy $E_{0}^{}=E_{F}^{}+m\alpha_{}^{2}$ corresponds to the Fermi level in the absence of SOI. Notice that the partial electron concentrations $n_{s}^{}=m(E_{0}^{}+s\Omega_{B}^{})/2\pi$ depend nonlinearly on the Fermi energy, in contrast to $n$ (\[eq:zero\_norm\]). Of course, the difference between $E_{0}^{}$ and $E_{F}^{}$ is small for weak SOI ($m\alpha_{}^{2}\ll E_{F}^{}$), but it should be taken into account when analyzing the SdH oscillations that are very sensitive to the electron spectrum character. The representation (\[eq:dos\_scba\]) allows one to obtain a simple analytical experssion for the DOS that holds good up to the quantizing fields region ($\omega_{c}^{}\tau\gtrsim 1$). Indeed, the DOS of a spinless electron in the large filling factors region ($E\gg\omega_{c}^{}$) has the form $$\label{eq:dos_spinless} n_{}^{(0)}(E)=\frac{m}{2\pi}\frac{\sinh{\displaystyle}\frac{\pi}{\omega_{c}^{}\tau}} {\cosh{\displaystyle}\frac{\pi}{\omega_{c}^{}\tau}+\cos 2\pi{\displaystyle}\frac{E}{\omega_{c}^{}}}$$ Inserting Eq. (\[eq:dos\_spinless\]) into Eq. (\[eq:dos\_scba\]), we obtain for the oscillating part of the DOS the following expression $$\begin{aligned} \label{eq:dos_sdh} \Delta n(E_{F}^{})=\frac{2m}{\pi}\exp\left(-\frac{\pi}{\omega_{c}^{}\tau} \right)\left[\cos 2\pi\frac{E_{0}^{}}{\omega_{c}^{}}\cos\pi\frac{\Omega_{B}^{}} {\omega_{c}^{}}-\right.\nonumber\\ -\left.\frac{2m\alpha_{}^{2}}{\Omega_{B}^{}}\sin 2\pi\frac{E_{0}^{}} {\omega_{c}^{}}\sin\pi\frac{\Omega_{B}^{}}{\omega_{c}}\right]\end{aligned}$$ that is valid in the magnetic fields region under consideration. It follows from (\[eq:dos\_sdh\]) that energy $E_{0}^{}$ (see Eq. (\[eq:zero\_norm\])) defines the main period of the SdH oscillations, and $\Omega_{B}^{}/2$ (see Eq. (\[eq:spin\_preces\])) defines their beating period that depends on the magnetic field (see Fig. 1(a)). The second term in Eq. (\[eq:dos\_sdh\]) appears due to the difference between the effective masses $m_{s}^{}$ (\[eq:eff\_mass\]). In the case of weak SOI ($\Omega\ll E$), the oscillations of the DOS are determined completely by the first term in Eq. (\[eq:dos\_sdh\]). Then, the location of the $k$-th node of beatings is determined by the condition $$\label{eq:node} \omega_{c}^{}=\frac{2\Omega}{\sqrt{(2k+1)_{}^{2}-(g-2)_{}^{2}}}\,.$$ This limit was considered in Ref. [@taras_etal_2002]. Unlike the results of that work, the above–obtained equations still stand in the case of strong SOI. In addition, we have taken into account the Zeeman splitting of the electron spectrum that allows to describe more correctly the oscillation pattern. For example, the Eq. (\[eq:node\]) allows to determine both the spin–orbit $\alpha$ and Zeeman $g$ couplings by measured locations of two different nodes (see upper curve in Fig. 1(a)). On the other hand, the spin precession frequency $\Omega_{B}^{}$ approaches $|\delta|\omega_{c}^{}$ as the magnetic field $B$ increases. Therefore, in this case a gradual transition from the beatings of the SdH oscillations to the familiar Zeeman splitting of the oscillating peaks should be observed. The beginning of this transition can be seen on the lower curve in Fig. 1(a). ![Plots of the SdH oscillations of the total DOS (upper panel) and of the difference of the partial DOS’s (bottom panel) of the $2D$ Rashba system at fixed $g=2.8$, and $k_{F}^{}l=35.0$, and different $\Omega\tau=3.0;\;1.5;\;0.75$ (up to down). The arrows point the nodes location that are calculated with Eq. (\[eq:node\]).](N.ps "fig:") ![Plots of the SdH oscillations of the total DOS (upper panel) and of the difference of the partial DOS’s (bottom panel) of the $2D$ Rashba system at fixed $g=2.8$, and $k_{F}^{}l=35.0$, and different $\Omega\tau=3.0;\;1.5;\;0.75$ (up to down). The arrows point the nodes location that are calculated with Eq. (\[eq:node\]).](DN.ps "fig:")  Another important characteristic of the one–electron states of the $2D$–Rashba system is the difference of the partial DOS’s with opposite spin projections onto the $OZ$–axis $$\label{eq:ddos} \delta n(E)=n_{{\uparrow\!\uparrow}}^{}(E)-n_{{\downarrow\!\downarrow}}^{}(E)=\frac{4\omega_{c}^{}\delta} {\Omega_{B}^{}}\sum_{s=\pm 1/2}sn_{s}^{(0)}(E)$$ This quantity is proportional to the derivative of the transverse spin magnetization with respect to energy $E$ and, therefore, it enters in the definitition of the effective concentrations of current carriers in the dissipative part of the $2D$–Rashba system conductivity in an orthogonal magnetic field (see the next section). Evidently, the difference of the partial DOS’s (\[eq:ddos\]) vanishes in the region of classical magnetic fields ($\omega_{c}^{}\tau\ll 1$), but it plays an important role in the SdH oscillations regime. In the case of large filling fsctors, the oscillating behavior of this quantity is described by the following expression $$\label{eq:ddos_sdh} \delta n(E_{F}^{})=\frac{2m}{\pi}\frac{2\omega_{c}^{}\delta}{\Omega_{B}^{}} \exp\left(-\frac{\pi}{\omega_{c}^{}\tau}\right)\sin 2\pi\frac{E_{0}^{}} {\omega_{c}^{}}\sin\pi\frac{\Omega_{B}^{}}{\omega_{c}^{}}\,.$$ Unlike the total DOS (\[eq:dos\_sdh\]), this expression contains just one oscillating term, because $\delta n(E)$ does not depend on the effective masses $m_{s}^{}$ (\[eq:eff\_mass\]). Indeed, the difference of the partial DOS’s $\delta n(E)$ is non zero, which is entirely due to the spin degrees of freedom of the electrons. The typical SdH oscillation patterns of $\delta n(E)$ are depicted in Fig. 1(b). Now, let us turn to the discussion of the electron life time $\tau_{s}^{}$ in the $s$–th spin–split subband which is defined, according to Eq. (\[eq:renorm\]), by the imaginary parts of the even and odd self–energies $\Sigma_{e(o)}^{}$. In other words, the total life time of the one–electron states $\tau_{s}^{}$ is determined by the sum of the weighted relaxation rates of the orbital and spin degrees of freedom. The first term in this expression is proportional to the above–considered total DOS, hence its magnetic–field dependence coincides up to the scale factor with the patterns shown in Fig. 1(a). Of particular interest is the last term in Eq. (\[eq:renorm\]) stemming from the Zeeman coupling renormalization. It is proportional to the difference of the partial DOS’s (\[eq:ddos\]) and, therefore, plays an important role in the SdH oscillation regime, as shown in Fig. 2. Notice that the beatings of the SdH oscillations get supressed with the increase of the relative magnitude of the second term in Eq. (\[eq:renorm\]). Indeed, Eq. (\[eq:node\]) determines the location of the beating loops of the oscillation instead of the nodes. Thus, the broadening of the Zeeman levels leads to observable supression of the beatings of the SdH oscillations, as in the case of the competition between the Rashba and Dresselhaus SOI’s [@averk_etal_2005]. Conductivity ============ The general expression for the conductivity (\[eq:kubo\]) consists of two different terms. The first of them describes the contribution of the electrons at the Fermi level, the second one contains the contribitions of all filled states below the Fermi level. We begin the calculation of the conductivity with the last term of (\[eq:kubo\]) $\sigma_{}^{II}$. First of all, it is pure imaginary and, therefore, makes a contribution in the Hall conductivity alone. Streda [@streda_etal] was first to show that, in the absence of SOI, this part of the conductivity is equal to $$\label{eq:streda} \sigma_{}^{II}=i|e|c\left(\frac{\partial n}{\partial B}\right)_{E}^{}\,,$$ where $n$ is the electron concentration. It should be pointed out that Eq. (\[eq:streda\]) is [*exact*]{}, and with the thermodynamic Maxwell relation $\sigma_{}^{II}$ can be expressed through $(\partial M/\partial E)_{B}^{}$, where $M$ is the orbital magnetization of the electron gas. Detailed discussion of $\sigma_{}^{II}$ and its physical interpretation can be found in survey [@pruisken]. This result is extended immediately to the electron systems with SOI. Following Streda’s argument, it can be shown that the part $\sigma_{}^{II}$ of the of $2D$ Rashba system conductivity is expressed as $$\begin{aligned} \label{eq:soi_streda} \sigma_{}^{II}&\!\!\!=&\!\!\!i|e|c\left[\left(\frac{\partial n}{\partial B} \right)_{E}^{}-\left(\frac{\partial M_{p}^{}}{\partial E}\right)_{B}^{}\right] \nonumber\\ &\!\!\!=&\!\!\! i|e|c\left[\left(\frac{\partial n}{\partial B}\right)_{E}^{}-\frac{g|e|} {4mc}\big[n_{{\uparrow\!\uparrow}}^{}(E)-n_{{\downarrow\!\downarrow}}^{}(E)\big]\right]\nonumber\\ &\!\!\!=&\!\!\!i\frac{|e|nc}{B}\big[n-N_{+}^{}-N_{-}^{}\big]\,,\end{aligned}$$ where $M_{p}^{}$ is the spin magnetization of the electron gas. The quantities $$\label{eq:n+-} N_{s}^{}=\left[E_{0}^{}+s\left(\Omega_{B}^{}+\frac{2\omega_{c}^{2}\delta} {\Omega_{B}^{}}\right)\right]n_{}^{(0)}\big[E_{0}+s(\Omega_{B}^{}- \omega_{c}^{})\big]$$ ($s=\pm 1/2$) are direct analogues of the familiar parameter $n_{\perp}^{} =En_{}^{(0)}(E)$ that stands for the current carrier concentration in the dissipative part of the conductivity tensor of spinless $2D$–electrons in a magnetic field in the SCBA [@gerhar_1975]. In the classical magnetic fields region, they are equal to the partial electron concentrations in the spin–split bands $n_{s}^{}=m(E_{0}^{}+s\Omega_{B}^{})/2\pi$. On the other hand, $N_{s}^{}$ (\[eq:n+-\]) approaches $(E-sg\omega_{c}^{}/2) n_{}^{(0)} (E-sg\omega_{c}^{}/2)$ in the limit $\alpha\to 0$, in which only the Zeeman energy splitting remains. The first two equalities in Eq. (\[eq:soi\_streda\]) are also [*exact*]{}. The last one is obtained in the SCBA at large filling factors ($E\gg\omega_{c}^{}$) using the expression (\[eq:dos\_scba\]) for the DOS of the $2D$ Rashba system in a magnetic field (for details see Appendix B). It should be emphasized that a similar SCBA expression is valid also for spinless electrons [@pruisken]. Now, we turn to the first term in the conductivity (\[eq:kubo\]). It is quite easy to show, by identical transformations, that $$\label{eq:rr_aa_part} -\frac{e_{}^{2}}{4\pi}{\mathop{\rm Re}\nolimits}{\mathop{\rm Tr}\nolimits}V_{+}^{}\Phi_{EE}^{AA}= \frac{e_{}^{2}}{2\pi m}{\mathop{\rm Re}\nolimits}{\mathop{\rm Tr}\nolimits}\big\langle\hat{G}_{}^{A}\big\rangle =\frac{e_{}^{2}}{2\pi m}\sum_{s}{\mathop{\rm Re}\nolimits}{\mathop{\rm Tr}\nolimits}\Big[\Phi_{s}^{R} \Phi_{s}^{A}\hat{G}_{}^{A}(E_{0}^{}+s\Omega_{B}^{})+ \Phi_{s}^{R}\Phi_{-s}^{A}\hat{G}_{}^{A}(E_{0}^{}-s\Omega_{B}^{})\Big]\,,$$ where the averaged GF’s of the “ideal” electron are defined in Eqs. (\[eq:ideal\_gf\]) and (\[eq:renorm\]). In obtaining the last equality in (\[eq:rr\_aa\_part\]), we used the immediately verified identities $$\label{eq:ident_1} \Phi_{s}^{R}=\Phi_{s}^{R}\Phi_{s}^{A}+\Phi_{s}^{R}\Phi_{-s}^{A}\,,\qquad \Phi_{s}^{A}=\Phi_{s}^{R}\Phi_{s}^{A}+\Phi_{-s}^{R}\Phi_{s}^{A}\,.$$ The main contribution to the dissipative part of the conductivity is proportional to the current vertex $\Phi_{EE}^{RA}$ in Eq. (\[eq:kubo\]). If we accept the SCBA (\[eq:scba\_def\]) for the electron self–energy $\hat\Sigma$, we must evaluate the conductivity (\[eq:kubo\]) in the ladder approximation in order to satisfy the particle conservation law. But as show the calculations, the relative magnitude of the ladder correction to the conductivity is second–order in the small parameter $1/(k_{F}^{}l)$, and it can be neglected in comparison with the “bare” part of the conductivity ($\Delta\sigma_{}^{\rm lad}/\sigma=0.01\div 0.001$ for typical values of $k_{F}^{}l=10\div 30$). Thus, it suffices to calculate the “bare” part of the conductivity that is obtained by replacement ${\mathop{\rm Tr}\nolimits}V_{+}^{}\Phi_{EE}^{RA}\to{\mathop{\rm Tr}\nolimits}V_{+}^{} \big\langle\hat{G}_{}^{R}\big\rangle V_{-}^{}\big\langle\hat{G}_{}^{A} \big\rangle$ in the first term in the right–hand side of Eq. (\[eq:kubo\]). We drop the details of calculations that can be found in Appendix B and proceed to the results. The overall contribution of the three above–considered parts has the usual Drude — Boltzmann form $$\label{eq:bare_sigma} \sigma=i\frac{|e|}{B}\left[n-\sum_{s}\frac{\tilde{N}_{s}^{}}{1-i\mu_{s}^{}B} \right]+\Delta\sigma$$ (it is meant here and below that $B\to B/c$). Here, the first two terms represent the sum of the partial conductivities of the electrons of two subbands with different mobilities $\mu_{s}^{}=|e|\tau_{s}^{}/m$, and effective concentrations $$\label{eq:new_n+-} \tilde{N}_{s}^{}=N_{s}^{} +\frac{4m\alpha_{}^{2}}{\tau_{s}^{}\Omega_{B}^{2}} \frac{m}{(2\pi)_{}^{2}}(E-s\Omega_{B}^{})\mu_{s}^{2}B_{}^{2}\,.$$ This expression differs from $N_{s}^{}$ (see Eq. (\[eq:n+-\])) by the second term that originates from the principal values of the one–electron GF’s. The last term in Eq. (\[eq:bare\_sigma\]) represents the small correction to the conductivity due to the electron intersubband transitions. In the leading approximation in powers of the smallness parameters $\omega_{c}^{}/E$ and $\Omega_{B}^{}/E$, it has the form $$\label{eq:delta_sigma} \Delta\sigma=-i\frac{|e|}{B}\frac{2m\alpha_{}^{2}}{\Omega_{B}^{2}}\omega_{c}^{2} \delta\frac{\overline{n_{}^{(0)}}}{(1-i\omega_{c}^{}\tau_{e}^{})_{}^{2}+ \Omega_{B}^{2}\tau_{e}^{2}}\,,$$ where $\overline{n_{}^{(0)}}=\sum_{s}n_{s}^{(0)}(E)$ is the DOS at the Fermi level averaged over the electron subbands. The relative contribution of this correction to the full conductivity (\[eq:bare\_sigma\]) is of the order of magnitude $(\omega_{c}^{}/E)_{}^{2}$ and can be neglected in the large filling factors ($E\gg\omega_{c}^{}$) region. We emphasize that the contribution to the conductivity from intersubband transitions vanishes as the magnetic field approaches zero. This would be expected, because the conductivity tensor in the absence of a magnetic field is diagonal in the original spin space ($\sigma_{{\uparrow\!\downarrow}}^{}=\sigma_{{\downarrow\!\uparrow}}^{} \equiv 0$) by virtue of the momentum parity of the GF’s, and the full conductivity is equal to $\sigma=\sigma_{{\uparrow\!\uparrow}}^{}+\sigma_{{\downarrow\!\downarrow}}^{}$ [@inoue_etal]. In fact, the case in point concerns the time inversion symmetry. Using a unitary transformation, it can be turned into a matrix $s$–representation in which the one–electron GF’s are diagonal and, therefore, $\sigma=\sigma_{{\uparrow\!\uparrow}}^{}+\sigma_{{\downarrow\!\downarrow}}^{}=\sigma_{+}^{}+\sigma_{-}^{}$ due to the trace conservation. Applying an external magnetic field breaks the above–mentioned symmetry that is responsible for the appearance of the intersubband transition–induced conductivity $\Delta\sigma$. Results and discussion ====================== First of all, let us summarize briefly the main results obtained in this work. We have shown that the eigenstates of the $2D$ Rashba electron in an orthogonal magnetic field are characterized by a special motion integral (\[eq:helic\_oper\]) that generalizes the notion of [*helicity*]{} [@edelst_1990]. Using this fact, we have found the relation (\[eq:gf\_repres\]) between the GF’s of the $2D$ Rashba electron and the “ideal” one that holds good for arbitrary orthogonal magnetic fields as well as for the strong spin–orbit coupling. With the help of this relation, we have obtained, in contrast to Refs. [@wang_etal_2003; @lange_etal_2004; @wang_etal_2005], the analytical SCBA expressions for the DOS (\[eq:dos\_scba\]) and magnetoconductivity (\[eq:bare\_sigma\]) of the $2D$ Rashba system that are valid in a wide range from the classical magnetic fields up to the quantizing ones ($\omega_{c}^{}\tau \gtrsim 1$). They permit a simple interpretation in the framework of the model of two types of current carriers with different concentrations and mobilities. The spin–orbit as well as the Zeeman splitting of the electron energy are properly allowed for in these expressions, unlike the results of Ref. [@taras_etal_2002]. We have shown that the competition of the relaxation rates of the orbital and spin degrees of freedom in the total inverse life time $1/\tau_{s}^{}$ of the one–electron states in the $s$–th subband leads to the supression of beatings of the SdH oscillations as does the competition of the Rashba and Dresselhaus SOI’s [@averk_etal_2005]. Finally, we have shown that the breaking of the time inversion symmetry in a magnetic field leads to the appearance of the intersubband term in the $2D$ Rashba system conductivity (\[eq:bare\_sigma\]). We start the discussion of the results with the conductivity in zero magnetic field. In this case, it follows immediately from (\[eq:bare\_sigma\]) that $$\label{eq:zero_cond} \sigma=|e|(n_{+}^{}\mu_{+}^{}+n_{-}^{}\mu_{-}^{})=\sigma_{D}^{}\left[1- 2\left(\frac{m\alpha}{k_{F}^{}}\right)_{}^{2}\right].$$ Thus, the Rashba spin–orbit interaction leads to a decrease in conductivity, and not to its increase, as it was claimed in Ref. [@inoue_etal]. Let us note that the authors of that work ignored the difference in mobility between the electrons of different subbands. As a result, thay obtained a correction to the conductivity of opposite sign compared to (\[eq:zero\_cond\]), which is actually unobservable, because it is absorbed by normalization condition (\[eq:zero\_norm\]). ![The smooth magnetic–field dependence of the resistance calculated with Eqs. \[eq:bare\_sigma\] —\[eq:delta\_sigma\] at fixed $g=3.5$, and $k_{F}^{}l=32$, and different $\Omega\tau=1.5;\;1.0;\;0.5$ (up to down).](PR.ps) Let us proceed now to the discussion of the magnetotransport in the $2D$ Rashba system. It is well known that in the two–subband conductors the classical positive magnetoresistance and the magnetic–field–dependent Hall coefficient are observed (see, for example, [@ziman_1972]). The considered system differs from a classical two–subband conductor in two points. Firstly, the mobilities and effective concentrations of current carriers (\[eq:n+-\]) depend on the magnetic field. Secondly, the full conductivity of the $2D$ Rashba system in an orthogonal magnetic field is not an additive sum of the intrasubband contributions, but it contains a nonadditive intersubband term (\[eq:bare\_sigma\]). However, all these factors lead to very slight magnetic field dependences of kinetic coefficients due to small differences between concentrations and mobilities of current carriers. For example, the relative magnitude of the classical magnetoresistive effect comes to only $1\div 2\%$ for typical values of parameters (see Fig. 3). In discussing the SdH oscillations, we restrict ourselves to the consideration of the large filling factors $(E\gg\omega_{c}^{})$ region, where the SCBA is applicable to the description of the one–electron states and kinetic phenomena. As usual, we extract in the linear approximation the oscillating parts of the conductivity that enter through partial DOS’s into the effective concentrations $N_{s}^{}$ (\[eq:n+-\]) and mobilities $\mu_{s}^{}$. Neglecting the small differences between concentrations and mobilities of current carriers in the smooth parts of conductivity, we obtain the expressions for the oscillating parts of the longitudinal $\rho$ resistance and Hall coefficient $R_{H}^{}$ \[eq:final\_sdh\] $$\begin{aligned} \label{eq:rho_sdh} \frac{\Delta\rho(B)}{\rho(0)}&\!\!\!=&\!\!\!4\exp\left(-\frac{\pi} {\omega_{c}^{}\tau}\right)\cos 2\pi\frac{E_{0}^{}}{\omega_{c}^{}} \cos\pi\frac{\Omega_{B}^{}}{\omega_{c}^{}}\,,\\ \label{eq:rhoh_sdh} \frac{\Delta R_{H}^{}}{R_{H}^{0}(0)}&\!\!\!=&\!\!\!\frac{2}{\mu_{}^{2}B_{}^{2}} \exp\left(-\frac{\pi}{\omega_{c}^{}\tau}\right)\cos 2\pi\frac{E_{0}^{}} {\omega_{c}^{}}\cos\pi\frac{\Omega_{B}^{}}{\omega_{c}^{}}\,.\nonumber\\\end{aligned}$$ Here, $\rho(0)=1/\sigma(0)$ and $R_{H}^{0}\approx-1/|e|nc$ are is the resistance (see Eq. (\[eq:zero\_cond\])) and Hall coefficient in zero magnetic field respectively. Results of numerical calculations of the SdH oscillations are performed using total expessions (\[eq:bare\_sigma\])—(\[eq:delta\_sigma\]) are showed in Fig. 4. Up to definition of the beatings period, the Eqs. (\[eq:final\_sdh\]) agree with expressions for the longitudinal and Hall conductivities obtained in Ref. [@taras_etal_2002]. As pointed out above, the spin precession frequency $\Omega_{B}^{}$ (\[eq:spin\_preces\]) and, therefore, beatings period (see, for example, (\[eq:dos\_sdh\])) depend on the magnetic field due to allowing for Zeeman coupling. As result, the measurement of two different nodes location permits to determine both the spin–orbit $\alpha$ and Zeeman $g$ couplings using the Eq. (\[eq:node\]). ![Plots of the SdH oscillations of the longitudinal magnetoresistance (upper panel) and Hall coefficient (bottom panel) of the $2D$ Rashba system calculated with Eqs. \[eq:bare\_sigma\] —\[eq:delta\_sigma\] at fixed $g=3.5$, and $k_{F}^{}l=32$, and different $\Omega\tau=1.5;\;1.0;\;0.5$ (up to down).](R.ps "fig:") ![Plots of the SdH oscillations of the longitudinal magnetoresistance (upper panel) and Hall coefficient (bottom panel) of the $2D$ Rashba system calculated with Eqs. \[eq:bare\_sigma\] —\[eq:delta\_sigma\] at fixed $g=3.5$, and $k_{F}^{}l=32$, and different $\Omega\tau=1.5;\;1.0;\;0.5$ (up to down).](Rh.ps "fig:") We thank A.K. Arzhnikov, A.V. Germanenko, G.I Kharus, G.M. Minkov and V.I. Okulov for helpful discussions of results of this work. This work was supported by the RFBR, grant 04–02–16614 Some useful identities ====================== We obtain here several identities for the off–diagonal matrix elements $\big\langle G_{{\uparrow\!\downarrow}({\downarrow\!\uparrow})}^{}\big\rangle$ of the one–particle GF (\[eq:gf\_def\]) that are necessary for calculation of the kinetic coefficients (see, for example, Eq. (\[eq:bare\])). The matrix of the one–electron GF satisfies the Dyson equation $$\label{eq:dyson} \left[E-\Sigma_{e}^{}-\frac{{\mbox{\boldmath $\pi$}}_{}^{2}}{2m}-\frac{1}{4}g\omega_{c}^{} \sigma_{z}^{}\right]\langle\hat{G}\rangle-\alpha{\bf n}\cdot ({\mbox{\boldmath $\sigma$}}\times{\mbox{\boldmath $\pi$}}) \langle\hat{G}\rangle=\hat{I}\,,$$ where $\hat{I}$ is the unit $2\times 2$–matrix. It is assumed that the odd part of the electron self–energy $\Sigma_{o}^{}$ is included into the effective Zeeman coupling $g$ (see (\[eq:substitution\])). The ${\uparrow\!\uparrow}$ matrix element of Eq. (\[eq:dyson\]) has the form $$\begin{aligned} \label{eq:dyson_uu_dd} \left[E-\Sigma_{e}^{}-\frac{{\mbox{\boldmath $\pi$}}_{}^{2}}{2m}-\frac{1}{4}g\omega_{c}^{}\right] \langle G_{{\uparrow\!\uparrow}}^{}\rangle-i\alpha\pi_{-}^{}\langle G_{{\downarrow\!\uparrow}}^{}\rangle=1\,, \nonumber\\ \left[E-\Sigma_{e}^{}-\frac{{\mbox{\boldmath $\pi$}}_{}^{2}}{2m}+\frac{1}{4}g\omega_{c}^{}\right] \langle G_{{\downarrow\!\downarrow}}^{}\rangle+i\alpha\pi_{+}^{}\langle G_{{\uparrow\!\downarrow}}^{}\rangle=1\,.\end{aligned}$$ Analogous relations can be obtained from the conjugated Dyson equation (\[eq:dyson\]). From their comparison with (\[eq:dyson\_uu\_dd\]) it follows that $$\begin{aligned} \label{eq:ud_du_ident} \pi_{-}^{}\big\langle G_{{\downarrow\!\uparrow}}^{}\big\rangle=-\big\langle G_{{\uparrow\!\downarrow}}^{}\big\rangle \pi_{+}^{}\,,\quad\pi_{+}^{}\big\langle G_{{\uparrow\!\downarrow}}^{}\big\rangle=-\big\langle G_{{\downarrow\!\uparrow}}^{}\big\rangle \pi_{-}^{}\,.\end{aligned}$$ Now, we replace in (\[eq:dyson\_uu\_dd\]) the matrix elements $\langle G_{{\uparrow\!\uparrow}({\downarrow\!\downarrow})}^{}\rangle$ with their expressions through the one–particle GF of a “ideal” electron (see Eq. (\[eq:gf\_repres\])). As a result, we have the following useful relations between $\langle G_{{\uparrow\!\downarrow}({\downarrow\!\uparrow})}^{}\rangle$ and the one–particle GF of the “ideal” electron $$\label{eq:r_ud_identity} \pi_{\mp}^{}\big\langle G_{{\downarrow\!\uparrow}({\uparrow\!\downarrow})}^{}\big\rangle=\pm im\alpha\sum_{s=\pm 1/2} \frac{E_{0}^{}+s\Omega_{B}^{}}{s\Omega_{B}^{}}G_{{\uparrow\!\uparrow}({\downarrow\!\downarrow})}^{}(E_{0}^{}+ s\Omega_{B}^{})\,.$$ By combining these identities with the corresponding diagonal matrix elements of Eq. (\[eq:gf\_repres\]), we obtain $$\begin{aligned} \label{eq:r_ud_plus} \pi_{\mp}^{}\big\langle G_{{\downarrow\!\uparrow}({\uparrow\!\downarrow})}^{}\big\rangle&\!\!\!\mp&\!\!\! 2im\alpha \big\langle G_{{\uparrow\!\uparrow}({\downarrow\!\downarrow})}^{}\big\rangle\nonumber\\ =\pm 4im &\!\!\!\alpha &\!\!\!\frac{E\pm\omega_{c}^{}\delta}{\Omega_{B}^{}}\!\! \sum_{s=\pm 1/2}\!\!sG_{{\uparrow\!\uparrow}({\downarrow\!\downarrow})}^{}(E_{0}^{}+s\Omega_{B}^{})\,.\end{aligned}$$ Calculation of conductivity =========================== We first consider the derivation of the last equality in Eq. (\[eq:soi\_streda\]). The immediate differentiation of the electron concentration with respect to the magnetic field gives the following result $$\label{eq:streda_1} \left(\frac{\partial n}{\partial B}\right)_{E}^{}=\frac{n}{B}-\frac{1}{\pi} {\mathop{\rm Im}\nolimits}{\mathop{\rm Tr}\nolimits}\big\langle\hat{G}_{}^{A}\big\rangle\frac{\partial\cal H}{\partial B}\,.$$ Now we should calculate the last term in this expression. Using the representations (\[eq:gf\_repres\]), (\[eq:ideal\_gf\]) for the averaged GF, we write down $$\begin{aligned} \label{eq:streda_2} {\mathop{\rm Tr}\nolimits}\big\langle\hat{G}\big\rangle\frac{\partial\cal H}{\partial B}=\frac{1}{2} \sum_{s=\pm1/2}{\mathop{\rm Tr}\nolimits}\hat{G}(E_{0}^{}+s\Omega_{B}^{})\left[1+\phantom{\frac{4}{4}} \right.\nonumber\\ \left. +4s\frac{m\alpha_{}^{2}-\alpha({\mbox{\boldmath $\sigma$}}\times{\mbox{\boldmath $\pi$}})\cdot{\bf n}-\omega_{c}^{} \delta\sigma_{z}^{}}{\Omega_{B}^{}}\right]\frac{\partial\cal H} {\partial B}\,,\end{aligned}$$ Let us multiply together the expression in square brackets and the derivative of Hamiltonian ${\cal H}$ (\[eq:helic\_oper\]), (\[eq:h\_rd\_connect\]), keeping the terms diagonal in the spin space and neglecting the terms linear in $\sigma_{z}^{}$. These latter make contributions proportional to $n_{}^{(0)}(E)-n_{}^{(0)}(E\pm\omega_{c}^{})$ and vanish in the magnetic field region of interest to us. As a result, Eq. (\[eq:streda\_2\]) takes the form $$\begin{aligned} \label{eq:streda_3} {\mathop{\rm Tr}\nolimits}\big\langle\hat{G}\big\rangle\frac{\partial\cal H}{\partial B}= &\!\!\!{\displaystyle}\frac{1}{2}&\!\!\!\sum_{s=\pm1/2}\times\nonumber\\ \times{\mathop{\rm Tr}\nolimits}\hat{G}(E_{0}^{} &\!\!\!+&\!\!\! s\Omega_{B}^{})\left[\frac {\partial{\cal H}_{0}^{}}{\partial B}-\frac{1}{B}\frac{4s}{\Omega_{B}^{}} \omega_{c}^{2}\delta_{}^{2}\right]\,,\end{aligned}$$ where ${\cal H}_{0}^{}$ is the Hamiltonian of the “ideal” electron (\[eq:ideal\_h\]). According to the well known theorem of quantum mechanics, there is the identity $(\partial H/\partial\lambda)_{nn}^{}=\partial E_{n}^{}/ \partial\lambda$, where $E_{n}^{}$ is the $n$–th eigenvalue of the Hermitian operator $H$. This make it possible to perform the following substitution $\partial{\cal H}_{0}^{}/\partial B\to{\cal H}_{0}^{}/ B$ in Eq. (\[eq:streda\_3\]). Then, on simple rearrangements, Eq. (\[eq:streda\_1\]) takes the form $$\begin{aligned} \label{eq:streda_4} \left(\frac{\partial n}{\partial B}\right)_{E}^{}&\!\!\!=&\!\!\!\frac{n}{B}- \frac{1}{2\pi B}\sum_{s=\pm 1/2}\times\nonumber\\ &\!\!\!\times &\!\!\!{\mathop{\rm Im}\nolimits}{\mathop{\rm Tr}\nolimits}\left[E_{0}^{}+s\Omega_{B}^{}-\frac{4s} {\Omega_{B}^{}}\omega_{c}^{2}\delta_{}^{2}\right]\hat{G}_{}^{A}(E_{0}^{}+ s\Omega_{B}^{})\,.\nonumber\\\end{aligned}$$ Eqs. (\[eq:soi\_streda\]), (\[eq:n+-\]) are derived immediately from this equation. Now, let us proceed to the calculation of the dissipative part of the conductivity that is proportional to ${\mathop{\rm Tr}\nolimits}V_{+}^{}\big\langle\hat{G}_{}^{R} \big\rangle V_{-}^{}\big\langle\hat{G}_{}^{A}\big\rangle$ in the SCBA. Performing the trace over the spin degrees of freedom and taking into account the relations (\[eq:ud\_du\_ident\]), we write it down in the form $$\begin{aligned} \label{eq:bare} {\mathop{\rm Tr}\nolimits}V_{+}^{}\big\langle\hat{G}_{}^{R}\big\rangle V_{-}^{}\big\langle \hat{G}_{}^{A}\big\rangle =\frac{1}{m_{}^{2}}{\mathop{\rm Sp}\nolimits}\Big[\pi_{+}^{} \big\langle G_{{\uparrow\!\uparrow}}^{R}\big\rangle\pi_{-}^{}\big\langle G_{{\uparrow\!\uparrow}}^{A}\big\rangle +\pi_{+}^{}\big\langle G_{{\downarrow\!\downarrow}}^{R}\big\rangle\pi_{-}^{}\big\langle G_{{\downarrow\!\downarrow}}^{A}\big\rangle-4m_{}^{2}\alpha_{}^{2} \langle G_{{\downarrow\!\downarrow}}^{R}\big\rangle\big\langle G_{{\uparrow\!\uparrow}}^{A}\big\rangle+\nonumber\\ +2\big(\pi_{+}^{}\big\langle G_{{\uparrow\!\downarrow}}^{R}\big\rangle+2im\alpha\big\langle G_{{\downarrow\!\downarrow}}^{R}\big\rangle\big)\big(\pi_{-}^{}\big\langle G_{{\downarrow\!\uparrow}}^{A}\big\rangle -2im\alpha\big\langle G_{{\uparrow\!\uparrow}}^{A}\big\rangle\big)\Big]\,.\end{aligned}$$ Two last terms in the right–hand side of this equation are calculated using identities (\[eq:gf\_repres\]), (\[eq:r\_ud\_plus\]). We present a more detaled calculation of one of the two first terms in (\[eq:bare\]). For example, let us substitute, in the first term in Eq. (\[eq:bare\]), the diagonal matrix elements $({\uparrow\!\uparrow})$ of (\[eq:gf\_repres\]) for $\big\langle\hat{G}\big\rangle$, and use the identity $$\label{eq:bare_1} G_{{\uparrow\!\uparrow}}^{R}(E_{0}^{}+s\Omega_{B}^{})\pi_{-}^{}G_{{\uparrow\!\uparrow}}^{A}(E_{0}^{} +s_{}'\Omega_{B}^{})=\frac{\pi_{-}^{}G_{{\uparrow\!\uparrow}}^{A}(E_{0}^{}+s_{}'\Omega_{B}^{})- G_{{\uparrow\!\uparrow}}^{R}(E_{0}^{}+s\Omega_{B}^{})\pi_{-}^{}}{{\displaystyle}\frac{i}{\tau_{e}^{}}+ \omega_{c}^{}+s\Omega_{B}^{R}-s_{}'\Omega_{B}^{A}}\,.$$ Then, after some simple but cumbersome algebra, we can rewrite the contribution of this term to the conductivity in the following form $$\begin{aligned} \label{eq:bare_2} \sigma\,\to\,\frac{e_{}^{2}}{\pi m}\sum_{s}\Phi_{s,{\uparrow\!\uparrow}}^{R}\Phi_{s,{\uparrow\!\uparrow}}^{A} \left\{\frac{\left(E_{0}^{}+s\Omega_{B}^{}-{\displaystyle}\frac{\omega_{c}^{}}{2}\right) \tau_{s}^{}}{1-i\omega_{c}^{}\tau_{s}^{}}{\mathop{\rm Im}\nolimits}{\mathop{\rm Sp}\nolimits}G_{{\uparrow\!\uparrow}}^{A}(E_{0}^{}+s \Omega_{B}^{})-\frac{1}{2}{\mathop{\rm Re}\nolimits}{\mathop{\rm Sp}\nolimits}G_{{\uparrow\!\uparrow}}^{A}(E_{0}^{}+s\Omega_{B}^{})\big] \right\}+\nonumber\\ +\frac{e_{}^{2}}{2\pi im}\sum_{s}\Phi_{s,{\uparrow\!\uparrow}}^{R}\Phi_{-s,{\uparrow\!\uparrow}}^{A}\left\{ \frac{\left(E_{0}^{}+{\displaystyle}\frac{s}{2}(\Omega_{B}^{R}-\Omega_{B}^{A})-{\displaystyle}\frac {\omega_{c}^{}}{2}\right)\tau_{e}^{}}{1-i(\omega_{c}^{}+2s\Omega_{B}^{}) \tau_{e}^{}}{\mathop{\rm Sp}\nolimits}\big[G_{{\uparrow\!\uparrow}}^{A}(E_{0}^{}-s\Omega_{B}^{})-G_{{\uparrow\!\uparrow}}^{R}(E_{0}^{}+ s\Omega_{B}^{})\big]+\right.\nonumber\\ \left.\phantom{\frac{{\displaystyle}\frac{s}{2}}{C}}+\frac{1}{2i}{\mathop{\rm Sp}\nolimits}\big[G_{{\uparrow\!\uparrow}}^{A} (E_{0}^{}-s\Omega_{B}^{})+G_{{\uparrow\!\uparrow}}^{R}(E_{0}^{}+s\Omega_{B}^{})\big]\right\}\,.\end{aligned}$$ The last terms in curly brackets are cancelled exactly by the corresponding terms from Eq. (\[eq:rr\_aa\_part\]). The contribution of the second term from Eq. (\[eq:bare\]) can be transformed in a similar way. Of course, it is necessary to perform a set of unwieldy some transformations to obtain Eqs. (\[eq:bare\_sigma\]), (\[eq:new\_n+-\]), and (\[eq:delta\_sigma\]). However, further calculations have a purely technical character and we omit them. [00]{} I. Zutic, J. Fabian, S. Das Sarma, Rev. Mod. Phys. [**76**]{}, 323 (2004). G. Dresselhaus, Phys. Rev. [**100**]{}, 580 (1955). E.I. Rashba, Fiz. Twerd. Tela [**2**]{}, 1224 (1960) \[Sov. Phys. Solid. State [**2**]{}, 1109 (1060)\]; Yu.A. Bychkov, E.I. Rashba, Pis’ma ZhETF [**39**]{}, 66 (1984) \[JETP Lett. [**39**]{}, 78 (1984)\]; Yu.A. Bychkov, E.I. Rashba, J. Phys. C: Solid State Phys., [**17**]{}, 6039 (1984). J. Luo, H. Munekata, F.F. Fang, P.J. Stiles, Phys. Rev. B[**38**]{}, 10142 (1988); Phys. Rev. B[**41**]{}, 7685 (1990). S.V. Iordanskii, Yu.B. Lyanda–Geller, G.E. Pikus, Pis’ma ZhETF [**60**]{}, 199 (1994) \[JETP Lett. [**60**]{}, 206 (1994)\]. F.G. Pikus, G.E. Pikus, Phys. Rev. B[**51**]{}, 16928 (1995). W. Knap, C. Skierbiszewski, A. Zduniak, et al., Phys. Rev. B [**53**]{}, 3912 (1996). L.E. Golub, Phys. Rev. B [**71**]{}, 235310 (2005). L.S. Levitov, Yu.V. Nazarov, G.M. Eliashberg, Zh. Eksper. Teor. Fiz., [**88**]{} 229 (1985) \[Sov. Phys. JETP [**61**]{}, 133 (1985)\]. A.G. Aronov, Yu.B. Lyanda–Geller, Pis’ma ZhETF, [**50**]{}, 398 (1989) \[JETP Lett., [**50**]{}, 431 (1989)\]. V.M. Edelstein, Solid State Commun. [**73**]{}, 233 (1990). M.I. Dyakonov, V.I. Perel’, Pis’ma ZhETF, [**13**]{}, 657 (1971) \[JETP Lett. [**13**]{}, 467 (1971)\]. S. Murakami, N. Nagaosa, S.C. Zhang, Science [**301**]{}, 1348 (2003). X.F. Wang, P. Vasilopoulos, Phys. Rev. B[**67**]{}, 085313 (2003). M. Langenbuch, M. Suhrke, U. Rössler, Phys. Rev. B[**69**]{}, 125303 (2004). N.S. Averkiev, M.M. Glazov, S.A. Tarasenko, Solid State Commun., [**133**]{}, 543 (2005). X.F. Wang, P. Vasilopoulos, preprint LANL cond-mat/0501214. J. Nitta, T. Akazaki, H. Takayanagi, T. Enoki, Phys. Rev. Lett., [**78**]{}, 1335 (1997); T. Koga, J. Nitta, T. Akazaki, H. Takayanagi, Phys. Rev. Lett., [**89**]{}, 046801 (2002). R.R. Gerhardts, Z. Phys. B[**22**]{}, 327 (1975). M.I. Dyakonov, V.I. Perel’, Fiz. Twerd. Tela, [**13**]{}, 3581 (1971) \[Sov. Phys. Solid State, [**13**]{}, 3023 (1971)\]. I.M. Tsidil’kovskii [*Zonnaya struktura poluprovodnikov*]{}, Наука, М. (1978); \[I.M. Tsidilkovski, [*Band Structure of Semiconductors*]{}, Oxford etc.; Pergamon Press (1982)\]. S.A. Tarasenko, N.S. Averkiev, Pis’ma ZhETF, [**75**]{}, 669 (2002) \[JETP Lett. [**75**]{}, 562 (2002)\]. L. Smrcka, P. Streda, J.Phys. C: Solid State Phys., [**10**]{}, 2153 (1977); P. Streda, J.Phys. C: Solid State Phys., [**15**]{}, L717 (1982). A.M.M. Pruisken, In [*The Quantum Hall Effect*]{}, ed. by R.E. Prange, and S.M. Girvin, Springer–Verlag, New York e.a. (1990). J. Inoue, G.E.W. Bauer, L.W.  Molenkamp, Phys. Rev. B[**67**]{}, 033104 (2003); Phys. Rev. B[**70**]{}, 041303 (2004). J.M. Ziman, [*Principles of the Theory of Solids*]{}, Cambridge, University Pres (1972).

--- abstract: 'Security is an important issue in wireless sensor networks (WSNs), which are often deployed in hostile environments. The $q$-composite key predistribution scheme has been recognized as a suitable approach to secure WSNs. Although the $q$-composite scheme has received much attention in the literature, there is still a lack of rigorous analysis for secure WSNs operating under the $q$-composite scheme in consideration of the unreliability of links. One main difficulty lies in analyzing the network topology whose links are not independent. Wireless links can be unreliable in practice due to the presence of physical barriers between sensors or because of harsh environmental conditions severely impairing communications. In this paper, we resolve the difficult challenge and investigate topological properties related to node degree in WSNs operating under the $q$-composite scheme with unreliable communication links modeled as independent on/off channels. Specifically, we derive the asymptotically exact probability for the property of minimum degree being at least $k$, present the asymptotic probability distribution for the minimum degree, and demonstrate that the number of nodes with an arbitrary degree is in distribution asymptotically equivalent to a Poisson random variable. We further use the theoretical results to provide useful design guidelines for secure WSNs. Experimental results also confirm the validity of our analytical findings.' author: - 'Jun Zhao, [^1]' title: 'Topological Properties of Secure Wireless Sensor Networks under the $q$-Composite Key Predistribution Scheme with Unreliable Links' --- [ZHAO: Topological Properties of Secure Sensor Networks under $q$-Composite Key Predistribution with Unreliable Links]{} Security, key predistribution, wireless sensor networks, random graphs, topological properties. Introduction ============ Wireless sensor networks (WSNs) enable a broad range of applications including military surveillance, home automation, and patient monitoring [@adrian]. In many scenarios, since WSNs are deployed in adversarial environments, security becomes an important issue. To this end, key predistribution has been recognized as a typical solution to secure WSNs [@virgil]. The idea is to randomly assign cryptographic keys to sensors before network deployment. Various key predistribution schemes have been studied in the literature [@ZhaoYaganGligor; @Rybarczyk; @yagan_onoff; @zhao2016resilience; @adrian; @virgil; @Krzywdzi; @ryb3; @zhao2015threshold; @yagan; @zhao2015resilience; @nikoletseas2015some; @ICASSP17-social]. The $q$-composite key predistribution scheme proposed by Chan *et al.* [@adrian] as an extension of the Eschenauer-Gligor scheme [@virgil] (the $q$-composite scheme in the case of $q=1$) has received much interest [@yagan; @zhao2015resilience; @nikoletseas2015some; @ICASSP17-social; @bloznelis2013; @GlobalSIP15-parameter; @ISIT_RKGRGG; @ICASSP17-design] since its introduction. The $q$-composite scheme when $q\geq 2$ outperforms the Eschenauer-Gligor scheme in terms of the strength against small-scale network capture attacks while trading off increased vulnerability in the face of large-scale attacks. The $q$-composite scheme [@adrian] works as follows. For a WSN with $n$ sensors, prior to deployment, each sensor is independently assigned $K_n$ different keys which are selected uniformly at random from a pool of $P_n$ keys, where $K_n$ and $P_n$ are both functions of $n$, with $K_n \leq P_n$. Then two sensors establish a link in between after deployment if and only if they share at least $q$ keys *and* the physical link constraint between them is satisfied. Examples of physical link constraints include the reliability of the transmission channel [@ZhaoYaganGligor; @yagan_onoff] and the requirement that the distance between two sensors should be close enough for communication [@ISIT_RKGRGG]. Communication links between sensor nodes may not be available due to the presence of physical barriers between nodes or because of harsh environmental conditions severely impairing transmission. To represent unreliable links, we use the *on*/*off* channel model where each link is either [*on*]{} (i.e., [*active*]{}) with probability $p_n$ or [*off*]{} (i.e., [*inactive*]{}) with probability $(1-p_n)$, where $p_n$ is a function of $n$ for generality. In addition to link failure, sensor nodes are also prone to failure in WSNs deployed in hostile environments. To ensure reliability against the failure of sensors, we study the property of minimum degree being at least $k$ so that each sensor is directly connected to at least $k$ other sensors. This means that a sensor may still be connected to a sufficient number of sensors even if some neighbors fail. Note that the degree of a node $v$ is the number of nodes having links with $v$; and the minimum (node) degree of a network is the least among the degrees of all nodes. Another related graph property is $k$-connectivity, which is stronger than the property of minimum degree being at least $k$. A network (or a graph) is said to be $k$-connected if it remains connected despite the deletion of any $(k - 1)$ nodes [@shahrivar2015robustness; @dibaji2015consensus]; a network is simply deemed connected if it is $1$-connected. Hence, $k$-connectivity provides a guarantee of network reliability against the failure of $(k - 1)$ sensors due to adversarial attacks, battery depletion, harsh environmental conditions, etc. In view of the above, we investigate topological properties related to node degree in WSNs employing the $q$-composite key predistribution scheme under the *on*/*off* channel model as the physical link constraint comprising independent channels which are either *on* or *off*. Specifically, we derive the asymptotically exact probabilities for the property of minimum degree being at least $k$, establish the asymptotic probability distribution for the minimum degree, and show that the number of nodes with an arbitrary degree is in distribution asymptotically equivalent to a Poisson random variable. Our results are useful for designing secure WSNs under link and node failure. We summarize our contributions in the following two subsections. We first present our results on node degree for a secure WSN employing the $q$-composite key predistribution scheme under the on/off channel model. Then we use the results to provide useful design guidelines for secure WSNs. **Results** ----------- For $\mathbb{G}_q\iffalse_{on}\fi$ denoting a secure sensor network with the $q$-composite key predistribution scheme under the on/off channel model, we present several results related to node degree, by considering the conditions on $p_{e, q}$, which denotes the probability of a secure link between two sensors. The secure link probability $p_{e, q}$ is given by $p_{e, q} = p_n \cdot \left[1- \sum_{u=0}^{q-1} \frac{\binom{K_n}{u}\binom{P_n-K_n}{K_n-u}}{\binom{P_n}{K_n}}\right]$, as shown in Equation (\[psq2cijFC3\]) on Page later. For the network $\mathbb{G}_q\iffalse_{on}\fi$, we now present the results, which are further elaborated in Section \[sec-results-Gq\]. - First, we derive the asymptotically exact probabilities for the property of minimum degree being at least $k$. Specifically, if $p_{e, q} = \frac{\ln n + {(k-1)} \ln \ln n + {\alpha_n}}{n}$ for a constant integer $k \geq 1$ and a sequence $\alpha_n$ satisfying $\lim_{n \to \infty} \alpha_n \in [-\infty, \infty]$, then the probability that $\mathbb{G}_q\iffalse_{on}\fi$ has a minimum degree at least $k$ converges to $e^{- \frac{e^{-\lim_{n \to \infty} \alpha_n }}{(k-1)!}}$, which equals (i) $e^{- \frac{e^{-\alpha ^*}}{(k-1)!}}$ if $\lim_{n \to \infty} \alpha_n = \alpha ^* \in (-\infty, \infty)$, (ii) $1$ if $\lim_{n \to \infty} \alpha_n = \infty$, and (iii) $0$ if $\lim_{n \to \infty} \alpha_n = -\infty$. - We extend the above result to provide the asymptotic probability distribution for the minimum degree. Specifically, when $\alpha_n$ above can be written as $\alpha_n = b \ln \ln n + \beta_n$ for a constant integer $b$ and a sequence $\beta_n$ satisfying $-1 < \liminf_{n \to \infty} \frac{\beta_n}{\ln \ln n} \leq \limsup_{n \to \infty} \frac{\beta_n}{\ln \ln n}< 1$ (i.e., $c_1 \ln \ln n \leq \beta_n \leq c_2 \ln \ln n $ for constants $-1<c_1 \leq c_2 < 1$), we have the following: - if $k+b \geq 1$ and $\lim_{n \to \infty} \beta_n \in [-\infty, \infty]$, then the minimum degree of $\mathbb{G}_q\iffalse_{on}\fi$ in the asymptotic sense equals (i) $k+b$ with probability $e^{- \frac{e^{-\lim_{n \to \infty} \beta_n }}{(k-1)!}}$, (ii) $k+b-1$ with probability $1-e^{- \frac{e^{-\lim_{n \to \infty} \beta_n }}{(k-1)!}}$, and (iii) other values with probability $0$; - if $k+b \leq 0$, then the minimum degree of $\mathbb{G}_q\iffalse_{on}\fi$ in the asymptotic sense equals $0$ with probability $1$. - Our results on minimum degree are obtained by analyzing the number of nodes with a fixed degree. Specifically, we show that for a non-negative constant integer $h$, the number of nodes in $\mathbb{G}_q\iffalse_{on}\fi$ with degree $h$ is in distribution asymptotically equivalent to a Poisson random variable with mean $ n (h!)^{-1}(n p_{e, q})^h e^{-n p_{e, q}}$. **Design guidelines for secure sensor networks** ------------------------------------------------ Based on the above results, for $\mathbb{G}_q\iffalse_{on}\fi$ denoting a secure sensor network employing the $q$-composite key predistribution scheme under the on/off channel model, we obtain several guidelines below for choosing parameters to ensure that the network $\mathbb{G}_q\iffalse_{on}\fi$ has certain minimum node degree. The guidelines are given by enforcing conditions on $p_{e, q}$, the probability of a secure link between two sensors. Note that $p_{e, q} = p_n \cdot \left[1- \sum_{u=0}^{q-1} \frac{\binom{K_n}{u}\binom{P_n-K_n}{K_n-u}}{\binom{P_n}{K_n}}\right]$; see Equation (\[psq2cijFC3\]) later. For the network $\mathbb{G}_q\iffalse_{on}\fi$, we now present the design guidelines, which are further explained in Section \[sec-design-guidelines\]. - First, to ensure that the network $\mathbb{G}_q\iffalse_{on}\fi$ has a minimum degree no less than $k$ (i.e., to ensure that each sensor is directly connected to at least $k$ other sensors), we can choose network parameters to have $$\begin{aligned} p_{e, q} \geq \frac{\ln n + {(k+c_1-1)} \ln \ln n }{n} \text{ for a constant $c_1 > 0$,} \label{peq1sbsc-critical1-repeat}$$ where the positive constant $c_1$ can be arbitrarily small. - Second, to guarantee that the network $\mathbb{G}_q\iffalse_{on}\fi$ has a minimum degree at least $k$ with probability no less than $\rho$, we choose parameters to have $$\begin{aligned} p_{e, q} \geq \frac{\ln n + {(k-1)} \ln \ln n - \ln [ (k-1)! \ln \frac{1}{\rho}] }{n} \label{peq1sbsc-critical1-with-prob-rho-repeat}.$$   - Third, to ensure that the network $\mathbb{G}_q\iffalse_{on}\fi$ has a minimum degree being $k$ *exactly*, we can choose network parameters to have $$\begin{aligned} p_{e, q} = \frac{\ln n + {(k+c_2-1)} \ln \ln n }{n} \text{ for a constant $0<c_2 <1$,} \label{peq1sbsc-critical2-repeat}$$ where the positive constant $c_2$ can be arbitrarily small. Roadmap ------- We organize the rest of the paper as follows. Section \[sec:SystemModel\] describes the system model in detail. Afterwards, we elaborate and discuss the results in Section \[sec:res\]. In Section \[sec-prove-first-two-theorems-mnd\], we prove Theorems \[thm:exact\_qcomposite\] and \[thm:exact\_qcomposite-more-fine-grained\] using Theorem \[thm:exact\_qcomposite2\]. In Section \[sec\_est\], we detail the steps of establishing Theorem \[thm:exact\_qcomposite2\] through Lemma \[LEM1\]. Section \[secprf:lem\_pos\_exp\] provides the proof of Lemma \[LEM1\] by the help of Propositions \[PROP\_ONE\] and \[PROP\_SND\], which are proved in Sections \[sec:PROP\_ONE\] and \[sec:PROP\_SND\], respectively. Subsequently, we present experiments in Section \[sec:expe\] to confirm our analytical results. Section \[related\] is devoted to relevant results in the literature. Next, we conclude the paper and identify future research directions in Section \[sec:Conclusion\], followed by the Appendix. System Model {#sec:SystemModel} ============ Our approach to the analysis is to explore the induced random graph models of the WSNs. As will be clear soon, the graph modeling a WSN under $q$-composite scheme and the on/off channel model is an intersection of two graphs belonging to different kinds, which renders the analysis challenging due to the intertwining of the two distinct types of random graphs [@yagan_onoff; @ISIT]. We elaborate the graph modeling of a WSN with $n$ sensors, which employs the $q$-composite key predistribution scheme and works under the [on/off]{} channel model. We consider a node set $\mathcal {V} = \{v_1, v_2, \ldots, v_n \}$ to represent the $n$ sensors (a sensor is also referred to as a node). For each node $v_i \in \mathcal {V}$, the set of its $K_n$ different keys is denoted by $S_i$, which is uniformly distributed among all $K_n$-size subsets of a key pool of $P_n$ keys, and is referred to as the key ring of node $v_i$. The $q$-composite key predistribution scheme is modeled by a graph denoted by $G_q(n,K_n,P_n)$, which is defined on the vertex set $\mathcal{V}$ such that any two different nodes $v_i$ and $v_j$ sharing at least $q$ keys (such event is denoted by $\Gamma_{ij}$) have an edge in between. With $S_{ij} : = S_{i} \cap S_{j}$, event $\Gamma_{ij}$ equals $\big[ |S_{ij}| \geq q \big]$, where $|A|$ with $A$ as a set means the cardinality of $A$. As discussed, under the [on/off]{} channel model, each node-to-node channel independently has probability $p_n $ of being [*on*]{} and probability $(1-p_n)$ of being [*off*]{}, where $p_n$ is a function of $n$. Denoting by ${B}_{i j}$ the event that the channel between distinct nodes $v_i$ and $v_j$ is [*on*]{}, we have ${{\mathbb{P}}\left[{C_{ij}}\right]} = p_n$, where $\mathbb{P}[\mathcal {E}]$ denotes the probability that event $\mathcal {E}$ happens, throughout the paper. The [on/off]{} channel model is represented by an Erdős-Rényi graph $G(n, p_n)$ [@citeulike:4012374] defined on the node set $\mathcal{V}$ such that $v_i$ and $v_j$ have an edge in between if event $C_{ij}$ happens. Finally, we denote by $\mathbb{G}_q\iffalse_{on}\fi (n, K_n, P_n, p_n)$ the underlying graph of the $n$-node WSN operating under the $q$-composite key predistribution scheme and the on/off channel model. We often write $\mathbb{G}_q$ rather than $\mathbb{G}_q(n, K_n, P_n, p_n)$ for notation brevity. Graph $\mathbb{G}_q\iffalse_{on}\fi$ is defined on the node set $\mathcal{V}$ such that there exists an edge between nodes $v_i$ and $v_j$ if events $\Gamma_{ij}$ and $C_{ij}$ happen at the same time. We set event $E_{ij} : = \Gamma_{ij} \cap C_{ij}$ and also write $E_{ij} $ as $E_{v_i v_j} $ when necessary. It is clear that $\mathbb{G}_q\iffalse_{on}\fi$ can be seen as the intersection of $G_q(n, K_n, P_n)$ and $G(n, p_n)$, meaning $$\mathbb{G}_q\iffalse_{on}\fi = G_q(n, K_n, P_n) \cap G(n, p_n). \label{eq:G_on_is_RKG_cap_ER_oyton}$$ We define $p_{s,q} $ as the probability that two different nodes share at least $q$ keys and $p_{e,q} $ as the probability that two distinct nodes have a link in between, where the subscripts “s” and “e” are short for “secure” and “edge”, respectively. $p_{s,q} $ and $p_{e,q}$ both rely on $K_n, P_n$ and $q$, while $p_{e,q}$ also depends on $p_n$. Under $P_n \geq 2 K_n$, we determine $p_{s,q}$ through $$\begin{aligned} p_{s,q}= \mathbb{P} [\Gamma_{i j} ] & = \sum_{u=q}^{K_n} \mathbb{P}[|S_{i} \cap S_{j}| = u] \nonumber \\ & = 1 - \sum_{u=0}^{q-1} \mathbb{P}[|S_{i} \cap S_{j}| = u] , \label{psq1}\end{aligned}$$ where $$\begin{aligned} & \mathbb{P}[|S_{i} \cap S_{j}| = u] = \frac{\binom{K_n}{u}\binom{P_n-K_n}{K_n-u}}{\binom{P_n}{K_n}} , \textrm{ for } u = 0,1, \ldots, K_n, \label{psq2}\end{aligned}$$ since $S_{i}$ and $S_{j}$ are independently and uniformly selected from all $K_n$-size subsets of a key pool with size $P_n$. Then by the independence of events ${C}_{i j} $ and $ \Gamma_{i j} $, we obtain $$\begin{aligned} {p_{e,q}} & = \mathbb{P} [E_{i j} ] = \mathbb{P} [{C}_{i j} ] \cdot \mathbb{P} [\Gamma_{i j} ] = p_n\iffalse_{on}\fi \cdot p_{s,q}. \label{eq_pre}\end{aligned}$$ Summarizing (\[psq1\]) (\[psq2\]) (\[eq\_pre\]), we derive that under $P_n \geq 2 K_n$, the link probability $p_{e,q}$ is given by $$\begin{aligned} p_{e,q} & = p_n \cdot \left[1- \sum_{u=0}^{q-1} \frac{\binom{K_n}{u}\binom{P_n-K_n}{K_n-u}}{\binom{P_n}{K_n}}\right] .\label{psq2cijFC3} \end{aligned}$$ The Results and Discussion {#sec:res} ========================== We present and discuss the results in this section. Throughout the paper, $q$ is a positive integer and does not scale with $n$; $\mathbb{N}_0 $ stands for the set of all positive integers; $\mathbb{R}$ is the set of all real numbers; $e$ is the base of the natural logarithm function, $\ln$; and the floor function $\lfloor x \rfloor$ is the largest integer not greater than $x$. We consider $e^{\infty} = \infty$ and $e^{-\infty} = 0$. The term “for all $n$ sufficiently large” means “for any $n \geq N$, where $N \in \mathbb{N}_0$ is selected appropriately”. As already mentioned, all asymptotic statements are understood with $n \to \infty$, and we use the standard asymptotic notation $o(\cdot), O(\cdot), \omega(\cdot), \Omega(\cdot),\Theta(\cdot), \sim$; see [@ZhaoYaganGligor Page 2-Footnote 1]. In particular, for two positive sequences $f_n$ and $g_n$, $f_n \sim g_n$ signifies $\lim_{n \to \infty}\frac{{f_n}}{g_n}=1$; namely, $f_n$ and $g_n$ are asymptotically equivalent. [^2] if (\[peq1sbsc\]) holds for a constant integer $k>0$ and a sequence $\alpha_n$ satisfying The Results of Graph $\mathbb{G}_q\iffalse_{on}\fi$ {#sec-results-Gq} --------------------------------------------------- We now present the results of graph $\mathbb{G}_q\iffalse_{on}\fi$ below. Theorem \[thm:exact\_qcomposite\] provides the probability of minimum degree being at least $k$ in $\mathbb{G}_q\iffalse_{on}\fi$. \[thm:exact\_qcomposite\] For graph $\mathbb{G}_q\iffalse_{on}^{(q)}\fi$ with $ K_n = \omega(1)$ and $\frac{{K_n}^2}{P_n} = o(1)$, if there exist a constant integer $k \geq 1$ and a sequence $\alpha_n$ satisfying $\lim\limits_{n \to \infty} \alpha_n \in [-\infty, \infty]$ such that $$\begin{aligned} p_{e, q} = \frac{\ln n + {(k-1)} \ln \ln n + {\alpha_n}}{n}, \label{peq1sbsc}$$ then with $\delta$ denoting the minimum degree of $\mathbb{G}_q\iffalse_{on}\fi$, we have $$\begin{aligned} \hspace{-114pt}\lim\limits_{n \to \infty}{{\mathbb{P}}\left[{\delta \geq k}\right]} \nonumber \end{aligned}$$   [ = ]{} e\^[- ]{}, & $\lim\limits_{n \to \infty} \alpha_n\hspace{-1pt} =\hspace{-1pt} \alpha ^* \in (-\infty, \infty)$, \[thm-mnd-alpha-finite\]\ 1, & $\lim\limits_{n \to \infty} \alpha_n \hspace{-1pt}=\hspace{-1pt} \infty$, \[thm-mnd-alpha-infinite\]\ 0, & $\lim\limits_{n \to \infty} \alpha_n \hspace{-1pt}= \hspace{-1pt}- \infty$. \[thm-mnd-alpha-minus-infinite\] \[thm:exact\_qcomposite-rem\] The results (\[thm-mnd-alpha-finite\]) (\[thm-mnd-alpha-infinite\]) (\[thm-mnd-alpha-minus-infinite\]) can be compactly summarized as $\lim_{n \to \infty}{{\mathbb{P}}\left[{\delta \geq k}\right]}=e^{- \frac{e^{-\lim_{n \to \infty} \alpha_n }}{(k-1)!}}$. **Interpreting Theorem \[thm:exact\_qcomposite\].** Theorem \[thm:exact\_qcomposite\] for graph $\mathbb{G}_q\iffalse_{on}\fi$ presents the asymptotically exact probability and a zero–one law for the event that $\mathbb{G}_q\iffalse_{on}\fi$ has a minimum degree no less than $k$, where a zero–one law means that the probability of a graph having a certain property asymptotically converges to $0$ under some conditions and to $1$ under some other conditions. To establish Theorem \[thm:exact\_qcomposite\], we explain the basic ideas in Section \[sec-basic-proof-ideas\], and more technical details in Section \[sec-prove-first-two-theorems-mnd\]. While Theorem \[thm:exact\_qcomposite\] above is for the property of minimum degree being at least some value, we now present Theorem \[thm:exact\_qcomposite-more-fine-grained\] below, which gives a more fine-grained result to provide the asymptotic probability distribution for the minimum degree. \[thm:exact\_qcomposite-more-fine-grained\] Under the conditions of Theorem \[thm:exact\_qcomposite\], if $\alpha_n$ in Equation (\[peq1sbsc\]) can be written as $$\begin{aligned} \alpha_n = b \ln \ln n + \beta_n \label{alpha-n-written-beta-n}\end{aligned}$$ for a constant integer $b$ and a sequence $\beta_n$ satisfying $$\begin{aligned} -1 < \liminf_{n \to \infty} \frac{\beta_n}{\ln \ln n} \leq \limsup_{n \to \infty} \frac{\beta_n}{\ln \ln n}< 1, \label{liminfbetan}\end{aligned}$$ then with $\delta$ denoting the minimum degree of $\mathbb{G}_q\iffalse_{on}\fi$, the properties – below follow: - for ${k+b \leq 0}$ (which implies $b \leq -k \leq - 1$ given $k \geq 1$), we have \_[n ]{} =1, \[thm-mnd-delta-0\]\ \_[n ]{} =0; \[thm-mnd-delta-non-0\] - - and for ${k+b \geq 1}$ (i.e., $b \geq 1-k$), we obtain properties –: - $\lim\limits_{n \to \infty}{{\mathbb{P}}\left[{(\delta = k+b) \text{ or } (\delta = k+b-1)}\right]}=1$; - if $\lim\limits_{n \to \infty} \beta_n = \beta ^* \in (-\infty, \infty)$, then \_[n ]{} =e\^[- ]{}, \[thm-mnd-beta-finite-1\]\ \_[n ]{} =1-e\^[- ]{}; \[thm-mnd-beta-finite-1\] - if $ \lim\limits_{n \to \infty} \beta_n = \infty$, then \_[n ]{} =1, \[thm-mnd-beta-infinite-1\]\ \_[n ]{} =0; \[thm-mnd-beta-infinite-2\] - if $ \lim\limits_{n \to \infty} \beta_n = - \infty$, then \_[n ]{} =1, \[thm-mnd-beta-minus-infinite-1\]\ \_[n ]{} =0.\[thm-mnd-beta-minus-infinite-2\] \[thm:exact\_qcomposite-more-fine-grained-rem\] The above results – for $k+b \geq 1$ and $\lim_{n \to \infty} \beta_n \in [-\infty, \infty]$ can be compactly summarized as that the minimum degree of $\mathbb{G}_q\iffalse_{on}\fi$ in the asymptotic sense equals (i) $k+b$ with probability $e^{- \frac{e^{-\lim_{n \to \infty} \beta_n }}{(k-1)!}}$, (ii) $k+b-1$ with probability $1-e^{- \frac{e^{-\lim_{n \to \infty} \beta_n }}{(k-1)!}}$, and (iii) other values with probability $0$, while results says that if $k+b \leq 0$, then the minimum degree of $\mathbb{G}_q\iffalse_{on}\fi$ in the asymptotic sense equals $0$ with probability $1$. **Interpreting Theorem \[thm:exact\_qcomposite-more-fine-grained\].** Theorem \[thm:exact\_qcomposite-more-fine-grained\] presents the asymptotic probability distribution for the minimum degree. We explain that Theorem \[thm:exact\_qcomposite-more-fine-grained\] is more fine-grained than Theorem \[thm:exact\_qcomposite\]. We discuss first Theorem \[thm:exact\_qcomposite-more-fine-grained\]’s result and then its results –. - In result above, $b \leq -k \leq - 1$ follows from $k+b \leq 0$ and $k \geq 1$. Using $b \leq - 1$ and (\[liminfbetan\]) in (\[alpha-n-written-beta-n\]), we have $\lim_{n \to \infty} \alpha_n = - \infty$, so we use (\[thm-mnd-alpha-minus-infinite\]) of Theorem \[thm:exact\_qcomposite\] to obtain $\delta<k$ almost surely (an event happens *almost surely* if its probability converges $1$ as $n \to \infty$), where $\delta$ denotes the minimum degree of $\mathbb{G}_q\iffalse_{on}\fi$. For comparison, (\[thm-mnd-delta-0\]) of Theorem \[thm:exact\_qcomposite-more-fine-grained\] presents the stronger result that $\delta=0$ almost surely. - In the above results – where ${k+b \geq 1}$ holds (i.e., $b \geq 1-k$) , we derive from (\[alpha-n-written-beta-n\]) and (\[liminfbetan\]) that [\_[n ]{} \_n =]{} , & \[b-geq-1\]\ \^\* , & \[b-eq-0-beta-finite\]\ , & \[b-eq-0-beta-infinite\]\ -, & \[b-eq-0-beta-minus-infinite\]\ -, & \[b-leq-minus-1\] Below we discuss (\[b-geq-1\])–(\[b-leq-minus-1\]), respectively. - For [(\[b-geq-1\])]{} above, (\[thm-mnd-alpha-infinite\]) of Theorem \[thm:exact\_qcomposite\] says $\delta\geq k$ almost surely, while of Theorem \[thm:exact\_qcomposite-more-fine-grained\] presents the stronger result that $\delta$ equals $k+b$ or $k+b-1$ almost surely (note $b\geq 1$ in (\[b-geq-1\])). - For [(\[b-eq-0-beta-finite\])]{} above, (\[thm-mnd-alpha-finite\]) of Theorem \[thm:exact\_qcomposite\] says $\delta\geq k$ with probability $e^{- \frac{e^{-\beta ^*}}{(k+b-1)!}}$ asymptotically, while of Theorem \[thm:exact\_qcomposite-more-fine-grained\] presents the stronger result that $\delta$ equals $k$ (note $b=0$ in (\[b-eq-0-beta-finite\]) here) with probability $e^{- \frac{e^{-\beta ^*}}{(k+b-1)!}}$ asymptotically, and equals $k-1$ with probability $1-e^{- \frac{e^{-\beta ^*}}{(k+b-1)!}}$ asymptotically (note $b=0$ in (\[b-eq-0-beta-finite\]) here). - For [(\[b-eq-0-beta-infinite\])]{} above, (\[thm-mnd-alpha-infinite\]) of Theorem \[thm:exact\_qcomposite\] says $\delta\geq k$ almost surely, while of Theorem \[thm:exact\_qcomposite-more-fine-grained\] presents the stronger result that $\delta=k$ almost surely (note $b=0$ in (\[b-eq-0-beta-infinite\]) here). - For [(\[b-eq-0-beta-minus-infinite\])]{} above, (\[thm-mnd-alpha-minus-infinite\]) of Theorem \[thm:exact\_qcomposite\] says $\delta < k$ almost surely, while of Theorem \[thm:exact\_qcomposite-more-fine-grained\] presents the stronger result that $\delta=k-1$ almost surely (note $b=0$ in (\[b-eq-0-beta-minus-infinite\]) here). - For [(\[b-leq-minus-1\])]{} above, (\[thm-mnd-alpha-minus-infinite\]) of Theorem \[thm:exact\_qcomposite\] says $\delta < k$ almost surely, while of Theorem \[thm:exact\_qcomposite-more-fine-grained\] presents the stronger result that $\delta$ equals $k+b$ or $k+b-1$ almost surely (note $b\leq -1$ in (\[b-leq-minus-1\])). Summarizing the above, compared with Theorem \[thm:exact\_qcomposite\], Theorem \[thm:exact\_qcomposite-more-fine-grained\] presents a more fine-grained result for minimum degree in $\mathbb{G}_q\iffalse_{on}\fi$. To prove Theorem \[thm:exact\_qcomposite-more-fine-grained\], we provide the basic ideas in Section \[sec-basic-proof-ideas\], and more technical details in Section \[sec-prove-first-two-theorems-mnd\]. Theorems \[thm:exact\_qcomposite\] and \[thm:exact\_qcomposite-more-fine-grained\] above are for the property of minimum degree being at least $k$. We now consider a stronger graph/network property, namely $k$-connectivity. **Extension to $k$-connectivity.** We can extend Theorem \[thm:exact\_qcomposite\] to obtain the probability of $k$-connectivity in $\mathbb{G}_q\iffalse_{on}\fi$. Specifically, we can replace $\lim_{n \to \infty}{{\mathbb{P}}\left[{\delta \geq k}\right]}$ by $\lim_{n \to \infty}{{\mathbb{P}}\left[{\text{$\mathbb{G}_q\iffalse_{on}^{(q)}\fi$ is $k$-connected.}}\right]}$, at the cost of replacing $ K_n = \omega(1)$ and $\frac{{K_n}^2}{P_n} = o(1)$ by a stronger condition set $ \frac{{K_n}^2}{P_n} = o\left( \frac{1}{\ln n} \right)$, $ \frac{K_n}{P_n} = o\left( \frac{1}{n\ln n} \right)$ and $ K_n = \Omega(n^{\epsilon})$ for a positive constant $\epsilon$. Due to space limitation, we present the proof in the full version [@full]. \[findbeta\] Equations (\[e1\]) and (\[e2\]) are determined by finding $\ell$ and $\alpha_n$ with $$\begin{aligned} -\frac{1}{2} \ln \ln n \leq \alpha_n < \frac{1}{2} \ln \ln n \label{betaineq} \end{aligned}$$ such that $$\begin{aligned} p_{e, q} & = \frac{\ln n + {(\ell-1)} \ln \ln n + {\alpha_n}}{n}. \label{peqexpr} \end{aligned}$$ In fact, it is clear that (\[peqexpr\]) follows from (\[e2\]); and with (\[e1\]), it holds that $$\begin{aligned} & \frac{np_{e, q} - \ln n + (\ln \ln n) / 2}{\ln \ln n} < \ell \leq \frac{np_{e, q} - \ln n + (\ln \ln n) / 2}{\ln \ln n} + 1,\nonumber\end{aligned}$$ which along with (\[e2\]) further leads to (\[betaineq\]) in view of $$\begin{aligned} \alpha_n & = np_{e, q} - \ln n - (\ell-1)\ln\ln n, \nonumber \\ & < np_{e, q} - \ln n- [np_{e, q} - \ln n + (\ln \ln n) / 2] + \ln\ln n \nonumber \\ & = (\ln \ln n) / 2, \nonumber\end{aligned}$$ and $$\begin{aligned} \alpha_n & = np_{e, q} - \ln n - (\ell-1)\ln\ln n, \nonumber \\ & \geq np_{e, q} - \ln n- [np_{e, q} - \ln n + (\ln \ln n) / 2] \nonumber \\ & = - (\ln \ln n) / 2. \nonumber\end{aligned}$$ [Design guidelines for secure sensor networks]{} {#sec-design-guidelines} ------------------------------------------------ Based on the above results, now we provide several design guidelines of secure sensor networks for achieving certain strength of minimum degree. - First, to ensure that $\mathbb{G}_q\iffalse_{on}\fi$ has a minimum degree no less than $k$, we can choose network parameters to set $$\begin{aligned} p_{e, q} \geq \frac{\ln n + {(k+c_1-1)} \ln \ln n }{n} \text{ for a constant $c_1 > 0$,} \label{peq1sbsc-critical1}$$ where the positive constant $c_1$ can be arbitrarily small. To see this, since (\[peq1sbsc-critical1\]) implies that $\alpha_n$ defined by (\[peq1sbsc\]) (i.e., $p_{e, q} = \frac{\ln n + {(k-1)} \ln \ln n + {\alpha_n}}{n}$) satisfies $\lim_{n \to \infty} \alpha_n = \infty$, we use Theorem \[thm:exact\_qcomposite\] to have ${{\mathbb{P}}\left[{\text{$\mathbb{G}_q\iffalse_{on}^{(q)}\fi$ has a minimum degree at least $k$.}}\right]}=1$. - Second, to guarantee that $\mathbb{G}_q\iffalse_{on}\fi$ has a minimum degree at least $k$ with probability no less than $\rho$, we choose parameters to ensure $$\begin{aligned} p_{e, q} \geq \frac{\ln n + {(k-1)} \ln \ln n - \ln [ (k-1)! \ln \frac{1}{\rho}] }{n} \label{peq1sbsc-critical1-with-prob-rho}.$$ To see this, since (\[peq1sbsc-critical1-with-prob-rho\]) implies that $\alpha_n$ defined by (\[peq1sbsc\]) (i.e., $p_{e, q} = \frac{\ln n + {(k-1)} \ln \ln n + {\alpha_n}}{n}$) satisfies $\alpha_n \geq - \ln [ (k-1)! \ln \frac{1}{\rho}] $, we use Theorem \[thm:exact\_qcomposite\] to obtain ${{\mathbb{P}}\left[{\text{$\mathbb{G}_q\iffalse_{on}^{(q)}\fi$ has a minimum degree at least $k$.}}\right]} \geq e^{- \frac{e^{\ln [ (k-1)! \ln \frac{1}{\rho}]}}{(k-1)!}} = \rho$. - Third, to ensure that $\mathbb{G}_q\iffalse_{on}\fi$ has a minimum degree being $k$ *exactly*, we can choose network parameters to have $$\begin{aligned} p_{e, q} = \frac{\ln n + {(k+c_2-1)} \ln \ln n }{n} \text{ for a constant $0<c_2 <1$,} \label{peq1sbsc-critical2}$$ where the positive constant $c_2$ can be arbitrarily small. To see this, (\[peq1sbsc-critical2\]) implies that $\alpha_n$ defined by (\[peq1sbsc\]) (i.e., $p_{e, q} = \frac{\ln n + {(k-1)} \ln \ln n + {\alpha_n}}{n}$) equals $c_2 \ln \ln n $, so $b$ in (\[alpha-n-written-beta-n\]) is $0$ with $\beta_n$ satisfying (\[alpha-n-written-beta-n\]) and $ \lim_{n \to \infty} \beta_n = \infty$. Then we use Theorem \[thm:exact\_qcomposite\]-Result to obtain ${{\mathbb{P}}\left[{\text{Minimum degree of $\mathbb{G}_q\iffalse_{on}^{(q)}\fi$ equals $k$ \textit{exactly}.}}\right]}=1$. [Basic Ideas to Establish Theorems \[thm:exact\_qcomposite\] and \[thm:exact\_qcomposite-more-fine-grained\]]{} {#sec-basic-proof-ideas} --------------------------------------------------------------------------------------------------------------- We establish Theorems \[thm:exact\_qcomposite\] and \[thm:exact\_qcomposite-more-fine-grained\] for minimum degree in graph $\mathbb{G}_q\iffalse_{on}^{(q)}\fi$ by analyzing the number of nodes with a fixed degree, for which we present Theorem \[thm:exact\_qcomposite2\] below. The details of using Theorem \[thm:exact\_qcomposite2\] to prove Theorems \[thm:exact\_qcomposite\] and \[thm:exact\_qcomposite-more-fine-grained\] are given in Section \[sec-prove-first-two-theorems-mnd\]. \[thm:exact\_qcomposite2\] For graph $\mathbb{G}_q\iffalse_{on}^{(q)}\fi$ with $ K_n = \omega(1)$ and $\frac{{K_n}^2}{P_n} = o(1)$, if $$\begin{aligned} p_{e, q} & = \frac{\ln n \pm O(\ln \ln n)}{n} \label{peq1}\end{aligned}$$ (i.e., $\frac{n p_{e, q} - \ln n }{\ln \ln n}$ is bounded), then for a non-negative constant integer $h$, the number of nodes in $\mathbb{G}_q\iffalse_{on}\fi$ with degree $h$ is in distribution asymptotically equivalent to a Poisson random variable with mean $\lambda_{n,h} : = n (h!)^{-1}(n p_{e, q})^h e^{-n p_{e, q}}$; i.e., as $n \to \infty$, $$\begin{aligned} & {{\mathbb{P}}\left[{\hspace{-3pt}\begin{array}{l}\text{The number of nodes in $\mathbb{G}_q\iffalse_{on}\fi$}\\\text{with degree $h$ equals $\ell$.}\end{array}\hspace{-3pt}}\right]}\Big/\Big[(\ell !)^{-1}{\lambda_{n,h}} ^{\ell}e^{-\lambda_{n,h}}\Big] \to 1, \nonumber \\ & \text{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~for $\ell = 0,1, \ldots$} \label{eq-Poisson-lemma}\end{aligned}$$ **Interpreting Theorem \[thm:exact\_qcomposite2\].** Theorem \[thm:exact\_qcomposite2\] for graph $\mathbb{G}_q\iffalse_{on}\fi$ shows that the number of nodes with a fixed degree follows a Poisson distribution asymptotically. We now explain the steps of proving Corollary \[thm:exact\_qcomposite\] through Theorem \[thm:exact\_qcomposite2\]. Establishing Theorem \[thm:exact\_qcomposite\] Given Theorem \[thm:exact\_qcomposite2\] --------------------------------------------------------------------------------------- Given (\[peq2\]) (a condition in Corollary \[thm:exact\_qcomposite\]), we determine $\ell$ and $\alpha_n$ through (\[e1\]) and (\[e2\]) in Theorem \[thm:exact\_qcomposite2\]. Then $$\begin{aligned} \ell & = \bigg\lfloor \frac{ {(k-1)} \ln \ln n + {\alpha_n} + (\ln \ln n) / 2}{\ln \ln n} \bigg\rfloor + 1 \nonumber \\ & = k + \bigg\lfloor \frac{\alpha_n}{\ln \ln n} + \frac{1}{2} \bigg\rfloor , \label{e3}\end{aligned}$$ and $$\begin{aligned} \alpha_n & = {(k-1)} \ln \ln n + {\alpha_n} - \hspace{-1pt} \bigg(\hspace{-1pt} k + \bigg\lfloor \frac{\alpha_n}{\ln \ln n} \hspace{-1pt} + \frac{1}{2} \hspace{-1pt} \bigg\rfloor-1 \hspace{-1pt} \bigg) \hspace{-1pt} \ln\ln n, \nonumber \\ & = \alpha_n - \bigg\lfloor \frac{\alpha_n}{\ln \ln n} + \frac{1}{2} \bigg\rfloor \ln \ln n . \label{e5}\end{aligned}$$ Given condition $\lim_{n \to \infty} \alpha_n = \alpha ^* \in [-\infty, \infty]$ in Theorem \[thm:exact\_qcomposite\], we consider the following three cases: $-\frac{1}{2} \ln \ln n \leq \alpha_n < \frac{1}{2} \ln \ln n$, $ \alpha_n \geq \frac{1}{2} \ln \ln n$ and $ \alpha_n < - \frac{1}{2} \ln \ln n$. **Case** : $-\frac{1}{2} \ln \ln n \leq \alpha_n < \frac{1}{2} \ln \ln n$. Then from (\[e3\]) and (\[e5\]), we obtain $\ell = k$ and $\alpha_n = \alpha_n $. It further holds that $\lim_{n \to \infty} \alpha_n = \lim_{n \to \infty} \alpha_n = \alpha ^* \in [-\infty, \infty]$. Therefore, by Theorem \[thm:exact\_qcomposite2\], $$\begin{aligned} \mathbb{P}[\delta \geq k] & \to \begin{cases} 1, \textrm{~if }\alpha ^* = \infty, \\ 0, \textrm{~if }\alpha ^* = -\infty, \\ e^{- \frac{e^{-\alpha ^*}}{(k-1)!}},\textrm{~if } \alpha ^* \in (-\infty, \infty). \end{cases} \nonumber \end{aligned}$$ Then with $e^{\infty} = \infty$ and $e^{-\infty} = 0$, (\[eqn\_minnodeq10\]) follows in case . **Case** : $ \alpha_n \geq \frac{1}{2} \ln \ln n$. Then from (\[e3\]) and (\[e5\]), it holds that $\ell \geq k+1$. Hence, $\mathbb{P}[\delta \geq k] \to 1$ by Theorem \[thm:exact\_qcomposite2\], leading to (\[eqn\_minnodeq10\]) in case . **Case** : $ \alpha_n < -\frac{1}{2} \ln \ln n$. Then from (\[e3\]) and (\[e5\]), it holds that $\ell \leq k-1$. Consequently, $\mathbb{P}[\delta \geq k] \to 0$ by Theorem \[thm:exact\_qcomposite2\], resulting in (\[eqn\_minnodeq10\]) in case . Summarizing cases and above, Theorem \[thm:exact\_qcomposite\] holds by Theorem \[thm:exact\_qcomposite2\]. Analogs of Theorem \[thm:exact\_qcomposite2\] and Theorem \[thm:exact\_qcomposite\] with an Approximation of $p_{e,q}$ {#subapp} ----------------------------------------------------------------------------------------------------------------------   Analogous results of Theorem \[thm:exact\_qcomposite2\] and Theorem \[thm:exact\_qcomposite\] can be given with $p_{e,q} $ in $\mathbb{G}_q$ substituted by a quantity expressed by $K_n, P_n$ and $q$; i.e., with $p_{s,q}$ replaced by $\frac{1}{q!} \big( \frac{{K_n}^2}{P_n} \big)^{q}$ given Lemma \[lem\_eval\_psq\]-Property (i) in the Appendix?, and hence with $p_{e,q} $ replaced by $p_n \cdot \frac{1}{q!} \big( \frac{{K_n}^2}{P_n} \big)^{q}$ due to ${p_{e,q}} = p_n \cdot p_{s,q}$ from (\[eq\_pre\]). Thus, with (\[peq1\]) (resp., (\[peq2\])) replaced by $p_n \cdot \frac{1}{q!} \big( \frac{{K_n}^2}{P_n} \big)^{q} = \frac{\ln n \pm O(\ln \ln n)}{n}$ (resp., $p_n \cdot \frac{1}{q!} \big( \frac{{K_n}^2}{P_n} \big)^{q} = \frac{\ln n + {(k-1)} \ln \ln n + {\alpha_n}}{n}$), and keeping all the conditions in Theorem \[thm:exact\_qcomposite2\] (resp., Corollary \[thm:exact\_qcomposite\]), we demonstrate below that the properties (a) and (b) in Theorem \[thm:exact\_qcomposite2\] (resp., (\[eqn\_minnodeq10\]) in Corollary \[thm:exact\_qcomposite\]) still hold. To do this, first, if $p_n \cdot \frac{1}{q!} \big( \frac{{K_n}^2}{P_n} \big)^{q} = \frac{\ln n \pm O(\ln \ln n)}{n}$, then from Lemma \[lem\_eval\_psq\]-Property (i), we obtain $$\begin{aligned} p_{e,q} & = p_n \cdot p_{s,q} = p_n \cdot \frac{1}{q!} \bigg( \frac{{K_n}^2}{P_n} \bigg)^{q} \cdot [1 \pm o(1)] \nonumber \\ & = \frac{\ln n \pm O(\ln \ln n)}{n} \cdot [1 \pm o(1)] = \frac{\ln n \pm O(\ln \ln n)}{n} \nonumber; \end{aligned}$$ second, with $p_n \cdot \frac{1}{q!} \big( \frac{{K_n}^2}{P_n} \big)^{q} = \frac{\ln n + {(k-1)} \ln \ln n + {\alpha_n}}{n}$ replacing (\[peq2\]) in Theorem \[thm:exact\_qcomposite\], in proving (\[eqn\_minnodeq10\]), we introduce an extra condition $|\alpha_n| \leq \ln \ln n$ by a coupling argument, the explanation of which is deferred to the next paragraph. Hence, from $p_n \cdot \frac{1}{q!} \big( \frac{{K_n}^2}{P_n} \big)^{q} = \frac{\ln n + {(k-1)} \ln \ln n + {\alpha_n}}{n} $, $|\alpha_n| \leq \ln \ln n$ and Lemma \[lem\_eval\_psq\]-Property (i), it holds that $p_{e,q} = \frac{\ln n + {(k-1)} \ln \ln n + {\alpha_n} \pm O(1)}{n} $. We now present the coupling argument which confines $\alpha_n$ as $|\alpha_n| \leq \ln \ln n$ to establish Corollary \[thm:exact\_qcomposite\] with (\[peq2\]) replaced by $p_n \cdot \frac{1}{q!} \big( \frac{{K_n}^2}{P_n} \big)^{q} = \frac{\ln n + {(k-1)} \ln \ln n + {\alpha_n}}{n}$. First, if $\lim_{n \to \infty} \alpha_n = \alpha ^* \in (-\infty, \infty)$ (i.e., $-\infty < \alpha ^* < \infty$), it is clear that $\alpha_n $ is bounded; namely, $|\alpha_n | = O(1)$, so $|\alpha_n| \leq \ln \ln n$ follows in this case. Therefore, in deriving (\[eqn\_minnodeq10\]), we only need to consider $\lim_{n \to \infty} \alpha_n = \alpha ^* = \infty $ and $\lim_{n \to \infty} \alpha_n = \alpha ^* = -\infty $. Setting $\widehat{\alpha}_n $ as $\min\{\alpha_n, \ln \ln n \} $ and $\widetilde{\alpha}_n $ as $ \max\{\alpha_n, - \ln \ln n \}$, we define $\widehat{p}_n$ and $\widetilde{p}_n $ through $$\begin{aligned} \widehat{p}_n \cdot \frac{1}{q!} \bigg( \frac{{K_n}^2}{P_n} \bigg)^{q} = \frac{\ln n + (k-1) \ln \ln n + \widehat{\alpha}_n}{n}, \nonumber\end{aligned}$$ and $$\begin{aligned} \widetilde{p}_n \cdot \frac{1}{q!} \bigg( \frac{{K_n}^2}{P_n} \bigg)^{q} = \frac{\ln n + (k-1) \ln \ln n + \widetilde{\alpha}_n}{n}. \nonumber\end{aligned}$$ With $\widehat{p}_n \leq p_n \leq \widetilde{p}_n$, similar to the argument in Section V-B in our work [@ZhaoYaganGligor] (we omit the details here due to the space limitation), there exist graph couplings such that $\mathbb{G}_q ( n, {K}_n, {P}_n,{ \widehat{p}_n} )$ is a spanning subgraph of $\mathbb{G}_q ( n, {K}_n, {P}_n,{{p}_n} )$, which is further a spanning subgraph of $\mathbb{G}_q ( n, {K}_n, {P}_n, \widetilde{p}_n )$. Clearly, if $\lim_{n \to \infty} \alpha_n = \infty $, then $\lim_{n \to \infty} \widehat{\alpha}_n = \infty $ and $|\widehat{\alpha}_n| \leq \ln \ln n$ for all $n$ sufficiently large; and if $\lim_{n \to \infty} \alpha_n = -\infty $, then $\lim_{n \to \infty} \widetilde{\alpha}_n = -\infty $ and $ |\widetilde{\alpha}_n |\leq \ln \ln n $ for all $n$ sufficiently large. Then by the fact that the probability that a graph has a minimum node degree at least $k$ is no less than the probability that an arbitrary spanning subgraph has a minimum node degree at least $k$, we can introduce the condition $|\alpha_n| \leq \ln \ln n$ in establishing (\[eqn\_minnodeq10\]). The Practicality of the Theorem Conditions {#sec-Conditions-Practicality} ------------------------------------------ We check the practicality of the conditions in Theorem \[thm:exact\_qcomposite\]: $ K_n = \omega(1)$ and $\frac{{K_n}^2}{P_n} = o(1)$. The condition $ K_n = \omega(1)$ means that the key ring size $K_n$ on a sensor grows with the number $n$ of sensors and thus it follows trivially in secure wireless sensor networks [@DiPietro:2008:RSN:1341731.1341734; @yagan; @zhao2016topology]. For $k$-connectivity, the condition on $K_n$ (i.e., $ K_n = \Omega(n^{\epsilon})$ is less appealing but is not much a problem because $\epsilon$ can be arbitrarily small. In addition, $\frac{{K_n}^2}{P_n}= o\left( \frac{1}{\ln n} \right)=o(1)$ and $ \frac{K_n}{P_n} = o\left( \frac{1}{n\ln n} \right)$ hold in practice since the key pool size $P_n$ is expected to be several orders of magnitude larger than the key ring size $K_n$ (see [@virgil Section 2.1] and [@yagan_onoff Section III-B]). The Proof of Theorem \[thm:exact\_qcomposite2\] {#secprflem} =============================================== Recalling $\delta$ as the minimum node degree of graph $\mathbb{G}_q\iffalse_{on}\fi$, we have the following properties and . - Event $(\delta \geq \xi)$ equals event $ \bigcap_{h=0}^{\xi-1} (\Phi_{n,h} = 0) $ (i.e., no node has degree falling in $\{0,1,\ldots, \xi-1\}$). - Event $(\delta \leq \xi)$ is equivalent to event $ \bigcup_{h=0}^{\xi} (\Phi_{n,h} \neq 0) $ (i.e., there is at least one node with degree at most $\xi$). Therefore, $$\begin{aligned} \mathbb{P}[\delta \geq k +1] & \leq \mathbb{P}[\Phi_{n,k} = 0] \textrm{~(by property \ding {192})}, \label{eqpd1} \\ \mathbb{P}[\delta \leq k -2] & \leq \mathbb{P}\bigg[ \bigcup_{h=0}^{k-2} (\Phi_{n,h} \neq 0)\bigg] \textrm{~(by property \ding {193})} \nonumber \\& \leq \sum_{h=0}^{k-2} \mathbb{P}[ \Phi_{n,h} \neq 0]\textrm{~~~~~(by the union bound)}, \label{eqpd2} \\ \mathbb{P}[\delta \geq k] & \leq \mathbb{P}[\Phi_{n,k-1} = 0] \textrm{~(by property \ding {192})}, \label{eqn_1mindel2} \end{aligned}$$ and $$\begin{aligned} \mathbb{P}[\delta \geq k] & = 1 - \mathbb{P}\bigg[\bigcup_{h=0}^{k-1} (\Phi_{n,h} \neq 0)\bigg] \textrm{~(by property \ding {192})} \nonumber \\ & \geq 1 - \sum_{h=0}^{k-1} \mathbb{P}[\Phi_{n,h} \neq 0]\textrm{ (by the union bound)} \nonumber \\ & = \mathbb{P}[\Phi_{n,k-1} = 0] - \sum_{h=0}^{k-2} \mathbb{P}[ \Phi_{n,h} \neq 0] . \label{eqn_1min} \end{aligned}$$ To use (\[eqpd1\]-\[eqn\_1min\]), we compute $\mathbb{P}[ \Phi_{n,h} \neq 0]$ for $h=0,1,\ldots$ given (\[eqn\_phihell\]) and thus evaluate $\lambda_{n,h}$ defined in (\[eqn\_labmdah\]). To calculate $\lambda_{n,h}$, we apply probability $p_{e,q}$ given by (\[thm\_eq\_pe\]). From (\[thm\_eq\_pe\]) (\[eqn\_labmdah\]) and (\[eq\_pe\_lnnn\]), we compute $\lambda_{n,h}$ by $$\begin{aligned} \lambda_{n,h} & = n (h!)^{-1}(n p_{e,q})^h e^{-n p_{e,q}} \nonumber \\ & \sim n (h!)^{-1} (\ln n)^h \times e^{-\ln n - (k-1)\ln \ln n - \alpha_n} \nonumber \\ & = (h!)^{-1} (\ln n)^{h+1-k} e^{-\alpha_n} \nonumber \\ & \begin{cases} \to 0,\textrm{ for }h = 0, 1, \ldots, k-2; \\ \sim \frac{e^{-\alpha_n}}{(k-1)!},\textrm{ for }h = k-1; \\ \to \infty ,\textrm{ for }h = k, k+1, \ldots. \end{cases} \label{eqn_lbdh} \end{aligned}$$ By (\[eqn\_phihell\]) and (\[eqn\_lbdh\]), $$\begin{aligned} \mathbb{P}[\Phi_{n,h} = 0] &\sim e^{-\lambda_{n,h}} \begin{cases}\to 1,\textrm{ for }h = 0, 1, \ldots, k-2; \\ \sim e^{-\frac{e^{-\alpha_n}}{(k-1)!}},\textrm{ for }h = k-1; \\ \to 0 ,\textrm{ for }h = k, k+1, \ldots.\end{cases}\label{eqn_expr_lahkk1} \end{aligned}$$ From (\[eqpd1\]-\[eqn\_1min\]) and (\[eqn\_expr\_lahkk1\]), $$\begin{aligned} \mathbb{P}[\delta \geq k +1] & = o(1), \label{eqn_delta_k1}\\ \mathbb{P}[\delta \leq k -2] & = o(1), \label{eqn_delta_k2}\end{aligned}$$ and $$\begin{aligned} \mathbb{P}[\delta \geq k] & \sim e^{- \frac{e^{-\alpha_n}}{(k-1)!}} \to e^{- \frac{e^{-\alpha ^*}}{(k-1)!}}. \nonumber\end{aligned}$$ Then $$\begin{aligned} \mathbb{P}[\delta = k] & = \mathbb{P}[\delta \geq k] - \mathbb{P}[\delta \geq k +1] \to e^{- \frac{e^{-\alpha ^*}}{(k-1)!}}, \label{prob_delta_k1a} \\ \hspace{-4pt}\mathbb{P}[\delta = k-1] & = 1- \mathbb{P}[\delta \leq k -2] - \mathbb{P}[\delta \geq k] \to 1 - e^{- \frac{e^{-\alpha ^*}}{(k-1)!}} , \label{prob_delta_k1} \end{aligned}$$ and $$\begin{aligned} \mathbb{P}[(\delta \neq k) \cap (\delta \neq k-1)] & = \mathbb{P}[\delta \geq k +1] + \mathbb{P}[\delta \leq k -2] \nonumber\\ & = o(1). \label{prob_delta_k1b} \end{aligned}$$ Finally, by (\[prob\_delta\_k1a\]-\[prob\_delta\_k1b\]), we obtain property ($1^{\circ}$) and hence property (a) of Theorem \[thm:exact\] and property (a) of Theorem \[thm:exact2\]. The Proof of Lemma \[LEM1\] {#secprf:lem_pos_exp} =========================== Lemmas on Graph $\mathbb{G}_q\iffalse_{on}^{(q)}\fi$ ---------------------------------------------------- \[lem\_psukn\_qcmp\] In graph $\mathbb{G}_q\iffalse_{on}^{(q)}\fi$, for three distinct nodes $v_i, v_j$ and $w$ in $\mathcal{V}_n$, let $K_{w v_i }^{(q)}$ (resp., $K_{w v_j }^{(q)}$) be the event that the key rings of nodes $w$ and $v_i$ (resp., $v_j$) share at least $q$ key(s). If $\frac{{K_n}^2}{P_n} = o(1)$, then for sufficiently large $n$, $$\begin{aligned} & \mathbb{P}[K_{{ wv_i}}^{(q)}\cap K_{ wv_j}^{(q)} \boldsymbol{\mid} |S_{i} \cap S_{j}| = u] \nonumber \\ & \quad\leq \frac{ p_{s,q}^{(q)} u}{K_n}+ (q+2)! \cdot \big({p_{s,q}^{(q)}\big)}^{\frac{q+1}{q}} . \nonumber\end{aligned}$$ \[lem\_evalprob\_qcmp\] In graph $\mathbb{G}_q\iffalse_{on}^{(q)}\fi$, let $w$ be any given node in $\mathcal{V}_n \setminus \{v_1, v_2, \ldots, v_m\}$. Given any $\mathcal {T}_m^{*} = (S_1, S_2, \ldots, S_m) \in \mathbb{T}_m$, we have $$\begin{aligned} & \mathbb{P} [w \in M_{0^m} \boldsymbol{\mid} \mathcal {T}_m = \mathcal {T}_m^{*} ] \geq 1 - m p_{e, q} , \label{eq_evalprob_3_qcmp}\end{aligned}$$ and for any $i = 1,2,\ldots,m $, $$\begin{aligned} & \mathbb{P}\big[w \in M_{0^{i-1}, 1, 0^{m-i}} \boldsymbol{\mid} \mathcal {T}_m = \mathcal {T}_m^{*} \big] \leq p_{e, q}; \label{eq_evalprob_1_qcmp}\end{aligned}$$ and if $\frac{{K_n}^2}{P_n} = o(1)$, the following (\[eq\_evalprob\_4\_qcmp\]) and (\[eq\_evalprob\_2\_qcmp\]) hold: $$\begin{aligned} & \mathbb{P} [w \in M_{0^m} \boldsymbol{\mid} \mathcal {T}_m = \mathcal {T}_m^{*} ] \nonumber \\ & \quad \leq e^{- m p_{e, q} + (q+2)! \binom{m}{2}{(p_{e,q})}^{\frac{q+1}{q}} + \frac{p_{e, q} p_n}{K_n}\sum_{1\leq i <j \leq m}|S_{i j}|}, \label{eq_evalprob_4_qcmp}\end{aligned}$$ and $$\begin{aligned} & \mathbb{P}\big[w \in M_{0^{i-1}, 1, 0^{m-i}} \boldsymbol{\mid} \mathcal {T}_m = \mathcal {T}_m^{*} \big] \nonumber \\ & \quad\geq p_{e, q} \bigg[ 1 - (q+2)!m{(p_{e,q})}^{\frac{1}{q}} \nonumber \\ & \quad \hspace{70pt} - \frac{p_n}{K_n} \sum_{j\in\{1,2,\ldots,m\} \setminus\{i\}} |S_i \cap S_j| \bigg], \label{eq_evalprob_2_qcmp}\end{aligned}$$ for any $i = 1,2,\ldots,m $. The Proof of Lemma \[lem\_evalprob\_qcmp\] ------------------------------------------ The Proof of Lemma \[lem\_evalprob\_qcmp\] is similar to that of Lemma \[lem\_evalprob\]. First, we establish (\[eq\_evalprob\_3\_qcmp\]) and (\[eq\_evalprob\_1\_qcmp\]) in a similar manner of proving (\[eq\_evalprob\_3\]) and (\[eq\_evalprob\_1\]). Then similar to the steps of showing (\[eq\_evalprob\_4\]) and (\[eq\_evalprob\_2\]) by Lemma \[lem\_psukn\], we demonstrate (\[eq\_evalprob\_4\_qcmp\]) and (\[eq\_evalprob\_2\_qcmp\]) by Lemma \[lem\_psukn\_qcmp\]. \[lem\_evalprob\_exp\_qcmp\] In graph $\mathbb{G}_q\iffalse_{on}^{(q)}\fi$, let $w$ be any given node in $\mathcal{V}_n \setminus \{v_1, v_2, \ldots, v_m\}$. Then given (\[thm\_eq\_pe\_qcmp\]) with $|\alpha_n| = \Omega(\ln \ln n)$, $ K_n = \omega(1)$ and $\frac{{K_n}^2}{P_n} = o(1)$, $$\begin{aligned} & \sum_{ \mathcal {T}_m^{*} \in \mathbb{T}_m } \mathbb{P}[\mathcal {T}_m = \mathcal {T}_m^{*}] \mathbb{P} [w \in M_{0^m} \boldsymbol{\mid} \mathcal {T}_m = \mathcal {T}_m^{*} ]^{n - m - hm} \nonumber \\ & \quad \leq e^{- m n p_{e, q}} \cdot [1+o(1)] , \label{eq_evalprob_exp_1_qcmp} \\ & \sum_{ \mathcal {T}_m^{*} \in \mathbb{T}_m^{(0)} } \mathbb{P}[\mathcal {T}_m = \mathcal {T}_m^{*}] \mathbb{P} [w \in M_{0^m} \boldsymbol{\mid} \mathcal {T}_m = \mathcal {T}_m^{*} ]^{n - m - hm} \nonumber \\ & \quad \leq e^{- m n p_{e, q}} \cdot [1+o(1)] , \label{eq_evalprob_exp_1_qcmp_tm0} \\ \hspace{-5pt} & \sum_{ \mathcal {T}_m^{*} \in \mathbb{T}_m }\Big\{ \mathbb{P}[\mathcal {T}_m = \mathcal {T}_m^{*}] \prod_{i=1}^{m} \{ \mathbb{P}\big[w \in M_{0^{i-1}, 1, 0^{m-i}}^{(0)} \boldsymbol{\mid} \mathcal {T}_m = \mathcal {T}_m^{*} \big]^h \}\Big\} \nonumber \\ & \quad \geq \big(p_{e, q}\big)^{hm} \cdot [1-o(1)] .\label{eq_evalprob_exp_2_qcmp}\end{aligned}$$ and $$\begin{aligned} & \hspace{-5pt} \sum_{ \mathcal {T}_m^{*} \in \mathbb{T}_m^{(0)} }\Big\{ \mathbb{P}[\mathcal {T}_m = \mathcal {T}_m^{*}] \prod_{i=1}^{m} \{ \mathbb{P}\big[w \in M_{0^{i-1}, 1, 0^{m-i}}^{(0)} \boldsymbol{\mid} \mathcal {T}_m = \mathcal {T}_m^{*} \big]^h \}\Big\} \nonumber \\ & \quad \geq \big(p_{e, q}\big)^{hm} \cdot [1-o(1)] .\label{eq_evalprob_exp_2_qcmp_tm0}\end{aligned}$$ \[lem\_MmMm0\_qcmp\] In graph $\mathbb{G}_q\iffalse_{on}^{(q)}\fi$, given (\[thm\_eq\_pe\]) with $|\alpha_n| = \Omega(\ln \ln n)$ and $K_n = \omega(p_n)$, we have $$\begin{aligned} \mathbb{P} \big[ \mathcal {M}_m = \mathcal{M}_m^{(0)} \big] & \sim (h!)^{-m} \big(n p_{e, q}\big)^{hm} e^{-m n p_{e, q}} \label{eqn_prMm_qcmp},\end{aligned}$$ and $$\begin{aligned} \mathbb{P} \big[ \big( \mathcal {M}_m = \mathcal{M}_m^{(0)} \big) \boldsymbol{\mid} \big( \mathcal {T}_m \in \mathbb{T}_m^{(0)} \big)\big]& \sim (h!)^{-m} \big(n p_{e, q}\big)^{hm} e^{-m n p_{e, q}} .\label{prob_MmMm_sim_qcmp}\end{aligned}$$ The Proof of Lemma \[lem\_evalprob\_exp\_qcmp\] ----------------------------------------------- ### Proof of (\[eq\_evalprob\_exp\_1\_qcmp\]) The proof of (\[eq\_evalprob\_exp\_1\_qcmp\]) in Lemma \[lem\_evalprob\_exp\_qcmp\] here is similar to that of (\[eq\_evalprob\_exp\_1\]) in Lemma \[lem\_evalprob\_exp\]. First, similar to the establishment of (\[eqn\_sum\_Tm\]) using (\[eq\_evalprob\_3\]) and (\[eq\_evalprob\_4\]), we show (\[eqn\_sum\_Tm\]) with $p_{e,q}$ replaced by $p_{e, q}$ using (\[eq\_evalprob\_3\_qcmp\]) and (\[eq\_evalprob\_4\_qcmp\]). Then similar to Lemma \[lem\_evalprob\_exp\], we also define $ {G}_{n,m}$ by (\[eqn\_def\_Gnm\]) with $p_{e,q}$ replaced by $p_{e, q}$, and define $ {H}_{n,m}$ by (\[eqn\_Hnm\]); i.e., $$\begin{aligned} {G}_{n,m} & : = \sum_{ \mathcal {T}_m^{*} \in \mathbb{T}_m \setminus \mathbb{T}_m^{(0)}} \mathbb{P}[\mathcal {T}_m = \mathcal {T}_m^{*}] e^{\frac{n p_{e, q} p_n}{K_n} \sum_{1\leq i <j \leq m}|S_{ij}|}, \nonumber $$ and $$\begin{aligned} H_{n,m} & : = \mathbb{P}[\mathcal {T}_m^{*} \in \mathbb{T}_m \setminus \mathbb{T}_m^{(0)} ] \nonumber.\end{aligned}$$ By similar argument in Lemma \[lem\_evalprob\_exp\], (\[eq\_evalprob\_exp\_1\_qcmp\]) follows once we show $G_{n,m} \leq H_{n,m} + o(1)$. Similar to the proof in Lemma \[lem\_evalprob\_exp\], here we also evaluate $G_{n,m}$ by an iterative approach (i.e., by relating $G_{n,m}$ to $G_{m-1}$). Note that here event $\big(\mathcal {T}_m^{*} \in \mathbb{T}_m \setminus \mathbb{T}_m^{(0)}\big)$ can be divided into the following two disjoint subevents - $\big(\mathcal {T}_{m-1}^{*} \in \mathbb{T}_{m-1} \setminus \mathbb{T}_{m-1}^{(0)}\big) \cap \big(S_m^* \in \mathbb{S}_m \big) ,$ and - $\big(\mathcal {T}_{m-1} \in \mathbb{T}_{m-1}^{(0)}) \cap \big[ \bigcup_{i =1}^{m-1} \big(|S_m^* \cap S_{i}^{*}| \geq q\big)\big]$. Similar to the proof in Lemma \[lem\_evalprob\_exp\], here we also consider the following two cases: **a)** $p_n < n^{-\delta} (\ln n)^{-1}$ and **b)** $p_n \geq n^{-\delta} (\ln n)^{-1}$ for any fixed $n$ ($n$ is also set to be sufficiently large whenever necessary). For case **a)** here, its proof is very much similar to that of case **a)** in Lemma \[lem\_evalprob\_exp\]. For brevity, we do not repeat the details. At the end, (\[eqngnmhnm\]) Lemma \[lem\_evalprob\_exp\] also holds here; i.e., $$\begin{aligned} G_{n,m} & \leq e^{(m^2 + m) n^{-\delta}} H_{n,m}. \label{eqn_gnmhnm}\end{aligned}$$ For case **b)** here, its proof is also similar to that of case **b)** in Lemma \[lem\_evalprob\_exp\] with some modifications. First, (\[eqn\_tmtm-1\]) still holds; and with the range of the summation $\big[S_m^*:~S_m^* \cap (\bigcup_{i =1}^{m-1}S_{i }^{*}) \neq \emptyset\big]$ in (\[eqn\_tmtm-1p2\_val\]) replaced by $\big[S_m^*:~\bigcup_{i =1}^{m-1} \big(|S_m^* \cap S_{i}^{*}| \geq q\big)\big]$ , (\[eqn\_tmtm-1p2\]) also follows. Similar to (\[eqps0\]), given $p_n \geq n^{-\delta} (\ln n)^{-1}$, we have $$\begin{aligned} p_{s,q}^{(q)} & = \frac{p_{e,q}}{p_n} \leq \frac{2\ln n}{n p_n} \leq 2 n^{\delta-1} (\ln n)^2. \label{eqps0_qcmp}\end{aligned}$$ Different from the bound for $\frac{{K_n}^2}{P_n-K_n}$ given in (\[eqn\_knpn\]), here by Lemma \[lem\_eval\_psq\]-Property (i) and (\[eqps0\_qcmp\]), we have $$\begin{aligned} \frac{{K_n}^2}{P_n-K_n} & \leq \big( q! p_{s,q}^{(q)} \big)^{\frac{1}{q}} \cdot [1+o(1)] \leq 3 q \cdot n^{\frac{\delta-1}{q}} (\ln n)^{\frac{2}{q}}.\label{eqn_knpn_qcmp}\end{aligned}$$ Here given $K_n = \omega(1) $, it still holds that $K_n = \omega(p_n) $ as in Lemma \[lem\_evalprob\_exp\]. Then for arbitrary constant $c > q$, we get $\frac{K_n}{p_n} \geq \frac{2cq \cdot m}{(c-q)(1-\delta)} $ for all sufficiently large $n$. Similar to (\[ja1\]), we obtain $$\begin{aligned} e^{\frac{2 m p_n \ln n}{K_n}} & \leq e^{ \frac{(c-q)(1-\delta)}{cq} \ln n} = n^{\frac{(c-q)(1-\delta)}{cq}} \label{ja1_qcmp}\end{aligned}$$ for all sufficiently large $n$. From (\[eqn\_knpn\_qcmp\]) and (\[ja1\_qcmp\]), $$\begin{aligned} \frac{m {K_n}^2}{P_n-K_n}\cdot e^{\frac{2 m p_n \ln n}{K_n}} & \leq 3 mq \cdot n^{\frac{\delta-1}{q}} (\ln n)^{\frac{2}{q}} \cdot n^{\frac{(c-q)(1-\delta)}{cq}} \nonumber \\ & = 3 mq \cdot n^{\frac{\delta-1}{c}} (\ln n)^{\frac{2}{q}}. \nonumber\end{aligned}$$ Then similar to the establishment of (\[gnmgnm-1\]) and (\[eqn\_gn2\_2\]), we obtain $$\begin{aligned} G_{n,m} + 1 & \leq \Big(e^{3 q \cdot n^{\frac{\delta-1}{c}} (\ln n)^{\frac{2}{q}}} \Big)^m \left( G_{n,m-1} + 1\right), \nonumber\end{aligned}$$ and $$\begin{aligned} G_{n,2} & \leq e^{6 q \cdot n^{\frac{\delta-1}{c}} (\ln n)^{\frac{2}{q}}}-1 \nonumber,\end{aligned}$$ together leading to $$\begin{aligned} G_{n,m} + 1 & \leq \Big(e^{3 q \cdot n^{\frac{\delta-1}{c}} (\ln n)^{\frac{2}{q}}} \Big)^{m+(m-1) + \ldots + 3} \cdot e^{6 q \cdot n^{\frac{\delta-1}{c}} \ln n} \nonumber \\ & = e^{\frac{3}{2}q(m^2+m+6) n^{\frac{\delta-1}{c}} (\ln n)^{\frac{2}{q}}} , \label{eqn_gnmhnm2}\end{aligned}$$ for all sufficiently large $n$. Summarizing cases a) and b), by (\[eqn\_gnmhnm\]) (\[eqn\_gnmhnm2\]) and $0 \leq H_{n,m} \leq 1$, it follows that $$\begin{aligned} & \left( G_{n,m} - H_{n,m} \right) & \nonumber \\ & \quad \leq \max\left\{e^{(m^2 + m) n^{-\delta}} , e^{\frac{3}{2}q(m^2+m+6) n^{\frac{\delta-1}{c}} (\ln n)^{\frac{2}{q}}}\right\} - 1. \nonumber\end{aligned}$$ Letting $n \to \infty$, we finally obtain $G_{n,m} \leq H_{n,m} + o(1)$. Then (\[eq\_evalprob\_exp\_1\_qcmp\]) follows by argument noted above. ### Proof of (\[eq\_evalprob\_exp\_1\_qcmp\_tm0\]) Similar to (\[eqn\_sum\_Tm\]), $$\begin{aligned} & \sum_{ \mathcal {T}_m^{*} \in \mathbb{T}_m^{(0)} } \mathbb{P}[\mathcal {T}_m = \mathcal {T}_m^{*}] \mathbb{P} [w \in M_{00 \ldots 0}^{*} \boldsymbol{\mid} \mathcal {T}_m = \mathcal {T}_m^{*} ]^{n - m - hm} \nonumber \\ & \quad \leq e^{- m n p_{e, q}} \cdot [1+o(1)] \nonumber \\ & \quad\quad \times \sum_{ \mathcal {T}_m^{*} \in \mathbb{T}_m^{(0)} } \mathbb{P}[\mathcal {T}_m = \mathcal {T}_m^{*}] e^{\frac{n p_{e, q} p_n}{K_n}\sum_{1\leq i <j \leq m}|S_{ij}|}. \label{eqn_sum_Tm_qcmp_tm0}\end{aligned}$$ ### Proof of (\[eq\_evalprob\_exp\_2\_qcmp\]) The proof of (\[eq\_evalprob\_exp\_2\_qcmp\]) in Lemma \[lem\_evalprob\_exp\_qcmp\] here is similar to that of (\[eq\_evalprob\_exp\_2\]) in Lemma \[lem\_evalprob\_exp\]. The Proof of Lemma \[lem\_MmMm0\_qcmp\] --------------------------------------- ### The Proof of (\[eqn\_prMm\_qcmp\]) The proof of (\[eqn\_prMm\_qcmp\]) in Lemma \[lem\_MmMm0\_qcmp\] here is similar to that of (\[eqn\_prMm\]) in Lemma \[lem\_MmMm0\]. (\[eq\_evalprob\_3\]) (\[eq\_evalprob\_1\]) (\[eq\_evalprob\_exp\_1\]) and (\[eq\_evalprob\_exp\_2\]) used to establish (\[eqn\_prMm\]) are replaced by (\[eq\_evalprob\_3\_qcmp\]) (\[eq\_evalprob\_1\_qcmp\]) (\[eq\_evalprob\_exp\_1\_qcmp\]) and (\[eq\_evalprob\_exp\_2\_qcmp\]) here to show (\[eqn\_prMm\_qcmp\]). ### The Proof of (\[prob\_MmMm\_sim\_qcmp\]) We have $$\begin{aligned} & \mathbb{P} \big[ \big( \mathcal {M}_m = \mathcal{M}_m^{(0)} \big) \boldsymbol{\mid} \big( \mathcal {T}_m \in \mathbb{T}_m^{(0)} \big)\big] \nonumber \\ & \quad = \sum_{\mathcal {T}_m^{*} \in \mathbb{T}_m^{(0)}} f\big(n-m , \mathcal{M}_m^{(0)}\big) \nonumber \\ & \quad \quad \times \prod_{i=1}^{m}\big\{ \mathbb{P}[w \in M_{0^{i-1}, 1, 0^{m-i}} \boldsymbol{\mid}\mathcal {T}_m = \mathcal {T}_m^{*} ]^{h} \big\}\nonumber \\ & \quad \quad \times \mathbb{P} [w \in M_{0^m} \boldsymbol{\mid}\mathcal {T}_m = \mathcal {T}_m^{*} ]^{n-m-hm} \label{eqn_probMm_qcmp}\end{aligned}$$ ${p_{e,q}}^{(1)} = p_{e, q}|_{q=1} = {p_{e,q}}$ From Lemma 7 in [@ZhaoYaganGligor], we know that ${p_{e,q}} \sim \frac{\ln n}{n}$ and $P_n \geq \sigma n$ lead to $K_n = \Omega(\sqrt{\ln n}) \to \infty$ as $n \to \infty$. Proof of Lemma \[lem\_psukn\_qcmp\] ----------------------------------- To compute the probability of the event $\Gamma_{it}\cap \Gamma_{jt}$ which is equivalent to the event $$(|S_t \cap S_i | \geq q) \cap (|S_t \cap S_j | \geq q),$$ we specify all the possible cardinalities of sets $S_t \cap (S_i \setminus S_j)$, $S_t \cap (S_j \setminus S_i)$, and $S_t \cap (S_i \cap S_j)$. We define event $F(a,b,d)$ as $$\begin{aligned} & \big[|S_t \cap (S_i \setminus S_j)| = a\big] \hspace{3pt} \cap \hspace{3pt} \big[|S_t \cap (S_j \setminus S_i)| = b\big]\nonumber \\ & \hspace{96pt} \cap \hspace{3pt} [|S_t \cap (S_i \cap S_j)| = d], \nonumber\end{aligned}$$ where $(a,b,d) \in \Lambda$ with $\Lambda$ defined as the set of all possible $(a,b,d) $ such that event $K_{{ wv_i}}^{(q)}\cap K_{ wv_j}^{(q)}$ happens. Therefore, $$\begin{aligned} K_{{ wv_i}}^{(q)}\cap K_{ wv_j}^{(q)} & = \bigcup_{(a,b,d) \in \Lambda} F(a,b,d) . \nonumber\end{aligned}$$ It’s clear that any $(a,b,d) $ in $ \Lambda$ satisfies $$\begin{aligned} & a, b, d \geq 0, \nonumber \\ & a, b \leq K_n - u, \nonumber \\ & d \leq u, \nonumber \\ & a+b+d \leq K_n, \nonumber \\ & a + d \geq q,\textrm{ and} \nonumber \\ & b + d \geq q. \nonumber\end{aligned}$$ Then $\Lambda$ is the set of all possible $(a,b,d) $ with $$\begin{aligned} 0 & \leq d \leq u, \nonumber \\ \max\{0, q-d \} & \leq a \leq K_n - u ,\textrm{ and }\nonumber \\ \max\{0, q-d \} & \leq b \leq \min\{ K_n - u, K_n - a -d \}. \nonumber\end{aligned}$$ By setting $$\begin{aligned} g(a,b,d) &: = \mathbb{P}\big[F(a,b,d)\big] , \label{def_gabd}\end{aligned}$$ we further obtain $$\begin{aligned} \mathbb{P}\big[K_{{ wv_i}}^{(q)}\cap K_{ wv_j}^{(q)} \boldsymbol{\mid} \big(|S_{i} \cap S_{j}| = u \big)\big] & = \sum_{(a,b,d) \in \Lambda} g(a,b,d). \label{sumgabd} $$ We compute $g(a,b,d)$ given its definition in (\[def\_gabd\]). We have $$\begin{aligned} & g(a,b,d) \nonumber \\ & \quad : = \mathbb{P}\big[ [|S_t \cap (S_i \setminus S_j)| = a] \hspace{3pt} \cap \nonumber \\ & \quad \hspace{26pt} [|S_t \cap (S_j \setminus S_i)| = b] ] \hspace{3pt} \cap \nonumber \\ & \quad \hspace{26pt} [|S_t \cap (S_i \cap S_j)| = d] \hspace{6pt} \boldsymbol{\mid} \hspace{4pt} (|S_{i} \cap S_{j}| = u)\big] \nonumber \\ & \quad = \frac{\binom{u}{d}\binom{K_n-u}{a}\binom{K_n-u}{b} \binom{P_n-2K_n+u}{K_n-a-b-d}}{\binom{P_n}{K_n}}. \label{eqn_gabd}\end{aligned}$$ For integers $x$ and $y$ with $x \geq y \geq 0$, given $\binom{x}{y} = \frac{x!}{y!(x-y)!}$, it is easy to check by direct inspection that $ \frac{(x-y)^y}{y!} \leq \binom{x}{y} \leq \frac{x^y}{y!}$. Then with $\frac{{K_n}^2}{P_n-K_n}$ denoted by $\xi$ for the brevity of notation, we get $$\begin{aligned} & g(a,b,d) \nonumber \\ & \quad \leq \frac{u^d}{d!} \cdot \frac{(K_n-u)^a}{a!} \cdot\frac{(K_n-u)^b}{b!} \nonumber \\ & \quad \quad \times \frac{(P_n-2K_n+u)^{K_n-a-b-d}}{(K_n-a-b-d)!} \cdot \frac{ K_n !}{(P_n-K_n)^{K_n }} \nonumber \\ & \quad \leq \frac{ 1 }{a!b!d! } u^d{K_n}^{2(a+b)+d} {(P_n-K_n)}^{-(a+b+d)} \nonumber \\ & \quad = \frac{ 1 }{a!b!d! } \bigg(\frac{u}{K_n}\bigg)^{d} {\xi}^{a+b+d}. \label{eq_gabd_ine}\end{aligned}$$ We define $\Lambda_1$ as the set of all possible $(a,b,d) $ satisfying $a \geq 0, b \geq 0, d \geq 0$, $a+d \geq q$ and $b+d \geq q$. Clearly, $\Lambda \subseteq \Lambda_1$. All $(a,b,d) $ in $\Lambda_1$ can be divided into the following cases: - $d = q, a = 0, b = 0$; - $d = q, a \geq 1, b \geq 0$; - $d > q, a \geq 0, b \geq 0$; and - $d < q, a \geq q-d, b \geq q-d$. Setting $\Lambda_2: = \Lambda_1 \setminus \{(0,0,q)\}$, we have $ \Lambda \setminus \{(0,0,q)\} \subseteq \Lambda_2$ given $\Lambda \subseteq \Lambda_1$. By definition of set $\Lambda$, it is straightforward to check - if $q \leq u$, then $(0,0,q) \in \Lambda$; and - if $q > u$, then $(0,0,q) \notin \Lambda$. Hence, from (\[sumgabd\]), $$\begin{aligned} & \mathbb{P}\big[K_{{ wv_i}}^{(q)}\cap K_{ wv_j}^{(q)} \boldsymbol{\mid} \big(|S_{i} \cap S_{j}| = u \big)\big] - \boldsymbol{1}[q \leq u] \cdot g(0, 0, q) \nonumber \\ & \quad = \sum_{(a,b,d) \in \Lambda \setminus \{(0,0,d)\}} g(a,b,d). \label{sumgabd_00q}\end{aligned}$$ From (\[eq\_gabd\_ine\]) (\[sumgabd\_00q\]) and $ \Lambda \setminus \{(0,0,d)\} \subseteq \Lambda_2$, $$\begin{aligned} & \mathbb{P}\big[K_{{ wv_i}}^{(q)}\cap K_{ wv_j}^{(q)} \boldsymbol{\mid} \big(|S_{i} \cap S_{j}| = u \big)\big] - \boldsymbol{1}[q \leq u] \cdot g(0, 0, q) \nonumber \\ & \quad \leq \sum_{a,b,d:\hspace{2pt} (a,b,d)\in \Lambda_2} \frac{ 1 }{a!b!d! } \bigg(\frac{u}{K_n}\bigg)^{d} \xi^{a+b+d} \nonumber \\ & \quad \leq \frac{ 1 }{q! } \bigg(\frac{u \xi}{K_n}\bigg)^{q} \sum_{a=1}^{\infty} \frac{ \xi^{a} }{a! } \sum_{b=0}^{\infty} \frac{ \xi^{b} }{b! } \nonumber \\ & \quad\quad + \sum_{d=q+1}^{\infty} \frac{ 1 }{d! } \bigg(\frac{u \xi}{K_n}\bigg)^{d} \sum_{a=0}^{\infty} \frac{ \xi^{a} }{a! } \sum_{b=0}^{\infty} \frac{ \xi^{b} }{b! } \nonumber \\ & \quad\quad + \sum_{d=0}^{q-1} \frac{ 1 }{d! } \bigg(\frac{u \xi}{K_n}\bigg)^{d} \sum_{a=q-d}^{\infty} \frac{ \xi^{a} }{a! } \sum_{b=q-d}^{\infty} \frac{ \xi^{b} }{b! }. \label{sumabd}\end{aligned}$$ From $\frac{{K_n}^2}{P_n} = o(1)$, we have $P_n = \omega(K_n)$ and further obtain $$\begin{aligned} \xi & = \frac{{K_n}^2}{P_n-K_n} \sim \frac{{K_n}^2}{P_n} = o(1). \label{eq_gammao1}\end{aligned}$$ For any non-negative integer $\phi$, by $\xi = o(1)$ in (\[eq\_gammao1\]), $$\begin{aligned} \sum_{t=\phi}^{\infty} \frac{{\xi}^t}{t!} & = {\xi}^{\phi} \sum_{\tau=0}^{\infty} \frac{{\xi}^{\tau}}{(\tau+\phi)!} \quad (\textrm{by setting } \tau = t - \phi ) \nonumber \\ & \leq {\xi}^{\phi} \sum_{\tau=0}^{\infty} \frac{1}{\tau! \phi!} {\xi}^{\tau} = \frac{{\xi}^{\phi}}{\phi!} \cdot e^{{\xi}} \leq \frac{{\xi}^{\phi}}{\phi!} \cdot [1+o(1)]. \label{exp_bound}\end{aligned}$$ Applying (\[exp\_bound\]) to (\[sumabd\]), $$\begin{aligned} & \mathbb{P}\big[K_{{ wv_i}}^{(q)}\cap K_{ wv_j}^{(q)} \boldsymbol{\mid} \big(|S_{i} \cap S_{j}| = u \big)\big] - \boldsymbol{1}[q \leq u] \cdot g(0, 0, q) \nonumber \\ & \quad \leq \frac{ 1 }{q! } \bigg(\frac{u \xi}{K_n}\bigg)^{q} \xi \cdot [1+o(1)] + \frac{{\xi}^{q+1}}{(q+1)!} \cdot [1+o(1)] \nonumber \\ & \quad\quad + \sum_{d=0}^{q-1} \frac{ 1 }{d! } \bigg(\frac{u \xi}{K_n}\bigg)^{d} \frac{{\xi}^{2(q-d)}}{[(q-d)!]^2} \cdot [1+o(1)]. \label{kwvikwvj}\end{aligned}$$ To bound the last term in (\[kwvikwvj\]), we have $$\begin{aligned} & \sum_{d=0}^{q-1} \frac{ 1 }{d! } \bigg(\frac{u \xi}{K_n}\bigg)^{d} \frac{{\xi}^{2(q-d)}}{[(q-d)!]^2} \nonumber \\ & \quad \leq \sum_{d=0}^{q-1} {\xi}^{2(q-d)} \leq \sum_{d=0}^{q-1} {\xi}^{q+1} = q {\xi}^{q+1} \label{kwvikwvj_last}\end{aligned}$$ Then using (\[kwvikwvj\_last\]) in (\[kwvikwvj\]), $$\begin{aligned} & \mathbb{P}\big[K_{{ wv_i}}^{(q)}\cap K_{ wv_j}^{(q)} \boldsymbol{\mid} \big(|S_{i} \cap S_{j}| = u \big)\big] - \boldsymbol{1}[q \leq u] \cdot g(0, 0, q) \nonumber \\ & \quad \leq \frac{ \xi^{q+1} }{q! } \cdot [1+o(1)] + \frac{{\xi}^{q+1}}{(q+1)!} \cdot [1+o(1)] \nonumber \\ & \quad\quad + q {\xi}^{q+1} \cdot [1+o(1)] \nonumber \\ & \quad \leq \bigg[ q+ \frac{ 1 }{q! } + \frac{1}{(q+1)!} \bigg] \xi^{q+1} \cdot [1+o(1)] \nonumber \\ & \quad \leq (q+2) \xi^{q+1}.\label{kwiwjq2}\end{aligned}$$ From Lemma \[lem\_eval\_psq\]-Property (i) and $ \xi \sim \frac{{K_n}^2}{P_n} $ established in (\[eq\_gammao1\]), $$\begin{aligned} p_{s,q}^{(q)} & \sim \frac{1}{q!} \bigg( \frac{{K_n}^2}{P_n} \bigg)^{q} \sim \frac{\xi^{q}}{q!}. \nonumber\end{aligned}$$ Hence, for any constant $c>1$, for sufficiently large $n$, $$\begin{aligned} \frac{\xi^{q}}{q!} & \leq c p_{s,q}^{(q)} \nonumber\end{aligned}$$ Then by setting $1<c\leq \big(\frac{q+1}{q}\big)^{\frac{q}{q+1}}$, for sufficiently large $n$, we obtain $$\begin{aligned} (q+2) \xi^{q+1} & \leq (q+2) \big( c p_{s,q}^{(q)} \cdot q! \big)^{\frac{q+1}{q}} \nonumber \\ & = (q+2) c^{\frac{q+1}{q}} (q!)^{\frac{q+1}{q}} \big( p_{s,q}^{(q)} \big)^{\frac{q+1}{q}} \nonumber \\ & \leq (q+2) \cdot (q+1)/q \cdot (q!) \cdot q \cdot \big( p_{s,q}^{(q)} \big)^{\frac{q+1}{q}} \nonumber \\ & = (q+2)! \cdot ( p_{s,q}^{(q)} \big)^{\frac{q+1}{q}} \label{eqnq2psq}\end{aligned}$$ From (\[eqn\_gabd\]) and (\[psijq\_eq\]), we have $$\begin{aligned} g(0,0,q) & = \frac{\binom{u}{q} \binom{P_n-2K_n+u}{K_n-q}}{\binom{P_n}{K_n}} \nonumber\end{aligned}$$ and further obtain $$\begin{aligned} \lefteqn{\frac{g(0,0,q)}{\mathbb{P}[|S_{i} \cap S_{j}| = q]}} \nonumber \\ & = \frac{\binom{u}{q} \binom{P_n-2K_n+u}{K_n-q}}{\binom{K_n}{q}\binom{P_n-K_n}{K_n-q}} \nonumber \\ & = \frac{~~~~\frac{\prod_{i=0}^{q-1}(u-i)}{q!} \cdot \frac{\prod_{i=0}^{K_n-q-1}(P_n-2K_n+u-i)}{(K_n-q)!}~~~~} {~~~~\frac{\prod_{i=0}^{q-1}(K_n-i)}{q!} \cdot \frac{\prod_{i=0}^{K_n-q-1}(P_n-K_n-i)}{(K_n-q)!}~~~~} \nonumber \\ & = \bigg( \prod_{i=0}^{q-1} \frac{u-i}{K_n-i} \bigg) \bigg( \prod_{i=0}^{q-1} \frac{P_n-2K_n+u-i}{P_n-K_n-i} \bigg) \nonumber \\ & = \bigg[\prod_{i=0}^{q-1}\bigg(\frac{u}{K_n}-\frac{i(K_n-u)}{K_n(K_n-i)}\bigg) \bigg] \bigg[\prod_{i=0}^{q-1}\bigg(1-\frac{K_n-u}{P_n-K_n-i}\bigg) \bigg] \nonumber \\ & \leq \Big(\frac{u}{K_n}\Big)^q. \label{eqnuknq}\end{aligned}$$ From (\[eqnuknq\]) and $p_{s,q}^{(q)} = \sum_{u=q}^{K_n} \mathbb{P}[|S_{i} \cap S_{j}| = u]$ by definition of $p_{s,q}^{(q)}$, $$\begin{aligned} g(0,0,q) & \leq \Big(\frac{u}{K_n}\Big)^q \cdot p_{s,q}^{(q)} \leq \frac{u}{K_n} \cdot p_{s,q}^{(q)}. \label{eq_g00qpsq}\end{aligned}$$ The proof of Lemma \[lem\_psukn\_qcmp\] is completed by substitution of (\[eqnq2psq\]) and (\[eq\_g00qpsq\]) into (\[kwiwjq2\]). Proofs of Theorems \[thm:exact\_qcomposite\] and \[thm:exact\_qcomposite-more-fine-grained\] Using Theorem \[thm:exact\_qcomposite2\] {#sec-prove-first-two-theorems-mnd} ===================================================================================================================================== As explained in Section \[sec-basic-proof-ideas\], we establish Theorems \[thm:exact\_qcomposite\] and \[thm:exact\_qcomposite-more-fine-grained\] based on Theorem \[thm:exact\_qcomposite2\]. Theorems \[thm:exact\_qcomposite\] and \[thm:exact\_qcomposite-more-fine-grained\] present results of $\delta$, where $\delta$ denotes the minimum degree of $\mathbb{G}_q\iffalse_{on}\fi$. With $\Phi_{n,h}$ denoting the number of nodes with degree $h$ in $\mathbb{G}_{q}\iffalse\fi$, Theorem \[thm:exact\_qcomposite2\] provides the asymptotic distribution of $\Phi_{n,h}$. To use Theorem \[thm:exact\_qcomposite2\] for proving Theorems \[thm:exact\_qcomposite\] and \[thm:exact\_qcomposite-more-fine-grained\], we now discuss the relationship between $\delta$ and $\Phi_{n,h}$. For non-negative integer $\mu$, it is straightforward to see properties and below. - The event $(\delta \geq \mu)$ (i.e., the event that the minimum node degree of graph $\mathbb{G}_q\iffalse_{on}\fi$ is at least $\mu$) is equivalent to the event $ \bigcap_{h=0}^{\mu-1} (\Phi_{n,h} = 0) $ (i.e., no node has degree falling in $\{0,1,\ldots, \mu-1\}$). - The event $(\delta \leq \mu)$ (i.e., the event that the minimum node degree of graph $\mathbb{G}_q\iffalse_{on}\fi$ is at most $\mu$) and the event $ \bigcup_{h=0}^{\mu} (\Phi_{n,h} \neq 0) $ (i.e., there is at least one node with degree at most $\mu$) are equivalent. Therefore, for any integer $\xi$, we obtain $$\begin{aligned} &\mathbb{P}[\delta \geq \xi +1] \nonumber \\&= \mathbb{P}\bigg[\bigcap_{h=0}^{\xi } (\Phi_{n,h} = 0)\bigg] \textrm{~(by property \ding {202})} \nonumber \\& \leq \mathbb{P}[\Phi_{n,\xi} = 0],\textrm{ if }\xi \geq 0 , \label{eqpd1} \end{aligned}$$$$\begin{aligned} & \mathbb{P}[\delta \leq \xi -2] \nonumber \\&\leq \mathbb{P}\bigg[ \bigcup_{h=0}^{\xi-2} (\Phi_{n,h} \neq 0)\bigg] \textrm{~(by property \ding {203})} \nonumber \\& \leq \sum_{h=0}^{\xi-2} \mathbb{P}[ \Phi_{n,h} \neq 0]\, \textrm{~(by the union bound)},\textrm{ if }\xi \geq 2, \label{eqpd2}\end{aligned}$$$$\begin{aligned} & \mathbb{P}[\delta \geq \xi]\nonumber \\& = \mathbb{P}\bigg[\bigcap_{h=0}^{\xi-1} (\Phi_{n,h} = 0)\bigg] \textrm{~(by property \ding {202})} \nonumber \\& \leq \mathbb{P}[\Phi_{n,\xi-1} = 0],\textrm{ if }\xi \geq 1, \label{eqn_1mindel2} \end{aligned}$$ and $$\begin{aligned} \mathbb{P}[\delta \geq \xi] & = \mathbb{P}\bigg[\bigcap_{h=0}^{\xi-1} (\Phi_{n,h} = 0)\bigg] \textrm{~(by property \ding {202})} \nonumber \\ & = 1 - \mathbb{P}\bigg[\bigcup_{h=0}^{\xi-1} (\Phi_{n,h} \neq 0)\bigg] \nonumber \\ & \geq 1 - \sum_{h=0}^{\xi-1} \mathbb{P}[\Phi_{n,h} \neq 0]\textrm{ (by the union bound)} \nonumber \\ & = \mathbb{P}[\Phi_{n,k-1} = 0] - \boldsymbol{1}[k\geq 2]\times \sum_{h=0}^{k-2} \mathbb{P}[ \Phi_{n,h} \neq 0] , \label{eqn_1min}\end{aligned}$$ where the indicator variable $\boldsymbol{1}[k\geq 2]$ equals $1$ if $k \geq 2$ and $0$ if $k<2$. To use (\[eqpd1\])–(\[eqn\_1min\]), we will compute $\mathbb{P}[ \Phi_{n,h} = 0]$ and $\mathbb{P}[ \Phi_{n,h} \neq 0]$ for $h=0,1,\ldots$ To this end, we use Theorem \[thm:exact\_qcomposite2\], which shows that $\Phi_{n,h}$ is in distribution asymptotically equivalent to a Poisson random variable with mean $\lambda_{n,h} $ specified by $$\begin{aligned} \lambda_{n,h} & : = n (h!)^{-1}(n p_{e, q})^h e^{-n p_{e, q}}; \label{eqn_labmdahnew} \end{aligned}$$ i.e., $$\begin{aligned} \mathbb{P}[\Phi_{n,h} = \ell] & \sim (\ell !)^{-1}{\lambda_{n,h}} ^{\ell} e^{-\lambda_{n,h}} . \label{eqn_phihellnew} \end{aligned}$$ To assess $\lambda_{n,h}$ in (\[eqn\_labmdahnew\]), we use (\[peq1sbsc\]) about $p_{e, q}$ (i.e., $p_{e, q} = \frac{\ln n + {(k-1)} \ln \ln n + {\alpha_n}}{n}$). While $\alpha_n$ in Theorem \[thm:exact\_qcomposite-more-fine-grained\] is given by (\[alpha-n-written-beta-n\]) and satisfies $ |\alpha_n| = O(\ln \ln n) = o(\ln n)$ for a constant integer $b$ and a sequence $\beta_n$ under (\[liminfbetan\]), $\alpha_n$ in Theorem \[thm:exact\_qcomposite\] may not satisfy $|\alpha_n| = o(\ln n)$. However, we can still introduce the additional condition $|\alpha_n| = o(\ln n)$ in proving Theorem \[thm:exact\_qcomposite\], as explained in Appendix E of the full version [@full]. The idea is to show that whenever Theorem \[thm:exact\_qcomposite\] with $|\alpha_n| = o(\ln n)$ holds, then Theorem \[thm:exact\_qcomposite\] regardless of $|\alpha_n| = o(\ln n)$. Now under $|\alpha_n| = o(\ln n)$ in Theorem \[thm:exact\_qcomposite\], we obtain $$\begin{aligned} p_{e, q} & \sim \frac{\ln n}{n},\label{eq_pe_lnnn-tonnew}\end{aligned}$$ where $f_n \sim g_n$ for two positive sequences $f_n$ and $g_n$ means $\lim_{n \to \infty} {{f_n}}/{g_n}=1$; i.e., (\[eq\_pe\_lnnn-tonnew\]) means $\lim_{n \to \infty} {p_{e, q}}\big/\big(\frac{\ln n}{n}\big)=1$. Then we substitute (\[peq1sbsc\]) and (\[eq\_pe\_lnnn-tonnew\]) into (\[eqn\_labmdahnew\]) to derive $$\begin{aligned} \lambda_{n,h} & = n (h!)^{-1}(n p_{e,q})^h e^{-n p_{e,q}} \nonumber \\ & \sim n (h!)^{-1} (\ln n)^h \times e^{-\ln n - (k-1)\ln \ln n - \alpha_n} \nonumber \\ & = (h!)^{-1} (\ln n)^{h+1-k} e^{-\alpha_n}. \label{liminfbetan3}\end{aligned}$$ We now [use Theorem \[thm:exact\_qcomposite2\] (i.e., (\[eqn\_phihellnew\])) to prove Theorem \[thm:exact\_qcomposite\]]{} under the additional condition $|\alpha_n| = o(\ln n)$, which we can introduce based on the above discussion. Then we evaluate $\mathbb{P}[\delta \geq k ].$ Given $k \geq 1$, we know from (\[eqn\_1mindel2\]) and (\[eqn\_phihellnew\]) that $$\begin{aligned} \mathbb{P}[\delta \geq k ] & \leq e^{-\lambda_{n,k-1}} \times [1+o(1)], \label{eqn_phihell-ton-part1}\end{aligned}$$ and know from (\[eqn\_1min\]) and (\[eqn\_phihellnew\]) that $$\begin{aligned} &\hspace{-8pt} \mathbb{P}[\delta \geq k ] \nonumber \geq \\ & \hspace{-8pt} e^{-\lambda_{n,k-1}} \hspace{-2pt}\times \hspace{-2pt}[1\hspace{-2pt}-\hspace{-2pt}o(1)] \hspace{-1pt}- \hspace{-1pt}\boldsymbol{1}[ k\hspace{-2pt}\geq \hspace{-2pt}2]\hspace{-2pt}\times \hspace{-2pt} \sum_{h=0}^{k-2} \big\{ \big(1- e^{-\lambda_{n,h}} \big)\hspace{-2pt} \times\hspace{-2pt} [1\hspace{-2pt}+\hspace{-2pt}o(1)]\big\} . \label{eqn_phihell-ton-part2}\end{aligned}$$ Based on (\[eqn\_phihell-ton-part1\]) and (\[eqn\_phihell-ton-part2\]), we discuss the following cases. - If $\lim_{n \to \infty} \alpha_n = \alpha ^* \in (-\infty, \infty)$, (\[liminfbetan3\]) implies for $k\geq 1$ that [\_[n,h]{} ]{} 0,&$h = 0, 1, \ldots, k-2,$, \[eqn-prove-thm1-alpha-n-finite-1\]\ ,&.\[eqn-prove-thm1-alpha-n-finite-2\] Applying (\[eqn-prove-thm1-alpha-n-finite-2\]) to (\[eqn\_phihell-ton-part1\]), and applying (\[eqn-prove-thm1-alpha-n-finite-1\]) (\[eqn-prove-thm1-alpha-n-finite-2\]) to (\[eqn\_phihell-ton-part2\]), we have $e^{- \frac{e^{-\alpha ^*}}{(k-1)!}} \times [1-o(1)] \leq \mathbb{P}[\delta \geq k] \leq e^{- \frac{e^{-\alpha ^*}}{(k-1)!}} \times [1+o(1)]$ so that $\lim_{n \to \infty} \mathbb{P}[\delta \geq k] = e^{- \frac{e^{-\alpha ^*}}{(k-1)!}}$; i.e., (\[thm-mnd-alpha-finite\]) is proved. - If $\lim_{n \to \infty} \alpha_n = \infty$, then (\[liminfbetan3\]) implies for $k\geq 1$ that $$\begin{aligned} \lambda_{n,h} \to 0 \textrm{~~~for } h = 0, 1, \ldots, k-1. \label{eqn-prove-thm1-alpha-n-infinite}\end{aligned}$$ Substituting (\[eqn-prove-thm1-alpha-n-infinite\]) into (\[eqn\_phihell-ton-part1\]), and substituting (\[eqn-prove-thm1-alpha-n-infinite\]) into (\[eqn\_phihell-ton-part2\]), we obtain $ [1-o(1)] \leq \mathbb{P}[\delta \geq k] \leq [1+o(1)]$ so that $\lim_{n \to \infty} \mathbb{P}[\delta \geq k] = 1$; i.e., (\[thm-mnd-alpha-infinite\]) is proved. - If $\lim_{n \to \infty} \alpha_n = - \infty$, then (\[liminfbetan3\]) implies for $k\geq 1$ that $$\begin{aligned} \lambda_{n,k-1} \to \infty. \label{eqn-prove-thm1-alpha-n-minus-infinite}\end{aligned}$$ Using (\[eqn-prove-thm1-alpha-n-minus-infinite\] ) in (\[eqn\_phihell-ton-part1\]), we have $ \mathbb{P}[\delta \geq k] \leq o(1)$ so that $\lim_{n \to \infty} \mathbb{P}[\delta \geq k] = 0$; i.e., (\[thm-mnd-alpha-minus-infinite\]) is proved. - We now [use Theorem \[thm:exact\_qcomposite2\] (i.e., (\[eqn\_phihellnew\])) to prove Theorem \[thm:exact\_qcomposite-more-fine-grained\]]{}. The condition (\[liminfbetan\]) on $\beta_n$ implies that there are constants $c_1$ and $c_2$ with $-1<c_1\leq c_2 <1$ such that $$\begin{aligned} c_1 \ln \ln n & \leq \beta_n \leq c_2 \ln \ln n, \text{ for all $n$ sufficiently large}, \label{liminfbetan2-pre}\end{aligned}$$ which implies $$\begin{aligned} \hspace{-10pt} (\ln n)^{-c_2} & \leq e^{-\beta_n} \leq (\ln n)^{-c_1}, \text{ for all $n$ sufficiently large}. \label{liminfbetan2}\end{aligned}$$ Using (\[alpha-n-written-beta-n\]) in (\[peq1sbsc\]), we have $p_{e, q} = \frac{\ln n + {(k+b-1)} \ln \ln n + {\beta_n}}{n}$, which along with (\[liminfbetan2-pre\]) implies $p_{e, q} \sim \frac{\ln n}{n}$. Then similar to (\[liminfbetan3\]), we derive $ \lambda_{n,h}\sim (h!)^{-1} (\ln n)^{h+1-(k+b)} e^{-\beta_n}$. Applying (\[liminfbetan2\]) to this result and noting $-1<c_1\leq c_2 <1$, we find $$\begin{aligned} \hspace{-9pt} \lambda_{n,h} & \begin{cases} \to 0,&\hspace{-22pt}\textrm{for }h = 0, 1, \ldots, k+b-2,\textrm{ if }k+b \geq 2; \\ \sim \frac{e^{-\beta_n}}{(k+b-1)!},&\hspace{-6pt}\textrm{for }h = k+b-1,\textrm{ if }k+b \geq 1; \\ \to \infty ,&\hspace{-22pt}\textrm{for }h = \max\{ k+b, 0\}, \max\{ k+b, 0\}+1, \ldots. \end{cases} \label{eqn_lbdh} \end{aligned}$$ Using (\[eqn\_lbdh\]) in (\[eqn\_phihellnew\]), we get $$\begin{aligned} \hspace{-129pt} \mathbb{P}[\Phi_{n,h} = 0] \sim e^{-\lambda_{n,h}} \nonumber \end{aligned}$$ 1, h = 0, 1, …, k+b-2,k+b 2; \[eqn\_expr\_lahkk1case1\]\ \~e\^[-]{}, h = k+b-1,k+b 1; \[eqn\_expr\_lahkk1case2\]\ 0 , h = { k+b, 0}, { k+b, 0}+1, …\[eqn\_expr\_lahkk1case3\] If $k+b \leq 0$, then (\[eqn\_expr\_lahkk1case3\]) gives $ \mathbb{P}[\Phi_{n,0} = 0] \to 0$, which along with (\[eqn\_1mindel2\]) and (\[eqn\_1min\]) yields $ \mathbb{P}[\delta \geq 1] = \mathbb{P}[\Phi_{n,0} = 0] \to 0$ so that we further obtain $\lim\limits_{n \to \infty}{{\mathbb{P}}\left[{\delta = 0}\right]}=1$ and $\lim\limits_{n \to \infty}{{\mathbb{P}}\left[{\delta > 0}\right]}=0$. Hence, property of Theorem \[thm:exact\_qcomposite-more-fine-grained\] is proved. Below we consider the case of $k+b \geq 1$ to prove properties – of Theorem \[thm:exact\_qcomposite-more-fine-grained\]. Given $k+b \geq 1$, we derive from (\[eqpd2\]) and (\[eqn\_expr\_lahkk1case1\]) that $$\begin{aligned} \hspace{-5pt}\mathbb{P}[\delta \leq k+b -2] \begin{cases} \leq \sum_{h=0}^{k+b-2} \mathbb{P}[ \Phi_{n,h} \neq 0] \to 0,\textrm{ if }k+b \geq 2,\\ =0,\textrm{ if }k+b =1, \end{cases} \nonumber $$ which implies $$\begin{aligned} \mathbb{P}[\delta \leq k+b -2] & = o(1). \label{eqn_delta_k2}\end{aligned}$$ Given $k+b \geq 1$, we obtain from (\[eqpd1\]) and (\[eqn\_expr\_lahkk1case3\]) that $$\begin{aligned} \mathbb{P}[\delta \geq k+b +1] & \leq \mathbb{P}[\Phi_{n,k+b} = 0] = o(1). \label{eqn_delta_k1}\end{aligned}$$ Given $k+b \geq 1$, we show from (\[eqn\_1mindel2\]) and (\[eqn\_expr\_lahkk1case2\]) that $$\begin{aligned} \mathbb{P}[\delta \geq k+b ] & \leq \mathbb{P}[\Phi_{n,k+b-1} = 0] \sim e^{- \frac{e^{-\beta_n}}{(k+b-1)!}}, \label{eqn_delta_k1given1}\end{aligned}$$ and show from (\[eqn\_1min\]) and (\[eqn\_expr\_lahkk1case1\]) that $$\begin{aligned} & \mathbb{P}[\delta \geq k+b ] \nonumber \\ & \geq \mathbb{P}[\Phi_{n,k+b-1} = 0] - \boldsymbol{1}[k+b\geq 2]\cdot \sum_{h=0}^{k+b-2} \mathbb{P}[ \Phi_{n,h} \neq 0] \nonumber \\ & \sim e^{- \frac{e^{-\beta_n}}{(k+b-1)!}}.\label{eqn_delta_k1given2}\end{aligned}$$ Then (\[eqn\_delta\_k1given1\]) and (\[eqn\_delta\_k1given2\]) together induce $$\begin{aligned} \mathbb{P}[\delta \geq k+b] & \sim e^{- \frac{e^{-\beta_n}}{(k+b-1)!}} . \label{eqn_delta_k3}\end{aligned}$$ From (\[eqn\_delta\_k2\]) and (\[eqn\_delta\_k1\]), we have $$\begin{aligned} & \mathbb{P}[(\delta \neq k+b) \cap (\delta \neq k+b-1)]\nonumber \\ & = \mathbb{P}[\delta \geq k+b +1] + \mathbb{P}[\delta \leq k+b -2] = o(1); \label{eqn_delta_k4} \end{aligned}$$ from (\[eqn\_delta\_k1\]) and (\[eqn\_delta\_k3\]), we obtain $$\begin{aligned} \mathbb{P}[\delta = k+b] & = \mathbb{P}[\delta \geq k+b] - \mathbb{P}[\delta \geq k+b +1] \nonumber \\ & \sim e^{- \frac{e^{-\beta_n}}{(k+b-1)!}} \nonumber \\ & \to \begin{cases} e^{- \frac{e^{-\beta ^*}}{(k+b-1)!}} , &\hspace{-8pt}\textrm{if $\lim_{n \to \infty} \beta_n = \beta ^* \in (-\infty, \infty)$,} \\ 1, &\hspace{-8pt}\textrm{if $ \lim_{n \to \infty} \beta_n = \infty$}, \\ 0, &\hspace{-8pt}\textrm{if $ \lim_{n \to \infty} \beta_n = - \infty$}; \end{cases} \label{eqn_delta_k5} \end{aligned}$$ and from (\[eqn\_delta\_k4\]) and (\[eqn\_delta\_k5\]), we conclude $$\begin{aligned} & \mathbb{P}[\delta = k+b-1] \nonumber \\ & = 1- \mathbb{P}[(\delta \neq k+b) \cap (\delta \neq k+b-1)] - \mathbb{P}[\delta = k+b] \nonumber \\ & \to \begin{cases} 1 - e^{- \frac{e^{-\beta ^*}}{(k+b-1)!}} , &\textrm{if $\lim_{n \to \infty} \beta_n = \beta ^* \in (-\infty, \infty)$,} \\ 0, &\textrm{if $ \lim_{n \to \infty} \beta_n = \infty$}, \\ 1, &\textrm{if $ \lim_{n \to \infty} \beta_n = - \infty$}. \end{cases} \label{eqn_delta_k6} \end{aligned}$$ Properties – of Theorem \[thm:exact\_qcomposite2\] follow from (\[eqn\_delta\_k4\])–(\[eqn\_delta\_k6\]). To summarize, We have used Theorem \[thm:exact\_qcomposite2\] (proved in Section \[sec\_est\] later) to establish Theorem \[thm:exact\_qcomposite\] under the additional condition $|\alpha_n| = o(\ln n)$, and to establish Theorem \[thm:exact\_qcomposite-more-fine-grained\]. In Appendix E of the full version [@full], we explain that whenever Theorem \[thm:exact\_qcomposite\] with $|\alpha_n| = o(\ln n)$ holds, then Theorem \[thm:exact\_qcomposite\] regardless of $|\alpha_n| = o(\ln n)$. [$\blacksquare$]{} Establishing Theorem \[thm:exact\_qcomposite2\] {#sec_est} =============================================== Method of Moments ----------------- For $h = 0,1, \ldots$, with $\Phi_{n,h}$ counting the number of nodes with degree $h$ in $\mathbb{G}_q\iffalse_{on}\fi$, we will show that $\Phi_{n,h} $ asymptotically follows a Poisson distribution with mean $\lambda_{n,h}$. This is done by using the method of moments; specifically, in view of [@2008asymptotic Theorem 2.13], we will obtain the desired result upon establishing $$\begin{aligned} \mathbb{P} [\textrm{Nodes }v_{1}, v_{2}, \ldots, v_{m}\textrm{ have degree }h] & \sim {\lambda_{n,h}}^m / n^m. \label{eqn_node_v12n}\end{aligned}$$ Therefore, if Lemma \[LEM1\] below holds, then the proof of property (a) in Theorem \[thm:exact\_qcomposite2\] is completed; in particular, we will have that for any integers $h \geq 0$ and $\ell \geq 0$, $$\begin{aligned} \mathbb{P}[\phi_h = \ell] & \sim (\ell !)^{-1}{\lambda_h} ^{\ell}e^{-\lambda_{n,h}} . \label{eqn_phihell} \end{aligned}$$ \[LEM1\] Given (\[peq1\]) (i.e., $p_{e, q} = \frac{\ln n \pm O(\ln \ln n)}{n}$), $ K_n = \omega(1)$ and $\frac{{K_n}^2}{P_n} = o(1)$, then for any integers $m \geq 1$ and $h \geq 0$, we have $$\begin{aligned} & \mathbb{P} [\textrm{Nodes }v_{1}, v_{2}, \ldots, v_{m}\textrm{ have degree }h] \nonumber \\ & \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad~ \sim (h!)^{-m} (n p_{e, q})^{hm} e^{-m n p_{e, q}};\nonumber\end{aligned}$$ i.e., (\[eqn\_node\_v12n\]) follows with $\lambda_{n,h} $ set by $$\begin{aligned} \lambda_{n,h} & = n (h!)^{-1}(n p_{e, q})^h e^{-n p_{e, q}}. \label{eqn_labmdah} \end{aligned}$$ Section \[secprf:lem\_pos\_exp\] details the proof of Lemma \[LEM1\]. Given (\[peq1\]), we obtain the following two results, which are frequently used in the rest of the paper: $$\begin{aligned} p_{e, q} & \sim \frac{\ln n}{n},\label{eq_pe_lnnn}\end{aligned}$$ and $$\begin{aligned} p_{e, q} & \leq \frac{2\ln n}{n}\textrm{ for all $n$ sufficiently large}. \label{eq_pe_upper}\end{aligned}$$ The Proof of Lemma \[LEM1\] {#secprf:lem_pos_exp} =========================== To start with, we consider several notation that will be used throughout. We recall that ${C}_{i j}$ is the event that the communication channel between distinct nodes $v_i$ and $v_j$ is [*on*]{}. Then we set $\boldsymbol{1}[C_{ij}]$ as the indicator variable of event ${C}_{i j}$ by $$\begin{aligned} \hspace{-2pt} \boldsymbol{1}[C_{ij}]& \hspace{-2pt} := \hspace{-2pt} \begin{cases} 1,~ \textrm{if the channel between }v_i\textrm{ and }v_j\textrm{ is \textit{on}}; \\ 0,~ \textrm{if the channel between }v_i\textrm{ and }v_j\textrm{ is \textit{off}}. \end{cases} \nonumber\end{aligned}$$ We denote by $\mathcal {C}_m$ a $\binom{m}{2}$-tuple consisting of all possible $\boldsymbol{1}[C_{ij}]$ with $1 \leq i < j \leq m$ as follows: $$\begin{aligned} \mathcal {C}_m : = ( &\boldsymbol{1}[C_{12}], ,\ldots,\boldsymbol{1}[C_{1m}],~~~\boldsymbol{1}[C_{23}], ,\ldots,\boldsymbol{1}[C_{2m}], \nonumber \\ & \boldsymbol{1}[C_{34}], \ldots,\boldsymbol{1}[C_{3m}],~~~\ldots,~~~ \boldsymbol{1}[C_{(m-1),m}]). \nonumber\end{aligned}$$ Recalling $S_i$ as the key set on node $v_i$, we define a $m$-tuple $\mathcal {T}_m$ through $$\begin{aligned} \mathcal {T}_m & : = (S_1, S_2, \ldots, S_m). \nonumber\end{aligned}$$ Then we define $\mathcal {L}_m$ as $$\begin{aligned} \mathcal {L}_m & : = (\mathcal {C}_m, \mathcal {T}_m). \nonumber\end{aligned}$$ With $\mathcal {L}_m$, we have the *on*/*off* states of all channels between nodes $v_1, v_2, \ldots, v_m$ and the key sets $S_1, S_2, \ldots, S_m$ on these $m$ nodes, so all edges between these nodes in graph $\mathbb{G}_q\iffalse_{on}\fi$ are determined. Let $\mathbb{C}_m, \mathbb{T}_m$ and $\mathbb{L}_m$ be the sets of all possible $\mathcal {C}_m, \mathcal {T}_m$ and $\mathcal {L}_m$, respectively. We define $\mathbb {L}_m^{(0)}$ such that $\big(\mathcal {L}_m \in \mathbb {L}_m^{(0)}\big)$ is the event that there is no edge between any two of nodes $ v_1, v_2, \ldots, v_m $; i.e., $$\begin{aligned} \mathbb{L}_m^{(0)} := \{\mathcal {L}_m \boldsymbol{\mid} & ~(|S_i \cap S_j| < q) \textrm{ \textit{or} }(\boldsymbol{1}[C_{ij}] = 0), \nonumber \\ & \quad\quad~~~~~ \forall i, j\textrm{ with }1 \leq i < j \leq m.\}. \label{def_Lm0}\end{aligned}$$ We define $N_i$ as the neighborhood set of node $v_i$ for $i=1,2,\ldots,m$, and define the node set $M_{j_1 j_2 \ldots j_m}$ for all $j_1, j_2, \ldots, j_m \in \{0,1\}$ by $$\begin{aligned} & M_{j_1 j_2 \ldots j_m} \nonumber \\ & := \mathlarger{\Bigg\{}w\mathlarger{\Bigg|} \begin{array}{l} w \in \mathcal {V} \setminus \{v_1, v_2, \ldots, v_m\};\textrm{ and} \\ \textrm{for }i=1,2,\ldots,m, \begin{cases} w \in N_i\textrm{ if }j_i = 1; \\ w \notin N_i\textrm{ if }j_i =0.\end{cases} \end{array}\hspace{-11pt}\mathlarger{\Bigg\}}. \nonumber\end{aligned}$$ Clearly, the sets $M_{j_1 j_2 \ldots j_m}$ for $j_1, j_2, \ldots, j_m \in \{0,1\}$ are mutually disjoint. Setting $\mathcal {V}_m : = \{v_1, v_2, \ldots, v_m\}$ and $\overline{\mathcal {V}_m} : = \mathcal {V} \setminus \mathcal {V}_m $, we obtain $$\begin{aligned} \hspace{-5pt} \bigcup_{j_1, j_2, \ldots, j_m \in \{0,1\}}\hspace{-5pt}|M_{j_1 j_2 \ldots j_m} | = \overline{\mathcal {V}_m} ,\label{eqn_nodevi_h3}\end{aligned}$$ and $$\begin{aligned} \hspace{-5pt} \bigcup_{\begin{subarray}{c}j_1, j_2, \ldots, j_m \in \{0,1\}: \\ \sum_{i=1}^{m}j_i \geq 1. \end{subarray}}\hspace{-5pt} |M_{j_1 j_2 \ldots j_m} | = \bigg( \bigcup_{i=1}^m N_i \bigg) \hspace{1pt} \mathlarger{\cap} \hspace{2pt} \overline{\mathcal {V}_m} .\label{eqn_nodevi_h2}\end{aligned}$$ We define $2^m$-tuple $\mathcal {M}_m$ through[^3] $$\begin{aligned} \hspace{-1pt} \mathcal {M}_m &\hspace{-2pt} := \hspace{-2pt} \big( | M_{j_1 j_2 \ldots j_m} | \boldsymbol{\mid} j_1, j_2, \ldots, j_m \in \{0,1\} \big) \nonumber \\ & \hspace{-2pt} = \hspace{-2pt} \big( | M_{0^m} |, |M_{0^{m-1}1}|, |M_{0^{m-2}1,0}| , |M_{0^{m-2}1,1}| , \ldots\big).\nonumber\end{aligned}$$ Let $\mathcal {E}$ be the event that each of $v_1, v_2, \ldots, v_m$ has a degree of $h$. Given $\mathcal {L}_m \in \mathbb {L}_m $, we define $\mathbb{M}_m(\mathcal {L}_m)$ as the set of $\mathcal {M}_m$ under the condition that $\mathcal {E}$ occurs. Then it’s straightforward to compute $\mathbb{P} [\mathcal {E}] $ via $$\begin{aligned} \mathbb{P} [\mathcal {E}] & = \hspace{-3pt} \sum_{\begin{subarray}{c}\mathcal {L}_m^{*} \in \mathbb{L}_m, \\ \mathcal {M}_m^{*} \in \mathbb{M}_m (\mathcal {L}_m^{*}). \end{subarray}} \hspace{-3pt} \mathbb{P} \big[ \big( \mathcal {L}_m = \mathcal {L}_m^{*} \big) \cap \big( \mathcal {M}_m = \mathcal {M}_m^{*} \big) \big]. \label{eqr_probe}\end{aligned}$$ Given that event $\mathcal {E}$ happens, if any two of nodes $v_1, v_2, \ldots, v_m$ do not have any common neighbor in $\overline{\mathcal {V}_m}= \mathcal {V} \setminus \{v_1, v_2, \ldots, v_m\}$, then $\mathcal {M}_m$ is determined and denoted by $\mathcal {M}_m^{(0)}$ which satisfies $$\begin{aligned} \begin{cases} |M_{0^{i-1}, 1, 0^{m-i}}| = h,& \textrm{for }i=1,2,\ldots,m; \\ |M_{j_1 j_2 \ldots j_m}| = 0,&\textrm{for } \sum_{i=1}^m j_i > 1;\\|M_{0^m}| = n - m - hm . \end{cases} \nonumber\end{aligned}$$ By (\[eqr\_probe\]), we further write $\mathbb{P} [\mathcal {E} ]$ as the sum of $$\begin{aligned} \sum_{\begin{subarray}{c}\mathcal {L}_m^{*} \in \mathbb{L}_m, \\ \mathcal {M}_m^{*} \in \mathbb{M}_m (\mathcal {L}_m^{*}): \\ \left(\mathcal {L}_m^{*} \notin \mathbb{L}_m^{(0)}\right) \\ \textrm{ or }\left(\mathcal {M}_m^{*} \neq \mathcal{M}_m^{(0)}\right) \end{subarray}} \hspace{-2pt} \mathbb{P} \big[ \big( \mathcal {L}_m \hspace{-2pt} = \hspace{-1pt} \mathcal {L}_m^{*} \big) \cap \big( \mathcal {M}_m \hspace{-2pt} = \hspace{-1pt} \mathcal {M}_m^{*} \big) \big]\label{term1}\end{aligned}$$ and $$\begin{aligned} & \mathbb{P} \big[ \big( \mathcal {L}_m \in \mathbb{L}_m^{(0)} \big) \cap \big( \mathcal {M}_m = \mathcal{M}_m^{(0)} \big) \big]. \label{term2}\end{aligned}$$ Consequently, Lemma \[LEM1\] holds after we prove the following Propositions 1 and 2. In the rest of the paper, we will often use $1+x \leq e^x$ for any $x \in \mathbb{R}$ and $1 - xy \leq (1-x)^y \leq 1 - xy + \frac{1}{2} x^2 y ^2$ for $0\leq x <1$ and $y = 0 , 1, 2, \ldots$ (Fact 2 in [@ZhaoYaganGligor]). \[PROP\_ONE\] Given (\[peq1\]) (i.e., $p_{e, q} = \frac{\ln n \pm O(\ln \ln n)}{n}$), $ K_n = \omega(1)$ and $\frac{{K_n}^2}{P_n} = o(1)$, we have $$\begin{aligned} & (\ref{term1}) = o \left((h!)^{-m} (n p_{e, q})^{hm} e^{-m n p_{e, q}}\right). \nonumber\end{aligned}$$ \[PROP\_SND\] Given (\[peq1\]) (i.e., $p_{e, q} = \frac{\ln n \pm O(\ln \ln n)}{n}$), $ K_n = \omega(1)$ and $\frac{{K_n}^2}{P_n} = o(1)$, we have $$\begin{aligned} &(\ref{term2}) \sim (h!)^{-m} (n p_{e, q})^{hm} e^{-m n p_{e, q}}. \nonumber\end{aligned}$$ The Proof of Proposition \[PROP\_ONE\] {#sec:PROP_ONE} ====================================== We embark on the evaluation of (\[term1\]) by computing $$\begin{aligned} \mathbb{P} \big[ \big( \mathcal {M}_m = \mathcal {M}_m^{*} \big) \boldsymbol{\mid} \mathcal {L}_m = \mathcal {L}_m^{*} \big]. \label{eq_MmMm}\end{aligned}$$ With $\mathcal {C}_m ^{*}$ and $\mathcal {T}_m ^{*}$ defined such that $\mathcal {L}_m^{*} = (\mathcal {C}_m^{*}, \mathcal {T}_m^{*})$, event $\big(\mathcal {L}_m = \mathcal {L}_m^{*}\big)$ is the union of events $\big(\mathcal {C}_m = \mathcal {C}_m^{*}\big)$ and $\big(\mathcal {T}_m = \mathcal {T}_m^{*}\big)$. Since $( \mathcal {C}_m \hspace{-1pt} = \hspace{-1pt} \mathcal {C}_m^{*} )$ and $( \mathcal {M}_m \hspace{-1pt} = \hspace{-1pt} \mathcal {M}_m^{*} )$ are independent, we obtain $$\begin{aligned} (\ref{eq_MmMm}) & = \mathbb{P} \big[ \big( \mathcal {M}_m = \mathcal {M}_m^{*} \big) \boldsymbol{\mid} \big( \mathcal {T}_m = \mathcal {T}_m^{*} \big) \big] \nonumber .\end{aligned}$$ For any $j_1, j_2, \ldots, j_m \in \{0,1\},$ for any distinct nodes $w_1 \hspace{-2pt} \in \hspace{-2pt} \overline{\mathcal {V}_m} $ and $ w_2 \hspace{-2pt} \in \hspace{-2pt} \overline{\mathcal {V}_m} $, events $(w_1 \hspace{-2pt} \in \hspace{-2pt} M_{j_1 j_2 \ldots j_m})$ and $(w_2 \in M_{j_1 j_2 \ldots j_m})$ are not independent [@ryb3], but are conditionally independent given $(\mathcal {T}_m = \mathcal {T}_m^{*})$ (with the key sets $S_1, S_2, \ldots, S_m$ specified as $S_1^{*}, S_2^{*}, \ldots, S_m^{*}$, respectively). Therefore, $$\begin{aligned} & \hspace{-2pt} (\ref{eq_MmMm}) = f(n-m , \mathcal {M}_m^{*})\mathbb{P} [w \in M_{0^m}^{*} \hspace{-2pt} \boldsymbol{\mid} \hspace{-2pt} \mathcal {T}_m = \mathcal {T}_m^{*} ]^{|M_{0^m}^{*} |} \times \nonumber \\ & \hspace{-2pt} \prod_{\begin{subarray}{c}j_1, j_2, \ldots, j_m \in \{0,1\}: \\ \sum_{i=1}^{m}j_i \geq 1. \end{subarray}} \hspace{-2pt} \mathbb{P}[w \in M_{j_1 j_2 \ldots j_m}^{*} \hspace{-2pt} \boldsymbol{\mid} \hspace{-2pt} \mathcal {T}_m = \mathcal {T}_m^{*}]^{|M_{j_1 j_2 \ldots j_m}^{*} |}, \label{eqn_epsilonmmmm}\end{aligned}$$ where $f\big(\sum_{i=1}^{\ell} x_i , (x_1, x_2, \ldots, x_{\ell})\big)$ for integers $\ell \geq 1$ and $x_i \geq 0$ with $i = 1,2 , \ldots, \ell$ is determined by $$\begin{aligned} & f\bigg(\sum_{i=1}^{\ell} x_i , (x_1, x_2, \ldots, x_{\ell})\bigg) \nonumber \\ & \quad : = \binom{\sum_{i=1}^{\ell} x_i }{x_1 } \binom{\sum_{i=2}^{\ell} x_i }{x_2 } \ldots \binom{\sum_{i=\ell-1}^{\ell} x_i }{x_{\ell-1} } \binom{x_{\ell} }{x_{\ell} } \nonumber \\ & \quad \hspace{2pt} = \frac{\big(\sum_{i=1}^{\ell} x_i \big)!}{x_1 ! x_2 ! \ldots x_{\ell} !}. \label{funcf}\end{aligned}$$ From (\[funcf\]) and $$\begin{aligned} \sum_{j_1, j_2, \ldots, j_m \in \{0,1\}}|M_{j_1 j_2 \ldots j_m}^{*}| = n - m \label{eqn_nodevi_h5}\end{aligned}$$ which holds by (\[eqn\_nodevi\_h3\]), we have $$\begin{aligned} & f(n-m , \mathcal {M}_m^{*}) \nonumber \\ & \hspace{-2pt} = \hspace{-2pt} \frac{ ( \sum_{j_1, j_2, \ldots, j_m \in \{0,1\}}|M_{j_1 j_2 \ldots j_m}^{*}| ) !}{\prod_{j_1, j_2, \ldots, j_m \in \{0,1\}} (|M_{j_1 j_2 \ldots j_m}^{*}|! )} \nonumber \\ & \hspace{-2pt} = \hspace{-2pt} \frac{ (n \hspace{-2pt} -\hspace{-2pt} m ) ! \hspace{-2pt} \Big/ \hspace{-2pt} \Big(n\hspace{-2pt} -\hspace{-2pt} m - \hspace{-3pt} \sum_{\begin{subarray}{c}j_1, j_2, \ldots, j_m \in \{0,1\}: \\ \sum_{i=1}^{m}j_i \geq 1. \end{subarray}}|M_{j_1 j_2 \ldots j_m}^{*}|\hspace{-1pt}\Big)!} {\prod_{\begin{subarray}{c}j_1, j_2, \ldots, j_m \in \{0,1\}: \\ \sum_{i=1}^{m}j_i \geq 1. \end{subarray}} (|M_{j_1 j_2 \ldots j_m}^{*}|! )} \label{eqn_fnexpr} \\ & \hspace{-2pt} \leq \hspace{-2pt} n^{\sum_{\begin{subarray}{c}j_1, j_2, \ldots, j_m \in \{0,1\}: \\ \sum_{i=1}^{m}j_i \geq 1. \end{subarray}}|M_{j_1 j_2 \ldots j_m}^{*}|}. \label{eqn_fnm}\end{aligned}$$ Denoting $\sum_{\begin{subarray}{c}j_1, j_2, \ldots, j_m \in \{0,1\}: \\ \sum_{i=1}^{m}j_i \geq 1. \end{subarray}}|M_{j_1 j_2 \ldots j_m}^{*}|$ by $\Lambda$, we prove $\Lambda \leq hm - 1$ below if $\big(\mathcal {L}_m^{*} \notin \mathbb{L}_m^{(0)} \big)$ or $\big(\mathcal {M}_m^{*} \neq \mathcal{M}_m^{(0)} \big)$. On the one hand, assuming $\mathcal {L}_m^{*} \notin \mathbb{L}_m^{(0)}$, there exist $i_1$ and $i_2$ with $1 \leq i_1 < i_2 \leq m$ such that nodes $v_{i_1}$ and $v_{i_2}$ are neighbors with each other. Hence, $ \{v_{i_1}, v_{i_2}\} \subseteq [( \bigcup_{i=1}^m N_i ) \bigcap \mathcal {V}_m ]$. Then from (\[eqn\_nodevi\_h2\]), $$\begin{aligned} & \Lambda = \bigg|\bigcup_{i=1}^m N_i\bigg| - \bigg|\bigg( \bigcup_{i=1}^{m} N_i \bigg) \hspace{2pt} \mathlarger{\cap} \hspace{2pt} \mathcal {V}_m\bigg| \leq hm - 2. \nonumber $$ On the other hand, assuming $\mathcal {M}_m^{*} \neq \mathcal{M}_m^{(0)}$, there exist $i_3 $ and $ i_4$ with $1 \leq i_3 < i_4 \leq m$ such that $N_{i_3} \cap N_{i_4} \neq \emptyset$. Then from (\[eqn\_nodevi\_h2\]), $$\begin{aligned} & \Lambda \leq \bigg|\bigcup_{i=1}^m N_i\bigg| \leq \bigg(\sum_{i=1}^m |N_i|\bigg) - |N_{i_3} \cap N_{i_4}| \leq hm - 1. \nonumber $$ To summarize, if $\big(\mathcal {L}_m^{*} \notin \mathbb{L}_m^{(0)} \big)$ or $\big(\mathcal {M}_m^{*} \neq \mathcal{M}_m^{(0)} \big)$, we have $$\begin{aligned} \Lambda \leq hm - 1 , \label{lambda}\end{aligned}$$ along with (\[eqn\_nodevi\_h5\]) leading to $$\begin{aligned} | M_{0^m}^{*} | &= n - m - \Lambda > n - m - hm. \label{eqn_M0m2n}\end{aligned}$$ For any $j_1, j_2, \ldots, j_m \in \{0,1\}$ with $\sum_{i=1}^{m}j_i \geq 1$, there exists $t \in \{0,1,\ldots, m\}$ such that $j_t = 1$, so $$\begin{aligned} & \mathbb{P}\big[w \in M_{j_1 j_2 \ldots j_m} \boldsymbol{\mid} \mathcal {T}_m = \mathcal {T}_m^{*} \big] \nonumber \\ & \leq \mathbb{P}[E_{w v_t} \boldsymbol{\mid} \mathcal {T}_m = \mathcal {T}_m^{*}] = \mathbb{P}[E_{w v_t} ] = p_{e, q}, \label{eqn_pe_not00}\end{aligned}$$ where $E_{w v_t}$ is the event that there exists an edge between nodes $w$ and $v_t$ in graph $\mathbb{G}_q$. Substituting (\[eqn\_fnm\]-\[eqn\_pe\_not00\]) into (\[eqn\_epsilonmmmm\]), we obtain that if\ $\big(\mathcal {L}_m^{*} \notin \mathbb{L}_m^{(0)} \big)$ or $\big(\mathcal {M}_m^{*} \neq \mathcal{M}_m^{(0)} \big)$, then $$\begin{aligned} \hspace{-1pt} (\ref{eq_MmMm}) & < (np_{e,q})^{hm - 1} \hspace{-1pt} \times \hspace{-1pt} \mathbb{P} [w \in M_{0^m} \hspace{-1pt} \boldsymbol{\mid} \hspace{-1pt} \mathcal {T}_m = \mathcal {T}_m^{*} ]^{n - m - hm} . \label{eq_pmmll2}\end{aligned}$$ Applying (\[eq\_MmMm\]) and (\[eq\_pmmll2\]) to (\[term1\]), we get $$\begin{aligned} (\ref{term1}) & < \sum_{\mathcal {L}_m^{*} \in \mathbb{L}_m} \Big\{ |\mathbb{M}_m (\mathcal {L}_m^{*})| \nonumber \\ & \quad \times \textrm{R.H.S. of (\ref{eq_pmmll2})} \times \mathbb{P} \big[ \mathcal {L}_m = \mathcal {L}_m^{*} \big] \Big\}. \label{eqn_TmCmt}\end{aligned}$$ To bound $|\mathbb{M}_m (\mathcal {L}_m^{*})|$, note that $\mathcal {M}_m$ is a $2^m$-tuple. Among the $ 2^m $ elements of the tuple, each of $|M_{j_1 j_2 \ldots j_m} |\big|_{\begin{subarray}{c}j_1, j_2, \ldots, j_m \in \{0,1\}: \\ \sum_{i=1}^{m}j_i \geq 1. \end{subarray}}$ is at least 0 and at most $h$; and the remaining element $| M_{0^m} |$ can be determined by (\[eqn\_nodevi\_h5\]). Then it’s straightforward that $$\begin{aligned} |\mathbb{M}_m (\mathcal {L}_m^{*})| & \leq (h+1)^{2^m-1}. \label{eqn_MmLm}\end{aligned}$$ Using (\[eqn\_MmLm\]) in (\[eqn\_TmCmt\]), and considering $\big(\mathcal {L}_m = \mathcal {L}_m^{*}\big)$ is the union of independent events $\big(\mathcal {T}_m = \mathcal {T}_m^{*}\big)$ and $\big(\mathcal {C}_m \hspace{-1pt} = \hspace{-1pt} \mathcal {C}_m^{*}\big)$, and $\sum_{\mathcal {C}_m^{*} \in \mathbb{C}_m} \hspace{-1pt} \mathbb{P} \big[ \mathcal {C}_m \hspace{-1pt} = \hspace{-1pt} \mathcal {C}_m^{*} \big] \hspace{-1pt} = \hspace{-1pt} 1$, we derive $$\begin{aligned} (\ref{term1}) & < (h+1)^{2^m-1} (np_{e,q})^{hm-1} \hspace{-2pt} \times \hspace{-2pt}\sum_{\mathcal {T}_m^{*} \in \mathbb{T}_m} \hspace{-4pt} \Big\{ \mathbb{P}\big[ \mathcal {T}_m = \mathcal {T}_m^{*} \big] \nonumber \\ & \quad \times \mathbb{P} [w \in M_{0^m} \boldsymbol{\mid} \mathcal {T}_m = \mathcal {T}_m^{*} ]^{n - m - hm} \Big\} . \label{prop_prf}\end{aligned}$$ From (\[prop\_prf\]) and $\lim_{n \to \infty} n p_{e, q} = \infty $ by (\[eq\_pe\_lnnn\]), the proof of Proposition \[PROP\_ONE\] is completed once we show $$\begin{aligned} & \sum_{ \mathcal {T}_m^{*} \in \mathbb{T}_m } \mathbb{P}[\mathcal {T}_m = \mathcal {T}_m^{*}] \mathbb{P} [w \in M_{0^m} \boldsymbol{\mid} \mathcal {T}_m = \mathcal {T}_m^{*} ]^{n - m - hm} \nonumber \\ & \quad \leq e^{- m n p_{e, q}} \cdot [1+o(1)] . \label{EQ}\end{aligned}$$ Establishing (\[EQ\]) --------------------- From (\[eq\_evalprob\_3\_qcmp\]) and (\[eq\_evalprob\_4\_qcmp\]) (viz., Lemma \[lem\_evalprob\_qcmp\] in the Appendix), it holds that $$\begin{aligned} & \mathbb{P} [w \in M_{0^m}^{*} \boldsymbol{\mid} \mathcal {T}_m = \mathcal {T}_m^{*} ]^{n - m - hm } \nonumber \\ & = \mathbb{P} [w \in M_{0^m}^{*} \boldsymbol{\mid} \mathcal {T}_m = \mathcal {T}_m^{*} ]^{ n} \mathbb{P} [w \in M_{0^m}^{*} \boldsymbol{\mid} \mathcal {T}_m = \mathcal {T}_m^{*} ]^{-m - h m} \nonumber \\ & \leq e^{- m n p_{e, q} + (q+2)! \binom{m}{2} n{(p_{e, q})}^{\frac{q+1}{q}} + \frac{n p_{e, q} p_n}{K_n} \sum_{1\leq i <j \leq m} |S_{ij}^{*}|} \nonumber \\ & \quad \times (1 - m p_{e, q})^{-m - h m}, \label{eqn_prbwM}\end{aligned}$$ where $S_{ij}^{*} = S_{i}^{*} \cap S_{j}^{*}$. With (\[eq\_pe\_lnnn\]) (i.e., $p_{e, q} \sim \frac{\ln n}{n}$), we have $m^2 n {p_{e, q}}^2 = o(1)$ and $m p_{e,q} = o(1)$, which are substituted into (\[eqn\_prbwM\]) to induce (\[EQ\]) once we prove$$\begin{aligned} \sum_{ \mathcal {T}_m^{*} \in \mathbb{T}_m } \mathbb{P}[\mathcal {T}_m = \mathcal {T}_m^{*}] e^{\frac{n p_{e, q} p_n}{K_n}\sum_{1\leq i <j \leq m}|S_{ij}^{*}|} & \leq 1+o(1). \label{eqn_sumTmst}\end{aligned}$$ L.H.S. of (\[eqn\_sumTmst\]) is denoted by $H_{n,m}$ and evaluated below. For each fixed and sufficiently large $n$, we consider: [a)]{} [${ p_n < n^{-\delta} (\ln n)^{-1}}$]{} and [b)]{} [${ p_n \geq n^{-\delta} (\ln n)^{-1}}$]{}, where $\delta$ is an arbitrary constant with $0<\delta<1$. **a)** $\boldsymbol{ p_n < n^{-\delta} (\ln n)^{-1}}$ From $p_n < n^{-\delta} (\ln n)^{-1}$, (\[eq\_pe\_upper\]) (namely, $p_{e,q} \leq \frac{2\ln n}{n}$) and $|S_{ij}^{*}| \leq K_n$ for $1\leq i <j \leq m$, it holds that $$\begin{aligned} e^{\frac{n p_{e,q} p_n}{K_n} \sum_{i =1}^{m-1}|S_{i m}^*|} & < e^{2 \ln n \cdot n^{-\delta} (\ln n)^{-1} \cdot \binom{m}{2}} < e^{ m^2 n^{-\delta}},\nonumber\end{aligned}$$ which is substituted into $H_{n,m}$ to bring about $$\begin{aligned} & H_{n,m} < e^{ m^2 n^{-\delta}} \sum_{ \mathcal {T}_m^{*} \in \mathbb{T}_m } \mathbb{P}[\mathcal {T}_m = \mathcal {T}_m^{*}] = e^{ m^2 n^{-\delta}} , \nonumber\end{aligned}$$ **b)** $\boldsymbol{ p_n \geq n^{-\delta} (\ln n)^{-1}}$ We relate $H_{n,m}$ to $H_{n,m-1}$ and assess $H_{n,m}$ iteratively. First, with $\mathcal {T}_m^{*} = (S_1^*, S_2^*, \ldots, S_m^*)$, event $(\mathcal {T}_m = \mathcal {T}_m^{*})$ is the intersection of independent events: $(\mathcal {T}_{m-1} = \mathcal {T}_{m-1}^{*})$ and $(S_m = S_m^*)$. Then we have $$\begin{aligned} & H_{n,m} \nonumber \\ & = \sum_{ \begin{subarray} ~\mathcal {T}_{m-1}^* \in \mathbb{T}_{m-1} , \\ \hspace{9pt}S_m^* \in \mathbb{S}_m \end{subarray} } \Big( \mathbb{P}[(\mathcal {T}_{m-1} = \mathcal {T}_{m-1}^{*}) \cap (S_m = S_m^*)] \times \nonumber \\ & \quad\quad \quad\quad e^{\frac{n p_{e,q} p_n}{K_n} \sum_{1\leq i <j \leq m-1}|S_{ij}^*|} e^{\frac{n p_{e,q} {p_n} }{K_n} \sum_{i =1}^{m-1}|S_{i m}^*|} \Big) \nonumber \\ & = H_{n,m-1} \cdot \sum_{S_m^* \in \mathbb{S}_m} \mathbb{P}[ S_m = S_m^* ] e^{\frac{n p_{e,q} p_n}{K_n} \sum_{i =1}^{m-1}|S_{i m}^*|} . \label{HnmHnm1}\end{aligned}$$ By $ \sum_{i =1}^{m-1} \hspace{-2pt} |S_{i m}^*| \hspace{-3pt} = \hspace{-3pt} \sum_{i =1}^{m-1} \hspace{-2pt} |S_i^* \cap S_m^*| \hspace{-3pt} \leq \hspace{-3pt} m \big|S_m^* \hspace{-1pt} \cap \hspace{-1pt} \big(\hspace{-2pt}\bigcup_{i =1}^{m-1} \hspace{-2pt}S_{i }^* \big) \hspace{-1pt} \big|$ and (\[eq\_pe\_upper\]) (i.e., $ p_{e,q} \leq \frac{2\ln n}{n}$), we get $$\begin{aligned} & e^{\frac{n p_{e,q} p_n }{K_n} \sum_{i =1}^{m-1}|S_{i m}^*|} \leq e^{ \frac{2m p_n \ln n}{K_n} |S_m^* \cap (\bigcup_{i =1}^{m-1}S_{i }^* ) |} , \nonumber\end{aligned}$$ further leading to $$\begin{aligned} & H_{n,m} / H_{n,m-1} \nonumber \\ & \leq \sum_{u=0}^{K_n} \mathbb{P}\bigg[\bigg|S_m^* \bigcap \bigg(\bigcup_{i =1}^{m-1}S_{i }^*\bigg)\bigg| = u \bigg] e^{\frac{2 u m {p_n} \ln n}{K_n}} .\label{eqn_tmtm-1}\end{aligned}$$ Denoting $\big|\bigcup_{i=1}^{m-1}S_{i}^*\big|$ by $v$, then we obtain that for $u \in [\max\{0, K_n + v - P_n\} , K_n] $, $$\begin{aligned} \mathbb{P}\bigg[\bigg|S_m^* \bigcap \bigg(\bigcup_{i=1}^{m-1}S_{i}^*\bigg)\bigg| = u \bigg] & = \frac{\binom{v}{u} \binom{P_n - v}{K_n - u}}{\binom{P_n}{K_n}}, \label{probsm}\end{aligned}$$ which together with $ K_n \leq v \leq m K_n$ yields $$\begin{aligned} & \textrm{L.H.S. of (\ref{probsm})} \nonumber \\& \quad \leq \frac{(m K_n)^u}{u!} \cdot \frac{(P_n - K_n)^{K_n - u}}{(K_n - u)!} \cdot \frac{K_n !}{(P_n - K_n)^{K_n}} \nonumber \\& \quad \leq \frac{1}{u!} \bigg( \frac{m {K_n}^2}{P_n - K_n}\bigg)^u. \label{probsm2}\end{aligned}$$ For $u \notin [\max\{0, K_n + v - P_n\} , K_n] $, L.H.S. of (\[probsm\]) equals 0. Then from (\[eqn\_tmtm-1\]) and (\[probsm2\]), $$\begin{aligned} \textrm{R.H.S. of (\ref{eqn_tmtm-1})} & \leq \sum_{u=0}^{K_n} \frac{1}{u!} \bigg( \frac{m {K_n}^2}{P_n - K_n} \cdot e^{\frac{2 m {p_n} \ln n}{K_n}}\bigg)^u \nonumber \\& \quad \leq e^{\frac{m {K_n}^2}{P_n - K_n} \cdot e^{\frac{2 m {p_n} \ln n}{K_n}}}. \label{umKnPN}\end{aligned}$$ By $\frac{{K_n}^2}{P_n} = o(1)$ and Lemma \[lem\_eval\_psq\]-Property (i), $$\begin{aligned} \frac{{K_n}^2}{P_n-K_n} & \leq \frac{{K_n}^2}{P_n} \cdot [1+o(1)] \leq \big( q! p_{s,q} \big)^{\frac{1}{q}} \cdot [1+o(1)] .\label{eqn_knpn_qcmp}\end{aligned}$$ For $n$ sufficiently large, from $p_n \geq n^{-\delta} (\ln n)^{-1}$ and (\[eq\_pe\_upper\]) (i.e., $p_{e,q} =p_n p_{s,q} \leq \frac{2\ln n}{n}$), we have $$\begin{aligned} p_{s,q} & = {p_n} ^{-1} {p_{e,q}} \leq {p_n} ^{-1} \cdot 2n^{-1}\ln n \leq 2 n^{\delta-1} (\ln n)^2. \label{eqps0}\end{aligned}$$ From (\[eqn\_knpn\_qcmp\]) and (\[eqps0\]), $$\begin{aligned} \frac{{K_n}^2}{P_n-K_n} & \leq [ q! \cdot 2 n^{\delta-1} (\ln n)^2]^{\frac{1}{q}} \cdot [1+o(1)] \nonumber \\& \leq 3 q \cdot n^{\frac{\delta-1}{q}} (\ln n)^{\frac{2}{q}}. \label{eqn_knpn2x}\end{aligned}$$ Given $K_n = \omega(1) $, for arbitrary constant $c > q$ and for all $n$ sufficiently large, $\frac{K_n}{p_n} \geq \frac{2cq \cdot m}{(c-q)(1-\delta)} $ holds. Then $$\begin{aligned} e^{\frac{2 m p_n \ln n}{K_n}} & \leq e^{ \frac{(c-q)(1-\delta)}{cq} \ln n} = n^{\frac{(c-q)(1-\delta)}{cq}} .\label{ja1}\end{aligned}$$ The use of (\[umKnPN\]) (\[eqn\_knpn2x\]) and (\[ja1\]) in (\[eqn\_tmtm-1\]) yields $$\begin{aligned} & H_{n,m} / H_{n,m-1} \leq \textrm{R.H.S. of (\ref{eqn_tmtm-1})} \nonumber \\ & \leq e^{ 3 qm \cdot n^{\frac{\delta-1}{q}} (\ln n)^{\frac{2}{q}} \cdot n^{\frac{(c-q)(1-\delta)}{cq}} } \leq \Big(e^{3 q \cdot n^{\frac{\delta-1}{c}} (\ln n)^{\frac{2}{q}}} \Big)^m. \label{gnmgnm-1}\end{aligned}$$ To derive $H_{n,m}$ iteratively based on (\[gnmgnm-1\]), we compute $H_{n,2}$ below. By definition, setting $m=2$ in L.H.S. of (\[eqn\_sumTmst\]) and considering the independence between events $(S_1 = S_1^*)$ and $(S_2 = S_2^*)$, we gain $$\begin{aligned} H_{n,2} & = \sum_{S_1^* \in \mathbb{S}_m} \mathbb{P}[ S_1 = S_1^* ] \sum_{S_2^* \in \mathbb{S}_m} \mathbb{P}[ S_2 = S_2^* ] e^{\frac{n p_{e,q} p_n}{K_n} |S_1^* \cap S_2^*|}. \label{eqn_gn2}\end{aligned}$$ Clearly, $\sum_{S_2^* \in \mathbb{S}_m} \hspace{-3pt} \mathbb{P}[ S_2 \hspace{-1pt} = \hspace{-1pt} S_2^* ] e^{\frac{n p_{e,q} p_n}{K_n} |S_1^* \cap S_2^*|} $ equals R.H.S. of (\[eqn\_tmtm-1\]) with $m = 2$. Then from (\[gnmgnm-1\]) and (\[eqn\_gn2\]), $$\begin{aligned} H_{n,2} & \leq \sum_{S_1^* \in \mathbb{S}_m} \mathbb{P}[ S_1 = S_1^* ] e^{6 q \cdot n^{\frac{\delta-1}{c}} (\ln n)^{\frac{2}{q}}} = e^{6 q \cdot n^{\frac{\delta-1}{c}} (\ln n)^{\frac{2}{q}}}. \label{hn2}\end{aligned}$$ Therefore, it holds via (\[gnmgnm-1\]) and (\[hn2\]) that $$\begin{aligned} H_{n,m} & \leq \Big(e^{3 q \cdot n^{\frac{\delta-1}{c}} (\ln n)^{\frac{2}{q}}} \Big)^{m+(m-1) + \ldots + 3} \cdot e^{6 q \cdot n^{\frac{\delta-1}{c}} (\ln n)^{\frac{2}{q}}} \nonumber \\ & = e^{\frac{3}{2}q(m^2+m-2) n^{\frac{\delta-1}{c}} (\ln n)^{\frac{2}{q}}} \nonumber.\end{aligned}$$ Finally, summarizing cases a) and b), we report $$\begin{aligned} H_{n,m} & \leq \max\left\{e^{ m^2 n^{-\delta}} , e^{\frac{3}{2}q(m^2+m-2) n^{\frac{\delta-1}{c}} (\ln n)^{\frac{2}{q}}}\right\} . \nonumber\end{aligned}$$ With $n \to \infty$, $ H_{n,m} \leq 1+ o(1)$ (i.e., (\[eqn\_sumTmst\])) follows. The Proof of Proposition \[PROP\_SND\] {#sec:PROP_SND} ====================================== We define $\mathcal{C}_m^{(0)}$ and $\mathbb{T}_m^{(0)}$ by $$\begin{aligned} \mathcal {C}_m^{(0)} & = ( \underbrace{0, 0, \ldots, 0}_{\binom{m}{2} \textrm{ number of ``}0\textrm{''}} ), \nonumber\end{aligned}$$ and $$\begin{aligned} \mathbb{T}_m^{(0)} & = \{\mathcal {T}_m \boldsymbol{\mid} | S_i \cap S_j | < q, ~ \forall i, j\textrm{ with }1 \leq i < j \leq m.\}. \nonumber\end{aligned}$$ Clearly, $\big(\mathcal {C}_m = \mathcal{C}_m^{(0)}\big)$ or $\big(\mathcal {T}_m \in \mathbb{T}_m^{(0)}\big)$ each implies $\big( \mathcal {L}_m \in \mathbb{L}_m^{(0)} \big)$. Also, $\big(\mathcal {C}_m = \mathcal{C}_m^{(0)}\big)$ and $\big(\mathcal {M}_m = \mathcal{M}_m^{(0)}\big)$ are independent with each other. Therefore, with $(\ref{term2}) = \mathbb{P} \big[ \big( \mathcal {L}_m \in \mathbb{L}_m^{(0)} \big) \cap \big( \mathcal {M}_m = \mathcal{M}_m^{(0)} \big) \big]$, we get $$\begin{aligned} & (\ref{term2}) \geq \mathbb{P} \big[ \mathcal {C}_m = \mathcal{C}_m^{(0)}\big] \mathbb{P} \big[ \mathcal {M}_m = \mathcal{M}_m^{(0)} \big], \label{prcm}\end{aligned}$$ and $$\begin{aligned} & (\ref{term2}) \geq \mathbb{P} \big[ \mathcal {T}_m \hspace{-1pt} \in \hspace{-1pt} \mathbb{T}_m^{(0)} \big] \mathbb{P} \big[ \big( \mathcal {M}_m \hspace{-1pt} = \hspace{-1pt} \mathcal{M}_m^{(0)} \big) \hspace{-2pt} \boldsymbol{\mid} \hspace{-2pt} \big( \mathcal {T}_m \hspace{-1pt} \in \mathbb{T}_m^{(0)} \hspace{-1pt} \big)\big]. \label{eqn_tmtmst}\end{aligned}$$ Given $\big(\mathcal {C}_m = \mathcal{C}_m^{(0)}\big) = \overline{\bigcup_{ 1 \leq i < j \leq m} {C_{ij}}} $ and\ $\big(\mathcal {T}_m \in \mathbb{T}_m^{(0)}\big) = \overline{\bigcup_{ 1 \leq i < j \leq m} {\Gamma_{ij}} }$, applying the union bound, we obtain $$\begin{aligned} & \mathbb{P} \big[ \mathcal {C}_m = \mathcal{C}_m^{(0)} \big]\geq 1 - \sum_{ 1 \leq i < j \leq m}\mathbb{P}[ C_{ij} ] \geq 1- m^2 p_n / 2, \label{prcmpn}\end{aligned}$$ and $$\begin{aligned} & \mathbb{P}\big[\mathcal {T}_m \in \mathbb{T}_m^{(0)}\big] \geq 1 - \sum_{ 1 \leq i < j \leq m}\mathbb{P}[ \Gamma_{ij} ] \geq 1 - m^2 p_{s,q} / 2.\label{mthbbP} $$ In the following two subsections, we will prove $$\begin{aligned} \mathbb{P} \big[ \mathcal {M}_m = \mathcal{M}_m^{(0)} \big] & \sim (h!)^{-m} (n p_{e,q})^{hm} e^{-m n p_{e,q}} \label{eqn_prMm},\end{aligned}$$ and $$\begin{aligned} & \mathbb{P} \big[ \big( \mathcal {M}_m = \mathcal{M}_m^{(0)} \big) \boldsymbol{\mid} \big( \mathcal {T}_m \in \mathbb{T}_m^{(0)} \big)\big] \nonumber \\ & \quad \geq (h!)^{-m} (n p_{e,q})^{hm} e^{-m n p_{e,q}} \cdot [1-o(1)] .\label{prob_MmMm_sim}\end{aligned}$$ Substituting (\[prcmpn\]) and (\[eqn\_prMm\]) into (\[prcm\]), and applying (\[mthbbP\]) and (\[prob\_MmMm\_sim\]) to (\[eqn\_tmtmst\]), we have $$\begin{aligned} & \frac{(\ref{term2})}{(h!)^{-m} (n p_{e,q})^{hm} e^{-m n p_{e,q}}} \nonumber \\ & ~ \geq ( 1 - \min\{ p_{s,q}, p_n \} \cdot m^2 / 2)\cdot [1-o(1)] . \label{pro2_pt1}\end{aligned}$$ From (\[eqn\_prMm\]), we get $$\begin{aligned} (\ref{term2}) & \leq \mathbb{P} \big[ \mathcal {M}_m \in \mathbb{M}_m^{(0)} \big] \nonumber \\ & \leq (h!)^{-m} (n p_{e,q})^{hm} e^{-m n p_{e,q}} \cdot [1+o(1)]. \label{pro2_pr}\end{aligned}$$ Combining (\[pro2\_pt1\]) and (\[pro2\_pr\]), and using $\min\{ p_{s,q}, p_n \} \leq \sqrt{p_{s,q} p_n} = \sqrt{p_{e,q}} \leq \sqrt{\frac{2\ln n}{n}} = o(1)$ which holds from $p_{e,q} = p_{s,q} p_n$ and (\[eq\_pe\_upper\]), Proposition 2 follows. Below we detail the proofs of (\[eqn\_prMm\]) and (\[prob\_MmMm\_sim\]). Establishing (\[eqn\_prMm\]) ---------------------------- We have $$\begin{aligned} &\mathbb{P} \big[ \mathcal {M}_m = \mathcal{M}_m^{(0)} \big] \nonumber \\ & \sum_{\mathcal {T}_m^{*} \in \mathbb{T}_m} \Big\{ \mathbb{P} \big[ \mathcal {T}_m = \mathcal {T}_m^{*} \big] \mathbb{P} \big[ \big( \mathcal {M}_m = \mathcal{M}_m^{(0)} \big) \boldsymbol{\mid} \big( \mathcal {T}_m = \mathcal {T}_m^{*} \big)\big] \Big\},\nonumber\end{aligned}$$ where $$\begin{aligned} & \mathbb{P} \big[ \big( \mathcal {M}_m = \mathcal{M}_m^{(0)} \big) \boldsymbol{\mid} \big( \mathcal {T}_m = \mathcal {T}_m^{*} \big)\big] \nonumber \\ & = f\big(n-m , \mathcal{M}_m^{(0)}\big) \mathbb{P} [w \in M_{0^m} \boldsymbol{\mid}\mathcal {T}_m = \mathcal {T}_m^{*} ]^{n-m-hm} \nonumber \\ & \quad \quad \times \prod_{i=1}^{m} \mathbb{P}[w \in M_{0^{i-1}, 1, 0^{m-i}} \boldsymbol{\mid}\mathcal {T}_m = \mathcal {T}_m^{*} ]^{h}, \label{pMmexpr}\end{aligned}$$ with function $f $ specified in (\[funcf\]). From (\[eqn\_fnexpr\]), $$\begin{aligned} & f\big(n\hspace{-1pt}-\hspace{-1pt}m , \mathcal{M}_m^{(0)} \hspace{-1pt}\big) \hspace{-2pt} = \hspace{-2pt} \frac{ (n \hspace{-1pt}- \hspace{-1pt}m) ! }{(n \hspace{-1pt}-\hspace{-1pt} m\hspace{-1pt} -\hspace{-1pt} hm)!(h!)^m} \hspace{-3pt} \sim \hspace{-2pt} (h!)^{-m}n^{hm}. \label{eqn_f00}\end{aligned}$$ We will establish $$\begin{aligned} &\hspace{-4pt} \sum_{ \mathcal {T}_m^{*} \in \mathbb{T}_m } \hspace{-7pt}\Big\{ \hspace{-1pt} \mathbb{P}[\mathcal {T}_m \hspace{-2pt} = \hspace{-2pt} \mathcal {T}_m^{*}] \hspace{-2pt} \prod_{i=1}^{m} \{ \mathbb{P}\big[w \hspace{-2pt} \in \hspace{-2pt} M_{0^{i-1}, 1, 0^{m-i}}^{(0)} \hspace{-2pt}\boldsymbol{\mid} \hspace{-2pt} \mathcal {T}_m \hspace{-2pt} = \hspace{-2pt} \mathcal {T}_m^{*} \big]^h \} \hspace{-2pt}\Big\} \nonumber \\ & \quad \geq {p_{e,q}}^{hm} \cdot [1-o(1)] .\label{eq_evalprob_exp_2}\end{aligned}$$ We use (\[eqn\_f00\]) and (\[eq\_evalprob\_exp\_2\]) as well as (\[eq\_evalprob\_3\_qcmp\]) (viz., Lemma \[lem\_evalprob\_qcmp\] in the Appendix) in evaluating $\mathbb{P} \big[ \mathcal {M}_m = \mathcal{M}_m^{(0)} \big]$ above. Then $$\begin{aligned} & \mathbb{P} \big[ \mathcal {M}_m = \mathcal{M}_m^{(0)} \big] \nonumber \\ & \geq (h!)^{-m}n^{hm} \cdot [1-o(1)] \cdot (1-m p_{e,q})^{n} \times \nonumber \\ & \hspace{-3pt} \sum_{\mathcal {T}_m^{*} \in \mathbb{T}_m} \hspace{-4pt} \mathbb{P}[\mathcal {T}_m \hspace{-2pt} = \hspace{-2pt} \mathcal {T}_m^{*}] \prod_{i=1}^{m}\big\{ \mathbb{P}[w \hspace{-2pt} \in \hspace{-2pt} M_{0^{i-1}, 1, 0^{m-i}} \hspace{-2pt} \boldsymbol{\mid} \hspace{-2pt} \mathcal {T}_m \hspace{-2pt} = \hspace{-2pt} \mathcal {T}_m^{*} ]^{h} \big\} \nonumber \\ & \geq (h!)^{-m} (n p_{e,q})^{hm} e^{-m n p_{e,q}} \cdot [1-o(1)] . \label{eqn_prMm_pt2}\end{aligned}$$ Substituting (\[EQ\]) (\[eqn\_f00\]) above and (\[eq\_evalprob\_1\_qcmp\]) in Lemma \[lem\_evalprob\_qcmp\] into the computation of $\mathbb{P} \big[ \mathcal {M}_m = \mathcal{M}_m^{(0)} \big]$ yields $$\begin{aligned} & \mathbb{P} \big[ \mathcal {M}_m = \mathcal{M}_m^{(0)} \big] \nonumber \\ & \leq (h!)^{-m}n^{hm} {p_{e,q}}^{hm} \times [1+o(1)] \times \nonumber \\ & \sum_{\mathcal {T}_m^{*} \in \mathbb{T}_m} \mathbb{P}[\mathcal {T}_m = \mathcal {T}_m^{*}] \mathbb{P} [w \in M_{0^m} \boldsymbol{\mid} \mathcal {T}_m = \mathcal {T}_m^{*} ]^{n-m-hm} \nonumber \\ & \sim (h!)^{-m} (n p_{e,q})^{hm} e^{-m n p_{e,q}} . \label{eqn_prMm_pt1}\end{aligned}$$ Then (\[eqn\_prMm\]) follows from (\[eqn\_prMm\_pt2\]) and (\[eqn\_prMm\_pt1\]). Namely, (\[eqn\_prMm\]) holds upon the establishment of (\[eq\_evalprob\_exp\_2\]), which is proved below. First, from (\[eq\_evalprob\_2\_qcmp\]) in Lemma \[lem\_evalprob\_qcmp\], with $\mathcal {T}_m^{*} = (S_1^{*} , S_2^{*} , \ldots, S_m^{*} ) $ and $S_{ij}^{*} = S_{i}^{*} \cap S_{j}^{*}$, we get $$\begin{aligned} & \hspace{-2pt} \prod_{i=1}^{m} \mathbb{P}\big[w \in M_{0^{i-1}, 1, 0^{m-i}}^{(0)} \boldsymbol{\mid} \mathcal {T}_m = \mathcal {T}_m^{*} \big]^h \nonumber \\ & \hspace{-2pt} \geq \hspace{-3pt} { {p_{e,q}}^{hm} \hspace{-2pt} \prod_{i=1}^{m} \hspace{-2pt} \bigg[ \hspace{-2pt} 1 \hspace{-2pt} - \hspace{-3pt} \bigg( \hspace{-2pt} (q+2)!m{(p_{e,q})}^{\frac{1}{q}} \hspace{-2pt} + \hspace{-2pt} \frac{p_n}{K_n} \hspace{-3pt} \sum_{j\in\{1,2,\ldots,m\}\setminus\{i\}} \hspace{-4pt} |S_{ij}^{*}|\hspace{-2pt} \bigg)\hspace{-2pt} \bigg]\hspace{-2pt} }^h\nonumber \\ & \hspace{-2pt} \geq \hspace{-3pt} {p_{e,q}}^{hm} \bigg( 1 - (q+2)! h m^2 (p_{e,q})^{\frac{1}{q}} - \frac{2 hp_n}{K_n} \sum _{1\leq i <j \leq m} |S_{ij}^{*}|\bigg). \nonumber\end{aligned}$$ With $p_{e,q} = o(1)$ by (\[eq\_pe\_lnnn\]), we obtain (\[eq\_evalprob\_exp\_2\]) once proving $$\begin{aligned} \frac{ p_n}{K_n} \hspace{-2pt} \sum_{ \mathcal {T}_m^{*} \in \mathbb{T}_m } \hspace{-2pt} \hspace{-2pt} \bigg( \mathbb{P}[\mathcal {T}_m = \mathcal {T}_m^{*}] \hspace{-2pt} \sum _{1\leq i <j \leq m} \hspace{-2pt}|S_{ij}^{*}| \hspace{-1pt} \bigg) & = o(1). \hspace{-2pt} \label{prfhpnkn}\end{aligned}$$ Clearly, $| S_{ij}^{*} | \leq K_n$. If $\mathcal {T}_m^{*} \in \mathbb{T}_m^{(0)}$, it further holds that $| S_{ij}^{*} | < q $. Consequently, from (\[mthbbP\]), $K_n = \omega(1)$ and $p_n p_{s,q} = p_{e,q} \leq \frac{2\ln n}{n}$, the proof of (\[prfhpnkn\]) becomes evident by $$\begin{aligned} & \textrm{L.H.S. of (\ref{prfhpnkn})} \nonumber \\ & ~ \leq \binom{m}{2} p_n \cdot \mathbb{P}[\mathcal {T}_m^{*} \in \mathbb{T}_m \setminus \mathbb{T}_m^{(0)}] + \frac{q}{K_n} \cdot p_n \cdot \mathbb{P}[\mathcal {T}_m^{*} \in \mathbb{T}_m^{(0)}] \nonumber \\ & ~ \leq m^2 /2 \cdot p_n \cdot m^2 p_{s,q} / 2 + \frac{q}{K_n} \nonumber \\ & ~ \leq m^4 n^{-1}\ln n / 2 + o(1) \nonumber \\ & ~\to 0,\textrm{ as }n \to \infty. \nonumber\end{aligned}$$ Establishing (\[prob\_MmMm\_sim\]) ---------------------------------- We have $$\begin{aligned} &\mathbb{P} \big[ \big( \mathcal {M}_m = \mathcal{M}_m^{(0)} \big) \mathlarger{\cap} \big( \mathcal {T}_m \in \mathbb{T}_m^{(0)} \big) \big] \nonumber \\ & = \sum_{\mathcal {T}_m^{*} \in \mathbb{T}_m^{(0)}} \Big\{ \mathbb{P} \big[ \mathcal {T}_m = \mathcal {T}_m^{*} \big] \mathbb{P} \big[ \big( \mathcal {M}_m = \mathcal{M}_m^{(0)} \big) \boldsymbol{\mid} \big( \mathcal {T}_m = \mathcal {T}_m^{*} \big)\big] \Big\},\nonumber\end{aligned}$$ where $\mathbb{P} \big[ \big( \mathcal {M}_m = \mathcal{M}_m^{(0)} \big) \boldsymbol{\mid} \big( \mathcal {T}_m = \mathcal {T}_m^{*} \big)\big]$ as given by (\[pMmexpr\]) equals$$\begin{aligned} &f\big(n - m , \mathcal{M}_m^{(0)}\big) \mathbb{P} [w \in M_{0^m} \boldsymbol{\mid} \mathcal {T}_m = \mathcal {T}_m^{*} ]^{n-m-hm}\nonumber \\ & ~~\times \prod_{i=1}^{m}\big\{ \mathbb{P}[w \in M_{0^{i-1}, 1, 0^{m-i}} \boldsymbol{\mid}\mathcal {T}_m = \mathcal {T}_m^{*} ]^{h} \big\},\label{eqn_probMm}\end{aligned}$$ with $f\big(n-m , \mathcal{M}_m^{(0)}\big)$ computed in (\[eqn\_f00\]). For $\mathcal {T}_m^{*} \in \mathbb{T}_m^{(0)}$, from $|S_{ij}^{*}| < q$ and (\[eq\_evalprob\_2\_qcmp\]) in Lemma \[lem\_evalprob\_qcmp\], we derive $$\begin{aligned} &\mathbb{P}\big[w \in M_{0^{i-1}, 1, 0^{m-i}} \boldsymbol{\mid} \mathcal {T}_m = \mathcal {T}_m^{*} \big] \nonumber \\ & \quad \geq p_{e, q} \bigg[ 1 - (q+2)!m{(p_{e, q})}^{\frac{1}{q}} - \frac{qp_n }{K_n} \bigg] \label{eqn_wM0} .\end{aligned}$$ Substituting (\[eqn\_f00\]) (\[eqn\_wM0\]) above and (\[eq\_evalprob\_3\_qcmp\]) in Lemma \[lem\_evalprob\_qcmp\] into (\[eqn\_probMm\]), and using $p_{e,q} = o(1)$ and $K_n = \omega(1)$, we conclude that $$\begin{aligned} &\mathbb{P} \big[ \big( \mathcal {M}_m = \mathcal{M}_m^{(0)} \big) \mathlarger{\cap} \big( \mathcal {T}_m \in \mathbb{T}_m^{(0)} \big) \big] \nonumber \\ & \quad \geq \mathbb{P}[\mathcal {T}_m \in \mathbb{T}_m^{(0)}] \cdot (h!)^{-m}n^{hm} \cdot [1-o(1)] \nonumber \\ & \quad \quad \times (1 - m p_{e,q})^{n-m-hm} {p_{e,q}}^{hm} \nonumber \\ & \quad \quad \times \bigg[ 1 - (q+2)!m{(p_{e,q})}^{\frac{1}{q}} - \frac{qp_n }{K_n} \bigg]^{hm} \nonumber \\ & \quad \sim (h!)^{-m} (n p_{e,q})^{hm} e^{-m n p_{e,q}}. \nonumber\end{aligned}$$ Experimental Results {#sec:expe} ==================== To confirm the theoretical results, we now provide experiments in the non-asymptotic regime; i.e., when parameter values are set according to real-world sensor network scenarios. As we will see, the experimental observations are in agreement with our theoretical findings. ![A plot of the probability that graph $\mathbb{G}_q(n,K,P,p)$ has a minimum node degree at least $k$ as a function of $K$ for $k=4$ and $k=8$ with $n=2,000$, $q=2$, $P=10,000$, and $p=0.8$. []{data-label="f5"}](f1.eps){height="23.60000%"} ![A plot of the probability that graph $\mathbb{G}_q(n,K,P,p)$’s minimum node degree equals $k$ exactly as a function of $K$ for $k=0,1,2,3,4,5$ with $n=3000$, $q = 2$, $P=10000$, and $p=0.5$. []{data-label="fig5"}](f2.eps){width="49.00000%"} ![A plot of the probability distribution for the number of nodes with degree $h$ for $h=0,1,2 $ in graph $\mathbb{G}_q(n,K,P,p)$ with $n=3000$, $q = 2$, $P=10000$, $K = 35 $ and $p = 0.5 $. []{data-label="fig2"}](f3.eps){height="35.20000%"} In Figure \[f5\], we depict the probability that graph $\mathbb{G}_q(n,K,P,p)$ has a minimum node degree at least $k$ from both the simulation and the analysis, for $k = 4,8$ and $K$ varying from 29 to 36 (we set $n=2,000$, $q = 2$ and $P=10,000$ and $p=0.8$). On the one hand, for the experimental curves in all figures, we generate $2,000$ independent samples of $\mathbb{G}_q(n,K,P,p)$ given a parameter set and record the count (out of a possible $2,000$) that the minimum degree of graph $\mathbb{G}_q(n,K,P,p)$ is no less than $k$. Then the empirical probabilities are obtained by dividing the counts by $2,000$. On the other hand, we approximate the analytical curves of Figure \[f5\] by the asymptotic results as explained below. First, we compute the corresponding probability of $p_{e,q}$ in $\mathbb{G}_q(n,K,P,p)$ through $p_{e,q} = p \cdot \sum_{u=q}^{K} \big[{\binom{K}{u}\binom{P-K}{K-u}}\big/{\binom{P}{K}}\big] $ given (\[psq2cijFC3\]) and $P > 2 K$. Then we determine $\alpha$ by (\[peq1sbsc\]) (we write $\alpha_n$ as $\alpha $ here as $n$ is fixed); i.e., $p_{e, q} = \frac{\ln n + {(k-1)} \ln \ln n + {\alpha}}{n}$. Then given Remark \[thm:exact\_qcomposite-rem\] after Theorem \[thm:exact\_qcomposite\], we plot the analytical curves by considering that the minimum degree of $\mathbb{G}_q(n,K,P,p)$ is at least $k $ with probability $e^{- \frac{e^{-\alpha}}{(k-1)!}}$. The observation that the simulation and the analytical curves in Figure \[f5\] are close is in accordance with Theorem \[thm:exact\_qcomposite\]. In Figures \[fig5\] and \[fig2\], the curves with legends labelled “(E)” are *experimental* curves produced from experiments, while the curves with legends labelled “(A)” are *analytical* curves generated from theoretical analysis. In Figure \[fig5\], we depict the probability that graph $\mathbb{G}_q(n,K,P,p)$’s minimum node degree equals $k$ exactly as a function of $K$ for $k=0,1,2,3,4,5$. We set $n=3000$, $q = 2$, and $P=10000$, and $p=0.5$. For the experimental curves, we generate $2000$ independent samples of graph $\mathbb{G}_q(n,K,P,p)$ and record the count that the minimum degree of graph $\mathbb{G}_q(n,K,P,p)$ is exactly $k$; and the empirical probability of $\mathbb{G}_q(n,K,P,p)$ having a minimum degree of $k$ is derived by averaging over the $2000$ experiments. The analytical curves are produced as follows. First, we compute the corresponding probability of $p_{e,q}$ in $\mathbb{G}_q(n,K,P,p)$ through the aforementioned expression $p_{e,q} = p \cdot \big\{1- \sum_{u=0}^{q-1} \big[{\binom{K}{u}\binom{P-K}{K-u}}\big/{\binom{P}{K}}\big]\big\} $. Then we select $\ell^*$ such that $\big|p_{e, q} - \frac{\ln n + (\ell-1) \ln \ln n }{n}\big|$ is minimized for integer $\ell$ (i.e., $\ell^*={\operatornamewithlimits{argmin}}_{\textrm{integer }\ell}\big|p_{e, q} - \frac{\ln n + (\ell-1) \ln \ln n }{n}\big|$) and further define $\gamma^*$ such that $p_{e, q} = \frac{\ln n + (\ell-1) \ln \ln n +\gamma^*}{n}$. Given Remark \[thm:exact\_qcomposite-more-fine-grained-rem\] after Theorem \[thm:exact\_qcomposite-more-fine-grained\], we plot the analytical curves by considering that i) if $\ell^* > 0$, then ${{\mathbb{P}}\left[{\delta = k}\right]}$ equals $e^{- \frac{e^{-\gamma ^*}}{(\ell^*-1)!}}$ for $k=\ell^*$, equals $1-e^{- \frac{e^{-\gamma ^*}}{(\ell^*-1)!}}$ for $k=\ell^*-1$, and equals $0$ for $k\neq\ell^*$ and $k\neq\ell^*-1$, and ii) if $\ell^* \leq 0$, then ${{\mathbb{P}}\left[{\delta = k}\right]}$ equals $1$ for $k=0$, and equals $0$ for $k\neq0$. The observation that the curves generated from the experimental and the analytical curves are close to each other confirms the result on the distribution of the minimum degree in Theorem \[thm:exact\_qcomposite-more-fine-grained\]. In Figure \[fig2\], we plot the probability distribution for the number of nodes with degree $h$ in graph $\mathbb{G}_q(n,K,P,p)$ for $h=0,1,2$ from both the experiments and the analysis. We set $n=3000$, $q = 2$, $K = 35 $, $P=10000$, and $p=0.5$. On the one hand, for the experiments, we generate $2000$ independent samples of $\mathbb{G}_q(n,K,P,p)$ and record the count (out of a possible $2000$) that the number of nodes with degree $h$ for each $h$ equals a particular non-negative number $M$. Then the empirical probabilities are obtained by dividing the counts by $2000$. On the other hand, we approximate the analytical curves by the asymptotic results as explained below. In Theorem \[thm:exact\_qcomposite2\], we establish that the number of nodes in $\mathbb{G}_q(n,K_n,P_n,p_n)\iffalse_{on}\fi$ with degree $h$ approaches to a Poisson distribution with mean $\lambda_{n,h}=n (h!)^{-1}(n p_{e,q})^h e^{-n p_{e,q}}$ as $n \to \infty$. We derive $\lambda_{n,h}$ by computing the corresponding probability of $p_{e,q}$ in $\mathbb{G}_q(n,K,P,p)$ through $p_{e,q} = p \cdot \big\{1- \sum_{u=0}^{q-1} \big[{\binom{K}{u}\binom{P-K}{K-u}}\big/{\binom{P}{K}}\big]\big\} $ as explained above. Then for each $h$, we plot a Poisson distribution with mean $\lambda_{n,h}$ as the curve corresponding to the analysis. In Figure \[fig2\], we observe that the curves generated from the experiments and those obtained by the analysis are close to each other, confirming the result on asymptotic Poisson distribution in Theorem \[thm:exact\_qcomposite2\]. Results for Graph $\mathbb{G}_1$ {#sec:g1} ================================ For graph $\mathbb{G}_1$ (i.e., $\mathbb{G}_q$ in the special case of $q=1$), we have derived asymptotically exact probabilities for $k$-connectivity and the property that the minimum node degree is at least $k$ with arbitrary $k$. Compared with Theorem \[thm:exact\_qcomposite2\] and Theorem \[thm:exact\_qcomposite\] for $\mathbb{G}_q$, our results for $\mathbb{G}_1$ as presented in the following Theorem \[thm:mobihocQ1\] does not need the condition $\frac{{K_n}^2}{P_n} = o(1)$, and only requires a weaker condition: $P_n \geq 3K_n $ for all $n$ sufficiently large. \[thm:mobihocQ1\] Consider a positive integer $k$ and scalings $K \hspace{-3pt}: \mathbb{N}_0 \rightarrow \mathbb{N}_0,P \hspace{-3pt}: \mathbb{N}_0 \rightarrow \mathbb{N}_0$ and $p \hspace{-2pt}: \mathbb{N}_0 \rightarrow (0,1]$, with $P_n \geq 3K_n $ for all $n$ sufficiently large. Let the sequence $\gamma\hspace{-2pt}: \hspace{1pt}\mathbb{N}_0 \rightarrow \mathbb{R}$ be defined through $$\begin{aligned} p_{e,1} & = \frac{\ln n + {(k-1)} \ln \ln n + {\gamma_n}}{n}. \nonumber\end{aligned}$$ For $\lim_{n \to \infty} \gamma_n = \gamma ^* \in [-\infty, \infty]$, the properties (a) and (b) below hold. [(a)]{} If $ K_n = \omega(1)$, then as $n \to \infty$, $$\begin{aligned} \mathbb{P}\left[ \begin{array}{c} \textrm{The minimum~node~degree} \\ \mbox{of graph $\mathbb{G}_q\iffalse_{on}\fi$ is at least }k. \end{array} \right] & \to e^{- \frac{e^{-\gamma ^*}}{(k-1)!}} . \nonumber \end{aligned}$$ If $P_n = \Omega (n)$, then as $n \to \infty$, $$\begin{aligned} \mathbb{P} \left[\textrm{Graph }\mathbb{G}_q\iffalse_{on}\fi \textrm{ is $k$-connected}.\hspace{2pt}\right] & \to e^{- \frac{e^{-\gamma ^*}}{(k-1)!}} . \nonumber \end{aligned}$$ We provide the proof of Theorem \[thm:mobihocQ1\] in Appendix B. Note that $p_{e,1}$ is $p_{e,q}$ with $q=1$, and is the probability that two nodes have a link in between in graph $\mathbb{G}_1$. In establishing Theorem \[thm:mobihocQ1\], as given within its proof in Appendix B, we have shown that the number of nodes with an arbitrary degree in graph $\mathbb{G}_1$ asymptotically converges to a Poisson distribution. Using the idea similar to that of proving property (b) of Theorem \[thm:exact\_qcomposite2\] in this paper, we also establish the asymptotic probability distribution for the minimum node degree and for the connectivity of graph $\mathbb{G}_1$. Therefore, we present the following theorem on graph $\mathbb{G}_1$, which is an analog of Theorem \[thm:exact\_qcomposite2\]: Consider scalings $K: \mathbb{N}_0 \rightarrow \mathbb{N}_0,P: \mathbb{N}_0 \rightarrow \mathbb{N}_0$ and $p: \mathbb{N}_0 \rightarrow (0,1]$ with $P_n \geq 3K_n $ for all $n$ sufficiently large. For $$\begin{aligned} p_{e,1} & = \frac{\ln n \pm O(\ln \ln n)}{n},\nonumber\end{aligned}$$ (i.e., $\frac{n p_{e,1} - \ln n}{\ln \ln n}$ is bounded for all $n$), the following properties (a) and (b) for graph $\mathbb{G}_1\iffalse_{on}^{(q)}\fi$ hold. **(a)** If $ K_n = \omega(1)$, the number of nodes in $\mathbb{G}_1\iffalse_{on}\fi$ with an arbitrary degree converges to a Poisson distribution as $n \to \infty$. **(b)** Defining $\ell$ and $\gamma_n$ by $$\begin{aligned} \ell : = \bigg\lfloor \frac{np_{e, 1} - \ln n + (\ln \ln n) / 2}{\ln \ln n} \bigg\rfloor + 1, \nonumber\end{aligned}$$ and $$\begin{aligned} \gamma_n : = np_{e, 1} - \ln n - (\ell-1)\ln\ln n, \nonumber\end{aligned}$$ we obtain that if $ K_n = \omega(1)$, with $\mu$ denoting the minimum node degree of graph $\mathbb{G}_1$, - $(\mu \neq \ell)\cap (\mu \neq \ell-1)$ 0 as $n \to \infty$; - if $\lim_{n \to \infty} \gamma_n = \gamma ^* \in (-\infty, \infty)$, then as $n \to \infty$, $$\begin{aligned} \begin{cases} \mu = \ell \textrm{ with a probability converging to } e^{- \frac{e^{-\gamma ^*}}{(k-1)!}}, \\ \mu = \ell - 1\textrm{ with a probability tending to }\Big( \hspace{-1pt} 1 \hspace{-2pt} - \hspace{-2pt} e^{- \frac{e^{-\gamma ^*}}{(k-1)!}} \hspace{-1pt} \Big);\nonumber \end{cases}\end{aligned}$$ - if $ \lim_{n \to \infty} \gamma_n = \infty$, then as $n \to \infty$, $$\begin{aligned} \hspace{-3pt}\begin{cases} \mu = \ell\textrm{ with a probability approaching to }1, \\ \mu \neq \ell\textrm{ with a probability going to }0; \end{cases}\nonumber \hspace{20pt}\textrm{and}\end{aligned}$$ - if $ \lim_{n \to \infty} \gamma_n = - \infty$, then as $n \to \infty$, $$\begin{aligned} \hspace{-27pt}\begin{cases} \mu = \ell - 1\textrm{ with a probability tending to }1, \\ \mu \neq \ell - 1\textrm{ with a probability converging to }0; \end{cases}\nonumber\end{aligned}$$ and that if $P_n = \Omega (n)$, with $\nu$ denoting the connectivity of graph $\mathbb{G}_1$, - $(\nu \neq \ell)\cap (\nu \neq \ell-1)$ 0 as $n \to \infty$; - if $\lim_{n \to \infty} \gamma_n = \gamma ^* \in (-\infty, \infty)$, then as $n \to \infty$, $$\begin{aligned} \begin{cases} \nu = \ell \textrm{ with a probability converging to } e^{- \frac{e^{-\gamma ^*}}{(k-1)!}}, \\ \nu = \ell - 1\textrm{ with a probability tending to }\Big( \hspace{-1pt} 1 \hspace{-2pt} - \hspace{-2pt} e^{- \frac{e^{-\gamma ^*}}{(k-1)!}} \hspace{-1pt} \Big);\nonumber \end{cases}\end{aligned}$$ - if $ \lim_{n \to \infty} \gamma_n = \infty$, then as $n \to \infty$, $$\begin{aligned} \hspace{-3pt}\begin{cases} \nu = \ell\textrm{ with a probability approaching to }1, \\ \nu \neq \ell\textrm{ with a probability going to }0; \end{cases}\nonumber \hspace{20pt}\textrm{and}\end{aligned}$$ - if $ \lim_{n \to \infty} \gamma_n = - \infty$, then as $n \to \infty$, $$\begin{aligned} \hspace{-27pt}\begin{cases} \nu = \ell - 1\textrm{ with a probability tending to }1, \\ \nu \neq \ell - 1\textrm{ with a probability converging to }0. \end{cases}\nonumber\end{aligned}$$ under the as the physical link constraint comprising independent channels which are either *on* or *off*. The degree of a node $v$ is the number of nodes having links with $v$; and the minimum (node) degree of a network is the least among the degrees of all nodes. Specifically, we demonstrate that the number of nodes with an arbitrary degree asymptotically converges to a Poisson distribution, establish the asymptotic probability distribution for the minimum degree of the network, and derive the asymptotically exact probability for the property that the minimum degree is no less than an arbitrary value. Yağan [@yagan_onoff] and Zhao *et al.* [@ZhaoYaganGligor; @ISIT] consider the WSNs with $q=1$ and show results for several topological properties, Related Work {#related} ============ Erdős and Rényi [@citeulike:4012374] propose the random graph model $G(n,p_n)$ defined on a node set with size $n$ such that an edge between any two nodes exists with probability $p_n$ *independently* of all other edges. For graph $G(n,p_n)$, Erdős and Rényi [@citeulike:4012374] derive the asymptotically exact probabilities for connectivity and the property that the minimum degree is at least $1$, by proving first that the number of isolated nodes converges to a Poisson distribution as $n \to \infty$. Later, they extend the results to general $k$ in [@erdos61conn], obtaining the asymptotic Poisson distribution for the number of nodes with any degree and the asymptotically exact probabilities for $k$-connectivity and the event that the minimum degree is at least $k$, where $k$-connectivity is defined as the property that the network remains connected in spite of the removal of any $(k-1)$ nodes. Recall that graph $\mathbb{G}_q(n, K_n, P_n)$ models the topology of the $q$-composite key predistribution scheme [@ANALCO; @farrell2015hyperbolicity; @6875009]. For graph $\mathbb{G}_q(n, K_n, P_n)$, Bloznelis *et al.* [@Rybarczyk] demonstrate that a connected component with at at least a constant fraction of $n$ emerges asymptotically when probability $p_{e,q}$ exceeds $1/n$. Recently, still for $G_q(n, K_n, P_n)$, Bloznelis [@bloznelis2013] establishes the asymptotic Poisson distribution for the number of nodes with any degree. Our results in Theorem \[thm:exact\_qcomposite2\] by setting $p_n$ as $1$ imply his result; in particular, the result that he obtains is a special case of property (a) in our Theorem \[thm:exact\_qcomposite2\]. Yağan [@yagan_onoff] presents zero-one laws in graph $\mathbb{G}_1$ (our graph $\mathbb{G}_q$ in the case of $q=1$) for connectivity and for the property that the minimum degree is at least $1$. Zhao *et al.* extend Yağan’s results to general $k$ for $\mathbb{G}_1$ in [@ZhaoYaganGligor; @ISIT]. Our results in this paper apply to general $q$, yet the corresponding results for $q=1$ are already stronger than those in [@yagan_onoff; @ISIT; @ZhaoYaganGligor]. Krishnan *et al.* [@ISIT_RKGRGG] and Krzywdziński and Rybarczyk [@Krzywdzi] describe results for the probability of connectivity asymptotically converging to 1 in WSNs employing the $q$-composite key predistribution scheme with $q=1$ (i.e., the Eschenauer-Gligor key predistribution scheme), not under the on/off channel model but under the well-known disk model [@ISIT_RKGRGG; @Krzywdzi; @ZhaoAllerton; @6909183], where nodes are distributed over a bounded region of a Euclidean plane, and two nodes have to be within a certain distance for communication. Simulation results in our work [@ZhaoYaganGligor] indicate that for WSNs under the key predistribution scheme with $q=1$, when the on-off channel model is replaced by the disk model, the performances for $k$-connectivity and for the property that the minimum degree is at least $k$ do not change significantly. Conclusion and Future Work {#sec:Conclusion} ========================== In this paper, we analyze topological properties in WSNs operating under the $q$-composite key predistribution scheme with on/off channels. Experiments are shown to be in agreement with our theoretical findings. A future research direction is to consider communication models different from the on/off channel model. [10]{} H. Chan, A. Perrig, and D. Song, “Random key predistribution schemes for sensor networks,” in [*IEEE Symposium on Security and Privacy*]{}, May 2003. L. Eschenauer and V. Gligor, “A key-management scheme for distributed sensor networks,” in [*ACM Conference on Computer and Communications Security (CCS)*]{}, 2002. J. Zhao, O. Yağan, and V. Gligor, “$k$-[C]{}onnectivity in random key graphs with unreliable links,” [*IEEE Transactions on Information Theory*]{}, vol. 61, pp. 3810–3836, July 2015. M. Bloznelis, J. Jaworski, and K. Rybarczyk, “Component evolution in a secure wireless sensor network,” [*Networks*]{}, vol. 53, pp. 19–26, January 2009. O. Yağan, “Performance of the [Eschenauer–Gligor]{} key distribution scheme under an on/off channel,” [*IEEE Transactions on Information Theory*]{}, vol. 58, pp. 3821–3835, June 2012. J. Zhao, “On resilience and connectivity of secure wireless sensor networks under node capture attacks,” [*IEEE Transactions on Information Forensics and Security*]{}, 2016. K. Krzywdziński and K. Rybarczyk, “Geometric graphs with randomly deleted edges – [C]{}onnectivity and routing protocols,” [*Mathematical Foundations of Computer Science*]{}, vol. 6907, pp. 544–555, 2011. K. Rybarczyk, “Diameter, connectivity and phase transition of the uniform random intersection graph,” [*Discrete Mathematics*]{}, vol. 311, 2011. J. Zhao, “Threshold functions in random s-intersection graphs,” in [*2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton)*]{}, pp. 1358–1365, 2015. O. Yağan and A. M. Makowski, “Zero–one laws for connectivity in random key graphs,” [*IEEE Transactions on Information Theory*]{}, vol. 58, pp. 2983–2999, May 2012. J. Zhao, “On the resilience to node capture attacks of secure wireless sensor networks,” in [*Allerton Conference on Communication, Control, and Computing (Allerton)*]{}, pp. 887–893, 2015. S. E. Nikoletseas and C. L. Raptopoulos, “On some combinatorial properties of random intersection graphs,” in [*Algorithms, Probability, Networks, and Games*]{}, pp. 370–383, 2015. J. Zhao, “Modeling interest-based social networks: [S]{}uperimposing [E]{}rdos–[R]{}enyi graphs over random intersection graphs,” in [*IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)*]{}, 2017. M. Bloznelis, “Degree and clustering coefficient in sparse random intersection graphs,” [*The Annals of Applied Probability*]{}, vol. 23, no. 3, pp. 1254–1289, 2013. J. Zhao, “Parameter control in predistribution schemes of cryptographic keys,” in [*IEEE Global Conference on Signal and Information Processing (GlobalSIP)*]{}, pp. 863–867, 2015. B. Krishnan, A. Ganesh, and D. Manjunath, “On connectivity thresholds in superposition of random key graphs on random geometric graphs,” in [*IEEE International Symposium on Information Theory (ISIT)*]{}, pp. 2389–2393, 2013. J. Zhao, “Designing secure networks under $q$-composite key predistribution with unreliable links,” in [*IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)*]{}, 2017. E. M. Shahrivar, M. Pirani, and S. Sundaram, “Robustness and algebraic connectivity of random interdependent networks,” [*IFAC-PapersOnLine*]{}, vol. 48, no. 22, pp. 252–257, 2015. S. M. Dibaji and H. Ishii, “Consensus of second-order multi-agent systems in the presence of locally bounded faults,” [*Systems & Control Letters*]{}, vol. 79, pp. 23–29, 2015. J. Zhao, O. Yağan, and V. Gligor, “Secure $k$-connectivity in wireless sensor networks under an on/off channel model,” in [*IEEE International Symposium on Information Theory (ISIT)*]{}, pp. 2790–2794, 2013. P. Erdős and A. Rényi, “On random graphs, [I]{},” [*Publicationes Mathematicae (Debrecen)*]{}, vol. 6, pp. 290–297, 1959. J. Zhao, “Topological properties of secure wireless sensor networks under the $q$-composite key predistribution scheme with unreliable links,” 2016. Full version of this paper, available online at\ [https://sites.google.com/site/workofzhao/qcomposite.pdf ]{} R. Di Pietro, L. V. Mancini, A. Mei, A. Panconesi, and J. Radhakrishnan, “Redoubtable sensor networks,” [*ACM Transactions on Information and System Security (TISSEC)*]{}, vol. 11, pp. 13:1–13:22, March 2008. J. Zhao, “Topology control in secure wireless sensor networks,” in [*Cyber Security for Industrial Control Systems: From the Viewpoint of Close-Loop*]{} (P. Cheng, H. Zhang, and J. Chen, eds.), ch. 8, pp. 183–224, CRC Press, 2016. A. DasGupta, [*Asymptotic Theory of Statistics and Probability*]{}, vol. XVII. Springer Texts in Statistics, 2008. P. Erdős and A. Rényi, “On the strength of connectedness of random graphs,” [*Acta Mathematica Academiae Scientiarum Hungarice*]{}, pp. 261–267, 1961. J. Zhao, O. Yağan, and V. Gligor, “On $k$-connectivity and minimum vertex degree in random $s$-intersection graphs,” in [*ACM-SIAM Meeting on Analytic Algorithmics and Combinatorics (ANALCO)*]{}, pp. 1–15, January 2015. M. Farrell, T. D. Goodrich, N. Lemons, F. Reidl, F. S. Villaamil, and B. D. Sullivan, “Hyperbolicity, degeneracy, and expansion of random intersection graphs,” in [*International Workshop on Algorithms and Models for the Web-Graph*]{}, pp. 29–41, 2015. J. Zhao, O. Yağan, and V. Gligor, “On topological properties of wireless sensor networks under the q-composite key predistribution scheme with on/off channels,” in [*IEEE International Symposium on Information Theory (ISIT)*]{}, pp. 1131–1135, June 2014. J. Zhao, O. Yağan, and V. Gligor, “Connectivity in secure wireless sensor networks under transmission constraints,” in [*Allerton Conference on Communication, Control, and Computing*]{}, pp. 1294–1301, 2014. J. Pandey and B. Gupta, “Distribution of $k$-connectivity in the secure wireless sensor network,” in [*International Conference on Recent Advances and Innovations in Engineering (ICRAIE)*]{}, pp. 1–6, May 2014. Additional Lemmas ----------------- \[lem\_eval\_psq\] The following two properties hold, where $p_{s,q} $ denotes the probability that two nodes in graph $\mathbb{G}_q\iffalse_{on}^{(q)}\fi$ share at least $q$ keys: - If $K_n = \omega(1)$ and $\frac{{K_n}^2}{P_n} = o(1)$, then\ $p_{s,q} = \frac{1}{q!} \big( \frac{{K_n}^2}{P_n} \big)^{q} \times [1\pm o(1)]$; i.e., $p_{s,q} \sim \frac{1}{q!} \big( \frac{{K_n}^2}{P_n} \big)^{q}$. - If $K_n = \omega(\ln n)$ and $\frac{{K_n}^2}{P_n} = o\big(\frac{1}{\ln n}\big)$, then\ $p_{s,q} = \frac{1}{q!} \big( \frac{{K_n}^2}{P_n} \big)^{q} \times [1\pm o\big(\frac{1}{\ln n}\big)]$. \[lem\_evalprob\_qcmp\] In graph $\mathbb{G}_q\iffalse_{on}^{(q)}\fi$, with $p_{e,q} $ denoting the probability that two distinct nodes have a secure link in between, for any $\mathcal {T}_m^{*} = (S_1^{*} , S_2^{*} , \ldots, S_m^{*} ) \in \mathbb{T}_m$ and any node $w \in \overline{\mathcal {V}_m} $, we obtain $$\begin{aligned} & \mathbb{P} [w \in M_{0^m} \boldsymbol{\mid} \mathcal {T}_m = \mathcal {T}_m^{*} ] \geq 1 - m p_{e, q} , \label{eq_evalprob_3_qcmp}\end{aligned}$$ and for any $i = 1,2,\ldots,m $, $$\begin{aligned} & \mathbb{P}\big[w \in M_{0^{i-1}, 1, 0^{m-i}} \boldsymbol{\mid} \mathcal {T}_m = \mathcal {T}_m^{*} \big] \leq p_{e, q}; \label{eq_evalprob_1_qcmp}\end{aligned}$$ and if $\frac{{K_n}^2}{P_n} = o(1)$, the following (\[eq\_evalprob\_4\_qcmp\]) and (\[eq\_evalprob\_2\_qcmp\]) hold: $$\begin{aligned} & \mathbb{P} [w \in M_{0^m} \boldsymbol{\mid} \mathcal {T}_m = \mathcal {T}_m^{*} ] \nonumber \\ & \quad \leq e^{- m p_{e, q} + (q+2)! \binom{m}{2}{(p_{e, q})}^{\frac{q+1}{q}} + \frac{p_{e, q} p_n}{K_n}\sum_{1\leq i <j \leq m}|S_{i j}^{*}|}, \label{eq_evalprob_4_qcmp}\end{aligned}$$ and for any $i = 1,2,\ldots,m $, $$\begin{aligned} & \mathbb{P}\big[w \in M_{0^{i-1}, 1, 0^{m-i}} \boldsymbol{\mid} \mathcal {T}_m = \mathcal {T}_m^{*} \big] \nonumber \\ & \quad \geq p_{e, q} \bigg[ 1 \hspace{-.5pt}-\hspace{-.5pt} (q\hspace{-.5pt}+\hspace{-.5pt}2)!m{(p_{e,q})}^{\frac{1}{q}} \hspace{-.5pt}-\hspace{-.5pt} \frac{p_n}{K_n} \hspace{-.5pt}\sum_{j\in\{1,2,\ldots,m\} \setminus\{i\}}\hspace{-.5pt} |S_{i j}^{*}| \bigg], \label{eq_evalprob_2_qcmp}\end{aligned}$$ where $S_{ij}^{*} = S_{i}^{*} \cap S_{j}^{*}$. \[lem\_psukn\_qcmp\] In graph $\mathbb{G}_q\iffalse_{on}^{(q)}\fi$, if $\frac{{K_n}^2}{P_n} = o(1)$, then for any three distinct nodes $v_i, v_j$ and $v_t$ and for any $u = 0, 1, \ldots, K_n$, we obtain that with sufficiently large $n$, $$\begin{aligned} \mathbb{P}[({\Gamma}_{i t} \cap {\Gamma}_{j t} \boldsymbol{\mid} (|S_{ij}| = u)] \leq \frac{ p_{s,q} u}{K_n}+ (q+2)! \cdot ({p_{s,q})}^{\frac{q+1}{q}} . \nonumber\end{aligned}$$ Due to space limitation, we provide the proofs of Lemmas \[lem\_eval\_psq\], \[lem\_evalprob\_qcmp\], \[lem\_psukn\_qcmp\] in Appendices B, C, D of the full version [@full]. [^1]: Manuscript received March 16, 2016; revised October 24, 2016; accepted January 5, 2017; approved by <span style="font-variant:small-caps;"></span> Editor Y. Yi. Date of publication February 1, 2017; date of current version February 1, 2017. J. Zhao was with the Cybersecurity Lab (CyLab) at Carnegie Mellon University, Pittsburgh, PA 15213, USA. He is now with Arizona State University, Tempe, AZ 85281, USA (Email: junzhao@alumni.cmu.edu). A preliminary version of this paper appeared as: J. Zhao, O. Yağan, and V. Gligor, “On Topological Properties of Wireless Sensor Networks under the $q$-Composite Key Predistribution Scheme with On/Off Channels,” IEEE International Symposium on Information Theory, Hawaii, HI, USA, June 2014.This research was supported in part by Arizona State University, by the U.S. National Science Foundation under Grant CNS-1422277, and by the U.S. Defense Threat Reduction Agency under Grant HDTRA1-13-1-0029. This research was also supported in part by CyLab and Department of Electrical & Computer Engineering at Carnegie Mellon University. [^2]: Specifically, given two positive functions $f(n)$ and $g(n)$, 1. $f(n) = o \left(g(n)\right)$ signifies $\lim_{n \to \infty}[{f(n)}/{g(n)}]=0$. 2. $f(n) = \omega \left(g(n)\right)$ means $\lim_{n \to \infty}[{f(n)}/{g(n)}]=\infty$; i.e., $g(n) = o \left(f(n)\right)$. 3. $f(n) = O \left(g(n)\right)$ signifies that there exists a positive constant $c$ such that $f(n) \leq c g(n)$ for all $n$ sufficiently large. 4. $f(n) \sim g(n)$ means $\lim_{n \to \infty}[ {f(n)}/{g(n)}]=1$; namely, $f(n)$ and $g(n)$ are asymptotically equivalent. [^3]: For a non-negative integer $x$, the term $0^{x}$ is short for $\underbrace{00 \ldots 0}_{\textrm{``}x\textrm{''} \textrm{ number of ``}0\textrm{''}}$.

--- abstract: 'Transient optical heating provides an efficient way to trigger phase transitions in naturally occurring media through ultrashort laser pulse irradiation. A similar approach could be used to induce topological phase transitions in the photonic response of suitably engineered artificial structures known as metamaterials. Here, we predict a topological transition in the isofrequency dispersion contours of a layered graphene metamaterial under optical pumping. We show that the contour topology transforms from elliptic to hyperbolic within a subpicosecond timescale by exploiting the extraordinary photothermal properties of graphene. This new phenomenon allows us to theoretically demonstrate applications in engineering the decay rate of proximal optical emitters, ultrafast beam steering, and dynamical far-field subwavelength imaging. Our study opens a disruptive approach toward ultrafast control of light emission, beam steering, and optical image processing.' author: - Renwen Yu - Rasoul Alaee - 'Robert W. Boyd' - 'F. Javier García de Abajo' title: Ultrafast Topological Engineering in Metamaterials --- Introduction {#introduction .unnumbered} ============ Transient heating induced by laser pulse absorption has been intensely studied to induce phase transitions of interest such as metal-insulator in VO$_2$ [@XBV11], amorphous-crystalline in Ge$_2$Sb$_2$Te$_5$ [@FWN00], and charge density waves in 1[*T*]{}-TaS$_2$ [@HEE16], all of which hold strong potential for applications in next-generation electronic and optical data storage [@WY07]. An interesting possibility arises when considering phase transitions in components of artificial structures known as metamaterials, which grant us access into a broader range of optical properties extending beyond those encountered in naturally occurring materials. Actually, fascinating applications have been demonstrated by using metamaterial properties, including negative refraction [@V1968; @P00], superlensing [@FLS05], optical cloaking [@SMJ06], and superfocusing [@LBP02]. More recently, hyperbolic metamaterials have been found to exhibit an effective uniaxial anisotropic permittivity tensor that produces peculiar hyperbolic topology in the isofrequency dispersion contours and enables engineering of the decay rates of optical emitters [@KJN12; @LKF14], as well as hyperbolic waveguiding [@PN05] and far-field subwavelength imaging [@JAN06; @LLX07], among other feats. Hyperbolic metamaterials can be realized, for example, by simply stacking layers composed by alternating dielectric and plasmonic materials [@LKF14; @KJN12; @PIB13]. Among the latter, graphene offers additional appealing properties compared with noble metals, such as remarkably low optical losses [@WLG15; @NMS18], exceptionally small electronic heat capacity [@ESG18], extraordinary photothermal response [@paper313], and the ability to tune its optical conductivity through electrical gating [@NGM04; @FV07; @FP07_2; @CGP09]. In fact, hyperbolic metamaterials based on layered graphene/dielectric stacks [@IMS13; @OGC13] are being intensely explored as an excellent platform for applications involving infrared light [@BPA19]. When subject to ultrafast optical pulse irradiation, the optical energy absorbed by graphene is first deposited in its conduction electrons, which can be heated to an elevated temperature that remains during $\sim0.5-1\,$ps before transferring a substantial fraction of heat to the lattice [@JUC13; @GPM13; @RWW10]. Notably, due to the small heat capacity of electrons in graphene [@paper286; @ESG18] compared with the lattice, the latter remains close to ambient temperature for optical pump-pulse fluences as high as $\sim2\,$mJ/cm$^2$ and peak electron temperatures of 1000’sK [@LMS10; @paper330], partially assisted by the weak electron-phonon coupling that characterizes this material [@ESG18; @paper313]. A strong thermo-optical response is then triggered thanks to the strong temperature dependence of the graphene optical conductivity. Consequently, we expect that topological transitions of isofrequency contours could be triggered in metamaterials formed by layered graphene/dielectric stacks under ultrafast optical pumping, similar to light-driven phase-transitions in natural materials. {width="85.00000%"} In this Letter, we theoretically investigate ultrafast photothermal manipulation of a topological transition in layered metamaterials composed by graphene/dielectric stacks, driven by light absorption in graphene. Based on a realistic description of the temperature-dependent optical properties of graphene, in combination with the spatiotemporal heat flow within its electron/lattice subsystems under ultrafast laser pumping, we predict a topological transition of the isofrequency dispersion contours of the metamaterial in the infrared domain, whereby the topology transforms from elliptic to hyperbolic within an ultrafast timescale. We show that the spatiotemporal dynamics of this topological transition can be probed by a delayed free electron beam, and further find the transient hyperbolic phase to last $\sim1\,$ps. Our results enable several exotic phenomena in the ultrafast regime, such as dynamical engineering of the decay rate of optical emitters in the vicinity of the metamaterial, as well as directional beam steering by carving the metamaterial into a cylindrical lens, which we show to be useful for subwavelength far-field image encoding/decoding. Results {#results .unnumbered} ======= In Fig. \[Fig1\](a), we sketch a metamaterial composed of graphene/dielectric stacks with graphene located in the center of the dielectric-graphene-dielectric unit cell (see inset). We take a unit cell thickness $d=20\,$nm, a permittivity of the dielectric material $\epsilon_{{\rm d}}=4$, and a graphene Fermi energy $E_{{\rm F}}=0.4\,$eV throughout this work. Given the small value of $d$ compared with the infrared light wavelengths considered in this work, an effective permittivity should accurately describe the dielectric response of the metamaterial. Because of the system symmetry, we then have an effective uniaxial anisotropic material with two different permittivities $\epsilon_{\parallel}$ and $\epsilon_{\perp}$ along in-plane ($x$-$y$ plane) and out-of-plane ($z$ axis) directions, respectively. Additionally, due to the two-dimensional (2D) nature of graphene, we have $\epsilon_{\perp}=\epsilon_{{\rm d}}$. It is well known that optical pulse pumping can elevate the temperature of graphene electrons up to $\sim5000\,$K within a ultrafast timescale [@JUC13; @GPM13; @TSJ13] without causing any damage to the material because of its exceptionally small electronic heat capacity [@ESG18]. We present the spectral dependence of the in-plane effective permittivity $\epsilon_{\parallel}=\epsilon_{{\rm d}}+\ii 4\pi\sigma/\omega d$ for different electron temperatures $\Te$ in Fig. \[Fig1\](b), where $\sigma$ is the temperature-dependent surface conductivity of graphene and $\omega$ is the angular frequency. It should be noted that the epsilon-near-zero frequency (${\rm Re}\{ \epsilon_{\parallel} \}=0$, see solid curves) is first redshifted and then blueshifted as $\Te$ is increased, which is a consequence of the nontrivial $\Te$-dependence of the graphene chemical potential [@paper313]. More specifically, the ${\rm Re}\{ \epsilon_{\parallel} \}=0$ condition leads to a frequency $\approx\sqrt{4e^2\mu^{\rm D}/\hbar^2\epsilon_{{\rm d}}d}$, where $\mu^{\rm D}$ is the temperature-dependent effective Drude weight in the graphene conductivity [@paper313]. Additionally, ${\rm Im}\{ \epsilon_{\parallel} \}$ (dashed curves) increases with $\Te$ as a consequence of the enhancement in the inelastic scattering rate of graphene electrons [@YYM19]. We now explore the isofrequency dispersion contours at different electron temperatures for p-polarized electromagnetic fields, which is shown in Fig. \[Fig1\](c) at a frequency of 42THz \[indicated by a vertical black dash-dotted line in Fig. \[Fig1\](b)\]. The figure reveals two distinct regimes represented by the topology of the isofrequency contour, which evolves from elliptic to hyperbolic as $\Te$ increases. Specifically, within the $\Te=300-3000\,$K range, the isofrequency contour remains elliptic, despite some variations in shape. When $\Te$ is further increased, a dramatic variation of the contour topology occurs, which results in an emerging hyperbolic metamaterial because ${\rm Re}\{ \epsilon_{\parallel} \}<0$ when $\Te>3000\,$K. These results demonstrate that controlling $\Te$ in graphene can be an efficient and ultrafast route toward inducing topological transitions through optical pulse pumping. Incidentally, we compare the isofrequency dispersion contour calculated with the transfer matrix method in Fig. \[Fig1\](c) (solid curves) with that obtained from the effective medium theory (dashed curves), as determined by $$\begin{aligned} \frac{k_{\parallel}^2}{\epsilon_{\perp}}+\frac{k_{\perp}^2}{\epsilon_{\parallel}}=k_0^2, \label{disp}\end{aligned}$$ where $k_0$ is the free-spade light wave vector, while $k_{\parallel}$ and $k_{\perp}$ are the wave vectors along in- and out-of plane directions, respectively. We attribute the discrepancies between these two methods observed at large values of $k_{\parallel}$ to the lack of validity of the effective medium model when $k_{\parallel}d \ll 1$ no longer holds. {width="85.00000%"} A promising application of hyperbolic metamaterials relates to their ability to manipulate the decay rate of proximal emitters by enhancing the local density of optical states (LDOS) [@LKF14], which is given by [@LK1977; @NH06] $$\begin{aligned} \frac{{\rm LDOS}_{\parallel}}{{\rm LDOS}_0}=1+\frac{3}{4}\int_{0}^{\infty}\frac{k_xdk_x}{k_0^3}{\rm Re} \left \{\left (\frac{k_0^2 r_s}{k_z}-k_z r_p \right ) {\rm e}^{2\ii k_z z_0}\right \} \label{LDOS1}\end{aligned}$$ and $$\begin{aligned} \frac{{\rm LDOS}_{\perp}}{{\rm LDOS}_0}=1+\frac{3}{2}\int_{0}^{\infty}\frac{k_x^3dk_x}{k_0^3}{\rm Re} \left \{ \frac{r_p}{k_z} {\rm e}^{2\ii k_z z_0} \right \} \label{LDOS2}\end{aligned}$$ for in- and out-of-plane polarization, respectively, where $k_z=\sqrt{k_0^2-k_x^2}$, $z_0$ is the separation distance between the emitter and the metamaterial surface, and $r_s$ and $r_p$ are the reflection coefficients at the metamaterial-air interface for s- and p-polarization. These expressions are normalized to the projected LDOS in free space ${\rm LDOS}_0=\omega^2/3\pi^2 c^3$, where $c$ is the speed of light. The temperature-dependence of the LDOS enhancement is presented in Fig. \[Fig2\] at a distance $z_0=50\,$nm above the upper surface of the metamaterial \[see inset in Fig. \[Fig2\](a)\]. A large LDOS enhancement ($>10^3$) is found as a result of strong confinement of the photonic modes supported by the metamaterial. In general, as the electron temperature $\Te$ increases, the spectral range with high LDOS extends to lower frequencies. More specifically, an enhancement of two orders of magnitude can be obtained at $\sim 48\,$THz when increasing $\Te$ (see color-coded labels) due to the topological transformation of the isofrequency contour described in Fig. \[Fig1\]. This means that one can control the emitter decay rate in an ultrafast manner through rising the electron temperature by means of optical pulse pumping. Our findings are robust against the number $N$ of vertical unit cells composing the metamaterial film, as shown in Fig. \[Fig2\](a-c) for a dipolar emitter polarized parallel to the surface, as calculated from Eq. \[LDOS1\]. Similar results and conclusions are found for out-of-plane polarization, as shown in Fig. \[Fig2\](d-f), calculated from Eq. \[LDOS2\]. The deviation between the results calculated by using the transfer matrix method (solid curves) or the effective medium model (dashed curves) are again related to the mismatch at large values of $k_{\parallel}$. ![(a) Schematic of a pump-probe configuration involving a normally-incident optical pump pulse and a probing free electron beam passing parallel to the surface at a distance of 50nm. We consider a metamaterial film consisting of $N=10$ vertical periods. (b) Calculated spatiotemporal dynamics of the graphene electron temperature assumed to be uniform across the thin metamaterial film. The spatial coordinate indicates the distance to the center of the axisymmetric optical Gaussian pulse (600nm beam width). (c,d) Spatiotemporal dynamics of the EELS signal for an electron frequency loss of 42THz, as obtained through full electromagnetic calculations (c) or using an effective medium model for the metamaterial (d). The occurrence of the topological transition of the isofrequency contour, taking place at $\Te \approx 3560\,$K for the optical frequency under consideration, is indicated by blue-dashed curves in (b-d), which enclose the spatiotemporal domain characterized by a hyperbolic response.[]{data-label="Fig3"}](Fig3){width="50.00000%"} In order to resolve the ultrafast spatial and temporal dynamics of the topological transition, an electron probe moving in free space, can be employed to spatially image the topological transition in the so-called aloof configuration [@paper149] after a short optical pulse pumping, as illustrated in Fig. \[Fig3\](a). This type of experiment can be performed with state-of-the-art ultrafast electron microscopes, relying on pulsed optical pumping and electron probing [@BFZ09; @FES15; @PLQ15; @paper311; @DNS19]. The probe electron can provide direct information about the topological transition through the electron energy-loss spectroscopy (EELS) signal, the probability of which is given by [@paper149] $$\begin{aligned} \frac{\Gamma(\omega)}{L}=\frac{2e^2}{\pi \hbar v^2}\int_{0}^{\infty}\frac{dk_y}{k_{\parallel}^2}{\rm Re} \left \{ \left( \frac{k_y^2 v^2}{k_z^2 c^2}r_s-r_p \right) k_z {\rm e}^{2\ii k_z z_0} \right \}, \label{EELS}\end{aligned}$$ where $L$ is the length of the electron trajectory, $v$ is the electron velocity, $z_0$ is the separation distance between the electron beam and the metamaterial surface, $k_{\parallel}=\sqrt{\omega^2/v^2+k_y^2}$, and $k_z=\sqrt{k_0^2-k_{\parallel}^2}$. We note that the loss probability given by Eq. (\[EELS\]) bears a close relation to the momentum decomposition of LDOS along the electron trajectory [@paper102]. In what follows, we set the electron velocity to $v=0.5c$ ($\approx100\,$keV energy) and the separation to $z_0=50\,$nm. We further consider a Gaussian pump pulse of 100fs duration, 2mJ/cm$^2$ fluence, and 600nm beam width, which is realistic for lasers in the visible range. Following optical pumping of a metamaterial film composed of $N=10$ vertical unit cells, we calculate the resulting spatiotemporal dynamics of the electron temperature using a two-temperature model [@paper313; @paper330]. The results are shown in Fig. \[Fig3\](b), where the in-plane radial distance is referred from the Gaussian beam center. The electron temperature is a maximum value $\sim 4000\,$K at the beam center immediately after pumping and then decreases as time evolves. This range of high electron temperatures is currently achievable in state-of-the-art experiments [@JUC13; @GPM13; @TSJ13]. In a way that is consistent with its intimate relation to the LDOS, the spatiotemporal dynamics of the EELS signal follows closely that of $\Te$ at a fixed frequency loss of 42THz, as shown in Fig. \[Fig3\](c,d), where the results obtained from the effective medium model (Fig. \[Fig3\](d)) match quite well those obtained from the full calculation (Fig. \[Fig3\](c)). The blue-dashed curves in Fig. \[Fig3\](b-d) enclose a spatiotemporal regime in which the isofrequency contour is transformed to be hyperbolic, giving rise to an enhanced EELS signal. Within $\sim1\,$ps timescale, the topology of the isofrequency contour transforms from elliptic to hyperbolic, and then back to elliptic, thus encompassing an ultrafast topological transition. ![(a) Spatial distribution of the optical magnetic field in a plane perpendicular to a cylindrical metamaterial lens (bounded by the two black semicircles of radii $\lambda/5$ and $6\lambda/5$, respectively) at room temperature $\Te=300\,$K when it is excited by two magnetic current sources (represented by two small black circles close to the inner lens surface) separated by a subwavelength distance $\lambda/4$. We consider a light wavelength $\lambda=7.8\,\mu$m (i.e., 38.2THz frequency). The permittivity of the medium inside the inner circle and outside the outer one is taken to be 1 and 4, respectively. (b) Same as (a) when the graphene electron temperature in the metamaterial is raised to $\Te=1000\,$K. (c) Normalized far-field emitted energy as a function of azimuthal angle, spanning a range from $-90^\circ$ to $90^\circ$, at the two electron temperatures considered in (a,b). We compare results obtained from full electromagnetic simulations (solid curves) and the effective medium model (dashed curves).[]{data-label="Fig4"}](Fig4){width="50.00000%"} As a final example of application of the ultrafast topological phase transition discussed above, we investigate light steering and super-resolution imaging. It has been demonstrated that a perfect lens can be realized through negative refraction and amplification of evanescent waves in negative-index metamaterials [@P00]. Hyperbolic media with nearly flat isofrequency dispersion contours are also capable of producing subwavelength imaging of a point source [@JAN06; @LLX07]. Here, we study the radiated power distribution emanating from two magnetic line current sources (black circles in Fig. \[Fig4\](a,b), separated by $\lambda/4$, where $\lambda=7.8\,\mu$m is the optical frequency corresponding to a 38.2THz frequency). These sources are placed in front of a cylindrical lens made of a curved version of the layered graphene/dielectric under consideration. At room temperature $\Te=300\,$K, the isofrequency contour is indeed hyperbolic at 38.2THz \[see Fig. \[Fig1\](b)\]. However, because of the curved nature of the isofrequency contour, several guided modes are excited with different wave vectors $k_{r}$ along the radial direction [@BS06]. As a result, multiple lobes show up in the spatial distribution of the magnetic fields \[Fig. \[Fig4\](a)\], which also transmit into the far-field \[red curves in Fig. \[Fig4\](c)\]. When $\Te$ is increased to 1000K, the so-called canalization condition [@BSI05; @SE06_2] ${\rm Re} \{ \epsilon_\theta \}\approx0$ is satisfied leading to a nearly flat isofrequency contour. Here, $\epsilon_\theta$ is the effective permittivity of the cylindrical lens along the azimuthal direction. This results in two highly directive lobes in the spatial distribution of the magnetic field \[Fig. \[Fig4\](b)\], thus demonstrating that the cylindrical lens is capable of ultrafast beam steering driven by the transient elevation of $\Te$-rising upon optical pulse pumping. Additionally, two clear angular peaks associated with those two individual point sources can be identified in the far-field regime \[see blue curves in Fig. \[Fig4\](c)\], further supporting the potential for far-field subwavelength imaging in a dynamical and ultrafast manner. Once more, the azimuthal distribution of the far-field signal obtained from full numerical simulations (solid curves) matches well the result obtained from the effective medium model (dashed curves), as shown in Fig. \[Fig4\](c). Our results further suggest a novel subwavelength image encoding/decoding mechanism, whereby an elevated electron temperature induced by optical pumping is the key to resolve encoded subwavelength images in the far-field regime. Conclusion {#conclusion .unnumbered} ========== In summary, we have shown that transient heating can significantly modify the topology of the isofrequency dispersion contours in metamaterials formed by layered graphene/dielectric stacks by exploiting the remarkably small electronic heat capacity of graphene, which allows us to efficiently elevate the electron temperature through ultrafast optical pumping. By examining the spatiotemporal dynamics of the EELS signal obtained by using an electron-beam probe, we have found that the contour topology can transform between elliptic and hyperbolic shapes within a sub-picosecond timescale, that is, the characteristic time over which the elevated electron temperature evolves in real space. We can thus manipulate the LDOS enhancement in the proximity of the metamaterial, reaching variations in the calculated LDOS of a few orders of magnitude at the frequency in which this topological transition appears. Additionally, this type of transition enables ultrafast beam steering, which we have illustrated by illuminating a cylindrical lens made of metamaterials with two line sources, in which dynamical far-field subwavelength imaging has been demonstrated. Our findings open a promising route toward ultrafast control of light emission, beam steering, and optical image processing. Acknowledgements {#acknowledgements .unnumbered} ================ This work has been supported in part by the ERC (Advanced Grant 789104-eNANO), the Spanish MINECO (MAT2017-88492-R and SEV2015-0522), the Catalan CERCA Program, and Fundació Privada Cellex. R.A. acknowledges the support of the Alexander von Humboldt Foundation through the Feodor Lynen Fellowship. R.W.B. acknowledges support through the Natural Sciences and Engineering Research Council of Canada, the Canada Research Chairs program, and the Canada First Research Excellence Fund. [46]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , , , , , , , ****, (). , , , , , ****, (). , , , , , , , ****, (). , ****, (). , ****, (). , ****, (). , , , , ****, (). , , , , , , , ****, (). , , , , , ****, (). , , , , , ****, (). , , , , ****, (). , ****, (). , , , ****, (). , , , , , ****, (). , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , ****, (). , , , , , , , , ****, (). , ****, (). , ****, (). , , , , , ****, (). , , , , , ****, (). , , , ****, (). , , , ****, (). , , , , , , , , , , , ****, (). , , , , , , , , , , ****, (). , , , , , , ****, (). , , , ****, (). , , , , ****, (). , , , , , , , ****, (). , , , , , , , , ****, (). , , , , , , , , , , , (), <https://doi.org/10.1021/acsphotonics.0c00028>. , ****, (). , ** (, , ). , ****, (). , , , ****, (). , , , , , , ****, (). , , , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , , , , , , , p. (). , ****, (). , ****, (). , , , ****, (). , ****, ().

--- abstract: 'This paper considers the impact of external noise sources, including interfering transmitters, on a diffusive molecular communication system, where the impact is measured as the number of noise molecules expected to be observed at a passive receiver. A unifying model for noise, multiuser interference, and intersymbol interference is presented, where, under certain circumstances, interference can be approximated as a noise source that is emitting continuously. The model includes the presence of advection and molecule degradation. The time-varying and asymptotic impact is derived for a series of special cases, some of which facilitate closed-form solutions. Simulation results show the accuracy of the expressions derived for the impact of a continuously-emitting noise source, and show how approximating old intersymbol interference as a noise source can simplify the calculation of the expected bit error probability of a weighted sum detector.' author: - 'Adam Noel, , Karen C. Cheung, and Robert Schober, [^1] [^2]' bibliography: - '../references/nano\_ref.bib' title: A Unifying Model for External Noise Sources and in Diffusive Molecular Communication --- Diffusion, intersymbol interference, molecular communication, multiuser interference, noise. Introduction ============ communication is a physical layer design strategy that could enable the deployment of nanonetworks by facilitating the sharing of information between individual devices with nanoscale functional components. It is envisioned that these networks will bring new applications to fields that require diagnostics or actions on a small physical scale, e.g., healthcare and manufacturing; see [@RefWorks:540; @RefWorks:608]. Molecular communication relies on molecules released by transmitters as information carriers, and is inspired by the common use of molecules for information transmission in biological systems; see [@RefWorks:750]. Passive molecular propagation methods do not require external energy for transport. The simplest such method, free diffusion, randomly moves molecules via collisions with other molecules, and does not require fixed connections between transceivers. However, the design of a communication network that is based on free diffusion faces a number of challenges. The propagation time increases and the reliability decreases as the distance between transceivers increases. Intersymbol interference ([ISI]{}) arises if there is no process to degrade information molecules or carry them away from the receiver. Furthermore, the deliberate release of molecules by the intended transmitter might not be the only local source of information molecules. We refer to other such sources as *external* molecule sources. External molecule sources can be expected in diffusive environments where nanonetworks may be deployed. Examples include: - Multiuser interference caused by molecules that are emitted by the transmitters of other communication links. This interference can be mitigated by using different molecule types for every communication link, but this might not be practical if there is a very large number of links and the individual transceivers share a common design. - Unintended leakage from vesicles (i.e., membrane-bound containers) where the information molecules are being stored by the transceivers. A rupture could result in a steady release of molecules or, if large enough, the sudden release of a large number of molecules; see [@RefWorks:755]. - The output from an unrelated biochemical process. The biocompatability of the nanonetwork may require the selection of a naturally-occurring information molecule. Thus, other processes that produce or release that type of molecule are effectively noise sources for communication. For example, calcium signalling is commonly used as a messenger molecule within cellular systems (see [@RefWorks:588 Ch. 16]), so selecting calcium as the information carrier in a new molecular communication network deployed in a biological environment would mean that the natural occurrence of calcium is a source of noise. - The unintended reception of other molecules that are sufficiently similar to the information molecules to be recognized by the receiver. For example, the receptors at the receiver might not be specific enough to only bind to the information molecules, or the other molecules might have a shape and size that is very similar to that of the information molecules; see [@RefWorks:588 Ch. 4]). Most existing literature on noise analysis in diffusive molecular communication has considered the noise in the communication link, i.e., via the noisiness of diffusion itself or chemical mechanisms at the receiver, cf. e.g. [@RefWorks:436; @RefWorks:469; @RefWorks:644; @RefWorks:687; @RefWorks:734; @RefWorks:737], without accounting for the impact of external noise sources. The impact of multiuser interference on capacity was evaluated numerically in [@RefWorks:656]. Wave theory was used to approximate both [ISI]{} and multiuser interference in [@RefWorks:677], where [ISI]{} was limited to one previous interval and only multiuser emissions in the current transmission interval were considered. In [@RefWorks:513], a stochastic model was proposed that included the spontaneous generation of information molecules in the propagation environment. In this paper, we propose a unifying model for external noise sources (including multiuser interference) and [ISI]{} in diffusive molecular communication. We consider an unbounded physical environment with steady uniform flow, based on a system model that we studied in [@RefWorks:747; @RefWorks:752] (but where we did not develop any detailed noise analysis; we only assumed that the asymptotic impact of the noise sources was known). The primary contributions of this paper are as follows: 1. We derive the expected asymptotic (and, wherever possible, time-varying) impact of a continuously-emitting noise source, given the location of the source and its rate of emission. By impact, we refer to the corresponding expected number of molecules observed at the receiver, and by asymptotic we refer to the source being active for infinite time. Closed-form solutions are available for a number of special cases; otherwise, the impact can only be found via numerical integration. 2. We use asymptotic noise from a source far from the receiver to approximate the impact of interfering transmitters, thus providing a simple expression for the molecules observed at the receiver due to multiuser interference without requiring the interfering transmitters’ data sequences. The accuracy of this approximation improves as the distance between the receiver and the interfering transmitters increases. 3. We approximate “old” [ISI]{} in the intended communication link as asymptotic interference from a continuously-emitting source. We decompose the received signal into molecules observed due to an emission in the current bit interval, molecules that were emitted in recent bit intervals, and molecules emitted in older intervals, where only the impact of the “old” emissions is approximated. Knowing the expected impact of a noise source enables us to model its effect on successful transmissions between the intended transmitter and receiver. For example, in [@RefWorks:747; @RefWorks:752] we assumed that we had knowledge of the expected impact of noise sources in order to evaluate the effect of external noise on the bit error probability at the intended receiver for a selection of detectors. The expected impact of noise sources can also be used to assess different methods to mitigate the effects of noise, e.g., via the degradation of noise molecules as we consider in this paper. Decomposing the signal received from the intended transmitter enables us to bridge all existing work on [ISI]{} by adjusting the number of “recent” bit intervals and deciding how we analytically model the “old” molecules. Most literature on diffusive molecular communication has accounted for only one recent bit interval and ignored the impact of old molecules; see [@RefWorks:677; @RefWorks:534; @RefWorks:574; @RefWorks:643]. More recently, it has become more common to account for *all* molecules released, i.e., treat all prior bit intervals as recent; see [@RefWorks:667; @RefWorks:668; @RefWorks:644; @RefWorks:687; @RefWorks:786] and our previous work in [@RefWorks:662; @RefWorks:747]. We introduce the number of recent bit intervals as a parameter that enables a trade-off between complexity and accuracy in analyzing receiver performance. Furthermore, modeling all older [ISI]{} as asymptotic noise will be shown to be a more accurate alternative to assuming that old [ISI]{} has no impact at all. In this paper, we also describe how an asymptotic model for old [ISI]{} simplifies the evaluation of the expected bit error probability of a weighted sum detector with equal weights. We proposed this detector as a member of the family of weighted sum detectors in [@RefWorks:747]. Other possible applications of an asymptotic model for old [ISI]{} include a simplified implementation of the optimal sequence detector (a detector that we also considered in [@RefWorks:747]), or simplifying the design of an adaptive weight detector, where the decision criteria are adjusted based on knowledge of the previously received information. The rest of this paper is organized as follows. The system model, including the physical environment and its representation in dimensionless form, is described in Section \[sec\_model\]. In Section \[sec\_noise\], we derive the time-varying and asymptotic impact of an external noise source on the receiver. In Section \[sec\_mui\], we consider the special case of a noise source that is an interfering transmitter. We adapt the noise analysis for asymptotic old [ISI]{} and use it to simplify detector performance evaluation in Section \[sec\_isi\]. Numerical and simulation results are described in Section \[sec\_num\], and conclusions are drawn in Section \[sec\_concl\]. System Model {#sec_model} ============ We consider an infinite 3-dimensional fluid environment of uniform constant temperature and viscosity. The receiver is a sphere with radius ${r_{obs}}$ and volume ${V_{obs}}$ (if the transmitter is sufficiently far from the receiver, then the precise shape is irrelevant and we are only interested in ${V_{obs}}$). As this paper focuses on the impact of unintended sources of information molecules on the observations made at the receiver, the receiver is centered at the origin. Without loss of generality, the intended transmitter is placed at coordinates $\{-{x}_1,0,0\}$. We assume there is steady uniform flow (or drift) in an arbitrary direction with a velocity component in each dimension, i.e., ${\vec{v}_{}} = \{{v_{{x}}},{v_{{y}}},{v_{{z}}}\}$. The receiver is a passive observer that does not impede diffusion or initiate chemical reaction (so that we can focus on the impact of the propagation environment). Its only interaction with the environment is the perfect counting of ${A}$ molecules if they are within ${V_{obs}}$; any other molecules that might be present are ignored. ${A}$ molecules are the information molecules that can be emitted by the transmitter or by some other sources. In practice, the receiver would observe ${A}$ molecules by having them bind to receptors that are on the surface of or inside ${V_{obs}}$. The expected local concentration of ${A}$ molecules at the point defined by vector ${\vec{r}_{}}$ and at time $t$ in ${\textnormal{molecule}}\cdot{\textnormal{m}}^{-3}$ is ${C_{{A}}({\vec{r}_{}},t)}$, and we write ${C_{{A}}}$ for compactness. All ${A}$ molecules diffuse independently with constant diffusion coefficient ${D_{{A}}}$, and they can degrade into a form that cannot be detected by the receiver via a reaction mechanism that can be described as $$\label{k1_mechanism} {A}\xrightarrow{{k_{}}} \emptyset,$$ where ${k_{}}$ is the reaction rate constant in ${\textnormal{s}}^{-1}$. If ${k_{}} = 0$, then this degradation is negligible. Eq. (\[k1\_mechanism\]) is a first-order reaction, but it can also be used to approximate higher-order reactions or reaction mechanisms with multiple steps. For example, in our previous work where we considered enzymes in the propagation environment to mitigate [ISI]{}, we implicitly used (\[k1\_mechanism\]) to derive a bound on the expected number of observed molecules; see [@RefWorks:631; @RefWorks:662]. First-order reactions have also been used to approximate higher-order reactions in a molecular communication context in [@RefWorks:737], where the reactions occurred only at the receiver. We emphasize that our assumptions include a passive receiver, first-order ${A}$ molecule degradation throughout the environment, and the constant diffusion of ${A}$ molecules. These assumptions make our model analytically tractable but ignore the impact of effects including anomalous diffusion, localized chemical reactions, and other interactions between molecules. We are interested in studying such complex systems in our future work. For clarity of exposition in the remainder of this paper, we convert our system model into dimensionless form. We have used dimensional analysis in our previous work, including [@RefWorks:752; @RefWorks:706], because it generalizes our model’s scalability and facilitates comparisons between different dimensional parameter sets. In this paper, dimensional analysis also provides clarity of exposition by reducing the number of parameters that appear in the equations. Unless otherwise noted, all variables that are described in this paper are assumed to be dimensionless (as denoted by a “$\star$” superscript), and they are equal to the dimensional variables scaled by the appropriate reference variables; see [@RefWorks:633] for more on dimensional analysis. We define reference distance ${L}$ in ${\textnormal{m}}$ and reference number of molecules ${N_{{A}_{REF}}}$. We also define reference concentration ${C_{0}} = {N_{{A}_{REF}}}/{L}^3$ in ${\textnormal{molecule}}\cdot{\textnormal{m}}^{-3}$. We then define the dimensionless concentration of ${A}$ molecules as ${{C_{{a}}}^\star} = {C_{{A}}}/{C_{0}}$, dimensionless time as ${t_{}^\star} = {D_{{A}}}t/{L}^2$, and the dimensionless reaction rate constant as ${{k_{}}^\star} = {L}^2{k_{}}/{D_{{A}}}$. The dimensionless coordinates along the three axes are $$\label{AUG12_43_coor} {x^\star}= \frac{{x}}{{L}}, \quad {y^\star}= \frac{{y}}{{L}}, \quad {z^\star}= \frac{{z}}{{L}},$$ such that they are the dimensional coordinates scaled (i.e., normalized) by the reference distance ${L}$. Advection is represented dimensionlessly with the Peclet number, ${v^\star_{}}$, written as [@RefWorks:750 Ch. 1] $$\label{EQ13_07_30} {v^\star_{}} = \frac{{v_{}}{L}}{{D_{{A}}}},$$ where ${v_{}} = |{\vec{v}_{}}|$ is the speed of the fluid. ${v^\star_{}}$ measures the relative impact of advection versus diffusion on molecular transport. If ${v^\star_{}} = 1$, then the typical time for a molecule to diffuse the reference distance ${L}$, i.e., ${L}^2/{D_{{A}}}$, is equal to the typical time for a molecule to move the same distance by advection alone. A value of ${v^\star_{}}$ much less or much greater than $1$ signals the dominance of diffusion or advection, respectively. We saw in [@RefWorks:752] that the impact of steady uniform flow on successful communication also depends on the *direction* of flow, so we define ${v^\star_{}}$ along each dimension as $$\label{EQ13_07_29} {{v^\star_{\scriptscriptstyle\parallel}}}= \frac{{v_{{x}}}{L}}{{D_{{A}}}}, \quad {v^\star_{\perp,1}} = \frac{{v_{{y}}}{L}}{{D_{{A}}}}, \quad {v^\star_{\perp,2}} = \frac{{v_{{z}}}{L}}{{D_{{A}}}}.$$ The dimensionless signal observed at the receiver, ${{{N_{{a}}}}_{obs}^\star\left({t_{}^\star}\right)}$, is the cumulative impact of all molecule emitters in the environment, including interfering transmitters and other noise sources. Due to the independence of the diffusion of all ${A}$ molecules, we can apply superposition to the impacts of the individual sources, such that the cumulative impact of multiple noise sources is the sum of the impacts of the individual sources. If we assume that there are ${U}-1$ sources of ${A}$ molecules that are not the intended transmitter (without specifying what kinds of sources these are, i.e., other transmitters or simply “leaking” ${A}$ molecules), then the complete observed signal can be written as $${{{N_{{a}}}}_{obs}^\star\left({t_{}^\star}\right)} = {{{N_{{a}}}}_{1}^\star\left({t_{}^\star}\right)} + \sum_{u=2}^{{U}}{{{N_{{a}}}}_{u}^\star\left({t_{}^\star}\right)}, \label{EQ13_05_28_obs}$$ where ${{{N_{{a}}}}_{1}^\star\left({t_{}^\star}\right)}$ is the signal from the intended transmitter. Without loss of generality (because the impact of multiple molecule sources can be superimposed), we can analyze each ${{{N_{{a}}}}_{u}^\star\left({t_{}^\star}\right)}$ term independently. Thus, for clarity, we assume that the $u$th source is placed at $\{-{x^\star}_u,0,0\}$, where ${x^\star}_u \ge 0$. Furthermore, for all molecule sources in (\[EQ13\_05\_28\_obs\]), the advection variables must be defined relative to the source’s corresponding coordinate frame, such that ${{v^\star_{\scriptscriptstyle\parallel}}}>0$ always represents flow from the source towards the receiver. By symmetry, and without loss of generality, we can set ${v^\star_{\perp,2}} = 0$ and write ${v^\star_{\perp,1}} = {{v^\star_{\perp}}}$, such that ${{v^\star_{\perp}}}$ represents flow perpendicular to the line between the source and receiver. In Table \[table\_overview\], we summarize where the different terms in (\[EQ13\_05\_28\_obs\]) are analyzed in the remainder of this paper and whether each type of source is treated as continuously-emitting. In Sections \[sec\_noise\] and \[sec\_mui\], we model ${{{N_{{a}}}}_{u}^\star\left({t_{}^\star}\right)}$ as a random noise source and as an interfering transmitter, respectively. In Section \[sec\_isi\], we decompose ${{{N_{{a}}}}_{1}^\star\left({t_{}^\star}\right)}$ to approximate old [ISI]{} as asymptotic noise. Source Type Section ---------------------------------------------------- ------------------------- ---------------- -------------------------- ------------------------------------------------------------------------ Random Noise \[sec\_noise\] (\[APR12\_42\_DMLS\]) Yes Interfering Transmitter \[sec\_mui\] (\[EQ13\_08\_02\]) Approximation in (\[EQ13\_08\_05\]) ${{{N_{{a}}}}_{1}^\star\left({t_{}^\star}\right)}$ Intended Transmitter \[sec\_isi\] (\[EQ13\_08\_08\_DMLS\]) Approximation of old ISI in (\[EQ13\_08\_09\])-(\[EQ13\_05\_18\_isi\]) : Description of the terms in (\[EQ13\_05\_28\_obs\]). \[table\_overview\] External Additive Noise {#sec_noise} ======================= In this section, we derive the impact of external noise sources on the receiver, given that we have some knowledge about the location of the noise sources and their mode of emission. First, we consider a single point noise source placed at $\{-{x^\star}_n,0,0\}$ where ${x^\star}_n$ is non-negative (we change the subscript of the source from $u$ to $n$ in order to emphasize that this source is random noise and not a transmitter of information). The source emits molecules according to the random process ${{N_{{{A}_{gen}}}}\left(t\right)}$, represented dimensionlessly as ${{N_{{{a}_{gen}}}}^\star\left({t_{}^\star}\right)} = {L}^2{{N_{{{A}_{gen}}}}\left(t\right)}/\left({D_{{A}}}{N_{{A}_{REF}}}\right)$. Assuming that the expected generation of molecules can be represented as a step function, i.e., ${\overline{{N_{{{A}_{gen}}}}}\left(t\right)} = {{N_{{{A}_{gen}}}}}, t \ge 0$, we then formulate the expected impact of the noise source at the receiver, ${\overline{{{N_{{a}}}}_{n}^\star}\left({t_{}^\star}\right)}$. In its most general form, we will not have a closed-form solution for the expected impact of the noise source. Next, we present either time-varying or asymptotic expressions for a number of relevant special cases, some of which are in closed form and others that facilitate numerical integration. While we are ultimately most interested in asymptotic solutions (particularly for extension to the analysis of interference), time-varying solutions are also of interest when they are available because they give us insight into how long a noise source must be “active” before we can model its impact as asymptotic. Time-varying solutions will also be useful when we consider old [ISI]{} in Section \[sec\_isi\]. As previously noted, we can use superposition to consider the cumulative impact of multiple noise sources, as given in (\[EQ13\_05\_28\_obs\]), where the advection variables ${{v^\star_{\scriptscriptstyle\parallel}}}$ and ${{v^\star_{\perp}}}$ must be defined for each source depending on its location. General Noise Model ------------------- First, we require the channel impulse response due to the noise source, i.e., the *expected* concentration of molecules observed at the receiver due to an emission of one molecule by the noise source at ${t_{}^\star} = 0$. This is analogous to the channel impulse response due to an intended transmitter at the same location. The reaction-diffusion differential equation describing the expected motion of ${A}$ molecules can be written by applying the principles of chemical kinetics (see [@RefWorks:585 Ch. 9]) to (\[k1\_mechanism\]) and including the advection terms (as in [@RefWorks:630 Ch. 4]), i.e., $$\label{JUN12_33_DMLS} {\frac{\partial {{C_{{a}}}^\star}}{\partial {t_{}^\star}}} = \nabla^2{{C_{{a}}}^\star} - {{v^\star_{\scriptscriptstyle\parallel}}}{\frac{\partial {{C_{{a}}}^\star}}{\partial {x^\star}}} - {{v^\star_{\perp}}}{\frac{\partial {{C_{{a}}}^\star}}{\partial {y^\star}}} - {{k_{}}^\star}{{C_{{a}}}^\star},$$ where $${\frac{\partial {{C_{{a}}}^\star}}{\partial {t_{}^\star}}} = {\frac{\partial {C_{{A}}}}{\partial t}}\frac{{L}^2}{{D_{{A}}}{C_{0}}}, \quad \nabla^2{{C_{{a}}}^\star} = \frac{{L}^2}{{C_{0}}}\nabla^2{C_{{A}}}, \quad {\frac{\partial {{C_{{a}}}^\star}}{\partial {x^\star}}} = {\frac{\partial {C_{{A}}}}{\partial {x}}}\frac{{L}}{{C_{0}}}, \quad {\frac{\partial {{C_{{a}}}^\star}}{\partial {y^\star}}} = {\frac{\partial {C_{{A}}}}{\partial {y}}}\frac{{L}}{{C_{0}}}, \label{AUG12_45_v}$$ $$\begin{aligned} {\frac{\partial {{C_{{a}}}^\star}}{\partial {t_{}^\star}}} = &\;{\frac{\partial {C_{{A}}}}{\partial t}}\frac{{L}^2}{{D_{{A}}}{C_{0}}}, \quad \nabla^2{{C_{{a}}}^\star} = \frac{{L}^2}{{C_{0}}}\nabla^2{C_{{A}}}, \nonumber \\ \label{AUG12_45_v} {\frac{\partial {{C_{{a}}}^\star}}{\partial {x^\star}}} = &\; {\frac{\partial {C_{{A}}}}{\partial {x}}}\frac{{L}}{{C_{0}}}, \quad {\frac{\partial {{C_{{a}}}^\star}}{\partial {y^\star}}} = {\frac{\partial {C_{{A}}}}{\partial {y}}}\frac{{L}}{{C_{0}}},\end{aligned}$$ and it is straightforward (using a moving reference frame) to show that the channel impulse response at the point $\{{x^\star},{y^\star},{z^\star}\}$ due to the noise source at $\{-{x^\star}_n,0,0\}$ is $$\label{APR12_22} {{C_{{a}}}^\star} = \frac{1}{(4\pi {t_{}^\star})^{3/2}}{\exp\left(\frac{-{|{{\vec{r}_{}}^\star}|}^2}{4 {t_{}^\star}} -{{k_{}}^\star}{t_{}^\star}\right)},$$ where ${|{{\vec{r}_{}}^\star}|}^2 = ({x^\star}+ {x^\star}_n - {{v^\star_{\scriptscriptstyle\parallel}}}{t_{}^\star})^2 + ({y^\star}- {{v^\star_{\perp}}}{t_{}^\star})^2 + ({z^\star})^2$ is the square of the time-varying *effective* distance between the noise source and the point $\{{x^\star},{y^\star},{z^\star}\}$. Unlike an intended transmitter, the noise source is emitting molecules as described by the general random process ${{N_{{{a}_{gen}}}}^\star\left({t_{}^\star}\right)}$. We are already averaging over the randomness of the diffusion channel (i.e., we have the expected channel impulse response), so we only consider the time-varying mean of the noise source emission process, i.e., ${\overline{{N_{{{a}_{gen}}}}^\star}\left({t_{}^\star}\right)}$. Thus, the expected impact of the noise source is found by multiplying (\[APR12\_22\]) by ${\overline{{N_{{{a}_{gen}}}}^\star}\left({t_{}^\star}\right)}$, integrating over ${{V_{obs}}^\star}$, and then integrating over all time up to ${t_{}^\star}$, i.e., $$\label{APR12_42_DMLS} {\overline{{{N_{{a}}}}_{n}^\star}\left({t_{}^\star}\right)} = \int\limits_{-\infty}^{{t_{}^\star}}\! \int\limits_0^{{r_{obs}^\star}} \int\limits_{0}^{2\pi} \int\limits_{0}^{\pi} {{r_{i}^\star}}^2{\overline{{N_{{{a}_{gen}}}}^\star}\left({\tau}\right)} {{C_{{a}}}^\star}\sin\theta d\theta d\phi d{r_{i}^\star}d{\tau},$$ where ${r_{i}^\star}$ is the magnitude of the distance from the origin to the point $\{{x^\star},{y^\star},{z^\star}\}$ within ${{V_{obs}}^\star}$. To solve (\[APR12\_42\_DMLS\]), we must also convert ${|{{\vec{r}_{}}^\star}|}^2$ from cartesian to spherical coordinates, which can be shown to be $$\begin{aligned} {|{{\vec{r}_{}}^\star}|}^2 = &\; {{r_{i}^\star}}^2 + {{x^\star}_0}^2 - 2{t_{}^\star}{x^\star}_0{{v^\star_{\scriptscriptstyle\parallel}}}+ 2{x^\star}_0{r_{i}^\star}\cos\phi\sin\theta + {{t_{}^\star}}^2\left({{{v^\star_{\scriptscriptstyle\parallel}}}}^2 + {{{v^\star_{\perp}}}}^2\right) \nonumber \\ & -2{t_{}^\star}{r_{i}^\star} \left({{v^\star_{\scriptscriptstyle\parallel}}}\cos\phi\sin\theta + {{v^\star_{\perp}}}\sin\phi\sin\theta\right), \label{EQ13_07_25}\end{aligned}$$ $$\begin{aligned} {|{{\vec{r}_{}}^\star}|}^2 = &\, {{r_{i}^\star}}^2 + {{x^\star}_0}^2 - 2{t_{}^\star}{x^\star}_0{{v^\star_{\scriptscriptstyle\parallel}}}+ 2{x^\star}_0{r_{i}^\star}\cos\phi\sin\theta \nonumber \\ & -2{t_{}^\star}{r_{i}^\star} \left({{v^\star_{\scriptscriptstyle\parallel}}}\cos\phi\sin\theta + {{v^\star_{\perp}}}\sin\phi\sin\theta\right) \nonumber \\ & + {{t_{}^\star}}^2\left({{{v^\star_{\scriptscriptstyle\parallel}}}}^2 + {{{v^\star_{\perp}}}}^2\right), \label{EQ13_07_25}\end{aligned}$$ where $\phi = \tan^{-1}\left({y^\star}/{x^\star}\right)$ and $\theta = \cos^{-1}\left({z^\star}/{r_{i}^\star}\right)$. Generally, (\[APR12\_42\_DMLS\]) does not have a known closed-form solution, even if we omit the integral over time. In the following subsection, we present a series of cases for which (\[APR12\_42\_DMLS\]) can be more easily solved (numerically or in closed form), for either arbitrary ${t_{}^\star}$ or as ${t_{}^\star} \to \infty$. In the asymptotic case, we write ${\overline{{{N_{{a}}}}_{n}^\star}\left({t_{}^\star}\right)}\big|_{{t_{}^\star} \to \infty} = {\overline{{{N_{{a}}}}_{n}^\star}}$ for compactness. The asymptotic case will also be useful to approximate multiuser interference and old intersymbol interference in Sections \[sec\_mui\] and \[sec\_isi\], respectively. Tractable Noise Analysis ------------------------ For tractability, we will assume throughout the remainder of this section that the *expected* noise source emission in (\[APR12\_42\_DMLS\]) can be described as a step function, i.e., ${\overline{{N_{{{a}_{gen}}}}^\star}\left({t_{}^\star}\right)} = {L}^2{{N_{{{A}_{gen}}}}}/\left({D_{{A}}}{N_{{A}_{REF}}}\right), {t_{}^\star} \ge 0$. Furthermore, we choose ${N_{{A}_{REF}}} = {L}^2{{N_{{{A}_{gen}}}}}/{D_{{A}}}$, so that ${\overline{{N_{{{a}_{gen}}}}^\star}\left({t_{}^\star}\right)} = 1, {t_{}^\star} \ge 0$ (the case where the noise source also “shuts off” at some future time, for example when a ruptured vesicle is depleted, is an interesting one that we leave for future work). We note that the emission of molecules by the noise source could then be deterministically uniform, such that the emission process ${{N_{{{a}_{gen}}}}^\star\left({t_{}^\star}\right)}$ is in fact ${\overline{{N_{{{a}_{gen}}}}^\star}\left({t_{}^\star}\right)}$ (e.g., via leakage from a vesicle that ruptured at ${t_{}^\star} = 0$), or it could be random with independent emission times (e.g., the stochastic output of a chemical reaction mechanism with a constant expected generation rate that was triggered to begin at ${t_{}^\star} = 0$). Strictly speaking, in the latter case the *expected* emission rate is $1$. This will not affect any of the following analysis because we are deriving the *expected* impact ${\overline{{{N_{{a}}}}_{n}^\star}\left({t_{}^\star}\right)}$. We emphasize that our analysis focuses on the expected impact and not the complete probability density function ([PDF]{}) of the impact. A case-by-case analysis of the noise release statistics would be needed to determine the time-varying [PDF]{} of the impact at the receiver. The solutions to (\[APR12\_42\_DMLS\]) that we present in the remainder of this section follow one of two general strategies. Both strategies reduce (\[APR12\_42\_DMLS\]) to a single integral, which can be solved numerically or reduced to closed form if additional assumptions are made. The first strategy is the uniform concentration assumption ([UCA]{}), where we assume that the expected concentration of ${A}$ molecules due to the noise source is uniform and equal to that expected at the center of the receiver (i.e., at the origin). This assumption is accurate if the noise source is sufficiently far from the receiver, such that the expected concentration of ${A}$ molecules will not vary significantly throughout the receiver. We studied the accuracy of this assumption for a transmitter using impulsive binary-coded modulation in the presence of steady uniform flow in [@RefWorks:752]. Here, applying the [UCA]{} means that we do not need to integrate over ${{V_{obs}}^\star}$ and (\[APR12\_42\_DMLS\]) becomes $$\label{EQ13_07_27_int} {\overline{{{N_{{a}}}}_{n}^\star}\left({t_{}^\star}\right)} = {{V_{obs}}^\star}\int\limits_{0}^{{t_{}^\star}} {{{{C_{{a}}}^\star}}({r_{eff}^\star},{\tau})} d{\tau},$$ where ${{r_{eff}^\star}}^2 = ({x^\star}_n - {{v^\star_{\scriptscriptstyle\parallel}}}{\tau})^2 + ({{v^\star_{\perp}}}{\tau})^2$ is the square of the *effective* distance from the noise source to the receiver, and the expected concentration at the receiver is $$\label{EQ13_07_26} {{{{C_{{a}}}^\star}}({r_{eff}^\star},{t_{}^\star})} = \frac{1}{(4\pi{t_{}^\star})^{3/2}} {\exp\left(-\frac{{{r_{eff}^\star}}^2} {4 {t_{}^\star}} - {{k_{}}^\star}{t_{}^\star}\right)}.$$ The second strategy for solving (\[APR12\_42\_DMLS\]) does not apply the [UCA]{}, so we include the integration over ${{V_{obs}}^\star}$. We considered that integration (but without the integration over time) for no molecule degradation in [@RefWorks:752], and for general advection we could only evaluate the integral over ${r_{i}^\star}$. However, a closed-form solution was possible if ${{v^\star_{\perp}}}= 0$, which we derived in [@RefWorks:752] from the no-flow case presented in [@RefWorks:706 Th. 2] via a change of variables. Including the integration over time and the impact of molecule degradation, where ${{N_{{{a}_{gen}}}}^\star\left({t_{}^\star}\right)} = 1, {t_{}^\star} \ge 0$, [@RefWorks:752 Eq. (17)] becomes $$\begin{aligned} {\overline{{{N_{{a}}}}_{n}^\star}\left({t_{}^\star}\right)} = &\; \int\limits_0^{{t_{}^\star}} \Bigg\{ \frac{1}{2}\left[{\operatorname{erf}\left(\frac{{r_{obs}^\star}\!- {{\vec{r}_{eff}}^\star}}{2{{\tau}}^\frac{1}{2}}\right)} + {\operatorname{erf}\left(\frac{{r_{obs}^\star}\!+{{\vec{r}_{eff}}^\star}}{2{{\tau}}^\frac{1}{2}}\right)}\right] \nonumber \\ & + \frac{1}{{{\vec{r}_{eff}}^\star}}\sqrt{\frac{{\tau}}{\pi}} \Bigg[{\exp\left(-\frac{({{\vec{r}_{eff}}^\star}+{r_{obs}^\star})^2}{4{\tau}}\right)} - {\exp\left(-\frac{({{\vec{r}_{eff}}^\star}-{r_{obs}^\star})^2}{4{\tau}}\right)}\Bigg] \Bigg\}{\exp\left(- {{k_{}}^\star}{\tau}\right)}d{\tau}, \label{EQ13_05_29}\end{aligned}$$ $$\begin{aligned} {\overline{{{N_{{a}}}}_{n}^\star}\left({t_{}^\star}\right)} = &\; \int\limits_0^{{t_{}^\star}} \Bigg\{ \frac{1}{2}\left[{\operatorname{erf}\left(\frac{{r_{obs}^\star}\!- {{\vec{r}_{eff}}^\star}}{2{{\tau}}^\frac{1}{2}}\right)} \!+ {\operatorname{erf}\left(\frac{{r_{obs}^\star}\!+{{\vec{r}_{eff}}^\star}}{2{{\tau}}^\frac{1}{2}}\right)}\!\right] \nonumber \\ & + \frac{1}{{{\vec{r}_{eff}}^\star}}\sqrt{\frac{{\tau}}{\pi}} \Bigg[ {\exp\left(-\frac{({{\vec{r}_{eff}}^\star}+{r_{obs}^\star})^2}{4{\tau}}\right)}\nonumber \\ & - {\exp\left(-\frac{({{\vec{r}_{eff}}^\star}-{r_{obs}^\star})^2}{4{\tau}}\right)}\Bigg] \Bigg\}{\exp\left(- {{k_{}}^\star}{\tau}\right)}d{\tau}, \label{EQ13_05_29}\end{aligned}$$ where ${{\vec{r}_{eff}}^\star} = -\left({x^\star}_n - {{v^\star_{\scriptscriptstyle\parallel}}}{\tau}\right)$ is the effective distance along the ${x^\star}$-axis from the noise source to the center of the receiver, and the error function is [@RefWorks:402 Eq. 8.250.1] $$\label{APR12_32} {\operatorname{erf}\left(a\right)} = \frac{2}{\sqrt{\pi}}\int_0^a {\exp\left(-b^2\right)} db.$$ Eq. (\[EQ13\_05\_29\]) can be evaluated numerically but, unlike (\[EQ13\_07\_27\_int\]), is valid for *any* ${x^\star}_n$ (although special consideration must be made if ${x^\star}_n = 0$, i.e., the “worst-case” location for the noise source, and we consider that case at the end of this subsection). The two strategies that we have presented reduce (\[APR12\_42\_DMLS\]) to a single integral (either (\[EQ13\_07\_27\_int\]) or (\[EQ13\_05\_29\])), thereby facilitating numerical evaluation. In the remainder of this subsection, we make additional assumptions that enable us to solve (\[APR12\_42\_DMLS\]) in closed form. ### Asymptotic Solutions In the asymptotic case, i.e., as ${t_{}^\star} \to \infty$, it is straightforward to show that (\[EQ13\_07\_27\_int\]) becomes $${\overline{{{N_{{a}}}}_{n}^\star}}= \frac{{{V_{obs}}^\star}}{4\pi{x^\star}_n} {\exp\left(\frac{{x^\star}_n{{v^\star_{\scriptscriptstyle\parallel}}}}{2} - \frac{{x^\star}_n}{2}\sqrt{{{{v^\star_{\scriptscriptstyle\parallel}}}}^2 + {{{v^\star_{\perp}}}}^2 +4{{k_{}}^\star}}\right)}, \label{EQ13_07_27}$$ where we apply [@RefWorks:402 Eq. 3.472.5] $$\int\limits_{0}^{\infty} \frac{1}{a^{3/2}}{\exp\left(-ba-\frac{c}{a}\right)}da = \sqrt{\frac{\pi}{c}}{\exp\left(-2\sqrt{bc}\right)}, \label{EQ13_05_20}$$ and recall that ${x^\star}_n$ is positive. \[remark\_far\_flow\] From (\[EQ13\_07\_27\]) it can be shown that, if there is no flow in the ${y}$-direction and no molecule degradation (i.e., ${{v^\star_{\perp}}}= 0$ and ${{k_{}}^\star} = 0$), then any *positive* flow along the ${x}$-direction (i.e., ${{v^\star_{\scriptscriptstyle\parallel}}}> 0$) will not change the asymptotic impact of the noise source. We had expected that this flow would increase the asymptotic impact in comparison to the no-flow case, so this is a somewhat surprising result. An asymptotic closed-form solution to (\[EQ13\_05\_29\]) is possible if we impose ${{v^\star_{\scriptscriptstyle\parallel}}}= 0$, such that we are restricted to the no-flow case. If the noise source is also close to the receiver, then this is another “worst-case” scenario because there is no advection to carry the noise molecules away. The result is presented in the following theorem: \[theorem\_no\_flow\] The expected asymptotic impact (i.e., as ${t_{}^\star} \to \infty$) of a noise source in the absence of flow, whose expected output is ${\overline{{N_{{{a}_{gen}}}}^\star}\left({t_{}^\star}\right)} = 1, {t_{}^\star} \ge 0$, is given by $$\begin{aligned} {\overline{{{N_{{a}}}}_{n}^\star}}= \frac{1}{2{{k_{}}^\star}}\! \left[\beta + 1 + \frac{1}{{x^\star}_n} {\exp\left(-{r_{sum}^\star}{{{k_{}}^\star}}^{\frac{1}{2}}\right)}\!\! \left({{{k_{}}^\star}}^{-\frac{1}{2}} + {r_{obs}^\star}\right)\! -\frac{1}{{x^\star}_n} {\exp\left(-|{r_{dif}^\star}|{{{k_{}}^\star}}^{\frac{1}{2}}\right)}\!\! \left({{{k_{}}^\star}}^{-\frac{1}{2}} + \beta{r_{obs}^\star}\right) \!\right]\!\!, \label{EQ13_05_53}\end{aligned}$$ $$\begin{aligned} {\overline{{{N_{{a}}}}_{n}^\star}}= &\; \frac{1}{2{{k_{}}^\star}}\bigg[1-\frac{1}{{x^\star}_n} {\exp\left(-|{r_{dif}^\star}|{{{k_{}}^\star}}^{\frac{1}{2}}\right)}\!\! \left({{{k_{}}^\star}}^{-\frac{1}{2}} + \beta{r_{obs}^\star}\right) \nonumber \\ &+\frac{1}{{x^\star}} {\exp\left(-{r_{sum}^\star}{{{k_{}}^\star}}^{\frac{1}{2}}\right)}\!\! \left({{{k_{}}^\star}}^{-\frac{1}{2}} + {r_{obs}^\star}\right)+\beta \bigg], \label{EQ13_05_53}\end{aligned}$$ where ${r_{dif}^\star} = {r_{obs}^\star}-{x^\star}_n$, ${r_{sum}^\star} = {r_{obs}^\star}+{x^\star}_n$, $\beta = {\operatorname{sgn}}({r_{obs}^\star}-{x^\star}_n)$, and ${\operatorname{sgn}}(\cdot)$ is the sign function. Please refer to Appendix \[app\_no\_flow\]. ### Absence of Flow and Molecule Degradation Time-varying solutions to both (\[EQ13\_07\_27\_int\]) and (\[EQ13\_05\_29\]) are only possible in the absence of flow and molecule degradation, i.e., if ${{v^\star_{\scriptscriptstyle\parallel}}}= {{v^\star_{\perp}}}= 0$ and ${{k_{}}^\star} = 0$. If we are using the [UCA]{}, then (\[EQ13\_07\_27\_int\]) can be combined with [@RefWorks:586 Eq. (3.5b)] and we can write $${\overline{{{N_{{a}}}}_{n}^\star}\left({t_{}^\star}\right)} = \frac{{{V_{obs}}^\star}}{4\pi{x^\star}_n} \left(1-{\operatorname{erf}\left(\frac{{x^\star}_n}{2\sqrt{{t_{}^\star}}}\right)}\right), \label{EQ13_05_18}$$ \[remark\_far\_no\_flow\] We see from (\[APR12\_32\]) that ${\operatorname{erf}\left(a\right)} \to 0$ as $a \to 0$, and that ${\operatorname{erf}\left(a\right)} \approx 0.056$ when $a = 0.05$. If the reference distance is chosen to be the distance of the noise source from the receiver, i.e., ${L}= {x}_n$, and if there is no advection or molecule degradation, then from (\[EQ13\_05\_18\]) we must wait until ${t_{}^\star} > 100$ before the impact is expected to be at least $95\,\%$ of the asymptotic impact. The time-varying solution to (\[EQ13\_05\_29\]) is presented in the following theorem: \[theorem\_no\_flow\_degrad\] The expected time-varying impact of a noise source, whose expected output is ${\overline{{N_{{{a}_{gen}}}}^\star}\left({t_{}^\star}\right)} = 1, {t_{}^\star} \ge 0$, is given in the absence of flow and molecule degradation by $$\begin{aligned} {\overline{{{N_{{a}}}}_{n}^\star}\left({t_{}^\star}\right)} = &\; {\operatorname{erf}\left(\frac{{r_{dif}^\star}}{2\sqrt{{t_{}^\star}}}\right)}\!\! \left[\frac{{{r_{dif}^\star}}^2}{4} + \frac{{t_{}^\star}}{2} + \frac{{{r_{dif}^\star}}^3}{6{x^\star}_n}\right] + {\operatorname{erf}\left(\frac{{r_{sum}^\star}}{2\sqrt{{t_{}^\star}}}\right)}\!\! \left[\frac{{{r_{sum}^\star}}^2}{4} + \frac{{t_{}^\star}}{2} - \frac{{{r_{sum}^\star}}^3}{6{x^\star}_n}\right] \nonumber \\ & - \frac{\beta{{r_{dif}^\star}}^2}{4} - \frac{{{r_{sum}^\star}}^2}{4} + \frac{{{r_{sum}^\star}}^3}{6{x^\star}_n} - \frac{|{r_{dif}^\star}|^3}{6{x^\star}_n} + \sqrt{\frac{{t_{}^\star}}{\pi}} {\exp\left(-\frac{{{r_{dif}^\star}}^2}{4{t_{}^\star}}\right)}\!\! \left[\frac{{r_{dif}^\star}}{2} - \frac{2{t_{}^\star}}{3{x^\star}_n} + \frac{{{r_{dif}^\star}}^2}{3{x^\star}_n}\right] \nonumber \\ & + \sqrt{\frac{{t_{}^\star}}{\pi}} {\exp\left(-\frac{{{r_{sum}^\star}}^2}{4{t_{}^\star}}\right)}\!\! \left[\frac{{r_{sum}^\star}}{2} + \frac{2{t_{}^\star}}{3{x^\star}_n} - \frac{{{r_{sum}^\star}}^2}{3{x^\star}_n}\right]. \label{EQ13_05_38}\end{aligned}$$ $$\begin{aligned} {\overline{{{N_{{a}}}}_{n}^\star}\left({t_{}^\star}\right)} = &\; {\operatorname{erf}\left(\frac{{r_{dif}^\star}}{2\sqrt{{t_{}^\star}}}\right)}\!\! \left[\frac{{{r_{dif}^\star}}^2}{4} + \frac{{t_{}^\star}}{2} + \frac{{{r_{dif}^\star}}^3}{6{x^\star}_n}\right] \nonumber \\ & + {\operatorname{erf}\left(\frac{{r_{sum}^\star}}{2\sqrt{{t_{}^\star}}}\right)}\!\! \left[\frac{{{r_{sum}^\star}}^2}{4} + \frac{{t_{}^\star}}{2} - \frac{{{r_{sum}^\star}}^3}{6{x^\star}_n}\right] \nonumber \\ & + \sqrt{\frac{{t_{}^\star}}{\pi}} {\exp\left(-\frac{{{r_{dif}^\star}}^2}{4{t_{}^\star}}\right)}\!\! \left[\frac{{r_{dif}^\star}}{2} - \frac{2{t_{}^\star}}{3{x^\star}_n} + \frac{{{r_{dif}^\star}}^2}{3{x^\star}_n}\right] \nonumber \\ & + \sqrt{\frac{{t_{}^\star}}{\pi}} {\exp\left(-\frac{{{r_{sum}^\star}}^2}{4{t_{}^\star}}\right)}\!\! \left[\frac{{r_{sum}^\star}}{2} + \frac{2{t_{}^\star}}{3{x^\star}_n} - \frac{{{r_{sum}^\star}}^2}{3{x^\star}_n}\right] \nonumber \\ & - \frac{\beta{{r_{dif}^\star}}^2}{4} - \frac{{{r_{sum}^\star}}^2}{4} + \frac{{{r_{sum}^\star}}^3}{6{x^\star}_n} - \frac{|{r_{dif}^\star}|^3}{6{x^\star}_n}. \label{EQ13_05_38}\end{aligned}$$ Please refer to Appendix \[app\_no\_flow\_degrad\]. Although (\[EQ13\_05\_38\]) is verbose, it can be evaluated for *any* non-zero values of ${x^\star}_n$ and ${t_{}^\star}$. We consider the case ${x^\star}_n=0$ at the end of this subsection. Here, we note that the asymptotic impact of a noise source, i.e., as ${t_{}^\star} \to \infty$, can be evaluated from (\[EQ13\_05\_38\]) using the properties of limits and l’Hôpital’s rule as $${\overline{{{N_{{a}}}}_{n}^\star}}= \frac{{{r_{sum}^\star}}^3}{6{x^\star}_n} - \frac{|{r_{dif}^\star}|^3}{6{x^\star}_n} - \frac{{{r_{sum}^\star}}^2}{4} - \frac{\beta{{r_{dif}^\star}}^2}{4}. \label{EQ13_05_60}$$ \[remark\_no\_flow\] Eq. (\[EQ13\_05\_60\]) simplifies to a single term if the noise source is outside of the receiver, i.e., if ${r_{dif}^\star} = {r_{obs}^\star}-{x^\star}_n < 0$. It can then be shown that ${\overline{{{N_{{a}}}}_{n}^\star}}= {{r_{obs}^\star}}^3/(3{x^\star}_n)$, which is equivalent to (\[EQ13\_07\_27\]) with a spherical receiver in the absence of advection and molecule degradation, i.e., ${{v^\star_{\scriptscriptstyle\parallel}}}= {{v^\star_{\perp}}}= {{k_{}}^\star} = 0$, even though (\[EQ13\_07\_27\]) was derived for a noise source that is *far* from the receiver. Thus, in the absence of advection and molecule degradation, the expected impact of a noise source *anywhere* outside the receiver increases with the inverse of the distance to the receiver. ### Worst-Case Noise Source Location Finally, we consider the special case where the noise source is located *at* the receiver, i.e., ${x^\star}_n=0$. Clearly, the [UCA]{} should not apply in this case, so we only consider the evaluation of (\[EQ13\_05\_29\]). Generally, we need to apply l’Hôpital’s rule to account for ${x^\star}_n=0$, and here we do so for three cases. First, if evaluating (\[EQ13\_05\_29\]) directly and ${{v^\star_{\scriptscriptstyle\parallel}}}= 0$, then l’Hôpital’s rule must be used to re-write the second term inside the curly braces in (\[EQ13\_05\_29\]) (i.e., the term with the two exponentials, including the scaling by $\sqrt{{\tau}/\pi}/{{\vec{r}_{eff}}^\star}$) as $$-\frac{{r_{obs}^\star}}{(\pi{\tau})^\frac{1}{2}}{\exp\left(-\frac{{{r_{obs}^\star}}^2}{4{\tau}}\right)}. \label{EQ13_05_29_x0}$$ Second, if evaluating (\[EQ13\_05\_53\]), which applies asymptotically in the absence of flow, then we can apply l’Hôpital’s rule in the limit of ${x^\star}_n \to 0$ and write (\[EQ13\_05\_53\]) as $$\lim_{{x^\star}_n \to 0} {\overline{{{N_{{a}}}}_{n}^\star}}= \frac{1}{{{k_{}}^\star}} - {\exp\left(-{r_{obs}^\star}\sqrt{{{k_{}}^\star}}\right)}\left(\frac{1}{{{k_{}}^\star}} + \frac{{r_{obs}^\star}}{\sqrt{{{k_{}}^\star}}}\right). \label{EQ13_05_54}$$ \[remark\_degradation\] From (\[EQ13\_05\_53\]) and (\[EQ13\_05\_54\]) we see that any increase in ${{k_{}}^\star}$ will result in a decrease in the expected number of noise molecules observed, even if the noise source is located at the receiver (i.e., ${x^\star}_n = 0$). Third, the time-varying impact of the “worst-case” noise source in the absence of flow and molecule degradation can be found using repeated applications of l’Hôpital’s rule to (\[EQ13\_05\_38\]) as $$\lim_{{x^\star}_n \to 0} {\overline{{{N_{{a}}}}_{n}^\star}\left({t_{}^\star}\right)} = {\operatorname{erf}\left(\frac{{r_{obs}^\star}}{2\sqrt{{t_{}^\star}}}\right)}\left[{t_{}^\star} - \frac{{{r_{obs}^\star}}^2}{2}\right] - {r_{obs}^\star}\sqrt{\frac{{t_{}^\star}}{\pi}}{\exp\left(-\frac{{{r_{obs}^\star}}^2}{4{t_{}^\star}}\right)} + \frac{{{r_{obs}^\star}}^2}{2}. \label{EQ13_05_39}$$ $$\begin{aligned} \lim_{{x^\star}_n \to 0} {\overline{{{N_{{a}}}}_{n}^\star}\left({t_{}^\star}\right)} = &\; {\operatorname{erf}\left(\frac{{r_{obs}^\star}}{2\sqrt{{t_{}^\star}}}\right)}\left[{t_{}^\star} - \frac{{{r_{obs}^\star}}^2}{2}\right] \nonumber \\ &- {r_{obs}^\star}\sqrt{\frac{{t_{}^\star}}{\pi}}{\exp\left(-\frac{{{r_{obs}^\star}}^2}{4{t_{}^\star}}\right)} + \frac{{{r_{obs}^\star}}^2}{2}. \label{EQ13_05_39}\end{aligned}$$ This subsection considered a number of solutions to (\[APR12\_42\_DMLS\]), where the expected molecule emission is described as a step function. In Table \[table\_noise\], we summarize precisely which conditions and assumptions apply to each equation. We will see the accuracy of these equations in comparison with simulated noise sources in Section \[sec\_num\]. In practice, these equations can enable us to more accurately assess the effect of noise sources on the bit error probability of the intended communication link (as we did in [@RefWorks:747; @RefWorks:752], where we only assumed that the expected impact of noise sources was known). In the remainder of this paper, we focus on using the noise analysis to approximate some or all of the signal observed by transmitters that release impulses of molecules. Eq. Closed Form? Asymptotic? ${x^\star}_n$ ${{k_{}}^\star}$ ${{v^\star_{\perp}}}$ ${{v^\star_{\scriptscriptstyle\parallel}}}$ ------------------------- -------------- ------------- --------------- ------------------ ----------------------- --------------------------------------------- (\[EQ13\_07\_27\_int\]) No No Far Any Any Any (\[EQ13\_05\_29\]) No No Any Any Any 0 (\[EQ13\_07\_27\]) Yes Yes Far Any Any Any (\[EQ13\_05\_53\]) Yes Yes $\neq 0$ Any 0 0 (\[EQ13\_05\_18\]) Yes No Far 0 0 0 (\[EQ13\_05\_38\]) Yes No $\neq 0$ 0 0 0 (\[EQ13\_05\_60\]) Yes Yes $\neq 0$ 0 0 0 (\[EQ13\_05\_54\]) Yes Yes $0$ Any 0 0 (\[EQ13\_05\_39\]) Yes No $0$ 0 0 0 : Summary of the equations for the impact of an external noise source and the conditions under which they can be used. By ${x^\star}_n =$ “Far”, we mean that the [UCA]{} is applied. \[table\_noise\] Multiuser Interference {#sec_mui} ====================== In this section, we consider the impact of transmitters that are using the same modulation scheme as the transmitter that is linked to the receiver of interest but are sending independent information. Thus, the ${A}$ molecules emitted by these unintended transmitters are effectively noise. We begin by presenting the complete model of the observations made at the receiver due to any number of transmitters (independent of whether the transmitters are linked to the receiver). This detailed model is the most comprehensive, so it enables the most accurate calculation of the bit error probability, but it requires knowledge of all transmitter sequences. Then, we apply our results in Section \[sec\_noise\] to simplify the analysis of an interfering transmitter. Complete Multiuser Model ------------------------ Consider from (\[EQ13\_05\_28\_obs\]) that all ${U}$ sources of ${A}$ molecules are transmitters with the same modulation scheme. Transmitter $u$ has independent binary sequence ${\mathbf{W_{u}}} = \{{W_{u}\left[1\right]},{W_{u}\left[2\right]},\dots\}$ to send to its intended receiver, where ${W_{u}\left[j\right]}$ is the $j$th information bit and $\Pr({W_{u}\left[j\right]} = 1) = {P_1}$. The only receiver that we are concerned with is the one at the origin. The transmitters do not coordinate their transmissions so they all transmit simultaneously, but for clarity of exposition we assume that the transmitters are initially synchronized and begin transmitting at ${t_{}^\star} = 0$. It is also straightforward to add an initial timing offset to each transmitter, but we omit that extension in this paper in order to focus on asymptotic multiuser interference. Transmitter $u$ has bit interval ${T_{int,u}}$ seconds and it releases ${{N_{{{A}_{EM,u}}}}}$ ${A}$ molecules at the start of the interval to send a binary $1$ and no molecules to send a binary $0$. We model instantaneous molecule releases as approximations of releases that are much shorter than the bit interval; we do not expect that instantaneous releases are practical. Furthermore, we define the dimensionless bit interval ${{T_{int,u}}^\star}$ and dimensionless number of emitted molecules ${{{N_{{{{a}}_{EM,u}}}}^\star}}$, where we scale the dimensional variables by ${D_{{A}}}/{L}^2$ and $1/{N_{{A}_{REF}}}$, respectively. We note that this binary modulation scheme can be easily extended to any pulse amplitude modulation scheme, where the $u$th transmitter encodes multiple bits in the number of molecules released at one time. The channel impulse response is the same as in the general noise source case, i.e., (\[APR12\_22\]), where we adjust the frame of reference for each transmitter so that it lies along the ${x^\star}$-axis. From (\[APR12\_42\_DMLS\]), we immediately have the expected number of molecules observed due to an emission by transmitter $u$ at time ${t_{}^\star} = 0$, ${\overline{{{N_{{a}}}}_{tx,u}^\star}\left({t_{}^\star}\right)}$, written as $$\label{EQ13_07_27_int_gen} {\overline{{{N_{{a}}}}_{tx,u}^\star}\left({t_{}^\star}\right)} = \int\limits_0^{{r_{obs}^\star}} \int\limits_{0}^{2\pi} \int\limits_{0}^{\pi} {{r_{i}^\star}}^2{{{N_{{{{a}}_{EM,u}}}}^\star}} {{C_{{a}}}^\star}\sin\theta d\theta d\phi d{r_{i}^\star},$$ which we have shown in [@RefWorks:662] can be accurately approximated as a time-varying Poisson random variable when in dimensional form. At any moment, the distribution of molecules observed at our intended receiver is the sum of molecules expected from all emissions made by all transmitters, as given in (\[EQ13\_05\_28\_obs\]). This is (dimensionally) a Poisson random variable (because it is a sum of independent Poisson random variables; see [@RefWorks:725 Ch. 5.2]) that has (dimensionless) mean $$\label{EQ13_08_02} {\overline{{{N_{{a}}}}_{obs}^\star}\left({t_{}^\star}\right)} = \sum_{u=1}^{{U}} \sum_{j=1}^{{\lfloor\frac{{t_{}^\star}}{{{T_{int,u}}^\star}}+1\rfloor}}\!\!\!\! {W_{u}\left[j\right]}{\overline{{{N_{{a}}}}_{tx,u}^\star}\left({t_{}^\star}-(j-1){{T_{int,u}}^\star}\right)}\!.$$ Asymptotic Interference ----------------------- Precise analysis of the performance of the receiver’s detector can be made using (\[EQ13\_08\_02\]), but we must have knowledge of every transmitter sequence ${\mathbf{W_{u}}}$. We propose simplifying the analysis by applying our results in Section \[sec\_noise\]. For widest applicability, i.e., to include molecule degradation and flow in any direction, we assume that interfering transmitters are sufficiently far away to apply the uniform concentration assumption (this makes sense; an interferer that is very close to the receiver would likely result in an error probability that is too high for communication with the intended transmitter to be practical). The corresponding closed-form analysis is asymptotic in time, but this is acceptable because we can assume that interferers were transmitting for a long time before the start of our intended transmission (we will see in Section \[sec\_num\] that this is an easy assumption to satisfy). The remainder of this section can also be easily extended to the other special cases in Section \[sec\_noise\]. Consider the asymptotic impact of a single interfering transmitter. The emissions of the $u$th transmitter must be approximated as a continuous function so that we can apply the results from our noise analysis. The effective emission rate is ${P_1}{{{N_{{{{a}}_{EM,u}}}}^\star}}$ molecules every ${{T_{int,u}}^\star}$ dimensionless time units. If we choose ${N_{{A}_{REF}}} = {L}^2{P_1}{{N_{{{A}_{EM,u}}}}}/({T_{int,u}}{D_{{A}}})$, then the emission function of the $u$th transmitter can be approximated as ${\overline{{N_{{{a}_{gen}}}}^\star}\left({t_{}^\star}\right)} = 1, {t_{}^\star} \ge 0$. From (\[EQ13\_07\_27\]), we immediately have the expected asymptotic impact of the noise source, ${\overline{{{N_{{a}}}}_{u}^\star}}$, written as $${\overline{{{N_{{a}}}}_{u}^\star}} = \frac{{{V_{obs}}^\star}}{4\pi{x^\star}_u} {\exp\left(\frac{{x^\star}_u{{v^\star_{\scriptscriptstyle\parallel}}}}{2} - \frac{{x^\star}_u}{2}\sqrt{{{{v^\star_{\scriptscriptstyle\parallel}}}}^2 + {{{v^\star_{\perp}}}}^2 +4{{k_{}}^\star}}\right)}, \label{EQ13_08_05}$$ where we recall that, in general, we have adjusted the reference coordinate frame so that the $u$th transmitter lies on the negative ${x^\star}$-axis. The time-varying impact can be found via numerical integration of (\[EQ13\_07\_27\_int\]), or, if ${{v^\star_{\scriptscriptstyle\parallel}}}= {{v^\star_{\perp}}}= 0$ and ${{k_{}}^\star} = 0$, i.e., if there is no advection or molecule degradation, via (\[EQ13\_05\_18\]). The complete asymptotic multiuser interference is given by adding (\[EQ13\_08\_05\]) for all ${U}-1$ interfering transmitters. We note that (\[EQ13\_08\_05\]) is a constant approximation of what is in practice a signal that is expected to oscillate over time. The channel impulse response given by (\[APR12\_22\]) has a definitive peak and tail. The interference can be envisioned as the most recent peak followed by all of the tails of prior transmissions. Even asymptotically, the expected impact at a given instant will depend on the time relative to the interferer’s transmission intervals. So, over time, (\[EQ13\_08\_05\]) will both overestimate and underestimate the impact of the interferer. However, we expect that, on average, (\[EQ13\_08\_05\]) will tend to overestimate the impact more often. This is because the approximation of molecule emission as a continuous function effectively makes the release of molecules later than they actually are by “spreading” emissions over the entire bit interval instead of releasing all of them at the start of the bit interval. We will visualize the accuracy of (\[EQ13\_08\_05\]) more clearly in Section \[sec\_num\]. Asymptotic ISI {#sec_isi} ============== In this section, we focus on characterizing the signal observed at the receiver due to the intended transmitter *only*. We seek a method to model some of the [ISI]{} asymptotically based on the previous analysis in this paper. Specifically, we model ${F}$ prior bits *explicitly* (and not as a signal from a continuously-emitting source), and the impact of all earlier bits is approximated asymptotically as a continuously-emitting source. The choice of ${F}$ enables a tradeoff between accuracy and computational efficiency. We describe the application of our model for asymptotic old [ISI]{} to simplify the evaluation of the expected bit error probability of weighted sum detectors, which in general requires finding the expected probability of error of all possible transmitter sequences and taking an average. Other applications of our model for asymptotic [ISI]{} are simplifying the implementation of the maximum likelihood detector and in the design of a weighted sum detector with adaptive weights. Adaptive weighting is physically realizable in biological systems; neurons sum inputs from synapses with dynamic weights (see [@RefWorks:587 Ch. 12]), but we leave the design of adaptive weighted sum detectors for future work. Decomposition of Received Signal -------------------------------- We now decompose the signal from the intended transmitter, i.e., ${{{N_{{a}}}}_{1}^\star\left({t_{}^\star}\right)}$ in (\[EQ13\_05\_28\_obs\]). To emphasize that source $1$ is the intended transmitter, we re-write its signal as ${{{N_{{a}}}}_{1}^\star\left({t_{}^\star}\right)} = {{{N_{{a}}}}_{tx}^\star\left({t_{}^\star}\right)}$ and also drop the $1$ subscript from its transmission parameters. If we apply the uniform concentration assumption for clarity of exposition, then the expected number of observed molecules due to the intended transmitter, ${\overline{{{N_{{a}}}}_{tx}^\star}\left({t_{}^\star}\right)}$, given the transmitter sequence ${\mathbf{W}}$, is $$\label{EQ13_04_02_DMLS} {\overline{{{N_{{a}}}}_{tx}^\star}\left({t_{}^\star}\right)} = {{{N_{{{{a}}_{EM}}}}^\star}}{{V_{obs}}^\star}\sum_{j=1}^{{\lfloor\frac{{t_{}^\star}}{{{T_{int}}^\star}}+1\rfloor}} {W\left[j\right]}{{{{C_{{a}}}^\star}}({r_{eff}^\star}(j),{\tau}(j))},$$ where here $({{r_{eff}^\star}}(j))^2 = ({x^\star}_1 - {{v^\star_{\scriptscriptstyle\parallel}}}{\tau}(j))^2 + ({{v^\star_{\perp}}}{\tau}(j))^2$ and ${\tau}(j) = {t_{}^\star}-(j-1){{T_{int}}^\star}$, i.e., $({{r_{eff}^\star}}(j))^2$ is the square of the effective distance between the receiver and the transmitter’s $j$th emission and ${\tau}(j)$ is the time elapsed since the beginning of the $j$th bit interval. For compactness in the remainder of this section, we write ${{{{C_{{a}}}^\star}}({r_{eff}^\star}(j),{\tau}(j))} = {{C_{{a}}}^\star}(j)$. We also note that it is straightforward to relax the uniform concentration assumption and re-write (\[EQ13\_04\_02\_DMLS\]) into a more general form, following our analysis in [@RefWorks:752]. Similarly to the discussion of ${{{N_{{a}}}}_{obs}^\star\left({t_{}^\star}\right)}$ in Section \[sec\_mui\], ${{{N_{{a}}}}_{tx}^\star\left({t_{}^\star}\right)}$ is (dimensionally) a Poisson random variable with time-varying mean ${\overline{{{N_{{a}}}}_{tx}^\star}\left({t_{}^\star}\right)}$. We decompose (\[EQ13\_04\_02\_DMLS\]) into three terms: molecules observed due to the current bit interval, ${{{N_{{a}}}}_{tx,cur}^\star\left({t_{}^\star}\right)}$, molecules observed that were released within ${F}$ intervals before the current interval, ${{{N_{{a}}}}_{tx,isi}^\star\left({t_{}^\star}\right)}$, and molecules observed that were released in any older bit interval, ${{{N_{{a}}}}_{old}^\star\left({t_{}^\star}\right)}$. Specifically, (\[EQ13\_04\_02\_DMLS\]) becomes $$\begin{aligned} \label{EQ13_08_08_DMLS} {\overline{{{N_{{a}}}}_{tx}^\star}\left({t_{}^\star}\right)} = &\; {{{N_{{{{a}}_{EM}}}}^\star}}{{V_{obs}}^\star}\left[ {W\left[j_c\right]}{{C_{{a}}}^\star}(j_c) + \sum_{j=j_c-{F}}^{j_c-1}{W\left[j\right]}{{C_{{a}}}^\star}(j) + \sum_{j=1}^{j_c-{F}-1}{W\left[j\right]}{{C_{{a}}}^\star}(j)\right] \\ = &\; {\overline{{{N_{{a}}}}_{tx,cur}^\star}\left({t_{}^\star}\right)} + {\overline{{{N_{{a}}}}_{tx,isi}^\star}\left({t_{}^\star}\right)} + {\overline{{{N_{{a}}}}_{old}^\star}\left({t_{}^\star}\right)}, \label{EQ13_08_08_summary}\end{aligned}$$ $$\begin{aligned} {\overline{{{N_{{a}}}}_{tx}^\star}\left({t_{}^\star}\right)} = &\; {{{N_{{{{a}}_{EM}}}}^\star}}{{V_{obs}}^\star}\Bigg[ {W\left[j_c\right]}{{C_{{a}}}^\star}(j_c) + \sum_{j=j_c-{F}}^{j_c-1}{W\left[j\right]}{{C_{{a}}}^\star}(j) \nonumber\\ \label{EQ13_08_08_DMLS} & + \sum_{j=1}^{j_c-{F}-1}{W\left[j\right]}{{C_{{a}}}^\star}(j)\Bigg] \\ = &\; {\overline{{{N_{{a}}}}_{tx,cur}^\star}\left({t_{}^\star}\right)} + {\overline{{{N_{{a}}}}_{tx,isi}^\star}\left({t_{}^\star}\right)} + {\overline{{{N_{{a}}}}_{old}^\star}\left({t_{}^\star}\right)}, \label{EQ13_08_08_summary}\end{aligned}$$ where $j_c = {\lfloor\frac{{t_{}^\star}}{{{T_{int}}^\star}}+1\rfloor}$ is the index of the *current* bit interval, and we emphasize that each term in (\[EQ13\_08\_08\_summary\]) is evaluated given the current transmitter sequence ${\mathbf{W}}$. The decomposition enables us to simplify the expression for the signal observed due to molecules released by the transmitter if we can write an *asymptotic* expression for the expected value of ${\overline{{{N_{{a}}}}_{old}^\star}\left({t_{}^\star}\right)}$, i.e., ${\overline{{{N_{{a}}}}_{old}^\star}}$. However, the analysis that we have derived in this paper for the asymptotic impact of signals is dependent on the on-going emission of molecules that began at time ${t_{}^\star}=0$. We present two methods to derive ${\overline{{{N_{{a}}}}_{old}^\star}}$. First, we begin with the asymptotic expression in (\[EQ13\_08\_05\]) for an interferer that is always emitting and then subtract the *unconditional* expected impact of molecules released within the last ${F}+1$ bit intervals. We then write the expected and *time-varying* but *asymptotic* expression for ${\overline{{{N_{{a}}}}_{old}^\star}\left({t_{}^\star}\right)}$ as $${\overline{{{N_{{a}}}}_{old}^\star}\left({t_{}^\star}\right)} = {\overline{{{N_{{a}}}}_{tx}^\star}} - {\overline{{{N_{{a}}}}_{tx,isi}^\star}\left({t_{}^\star}\right)} - {\overline{{{N_{{a}}}}_{tx,cur}^\star}\left({t_{}^\star}\right)}, \label{EQ13_08_09}$$ where ${\overline{{{N_{{a}}}}_{tx}^\star}}$ is in the same form as (\[EQ13\_08\_05\]), and here ${\overline{{{N_{{a}}}}_{tx,cur}^\star}\left({t_{}^\star}\right)}$ and ${\overline{{{N_{{a}}}}_{tx,isi}^\star}\left({t_{}^\star}\right)}$ do *not* depend on ${\mathbf{W}}$ because they are averaged over the 2 and $2^{{F}}$ possible corresponding bit sequences, respectively. Eq. (\[EQ13\_08\_09\]) is time-varying because the expected impact that we subtract depends on the time within the current bit interval. From (\[EQ13\_08\_09\]), ${{{N_{{a}}}}_{old}^\star\left({t_{}^\star}\right)}$ is asymptotically a cyclostationary process; the expected mean is periodic with period ${{T_{int}}^\star}$. Although (\[EQ13\_08\_09\]) is tractable, it is cumbersome to evaluate (because we need to average the expected impact of molecules released over all $2^{{F}+1}$ possible recent bit sequences, including the current bit) and is also not accurate (because (\[EQ13\_08\_05\]) is based on continuous emission while ${\overline{{{N_{{a}}}}_{tx,isi}^\star}\left({t_{}^\star}\right)}$ and ${\overline{{{N_{{a}}}}_{tx,cur}^\star}\left({t_{}^\star}\right)}$ are based on discrete emissions at the start of the corresponding bit intervals). A second method to derive ${\overline{{{N_{{a}}}}_{old}^\star}}$, which requires less approximation, is to start with (\[EQ13\_07\_27\_int\]) and change the limits of integration over time, i.e., $${\overline{{{N_{{a}}}}_{old}^\star}\left({t_{}^\star}\right)} = \int\limits_{{t_{}^\star} - (j_c-{F}-1){{T_{int}}^\star}}^{\infty} {{V_{obs}}^\star}{{{{C_{{a}}}^\star}}({r_{eff}^\star},{\tau})} d{\tau}, \label{EQ13_08_09_int}$$ where ${{r_{eff}^\star}}^2 = ({x^\star}_1 - {{v^\star_{\scriptscriptstyle\parallel}}}{\tau})^2 + ({{v^\star_{\perp}}}{\tau})^2$ and, if $j_c-{F}-1 \le 0$, then we do not yet have asymptotic [ISI]{}. Eq. (\[EQ13\_08\_09\_int\]) is also periodic with period ${{T_{int}}^\star}$. A special case of (\[EQ13\_08\_09\_int\]) occurs if we have ${{v^\star_{\scriptscriptstyle\parallel}}}= {{v^\star_{\perp}}}= 0$ and ${{k_{}}^\star} = 0$. In such an environment, we can subtract (\[EQ13\_05\_18\]) from the asymptotic expression in (\[EQ13\_07\_27\]). Specifically, we can write $${\overline{{{N_{{a}}}}_{old}^\star}\left({t_{}^\star}\right)} = \frac{{{V_{obs}}^\star}}{4\pi{x^\star}_1} {\operatorname{erf}\left(\frac{{x^\star}_1}{2\sqrt{{t_{}^\star} - (j_c-{F}-1){{T_{int}}^\star}}}\right)}. \label{EQ13_05_18_isi}$$ Depending on the environmental parameters and whether there is a preference for tractability or accuracy, either (\[EQ13\_08\_09\]), (\[EQ13\_08\_09\_int\]), or (\[EQ13\_05\_18\_isi\]) can be used for ${\overline{{{N_{{a}}}}_{old}^\star}\left({t_{}^\star}\right)}$ in (\[EQ13\_08\_08\_summary\]). This asymptotic [ISI]{} term is independent of the actual transmitter data sequence ${\mathbf{W}}$, so it can be pre-computed and used to assist in applications such as evaluating the expected bit error probability of a weighted sum detector. Weighted Sum Detection ---------------------- We focus on a single type of detector at the receiver as a detailed example of the application of an asymptotic model of old [ISI]{}. We proposed the family of weighted sum detectors in [@RefWorks:747] as detectors that can operate with limited memory and computational requirements. We envision such detectors to be physically practical because they can already be found in biological systems such as neurons; see [@RefWorks:587 Ch. 12]. Here, we consider weighted sum detectors where the receiver makes ${M}$ observations in every bit interval, and we assume that these observations are equally spaced such that the ${m}$th observation in the $j$th interval is made at time ${t_{}^\star}(j,{m}) = \left(j+\frac{{m}}{{M}}\right){{T_{int}}^\star}$, where $j = \{1,2,\ldots,{B_{}}\}, {m}= \{1,2,\ldots,{M}\}$. The dimensionless decision rule of the weighted sum detector in the $j$th bit interval is $${\hat{W}\left[j\right]} = \left\{ \begin{array}{rl} 1 & \text{if} \quad \sum_{{m}= 1}^{{M}}{w_{{m}}}{{{N_{{a}}}}_{obs}^\star\left({t_{}^\star}(j,{m})\right)} \ge {\xi^\star},\\ 0 & \text{otherwise}, \end{array} \right. \label{FEB13_33}$$ where ${{{N_{{a}}}}_{obs}^\star\left({t_{}^\star}(j,{m})\right)}$ is the ${m}$th observation as given by (\[EQ13\_05\_28\_obs\]), ${w_{{m}}}$ is the weight of the ${m}$th observation, and ${\xi^\star}$ is the binary decision threshold (we note that we do not need to make the weights dimensionless because they already are). We assume that a constant optimal ${\xi^\star}$ for the given environment (and for the given formulation of [ISI]{} when evaluating the expected performance) is found via numerical search. Given a particular transmitter sequence ${\mathbf{W}}$, we can calculate the expected error probability of a weighted sum detector. The expected probability of error of the $j$th bit, ${P_e\left[j | {\mathbf{W}}\right]}$, is $${P_e\left[j | {\mathbf{W}}\right]} = \quad\left\{ \begin{array}{rl} \Pr\left(\sum_{{m}= 1}^{{M}}{w_{{m}}}{{{N_{{a}}}}_{obs}^\star\left({t_{}^\star}(j,{m})\right)} < {\xi^\star}\right) & \text{if} \;{W\left[j\right]} = 1, \vspace*{2mm}\\ \Pr\left(\sum_{{m}= 1}^{{M}}{w_{{m}}}{{{N_{{a}}}}_{obs}^\star\left({t_{}^\star}(j,{m})\right)} \ge {\xi^\star}\right) & \text{if} \;{W\left[j\right]} = 0. \end{array} \right. \label{EQ13_06_15}$$ $$\begin{aligned} & {P_e\left[j | {\mathbf{W}}\right]} = \nonumber \\ & \quad\left\{ \begin{array}{rl} \!\Pr\left(\sum_{{m}= 1}^{{M}}{w_{{m}}}{{{N_{{a}}}}_{obs}^\star\left({t_{}^\star}(j,{m})\right)} < {\xi^\star}\right) & \text{if} \;{W\left[j\right]} = 1, \vspace*{2mm}\\ \!\Pr\left(\sum_{{m}= 1}^{{M}}{w_{{m}}}{{{N_{{a}}}}_{obs}^\star\left({t_{}^\star}(j,{m})\right)} \ge {\xi^\star}\right) & \text{if} \;{W\left[j\right]} = 0. \end{array} \right. \label{EQ13_06_15}\end{aligned}$$ In our previous work in [@RefWorks:747], we approximated the expected error probability for the $j$th bit averaged over all possible transmitter sequences, ${\overline{P}_e\left[j\right]}$, by averaging (\[EQ13\_06\_15\]) over a subset of all sequences. An error probability was determined for all ${B_{}}$ bit intervals of every considered sequence. This analysis can be greatly simplified by evaluating the probability of error of a single bit that is sufficiently “far” from the start of the sequence, i.e., $j \to \infty$, and then model only the most recent ${F}$ intervals of [ISI]{} explicitly and represent all older intervals with ${\overline{{{N_{{a}}}}_{old}^\star}\left({t_{}^\star}\right)}$. Furthermore, if the impacts of the external noise sources in (\[EQ13\_05\_28\_obs\]) are represented asymptotically (whether they are interferers or other noise sources), or if there are no external noise sources present, then we only need to evaluate the expected probability of error of the *last* bit in $2^{{F}+1}$ sequences. The evaluation of (\[EQ13\_06\_15\]) depends on the statistics of the weighted sum $\sum_{{m}= 1}^{{M}}{w_{{m}}}{{{N_{{a}}}}_{obs}^\star\left({t_{}^\star}(j,{m})\right)}$. For simplicity, we limit our discussion to the special case where the weights are all equal, i.e., ${w_{{m}}} = 1\, \forall {m}$, such that we can assume that the (dimensional) observations are independent Poisson random variables (we also considered the general case, where we must approximate the observations as Gaussian random variables, in [@RefWorks:747]). Then, the sum of observations is also a Poisson random variable. The [CDF]{} of the weighted sum in the $j$th bit interval is then [@RefWorks:747 Eq. 38] $$\Pr\left(\sum_{{m}= 1}^{{M}}{{{{N_{{A}}}}_{obs}}\!\left({t\left(j,{m}\right)}\right)} < {\xi}\right) = {\exp\left(-\sum_{{m}= 1}^{{M}}{\overline{{{N_{{A}}}}_{obs}}\left({t\left(j,{m}\right)}\right)}\right)} \sum_{i=0}^{{\xi}-1} \frac{\left(\sum\limits_{{m}= 1}^{{M}}{\overline{{{N_{{A}}}}_{obs}}\left({t\left(j,{m}\right)}\right)}\right)^i}{i!}, \label{EQ13_06_02}$$ $$\begin{aligned} & \Pr\left(\sum_{{m}= 1}^{{M}}{{{{N_{{A}}}}_{obs}}\!\left({t\left(j,{m}\right)}\right)} < {\xi}\right) = \nonumber \\ & \qquad{\exp\left(-\sum_{{m}= 1}^{{M}}{\overline{{{N_{{A}}}}_{obs}}\left({t\left(j,{m}\right)}\right)}\right)} \nonumber \\ & \qquad\times \sum_{i=0}^{{\xi}-1} \frac{\left(\sum\limits_{{m}= 1}^{{M}}{\overline{{{N_{{A}}}}_{obs}}\left({t\left(j,{m}\right)}\right)}\right)^i}{i!}, \label{EQ13_06_02}\end{aligned}$$ where, from (\[EQ13\_05\_28\_obs\]), $${\overline{{{N_{{A}}}}_{obs}}\left({t\left(j,{m}\right)}\right)} = {\overline{{{N_{{A}}}}_{TX}}\left({t\left(j,{m}\right)}\right)} + \sum_{u=2}^{{U}}{\overline{{N_{{A}_{u}}}}}, \label{EQ13_05_28_obs_avg}$$ and ${\overline{{{N_{{A}}}}_{TX}}\left(t\right)}$ and ${\overline{{N_{{A}_{u}}}}}$ are the *dimensional* forms of the number of molecules expected from the intended transmitter and $u$th noise source, i.e., ${\overline{{{N_{{a}}}}_{tx}^\star}\left({t_{}^\star}\right)}$ of ${\overline{{{N_{{a}}}}_{u}^\star}}$, respectively, and we emphasize that we represent the noise sources asymptotically. We write (\[EQ13\_06\_02\]) and (\[EQ13\_05\_28\_obs\_avg\]) in dimensional form to emphasize that the observations are discrete. For the corresponding simulations in Section \[sec\_num\], we only consider ${U}=1$ to focus on the accuracy of the asymptotic approximation of old [ISI]{}, and we evaluate the old [ISI]{} as given by (\[EQ13\_08\_09\_int\]) or (\[EQ13\_05\_18\_isi\]) for ${k_{}}\ne 0$ and ${k_{}}=0$, respectively. Numerical Results {#sec_num} ================= In this section, we present numerical and simulation results to verify the analysis of noise, multiuser interference, and [ISI]{} performed in this paper. To clearly show the accuracy of all equations derived in this paper, we simulate only *one* source at a time, measuring either 1) the impact of a noise source or an interfering transmitter, or 2) the receiver error probability when the intended transmitter is the only molecule source. Our simulations are executed in the particle-based stochastic framework that we introduced in [@RefWorks:631; @RefWorks:662]. The ${A}$ molecules are initialized at the corresponding source when they are released. The location of each molecule, as determined by the uniform flow and random diffusion, is updated every time step $\Delta t$, where diffusion along each dimension is simulated by generating a normal random variable with variance $2{D_{{A}}} \Delta t$. If there is molecule degradation, then every molecule has a chance of degrading in every time step with probability ${k_{}}\Delta t$. If there is no molecule degradation, then all molecules released are present indefinitely. The signal at the receiver is updated in every time step by counting the number of ${A}$ molecules that are within ${r_{obs}}$ of the origin. Constant environmental parameters are listed in Table \[table\_param\]. The chosen values are consistent with those that we considered in [@RefWorks:752], where we noted that the value of the diffusion coefficient ${D_{{A}}}$ is similar to that of many small molecules in water at room temperature (see [@RefWorks:742 Ch. 5]), and is also comparable to that of small biomolecules in blood plasma (see [@RefWorks:754]). Most of the results in this section have been non-dimensionalized with the reference distance ${L}$ depending on the distance from the source of molecules to the receiver. For reference, conversions between the dimensional variables that were simulated and their values in dimensionless form are listed in Table \[table\_dmls\]. Parameter Symbol Value ------------------------------------------------------ ------------------------------------------------ -------------------------------------------------------------------- Release rate of ideal noise source ${\overline{{N_{{{A}_{gen}}}}}\left(t\right)}$ $1.2\times10^6\, \frac{{\textnormal{molecule}}}{{\textnormal{s}}}$ Molecules per transmitter emission ${{N_{{{A}_{EM,u}}}}}$ $10^4$ Probability of transmitter binary $1$ ${P_1}$ $0.5$ Length of transmitter sequence ${B_{}}$ $100$ bits Transmitter bit interval ${T_{int,u}}$ $0.2\,$ms Diffusion coefficient [@RefWorks:742; @RefWorks:754] ${D_{{A}}}$ $10^{-9}\,{\textnormal{m}}^2/{\textnormal{s}}$ Radius of receiver ${r_{obs}}$ $50\,$nm Step size for continuous noise $\Delta {t_{}^\star}$ $0.1$ Step size for transmitters $\Delta t$ $2\,\mu{\textnormal{s}}$ : System parameters used for numerical and simulation results. \[table\_param\] ${x}_n$ \[nm\] ${L}$ \[nm\] $t$ \[$\mu{\textnormal{s}}$\] ${v_{}}$ \[$\frac{{\textnormal{m}}{\textnormal{m}}}{{\textnormal{s}}}$\] ${k_{}}$ \[${\textnormal{s}}^{-1}$\] ${N_{{A}_{REF}}}$ ---------------- -------------- ------------------------------- -------------------------------------------------------------------------- -------------------------------------- ------------------- 0 50 2.5 20 $4\times10^5$ 3 50 50 2.5 20 $4\times10^5$ 3 100 100 10 10 $1\times10^5$ 12 200 200 40 5 $2.5\times10^4$ 48 400 400 160 2.5 $6.25\times10^3$ 192 1000 1000 1000 1 $10^3$ 1200 : Conversion between dimensional and dimensionless variables. The values of $t$, ${v_{}}$, and ${k_{}}$ correspond to ${t_{}^\star}=1$, ${v^\star_{}} = 1$, and ${{k_{}}^\star} = 1$, respectively. \[table\_dmls\] Continuous Noise Source ----------------------- We first present the time-varying impact of the continuously-emitting noise source that we analyzed in Section \[sec\_noise\]. The times between the release of consecutive molecules from the noise source are simulated as a continuous Poisson process so that the times between molecule release are independent. The expected release rate, $1.2\times10^6\,\frac{{\textnormal{molecule}}}{{\textnormal{s}}}$, is chosen so that, asymptotically, one (dimensional) molecule is expected to be observed at the receiver due to a noise source placed $50\,$nm from the center of the receiver (this distance is actually at the edge of the receiver, cf. Table \[table\_param\]). To accommodate the range of distances considered, we adjust the simulation time step $\Delta t$ so that $10$ steps are made within every ${t_{}^\star} = 1$ time unit. Simulations are averaged over $10^5$ independent realizations. The specific equations used for calculating the expected values, both time-varying and asymptotically, were chosen as appropriate from Table \[table\_noise\]. In Fig. \[fig\_noise\_v0\_k0\], we show the time-varying impact of the noise source when there is no advection and no molecule degradation, i.e., ${{v^\star_{\scriptscriptstyle\parallel}}}= {{v^\star_{\perp}}}= 0$ and ${{k_{}}^\star} = 0$. Under these conditions, we have the expected time-varying and asymptotic impact in closed form. For every distance shown, the impact approaches the asymptotic value as ${t_{}^\star} \to 100$, as expected from Remark \[remark\_far\_no\_flow\]. The expected impact without the [UCA]{} is highly accurate for all time, and the expected impact with the [UCA]{} shows visible deviation only for ${t_{}^\star} < 1$ when ${x}_n < 200\,$nm, i.e., when the noise source is not far from the receiver. We also observe that the overall impact decreases as the noise source is placed further from the receiver; doubling the distance decreases ${\overline{{{N_{{a}}}}_{n}^\star}\left({t_{}^\star}\right)}$ by about a factor of $8$ while the corresponding value of ${N_{{A}_{REF}}}$, defined as ${N_{{A}_{REF}}} = {L}^2{{N_{{{A}_{gen}}}}}/{D_{{A}}}$ and used to convert ${\overline{{{N_{{a}}}}_{n}^\star}\left({t_{}^\star}\right)}$ into dimensional form, only increases by a factor of $4$ (see Table \[table\_dmls\]). The overall (dimensional) decrease in impact by a factor of $2$ is as expected by Remark \[remark\_no\_flow\]. ![The dimensionless number of noise molecules observed at the receiver as a function of time when ${{v^\star_{\scriptscriptstyle\parallel}}}={{v^\star_{\perp}}}=0$ and ${{k_{}}^\star}=0$, i.e., when there is no advection or molecule degradation.[]{data-label="fig_noise_v0_k0"}](simulations/sim13_04_23/sim13_04_23_21_v0_k0.eps){width="1\linewidth"} In Fig. \[fig\_noise\_v0\_k1\], we consider the same environment as in Fig. \[fig\_noise\_v0\_k0\] but we set the molecule degradation rate ${{k_{}}^\star} = 1$. The accuracy of the expected expressions is comparable to that observed in Fig. \[fig\_noise\_v0\_k0\], but here the asymptotic impact is approached about two orders of magnitude faster, as ${t_{}^\star} \to 2$. The asymptotic impact at any distance is also less than half of that observed in Fig. \[fig\_noise\_v0\_k0\] because of the molecule degradation. ![The dimensionless number of noise molecules observed at the receiver as a function of time when ${{v^\star_{\scriptscriptstyle\parallel}}}={{v^\star_{\perp}}}=0$ and ${{k_{}}^\star} = 1$.[]{data-label="fig_noise_v0_k1"}](simulations/sim13_04_23/sim13_04_23_21_k1.eps){width="1\linewidth"} In Fig. \[fig\_noise\_k\], we observe the impact of a noise source at the “worst-case” location, i.e., ${x}_n = 0$, and we vary the molecule degradation rate ${{k_{}}^\star}$. The expressions for the expected time-varying and asymptotic impact are both highly accurate. We see the general trend that the asymptotic impact decreases (as expected by Remark \[remark\_degradation\]) and is reached sooner as ${{k_{}}^\star}$ increases. Increasing ${{k_{}}^\star}$ also degrades the signal from the desired transmitter, but this can be good for reducing [ISI]{} as we will see in the following subsection. Furthermore, it is interesting that the impact of the noise source can be significantly reduced by increasing the rate of noise molecule degradation, even though the noise molecules are being emitted directly at the receiver. This implies that, if they were not degraded, significantly more noise molecules would have been observed by the receiver before diffusing away. ![The dimensionless number of noise molecules observed at the receiver as a function of time when ${x}_n = 0$, ${{v^\star_{\scriptscriptstyle\parallel}}}={{v^\star_{\perp}}}=0$, and the molecule degradation rate is ${{k_{}}^\star} = \{0,1,2,5,10,20,50\}$.[]{data-label="fig_noise_k"}](simulations/sim13_04_23/sim13_04_23_21_0nm_k.eps){width="1\linewidth"} In Figs. \[fig\_noise\_v1\] and \[fig\_noise\_v1\_far\], we consider the effect of advection on the impact of noise without molecule degradation. For clarity, we observe ${x}_n = \{0,100\}\,$nm in Fig. \[fig\_noise\_v1\] and ${x}_n = \{200,400\}\,$nm in Fig. \[fig\_noise\_v1\_far\]. When ${x}_n = 0$, only one flow direction is relevant because all flows are equivalent by symmetry. As with molecule degradation, we observe that the presence of advection reduces the time required for the impact of the noise source to become asymptotic, which here occurs by about ${t_{}^\star} = 4$. Flows that are not in the direction of a line from the noise source to the receiver, i.e., ${{v^\star_{\scriptscriptstyle\parallel}}}< 0$ or ${{v^\star_{\perp}}}\neq 0$ (which we termed “disruptive” flows in [@RefWorks:752]), decrease the asymptotic impact of the noise source. However, the flow ${{v^\star_{\scriptscriptstyle\parallel}}}= 1$ results in about the same asymptotic impact as the no-flow case when ${x}_n \ne 0\,$nm, which we expect from Remark \[remark\_far\_flow\], although it might not be an intuitive result. ![The dimensionless number of noise molecules observed at the receiver as a function of time when ${{k_{}}^\star}=0$, we vary ${{v^\star_{\scriptscriptstyle\parallel}}}$ or ${{v^\star_{\perp}}}$, and we consider ${x}_n = 0\,$nm and ${x}_n = 100\,$nm.[]{data-label="fig_noise_v1"}](simulations/sim13_04_23/sim13_04_23_21_v1_close.eps){width="1\linewidth"} ![The dimensionless number of noise molecules observed at the receiver as a function of time when ${{k_{}}^\star}=0$, we vary ${{v^\star_{\scriptscriptstyle\parallel}}}$ or ${{v^\star_{\perp}}}$, and we consider ${x}_n = 200\,$nm and ${x}_n = 400\,$nm.[]{data-label="fig_noise_v1_far"}](simulations/sim13_04_23/sim13_04_23_21_v1.eps){width="1\linewidth"} Interference and [ISI]{} ------------------------ We now assess the accuracy of approximating transmitters as continuously-emitting noise sources. First, we observe the impact of an interfering transmitter. Second, we assess the accuracy of evaluating the receiver error probability where we vary the number ${F}$ of symbols of [ISI]{} treated explicitly and approximate all older [ISI]{} as an asymptotic noise source. We consider transmitters with a common set of dimensional transmission parameters, as described in Table \[table\_param\]. In Fig. \[fig\_interferer\], we show the time-varying impact on the receiver of a single interferer using binary-encoded impulse modulation, both with and without molecule degradation, for the interferer placed ${x}_2 = 400\,$nm or $1\,\mu$m from the receiver (we emphasize that the *only* active molecule source is *not* the intended transmitter by using the subscript $2$). At both distances, the same bit interval is used (${T_{int,2}} = 0.2\,$ms). The expected time-varying and asymptotic curves are evaluating using (\[EQ13\_07\_27\_int\]) and (\[EQ13\_08\_05\]), respectively. The simulations are averaged over $10^5$ independent realizations, and in Fig. \[fig\_interferer\] we clearly observe oscillations in the simulated values above and below the expected curves. The relative amplitude of these oscillations is much greater when the interferer is closer to the receiver, and also greater when there is molecule degradation; when ${x}_2 = 400\,$nm and ${{k_{}}^\star} = 1$, the impact in the asymptotic regime varies from $4\times10^{-5}$ to over $6\times10^{-4}$, but when ${x}_2 = 1\,\mu$m and ${{k_{}}^\star} = 0$, the relative amplitude of the oscillations is an order of magnitude smaller. Thus, the impact of an interferer that is sufficiently far from the receiver can be accurately approximated with a non-oscillating function, and an interferer does not need to be transmitting for a very long time to assume that its impact is asymptotic ($8$ and $50$ bit intervals are shown in Fig. \[fig\_interferer\] for the interferers at $400\,$nm and $1\,\mu$m, respectively; the difference is due to plotting on a dimensionless time axis). We note that the relative amplitude of oscillations would also decrease if the interferer transmitted with a smaller bit interval. ![The dimensionless number of interfering molecules observed at the receiver as a function of time an interfering transmitter placed at ${x}_2 = 400$nm and ${x}_2 = 1\,\mu$m.[]{data-label="fig_interferer"}](simulations/sim13_04_23/sim13_04_23_22.eps){width="1\linewidth"} In Fig. \[fig\_isi\], we measure the average bit error probability of the equal weight detector when ${M}= 10$ samples are taken per bit interval and the optimal decision threshold is found numerically. The receiver is placed ${x}_1 = 400\,$nm from the transmitter and we vary ${{k_{}}^\star}$ to control the amount of [ISI]{} that we expect (since a faster molecule degradation rate means that emitted molecules are less likely to exist sufficiently long to interfere with future transmissions). We do not add any external noise or interference (i.e., there is only *one* source of information molecules), but we vary the number ${F}$ of bit intervals that are treated explicitly as [ISI]{}, i.e., the complexity of ${\overline{{{N_{{a}}}}_{tx,isi}^\star}\left({t_{}^\star}\right)}$, in evaluating the expected error probability. Simulations are averaged over $10^4$ independent realizations, and we ignore the decisions made within the first $50$ of the $100$ bits in each sequence in order to approximate the “old” [ISI]{} as asymptotic. The old [ISI]{}, ${{{N_{{a}}}}_{old}^\star\left({t_{}^\star}\right)}$, is found by evaluating (\[EQ13\_08\_09\_int\]) (or by using (\[EQ13\_05\_18\_isi\]) when ${{k_{}}^\star} = 0$), and to emphasize the benefit of including this term we also consider evaluating the expected error probability where we set ${\overline{{{N_{{a}}}}_{old}^\star}\left({t_{}^\star}\right)} = 0$. ![Receiver error probability as a function of ${F}$, the number of bit intervals of [ISI]{} treated explicitly, for varying molecule degradation rate ${{k_{}}^\star}$.[]{data-label="fig_isi"}](simulations/sim13_08_12/sim13_08_12_23.eps){width="1\linewidth"} We generally observe in Fig. \[fig\_isi\] that, as ${F}$ increases, the expected error probability becomes more accurate because we treat more of the [ISI]{} explicitly instead of as asymptotic noise via ${\overline{{{N_{{a}}}}_{old}^\star}\left({t_{}^\star}\right)}$. The exception to this is when ${{k_{}}^\star}=0$ and we calculate the expected value using ${\overline{{{N_{{a}}}}_{old}^\star}\left({t_{}^\star}\right)}$. The [ISI]{} in that case is much greater than when ${{k_{}}^\star} > 0$, such that the expected bit error probability is more sensitive to the approximation for ${\overline{{{N_{{a}}}}_{old}^\star}\left({t_{}^\star}\right)}$, which assumes that the release of molecules is continuous over the entire bit interval. This approximation means that the expected “old” [ISI]{} is overestimated and a higher expected bit error probability is calculated. When ${{k_{}}^\star} > 0$, the expected bit error probability tends to underestimate that observed via simulation because the evaluation of the expected bit error probability assumes that all observed samples are independent, but this assumption loses accuracy for larger ${M}$. Importantly, the expected bit error probability tends to that observed via simulation much faster when including ${\overline{{{N_{{a}}}}_{old}^\star}\left({t_{}^\star}\right)}$, even though it is an approximation. For all values of ${{k_{}}^\star}$ considered, it is sufficient to consider only $2$ or $3$ intervals of [ISI]{} explicitly while approximating all prior intervals with ${\overline{{{N_{{a}}}}_{old}^\star}\left({t_{}^\star}\right)}$. If we use ${\overline{{{N_{{a}}}}_{old}^\star}\left({t_{}^\star}\right)} = 0$, as is common in the existing literature, then many more intervals of explicit [ISI]{} are needed for comparable accuracy (${F}= 20$ is still not sufficient if ${{k_{}}^\star} = 0$, although ${F}= 5$ might be acceptable if ${{k_{}}^\star} = 0.2$). Since the computational complexity of evaluating the expected bit error probability increases exponentially with ${F}$ (because we need to evaluate the expected probability of error due to all $2^{{F}+1}$ bit sequences), approximating old [ISI]{} with ${\overline{{{N_{{a}}}}_{old}^\star}\left({t_{}^\star}\right)}$ provides an effective means with which to reduce the complexity without making a significant sacrifice in accuracy. Conclusion {#sec_concl} ========== In this paper, we proposed a unifying model to account for the observation of unintended molecules by a passive receiver in a diffusive molecular communication system, where the unintended molecules include those emitted by the intended transmitter in previous bit intervals, those emitted by interfering transmitters, and those emitted by other external noise sources that are continuously emitting molecules. We presented the general time-varying expression for the expected impact of a noise source that is emitting continuously, and then we considered a series of special cases that facilitate time-varying or asymptotic solutions. Knowing the expected impact of noise sources enables us to find the effect of those sources on the bit error probability of a communication link. We used the analysis for asymptotic noise to approximate the impact of an interfering transmitter, which we extended to the general case of multiuser interference. Finally, we decomposed the signal received from the intended transmitter so that we could approximate “old” [ISI]{} as asymptotic interference. We showed how this approximation could be used to simplify the evaluation of the expected bit error probability of a weighted sum detector. Our simulation results showed the high accuracy of our expressions for time-varying and asymptotic noise. We showed that an interfering transmitter placed sufficiently far from the receiver can be approximated as an asymptotic noise source soon after it begins transmitting, and that approximating old [ISI]{} as asymptotic noise is an effective method to reduce the computational complexity of evaluating the expected probability of error. Our future work includes investigating the expected impact of noise sources with random locations, which can be used to model the random generation of noise molecules anywhere in the propagation medium, and using the approximation for asymptotic [ISI]{} to design adaptive detectors, where the decision threshold is adjusted based on the knowledge of the previously received information. Proof of Theorem 1 {#app_no_flow} ------------------ The asymptotic integration (i.e, as ${t_{}^\star} \to \infty$) in (\[EQ13\_05\_29\]) to prove Theorem \[theorem\_no\_flow\] can be written as the summation of four integrals which can be found by solving the following two integrals: $$\begin{aligned} \label{EQ13_05_45} & \int\limits_0^\infty {\operatorname{erf}\left(\frac{a}{{{\tau}}^\frac{1}{2}}\right)} {\exp\left(-{{k_{}}^\star}{\tau}\right)}d{\tau}, \\ \label{EQ13_05_46} & \int\limits_0^\infty {{\tau}}^\frac{1}{2} {\exp\left(-\frac{b}{{\tau}}-{{k_{}}^\star}{\tau}\right)}d{\tau},\end{aligned}$$ where $a$ could be positive or negative and the latter occurs only when ${x^\star}_n > {r_{obs}^\star}$. To solve (\[EQ13\_05\_45\]) for $a > 0$, we apply the substitution $c = a/{{\tau}}^\frac{1}{2}$ and use the definite integral [@RefWorks:700 Eq. 4.3.28] $$\int\limits_0^\infty {\operatorname{erf}\left(b_1c\right)} {\exp\left(-\frac{b_2^2}{4c^2}\right)}\frac{dc}{c^3} = \frac{2}{b_2^2} \left(1-{\exp\left(-b_1b_2\right)}\right),$$ so that we can write (\[EQ13\_05\_45\]) as $$\label{EQ13_05_47} \frac{1}{{{k_{}}^\star}}\left(1-{\exp\left(-2a\sqrt{{{k_{}}^\star}}\right)}\right).$$ Recalling that ${\operatorname{erf}\left(\cdot\right)}$ is an odd function, i.e., ${\operatorname{erf}\left(-c\right)} = -{\operatorname{erf}\left(c\right)}$, we solve (\[EQ13\_05\_45\]) for $a < 0$ as $$\label{EQ13_05_49} \frac{1}{{{k_{}}^\star}}\left({\exp\left(2a\sqrt{{{k_{}}^\star}}\right)}-1\right).$$ We can solve (\[EQ13\_05\_46\]) by directly applying [@RefWorks:402 Eq. 3.471.16] as $$\label{EQ13_05_52} \frac{\sqrt{\pi}}{2{{k_{}}^\star}}{\exp\left(-2\sqrt{b{{k_{}}^\star}}\right)} \left({{{k_{}}^\star}}^{-\frac{1}{2}}+2\sqrt{b}\right).$$ By taking care to consider the sign of ${r_{obs}^\star}-{x^\star}_n$, it is straightforward to combine (\[EQ13\_05\_47\]), (\[EQ13\_05\_49\]), and (\[EQ13\_05\_52\]) to arrive at (\[EQ13\_05\_53\]). Proof of Theorem 2 {#app_no_flow_degrad} ------------------ Similarly to Appendix \[app\_no\_flow\], we can solve the integration in (\[EQ13\_05\_29\]) up to any ${t_{}^\star}$ by solving the following two integrals: $$\begin{aligned} \label{EQ13_05_35_int} & \int\limits_0^{{t_{}^\star}} {\operatorname{erf}\left(\frac{a}{{{\tau}}^\frac{1}{2}}\right)} d{\tau}, \\ \label{EQ13_05_37_int} & \int\limits_0^{{t_{}^\star}} {{\tau}}^\frac{1}{2} {\exp\left(-\frac{b}{{\tau}}\right)}d{\tau},\end{aligned}$$ where $a$ could be positive or negative and the latter occurs only when ${x^\star}_n > {r_{obs}^\star}$. To solve (\[EQ13\_05\_35\_int\]) for $a > 0$, we apply the substitution $c = a/{{\tau}}^\frac{1}{2}$ and use the indefinite integrals [@RefWorks:700 Eq. 4.1.14] $$\int {\operatorname{erf}\left(c\right)} \frac{dc}{c^3} = \frac{-{\operatorname{erf}\left(c\right)}}{2c^2} + \frac{1}{\sqrt{\pi}} \int {\exp\left(-c^2\right)}\frac{dc}{c^2}, \label{EQ13_05_30}$$ and [@RefWorks:402 Eq. 2.325.5] $$\int {\exp\left(-c^n\right)}\frac{dc}{c^m} = \frac{1}{m-1}\left[\frac{-{\exp\left(-c^n\right)}}{c^{m-1}} - n\int \frac{{\exp\left(-c^n\right)}}{c^{m-n}}dc\right], \label{EQ13_05_31}$$ $$\begin{gathered} \int {\exp\left(-c^n\right)}\frac{dc}{c^m} = \\ \frac{1}{m-1}\left[\frac{-{\exp\left(-c^n\right)}}{c^{m-1}} - n\int \frac{{\exp\left(-c^n\right)}}{c^{m-n}}dc\right], \label{EQ13_05_31}\end{gathered}$$ as well as the definition of the error function. It is then straightforward to show that (\[EQ13\_05\_35\_int\]) for $a > 0$ is solved as $$\left(2a^2 + {t_{}^\star}\right){\operatorname{erf}\left(\frac{a}{\sqrt{{t_{}^\star}}}\right)} + 2a\sqrt{\frac{{t_{}^\star}}{\pi}}{\exp\left(-\frac{a^2}{{t_{}^\star}}\right)}-2a^2. \label{EQ13_05_33}$$ Recalling that ${\operatorname{erf}\left(\cdot\right)}$ is an odd function, we solve (\[EQ13\_05\_35\_int\]) for $a < 0$ as $$\left(2a^2 + {t_{}^\star}\right){\operatorname{erf}\left(\frac{a}{\sqrt{{t_{}^\star}}}\right)} + 2a\sqrt{\frac{{t_{}^\star}}{\pi}}{\exp\left(-\frac{a^2}{{t_{}^\star}}\right)}+2a^2. \label{EQ13_05_34}$$ To solve (\[EQ13\_05\_37\_int\]), we apply the substitution $c = \sqrt{b{\tau}}$, apply (\[EQ13\_05\_31\]) twice, and use the definition of the error function. It is then straightforward to show that (\[EQ13\_05\_37\_int\]) is solved as $$\frac{2}{3}\sqrt{{t_{}^\star}}{\exp\left(-\frac{b}{{t_{}^\star}}\right)} ({t_{}^\star} -2b) + \frac{4}{3}b^{\frac{3}{2}}\sqrt{\pi} \left(1-{\operatorname{erf}\left(\sqrt{\frac{b}{{t_{}^\star}}}\right)}\right). \label{EQ13_05_37}$$ If we take care to consider the sign of ${r_{obs}^\star}-{x^\star}_n$, then we can arrive at (\[EQ13\_05\_38\]) by combining (\[EQ13\_05\_33\]), (\[EQ13\_05\_34\]), and (\[EQ13\_05\_37\]). [Adam Noel]{} (S’09) received the B.Eng. degree from Memorial University in 2009 and the M.A.Sc. degree from the University of British Columbia (UBC) in 2011, both in electrical engineering. He is now a Ph.D. candidate in electrical engineering at UBC, and in 2013 was a visiting researcher at the Institute for Digital Communications, Friedrich-Alexander-Universität Erlangen-Nürnberg. His research interests include wireless communications and how traditional communication theory applies to molecular communication. [Karen C. Cheung]{} received the B.S. and Ph.D. degrees in bioengineering from the University of California, Berkeley, in 1998 and 2002, respectively. From 2002 to 2005, she was a postdoctoral researcher at the Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland. She is now at the University of British Columbia, Vancouver, BC, Canada. Her research interests include lab-on-a-chip systems for cell culture and characterization, inkjet printing for tissue engineering, and implantable neural interfaces. [Robert Schober]{} (S’98, M’01, SM’08, F’10) received the Diplom (Univ.) and the Ph.D. degrees in electrical engineering from the University of Erlangen-Nuremberg in 1997 and 2000, respectively. Since May 2002 he has been with the University of British Columbia (UBC), Vancouver, Canada, where he is now a Full Professor. Since January 2012 he is an Alexander von Humboldt Professor and the Chair for Digital Communication at the Friedrich Alexander University (FAU), Erlangen, Germany. His research interests fall into the broad areas of Communication Theory, Wireless Communications, and Statistical Signal Processing. He is currently the Editor-in-Chief of the IEEE Transactions on Communications. Source Type Section ---------------------------------------------------- ------------------------- ---------------- -------------------------- ------------------------------------------------------------------------ Random Noise \[sec\_noise\] (\[APR12\_42\_DMLS\]) Yes Interfering Transmitter \[sec\_mui\] (\[EQ13\_08\_02\]) Approximation in (\[EQ13\_08\_05\]) ${{{N_{{a}}}}_{1}^\star\left({t_{}^\star}\right)}$ Intended Transmitter \[sec\_isi\] (\[EQ13\_08\_08\_DMLS\]) Approximation of old ISI in (\[EQ13\_08\_09\])-(\[EQ13\_05\_18\_isi\]) : Description of the terms in (\[EQ13\_05\_28\_obs\]). \[table\_overview\] Eq. Closed Form? Asymptotic? ${x^\star}_n$ ${{k_{}}^\star}$ ${{v^\star_{\perp}}}$ ${{v^\star_{\scriptscriptstyle\parallel}}}$ ------------------------- -------------- ------------- --------------- ------------------ ----------------------- --------------------------------------------- (\[EQ13\_07\_27\_int\]) No No Far Any Any Any (\[EQ13\_05\_29\]) No No Any Any Any 0 (\[EQ13\_07\_27\]) Yes Yes Far Any Any Any (\[EQ13\_05\_53\]) Yes Yes $\neq 0$ Any 0 0 (\[EQ13\_05\_18\]) Yes No Far 0 0 0 (\[EQ13\_05\_38\]) Yes No $\neq 0$ 0 0 0 (\[EQ13\_05\_60\]) Yes Yes $\neq 0$ 0 0 0 (\[EQ13\_05\_54\]) Yes Yes $0$ Any 0 0 (\[EQ13\_05\_39\]) Yes No $0$ 0 0 0 : Summary of the equations for the impact of an external noise source and the conditions under which they can be used. By ${x^\star}_n =$ “Far”, we mean that the [UCA]{} is applied. \[table\_noise\] Parameter Symbol Value ------------------------------------------------------ ------------------------------------------------ -------------------------------------------------------------------- Release rate of ideal noise source ${\overline{{N_{{{A}_{gen}}}}}\left(t\right)}$ $1.2\times10^6\, \frac{{\textnormal{molecule}}}{{\textnormal{s}}}$ Molecules per transmitter emission ${{N_{{{A}_{EM,u}}}}}$ $10^4$ Probability of transmitter binary $1$ ${P_1}$ $0.5$ Length of transmitter sequence ${B_{}}$ $100$ bits Transmitter bit interval ${T_{int,u}}$ $0.2\,$ms Diffusion coefficient [@RefWorks:742; @RefWorks:754] ${D_{{A}}}$ $10^{-9}\,{\textnormal{m}}^2/{\textnormal{s}}$ Radius of receiver ${r_{obs}}$ $50\,$nm Step size for continuous noise $\Delta {t_{}^\star}$ $0.1$ Step size for transmitters $\Delta t$ $2\,\mu{\textnormal{s}}$ : System parameters used for numerical and simulation results. \[table\_param\] ${x}_n$ \[nm\] ${L}$ \[nm\] $t$ \[$\mu{\textnormal{s}}$\] ${v_{}}$ \[$\frac{{\textnormal{m}}{\textnormal{m}}}{{\textnormal{s}}}$\] ${k_{}}$ \[${\textnormal{s}}^{-1}$\] ${N_{{A}_{REF}}}$ ---------------- -------------- ------------------------------- -------------------------------------------------------------------------- -------------------------------------- ------------------- 0 50 2.5 20 $4\times10^5$ 3 50 50 2.5 20 $4\times10^5$ 3 100 100 10 10 $1\times10^5$ 12 200 200 40 5 $2.5\times10^4$ 48 400 400 160 2.5 $6.25\times10^3$ 192 1000 1000 1000 1 $10^3$ 1200 : Conversion between dimensional and dimensionless variables. The values of $t$, ${v_{}}$, and ${k_{}}$ correspond to ${t_{}^\star}=1$, ${v^\star_{}} = 1$, and ${{k_{}}^\star} = 1$, respectively. \[table\_dmls\] ![The dimensionless number of noise molecules observed at the receiver as a function of time when ${{v^\star_{\scriptscriptstyle\parallel}}}={{v^\star_{\perp}}}=0$ and ${{k_{}}^\star}=0$, i.e., when there is no advection or molecule degradation.[]{data-label="fig_noise_v0_k0"}](simulations/sim13_04_23/sim13_04_23_21_v0_k0.eps){width="0.75\linewidth"} ![The dimensionless number of noise molecules observed at the receiver as a function of time when ${{v^\star_{\scriptscriptstyle\parallel}}}={{v^\star_{\perp}}}=0$ and ${{k_{}}^\star} = 1$.[]{data-label="fig_noise_v0_k1"}](simulations/sim13_04_23/sim13_04_23_21_k1.eps){width="0.7\linewidth"} ![The dimensionless number of noise molecules observed at the receiver as a function of time when ${x}_n = 0$, ${{v^\star_{\scriptscriptstyle\parallel}}}={{v^\star_{\perp}}}=0$, and the molecule degradation rate is ${{k_{}}^\star} = \{0,1,2,5,10,20,50\}$.[]{data-label="fig_noise_k"}](simulations/sim13_04_23/sim13_04_23_21_0nm_k.eps){width="0.7\linewidth"} ![The dimensionless number of noise molecules observed at the receiver as a function of time when ${{k_{}}^\star}=0$, we vary ${{v^\star_{\scriptscriptstyle\parallel}}}$ or ${{v^\star_{\perp}}}$, and we consider ${x}_n = 0\,$nm and ${x}_n = 100\,$nm.[]{data-label="fig_noise_v1"}](simulations/sim13_04_23/sim13_04_23_21_v1_close.eps){width="0.7\linewidth"} ![The dimensionless number of noise molecules observed at the receiver as a function of time when ${{k_{}}^\star}=0$, we vary ${{v^\star_{\scriptscriptstyle\parallel}}}$ or ${{v^\star_{\perp}}}$, and we consider ${x}_n = 200\,$nm and ${x}_n = 400\,$nm.[]{data-label="fig_noise_v1_far"}](simulations/sim13_04_23/sim13_04_23_21_v1.eps){width="0.7\linewidth"} ![The dimensionless number of interfering molecules observed at the receiver as a function of time an interfering transmitter placed at ${x}_2 = 400$nm and ${x}_2 = 1\,\mu$m.[]{data-label="fig_interferer"}](simulations/sim13_04_23/sim13_04_23_22.eps){width="0.7\linewidth"} ![Receiver error probability as a function of ${F}$, the number of bit intervals of [ISI]{} treated explicitly, for varying molecule degradation rate ${{k_{}}^\star}$.[]{data-label="fig_isi"}](simulations/sim13_08_12/sim13_08_12_23.eps){width="0.7\linewidth"} [^1]: Manuscript received October 21, 2013; revised April 21, 2014; accepted June 3, 2014. This work was supported by the Natural Sciences and Engineering Research Council of Canada, and a Walter C. Sumner Memorial Fellowship. Computing resources were provided by WestGrid and Compute/Calcul Canada. [^2]: The authors are with the Department of Electrical and Computer Engineering, University of British Columbia, Vancouver, BC, Canada, V6T 1Z4 (email: {adamn, kcheung, rschober}@ece.ubc.ca). R. Schober is also with the Institute for Digital Communication, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Erlangen, Germany (email: schober@lnt.de).

--- abstract: 'We study the asymptotic behavior of solutions of the two dimensional incompressible Euler equations in the exterior of a curve when the curve shrinks to a point. This work links two previous results: \[Iftimie, Lopes Filho and Nussenzveig Lopes, Two Dimensional Incompressible Ideal Flow Around a Small Obstacle, Comm. PDE, [**28**]{} (2003), 349-379\] and \[Lacave, Two Dimensional Incompressible Ideal Flow Around a Thin Obstacle Tending to a Curve, Ann. IHP, Anl **26** (2009), 1121-1148\]. The second goal of this work is to complete the previous article, in defining the way the obstacles shrink to a curve. In particular, we give geometric properties for domain convergences in order that the limit flow be a solution of Euler equations.' address: | Université Paris-Diderot (Paris 7)\ Institut de Mathématiques de Jussieu\ UMR 7586 - CNRS\ 175 rue du Chevaleret\ 75013 Paris\ France author: - 'C. Lacave' title: Two Dimensional Incompressible Ideal Flow Around a Small Curve --- Introduction ============ The purpose of this work is to study the influence of a material curve on the behavior of two-dimensional ideal flows when the size of the curve tends to zero. The study of the fluid flows in a singularly perturbed domains was initiated by Iftimie, Lopes Filho and Nussenzveig Lopes in [@ift_lop_euler], in the case of a smooth obstacle which shrinks to a point. For some initial data, they obtain a blow-up of the limit velocity like $1/|x|$ centered at the point where the obstacle disappears. For some other initial data, they prove that there is no blow-up. Six years later, the case of thin obstacles shrinking to a curve was treated in [@lac_euler]. It was shown that the limit velocity always blows up at the end-points of the curve like $1/\sqrt{|x|}$. In light of this two works a natural question arises: what happens in the case of small curves ? Our result can be stated as following: as the end-points get closer and closer, for some initial data, the two blow-ups like $1/\sqrt{|x|}$ combine in order to give $1/|x|$, and for other initial data, the blow-ups compensate each other and disappear. More precisely, we fix both an initial vorticity ${\omega}_0$, smooth and compactly supported outside the obstacle ${\Omega}$, and the circulation ${\gamma}$ of the initial velocity around the obstacle. We assume that the obstacle ${\Omega}$ is a bounded, connected, simply connected subset of the plane. Let us define the exterior domain $\Pi := {{\mathbb R}}^2 \setminus \overline{{\Omega}}$. Then, the vorticity and the circulation uniquely determine a vector field $u_0$ tangent to the obstacle such that: $${{\rm div}\,}u_0 = 0, \ {{\rm curl}\,}u_0 = {\omega}_0, \ \lim_{|x|\to \infty} u_0(x) = 0,\ \oint_{{\partial}{\Omega}} u_0\cdot {{\mathbf{ds}}}= {\gamma}.$$ When the obstacle ${\Omega}$ is smooth and open, it is proved by Kikuchi [@kiku] that there exists a unique global strong solution to the Euler equations in $\Pi$. If ${\Omega}$ is a smooth curve ${\Gamma}$ (with two end-points), we have to define what is a weak-solution. \[sol-curve\] Let ${\omega}_0\in L^1\cap L^\infty({{\mathbb R}^2})$ and ${\gamma}\in{{\mathbb R}}$. We say that $(u,{\omega})$ is a global weak solution of the Euler equations outside the curve ${\Gamma}$ with initial condition $({\omega}_0,{\gamma})$ if $${\omega}\in L^\infty({{\mathbb R}}^+;L^1\cap L^\infty({{\mathbb R}^2}))$$ and if we have in the sense of distributions $$\label{transport} \begin{cases} {\partial_t}{\omega}+{{\rm div}\,}(u {\omega})=0, \\ {\omega}(0)={\omega}_0, \end{cases}$$ where $u$ verifies $$\left\lbrace\begin{aligned} {{\rm div}\,}u &=0 &\text{ in } {\Pi} \\ {{\rm curl}\,}u &={\omega}&\text{ in } {\Pi} \\ u\cdot\hat{n}&=0 &\text{ on } {\Gamma}\\ \oint_{{\Gamma}} u \cdot {{\mathbf{ds}}}& = {\gamma}& \text{ for }t\in[0,\infty)\\ \lim_{|x|\to\infty}|u|&=0. \end{aligned}\right.$$ In this definition, $\Pi:= {{\mathbb R}}^2\setminus {\Gamma}$, and means that we have $$\int_0^\infty\int_{{{\mathbb R}}^2}{\varphi}_t{\omega}dxdt +\int_0^\infty \int_{{{\mathbb R}}^2}{\nabla}{\varphi}.u{\omega}dxdt+\int_{{{\mathbb R}}^2}{\varphi}(0,x){\omega}_0(x)dx=0,$$ for any test function ${\varphi}\in C^\infty_c([0,\infty)\times{{\mathbb R}}^2)$. In [@lac_euler], we prove the existence of a global weak solution in the sense of the previous definition. The idea of the previous paper is the following: for ${\Gamma}$ given, we manage to construct a sequence of smooth obstacles ${\Omega}_n$ (thanks to biholomorphisms), which shrink to the curve. Next, we consider the strong solution $(u^n,{\omega}^n)$ in smooth domain $\Pi_n := {{\mathbb R}}^2\setminus \overline{{\Omega}_n}$, and we pass to the limit. The details of this proof will be presented in Subsection \[thicken\]. However, we have constructed a special family of obstacles. The first goal here is to generalized [@lac_euler] in the case of a geometrical convergence of ${\Omega}_n$ to ${\Gamma}$. \[main 2\] Let $\{{\Omega}_n\}$ be a sequence of smooth, open obstacles containing ${\Gamma}$. If ${\Omega}_n \to {\Gamma}$ in the sense of Theorem \[theo-assump\], then there exists a subsequence $n=n_k\to 0$ such that - ${\Phi}^n u^n \to u$ strongly in $L^2_{{\operatorname{{loc}}}}({{\mathbb R}}^+\times{{\mathbb R}}^2)$; - ${\Phi}^n {\omega}^n \to {\omega}$ weak-$*$ in $L^\infty({{\mathbb R}}^+;L^4({{\mathbb R}}^2))$; - $(u,{\omega})$ is a global weak solution of the Euler equations around the curve ${\Gamma}$. In this result, ${\Phi}^n$ is a cut-off function of an $\frac1n$-neighborhood of ${\Omega}_n$. In particular, we will remark that this sense of convergence holds for smooth domain, i.e. if ${\Omega}_n\to {\Omega}$, where ${\Omega}$ is smooth, then the Euler solutions on ${\Omega}_n$ (respectively on the exterior domain $\Pi_n$) tends to the Euler solution on ${\Omega}$ (respectively on $\Pi$). Another consequence of proving a geometrical theorem (Theorem \[theo-assump\]) is the extension of [@lac_NS], which corresponds to the previous theorem in the viscous case (with Navier-Stokes equations instead to Euler equations). The second goal of this article is to study the behavior of the weak solution when the curve shrinks to a point. In [@ift_lop_euler], the authors fix a smooth obstacle ${\Omega}_0$, containing the origin, and choose ${\Omega}_{\varepsilon}:= {\varepsilon}{\Omega}_0$. For this homothetic convergence, they prove the following theorem. \[ift\_main\] There exists a subsequence ${\varepsilon}={\varepsilon}_k\to 0$ such that - ${\Phi}^{\varepsilon}u^{\varepsilon}\to u$ strongly in $L^1_{{\operatorname{{loc}}}}({{\mathbb R}}^+\times{{\mathbb R}}^2)$; - ${\Phi}^{\varepsilon}{{\omega}}^{\varepsilon}\to {{\omega}}$ weak $*$ in $L^\infty({{\mathbb R}}^+\times{{\mathbb R}}^2)$; - the limit pair $(u,{{\omega}})$ verify in the sense of distributions: $$\left\lbrace\begin{aligned} &{\partial}_t {{\omega}}+u\cdot {\nabla}{{\omega}}=0 &\text{ in } (0,\infty) \times {{\mathbb R}}^2 \\ &{{\rm div}\,}u =0 &\text{ in } (0,\infty) \times {{\mathbb R}}^2 \\ &{{\rm curl}\,}u ={{\omega}} + {\gamma}{\delta}_0 &\text{ in } (0,\infty) \times {{\mathbb R}}^2 \\ &\lim_{|x|\to\infty}|u|=0 & \text{ for }t\in[0,\infty)\\ &{{\omega}}(0,x)={{\omega}}_0(x) &\text{ in } {{\mathbb R}}^2 \end{aligned}\right.$$ with ${\delta}_0$ the Dirac function at $0$. In this result, ${\Phi}^{\varepsilon}$ is a cut-off function of an ${\varepsilon}$-neighborhood of ${\Omega}_{\varepsilon}$. Therefore, they obtain at the limit the Euler equations in the full plane, where a Dirac mass at the origin appears. This additional term is a reminiscense of the circulation ${\gamma}$ of the initial velocities around the obstacles, and we note that this term does not appear if ${\gamma}=0$. Actually, we can write the velocity as a sum of a smooth vector field and ${\gamma}\dfrac{x^\perp}{2\pi |x|^2}$. Additionally, [@lac_miot] proves that there exists at most one global solution of the previous limit system. Therefore, we can state that Theorem \[ift\_main\] holds true for all sequences ${\varepsilon}_k\to 0$, without extracting subsequences. In the exterior of the curve, we will note in Remark \[rem : prop-curve\] that the velocity, for any weak solution, is continuous up to the curve, with different values on each side of ${\Gamma}$, and blows up at the endpoints of the curve as the inverse of the square root of the distance. As it was said at the beginning of this introduction, we remark in this two results that the velocity blows up like $1/|x|$ in the case of a point, and like $1/\sqrt{|x|}$ near the end-points in the case of the curve. The problem here is to check that we find a result similar to Theorem \[ift\_main\] when a curve shrinks to a point. We fix a smooth open Jordan arc ${\Gamma}$, and we set ${\Gamma}_{\varepsilon}:= {\varepsilon}{\Gamma}$. Then there exists at least one weak solution of Euler equation outside the curve ${\Gamma}_{\varepsilon}$. Our goal is to prove the following theorem. \[main\] Let $(u^{\varepsilon},{\omega}^{\varepsilon})$ be a weak solution for Euler equation outside ${\Gamma}_{\varepsilon}$. Then, for all sequences ${\varepsilon}={\varepsilon}_k\to 0$, we have - $u^{\varepsilon}\to u$ strongly in $L^1_{{\operatorname{{loc}}}}({{\mathbb R}}^+ \times {{\mathbb R}}^2)$; - ${{\omega}}^{\varepsilon}\to {{\omega}}$ weak $*$ in $L^\infty({{\mathbb R}}^+ \times {{\mathbb R}}^2)$; - the limit pair $(u,{{\omega}})$ is the unique solution in the sense of distributions of: $$\left\lbrace\begin{aligned} &{\partial}_t {{\omega}}+u\cdot {\nabla}{{\omega}}=0 &\text{ in } (0,\infty) \times {{\mathbb R}}^2 \\ &{{\rm div}\,}u =0 &\text{ in } (0,\infty) \times {{\mathbb R}}^2 \\ &{{\rm curl}\,}u ={{\omega}} + {\gamma}{\delta}_0 &\text{ in } (0,\infty) \times {{\mathbb R}}^2 \\ &\lim_{|x|\to\infty}|u|=0 & \text{ for }t\in[0,\infty)\\ &{{\omega}}(0,x)={{\omega}}_0(x) &\text{ in } {{\mathbb R}}^2. \end{aligned}\right.$$ Although Theorems \[ift\_main\] and \[main\] appear to be similar, the estimations and arguments are different. In all these works, the main tool is the explicit formula of the Biot-Savart law (law giving the velocity in terms of vorticity), thanks to conformal mappings. In [@ift_lop_euler], the authors use the change of variables $y=x/{\varepsilon}$, in order to work in a fixed domain. Then they obtain $L^\infty$ estimates for a part of the velocity, thanks to the smoothness of the obstacle, and they pass to the limit using a Div-Curl Lemma. In [@lac_euler], the change of variables does not hold and we work on convergence of biholomorphisms when the domains change. Next, we take advantage of this convergence to pass directly to the limit in the Biot-Savart law, thanks to the dominated convergence theorem. In our case of the small curve, we lost the convergence of biholomorphism and [@lac_euler] cannot be applied directly. The curve being a non-smooth domain, because of the end-points, we cannot either apply directly the result from [@ift_lop_euler]. Indeed, we will see that we only have $L^p$ estimates of the velocity for $p<4$ instead of $L^\infty$. Actually, we will improve some estimates and we will manage to pass to the limit with the Div-Curl Lemma. The remainder of this work is organized in three sections. In Section \[sect : 2\], we recall some results on conformal mapping and Biot-Savart law. We show that [@lac_euler] gives us the existence of at least one global weak solution $(u^{\varepsilon},{\omega}^{\varepsilon})$ for the Euler equations outside the curve ${\Gamma}_{\varepsilon}$ (in the sense of Definition \[sol-curve\]). We take advantage of this part to prove the convergence of biholomorphisms when an obstacle shrinks to a curve, completing [@lac_euler; @lac_NS]. Theorem \[main 2\] will be a consequence of this section. There will be a general remark concerning the convergence of Euler solutions when the domain converges. In Section \[sect : 3\], we establish [*a priori*]{} estimates for the vorticity and the velocity, in order to pass to the limit in the last section. We emphasize that the techniques used here, and in [@ift_lop_euler; @lac_euler], are specific to the ideal flows in dimension two, around one obstacle. The study of several obstacles does not allow us to use Riemann mappings. Loosing the explicit formula of the Biot-Savart law, the author in [@milton] choose to work in a bounded domain, with several holes, where one hole shrinks to a point. In a bounded domain, he can use the maximum principle, and he obtains a theorem similar to Theorem \[ift\_main\]. In [@lac_lop], we work in an unbounded domain, with $n$ obstacles, of size ${\varepsilon}$, uniformly distributed on a imaginary curve ${\Gamma}$, and the goal is to determine if a chain of small islands ($n\to \infty$, ${\varepsilon}\to 0$) has the same effect as a wall ${\Gamma}$. Concerning the viscous flow, there is no control of the vorticity in a domain with some boundary. Therefore, we do not use ${\omega}$ and the Biot-Savart law, and we gain control on one derivative of the velocity thanks to the energy inequality. The case of a small obstacle in dimension two is studied in [@ift_lop_NS], and in dimension three in [@ift_kell]. The thin obstacle in dimension two is treated in [@lac_NS], and in dimension three in [@lac_3D]. We note that there does not exist any result for ideal flow in dimension three. Indeed, we cannot use the vorticity equation, and we do not have energy inequality. Then, a control of the derivative of the velocity is missing. For the sake of clarity, the main notations are listed in an appendix at the end of the paper. Conformal mapping {#sect : 2} ================= As it is mentioned in the introduction, complex analysis is an important tool for the study of two dimensional ideal flow outside one obstacle. Identifying ${{\mathbb R}}^2$ with the complex plane ${{\mathbb C}}$, the biholomorphism mapping the exterior of the obstacle to the exterior of the unit disk will be used to obtain an explicit formula for the Biot-Savart law. A key of this work, as in [@ift_lop_euler; @lac_euler], is to estimate these biholomorphisms when the size and the sickness of the obstacle go to zero. We begin this section by some reminders on thin obstacles (see [@lac_euler] for more details). Thin obstacles -------------- Let $D=B(0,1)$ and $S={\partial}D$. We begin by giving some basic definitions on the curve. We call a [*Jordan arc*]{} a curve $C$ given by a parametric representation $C:{\varphi}(t)$, $0\leq t\leq 1$ with ${\varphi}$ an injective ($=$ one-to-one) function, continuous on $[0,1]$. An [*open Jordan arc*]{} has a parametrization $C:{\varphi}(t)$, $0< t<1$ with ${\varphi}$ continuous and injective on $(0,1)$. We call a [*Jordan curve*]{} a curve $C$ given by a parametric representation $C:{\psi}(t)$, $t\in {{\mathbb R}}$, $1$-periodic, with ${\psi}$ an injective function on $[0,1)$, continuous on ${{\mathbb R}}$. Thus a Jordan curve is closed (${\varphi}(0)={\varphi}(1)$) whereas a Jordan arc has distinct endpoints. If $J$ is a Jordan curve in ${{\mathbb C}}$, then the Jordan Curve Theorem states that ${{\mathbb C}}\setminus J$ has exactly two components $G_0$ and $G_1$, and these satisfy ${\partial}G_0={\partial}G_1=J$. The Jordan arc (or curve) is of class $C^{n,{\alpha}}$ ($n\in{{\mathbb N}}^*,0<{\alpha}\leq 1$) if its parametrization ${\varphi}$ is $n$ times continuously differentiable, satisfying ${\varphi}'(t)\neq 0$ for all $t$, and if $|{\varphi}^{(n)}(t_1)-{\varphi}^{(n)}(t_2)|\leq C|t_1-t_2|^{\alpha}$ for all $t_1$ and $t_2$. Let ${\Gamma}: {\Gamma}(t),0\leq t\leq 1$ be a Jordan arc. Then the subset ${{\mathbb R}}^2\setminus{\Gamma}$ is connected and we will denote it by $\Pi$. The purpose of the following proposition is to obtain some properties of a biholomorphism $T: \Pi \to {{\rm int}\,}\ D^c$. After applying a homothetic transformation, a rotation and a translation, we can suppose that the endpoints of the curve are $-1={\Gamma}(0)$ and $1={\Gamma}(1)$. \[2.2\] If ${\Gamma}$ is a $C^2$ Jordan arc, such that the intersection with the segment $[-1,1]$ is a finite union of segments and points, then there exists a biholomorphism $T:\Pi\to {{\rm int}\,}\ D^c$ which verifies the following properties: - $T^{-1}$ and $DT^{-1}$ extend continuously up to the boundary, and $T^{-1}$ maps $S$ to ${\Gamma}$, - $DT^{-1}$ is bounded, - $T$ and $DT$ extend continuously up to ${\Gamma}$ with different values on each side of ${\Gamma}$, except at the endpoints of the curve where $T$ behaves like the square root of the distance and $DT$ behaves like the inverse of the square root of the distance, - $DT$ is bounded in the exterior of the disk $B(0,R)$, with ${\Gamma}\subset B(0,R)$, - $DT$ is bounded in $L^p(\Pi\cap B(0,R))$ for all $p<4$ and $R>0$. The behavior of $T$ and $DT$ gives us the behavior of the velocity around the curve (see Proposition \[remark\_lac\_euler\]). We rewrite also a remark from [@lac_euler] concerning behavior of biholomorphisms at infinity. \[2.5\]If we have a biholomorphism $H$ between the exterior of an open connected and simply connected domain $A$ and $D^c$, such that $H(\infty)=\infty$, then there exists a nonzero real number ${\beta}$ and a holomorphic function $h:\Pi\to {{\mathbb C}}$ such that: $$H(z)={\beta}z+ h(z).$$ with $$h'(z)=O\Bigl(\frac{1}{|z|^2}\Bigl), \text{ as }|z|\to\infty.$$ This property can be applied for the $T$ above. In [@lac_euler; @lac_NS], we consider a family of obstacles $\{{\Omega}_{\eta}\}$ which shrink to the curve ${\Gamma}$ in the following sense. If we denote by $T_{\eta}$ the biholomorphism between $\Pi_{\eta}:= {{\mathbb R}}^2\setminus \overline{{\Omega}_{\eta}}$ and $D^c$, then we supposed the following properties: \[assump\] The biholomorphism family $\{T_{\eta}\}$ verifies - $\|(T_{\eta}- T)/|T|\|_{L^\infty(\Pi_{\eta})}\to 0$ as ${\eta}\to 0$, - $\det(DT_{\eta}^{-1})$ is bounded on $D^c$ independently of ${\eta}$, - for any $R>0$, $\|DT_{\eta}- DT\|_{L^3(B(0,R)\cap \Pi_{\eta})}\to 0$ as ${\eta}\to 0$, - for $R>0$ large enough, there exists $C_R>0$ such that $|DT_{\eta}(x)|\leq C_R$ on $B(0,R)^c$. - for $R>0$ large enough, there exists $C_R>0$ such that $|D^2 T_{\eta}(x)|\leq \frac{C_R}{|x|}$ on $B(0,R)^c$. We can observe that property (iii) implies that for any $R$, $DT_{\eta}$ is bounded in $L^p(B(0,R)\cap \Pi_{\eta})$ independently of ${\eta}$, for $p\leq3$. Moreover, condition (i) means that $T_{\eta}\to T$ uniformly on $B(0,R)\cap \Pi_{\eta}$ for any $R>0$. Assumption \[assump\] corresponds to Assumption 3.1 in [@lac_euler], adding part (v) and strengthening property (i) therein. Concerning our problem of the small curve, we will not need to assume something. Indeed, in the following subsection, we will present a way of thicken the curve such that all the properties of Assumption \[assump\] are verified. However, an open problem raised by [@lac_euler; @lac_NS] is to prove that Assumption \[assump\] is also verified for more general geometrical convergences ${\Omega}_{\eta}\to {\Gamma}$. It is the purpose of Subsection \[assump-proof\]. Thicken the curve {#thicken} ----------------- Thanks to Proposition \[2.2\], for a curve ${\Gamma}$ given, we associate its biholomorphism $T$. Therefore, we have for $$\label{T_eps} {\Gamma}_{\varepsilon}:= {\varepsilon}{\Gamma},\ T_{\varepsilon}(x)=T(x/{\varepsilon}).$$ We recall that $T_{\varepsilon}$ maps $\Pi_{\varepsilon}:= {{\mathbb R}}^2\setminus {\Gamma}_{\varepsilon}$ to ${{\mathbb R}}^2 \setminus \overline{D}$ and ${\Gamma}_{\varepsilon}$ to ${\partial}D$. Let us define $${\Omega}_{{\varepsilon},{\eta}} := T_{\varepsilon}^{-1}(B(0,1+{\eta})\setminus D).$$ Knowing that $T_{\varepsilon}$ is a biholomorphism, we can state that ${\Omega}_{{\varepsilon},{\eta}}$ is a smooth, bounded, open, connected, simply connected subset of the plane containing ${\Gamma}_{\varepsilon}$. We can also remark that $$\label{T_eps_eta} T_{{\varepsilon},{\eta}}(x)=\frac{1}{1+{\eta}}T_{\varepsilon}(x)=\frac{1}{1+{\eta}}T(x/{\varepsilon})$$ is a biholomorphism mapping $\Pi_{{\varepsilon},{\eta}}:= {\Omega}_{{\varepsilon},{\eta}}^c$ to $D^c$ and ${\partial}{\Omega}_{{\varepsilon},{\eta}}$ to ${\partial}D$. We give here the shape of ${\Omega}_{{\varepsilon},{\eta}}$ in the special case of the segment. If ${\Gamma}:= [(-1,0);(1,0)]$, we have an explicit form for $T$. It is the inverse map of the Joukowski function: $$G(z)=\frac{1}2 (z+\frac1z) \text{ (see \cite{lac_euler} for more details about this function).}$$ In this case, we can easily compute that ${\Omega}_{{\varepsilon},{\eta}}$ is the interior of an ellipse parametrized by $$x({\theta})=\frac{{\varepsilon}}2 \bigl( (1+{\eta})+\frac{1}{1+{\eta}}\bigl) \cos {\theta},\ y({\theta})=\frac{{\varepsilon}}2 \bigl( (1+{\eta})-\frac{1}{1+{\eta}}\bigl) \cos {\theta}.$$ Then, for small ${\eta}$, the length of the ellipse is approximately (Taylor expansion of order 2) ${\varepsilon}(2+{\eta}^2)$ whereas the higher is ${\varepsilon}(2{\eta})$. We can also see that for ${\eta}=0$, we obtain the segment ${\Gamma}_{\varepsilon}=[(-{\varepsilon},0);({\varepsilon},0)]$. We note that, for ${\varepsilon}$ fixed, the family defined in verifies Assumption \[assump\]. Therefore, we can apply directly the result obtain in [@lac_euler]. Let ${\omega}_0$ be a smooth initial vorticity, compactly supported outside the obstacle. Let ${\gamma}$ be a real. The motion of an incompressible ideal flow in $\Pi_{{\varepsilon},{\eta}}$ is governed by the Euler equations: $$\left\lbrace\begin{aligned} &{\partial}_t u^{{\varepsilon},{\eta}}+u^{{\varepsilon},{\eta}}\cdot {\nabla}u^{{\varepsilon},{\eta}}=-{\nabla}p^{{\varepsilon},{\eta}} &\text{ in }(0,\infty) \times {\Pi_{{\varepsilon},{\eta}}} \\ &{{\rm div}\,}u^{{\varepsilon},{\eta}} =0 &\text{ in } [0,\infty) \times {\Pi_{{\varepsilon},{\eta}}} \\ &u^{{\varepsilon},{\eta}}\cdot \hat{n} =0 &\text{ on } [0,\infty) \times {{\partial}{\Omega}_{{\varepsilon},{\eta}}}\\ &\lim_{|x|\to\infty}|u^{{\varepsilon},{\eta}}|=0 & \text{ for }t\in[0,\infty)\\ &u^{{\varepsilon},{\eta}}(0,x)=u_0^{{\varepsilon},{\eta}}(x) &\text{ in } \Pi_{{\varepsilon},{\eta}} \end{aligned}\right.$$ where $p^{{\varepsilon},{\eta}}=p^{{\varepsilon},{\eta}}(t,x)$ is the pressure. In fact, to study the two dimensional ideal flows, it is more convenient to work on the vorticity equations ${{\omega}}^{{\varepsilon},{\eta}}:= {{\rm curl}\,}u^{{\varepsilon},{\eta}}(={\partial}_1 u^{{\varepsilon},{\eta}}_2-{\partial}_2 u^{{\varepsilon},{\eta}}_1)$ which are equivalent to the previous system: $$\label{euler} \left\lbrace\begin{aligned} &{\partial}_t {{\omega}}^{{\varepsilon},{\eta}}+u^{{\varepsilon},{\eta}}\cdot {\nabla}{{\omega}}^{{\varepsilon},{\eta}}=0 &\text{ in } (0,\infty) \times {\Pi_{{\varepsilon},{\eta}}} \\ &{{\rm div}\,}u^{{\varepsilon},{\eta}} = 0, \ {{\rm curl}\,}u^{{\varepsilon},{\eta}} = {\omega}^{{\varepsilon},{\eta}} &\text{ in } (0,\infty) \times {\Pi_{{\varepsilon},{\eta}}} \\ &u^{{\varepsilon},{\eta}}\cdot \hat{n} =0 &\text{ on } [0,\infty) \times {{\partial}{\Omega}_{{\varepsilon},{\eta}}}\\ &\lim_{|x|\to\infty}|u^{{\varepsilon},{\eta}}|=0 & \text{ for }t\in[0,\infty)\\ & \oint_{{\partial}{\Omega}_{{\varepsilon},{\eta}}} u^{{\varepsilon},{\eta}}\cdot {{\mathbf{ds}}}= {\gamma}& \text{ for }t\in[0,\infty)\\ &{{\omega}}^{{\varepsilon},{\eta}}(0,x)={{\omega}}_0(x) &\text{ in } \Pi_{{\varepsilon},{\eta}}. \end{aligned}\right.$$ The interest of such a formulation is that we recognize a transport equation. The transport nature allows us to conclude that the $L^p(\Pi_{{\varepsilon},{\eta}})$ norms of the vorticity are conserved, for $p\in [1,\infty]$, which gives us directly an estimate and a weak convergence in $L^\infty(L^p)$ for the vorticity. For all ${\varepsilon},{\eta}$, ${\Omega}_{{\varepsilon},{\eta}}$ is smooth and Kikuchi in [@kiku] states that there exists a unique pair $(u^{{\varepsilon},{\eta}},{\omega}^{{\varepsilon},{\eta}})$ which is a global strong solution of Euler equation in ${{\mathbb R}}^+\times \Pi_{{\varepsilon},{\eta}}$ verifying $${\omega}^{{\varepsilon},{\eta}}(0,\cdot)={\omega}_0,\ \int_{{\partial}{\Omega}_{{\varepsilon},{\eta}}} u^{{\varepsilon},{\eta}}(0,s)\cdot \tau \, ds ={\gamma}\text{ and } \lim_{|x|\to \infty} u^{{\varepsilon},{\eta}}(t,x)=0 \ \forall t.$$ A characteristic of this solution is the conservation of the velocity circulation on the boundary, and that $m:= \int {{\rm curl}\,}u^{{\varepsilon},{\eta}}= \int {\omega}_0$. Next, we apply the result of [@lac_euler] to define a pair $(u^{\varepsilon},{\omega}^{\varepsilon})$: \[lac\_euler\] If Assumption \[assump\] is verified, then there exists a subsequence ${\eta}_k\to 0$ with $${\Phi}^{{\varepsilon},{\eta}} {\omega}^{{\varepsilon},{\eta}} \rightharpoonup {\omega}^{\varepsilon}\text{ weak-$*$ in } L^\infty({{\mathbb R}};L^4({{\mathbb R}}^2)) \text{ and } {\Phi}^{{\varepsilon},{\eta}} u^{{\varepsilon},{\eta}} \to u^{\varepsilon}\text{ strongly in } L^2_{{\operatorname{{loc}}}}({{\mathbb R}}\times {{\mathbb R}}^2)$$ where the following properties are verified - $u^{\varepsilon}$ can be expressed in terms of ${\gamma}$, and ${\omega}^{\varepsilon}$: $$u^{\varepsilon}(x)=\dfrac{1}{2\pi} DT_{\varepsilon}^t(x) \int_{{{\mathbb R}}^2} \Bigl(\dfrac{(T_{\varepsilon}(x)-T_{\varepsilon}(y))^\perp}{|T_{\varepsilon}(x)-T_{\varepsilon}(y)|^2}-\dfrac{(T_{\varepsilon}(x)- T_{\varepsilon}(y)^*)^\perp}{|T_{\varepsilon}(x)- T_{\varepsilon}(y)^*|^2}\Bigl){\omega}^{\varepsilon}(t,y)dy + {\alpha}\dfrac{1}{2\pi} DT_{\varepsilon}^t(x) \dfrac{T_{\varepsilon}(x)^\perp}{|T_{\varepsilon}(x)|^2}$$ with ${\alpha}= \int {\omega}_0 + {\gamma}$; - $u^{\varepsilon}$ and ${\omega}^{\varepsilon}$ are weak solutions of $${\partial}_t {\omega}^{\varepsilon}+u^{\varepsilon}.{\nabla}{\omega}^{\varepsilon}=0 \text{ in }{{\mathbb R}}^2\times(0,\infty).$$ In this theorem, ${\Phi}^{{\varepsilon},{\eta}}$ denotes a cutoff function of an ${\eta}$-neighborhood of ${\Gamma}_{\varepsilon}$, and we write $x^*=\frac{x}{|x|^2}$. Moreover, we can find in [@lac_euler] the following properties. \[remark\_lac\_euler\] Let $u^{\varepsilon}$ be given as in Theorem \[lac\_euler\]. For fixed $t$, the velocity - is continuous on ${{\mathbb R}}^2\setminus {\Gamma}_{\varepsilon}$ and tends to zero at infinity. - is continuous up to ${\Gamma}_{\varepsilon}\setminus \{(-{\varepsilon},0);({\varepsilon},0)\}$, with different values on each side of ${\Gamma}_{\varepsilon}$. - blows up at the endpoints of the curve like $C/\sqrt{|x-({\varepsilon},0)||x+({\varepsilon},0)|}$, which belongs to $L^p_{{\operatorname{{loc}}}}$ for $p<4$. - is tangent to the curve, with circulation ${\gamma}$. - we have ${{\rm div}\,}u^{\varepsilon}=0$ and ${{\rm curl}\,}u^{\varepsilon}={\omega}^{\varepsilon}+g^{\varepsilon}_{{\omega}^{\varepsilon}}{\delta}_{{\Gamma}_{\varepsilon}}$ in the sense of distributions of ${{\mathbb R}}^2$. The function $g^{\varepsilon}_{{\omega}^{\varepsilon}}$ is continuous on ${\Gamma}_{\varepsilon}$ and blows up at the endpoints of the curve ${\Gamma}_{\varepsilon}$ as the inverse of the square root of the distance. One can also characterize $g^{\varepsilon}_{{\omega}^{\varepsilon}}$ as the jump of the tangential velocity across ${\Gamma}_{\varepsilon}$. There is a sharp contrast between the behavior of ideal flows around a small and a thin obstacle. In [@ift_lop_euler], the additional term due to the vanishing obstacle appears as a time-independent additional convection centered at $P$, whereas in the case of a thin obstacle, the correction term depends on the time. Moreover, if initially ${\gamma}=0$ we note that there is no singular term in the case of the small obstacle, whereas $g_{{\omega}}$ again appears in the case of a thin obstacle. Physically, it can be interpreted by the fact that a wall blocks the fluids, a point not. Therefore, for ${\Gamma}_{\varepsilon}$ given, we have constructed a weak solution $(u^{\varepsilon},{\omega}^{\varepsilon})$ for the Euler equation outside the curve ${\Gamma}_{\varepsilon}$ in the sense of Definition \[sol-curve\]. Actually, it misses one property which can be establish as follow. For all ${\eta}$, ${\omega}^{{\varepsilon},{\eta}}$ verifies the transport equation in a strong sense, then we have the classical estimates $\| {\omega}^{{\varepsilon},{\eta}}(t,\cdot)\|_{L^p} = \| {\omega}_0\|_{L^p}$ for all $p\in [1,\infty]$. Then, the weak limit allows us to state that: $$\|{\omega}^{{\varepsilon}}(t,\cdot)\|_{L^p(\Pi_{{\varepsilon}})}\leq \limsup_{{\eta}\to 0} \|{\omega}^{{\varepsilon},{\eta}}(t,\cdot)\|_{L^p(\Pi_{{\varepsilon}})}=\|{\omega}_0\|_{L^p({{\mathbb R}}^2)},$$ which means that ${\omega}^{\varepsilon}\in L^{\infty}(L^1\cap L^\infty)$. By thicken the curve, we prove here that there exists at least one weak solution of the Euler equations outside the curve. The goal of this paper is to study the limit of $(u^{\varepsilon},{\omega}^{\varepsilon})$, where $(u^{\varepsilon},{\omega}^{\varepsilon})$ is one of these solutions. An important future work will be the uniqueness of $(u^{\varepsilon},{\omega}^{\varepsilon})$. Indeed, until now, it is possible that this pair depends on the way of the obstacles shrink to the curve, and the convergence $(u^{{\varepsilon},{\eta}},{\omega}^{{\varepsilon},{\eta}})\to (u^{\varepsilon},{\omega}^{\varepsilon})$ holds true by extracting a subsequence. We have: $$T_{{\varepsilon},{\eta}}=T_{1,{\eta}}(x/{\varepsilon}).$$ This equality means that, for ${\eta}>0$ fixed, we are exactly in the case of [@ift_lop_euler] (see Theorem \[ift\_main\]). Then, we can apply directly their result to extract a subsequence ${\varepsilon}_k \to 0$ such that $${\Phi}^{{\varepsilon},{\eta}} {\omega}^{{\varepsilon},{\eta}} \rightharpoonup {\omega}^{\eta}\text{ weak-$*$ in } L^\infty({{\mathbb R}}\times {{\mathbb R}}^2) \text{ and } {\Phi}^{{\varepsilon},{\eta}} u^{{\varepsilon},{\eta}} \to u^{\eta}\text{ strongly in } L^2_{{\operatorname{{loc}}}}({{\mathbb R}}\times {{\mathbb R}}^2)$$ where the following properties are verified - $u^{\eta}$ and ${\omega}^{\eta}$ are weak solutions of ${\partial}_t {\omega}^{\eta}+u^{\eta}.{\nabla}{\omega}^{\eta}=0$ in ${{\mathbb R}}^2\times(0,\infty)$; - ${{\rm div}\,}u^{\eta}=0$ and ${{\rm curl}\,}u^{\eta}={\omega}^{\eta}+{\gamma}{\delta}_{0}$ in the sense of distributions of ${{\mathbb R}}^2$; - $|u^{\eta}|\to 0$ at infinity. Moreover, [@lac_miot] establishes the uniqueness of the previous problem, then we can state that the convergence holds true without extraction of a subsequence, and the limit does not depend on ${\eta}$: $$(u^{{\varepsilon},{\eta}},{\omega}^{{\varepsilon},{\eta}})\to (u,{\omega}) \text{ as } {\varepsilon}\to 0.$$ ### Comment on the previous remark We could take advantage of the previous remark thinking along this line: assuming that we can prove that $$\label{conv-unif} (u^{{\varepsilon},{\eta}},{\omega}^{{\varepsilon},{\eta}})\to (u^{\varepsilon},{\omega}^{\varepsilon}) \text{ as } {\eta}\to 0 \text{, uniformly in }{\varepsilon},$$ then, for all $\rho >0$, there exists a ${\eta}_\rho$, such that $$\| (u^{\varepsilon},{\omega}^{\varepsilon}) - (u^{{\varepsilon},{\eta}},{\omega}^{{\varepsilon},{\eta}}) \| \leq \rho/2,\ \forall {\varepsilon}.$$ For this ${\eta}_\rho$ fixed, we apply directly the result of [@ift_lop_euler] to find ${\varepsilon}_\rho$, such that for all ${\varepsilon}\leq {\varepsilon}_\rho$, we have $$\| (u^{{\varepsilon},{\eta}_\rho},{\omega}^{{\varepsilon},{\eta}_\rho}) - (u,{\omega}) \| \leq \rho/2.$$ The fact that we find the same $(u,{\omega})$ for any ${\eta}$ comes from [@lac_miot]. Therefore, we have found ${\varepsilon}_\rho$, such that for all ${\varepsilon}\leq {\varepsilon}_\rho$, we have $$\| (u^{\varepsilon},{\omega}^{\varepsilon}) - (u,{\omega}) \| \leq \| (u^{\varepsilon},{\omega}^{\varepsilon}) - (u^{{\varepsilon},{\eta}_\rho},{\omega}^{{\varepsilon},{\eta}_\rho}) \| + \| (u^{{\varepsilon},{\eta}_\rho},{\omega}^{{\varepsilon},{\eta}_\rho}) - (u,{\omega}) \| \leq \rho,$$ which is the desired result. To make this proof rigourous, we have to prove , i.e. to rewrite the article [@lac_euler] with adding the parameter ${\varepsilon}$, and to check carefully that the convergence is uniform in ${\varepsilon}$. It is possible to check that all the estimates are uniform in ${\varepsilon}$, but it is difficult to give a sense to uniform convergence for the weak-$*$ topology. We would have had to establish a uniform version of Banach-Alaoglu’s Theorem. Indeed, we always say: as $\| {\omega}^{{\varepsilon},{\eta}} \|_{L^p} = \| {\omega}_0 \|_{L^p}$, we can extract a subsequence such that $ {\omega}^{{\varepsilon},{\eta}} \rightharpoonup {\omega}^{\varepsilon}$ weak-$*$ in $L^\infty(L^p)$, as ${\eta}\to 0$. What does it mean and how can we prove that we can choose a subsequence such that this convergence weak-$*$ is uniform in ${\varepsilon}$ ? Actually, the general idea is to pass to the limit $(u^{\varepsilon},{\omega}^{\varepsilon})\to(u,{\omega})$ with similar arguments to [@ift_lop_euler]. This last article cannot be used directly and, as we will see, it is not obvious to adapt their arguments. An other possibility is to adapt the arguments of [@lac_euler]. However, in the present case of the small curve, some estimates becomes better than in the case of the thin obstacle, and we will see that we can apply Aubin-Lions and Div-Curl Lemmas, in order to pass to the limit. This idea appeared in [@ift_lop_euler], cannot be applied in [@lac_euler], and we choose here to use it in our case because it goes faster than using the arguments from [@lac_euler]. ### Biholomorphism estimate As it was focused in the introduction, we need to estimate the biholomorphism to estimate the velocity, thanks to the Biot-Savart law. \[biholo-est\] The biholomorphism family $\{T_{{\varepsilon}}\}$, defined in , verifies - ${\varepsilon}^{-2} \det(DT_{{\varepsilon}}^{-1})$ is bounded on $D^c$ independently of ${\varepsilon}$, - for any $R>0$, for all $p\in [1,4)$, ${\varepsilon}\|DT_{\varepsilon}\|_{L^p(B(0,R))}$ is bounded uniformly in ${\varepsilon}$, - for $R>0$ large enough, there exists $C_R>0$ such that $|DT_{{\varepsilon}}(x)|\leq \dfrac{C_R}{\varepsilon}$ on $B(0,R)^c$. The point (iii) is straightforward using and Proposition \[2.2\]. Concerning (i), we directly see that $$T_{{\varepsilon}}^{-1}(x) = {\varepsilon}T^{-1}(x),$$ hence $$\det(DT_{{\varepsilon}}^{-1}) (x) = {\varepsilon}^2 \det(DT^{-1}) (x),$$ which prove point (i), thanks to Proposition \[2.2\]. For the last point, we compute $$\begin{aligned} \Bigl( \int_{B(0,R)} \bigl| \frac1{\varepsilon}DT(\frac{x}{\varepsilon}) \bigl|^p\, dx\Bigl)^{1/p}= {\varepsilon}^{\frac2p - 1} \Bigl( \int_{B(0,R/{\varepsilon})} \bigl| DT(y) \bigl|^p\, dy\Bigl)^{1/p}.\end{aligned}$$ Using Proposition \[2.2\], we know that $DT$ belongs in $L^p (0,R_1)$ for all $R_1>0$ and all $p<4$. However, we should also take care of the behavior of $DT$ at infinity, because $\lim_{{\varepsilon}\to 0} R/{\varepsilon}=\infty$. Remark \[2.5\] allows us to pretend that there exists $\tilde R$ such that $$|DT(y)| \leq {\beta}+1 \text{ for } |y|\geq \tilde R.$$ Therefore, for $p<4$ $$\begin{aligned} \Bigl( \int_{B(0,R/{\varepsilon})} \bigl| DT(y) \bigl|^p\, dy\Bigl)^{1/p}&\leq&\Bigl( \int_{B(0,\tilde R)} \bigl| DT(y) \bigl|^p\, dy\Bigl)^{1/p}+\Bigl( \int_{B(0,R/{\varepsilon})\setminus B(0,\tilde R)} ( {\beta}+1 )^p\, dy\Bigl)^{1/p} \\ &\leq& C_p + \frac{C_{\beta}R^{2/p}}{{\varepsilon}^{2/p}},\end{aligned}$$ which means that, for ${\varepsilon}$ small enough, independently of $R$, $$\label{T_Lp} \|DT_{\varepsilon}\|_{L^p(B(0,R)\cap\Pi_{{\varepsilon},{\eta}})} \leq {\varepsilon}^{- 1} C(1 + R^{2/p}),$$ which ends the proof. Biot-Savart law {#sect:biot} ---------------   One of the key of the study for two dimensional ideal flow is to work with the vorticity equation, which is a transport equation. For example, in the case of a smooth obstacle, we choose initially ${\omega}_0 \in L^1\cap L^\infty$, then $\| {\omega}^{\varepsilon}(t,\cdot)\|_{L^p}= \| {\omega}_0\|_{L^p}$ for all $t$. Next, Banach-Alaoglu’s Theorem allows us to extract a subsequence such that ${\omega}^{\varepsilon}\rightharpoonup {\omega}$ weak-$*$. So, we have some estimates and weak-$*$ convergence for the vorticity, and the goal is to establish estimates and strong convergence for the velocity. For that, we introduce the Biot-Savart law, which gives the velocity in terms of the vorticity. Another advantage of the two dimensional space is that we have explicit formula, thanks to complex analysis and the identification of ${{\mathbb R}}^2$ and ${{\mathbb C}}$. Let ${\Omega}$ be a bounded, connected, simply connected subset of the plane. We denote by $\Pi$ the exterior domain: $\Pi:= {{\mathbb R}}^2\setminus\overline{{\Omega}}$, and let $T$ be a biholomorphism between $\Pi$ and $(\overline D)^c$ such that $T(\infty)=\infty$. We denote by $G_{\Pi}=G_{\Pi} (x,y)$ the Green’s function, whose the value is: $$\label{green} G_{\Pi}(x,y)=\frac{1}{2\pi}\ln \frac{|T(x)-T(y)|}{|T(x)-T(y)^*||T(y)|}$$ writing $x^*=\frac{x}{|x|^2}$. The Green’s function verifies: $$\left\lbrace \begin{aligned} &{\Delta}_y G_{\Pi}(x,y)={\delta}(y-x) \text{ for }x,y\in \Pi \\ &G_{\Pi}(x,y)=0 \text{ for }y\in{\Gamma}\\ &G_{\Pi}(x,y)=G_{\Pi}(y,x) \end{aligned} \right.$$ \[T-unique\] In fact, the Green’s function is unique, even in the case where ${\Omega}$ is a curve ${\Gamma}$. Indeed, let $F_1$ and $F_2$ be two biholomorphisms from $\Pi$ to the exterior of the unit disk. Then $F_1\circ F_2^{-1}$ maps $D^c$ to $D^c$ and we can apply the uniqueness result for Riemann mappings to conclude that there exists $a\in {{\mathbb C}}$ such that $|a|=1$ and $F_1=aF_2$. Moreover, changing $T$ by $aT$ in does not change the Green’s function. The uniqueness of the Green’s function outside the unit disk gives us the result. The kernel of the Biot-Savart law is $K_{\Pi}=K_{\Pi}(x,y) := {\nabla}_x^\perp G_{\Pi}(x,y)$. With $(x_1,x_2)^\perp=\begin{pmatrix} -x_2 \\ x_1\end{pmatrix}$, the explicit formula of $K_{\Pi}$ is given by $$K_{\Pi}(x,y)=\dfrac{1}{2\pi} DT^t(x)\Bigl(\dfrac{(T(x)-T(y))^\perp}{|T(x)-T(y)|^2}-\dfrac{(T(x)- T(y)^*)^\perp}{|T(x)- T(y)^*|^2}\Bigl).$$ We require information on far-field behavior of $K_{\Pi}$. We will use several times the following general relation: $$\label{frac} \Bigl| \frac{a}{|a|^2}-\frac{b}{|b|^2}\Bigl|=\frac{|a-b|}{|a||b|},$$ which can be easily checked by squaring both sides. We now find the following inequality: $$|K_{\Pi}(x,y)|\leq C \frac{|T(y)-T(y)^*|}{|T(x)-T(y)||T(x)-T(y)^*|}.$$ For $f\in C_c^\infty({\Pi})$, we introduce the notation $$K_{\Pi}[f]=K_{\Pi}[f](x):=\int_{\Pi} K_{\Pi}(x,y)f(y)dy,$$ and we have for large $|x|$ that $$\label{K-inf} |K_{\Pi}[f]|(x)\leq \dfrac{C_1}{|x|^2},$$ where $C_1$ depends on the size of the support of $f$. We have used here Remark \[2.5\]. The vector field $u=K_{\Pi}[f]$ is a solution of the elliptic system: $$\left\lbrace\begin{aligned} {{\rm div}\,}u &=0 &\text{ in } {\Pi} \\ {{\rm curl}\,}u &=f &\text{ in } {\Pi} \\ u\cdot \hat{n}&=0 &\text{ on } {\partial}{\Omega}\\ \lim_{|x|\to\infty}|u|&=0 \end{aligned}\right.$$ Let $\hat{n}$ be the unit normal exterior to $\Pi$. In what follows all contour integrals are taken in the counter-clockwise sense, so that $\int_{{\partial}{\Omega}} F\cdot {{\mathbf{ds}}}=-\int_{{\partial}{\Omega}} F\cdot \hat{n}^\perp ds$. Then the harmonic vector fields $$H_{\Pi} (x)=\frac{1}{2\pi}{\nabla}^\perp \ln |T(x)|= \frac{1}{2\pi} DT^t(x)\frac{T(x)^\perp}{|T(x)|^2}$$ is the unique[^1] vector field verifying $$\left\lbrace\begin{aligned} {{\rm div}\,}H_{\Pi} &=0 &\text{ in } {\Pi} \\ {{\rm curl}\,}H_{\Pi} &=0 &\text{ in } {\Pi} \\ H_{\Pi}\cdot \hat{n}&=0 &\text{ on } {\partial}{\Omega}\\ H_{\Pi} (x)\to 0 &\text{ as }|x|\to\infty\\ \int_{{\Gamma}} H_{\Pi} \cdot {{\mathbf{ds}}}&=1. \end{aligned}\right.$$ It is also known that $H_{\Pi}(x)=\mathcal{O}(1/|x|)$ at infinity. Therefore, putting together the previous properties, we obtain that the vector field $u^{\varepsilon}$: $$\label{biot} u^{\varepsilon}(x):= K_{\varepsilon}[{\omega}^{\varepsilon}](x) + ({\gamma}+\int {\omega}^{\varepsilon}) H_{\varepsilon}(x)$$ is the unique vector fields which verifies $$\left\lbrace\begin{aligned} {{\rm div}\,}u^{\varepsilon}&=0 &\text{ in } {\Pi_{\varepsilon}} \\ {{\rm curl}\,}u^{\varepsilon}&= {\omega}^{\varepsilon}&\text{ in } {\Pi_{\varepsilon}} \\ u^{\varepsilon}\cdot \hat{n}&=0 &\text{ on } {\Gamma}_{\varepsilon}\\ u^{\varepsilon}(x)\to 0 &\text{ as }|x|\to\infty\\ \int_{{\Gamma}_{\varepsilon}} u^{\varepsilon}\cdot {{\mathbf{ds}}}&= {\gamma}. \end{aligned}\right.$$ Concerning the uniqueness, we note that Remark \[T-unique\] allows us to apply the theory developed in [@ift_lop_euler] (see Lemma 3.1 therein, which is a consequence of Kelvin’s Circulation Theorem). \[rem : prop-curve\] This property means that the velocity of any weak solution (in the sense of Definition \[sol-curve\]) can be written as . Adding the fact that ${\omega}^{\varepsilon}$ belongs to $L^\infty(L^1\cap L^\infty)$, Proposition \[remark\_lac\_euler\] states that and the behavior of $T_{\varepsilon}$ (see Proposition \[2.2\]) imply some interesting properties on the velocity. For example, we infer that $u^{\varepsilon}$ has a jump across ${\Gamma}_{\varepsilon}$ and blows up near the end-points of the curve as the inverse of the square root of the distance. Before working on the convergence $(u^{\varepsilon},{\omega}^{\varepsilon})\to (u,{\omega})$, let us prove that Assumption \[assump\] is verified for good geometrical convergence ${\Omega}_{\eta}\to {\Gamma}$, which will complete [@lac_euler; @lac_NS]. Proof of Assumption \[assump\]. {#assump-proof} -------------------------------   For this subsection, we consider, as Pommerenke in [@pomm-1; @pomm-2], that a domain is an open, connected subset of ${{\mathbb C}}$. We identify also ${{\mathbb C}}$ and ${{\mathbb R}}^2$. Let ${\Gamma}$ be an open Jordan arc which verifies the assumptions of Proposition \[2.2\]. Let $\{{\Omega}_n\}$ be a sequence of bounded, open, connected, simply connected subset of the plane such that ${\Gamma}\subset {\Omega}_n$ and ${\partial}{\Omega}_n$ is a smooth Jordan curve. We denote his complementary by $\Pi_n:= {{\mathbb R}}^2\setminus \overline{{\Omega}_n}$. We recall that $D:= B(0,1)$, and we set ${\Delta}:= {{\mathbb R}}^2 \setminus \overline{D}$. For all $n$, we denote by $T_n$ the unique conformal mapping from $\Pi_n$ to ${\Delta}$ which sends $\infty$ to $\infty$, ${\partial}{\Omega}_n$ to ${\partial}D$ and such that $T_n'(\infty)\in {{\mathbb R}}^+_*$ (Riemann mapping theorem). By Remark \[T-unique\], we consider the unique $T$ which maps $\Pi:= {\Gamma}^c$ to ${\Delta}$, such that $T(\infty)=\infty$ and $T'(\infty)\in {{\mathbb R}}^+_*$. The properties of $T$ are listed in Proposition \[2.2\]. In order to apply the theory of domain convergence to smooth domains, we work here in the image of $T$, where there is no end-point. By continuity of $T$, we state that $\tilde \Pi_n:= T(\Pi_n)$ is an exterior domain, which means that $\tilde {\Omega}_n := {{\mathbb R}}^2\setminus \overline{\tilde \Pi_n}$ is a smooth, bounded, open, connected, simply connected subset of the plane, containing $\overline{D}$. {height="5.5cm"} [**PICTURE 1**]{}: image of $T$. By the Riemann mapping theorem, we have that $g_n:= T_n\circ T^{-1}$ is the unique univalent function (meromorphic and injective) mapping $\tilde \Pi_n$ to ${\Delta}$ and satisfying $g_n(\infty)=\infty$, arg $g_n'(\infty)=0$. As $g_n(\infty)=\infty$, it means that $g_n$ is analytic in ${\Delta}$. To apply directly some results from [@pomm-1], we also introduce Riemann mappings in bounded domains: $$f_n(z):= \frac{1}{g_n(1/z)},$$ which maps $\hat \Pi_n:= 1/T(\Pi_n)$ to $D$, verifying $f_n(0)=0$ and arg $f_n'(0)=0$ (see Picture 1). A convergence of ${\Omega}_n$ to ${\Gamma}$ can be translated by a convergence of $\tilde {\Omega}_n$ to $D$. The goal of this subsection is to define the geometric convergence in order that the properties cited in Assumption \[assump\] are verified. In other word, we have already an example of an obstacle family where they are verified: $$\tilde {\Omega}_n:= B(0,1+\frac{1}n),$$ (see ), and the issue here is to find more families where the results of [@lac_euler; @lac_NS] hold. Concerning univalent functions, the first convergence of domain was introduced by Carathéodory in 1912. Let $w_0\in {{\mathbb C}}$ be given and let $G_n$ be domains with $w_0\in G_n\subset {{\mathbb C}}$. We say that $G_n\to G$ as $n\to \infty$ with respect to $w_0$, in the sense of [*kernel convergence*]{} if - either $G=\{ w_0\}$, or $G$ is a domain $\neq {{\mathbb C}}$ with $w_0\in G$ such that some neighborhoods of every $w\in G$ lie in $G_n$ for large $n$; - for $w\in {\partial}G$ there exist $w_n\in {\partial}G_n$ such that $w_n\to w$ as $n\to \infty$. It is clear that the limit is unique. This notion is very general, and it follows the Carathéodory kernel theorem: Let the functions $h_n$ be analytic and univalent in $D$, and let $h_n(0)=0$, $h'_n(0)>0$, $H_n = h_n(D)$. Then $\{ h_n \}$ converges locally uniformly in $D$ if and only if $\{H_n\}$ converges to its kernel $H$ and if $H\neq{{\mathbb C}}$. Furthermore, the limit function maps $D$ onto $H$. We remark that this theorem does not concern uniform convergence of biholomorphism up to the boundary. Indeed, we need such a property for points (i), (ii), (iii) in Assumption \[assump\]. Moreover, convergence in the kernel sense allows some strange examples of families, as a fold on the boundary (see Picture 2). It seems unbelievable that $\{T_n\}$ verifies Assumption \[assump\] near the boundary. {height="4cm"} [**PICTURE 2**]{}: example of a family which converges in the kernel sense. To prevent such a case, we add another definition. A sequence $\{A_n\}$ of compact sets in ${{\mathbb C}}$ is called [*uniformly locally connected*]{} iff for every ${\varepsilon}>0$, there exists ${\delta}>0$ (independent of $n$) such that, if $a_n,b_n\in A_n$ and $|a_n-b_n|<{\delta}$, then we can find connected compact sets $B_n$ with $$a_n,b_n\in B_n \subset A_n, \ \textrm{diam } B_n < {\varepsilon}, \ \forall n.$$ This kind of definition does not allow the case of Picture 2. However, the examples of Pictures 3 and 4 are authorized, which are classical shapes in rugosity theory. {height="5cm"}   {height="5cm"} [**PICTURE 4**]{}  Thanks to this definition, we can cited Theorem 9.11 of [@pomm-1], which extends to the uniform convergence up to the boundary. \[theo 9.11\] Let the functions $g_n$ and $g$ be univalent in ${\Delta}$ and continuous in $\overline{{\Delta}}$, and let $g_n(\infty)=\infty$ and $E_n={{\mathbb C}}\setminus g_n({\Delta})$. Suppose that $g_n\to g$ locally uniformly in ${\Delta}$. Then this convergence is uniform in $\overline{{\Delta}}$ if and only if the sequence $(E_n)$ is uniformly locally connected. We apply these theorems to obtain a part of the properties of Assumption \[assump\]. \[prop-1\] Let us assume that there exists $R_0>1$ such that $\tilde {\Omega}_n \subset B(0,R_0)$ for all $n$. If $\tilde \Pi_n \to {\Delta}$ in the sense of kernel convergence with respect to $\infty$, such that $\{ \tilde {\Omega}_n \}$ is uniformly locally connected, then there exists $R_1>0$ such that - $\|(g_n(z)- z)/|z|\|_{L^\infty(\tilde \Pi_n)}\to 0$ as $n \to \infty$, - $(g_n^{-1})'$ is bounded on $B(0,R_1)^c$ independently of $n$, - $\|g_n'(z) - 1\|_{L^\infty(B(0,R_1)^c)}\to 0$ as $n \to \infty$, - there exists $C>0$ such that $|g_n'' (z)|\leq \frac{C}{|z|}$ on $B(0,R_1)^c$. After noting that $\hat \Pi_n \to D$ in the kernel sense with respect to $0$, we apply the Carathéodory kernel theorem to $f_n^{-1}$. Riemann mappings are biholomorphisms, then they are univalents and analytics. We deduce that $f_n^{-1}$ converges uniformly to Id in each compact subset of $D$. As $g_n^{-1}(z) =1/f_n^{-1}(1/z)$, we also know that $g_n^{-1}$ converges locally uniformly in ${\Delta}$. Moreover, Kellogg-Warschawski Theorem (see Theorem 3.6 of [@pomm-2]) allows us to state that $g_n^{-1}$ is continuous in $\overline{{\Delta}}$ because the boundary ${\partial}\tilde {\Omega}_n$ is smooth. Therefore, we use Theorem \[theo 9.11\] to state that $g_n^{-1}$ converges uniformly to Id in $\overline{{\Delta}}$. In particularly, $g_n^{-1}$ converges uniformly to Id in $B(0,2)\setminus D$, which means that $f_n^{-1}$ tends to Id uniformly in $\overline{D}\setminus B(0,1/2)$. Adding the fact that $f_n^{-1}\to$Id uniformly in $\overline{B(0,3/4)}$, we obtain that this convergence is uniformly in $\overline{D}$. A consequence of this uniform convergence, is that $f_n\to$Id uniformly. Indeed, for all $z\in \hat \Pi_n$, we have $$|f_n(z) - z|= |y_n - f_n^{-1}(y_n)|\leq \| \textrm{Id} - f_n^{-1}\|_{L^\infty (\overline{D})},$$ which means that $\|f_n - \textrm{Id} \|_{L^\infty (\hat \Pi_n)}\to 0$ as $n\to \infty$. Let us prove that it follows Point (a). Near the boundary, we easily compute that $$\Bigl\| \frac{|g_n(z)- z|}{|z|}\Bigl\|_{L^\infty(\tilde \Pi_n\cap B(0,3R_0))} = \Bigl\| \frac{|\frac1z- f_n(\frac1z)|}{|f_n(\frac1z)|}\Bigl\|_{L^\infty(\tilde \Pi_n\cap B(0,3R_0))} = \Bigl\| \frac{|y-f_n(y)|}{|f_n(y) |}\Bigl\|_{L^\infty(\hat \Pi_n \setminus B(0,\frac{1}{3 R_0}))}.$$ As, $\|f_n - \textrm{Id} \|_{L^\infty (\hat \Pi_n)}\to 0$ as $n\to \infty$, it means that there exists $N_1$ such that for all $n>N_1$, we have $|f_n(z)|> \frac{1}{4R_0}$ for all $z\in \hat \Pi_n\setminus B(0,\frac{1}{3R_0})$. Therefore, for all $n>N_1$, we obtain that $$\label{g-bd} \| |g_n(z)- z|/|z|\|_{L^\infty(\tilde \Pi_n\cap B(0,3R_0))} \leq 4R_0 \| \textrm{Id} -f_n \|_{L^\infty(\hat \Pi_n)}.$$ Far the boundary, first we prove that $f_n(z)/z$ converge uniformly to $1$ in $B(0,1/(2R_0))$. As $f_n(0)=0$, we know that the map $z\mapsto f_n(z)/z$ is holomorphic in $D$. After remarking that $1/(2R_0)<1/R_0<1$, then we can write the Cauchy formula to state that: $$\begin{aligned} \forall z\in B(0,1/(2R_0)), \ \frac{f_n(z)}{z} -1 &=& \frac{1}{2\pi i} \oint_{{\partial}B(0,1/R_0)} \frac{f_n(y)/y-1}{y-z}\, dy\\ \Bigl| \frac{f_n(z)}{z}-1\Bigl| &\leq & 2 R_0^2 \| f_n - \textrm{Id} \|_{L^\infty(\overline{B(0,1/R_0)})},\end{aligned}$$ which means that $f_n(z)/z$ converges uniformly to $1$ in $B(0,1/(2R_0))$. Finally, we write that $$\label{g-inf} \| |g_n(z)- z|/|z|\|_{L^\infty( B(0,2R_0)^c)} = \| \frac{y}{f_n(y)} -1 \|_{L^\infty( B(0,\frac{1}{2R_0}))}.$$ Putting together and , we end the proof of point (a). Now, we focus on uniform convergence of derivatives. As $f_n^{-1}$ is holomorphic in $D$, we can write the Cauchy formula for all $z\in B(0,1/2)$ $$\begin{aligned} (f_n^{-1})'(z)-1 &=& \frac{1}{2\pi i} \oint_{{\partial}B(0,3/4)} \frac{f_n^{-1}(y)-y}{(y-z)^2}\, dy\\ |( f_n^{-1})'(z)-1| &\leq & 16 \| f_n^{-1}-\textrm{Id} \|_{L^\infty(B(0,3/4))}.\end{aligned}$$ which gives that $(f_n^{-1})'$ converges also uniformly to $1$ in $B(0,1/2)$. Differentiating $g_n^{-1}(z) = 1/f_n^{-1}(1/z)$, we obtain $(g_n^{-1})'(z)=\frac{(f_n^{-1})'(1/z)}{(zf_n^{-1}(1/z))^2}$. Therefore $(g_n^{-1})'$ converges uniformly to $1$ in $B(0,2)^c$, which implies point (b). The other points can be established in the same way. Indeed, $f_n\to $Id uniformly in $B(0,1/R_0)$, then using again the Cauchy formula, we prove that $f_n'\to 1$ uniformly in $B(0,1/(2R_0))$. Using that $(g_n)'(z)=\frac{(f_n)'(1/z)}{(zf_n (1/z))^2}$, we pretend that $g_n'\to 1$ uniformly in $B(0,2R_0)^c$ which implies (c). Finally, we use again the Cauchy formula to get that $f_n'' \to 0$ uniformly in $B(0,1/(2R_0))$. Next, we differentiate once more the relation between $g_n$ and $f_n$, and we compute: $$g_n''(z)=-\frac{f_n''(1/z)+ 2 f_n'(1/z)(zf_n(1/z)-f_n'(1/z))}{z^4 (f_n(1/z))^3}.$$ Using all the uniform convergences of $f_n$, $f'_n$ and $f_n''$ in $B(0,1/(2R_0))$, we obtain (d) with $R_1=2 R_0$. This ends the proof. Using the relation, $T_n=g_n \circ T$, it will not be complicated to obtain properties (i), (iv) and (v) of Assumption \[assump\]. However, seeing points (ii) and (iii), we remark that it misses us uniform convergence of the derivatives up to the boundary. In the previous proof, we obtain uniform convergence of $(f_n^{-1})'$ in all the compact subsets, thanks to the Cauchy formula, finding a curve between the compacts and ${\partial}D$. Such a proof can not be used to obtain convergence up to the boundary. Therefore, we need to present the results of [@war]. In this article, it appears that we can except a uniform convergence of $(f^{-1}_n)'$ only if we assume some convergences of the tangent vectors of ${\partial}\hat \Pi_n$ to the tangent vector of ${\partial}D$. For that, we remark that $C_n:= {\partial}\hat \Pi_n$ denotes a closed Jordan curve which posses continuously turning tangents. We also put $C:= {\partial}D$. Let ${\tau}_n(s)$ and ${\tau}$ be their tangent angles, expressed as functions of the arc length. We gives now the assumptions needed to present the main theorem of [@war]. We assume that there exists ${\varepsilon}>0$ such that: 1. $C_n$ is in the ${\varepsilon}$-neighborhood of $C$, i.e. any point of $C_n$ is contained in a circle of radius ${\varepsilon}$ about some points of $C$. 2. If ${\Delta}s$ denotes the (shorter) arc of the curve $C_n$ between $w'$ and $w''$, then $$\frac{{\Delta}s}{|w'-w''|}\leq c.$$ 3. For any point $w_n\in C_n$, pertaining to the arc length, $0\leq s_n\leq L_n$, there corresponds a point $w\in C$, pertaining to the arc length ${\sigma}={\sigma}(s_n)$, such that $|w_n-w|\leq {\varepsilon}$ and that, for suitable choise of the branches, $$\sup_{0\leq s_n\leq L_n} |{\tau}_n(s_n) - {\tau}({\sigma}(s_n))| \leq q{\varepsilon}.$$ 4. ${\tau}_n$ is Hölder continuous, i.e. there exist $k>0$, ${\alpha}\in (0,1]$ such that $$|{\tau}_n(s_1)-{\tau}_n(s_2)| \leq k(s_1-s_2)^{\alpha}, \forall s_1,s_2 \in [0,L_n].$$ In (3), $L_n$ denotes the total length of $C_n$. In [@war], the author also assumes that $C$ and $C_n$ are contained in the ring $0<d\leq |w|\leq D$. This property is directly verified in our case, with $D=1$ and $d= 1-{\varepsilon}$. Moreover, he assumes that $C$ is in an ${\varepsilon}$-neighborhood of $C_n$. Indeed, in general case, point (ii) does not imply it. However, in our case we have $D\subset \tilde {\Omega}_n$, and we obtain such a property. \[rugo\] We see that point (2) prevents us the convergence as in Picture 3. Indeed, in the case of this picture, we have $\frac{{\Delta}s}{|w'-w''|}\approx C/{\varepsilon}$, which tends to $\infty$ as ${\varepsilon}\to 0$. In literature, a curve which verified (2) is called a chord-arc curve, and we will prove that the domains outside this kind of curve are uniformly locally connected. Moreover, we cannot either consider the case of Picture 4, because of point (3). Then we give Theorem IV of [@war]: \[war\] If $C_n$ satisfies hypotheses (1)-(4), then $$\sup_{|z|\leq 1 } |(f_n^{-1})'(z)-1|\leq K {\varepsilon}\ln \frac{\pi}{{\varepsilon}}$$ where $K$ depends only on $c$, $k$, ${\alpha}$ and $q$. Putting together Proposition \[prop-1\] and Theorem \[war\], we can prove the main theorem of this subsection: \[theo-assump\] Let ${\Gamma}$ be an open Jordan arc which verifies the assumptions of Proposition \[2.2\]. Let $\{{\Omega}_n\}$ be a sequence of smooth, bounded, open, connected, simply connected subset of the plane such that ${\Gamma}\subset {\Omega}_n$. For all $n$, we denote by $T_n$ the unique conformal mapping from $\Pi_n$ to ${\Delta}$ such that $T_n(\infty)=\infty$ and $T_n'(\infty)\in {{\mathbb R}}^+_*$. Let $T$ the biholomorphism constructed in Proposition \[2.2\], such that $T'(\infty)\in {{\mathbb R}}^+_*$. We also denote $\hat \Pi_n:= 1/T(\Pi_n)$. For all $n$, we assume that $C_n:= {\partial}\hat \Pi_n$ verifies hypotheses (1)-(4), with $c$, $k$, ${\alpha}$ and $q$ independent of $n$, and where ${\varepsilon}_n \to 0$ as $n\to 0$. Then, the family $\{T_n\}$ verifies Assumption \[assump\]. We use as before the conformal mappings $g_n$ and $f_n$. Let us first note that we can apply Proposition \[prop-1\]. Indeed, the condition (1) implies that $\hat \Pi_n \to D$ in the kernel sense with respect to $0$, which means that $\tilde \Pi_n \to {\Delta}$ in the kernel sense with respect to $\infty$. Moreover, choosing $R_0=1+\sup_n {\varepsilon}_n$, we also see that $\tilde {\Omega}_n \subset B(0,R_0)$ for all $n$. Finally, we can easily check that the condition of uniformly locally connected comes from the condition (2): for every ${\varepsilon}$, we choose ${\delta}= {\varepsilon}/c$. Indeed, for all $a_n$ $b_n$ such that $|a_n-b_n|<{\delta}$, (2) means that there exists $B_n\subset \tilde {\Omega}_n$ with $a_n,b_n\in B_n$ and diam$B_n\leq c |a_n-b_n|< {\varepsilon}$. Therefore, we use directly Proposition \[prop-1\]. Theorem \[war\] states that the convergence of $(f_n^{-1})'$ to $1$ is uniform in $\overline{D}$. Thanks to the relation between $(f_n^{-1})'$ and $(g_n^{-1})'$ (see the proof of Proposition \[prop-1\], it means that $(g_n^{-1})'\to 1$ uniformly in $\overline{{\Delta}}$. Adding that $g_n'(z)=1/(g_n^{-1})'(g_n(z))$, we conclude that Theorem \[war\] allows us to extend (b) and (c) up to the boundary, i.e. - $(g_n^{-1})'$ is bounded on ${\Delta}$ independently of $n$, - $\|g_n'(z) - 1\|_{L^\infty(\tilde \Pi_n)}\to 0$ as $n \to \infty$. In order to finish this proof, we just have to compose by $T$. As $T_n=g_n\circ T$, it is obvious that $$\Bigl\| \frac{T_n-T}{|T|}\Bigl\|_{L^\infty(\Pi_n)} = \Bigl\| \frac{g_n-\textrm{Id}}{|\textrm{Id}|}\Bigl\|_{L^\infty(\tilde \Pi_n)}$$ which tends to zero as $n\to \infty$ by (a) of Proposition \[prop-1\]. This proves Point (i) of Assumption \[assump\]. Differentiating the relation between $T_n^{-1}$ and $g_n^{-1}$, we obtain that $$(T_n^{-1})'(z)= (T^{-1})'(g_n^{-1}(z)) (g_n^{-1})' (z).$$ Then, it is easy to conclude that (ii) follows from (b’) and Proposition \[2.2\]. For any $R>0$, $$\begin{aligned} \| T'_n-T' \|_{L^3(B(0,R)\cap \Pi_n)} &=& \| g'_n(T(z)) T'(z) -T'(z) \|_{L^3(B(0,R)\cap \Pi_n)}\\ &\leq& \|g'_n(z)-1\|_{L^\infty(\tilde \Pi_n)} \| T' \|_{L^3(B(0,R)\cap \Pi_n)} \end{aligned}$$ which proves (iii), thanks to (c’) and Proposition \[2.2\]. Differentiating once the relation between $T_n$ and $g_n$, and using (c) and Proposition \[2.2\], we obtain (iv). Differentiating once more, we have $$T''_n(z)=g_n''(T(z)) (T'(z))^2+g_n'(T(z)) T''(z).$$ Using (d), Proposition \[2.2\], (c) and Remark \[2.5\], we finally get (v), which ends the proof. We easily see at the end of the previous proof that we can prove that: $$\text{for any $R>0$, $p<4$, $\|DT_n - DT\|_{L^p(B(0,R)\cap \Pi_n)}\to 0$ as $n \to \infty$.}$$ Theorem \[main 2\] follows from Theorem \[theo-assump\] and [@lac_euler] (see Subsection \[thicken\] for the details). ### Comment on this domain convergence {#comment-on-this-domain-convergence .unnumbered} This theorem completes [@lac_euler; @lac_NS] in the following sense: if $C_n:= {\partial}T( {\Omega}_n)$ verifies hypotheses (1)-(4), then solutions of the Euler equations (respectively Navier-Stokes) in the exterior of ${\Omega}_n$ converge to the solution of the Euler equations (respectively Navier-Stokes) outside the curve. To make this result more attractive, it should be better to give geometric properties on ${\partial}{\Omega}_n$ instead of ${\partial}T( {\Omega}_n)$, but this translation is not so easy. Of course, by continuity of $T$, condition (1) can be assumed on ${\partial}{\Omega}_n$, which is not the case for (2)-(4), where we give some properties of the tangent vectors. Conditions (2)-(4), prevent the pointed parts as in Picture 2. However, studying the example in , we see that there is a pointed part near the end-point $-1$, which is mapped by $T$ to the circle $B(0,1+{\varepsilon})$. In other word, ${\partial}{\Omega}_n$ cannot satisfy (2)-(4) near the end-points, but $T$ can straighten it to a curve which verifies (2)-(4). This straightening up should hold for some shapes of pointed part, but surely not for any shape. We have a big family of authorized shapes: $T^{-1}(C_n)$ where $C_n$ verifies (1)-(4). A possible result could be stated like that: ** - if there exists ${\delta}>0$ such that $({\partial}{\Omega}_n)\setminus (\cup B(\pm 1,{\delta}))$ satisfy hypothesis (1)-(4) (where we replace $C$ by ${\Gamma}$), - if ${\partial}{\Omega}_n$ corresponds to an “authorized shape” in $\cup B(\pm 1,{\delta})$ then Assumption \[assump\] is verified. ### Comment on the convergence to smooth domains {#comment-on-the-convergence-to-smooth-domains .unnumbered} If we replace the convergence of domain to a curve by a convergence to a smooth domain ${\Omega}$, we can adapt easily Theorem \[theo-assump\] and [@lac_euler] in order to establish that the solution of Euler equations in (or outside) ${\Omega}_n$ converges to the solution of Euler equations in (or outside) ${\Omega}$, if ${\Omega}_n\to{\Omega}$ in the sense of hypothesis (1)-(4). Then, a consequence of this work is that we have found a sense for the domain convergences (hypothesis (1)-(4)), in order that the limit solution is a solution of Euler equations. However, we see in Remark \[rugo\] that the classical shape in rugosity problems (Picture 4) is not allowed in our analysis. Concerning bounded domains (simply connected), Taylor in [@taylor] proves that we do not need so strong properties. Passing to the limit with a weaker domain convergence, he can show the existence of weak solutions for Euler equations in a non-smooth convex domain (with Lipschitz boundary). Therefore, our result does not improve the case of bounded domain, but it gives a new result in exterior domains (outside one obstacle). In exterior domain, the analysis is harder because of the behavior at infinity: the velocity is not square integrable. [*A priori*]{} estimates {#sect : 3} ======================== Vorticity estimate {#om-est} ------------------   The study of two dimensional ideal flow is based on velocity estimates thanks to vorticity estimates. In a domain with smooth boundaries, the pair $(u,{\omega})$ is a strong solution of the transport equation , which gives us the classical estimates for the vorticity: - $\|{\omega}(t,\cdot)\|_{L^p(\Pi)}=\|{\omega}_0\|_{L^p({{\mathbb R}}^2)}$ for $p\in[1,+\infty]$; - if ${\omega}_0$ is compactly supported in $B(0,R)$, then there exists $C>0$ such that ${\omega}(t,\cdot)$ is compactly supported in $B(0,R+Ct)$; - for any $t>0$, we have $\int_{\Pi} {\omega}(t,x)\, dx = \int_{\Pi} {\omega}_0(x)\, dx$. The goal of this subsection is to prove such properties for the weak solution $(u^{\varepsilon},{\omega}^{\varepsilon})$ defined in Definition \[sol-curve\]. The main point is to remark that this pair is a renormalized solution in the sense of DiPerna and Lions (see [@dip-li]) of the transport equation. Let us assume that ${\omega}_0$ is $L^\infty$ and compactly supported in $B(0,R)\cap B(0,r)$. Moreover we fix ${\varepsilon}$ small enough such that the support of ${\omega}_0$ does not intersect ${\Gamma}_{{\varepsilon}}$ (any ${\varepsilon}\leq r$). We consider equation as a linear transport equation with given velocity field $u^{\varepsilon}$. Our purpose here is to show that if ${\omega}^{\varepsilon}$ solves this linear equation, then so does $\beta({\omega}^{\varepsilon})$ for a suitable smooth function ${\beta}$. This follows from the theory developed in [@dip-li] (see also [@dej] for more details), where they need that the velocity field belongs to $L_{{\operatorname{{loc}}}}^1\left({{\mathbb R}}^+,W_{{\operatorname{{loc}}}}^{1,1}({{\mathbb R}^2})\right)$. Let us check that we are in this setting. Let $(u^{\varepsilon},{\omega}^{\varepsilon})$ be a weak-solution of the Euler equations outside the curve ${\Gamma}_{\varepsilon}$. As ${\omega}^{\varepsilon}\in L^\infty (L^1\cap L^\infty)$ then $$u^{\varepsilon}\in L^\infty \left({{\mathbb R}}^+,W_{{\operatorname{{loc}}}}^{1,1}({{\mathbb R}^2})\right).$$ Here, we are not looking for estimates uniformly in ${\varepsilon}$, as later (e.g. Lemma \[I\_est\]). Then, we fix ${\varepsilon}>0$, and we rewrite : $$\begin{aligned} u^{\varepsilon}(x)&=&\frac{1}{2\pi} DT_{\varepsilon}^t(x) \Bigl( \int_{\Pi_{\varepsilon}} \Bigl(\frac{T_{\varepsilon}(x)-T_{\varepsilon}(y)}{|T_{\varepsilon}(x)-T_{\varepsilon}(y)|^2}- \frac{T_{\varepsilon}(x)-T_{\varepsilon}(y)^*}{|T_{\varepsilon}(x)-T_{\varepsilon}(y)^*|^2}\Bigl)^\perp {\omega}^{\varepsilon}(y)\, dy + {\alpha}\frac{T_{\varepsilon}(x)^\perp}{|T_{\varepsilon}(x)|^2}\Bigl)\\ &:= & \frac{1}{2\pi} DT_{\varepsilon}^t(x) f(T_{\varepsilon}(x))\end{aligned}$$ where ${\alpha}$ is bounded by ${\gamma}+ \| {\omega}^{\varepsilon}\|_{L^\infty(L^1)}$. We start by treating $f$. We change variable ${\eta}=T_{\varepsilon}(y)$, and we obtain $$\begin{aligned} f(z)&=& \int_{B(0,1)^c} \Bigl(\frac{z-{\eta}}{|z-{\eta}|^2}- \frac{z-{\eta}^*}{|z-{\eta}^*|^2}\Bigl)^\perp {\omega}^{\varepsilon}(T_{\varepsilon}^{-1}({\eta})) |\det DT_{\varepsilon}^{-1}({\eta})| \, d{\eta}+ {\alpha}\frac{z^\perp}{|z|^2} \\ &=&\int_{B(0,1)^c}\frac{(z-{\eta})^\perp}{|z-{\eta}|^2} g({\eta}) \, d{\eta}- \int_{B(0,2)^c}\frac{(z-{\eta}^*)^\perp}{|z-{\eta}^*|^2} g({\eta}) \, d{\eta}\\ &&- \int_{B(0,2)\setminus B(0,1)}\frac{(z-{\eta}^*)^\perp}{|z-{\eta}^*|^2} g({\eta}) \, d{\eta}+ {\alpha}\frac{z^\perp}{|z|^2}\\ &:=& f_1(z)-f_2(z)-f_3(z)+f_4(z),\end{aligned}$$ with $g({\eta})={\omega}^{\varepsilon}(T_{\varepsilon}^{-1}({\eta})) |\det DT_{\varepsilon}^{-1}({\eta})|$ belongs[^2] to $L^\infty(L^1\cap L^\infty({{\mathbb R}}^2))$. As $|z|= |T_{\varepsilon}(x)|\geq 1$, we are looking for estimates in $D^c$. Obviously we have that $$f_4 \text{ belongs to } L^{\infty}(D^c) \text{ and } Df_4 \text{ belongs to } L^{\infty}(D^c).$$ Concerning $f_1$, we introduce $g_1:= g \chi_{D^c}$ where $\chi_S$ denotes the characteristic function on $S$. Hence $$f_1(z)=\int_{{{\mathbb R}}^2}\frac{(z-{\eta})^\perp}{|z-{\eta}|^2}g_1({\eta}) \, d{\eta}\text{ with } g_1\in L^\infty(L^1({{\mathbb R}}^2)\cap L^\infty({{\mathbb R}}^2)).$$ The standard estimates on Biot-Savart kernel in ${{\mathbb R}}^2$ (see e.g. Lemma \[ift\]) and Calderon-Zygmund inequality give that $$f_1 \text{ belongs to } L^\infty({{\mathbb R}}^+\times {{\mathbb R}}^2) \text{ and } Df_1 \text{ belongs to } L^\infty(L^p({{\mathbb R}}^2)),\ \forall p\in [1,\infty) .$$ For $f_2$, we can remark that for any ${\eta}\in B(0,2)^c$ we have $|z-{\eta}^*|\geq \frac12$. Therefore, the function $(z,{\eta})\mapsto \frac{(z-{\eta}^*)^\perp}{|z-{\eta}^*|^2} $ is smooth in $B(0,1)^c\times B(0,2)^c$, which gives us, by a classical integration theorem, that $$f_2 \text{ belongs to } L^\infty({{\mathbb R}}^+\times D^c) \text{ and } Df_2 \text{ belongs to } L^\infty({{\mathbb R}}^+\times D^c).$$ To treat the last term, we change variables $\theta = {\eta}^*$ $$f_3(z) = \int_{B(0,1)\setminus B(0,1/2)} \frac{(z-\theta)^\perp}{|z-\theta|^2} g(\theta^*) \frac{d\theta}{|\theta|^4}:= \int_{{{\mathbb R}}^2} \frac{(z-\theta)^\perp}{|z-\theta|^2} g_3(\theta)\, d\theta,$$ with $g_3(\theta)=\displaystyle \frac{g(\theta^*)}{|\theta|^4} \chi_{B(0,1)\setminus B(0,1/2)}(\theta)$ which belongs to $L^\infty(L^1({{\mathbb R}}^2)\cap L^\infty({{\mathbb R}}^2))$. Therefore, standard estimates on Biot-Savart kernel and Calderon-Zygmund inequality give that $$f_3 \text{ belongs to } L^\infty({{\mathbb R}}^+\times {{\mathbb R}}^2) \text{ and } Df_3 \text{ belongs to } L^\infty(L^p({{\mathbb R}}^2)),\ \forall p\in [1,\infty) .$$ Now, we come back to $u^{\varepsilon}$. As $u^{\varepsilon}(x) = \frac{1}{2\pi} DT_{\varepsilon}^t(x) f(T_{\varepsilon}(x))$, with $DT_{\varepsilon}$ belonging to $L^p_{{\operatorname{{loc}}}}({{\mathbb R}}^2)$ for $p<4$ and $f\circ T_{\varepsilon}$ uniformly bounded, we have that $$u^{\varepsilon}\text{ belongs to } L^{\infty}({{\mathbb R}}^+;L^1_{{\operatorname{{loc}}}} ({{\mathbb R}}^2)).$$ Moreover, we have $$\begin{aligned} |Du^{\varepsilon}(x)| &\leq& \frac{1}{2\pi}\Bigl( |D^2 T_{\varepsilon}(x)| |f(T_{\varepsilon}(x))| + |DT_{\varepsilon}(x)|^2 |(Df_1-Df_3) (T_{\varepsilon}(x))|\\ &&+|DT_{\varepsilon}(x)|^2 |(-Df_2+Df_4) (T_{\varepsilon}(x))|\Bigl). \end{aligned}$$ We see that the second right hand side term belongs to $L^{\infty}({{\mathbb R}}^+;L^1_{{\operatorname{{loc}}}} ({{\mathbb R}}^2))$ because $DT_{\varepsilon}$ belongs to $L^{3}_{{\operatorname{{loc}}}}({{\mathbb R}}^2)$ and $(Df_1-Df_3)(T_{\varepsilon}(x))$ belongs to $L^{\infty}({{\mathbb R}}^+;L^3 ({{\mathbb R}}^2))$. Similarly, the third right hand side term belongs to $L^{\infty}({{\mathbb R}}^+;L^1_{{\operatorname{{loc}}}} ({{\mathbb R}}^2))$ because $DT_{\varepsilon}$ belongs to $L^{2}_{{\operatorname{{loc}}}}({{\mathbb R}}^2)$ and $-Df_2+Df_4$ belongs to $L^{\infty}({{\mathbb R}}^+\times D^c)$. For the first right hand side term, we know that $f\circ T_{\varepsilon}$ is uniformly bounded, then we have to prove that $D^2 T_{\varepsilon}$ belongs to $L^1_{{\operatorname{{loc}}}}({{\mathbb R}}^2)$ in order to finish the proof. Keeping in mind that $DT_{\varepsilon}$ is smooth everywhere, except near the end-points where it behaves like the inverse of the square root of the distance, and noting that the map $x\mapsto 1/\sqrt{|x|}$ belongs to $W^{1,1}_{{\operatorname{{loc}}}}({{\mathbb R}}^2)$, it is natural to think that it holds true. However, this argument needs an estimate on $D^2 T$ up to the boundary. We have to check carefully in the proof of Proposition \[2.2\] (see [@lac_euler]) how to gain this control. Actually, we can add a point at Proposition \[2.2\]: - $D^2 T$ extends continuously up to ${\Gamma}$ with different values on each side of ${\Gamma}$, except at the endpoints of the curve where $D^2 T$ behaves like the inverse of the power $3/2$ of the distance, which implies that $D^2 T$ is bounded in $L^p(\Pi\cap B(0,R))$ for all $p<4/3$ and $R>0$. This extension allows us to finish the proof of this lemma. For completeness, this extension of Proposition \[2.2\] is proved in Annexe. Therefore, [@dip-li; @dej] imply that ${\omega}^{\varepsilon}$ is a renormalized solution. \[renorm1\] Let ${\omega}^{\varepsilon}$ be a solution of . Let $\beta:{{\mathbb R}}\rightarrow {{\mathbb R}}$ be a smooth function such that $$|\beta'(t)|\leq C(1+ |t|^p),\qquad \forall t\in {{\mathbb R}},$$ for some $p\geq 0$. Then for all test function $\psi \in \mathcal{D}({{\mathbb R}}^+ \times {{\mathbb R}^2})$, we have $$\frac{d}{dt}\int_{{{\mathbb R}^2}} \psi \beta({\omega}^{\varepsilon})\,dx=\int_{{{\mathbb R}^2}} \beta({\omega}^{\varepsilon}) ({\partial_t}\psi +u^{\varepsilon}\cdot \nabla \psi)\,dx \:\: \textrm{in }\: L_{{\operatorname{{loc}}}}^1({{\mathbb R}}^+).$$ Now, we write a remark from [@lac_miot], in order to establish the some desired properties for ${\omega}^{\varepsilon}$. \[remark : conserv\] (1) Lemma \[renorm1\] actually still holds when $\psi$ is smooth, bounded and has bounded first derivatives in time and space. In this case, we have to consider smooth functions $\beta$ which in addition satisfy $\beta(0)=0$, so that $\beta({\omega}^{\varepsilon})$ is integrable. This may be proved by approximating $\psi$ by smooth and compactly supported functions $\psi_n$ for which Lemma \[renorm1\] applies, and by letting then $n$ go to $+\infty$.\ (2) We apply the point (1) for $\beta(t)=t$ and ${\psi}\equiv 1$, which gives $$\label{om-est-1} \int_{{{\mathbb R}}^2} {\omega}^{\varepsilon}(t,x)\, dx = \int_{{{\mathbb R}}^2} {\omega}_0(x)\, dx \text{ for all }t>0.$$ (3) We let $1\leq p<+\infty$. Approximating $\beta(t)=|t|^p$ by smooth functions and choosing $\psi\equiv 1$ in Lemma \[renorm1\], we deduce that for an solution ${\omega}^{\varepsilon}$ to , the maps $t\mapsto \|{\omega}^{\varepsilon}(t)\|_{L^p({{\mathbb R}^2})}$ are continuous and constant. In particular, we have $$\label{om-est-2} \|{\omega}^{\varepsilon}(t)\|_{L^1({{\mathbb R}^2})}+\|{\omega}^{\varepsilon}(t)\|_{L^\infty({{\mathbb R}^2})}:= \|{\omega}_0\|_{L^1({{\mathbb R}^2})}+\|{\omega}_0\|_{L^\infty({{\mathbb R}^2})}.$$ Unfortunately, we cannot establish now that ${\omega}^{\varepsilon}$ is compactly supported uniformly in ${\varepsilon}$. For that, we need some estimates on the velocity. Velocity estimate -----------------   The goal of this subsection is to find a velocity estimate uniformly in ${\varepsilon}$, thanks to the explicit formula of $u^{{\varepsilon}}$ in function of ${\omega}^{{\varepsilon}}$ and ${\gamma}$ (Biot-Savart law). We will need the following lemma from [@ift]: \[ift\] Let $S\subset{{\mathbb R}}^2$ and $g:S\to{{\mathbb R}}^+$ be a function belonging in $L^1(S)\cap L^p(S)$, for $p\in (2;+\infty]$. Then $$\int_S \frac{g(y)}{|x-y|}dy\leq C\|g\|_{L^1(S)}^{\frac{p-2}{2(p-1)}}\|g\|_{L^p(S)}^{\frac{p}{2(p-1)}}.$$ For $h: \Pi_{\varepsilon}\to{{\mathbb R}}$ a function belonging in $L^1(S)\cap L^p(S)$, with $p\in (2;+\infty]$, we introduce $$I_1^{\varepsilon}[h](x)= \int_{\Pi_{{\varepsilon}}}\dfrac{(T_{{\varepsilon}}(x)-T_{{\varepsilon}}(y))^\perp}{|T_{{\varepsilon}}(x)-T_{{\varepsilon}}(y)|^2} h(y)dy,$$ and $$I_2^{\varepsilon}[h](x)=\int_{\Pi_{{\varepsilon}}}\dfrac{(T_{{\varepsilon}}(x)- T_{{\varepsilon}}(y)^*)^\perp}{|T_{{\varepsilon}}(x)- T_{{\varepsilon}}(y)^*|^2} h(y)dy.$$ Therefore, the Biot-Savart law can be written $$\label{u_e} u^{{\varepsilon}}(t,x)=\dfrac{1}{2\pi}DT_{{\varepsilon}}^t(x)(I_1^{\varepsilon}[{\omega}^{\varepsilon}(t,\cdot)](x) - I_2^{\varepsilon}[{\omega}^{\varepsilon}(t,\cdot)])(x)+({\gamma}+m) H_{{\varepsilon}}(x),$$ with $m:=\int {\omega}^{\varepsilon}(t,\cdot)= \int {\omega}_0$ by . In [@lac_euler], we manage to estimate directly $I_1^{\varepsilon}[{\omega}^{\varepsilon}(t,\cdot)](x)$ and $I_2[{\omega}^{\varepsilon}(t,\cdot)](x)$, uniformly in $x$ and ${\varepsilon}$, by $\| {\omega}^{\varepsilon}(t,\cdot)\|_{L^1\cap L^\infty}$. In [@ift_lop_euler], the authors obtain a uniform estimate in $x$ and ${\varepsilon}$ of $$\dfrac{1}{2\pi}DT_{{\varepsilon}}^t(x) I_1^{\varepsilon}[{\omega}^{\varepsilon}(t,\cdot)](x) \text{ and }- \dfrac{1}{2\pi}DT_{{\varepsilon}}^t(x) I_2^{\varepsilon}[{\omega}^{\varepsilon}(t,\cdot)] (x)+ m H_{{\varepsilon}}(x),$$ by $\| {\omega}^{\varepsilon}(t,\cdot)\|_{L^1\cap L^\infty}$. The advantage of this decomposition is that each term has zero circulation around the small obstacle. Later, they have to study independently the last part of the velocity ${\gamma}H_{{\varepsilon}}$. As the size of the curve tends to zero, we see here that we have to use the decomposition from [@ift_lop_euler]. Then, let us introduce $$\tilde I_2^{\varepsilon}[h](x)=-I_2^{\varepsilon}[h](x) + m_h \frac{T_{\varepsilon}(x)^\perp}{|T_{\varepsilon}(x)|^2},$$ with $m_h=\displaystyle \int_{\Pi^{\varepsilon}} h(y) \, dy.$ \[I\_est\] For any $p\in (2,\infty]$, there exists a constant $C_p>0$ depending only on the shape of ${\Gamma}$, such that $$|I_{1}^{\varepsilon}[h](x)| \leq C {\varepsilon}\| h \|_{L^1}^{\frac{p-2}{2(p-1)}} \| h \|_{L^p}^{\frac{p}{2(p-1)}} \text{\ and\ } | \tilde I_2^{\varepsilon}[h](x) | \leq C {\varepsilon}\| h \|_{L^1}^{\frac{p-2}{2(p-1)}} \| h \|_{L^p}^{\frac{p}{2(p-1)}},$$ for all $x\in {{\mathbb R}}^2$, ${\varepsilon}>0$. The proof is the same as [@ift_lop_euler], where you replace $DT_{{\varepsilon}}(x)$ by $1/{\varepsilon}$ and where you use Lemma \[ift\]. For sake of clarity, we write the details. We start by treating $I_1^{\varepsilon}$: $$|I_1^{\varepsilon}[h](x) | \leq \int_{\Pi_{\varepsilon}}\dfrac{|h(y)|}{|T(x/{\varepsilon})-T(y/{\varepsilon})|} dy.$$ We introduce $J=J({\xi}):= |\det(DT^{-1})({\xi})|$ and $z={\varepsilon}T(x/{\varepsilon})$. Changing the variables ${\eta}={\varepsilon}T(y/{\varepsilon})$, we find $$|I_1^{\varepsilon}[h] (x)|\leq {\varepsilon}\int_{|{\eta}|\geq{\varepsilon}}\dfrac{|h({\varepsilon}T^{-1}({\eta}/{\varepsilon}))| J({\eta}/{\varepsilon})}{|z-{\eta}|}\, d{\eta}.$$ Then, we denote $f^{\varepsilon}({\eta})=|h({\varepsilon}T^{-1}({\eta}/{\varepsilon}))| J({\eta}/{\varepsilon}){\chi}_{|{\eta}\geq{\varepsilon}}$, with ${\chi}_E$ the characteristic function of the set $E$. Changing variables back, we remark that $$\|f^{\varepsilon}\|_{L^1({{\mathbb R}}^2)}=\|h\|_{L^1},$$ and $$\|f^{\varepsilon}\|_{L^p({{\mathbb R}}^2)}\leq C_p\|h\|_{L^p},$$ using the second point of Proposition \[2.2\]. Now, we can use Lemma \[ift\] to state that $$| I_1^{\varepsilon}[h](x) |\leq {\varepsilon}\int_{{{\mathbb R}}^2}\frac{f^{\varepsilon}({\eta}) }{|z-{\eta}|}\, d{\eta}\leq C_p{\varepsilon}\|f^{\varepsilon}\|_{L^1}^\frac{p-2}{2(p-1)} \|f^{\varepsilon}\|_{L^p}^\frac{p}{2(p-1)},$$ which allows us to conclude for $I_1^{\varepsilon}$. Let us focus now on the second term: $$\tilde I_2^{\varepsilon}[h](x) = \int_{\Pi_{\varepsilon}}\left(-\dfrac{(T(x/{\varepsilon})-T(y/{\varepsilon})^*)^\perp}{|T(x/{\varepsilon})-T(y/{\varepsilon})^*|^2}+\dfrac{T(x/{\varepsilon})^\perp}{|T(x/{\varepsilon})|^2}\right) h(y)\, dy.$$ We use, as before, the notations $J$, $z$ and the change of variables ${\eta}$ $$\begin{aligned} \tilde I_2^{\varepsilon}[h](x) &=& {\varepsilon}\int_{|{\eta}|\geq {\varepsilon}} \left(-\dfrac{z-{\varepsilon}^2 {\eta}^*}{|z-{\varepsilon}^2{\eta}^*|^2}+\dfrac{z}{|z|^2} \right) h({\varepsilon}T^{-1}({\eta}/{\varepsilon})) J({\eta}/{\varepsilon})\, d{\eta}\\ |\tilde I_2^{\varepsilon}[h](x) | &\leq& {\varepsilon}\int_{|{\eta}|\geq {\varepsilon}} \frac{{\varepsilon}^2|{\eta}^*|}{|z||z-{\varepsilon}^2{\eta}^*|}|h({\varepsilon}T^{-1}({\eta}/{\varepsilon}))|J({\eta}/{\varepsilon})d{\eta}.\end{aligned}$$ using . As $z={\varepsilon}T(x/{\varepsilon})$, we have $|z|\geq {\varepsilon}$, hence $$|\tilde I_2^{\varepsilon}[h](x) | \leq {\varepsilon}\int_{|{\eta}|\geq {\varepsilon}} \frac{{\varepsilon}|{\eta}^*|}{|z-{\varepsilon}^2{\eta}^*|}|h({\varepsilon}T^{-1}({\eta}/{\varepsilon}))|J({\eta}/{\varepsilon})d{\eta}.$$ Next, we change variables ${\theta}={\varepsilon}{\eta}^*$, and we obtain: $$\begin{aligned} |\tilde I_2^{\varepsilon}[h](x) | &\leq& {\varepsilon}\int_{|{\theta}|\leq 1}\frac{|{\theta}|}{|z-{\varepsilon}{\theta}|}|h({\varepsilon}T^{-1}({\theta}^*))|J({\theta}^*)\frac{{\varepsilon}^2}{|{\theta}|^4}d{\theta}\\ &\leq& {\varepsilon}\left(\int_{|{\theta}|\leq 1/2}+\int_{1/2\leq|{\theta}|\leq 1}\right):= {\varepsilon}(I_{21}+I_{22}).\end{aligned}$$ We start with $I_{21}$. If $|{\theta}|\leq 1/2$ then $|z-{\varepsilon}{\theta}|\geq{\varepsilon}/2$. Hence $$\begin{aligned} I_{21} &\leq& \int_{|{\theta}|\leq 1/2}2{\varepsilon}|{\theta}||h({\varepsilon}T^{-1}({\theta}^*))|J({\theta}^*)\frac{d{\theta}}{|{\theta}|^4} \\ &=& 2 \int_{|{\eta}|\geq 2{\varepsilon}}\frac{|h({\varepsilon}T^{-1}({\eta}/{\varepsilon}))|J({\eta}/{\varepsilon})}{|{\eta}|}d{\eta}\leq 2\int_{{{\mathbb R}}^2} \frac{f^{\varepsilon}({\eta})}{|{\eta}|}d{\eta},\end{aligned}$$ with $f^{\varepsilon}$ defined above. Using again Lemma \[ift\], we can conclude for $I_{21}$. To treat $I_{22}$, we put $g^{\varepsilon}({\theta})=|g({\varepsilon}T^{-1}({\theta}^*))|J({\theta}^*)\frac{{\varepsilon}^2}{|{\theta}|^4}$. We have $$I_{22}=\int_{1/2\leq|{\theta}|\leq 1}\frac{|{\theta}|}{|z-{\varepsilon}{\theta}|}g^{\varepsilon}({\theta})d{\theta}.$$ Changing variables back, we remark that $$\|g^{\varepsilon}\|_{L^1(1/2\leq|{\theta}|\leq 1)}\leq \|h\|_{L^1}.$$ Moreover, it is easy to see that $$\|g^{\varepsilon}\|_{L^p(1/2\leq|{\theta}|\leq 1)} \leq C {\varepsilon}^{\frac{2p-2}{p}}\|h\|_{L^p}.$$ Next, we apply Lemma \[ift\] with $g^{\varepsilon}$: $$\begin{aligned} I_{22}&=&\frac{1}{{\varepsilon}}\int_{1/2\leq|{\theta}|\leq 1} \frac{|{\theta}|}{|z/{\varepsilon}-{\theta}|}g^{\varepsilon}({\theta})d{\theta}\\ &\leq& \frac{C}{{\varepsilon}} \|g^{\varepsilon}\|_{L^1}^{\frac{p-2}{2(p-1)}}\|g^{\varepsilon}\|_{L^p}^{\frac{p}{2(p-1)}} \leq C_1 \|h \|_{L^1}^{\frac{p-2}{2(p-1)}}\|h\|_{L^p}^{\frac{p}{2(p-1)}},\end{aligned}$$ which ends the proof. In [@ift_lop_euler], the authors use the estimate of $\frac{1}{2\pi}DT_{{\varepsilon}}^t I_1^{\varepsilon}[h]$ and $\frac{1}{2\pi}DT_{{\varepsilon}}^t \tilde I_2^{\varepsilon}[h]$ with $h={\omega}^{\varepsilon}(t,\cdot)$, and with $h= \zeta \cdot {\nabla}{\Phi}^{\varepsilon}$ (${\Phi}^{\varepsilon}$ denoting the cutoff function of an ${\varepsilon}$ neighborhood of ${\Omega}_{\varepsilon}$, see the proof of Corollary 4.1 therein), where there exist some $L^1$ and $L^\infty$ estimates for these two functions. In our case, we have again that ${\omega}^{\varepsilon}(t,\cdot)$ are uniformly bounded in $L^1\cap L^\infty$, but we will only obtain $L^p$ estimates for ${\nabla}{\Phi}^{\varepsilon}$, with $p<4$ (see Lemma \[4.4\]). It explains why we have to establish estimates for $h$ belonging in $L^1\cap L^p$ for $p\in (2,\infty]$. Using the previous lemma with $h={\omega}^{\varepsilon}(t,\cdot)$, $p=+\infty$, and thanks to , , , we can deduce directly the following theorem: \[4.2\] We denote $v^{{\varepsilon}}:= u^{{\varepsilon}} - {\gamma}H_{{\varepsilon}}$. For any $p<4$, $v^{{\varepsilon}}$ is bounded in $L^\infty({{\mathbb R}}^+,L^p_{{\operatorname{{loc}}}}(\Pi_{{\varepsilon}}))$ independently of ${{\varepsilon}}$. More precisely, there exists a constant $C_p>0$ depending only on the shape of ${\Gamma}$ and the initial conditions $\|{\omega}_0\|_{L^1}$, $\|{\omega}_0\|_{L^\infty}$, such that $$\|v^{{\varepsilon}}(t,\cdot)\|_{L^p(B(0,R) \cap \Pi_{{\varepsilon}})}\leq C_p(1+R^{2/p}), \text{\ for all\ } R>0, \ t\geq 0.$$ The difference with [@lac_euler] is that we have an estimate $L^p_{{\operatorname{{loc}}}}$ only on $v^{{\varepsilon}}$, then we will have to study independently $H_{{\varepsilon}}$. We note also that we cannot obtain $L^\infty$ estimates, and we have to check carefully that we can adapt the tools used in [@ift_lop_euler]. Compact support of the vorticity --------------------------------   Specifying our choice for $\beta$ in Lemma \[renorm1\], we are led to the following. \[compact\_vorticity\] Let ${\omega}^{\varepsilon}$ be a weak solution of such that $${\omega}_0 \text{ is compactly supported in } B(0,R_0)$$ for some positive $R_0$. Then there exists $C>0$ independent of ${\varepsilon}$ such that $${\omega}^{\varepsilon}(t,\cdot) \text{ is compactly supported in } B(0,R_0+Ct),$$ for any $t\geq 0$. The main computation of this proof can be found in [@lac_miot], but we have to write the details because the velocity has a different form and that we need that $C$ is independent of ${\varepsilon}$. We set $\beta(t)=t^2$ and use Lemma \[renorm1\] with this choice. Let ${\Phi}\in \mathcal{D}({{\mathbb R}}^+\times {{\mathbb R}^2})$. We claim that for all $T$ $$\int_{{{\mathbb R}^2}} {\Phi}(T,x) ({\omega}^{\varepsilon})^2(T,x)\,dx - \int_{{{\mathbb R}^2}} {\Phi}(0,x) ({\omega}^{\varepsilon})^2(0,x)\,dx =\int_0^T\int_{{{\mathbb R}^2}} ({\omega}^{\varepsilon})^2 ({\partial_t}{\Phi}+u^{\varepsilon}\cdot \nabla {\Phi})\,dx \, dt.$$ This is actually an improvement of Lemma \[renorm1\], in which the equality holds in $L^1_{{\operatorname{{loc}}}}({{\mathbb R}}^+)$. Indeed, we have ${\partial}_t {\omega}^{\varepsilon}=-{{\rm div}\,}(u^{\varepsilon}{\omega}^{\varepsilon})$ (in the sense of distributions) with ${\omega}^{\varepsilon}\in L^\infty$ and $u^{\varepsilon}\in L^\infty({{\mathbb R}}^+,L^q_{{\operatorname{{loc}}}}({{\mathbb R}^2}))$ for all $q<4$, which implies that ${\partial}_t {\omega}^{\varepsilon}$ is bounded in $L^1_{{\operatorname{{loc}}}}({{\mathbb R}}^+,W^{-1,q}_{{\operatorname{{loc}}}}({{\mathbb R}^2}))$. Hence, ${\omega}^{\varepsilon}$ belongs to $C({{\mathbb R}}^+,W^{-1,q}_{{\operatorname{{loc}}}}({{\mathbb R}^2}))\subset C_{w}({{\mathbb R}}^+, L^2_{{\operatorname{{loc}}}}({{\mathbb R}}^2))$, where $C_{w}(L_{{\operatorname{{loc}}}}^{2})$ stands for the space of maps $f$ such that for any sequence $t_n\to t$, the sequence $f(t_n)$ converges to $f(t)$ weakly in $L^2_{{\operatorname{{loc}}}}$. Since on the other hand $t \mapsto \|{\omega}^{\varepsilon}(t)\|_{L^2}$ is continuous by Remark \[remark : conserv\], we have ${\omega}^{\varepsilon}\in C({{\mathbb R}}^+,L^2({{\mathbb R}^2}))$. Therefore the previous integral equality holds for all $T$. Now, we choose a good test function. We let ${\Phi}_0$ be a non-decreasing function on ${{\mathbb R}}$, which is equal to $1$ for $s\geq 2$ and vanishes for $s\leq 1$ and we set ${\Phi}(t,x)={\Phi}_0(|x|/R(t))$, with $R(t)$ a smooth, positive and increasing function to be determined later on, such that $R(0)=R_0$. For this choice of ${\Phi}$, we have $({\omega}_0(x))^2 {\Phi}(0,x)\equiv 0$. We compute then $${\nabla}{\Phi}= \frac{x}{|x|}\frac{{\Phi}_0'}{R(t)}$$ and $${\partial}_t {\Phi}= -\frac{R'(t)}{R^2(t)}|x|{\Phi}_0'.$$ We obtain $$\begin{aligned} \int_{{{\mathbb R}^2}} {\Phi}(T,x) ({\omega}^{\varepsilon})^2(T,x)\,dx & =&\int_0^T \int_{{{\mathbb R}^2}} ({\omega}^{\varepsilon})^2 \frac{{\Phi}_0'(\frac{|x|}{R})}{R}\Bigl( u^{\varepsilon}(x) \cdot \frac{x}{|x|}-\frac{R'}{R}|x|\Bigl)\, dx\, dt\\ &\leq& \int_0^T \int_{{{\mathbb R}^2}} ({\omega}^{\varepsilon})^2 \frac{|{\Phi}_0'|(\frac{|x|}{R})}{R} (C -R')\, dx\, dt,\end{aligned}$$ where $C$ is independent of ${\varepsilon}$. Indeed, we have that $$u^{{\varepsilon}}(t,x)=\dfrac{1}{2\pi}DT_{{\varepsilon}}^t(x)(I_1^{\varepsilon}+ \tilde I_2^{\varepsilon}+{\gamma}\frac{T_{\varepsilon}(x)^\perp}{|T_{\varepsilon}(x)|^2} )$$ with $ |I_1^{\varepsilon}+ \tilde I_2^{\varepsilon}| \leq C_1 {\varepsilon}$ (see Lemma \[I\_est\]) and $DT_{\varepsilon}(x) = \frac{1}{{\varepsilon}} DT(\frac{x}{\varepsilon})$. Using Remark \[2.5\], we know that there exist some positives $C_3,C_4$ independent of ${\varepsilon}$, such that $$|DT(\frac{x}{\varepsilon}) | \leq C_2 |{\beta}| \text{ and } C_4 |{\beta}| \frac{|x|}{\varepsilon}\leq |T(\frac{x}{\varepsilon})|,$$ for all $|x|\geq R_0$. Putting together all these inequalities, we obtain $C=\frac{1}{2\pi}C_2( |{\beta}| C_1 + \frac{|{\gamma}|}{R_0 C_4})$. Taking $R(t) =R_0 + Ct$, we arrive at $$\int_{{{\mathbb R}^2}} {\Phi}(T,x) ({\omega}^{\varepsilon})^2(T,x)\,dx\leq 0,$$ which ends the proof. We only use in this paper that $(u^{\varepsilon},{\omega}^{\varepsilon})$ is a weak solution of the Euler equations outside the curve (see Definition \[sol-curve\]). If the uniqueness is proved, we could simplify the proofs of , and Proposition \[compact\_vorticity\]. Indeed, we would say by uniqueness that ${\omega}^{\varepsilon}$ is the weak-$*$ limit of ${\Phi}^{{\varepsilon},{\eta}} {\omega}^{{\varepsilon},{\eta}}$ with ${\Omega}^{{\varepsilon},{\eta}}$ defined in Subsection \[thicken\]. As $ {\omega}^{{\varepsilon},{\eta}}$ verifies the transport equation in a strong sense, we have: - for all ${\eta}$ and $t$, $\| {\omega}^{{\varepsilon},{\eta}}(t,\cdot) \|_{L^1\cap L^\infty} = \| {\omega}_0 \|_{L^1\cap L^\infty}$, which means that $\| {\omega}^{{\varepsilon}}(t,\cdot) \|_{L^1\cap L^\infty} \leq \| {\omega}_0 \|_{L^1\cap L^\infty}$ which is sufficient; - ${\omega}^{\varepsilon}$ is also the weak-$*$ limit of ${\chi}_{\Pi_{{\varepsilon},{\eta}}} {\omega}^{{\varepsilon},{\eta}}$, and $\int {\chi}_{\Pi_{{\varepsilon},{\eta}}} {\omega}^{{\varepsilon},{\eta}} = \int {\omega}_0$ for all $t$, so we obtain ; - it is easy to prove that there exists $C$ independent of ${\eta}$ and ${\varepsilon}$ such that $ {\omega}^{{\varepsilon},{\eta}}(t,\cdot)$ is compactly supported in $B(0,R_1+Ct)$, which proves Proposition \[compact\_vorticity\], using test functions supported in $B(0,R_1+Ct)^c$. Cutoff function {#cutoff} --------------- The function $u^{{\varepsilon}}$ is defined on ${{\mathbb R}}^2$, but we prefer to multiply it by an ${{\varepsilon}}$-dependent cutoff function for a neighborhood of ${\Omega}_{{\varepsilon}}$. Indeed, ${{\rm curl}\,}u^{\varepsilon}= {\omega}^{\varepsilon}+ g_{{\omega}^{\varepsilon}} {\delta}_{{\Gamma}_{\varepsilon}}$, so the cutoff function allows us to remove the dirac mass and the jump of the velocity through the curve. Let ${\Phi}\in C^\infty({{\mathbb R}})$ be a non-decreasing function such that $0\leq{\Phi}\leq 1$, ${\Phi}(s)=1$ if $s\geq 3$ and ${\Phi}(s)=0$ if $s\leq 2$. Then we introduce $${\Phi}^{{\varepsilon}}={\Phi}^{{\varepsilon}}(x)={\Phi}(|T_{{\varepsilon}}(x)|).$$ Clearly ${\Phi}^{{\varepsilon}}$ is $C^\infty({{\mathbb R}}^2)$ vanishing in a neighborhood of $\overline{{\Omega}_{{\varepsilon}}}$. We require some properties of ${\nabla}{\Phi}^{{\varepsilon}}$ which we collect in the following Lemma. \[4.4\] The function ${\Phi}^{{\varepsilon}}$ defined above has the following properties: - $H_{{\varepsilon}}\cdot {\nabla}{\Phi}^{{\varepsilon}}\equiv 0$ in $\Pi_{{\varepsilon}}$, - there exists a constant $C>0$ such that the Lebesgue measure of the support of ${\Phi}^{{\varepsilon}}-1$ is bounded by $C{{\varepsilon}^2}$. - for all $p<4$, there exists a constant $C_p>0$ such that $\|{\nabla}{\Phi}^{{\varepsilon}} \|_{L^p}\leq {\varepsilon}^{\frac2p -1} C_p$. First, we remark that $$H_{{\varepsilon}}(x)=\frac{1}{2\pi}{\nabla}^\perp \ln |T_{{\varepsilon}}(x)|=\frac{1}{2\pi|T_{{\varepsilon}}(x)|}{\nabla}^\perp |T_{{\varepsilon}}(x)|,$$ and $${\nabla}{\Phi}^{{\varepsilon}}={\Phi}'(|T_{{\varepsilon}}(x)|){\nabla}|T_{{\varepsilon}}(x)|$$ what gives us the first point. Concerning the second point, the support of ${\Phi}^{{\varepsilon}} -1$ is contained in the subset $\{ x\in\Pi_{{\varepsilon}} | 1 \leq |T_{{\varepsilon}}(x)| \leq 3\}$. By Proposition \[biholo-est\], the Lebesgue measure can be estimated as follows: $$\int_{ 1 \leq |T_{{\varepsilon}}(x)| \leq 3}dx=\int_{1 \leq |z| \leq 3} |\det(DT_{{\varepsilon}}^{-1})|(z) dz \leq C_1 {{\varepsilon}^2}.$$ Finally, we have $$|{\nabla}{\Phi}^{{\varepsilon}}(x)| \leq |{\Phi}'(|T_{\varepsilon}(x)|)| | DT_{\varepsilon}(x)|,$$ hence, $$\| {\nabla}{\Phi}^{{\varepsilon}} \|_{L^p}\leq C \|DT_{\varepsilon}(x) \|_{L^p(\{x| |T_{\varepsilon}(x)|\leq 3\})}.$$ Using, that $T(z)$ goes to infinity when $|z|\to \infty$, we can state that there exists $R_1>0$ such that $\{ y\in {{\mathbb R}}^2 | |T(y)|\leq 3\}=T^{-1}(B(0,3)\setminus B(0,1)) \subset B(0,R_1)$. We rewrite the computation made in the proof of Proposition \[biholo-est\]: $$\begin{aligned} \Bigl( \int_{\{x | |T(x/{\varepsilon})|\leq 3\}} \bigl| \frac1{\varepsilon}DT(\frac{x}{\varepsilon}) \bigl|^p\, dx\Bigl)^{1/p} &=& {\varepsilon}^{\frac2p - 1} \Bigl( \int_{\{y | |T(y)|\leq 3\}} \bigl| DT(y) \bigl|^p\, dy\Bigl)^{1/p} \\ &\leq& {\varepsilon}^{\frac2p - 1} \Bigl( \int_{B(0,R_1)} \bigl| DT(y) \bigl|^p\, dy\Bigl)^{1/p} \\ &\leq& {\varepsilon}^{\frac2p - 1} C_p,\end{aligned}$$ which ends the proof. \[rk-phi\] As $v^{\varepsilon}(x)=\frac{1}{2\pi} DT_{\varepsilon}(x) (I_1+\tilde I_2)$, using Lemma \[I\_est\] and the proof of point (c), we can state that for all $p<4$, there exists a constant $C_p>0$ such that $$\| v^{\varepsilon}(x) \|_{L^p(\{x| |T_{\varepsilon}(x)|\leq 3\})}\leq {\varepsilon}^{\frac2p} C_p.$$ In the case where the obstacle is smooth (see [@ift_lop_euler]), $DT$ is bounded, which implies that the norm $L^2$ of ${\nabla}{\Phi}^{{\varepsilon}}$ is bounded. Moreover, in their case, the part of velocity $v^{{\varepsilon}}$ is bounded independently of ${{\varepsilon}}$, so we can prove that the limits of $v^{{\varepsilon}} \cdot {\nabla}{\Phi}^{{\varepsilon}}$ and $v^{{\varepsilon}} \cdot {\nabla}^\perp {\Phi}^{{\varepsilon}}$ is bounded in $L^\infty(L^2_{{\operatorname{{loc}}}})$. As $L^2_{{\operatorname{{loc}}}}$ is compactly imbedded in $H^{-1}_{{\operatorname{{loc}}}}$, we can prove by Aubin-Lions Lemma that the divergence and the curl of ${\Phi}^{\varepsilon}v^{\varepsilon}$ is precompact in $C([0,T]; H^{-1}_{{\operatorname{{loc}}}} ({{\mathbb R}}^2))$. Finally the authors of [@ift_lop_euler] conclude thanks to the Div-Curl Lemma. In our case, let us show that we can apply this argument. We use Lemma \[4.4\] and Remark \[rk-phi\] with $p=3$, then $$\label{v-phi-1} \|v^{{\varepsilon}} \cdot {\nabla}^\perp {\Phi}^{{\varepsilon}}\|_{L^{3/2}}\leq \|v^{\varepsilon}(x) \|_{L^3({\operatorname{supp\,}}({\nabla}{\Phi}^{{\varepsilon}}))} \|{\nabla}{\Phi}^{{\varepsilon}} \|_{L^3({\operatorname{supp\,}}({\nabla}{\Phi}^{{\varepsilon}}))} \leq C{\varepsilon}^{1/3} .$$ Similarly, we have $$\label{v-phi-2} \|v^{{\varepsilon}} \cdot {\nabla}{\Phi}^{{\varepsilon}}\|_{L^{3/2}}\leq \|v^{\varepsilon}(x) \|_{L^3({\operatorname{supp\,}}({\nabla}{\Phi}^{{\varepsilon}}))} \|{\nabla}{\Phi}^{{\varepsilon}} \|_{L^3({\operatorname{supp\,}}({\nabla}{\Phi}^{{\varepsilon}}))} \leq C{\varepsilon}^{1/3} .$$ As $H^1({{\mathbb R}}^2)$ is imbedded in $L^{3}({{\mathbb R}}^2)$, so $L^{3/2}({{\mathbb R}}^2)$ is imbedded in $H^{-1}({{\mathbb R}}^2)$, and we could apply Aubin-Lions Lemma. This last computation is an improvement of a naive estimate. Indeed, we would have written that: $$\|v^{{\varepsilon}} \cdot {\nabla}^\perp {\Phi}^{{\varepsilon}}\|_{L^{1}} \leq C\|v^{{\varepsilon}}\|_{L^{4}} \|DT_{\varepsilon}\|_{L^{4}_{loc}} \|1 \|_{L^{2}({\operatorname{supp\,}}({\nabla}{\Phi}^{\varepsilon}))} \leq C_1\frac1{\varepsilon}{\varepsilon},$$ assuming that Theorem \[4.2\] and point (ii) of Proposition \[biholo-est\] could be applied for $p=4$, which is not true. Even in this limit case, we remark that we can only control the $L^1$ norm of $v^{{\varepsilon}} \cdot {\nabla}^\perp {\Phi}^{{\varepsilon}}$, which does not embed in $H^{-1}$ in dimension two. With this estimate, the argument from [@ift_lop_euler] falls down. Estimate was established thanks to point (c) of Lemma \[4.4\] and Remark \[rk-phi\]. Without this improvement, we would have adapted the arguments from [@lac_euler]. However, we choose here to use techniques from [@ift_lop_euler], because it is faster and it is less technical. As we decompose $u^{\varepsilon}=v^{\varepsilon}+{\gamma}H_{{\varepsilon}}$, we have to focus on the harmonic part. \[H-limit\] Let $H:= x^\perp / (2\pi |x|^2)$ and fix $R>0$. Then, $$H_{{\varepsilon}} \to H,$$ strongly in $L^p(B(0,R))$ as ${\varepsilon}\to 0$, for any $p<2$. The proof is similar than [@ift_lop_euler], because there is the same behaviour at infinity (see Remark \[2.5\]). However this lemma is stated in [@ift_lop_euler] only with $p=1$. We will see in the following subsection that we need for $p=3/2$. For this reason, we rewrite the proof here. We fix $p<2$ and we decompose: $$\begin{aligned} \| H_{{\varepsilon}}-H\|_{L^p(B(0,R))} &\leq& \| H_{{\varepsilon}}-H\|_{L^p(B(0,R)\cap\{|T_{\varepsilon}(x)|\geq 2\})} + \| H_{{\varepsilon}}\|_{L^p(\{|T_{\varepsilon}(x)|\leq 2\})} + \| H\|_{L^p(\{|T_{\varepsilon}(x)|\leq 2\})}\\ &:=& \mathcal{I}_1+\mathcal{I}_2+\mathcal{I}_3.\end{aligned}$$ From the proof of Lemma \[4.4\], we know that the Lebesgue measure of the set $\{|T_{\varepsilon}(x)|\leq 2\}$ tends to zero as ${\varepsilon}\to 0$. Having in mind that $H$ belongs in $L^q_{{\operatorname{{loc}}}}$ for $q\in (p,2)$, we can state that $\mathcal{I}_3\to 0$ as ${\varepsilon}\to 0$. Concerning $\mathcal{I}_2$, we change variables $y=x/{\varepsilon}$: $$\begin{aligned} \mathcal{I}_2 &=& \Bigl( \int_{\{|T(x/{\varepsilon})|\leq 2\}} \Bigl| \frac{1}{2{\varepsilon}\pi} DT(x/{\varepsilon})^t \frac{T(x/{\varepsilon})^\perp}{|T(x/{\varepsilon}) |^2} \Bigl|^p\, dx\Bigl)^{1/p} \\ &=& \Bigl( \int_{\{|T(y)|\leq 2\}} \Bigl| \frac{1}{2{\varepsilon}\pi} DT(y)^t \frac{T(y)^\perp}{|T(y) |^2} \Bigl|^p {\varepsilon}^2\, dy\Bigl)^{1/p}\\ &\leq& \frac{{\varepsilon}^{\frac{2-p}{p}}}{2\pi} \| DT \|_{L^p(\{|T(y)|\leq 2\})} \end{aligned}$$ which gives the result because $DT$ belongs to $L^q_{{\operatorname{{loc}}}}$ for $q<4$ (see Proposition \[2.2\]). For $\mathcal{I}_1$, we use Remark \[2.5\]: $T(y)={\beta}y+h(y)$, with ${\beta}\in {{\mathbb R}}^*$, and $h$ holomorphic such that $|Dh(y)|\leq C/|y|^2$. Changing variables as above, we find: $$\begin{aligned} \mathcal{I}_1 &=& \frac{{\varepsilon}^{\frac{2-p}{p}}}{2\pi} \Bigl( \int_{B(0,R/{\varepsilon})\cap\{|T(y)|\geq 2\}} \Bigl| DT(y)^t \frac{T(y)^\perp}{|T(y) |^2}-\frac{y^\perp}{|y|^2} \Bigl|^p\, dy \Bigl)^{1/p} \\ &=& \frac{{\varepsilon}^{\frac{2-p}{p}}}{2\pi} \Bigl( \int_{B(0,R/{\varepsilon})\cap\{|T(y)|\geq 2\}} \Bigl| ({\beta}\mathbb{I}+ Dh^t(y)) \frac{({\beta}y+h(y))^\perp}{|{\beta}y+h(y) |^2}-{\beta}\mathbb{I} \frac{{\beta}y^\perp}{|{\beta}y|^2} \Bigl|^p\, dy\Bigl)^{1/p} \\ &\leq& \frac{{\varepsilon}^{\frac{2-p}{p}}}{2\pi} \Bigl( \int_{B(0,R/{\varepsilon})\cap\{|T(y)|\geq 2\}} \Bigl| Dh^t(y) \frac{({\beta}y+h(y))^\perp}{|{\beta}y+h(y) |^2} \Bigl|^p\, dy\Bigl)^{1/p}\\ &&+ \frac{{\varepsilon}^{\frac{2-p}{p}}}{2\pi} \Bigl( \int_{B(0,R/{\varepsilon})\cap\{|T(y)|\geq 2\}} \Bigl| {\beta}\mathbb{I} \Bigl( \frac{({\beta}y+h(y))^\perp}{|{\beta}y+h(y) |^2}- \frac{{\beta}y^\perp}{|{\beta}y|^2}\Bigl) \Bigl|^p\, dy\Bigl)^{1/p}\\ &\leq& C {\varepsilon}^{\frac{2-p}{p}} \Bigl( \int_{\{|T(y)|\geq 2\}} \frac{1}{|y|^{3p}}\, dy\Bigl)^{1/p}+ C{\varepsilon}^{\frac{2-p}{p}} \Bigl( \int_{B(0,R/{\varepsilon})\cap\{|T(y)|\geq 2\}} \Bigl(\frac{|h(y)|}{|y||{\beta}y+h(y) |} \Bigl)^p\, dy\Bigl)^{1/p},\end{aligned}$$ using . If $p\in (1,2)$, we bound the right hand side term by $$C_1 {\varepsilon}^{\frac{2-p}{p}}+ C_2 {\varepsilon}^{\frac{2-p}{p}} \Bigl( \int_{\{|T(y)|\geq 2\}} \frac{1}{|y|^{2p}}\, dy\Bigl)^{1/p}\leq C_3 {\varepsilon}^{\frac{2-p}{p}}$$ which tends to zero if ${\varepsilon}\to 0$. If $p=1$ we bound the right hand side term by $$C_1 {\varepsilon}+ C_2 {\varepsilon}\ln (R/{\varepsilon})$$ which also tends to zero if ${\varepsilon}\to 0$. Now, we need some estimates of ${\omega}^{\varepsilon}_t$ and $v^{\varepsilon}_t$ in order to use Aubin-Lions Lemma. Temporal estimates ------------------ Although in our case, the vorticity equation is verify in the sense of distribution, we directly see that it also means that ${\omega}^{\varepsilon}_t$ is bounded in $L^\infty([0,T]; W^{-1,1}_{{\operatorname{{loc}}}}({{\mathbb R}}^2))$. Indeed, we have proved that $v^{\varepsilon}$ and $H_{{\varepsilon}}$ are bounded in $L^\infty(L^1_{{\operatorname{{loc}}}})$, whereas ${\omega}^{\varepsilon}$ is bounded in $L^\infty$. We recall from Proposition \[compact\_vorticity\], that for $T$ fixed, there exists $R_1>0$ such that ${\omega}^{\varepsilon}(t,\cdot)$ is compactly supported in $B(0,R_1)$ for all $0\leq t \leq T$ and ${\varepsilon}>0$. Additionally, ${\omega}^{\varepsilon}_t$ is also compactly supported in the same ball. Concerning $v^{\varepsilon}_t$, we have to prove that Proposition 4.1 and Corollary 4.1 of [@ift_lop_euler] hold true in our case. We introduce the stream function associated to ${\omega}^{\varepsilon}$ by ${\psi}^{\varepsilon}:= G_{\varepsilon}[{\omega}^{\varepsilon}]$, with $$G_{\varepsilon}[f](x) = \int_{\Pi_{\varepsilon}} G_{\varepsilon}(x,y) f(y)\, dy, \ \forall f\in C^\infty_c(\Pi_{\varepsilon})$$ (see for the explicit formula). We note that $K_{\varepsilon}[f]={\nabla}^\perp G_{\varepsilon}[f]$. For each $R,T>0$, there exists a constant $C>0$ independent of ${\varepsilon}$, such that $$\Bigl| \int_{\Pi_{\varepsilon}} {\varphi}(x) {\psi}^{\varepsilon}_t(t,x)\, dx \Bigl|\leq C(\| {\varphi}\|_{L^1}+\| {\varphi}\|_{L^1}^{1/4} \| {\varphi}\|_{L^3}^{3/4}),$$ for every ${\varphi}\in C_0(\Pi_{\varepsilon}\cap B(0,R))$ and for all $0\leq t\leq T$. We differentiate with respect of $t$ the stream function: ${\psi}^{\varepsilon}_t=G_{\varepsilon}[{\omega}^{\varepsilon}_t]$, which means that $${\Delta}{\psi}^{\varepsilon}_t = {\omega}_t^{\varepsilon}\text{ in } \Pi_{\varepsilon}, \text{ and } {\psi}_t^{\varepsilon}=0 \text{ on }{\Gamma}_{\varepsilon}.$$ To obtain information on the behavior of ${\psi}^{\varepsilon}_t$ at infinity, we use the same argument than to state that $$\label{psi} | {\psi}_t^{\varepsilon}(t,x)-L[{\omega}^{\varepsilon}_t(t,\cdot)](x)| = O(1/|x|) \text{ at infinity},$$ where the functional $L$ is defined by $$\zeta\mapsto L[\zeta]:= -\frac1{2\pi} \int_{\Pi_{\varepsilon}} \ln |T_{\varepsilon}(y)| \zeta(y)\, dy,$$ for any test function $\zeta$. The asymptotic behavior is not independent of ${\varepsilon}$, but we will only need that for ${\varepsilon}$ fixed. Moreover, we recall that gives $$\label{na-psi} |{\nabla}{\psi}_t^{\varepsilon}|=|K_{\varepsilon}[{\omega}_t^{\varepsilon}]| =O(1/|x|^2) \text{ at infinity.}$$ Let ${\varphi}$ be a fixed test function in $C_0(\Pi_{\varepsilon}\cap B(0,R))$, we define $${\eta}:= G_{\varepsilon}[{\varphi}] +\frac{m_{\varphi}}{2\pi} \ln |T_{\varepsilon}|,$$ where $m_{\varphi}= \int_{\Pi_{\varepsilon}} {\varphi}(x)\, dx$. As above, we can remark that ${\eta}$ satisfies $${\Delta}{\eta}= {\varphi}\text{ in } \Pi_{\varepsilon}, \text{ and } {\eta}=0 \text{ on }{\Gamma}_{\varepsilon},$$ $$\label{eta} {\eta}(x)= \frac{m_{\varphi}}{2\pi} \ln |T_{\varepsilon}|(x) + L[{\varphi}](x) + O(1/|x|) \text{ at infinity}$$ and $$\label{na-eta} |{\nabla}( {\eta}- \frac{m_{\varphi}}{2\pi} \ln |T_{\varepsilon}|)|=|K_{\varepsilon}[{\varphi}]| =O(1/|x|^2) \text{ at infinity.}$$ We compute $$\begin{aligned} \int_{\Pi_{\varepsilon}} {\varphi}(x) {\psi}^{\varepsilon}_t(t,x)\, dx &=& \int_{\Pi_{\varepsilon}} {\Delta}{\eta}(x) {\psi}^{\varepsilon}_t(t,x)\, dx\\ &=& \int_{\Pi_{\varepsilon}} {\eta}(x) {\Delta}{\psi}^{\varepsilon}_t(t,x)\, dx + \int_{{\partial}\Pi_{\varepsilon}} ( {\psi}^{\varepsilon}_t {\nabla}{\eta}- {\eta}{\nabla}{\psi}^{\varepsilon}_t)\cdot \hat{n} \, ds\\ &:=& I + J \end{aligned}$$ where the boundary terms include the terms at infinity. Using [^3], we begin by estimating $I$: $$I= \int_{\Pi_{\varepsilon}} {\eta}(x) {\omega}^{\varepsilon}_t(t,x)\, dx= \int_{\Pi_{\varepsilon}} {\nabla}{\eta}(x)\cdot (v^{\varepsilon}+ {\gamma}H_{{\varepsilon}}) {\omega}^{\varepsilon}\, dx,$$ then $$| I | \leq \| {\nabla}{\eta}\|_{L^3(B(0,R_1))} \|v^{\varepsilon}+ {\gamma}H_{{\varepsilon}}\|_{L^{3/2}(B(0,R_1))} \| {\omega}^{\varepsilon}\|_{L^\infty} \leq C \| {\nabla}{\eta}\|_{L^3(B(0,R_1))},$$ thanks to and using again Theorem \[4.2\] and Lemma \[H-limit\] with $p=3/2$. Moreover, as we have $${\nabla}^\perp {\eta}(x) =\frac1{2\pi} DT_{\varepsilon}^t(x) \bigl( I_1^{\varepsilon}[{\varphi}] - I_2^{\varepsilon}[{\varphi}] + m_{\varphi}\frac{T_{\varepsilon}(x)^\perp}{|T_{\varepsilon}(x)|^2 }\bigl),$$ we can use point (ii) of Proposition \[biholo-est\] for $p=1$ and Lemma \[I\_est\] for $p=3$ to conclude that $$| I | \leq C\frac{1}{\varepsilon}{\varepsilon}\| {\varphi}\|_{L^1}^{1/4} \| {\varphi}\|_{L^3}^{3/4}.$$ Concerning the boundary terms $J$, we note that the integrals on ${\Gamma}_{\varepsilon}$ vanish, because ${\eta}={\psi}_t^{\varepsilon}= 0$ on the curve. Thanks to and , we have $$\int_{{\partial}B(0,R)} {\eta}{\nabla}{\psi}^{\varepsilon}_t\, ds \leq C \frac{\ln R}R$$ which tends to zero as $R\to \infty$. Using now and , we obtain $$|J| \leq C m_{\varphi}|L[{\omega}^{\varepsilon}_t]|\leq C \| {\varphi}\|_{L^1}|L[{\omega}^{\varepsilon}_t]| .$$ To finish the proof, we have to estimate $|L[{\omega}^{\varepsilon}_t]|$. Keeping in mind that $H_{{\varepsilon}}(y)= {\nabla}^\perp (\ln |T_{\varepsilon}(y)|)$, we compute $$\begin{aligned} L[{\omega}^{\varepsilon}_t] &=& -\frac1{2\pi} \int_{\Pi_{\varepsilon}} {\nabla}(\ln |T_{\varepsilon}(y)|)\cdot (v^{\varepsilon}(t,y)+{\gamma}H_{{\varepsilon}}(y)) {\omega}^{\varepsilon}(t,y)\, dy \\ &=& -\frac1{2\pi} \int_{\Pi_{\varepsilon}} H_{{\varepsilon}}(y)^\perp \cdot v^{\varepsilon}(t,y) {\omega}^{\varepsilon}(t,y)\, dy \\ |L[{\omega}^{\varepsilon}_t]| &\leq& \| H_{{\varepsilon}}(y) \|_{L^{3/2}(B(0,R_1))} \| v^{\varepsilon}\|_{L^{3}(B(0,R_1))} \| {\omega}^{\varepsilon}\|_{L^{\infty}} \leq C\end{aligned}$$ using Theorem \[4.2\] with $p=3$ and Lemma \[H-limit\] with $p=3/2$. Putting together the estimates concludes the proof. In [@ift_lop_euler], it is sufficient to bound the integral by $ \| {\varphi}\|_{L^1}^{1/2} \| {\varphi}\|_{L^\infty}^{1/2}$. We will see in the following proposition that we need $\| {\varphi}\|_{L^p}$ for some $p<4$ instead of $\| {\varphi}\|_{L^\infty}$ (e.g. $p=3$). For this goal, we use in the previous proof Lemma \[4.2\] for $h\in L^p$ instead of $L^\infty$, which justifies the extension for $p\neq \infty$ in Lemma \[4.2\]. We also see at the end of the previous proof that we cannot write $\| v^{\varepsilon}\|_{L^\infty}$, so it explains why we need the extension for $p>1$ in Lemma \[H-limit\]. Thanks to this proposition, we can establish the main result of this subsection. \[v\_t\] Let $R,T>0$. Then there exists a constant $C=C(R,T)>0$ such that $$\|({\Phi}^{\varepsilon}v^{\varepsilon})_t(t,\cdot)\|_{H^{-3}(B(0,R))}\leq C,$$ for all ${\varepsilon}$ and $0\leq t\leq T$. Let $\zeta \in (H_0^3(B(0,R)))^2$. Applying twice the previous proposition, we compute $$\begin{aligned} | \langle \zeta, ({\Phi}^{\varepsilon}v^{\varepsilon})_t (\cdot, t)\rangle | &= & \Bigl| \int \zeta {\Phi}^{\varepsilon}{\nabla}^\perp {\psi}^{\varepsilon}_t(t,\cdot)\Bigl| = \Bigl| \int {{\rm curl}\,}(\zeta {\Phi}^{\varepsilon}) {\psi}^{\varepsilon}_t(t,\cdot)\Bigl|\\ &=& \Bigl| \int {{\rm curl}\,}(\zeta) {\Phi}^{\varepsilon}{\psi}^{\varepsilon}_t(t,\cdot) + \int \zeta\cdot {\nabla}^\perp {\Phi}^{\varepsilon}{\psi}^{\varepsilon}_t(t,\cdot)\Bigl| \\ &\leq& C(\| {{\rm curl}\,}(\zeta) {\Phi}^{\varepsilon}\|_{L^1}+\| {{\rm curl}\,}(\zeta) {\Phi}^{\varepsilon}\|_{L^1}^{1/4} \| {{\rm curl}\,}(\zeta) {\Phi}^{\varepsilon}\|_{L^3}^{3/4})\\ && + C(\| \zeta\cdot {\nabla}^\perp {\Phi}^{\varepsilon}\|_{L^1}+\| \zeta\cdot {\nabla}^\perp {\Phi}^{\varepsilon}\|_{L^1}^{1/4} \| \zeta\cdot {\nabla}^\perp {\Phi}^{\varepsilon}\|_{L^3}^{3/4})\\ &\leq & C (\| {{\rm curl}\,}\zeta\|_{L^\infty}+ \| \zeta\|_{L^\infty}),\end{aligned}$$ since $\| {\nabla}{\Phi}^{\varepsilon}\|_{L^1} \leq C {\varepsilon}$ and $\| {\nabla}{\Phi}^{\varepsilon}\|_{L^3} \leq C {\varepsilon}^{-1/3}$ (see point (c) of Lemma \[4.4\]). Sobolev embedding theorem allows us to end the proof. We understand now why we need all these estimates in terms of $\| h \|_{L^3}$ instead of $\| h \|_{L^\infty}$. Indeed, we use them with ${\nabla}{\Phi}^{\varepsilon}$, and we remark in Lemma \[4.4\] that we cannot obtain estimates in $L^p$ norm for $p\geq 4$, because of $DT$ which blows up at the end-points like the inverse of the square root of the distance. Passing to the limit ==================== Thanks to , and the previous corollary, we can exactly apply the arguments from [@ift_lop_euler]. In order to simplify the reading, we write the details. Strong compactness for the velocity ----------------------------------- The principal tool is a parameterized version of Tartar and Murat’s Div-Curl Lemma, whose proof can be found in [@div-curl]: \[div-curl\] Fix $T>0$ and let $\{F^{\varepsilon}(t,\cdot)\}$ and $\{G^{\varepsilon}(t,\cdot)\}$ be vector fields on ${{\mathbb R}}^2$ for $0\leq t\leq T$. Suppose that: - both $F^{\varepsilon}\to F$ and $G^{\varepsilon}\to G$ weak-$*$ in $L^\infty([0,T]; L^2_{{\operatorname{{loc}}}} ({{\mathbb R}}^2;{{\mathbb R}}^2))$ and also strongly in $C([0,T]; H^{-1}_{{\operatorname{{loc}}}} ({{\mathbb R}}^2;{{\mathbb R}}^2))$; - $\{ {{\rm div}\,}F^{\varepsilon}\}$ is precompact in $C([0,T]; H^{-1}_{{\operatorname{{loc}}}} ({{\mathbb R}}^2))$; - $\{ {{\rm curl}\,}G^{\varepsilon}\}$ is precompact in $C([0,T]; H^{-1}_{{\operatorname{{loc}}}} ({{\mathbb R}}^2;{{\mathbb R}}))$. Then $F^{\varepsilon}\cdot G^{\varepsilon}\rightharpoonup F\cdot G$ in $\mathcal{D}'([0,T]\times {{\mathbb R}}^n)$. We will use the Div-Curl Lemma with $F^{\varepsilon}=G^{\varepsilon}={\Phi}^{\varepsilon}v^{\varepsilon}$. For that, we check now that the three points of this lemma are verified. For point (a), we know from Theorem \[4.2\] that $\{ {\Phi}^{\varepsilon}v^{\varepsilon}\}$ is bounded in $L^\infty([0,T]; L^2_{{\operatorname{{loc}}}} ({{\mathbb R}}^2))$. Moreover, thanks to Corollary \[v\_t\] we know that $\{ {\Phi}^{\varepsilon}v^{\varepsilon}\}$ is equicontinuous from $[0,T]$ to $H^{-3}_{{\operatorname{{loc}}}}$. Then we can apply Aubin-Lions Lemma (see [@temam]) to state that $\{ {\Phi}^{\varepsilon}v^{\varepsilon}\}$ is precompact in $C([0,T]; H^{-1}_{{\operatorname{{loc}}}} ({{\mathbb R}}^2))$. Passing to a subsequence if necessary, we conclude that there exists $v\in L^\infty([0,T]; L^2_{{\operatorname{{loc}}}}) \cap C([0,T]; H^{-1}_{{\operatorname{{loc}}}} )$ such that $${\Phi}^{\varepsilon}v^{\varepsilon}\to v$$ weak-$*$ in $L^\infty([0,T]; L^2_{{\operatorname{{loc}}}})$ and strongly in $C([0,T]; H^{-1}_{{\operatorname{{loc}}}})$. For point (b), we start by remarking that $({{\rm div}\,}({\Phi}^{\varepsilon}v^{\varepsilon}))_t={{\rm div}\,}({\Phi}^{\varepsilon}v^{\varepsilon})_t$ is bounded in $L^\infty([0,T]; H^{-4}_{{\operatorname{{loc}}}})$ (see Corollary \[v\_t\]). Moreover, we know that $${{\rm div}\,}({\Phi}^{\varepsilon}v^{\varepsilon}) = v^{\varepsilon}\cdot {\nabla}{\Phi}^{\varepsilon}$$ is bounded in $L^\infty([0,T]; L^{3/2})$ (see ). Since $L^{3/2}_{{\operatorname{{loc}}}}$ is compactly imbedded in $H^{-1}_{{\operatorname{{loc}}}}$, we can again apply Aubin-Lions Lemma to conclude that the divergence is precompact in $C([0,T]; H^{-1}_{{\operatorname{{loc}}}})$. Finally, we do the same thing with the curl: - $({{\rm curl}\,}({\Phi}^{\varepsilon}v^{\varepsilon}))_t={{\rm curl}\,}({\Phi}^{\varepsilon}v^{\varepsilon})_t$ is bounded in $L^\infty([0,T]; H^{-4}_{{\operatorname{{loc}}}})$; - ${{\rm curl}\,}({\Phi}^{\varepsilon}v^{\varepsilon}) = {\Phi}^{\varepsilon}{\omega}^{\varepsilon}+ v^{\varepsilon}\cdot {\nabla}^\perp {\Phi}^{\varepsilon}$ is bounded in $L^\infty([0,T]; L^{3/2})$ then the curl is precompact in $C([0,T]; H^{-1}_{{\operatorname{{loc}}}})$. Therefore, we can apply Lemma \[div-curl\] to ensure that $| {\Phi}^{\varepsilon}v^{\varepsilon}|^2 \rightharpoonup |v|^2$ in $\mathcal{D}'$, which implies the following theorem. \[v-limit\] For all $T>0$, we can extract a subsequence ${\varepsilon}_k \to 0$ such that ${\Phi}^{\varepsilon}v^{\varepsilon}\to v$ strongly in $L^2_{{\operatorname{{loc}}}}([0,T]\times {{\mathbb R}}^2)$. By a diagonal extraction, we have a subsequence ${\varepsilon}_k \to 0$ such that the convergence holds in $L^2_{{\operatorname{{loc}}}}({{\mathbb R}}^+\times {{\mathbb R}}^2)$. The asymptotic vorticity equation ---------------------------------   We begin by observing that the sequence $\{{\Phi}^{\varepsilon}{\omega}^{\varepsilon}\}$ is bounded in $L^\infty({{\mathbb R}}^+\times {{\mathbb R}}^2)$, then, passing to a subsequence if necessary, we have $${\Phi}^{\varepsilon}{\omega}^{\varepsilon}\rightharpoonup {\omega}\text{, weak-$*$ in }L^\infty({{\mathbb R}}^+\times {{\mathbb R}}^2).$$ We already have a limit velocity: $u:= v+{\gamma}H$. The purpose of this section is to prove that $u$ and ${\omega}$ verify, in an appropriate sense, the system: $$\left\lbrace \begin{aligned} \label{tour_equa} &{\partial}_t {\omega}+u\cdot {\nabla}{\omega}=0, & \text{ in } (0,\infty) \times{{\mathbb R}}^2 \\ & {{\rm div}\,}u=0 \text{ and }{{\rm curl}\,}u={\omega}+{\gamma}{\delta}, &\text{ in }(0,\infty) \times{{\mathbb R}}^2 \\ & |u|\to 0, &\text{ as }|x|\to \infty \\ & {\omega}(0,x)={\omega}_0(x), &\text{ in }{{\mathbb R}}^2. \end{aligned} \right .$$ where ${\delta}$ is the Dirac function centered at the origin. The pair $(u,{\omega})$ is a weak solution of the previous system if - for any test function ${\varphi}\in C^\infty_c([0,\infty)\times{{\mathbb R}}^2)$ we have $$\int_0^\infty\int_{{{\mathbb R}}^2}{\varphi}_t{\omega}dxdt +\int_0^\infty \int_{{{\mathbb R}}^2}{\nabla}{\varphi}\cdot u{\omega}dxdt+\int_{{{\mathbb R}}^2}{\varphi}(0,x){\omega}_0(x)dx=0,$$ - we have ${{\rm div}\,}u=0$ and ${{\rm curl}\,}u={\omega}+{\gamma}{\delta}$ in the sense of distributions of ${{\mathbb R}}^2$, with $|u|\to 0$ at infinity. The pair $(u,{\omega})$ obtained at the beginning of this subsection is a weak solution of the previous system. The velocity $u$ satisfies $|u|\to 0$ at infinity because the convergence of ${\Phi}^{\varepsilon}u^{\varepsilon}$ to $u$ is uniform outside a ball containing the origin, as can be checked directly by the explicit expressions for $K_{\varepsilon}[{\omega}^{\varepsilon}]$ and $H_{{\varepsilon}}$, using the uniform compact support of ${\omega}^{\varepsilon}$. Moreover, using , , and ${{\rm div}\,}H=0$, ${{\rm curl}\,}H = {\delta}$, we obtain directly the point (b). Next, we introduce an operator $I_{\varepsilon}$, which for a function ${\varphi}\in C^\infty_0([0,\infty)\times {{\mathbb R}}^2)$ gives: $$I_{\varepsilon}[{\varphi}]:= \int_0^\infty\int_{{{\mathbb R}}^2} {\varphi}_t({\Phi}^{\varepsilon})^2{\omega}^{\varepsilon}dxdt+\int_0^\infty\int_{{{\mathbb R}}^2} {\nabla}{\varphi}\cdot ({\Phi}^{\varepsilon}u^{\varepsilon})({\Phi}^{\varepsilon}{\omega}^{\varepsilon}) dxdt.$$ To prove that $(u,{\omega})$ is a weak solution, we will show that - $I_{\varepsilon}[{\varphi}]+\int_{{{\mathbb R}}^2}{\varphi}(0,x){\omega}_0(x)dx\to 0$ as ${\varepsilon}\to 0$ - $I_{\varepsilon}[{\varphi}]\to \int_0^\infty\int_{{{\mathbb R}}^2}{\varphi}_t{\omega}dxdt +\int_0^\infty\int_{{{\mathbb R}}^2}{\nabla}{\varphi}\cdot u{\omega}dxdt$ as ${\varepsilon}\to 0$. Clearly these two steps complete the proof. We begin by showing (i). As $u^{\varepsilon}$ and ${\omega}^{\varepsilon}$ verify , it can be easily seen that $$\int_0^\infty\int_{{{\mathbb R}}^2} {\varphi}_t({\Phi}^{\varepsilon})^2{\omega}^{\varepsilon}dxdt =-\int_0^\infty\int_{{{\mathbb R}}^2} {\nabla}({\varphi}({\Phi}^{\varepsilon})^2)\cdot u^{\varepsilon}{\omega}^{\varepsilon}dxdt-\int_{{{\mathbb R}}^2} {\varphi}(0,x)({\Phi}^{\varepsilon})^2(x){\omega}_0(x) dx.$$ Thus we compute $$\begin{aligned} I_{\varepsilon}[{\varphi}] &=& -2\int_0^\infty\int_{{{\mathbb R}}^2} {\varphi}{\nabla}{\Phi}^{\varepsilon}\cdot u^{\varepsilon}({\Phi}^{\varepsilon}{\omega}^{\varepsilon}) dxdt-\int_{{{\mathbb R}}^2} {\varphi}(0,x)({\Phi}^{\varepsilon})^2(x){\omega}_0(x) dx \\ &=& -2\int_0^\infty\int_{{{\mathbb R}}^2} {\varphi}{\nabla}{\Phi}^{\varepsilon}\cdot v^{\varepsilon}({\Phi}^{\varepsilon}{\omega}^{\varepsilon}) dxdt-\int_{{{\mathbb R}}^2} {\varphi}(0,x)({\Phi}^{\varepsilon})^2(x){\omega}_0(x) dx \end{aligned}$$ because ${\nabla}{\Phi}^{\varepsilon}\cdot H_{{\varepsilon}} =0$ (see point (i) of Lemma \[4.4\]). By Lemma \[4.4\] and Theorem \[4.2\], we have $$\Bigl|I_{\varepsilon}[{\varphi}]+\int_{{{\mathbb R}}^2} {\varphi}(0,x)({\Phi}^{\varepsilon})^2(x){\omega}_0(x) dx \Bigl|\leq 2 \|{\Phi}^{\varepsilon}{\omega}^{\varepsilon}\|_{L^\infty (L^\infty)}\|{\varphi}\|_{L^1(L^\infty)}\| v^{\varepsilon}\|_{L^\infty(L^3)} \| {\nabla}{\Phi}^{\varepsilon}\|_{L^\infty(L^{3/2})}\leq C {\varepsilon}^{1/3},$$ which tends to zero as ${\varepsilon}\to 0$. This shows (i) for all ${\varepsilon}$ sufficiently small such that $({\Phi}^{\varepsilon})^2(x){\omega}_0={\omega}_0$, since the support of ${\omega}_0$ does not intersect the curve. For (ii), the linear term presents no difficulty. The second term consists of the weak-strong convergence of the pair vorticity-velocity: $$\begin{aligned} \Bigl|\int\int{\nabla}{\varphi}\cdot ({\Phi}^{\varepsilon}u^{\varepsilon})({\Phi}^{\varepsilon}{\omega}^{\varepsilon})-\int\int{\nabla}{\varphi}\cdot u{\omega}\Bigl| &\leq& \Bigl|\int\int{\nabla}{\varphi}\cdot ({\Phi}^{\varepsilon}u^{\varepsilon}-u)({\Phi}^{\varepsilon}{\omega}^{\varepsilon})\Bigl| \\ &&+\Bigl|\int\int{\nabla}{\varphi}\cdot u({\Phi}^{\varepsilon}{\omega}^{\varepsilon}-{\omega})\Bigl|.\end{aligned}$$ Writing ${\Phi}^{\varepsilon}H_{\varepsilon}-H=({\Phi}^{\varepsilon}-1)H_{\varepsilon}+(H_{\varepsilon}-H)$ and using Theorem \[v-limit\], Lemmas \[H-limit\] and \[4.4\], we can easily show that ${\Phi}^{\varepsilon}u^{\varepsilon}\to u$ strongly in $L^1_{{\operatorname{{loc}}}}({{\mathbb R}}^+\times {{\mathbb R}}^2)$. So the first term tends to zero because ${\Phi}^{\varepsilon}{\omega}^{\varepsilon}$ is bounded in $L^\infty({{\mathbb R}}^+\times {{\mathbb R}}^2)$ (see ). In the same way, the second term tends to zero because ${\Phi}^{\varepsilon}{\omega}^{\varepsilon}\rightharpoonup {\omega}\text{ weak-$*$ in }L^\infty({{\mathbb R}}^+\times {{\mathbb R}}^2)$ and $u\in L^1_{{\operatorname{{loc}}}}({{\mathbb R}}^+\times {{\mathbb R}}^2)$. Its ends the proof. Extracting again a subsequence, we can write the convergences without cutoff function. Indeed, $\|{\omega}^{\varepsilon}\|_{L^\infty({{\mathbb R}}^+\times {{\mathbb R}}^2)}$ is uniformly bounded, then we extract such that ${\omega}^{\varepsilon}\rightharpoonup {\omega}\text{, weak-$*$ in }L^\infty({{\mathbb R}}^+\times {{\mathbb R}}^2)$. Next, for any $T>0$ and $K$ compact set of ${{\mathbb R}}^2$, we write $$\begin{aligned} \| u^{\varepsilon}-u\|_{L^1([0,T]\times K)} &\leq& T \|1-{\Phi}^{\varepsilon}\|_{ L^2 ({{\mathbb R}}^2)} \| v^{\varepsilon}\|_{L^\infty ([0,T], L^2(K))} + C_K \| {\Phi}^{\varepsilon}v^{\varepsilon}-v\|_{L^1([0,T]\times K)}\\ &&+ T \| H_{{\varepsilon}}- H \|_{L^1(K)} \end{aligned}$$ which tends to zero by Lemma \[4.4\], Theorem \[4.2\], Theorem \[v-limit\] and Lemma \[H-limit\]. Therefore, it means that $u^{\varepsilon}\to u$ in $L^1_{{\operatorname{{loc}}}}({{\mathbb R}}^+\times {{\mathbb R}}^2)$. Moreover, [@lac_miot] establishes that the solution of is unique. Although we have extracted a subsequence, we can conclude that all the sequence $(u^{\varepsilon},{\omega}^{\varepsilon})$ tends to the unique pair $(u,{\omega})$ solution of . It ends the proof of Theorem \[main\]. For completeness, the reader should read Subsection 5.3 of [@ift_lop_euler], concerning the asymptotic velocity equation. Acknowledgments {#acknowledgments .unnumbered} =============== I want to thank the researchers from Nantes university for pointing out the interest of the small curve problem. I also want to warmly thank Donald E. Marshall for suggesting me [@war]. Annexe {#annexe .unnumbered} ====== Extension of Proposition \[2.2\] {#extension-of-proposition-2.2 .unnumbered} -------------------------------- We prove here an extension of Proposition 2.2 from [@lac_euler]: If ${\Gamma}$ is a $C^{3}$ Jordan arc, such that the intersection with the segment $[-1,1]$ is a finite union of segments and points, then there exists a unique biholomorphism $T:\Pi\to {{\rm int}\,}\ D^c$ which verifies the following properties: - $T(\infty)=\infty$ and $T'(\infty)\in {{\mathbb R}}^+_*$; - $T^{-1}$ and $DT^{-1}$ extend continuously up to the boundary, and $T^{-1}$ maps $S$ to ${\Gamma}$; - $T$ extends continuously up to ${\Gamma}$ with different values on each side of ${\Gamma}$; - $D T$ extends continuously up to ${\Gamma}$ with different values on each side of ${\Gamma}$, except at the endpoints of the curve where $D T$ behaves like the inverse of the square root of the distance; - $D^2 T$ extends continuously up to ${\Gamma}$ with different values on each side of ${\Gamma}$, except at the endpoints of the curve where $D^2 T$ behaves like the inverse of the power $3/2$ of the distance. We only give here the properties near the curve, because the behavior at infinity is given by Remark \[2.5\]. Let us work in ${{\mathbb C}}$. We follow the proof made in [@lac_euler]. [**First step: case where ${\Gamma}:= [-1,1]$.**]{} In the special case of the segment $[-1,1]$, we have an explicit formula of $T$, thanks to the Joukowski function $$G(z)= \frac{1}{2} (z+\frac{1}{z}).$$ This function maps the exterior of the disk to the exterior of the segment, and we only have to solve an equation of degree two: $$T(z) = z\pm \sqrt{z^2-1}$$ where you have to choose in a good way the sign $\pm$ (see [@lac_euler] for more details). Hence, we have $$\begin{aligned} T'(z)&=& 1\pm \frac{z}{\sqrt{z^2-1}}=1\pm \frac{z}{\sqrt{(z-1)(z+1)}}\\ T''(z)&=& \mp \frac{1}{(z^2-1)^{3/2}}=\mp \Bigl(\frac{1}{(z-1)(z+1)}\Bigl)^{3/2},\end{aligned}$$ which allows us to finish the proof in this case. [**Second step: general case.**]{} The natural idea is to want to straighten the curve to the segment by a biholomorphism which would be $C^2$ up to the boundary. Apply after the inverse of the Joukowski function would give the result. However, it is not well established that such a straightening up exists. Of course, we know how to straighten the curve to the segment by a biholomorphism, and how to straighten up by a $C^2$ function, but we do not know how to find an application which verify the two properties (see [@lac_thesis] for a discussion on this subject). The idea in [@lac_euler] is to apply first the inverse of the Joukowski function. Let assume[^4] that the end-points of ${\Gamma}$ are $-1$ and $1$, we consider the curve $$\tilde {\Gamma}:= G^{-1}({\Gamma}) = (z+\sqrt{z^2-1})({\Gamma}) \cup (z-\sqrt{z^2-1})({\Gamma}).$$ It is proved that $\tilde {\Gamma}$ is a $C^{1,1}$ Jordan curve. To gain estimate of one more derivative, the only thing to do is to show that $\tilde {\Gamma}$ is a $C^{2,1}$ Jordan curve. As it is said in [@lac_euler], the difficult part is to show that $\tilde {\Gamma}$ is $C^{2,1}$ at the points $-1$ and $1$, where we change the sign and where the square root is non smooth. Then, let us prove it at the point $-1$. We denote a parametrization of the curve ${\Gamma}$ by ${\Gamma}(t)$ (with ${\Gamma}(0)=-1$, ${\Gamma}(1)=1$), and ${\gamma}_1(t)= (z+\sqrt{z^2-1})({\Gamma}(t))$, ${\gamma}_2(t)= (z-\sqrt{z^2-1})({\Gamma}(t))$. We write the Taylor expansion of ${\Gamma}(t)= -1+at + O(t^2)$ with $a\in {{\mathbb C}}$. The aim is to compute the Taylor expansion of $\frac{{\gamma}_1'(t)}{|{\gamma}_1'(t)|}$ and of $\frac{{\gamma}_2'(t)}{|{\gamma}_2'(t)|}$. For that, we compute $$\begin{aligned} {\gamma}_1(t) &=& -1+\sqrt{-2a} \sqrt{t}+at+O(t\sqrt{t})\\ {\gamma}_2(t) &=& -1-\sqrt{-2a} \sqrt{t}+at+O(t\sqrt{t})\end{aligned}$$ hence, $$\begin{aligned} {\gamma}'_1(t) &=& \frac{\sqrt{-2a}}{2}\frac{1}{\sqrt{t}} +a+O(\sqrt{t})\\ {\gamma}'_2(t) &=& -\frac{\sqrt{-2a}}{2}\frac{1}{\sqrt{t}} +a+O(\sqrt{t}).\end{aligned}$$ Writing that $\dfrac{1}{|f(t)|}= \Bigl(f(t)\overline{f(t)}\Bigl)^{-1/2}$, we obtain $$\begin{aligned} \frac{1}{|{\gamma}_1'(t)|} &=& \sqrt{\frac2{|a|}} \sqrt{t}- \sqrt{\frac2{|a|^3}} {\rm Re}(\overline{a}\sqrt{-2a}) t + O(t\sqrt{t}) \\ \frac{1}{|{\gamma}_2'(t)|} &=& \sqrt{\frac2{|a|}} \sqrt{t}+ \sqrt{\frac2{|a|^3}} {\rm Re}(\overline{a}\sqrt{-2a}) t + O(t\sqrt{t}) ,\end{aligned}$$ which give $$\begin{aligned} \frac{{\gamma}_1'(t)}{|{\gamma}_1'(t)|} &=& \frac{\sqrt{-2a}}{|\sqrt{-2a}|} + \Bigl(a\sqrt{\frac2{|a|}}-\frac{\sqrt{-2a}}{|\sqrt{-2a}|}\frac1{|a|} {\rm Re}(\overline{a}\sqrt{-2a})\Bigl) \sqrt{t} +O(t\sqrt{t})\\ \frac{{\gamma}_2'(t)}{|{\gamma}_2'(t)|} &=& -\frac{\sqrt{-2a}}{|\sqrt{-2a}|} + \Bigl(a\sqrt{\frac2{|a|}}-\frac{\sqrt{-2a}}{|\sqrt{-2a}|}\frac1{|a|} {\rm Re}(\overline{a}\sqrt{-2a})\Bigl) \sqrt{t} +O(t\sqrt{t}).\end{aligned}$$ We denote $A=a\sqrt{\frac2{|a|}}-\frac{\sqrt{-2a}}{|\sqrt{-2a}|}\frac1{|a|} {\rm Re}(\overline{a}\sqrt{-2a})$. Let $s_1$, respectively $s_2$, the arclength coordinates associated to ${\gamma}_1$, respectively ${\gamma}_2$. The previous computation allows us to state that $$\begin{aligned} \frac{d{\gamma}_1(s)}{ds}& =&\frac{{\gamma}_1'(t)}{|{\gamma}_1'(t)|} \to \frac{\sqrt{-2a}}{|\sqrt{-2a}|} \\ \frac{d{\gamma}_2(s)}{ds}& =&\frac{{\gamma}_2'(t)}{|{\gamma}_2'(t)|} \to - \frac{\sqrt{-2a}}{|\sqrt{-2a}|},\end{aligned}$$ as $t\to 0$, which means that $\tilde {\Gamma}$ is $C^1$. Moreover, $$\frac{d^2{\gamma}_i(s)}{ds^2}= \frac{d\Bigl( \frac{d{\gamma}_i(s)}{ds} \Bigl)}{dt} \frac{1}{|{\gamma}_i'(t)|}= \frac{d\Bigl( \frac{{\gamma}_i'(t)}{|{\gamma}_1'(t)|} \Bigl)}{dt} \frac{1}{|{\gamma}_i'(t)|},$$ which implies that $$\begin{aligned} \frac{d^2{\gamma}_1(s)}{ds^2}& =&\frac{A}2\frac1{\sqrt{t}}\sqrt{\frac2{|a|}} \sqrt{t}+O(\sqrt{t}) \to \frac{A}2 \sqrt{\frac2{|a|}}\\ \frac{d^2{\gamma}_2(s)}{ds^2}& =&\frac{A}2\frac1{\sqrt{t}}\sqrt{\frac2{|a|}} \sqrt{t}+O(\sqrt{t}) \to \frac{A}2 \sqrt{\frac2{|a|}},\end{aligned}$$ as $t\to 0$, which means that $\tilde {\Gamma}$ is $C^2$. In the same way, we have $$\frac{d^3{\gamma}_i(s)}{ds^3}= \frac{d\Bigl( \frac{d^2{\gamma}_i(s)}{ds^2} \Bigl)}{dt} \frac{1}{|{\gamma}_i'(t)|}= O\Bigl(\frac1{\sqrt{t}}\Bigl)\sqrt{\frac2{|a|}} \sqrt{t}=O(1),$$ which implies that $\frac{d^2{\gamma}_i(s)}{ds^2}$ is lipschitz in a neighborhood of $-1$. Therefore, we have proved that $\tilde {\Gamma}$ is a $C^{2,1}$ Jordan curve. Now, we can conclude as in [@lac_euler]. For sake of clarity, we rewrite here this argument. We denote by $\tilde \Pi$ the unbounded connected component of ${{\mathbb R}}^2\setminus\tilde{\Gamma}$. Choosing well the $\pm$, we claim that we can construct $T_2$, a biholomorphism between $\Pi$ and $\tilde \Pi$, such that $T_2^{-1}=G$. Next, we just have to use the Riemann mapping theorem and we find a conformal mapping $F$ between $\tilde\Pi$ and $D^c$, such that $F(\infty)=\infty$ and $F'(\infty)\in {{\mathbb R}}^+_*$. Then $T:= F\circ T_2$ maps $\Pi$ to $D^c$ and $T(\infty)=\infty$, $T'(\infty)\in {{\mathbb R}}^+_*$. To finish the proof, we use the Kellogg-Warschawski theorem (see Theorem 3.6 of [@pomm-2], which can be applied for the exterior problems), to observe that $F$, $F'$ and $F''$ have a continuous extension up to the boundary, because $\tilde {\Gamma}$ is a $C^{2,1}$ Jordan curve . Therefore, the behavior near the curve of $DT$ and $D^2 T$ becomes from the behavior of $T_2$ which is the inverse of the Joukowski function. Then we find the same properties as in the segment case. The uniqueness of $T$ can be proved thanks to the uniqueness of the Riemann mapping from $D^c$ to $D^c$ (see Remark \[T-unique\]). List of notations {#list-of-notations .unnumbered} ================= Domains: {#domains .unnumbered} -------- $D:= B(0,1)$ the unit disk and $S:= {\partial}D$. ${\Gamma}$ is a Jordan arc (see Proposition \[2.2\]) and ${\Gamma}_{\varepsilon}:= {\varepsilon}{\Gamma}$. $\Pi_{\varepsilon}:= {{\mathbb R}}^2\setminus {\Gamma}_{\varepsilon}$. ${\Omega}_{n}$ is a bounded, open, connected, simply connected subset of the plane, where ${\partial}{\Omega}_{n}$ is a $C^\infty$ Jordan curve. $\Pi_n:= {{\mathbb R}}^2\setminus \overline{{\Omega}_n}$. Functions: {#functions .unnumbered} ---------- ${\omega}_0$ is the initial vorticity ($C^\infty_c(\Pi)$). ${\gamma}$ is the circulation of $u_0^{{\varepsilon}}$ around ${\Gamma}_{{\varepsilon}}$ (see Introduction). $(u^{{\varepsilon}},{\omega}^{{\varepsilon}})$ is the solution of the Euler equations on $\Pi_{{\varepsilon}}$ in the sense of Definition \[sol-curve\]. $T$ is a biholomorphism between $\Pi$ and ${{\rm int}\,}\ D^c$. $T_n$ is a biholomorphism between $\Pi_n$ and ${{\rm int}\,}\ D^c$. $K_{\varepsilon}$ and $H_{\varepsilon}$ are given in Subsection \[sect:biot\]. $K_{{\varepsilon}}[{\omega}^{{\varepsilon}}](x):= \int_{\Pi_{{\varepsilon}}} K_{\varepsilon}(x,y) {\omega}^{{\varepsilon}}(y)dy$. ${\Phi}^{{\varepsilon}}$ is a cutoff function for a ${{\varepsilon}}$-neighborhood of ${\Gamma}_{{\varepsilon}}$. [99]{} Desjardins B., [*A few remarks on ordinary differential equations*]{}, Commun. in Part. Diff. Eq. , 21:11, 1667-1703, 1996. DiPerna R. J. and Lions P. L., [*Ordinary differential equations, transport theory and Sobolev spaces*]{}, Invent. Math. 98, 511-547, 1989. Iftimie D., [*Evolution de tourbillon à support compact*]{}, Actes du Colloque de Saint-Jean-de-Monts, 1999. Iftimie D. et Kelliher J., [*Remarks on the vanishing obstacle limit for a 3D viscous incompressible fluid*]{}, Proc. Amer. Math. Soc. 137 (2009), no. 2, 685-694. Iftimie D., Lopes Filho M.C. and Nussenzveig Lopes H.J., [*Two Dimensional Incompressible Ideal Flow Around a Small Obstacle*]{}, Comm. Partial Diff. Eqns. 28 (2003), no. 1$\&$2, 349-379. Iftimie D., Lopes Filho M.C. and Nussenzveig Lopes H.J., [*Two Dimensional Incompressible Viscous Flow Around a Small Obstacle*]{}, Math. Annalen. 336 (2006), 449-489. Kikuchi K., [*Exterior problem for the two-dimensional Euler equation*]{}, J Fac Sci Univ Tokyo Sect 1A Math 1983; 30(1):63-92. Lacave C., [*Two Dimensional Incompressible Ideal Flow Around a Thin Obstacle Tending to a Curve*]{}, Annales de l’IHP, Anl **26** (2009), 1121-1148. Lacave C., [*Two Dimensional Incompressible Viscous Flow Around a Thin Obstacle Tending to a Curve*]{}, Proc. Roy. Soc. Edinburgh Sect. A, Vol. 139 (2009), No. 6, pp. 1138-1163. Lacave C., [*Fluide visqueux incompressible en dimension trois autour d’obstacles fins*]{}, Actes du séminaire GM3N à Caen, 2009. English translation in preparation. Lacave C., [*Fluides autour d’obstacles minces*]{}, thesis (2008), http://tel.archives-ouvertes.fr/tel-00345665/en/. Lacave C., Lopes Filho M.C. and Nussenzveig Lopes H.J., [*Homogenization of Two Dimensional Incompressible Ideal Flow in a Perforated Domain*]{}, in preparation. Lacave C. and Miot E., [*Uniqueness for the vortex-wave system when the vorticity is constant near the point vortex*]{}, SIAM Journal on Mathematical Analysis, Vol. 41 (2009), No. 3, pp. 1138-1163. Lopes Filho M.C., [*Vortex dynamics in a two dimensional domain with holes and the small obstacle limit*]{}, SIAM Journal on Mathematical Analysis, 39(2)(2007) : 422-436. Lopes Filho M.C., Nussenzveig Lopes H.J. and Tadmor E., [*Approximate solutions of the incompressible Euler equations with no concentrations*]{}, Annales de l’IHP, Anl **17** (2000), 371-412. Pommerenke C., [*Univalent functions*]{}, Vandenhoeck $\&$ Ruprecht, 1975. Pommerenke C., [*Boundary behaviour of conformal maps*]{}, Berlin New York: Springer-Verlag, 1992. Taylor M., [*Incompressible fluid flows on rough domains*]{}. Semigroups of operators: theory and applications (Newport Beach, CA, 1998), 320-334, Progr. Nonlinear Differential Equations Appl., 42, Birkhäuser, Basel, 2000. Temam R., [*Navier-Stokes Equations, Theory and Numerical Analysis*]{}, North-Holland, Amsterdam, 1979. Warschawski S.E., [*On the Distortion in Conformal Mapping of Variable Domains*]{}, Trans. of the A.M.S., Vol. 82, No. 2 (Jul., 1956), pp. 300-322. [^1]: see e.g. [@ift_lop_euler]. [^2]: This estimate is not uniform in ${\varepsilon}$. [^3]: this equality is given in $\mathcal{D}'({{\mathbb R}}^+)$, but it holds for all $t$ (see the proof of Proposition \[compact\_vorticity\]). [^4]: which is possible after homothety, translation and rotation.

--- abstract: 'Quasi-equilibrium models of uniformly rotating axisymmetric and triaxial quark stars are computed in general relativistic gravity scenario. The Isenberg-Wilson-Mathews (IWM) formulation is employed and the Compact Object CALculator ([<span style="font-variant:small-caps;">cocal</span>]{}) code is extended to treat rotating stars with finite surface density and new equations of state (EOSs). Besides the MIT bag model for quark matter which is composed of de-confined quarks, we examine a new EOS proposed by Lai and Xu that is based on quark clustering and results in a stiff EOS that can support masses up to $3.3M_\odot$ in the case we considered. We perform convergence tests for our new code to evaluate the effect of finite surface density in the accuracy of our solutions and construct sequences of solutions for both small and high compactness. The onset of secular instability due to viscous dissipation is identified and possible implications are discussed. An estimate of the gravitational wave amplitude and luminosity based on quadrupole formulas is presented and comparison with neutron stars is discussed.' author: - Enping Zhou - Antonios Tsokaros - Luciano Rezzolla - Renxin Xu - Kōji Uryū bibliography: - 'aeireferences.bib' title: 'Uniformly rotating, axisymmetric and triaxial quark stars in general relativity' --- Introduction ============ The recent gravitational-wave (GW) event GW170817 together with accompanying electromagnetic emission observations [@Abbott2017; @Abbott2017b] from a binary neutron star (BNS) merger has opened a brand new multi-messenger observation era for us to explore the Universe. Apart from enriching our knowledge on origins of short gamma-ray bursts [@Abbott2017d] and nucleosynthesis associated with BNS mergers [@Abbott2017c; @Baiotti2016], it also provides an effective way for us to constrain the equation of state (EOS) of neutron stars (NS). In addition to systems such as binary black-hole mergers and BNS mergers, rapidly rotating compact stars have also been considered as important candidates of GW sources [@Andersson:2009yt], which could be detected by ground-based GW observatories [@Abramovici92; @Punturo:2010; @Accadia2011_etal; @Kuroda2010; @Aso:2013] and help us understand the nature of strong interaction of dense matter. It has been long since the equilibrium models of self-gravitating, uniformly rotating, incompressible fluid stars were systematically studied in a Newtonian gravity scheme [@Chandrasekhar1969book]. Depending on the rotational kinetic energy, the configuration could be axisymmetric Maclaurin ellipsoids as well as nonaxisymmetric ellipsoids, such as Jacobian (triaxial) ellipsoid. For compact stars that we are interested in for GW astronomy, however, general relativity is required to replace Newtonian gravity. The field of relativistic rotating stars has been studied for many years [@Meinel:2008; @Friedman2012]. A rotating NS will spontaneously break its axial symmetry if the rotational kinetic energy to gravitational binding energy ratio, $T/|W|$ exceeds a critical value. This instability can either be of secular type [@shapiro98b; @Bonazzola1996b; @rosinska2002; @Bonazzola1998c] or dynamical [@Houser96; @Pickett96; @brown2000; @New2000; @Liu02; @Watts:2003nn; @Baiotti06b; @Manca07; @Corvino:2010], depending on the process driving the instability and with only small modifications if a magnetisation is present [@Camarda:2009mk; @Franci2013; @Muhlberger2014] (see [@Andersson03] for a review). A high $T/|W|$ ratio can also be reached for a newly born rotating compact star during a core collapse supernova or for a NS which is spun up by accretion [@Lai95; @Bildsten98; @woosley2005; @watts2008; @piro2012b]. Quasi-equilibrium figures of triaxially rotating NSs have also been created and studied in full general relativity [@Huang08; @Uryu2016a]. In this case, the bifurcation from an axisymmetric to triaxial configuration happens very close to the mass shedding limit, and, for soft NS EOSs or for NSs with large compactness, the triaxial sequence could totally vanish [@Jame64; @Bonazzola1996b; @Bonazzola1998c; @hachisu1982; @Lai93]. As a result, it is presently unclear whether triaxial configurations of NSs can actually be realized in practice. On the other hand, it is worth noting that the EOS of compact stars is still a matter of lively debate since astronomical observations are not sufficient to rule out many of the nuclear-physics EOSs that are compatible with the observations. As a result, besides the popular idea of NSs, other models for compact stars are possible and have been considered in the past. A particularly well developed literature is the one concerned with strange quark stars (QSs), since it was long conjectured that strange quark matter composed of de-confined up, down and strange quarks could be absolutely stable [@Bodmer1971; @Witten84]. There is also possible observational evidence indicating the existence of QSs (for a recent example, see [@Dai_ZG:2016]). Additionally, the small tidal deformability of QSs is favoured by the observation of GW170817 [@Lai2017b] and possible models with QS merger or QS formations are also suggested to explain the electromagnetic counterparts for a short gamma-ray burst (c.f. [@Li2016; @Lai2017b]). Following this possibility, a large effort has been developed to calculate equilibrium configurations of QSs, starting from the first attempts [@Itoh70; @Alcock86; @Haensel1986]. At present, both uniformly rotating [@rosinska2000b; @gourgoulhon1999; @Stergioulas99a] and differentially rotating QSs [@szkudlarek2012] have been studied in full general relativity. Unlike NSs, which are bound by self-gravity, QSs are self-bound by strong interaction. Consequently, rotating QSs can reach a much larger $T/|W|$ ratio compared with NSs and the triaxial instability could play a more important role [@rosinska2000a; @rosinska2000b; @rosinska2001]. The triaxial bar mode (Jacobi-like) instability for MIT bag-model EOS has been investigated in a general relativistic framework [@rosinska2003]. We here use the Compact Object CALculator code, [<span style="font-variant:small-caps;">cocal</span>]{}, to build general-relativistic triaxial QS solution sequences using different EOS models. [<span style="font-variant:small-caps;">cocal</span>]{} is a code to calculate general-relativistic equilibrium and quasi-equilibrium solutions for binary compact stars (black hole and NS) as well as rotating (uniformly or differentially) NSs [@Uryu2012; @Uryu:2012b; @Tsokaros2012; @Tsokaros2015; @Huang08; @Uryu2016a]. The part of the [<span style="font-variant:small-caps;">cocal</span>]{} code handling the calculation of the EOS was originally designed for piecewise polytropic EOSs. We have here extended the code to include polynomial type of EOSs, as those that can be used to describe QSs. In doing so the trivial relationship between the thermodynamic quantities for a piecewise polytrope ([e.g., ]{}see Eqs. (64)–(68) in [@Tsokaros2015]), is lost and now one has to apply root finding methods. Another issue is related to the surface fitted coordinates that are used in [<span style="font-variant:small-caps;">cocal</span>]{} to track the surface of the star. For NSs, the surface was identified as the place where the rest-mass density goes to zero or where the specific enthalpy becomes one. This is no longer generally true for a self-bound QS and a different approach needs to be developed. The nonlinear algebraic system that determines the angular velocity, the constant from the Euler equation, and the renormalization constant of the spherical grid has to be modified in order to accommodate the arbitrary surface enthalpy. We here compute solutions for both axisymmetric and triaxial rotating QSs with the new code, as well as sequences with various QS EOS and different compactnesses. We checked our new implementation for those cases were previous studies have been possible [@Huang08], and we confirm the accuracy of our new code. We discuss the astrophysical implications of the quantities of rotating QSs at the bifurcation point. For instance, the spin frequency at the bifurcation point could be a more realistic spin up limit for compact stars rather than the mass shedding limit, which relates to the fastest spinning pulsar we might be able to observe. The GW strain and luminosity estimates for our models are given, while full numerical simulations are left for the future (see [@Tsokaros2017] for recent simulations involving triaxial NSs). The structure of this paper is organized as follows: In Sec. \[sec:formandmethod\] we discuss the formulation we used and the field equations (Sec. \[sec:IWMformulation\]), the hydrodynamics (Sec. \[sec:fluidformulation\]) and the EOS part (Sec. \[sec:eos\]). In order to test the behavior of the modified code, we have performed convergence tests with five resolutions and compared with rotating NS solutions built by the original [<span style="font-variant:small-caps;">cocal</span>]{} code. These tests can be found in Sec. \[sec:codetest\]. Triaxially deformed rotating QS sequences for different compactnesses, are presented in Sec. \[sec:triasolution\], while the implications for the astrophysical observations of this work are presented in Sec. \[sec:disandconclu\]. Hereafter we use units with $G=c=M_\odot=1$ unless otherwise stated; a conversion table to the standard cgs units can be found, for instance, in [@Rezzolla_book:2013]. Formulation and numerical method {#sec:formandmethod} ================================ Field equations {#sec:IWMformulation} --------------- In order to solve the field equations numerically, the Isenberg-Wilson-Mathews (IWM) formulation [@Isenberg1980; @Isenberg08; @Wilson89] is employed. In a coordinate chart $\{t,x^i\}$, the $\mathrm{3+1}$ decomposition of the spacetime metric gives $$ds^2 = -\alpha^2 dt^2 + \psi^4 \delta_{ij} (dx^{i}+\beta^{i}dt) (dx^{j}+\beta^{j}dt)\,,$$ where $\alpha,\ \beta^i$ are the lapse and shift vector (the kinematical quantities), while $\gamma_{ij}=\psi^4\delta_{ij}$ is the IWM approximation for the three-metric. The extrinsic curvature of the foliation is defined by $$\Kabd \,:=\,-\frac1{2\alpha}\pa_t \gmabd +\frac1{2\alpha}\Lie_\beta \gmabd\,.$$ and a maximal slicing condition $K=0$ is assumed. Decomposing the Einstein equations with respect to the normal $n^\alpha$ of foliation, we get the following 5 equations in terms of the five metric coefficients $\{\psi, \beta^a, \alpha\}$ on the initial slice $\Sigma_0$: $$\begin{aligned} &&(\Gabd-8\pi\Tabd)\,n^\alpha n^\beta \ \,=\, 0, \label{eq:Ham}\\ &&(\Gabd-8\pi\Tabd)\,\gamma^{i\alpha} n^\beta \,=\, 0, \label{eq:Mom}\\ &&(\Gabd-8\pi\Tabd)\,\Big(\gamma^{\albe}+\frac12 n^\alpha n^\beta\Big) \,=\, 0, \label{eq:trG}\end{aligned}$$ where the first and second equations are the Hamiltonian and momentum constraints, respectively. Here $\gamma_{\alpha\beta} = g_{\alpha\beta}+n_\alpha n_\beta$ is the projection tensor onto the spatial slices. These equations can be written in the form of elliptic equations with the non-linear source terms, respectively, $$\begin{aligned} && \!\!\!\!\!\!\! \nabla^2\psi \,=\, - \frac{\psi^5}{8}A_{ab}A^{ab} - 2\pi\psi^5\rhoH, \label{eq:HaC_elip2} \\ && \!\!\!\!\!\!\! \nabla^2\beta^a + \frac13 \partial^a\partial_b\beta^b \,=\, -2\alpha A^{ab}\partial_b\ln\frac{\psi^6}{\alpha} + 16\pi\alpha j^a, \label{eq:MoC_elip2} \\ && \!\!\!\!\!\!\! \nabla^2(\alpha\psi) \,=\, \frac{7}{8}\alpha\psi^5A_{ab}A^{ab} + 2\pi\alpha\psi^5(\rhoH+2S). \label{eq:trG_elip2}\end{aligned}$$ where $A^{ij}=K^{ij}={\psi^{-4}}(\partial^i \beta^j + \partial^j\beta^i -\frac{2}{3}\delta^{ij}\partial_k\beta^k)/{2\alpha}$, and the source terms of matter are defined by $\rhoH:=\Tabd n^\alpha n^\beta$, $j^i:=-\Tabd \gamma^{i\alpha} n^\beta$, and $S:=\Tabd \gamma^{\alpha \beta}$. The above set of equations must be supplied with boundary conditions at infinity. Since we are working in the inertial frame and we impose asymptotic flatness, we must have $$\lim_{r\rightarrow\infty} \psi = 1\,,\qquad \lim_{r\rightarrow\infty} \alpha = 1\,,\qquad \lim_{r\rightarrow\infty} \beta^i = 0\,. \label{eq:bcpsal}$$ Hydrostatic equilibrium {#sec:fluidformulation} ----------------------- The hydrostatic equation for a perfect fluid in quasi-equilibrium can be derived from the relativistic Euler equation [@Rezzolla_book:2013] $$u^\beta \na_\beta(hu_\alpha) + \na_\alpha h\,=0\,,$$ where $u^\alpha=u^t(1,v^i)=u^t(1,\Omega\phi^i)$ is the 4-velocity of the fluid, $\phi^i=(-y,x,0)$, and $h$ is the specific enthalpy defined by $h:=(\epsilon+p)/\rho$ ($\rho$ is the rest-mass density and $\epsilon$ the total energy density). When the symmetry along a helical Killing vector $k^\alpha=t^\alpha + \Omega \phi^\alpha$ is imposed for the fluid variables, which is approximately true also in the case for a rotating nonaxisymmetric star in quasi-equilibrium, the integral of the Euler equation becomes $$\frac{h}{u^t}\,=\, \mathcal{E}\, , \label{eq:firstint}$$ where $\mathcal{E}$ is a constant. From the normalization of the four velocity $u_\alpha u^\alpha=-1$, one obtains $$u^t = \frac1{\sqrt{\alpha^2 - \omega_a \omega^a}} = \frac1{\sqrt{\alpha^2 - \psi^4 \delta_{ab}\,\omega^a \omega^b}}\,, \label{eq:ut}$$ where $\omega^a = \beta^a + \Omega \phi^a$. The fluid sources of Eqs. –\[eq:trG\_elip2\], [i.e., ]{}$\rho_{\mathrm{H}}$, $j_a$ and $S$, are defined in terms of the energy momentum tensor in the previous section. In terms of the fluid and field variables they can be written as [@Rezzolla_book:2013] $$\begin{aligned} &&\rho_{\mathrm{H}}=\rho[h(\alpha u^t)^2-q], \label{eq:rhoh}\\ && j^i=\rho h\alpha(u^t)^2\gamma^{i\alpha} u_\alpha, \label{eq:ja}\\ && S=\rho h (\alpha u^t)^2 - \rho h +3\rho q, \label{eq:s}\end{aligned}$$ in which $q:=p/\rho$ is the relativistic analogue of the Emden function. Here $u^t$ is related to $h$ through Eq. . Therefore in order to close the system, an additional relationship is needed between the specific enthalpy, the pressure and the rest-mass density of the fluid, [i.e., ]{}an EOS. Once such a relation is available, to solve the field equations \[Eqs. –\[eq:trG\_elip2\]\] and the hydrostatic equation \[Eq. \] one has to find the two constants $\{\Omega,\mathcal{E}\}$ that appear in all of them. This procedure is described in detail for example in Ref. [@Tsokaros2015]. Equation of State {#sec:eos} ----------------- In this work, we have considered two types of EOS for QSs. One of them is the MIT bag-model EOS [@chodos1974], since it is the most widely used EOS for QSs. In the case when strange quark mass is neglected, the pressure is related to total energy density according to $$p=\sigma (\epsilon-\epsilon_s) \, , \label{eq:mit_gen}$$ where $\sigma,\ \epsilon_s$ are two constants, the second being the total energy density at the surface. Related to $\epsilon_s$ is the so called bag constant, $B=\epsilon_s/4$. In this work, and following [@limousin2005], the simplest MIT bag-model EOS has been employed, where $\sigma=1/3$ and $B^{1/4}=138\,\mathrm{MeV}$. Besides the MIT bag-model EOS, we have also considered another QS EOS suggested by Lai and Xu [@lai2009], which we will refer to as the LX EOS hereafter. Unlike the conventional QS models ([e.g., ]{}the MIT bag-model EOS) which are composed of de-confined quarks, Lai and Xu [@lai2009] suggested that quark clustering is possible at the density of a cold compact star since the coupling of strong interaction is still decent at such energy scale. Due to the non-perturbative effect of strong interaction at low energy scales and the many-body problem, it is very difficult to derive the EOS of such a quark-cluster star[^1] from first principles. Lai and Xu attempted to approach the EOS of such quark-cluster star with phenomenological models, i.e., to compare the intercluster potential with the interaction between inert molecules (a similar approach has also been discussed in [@Guo2014]). They also take the lattice effects into account as the potential could be deep enough to trap the quark clusters. Combining the inter-cluster potential and the lattice thermodynamics, they have derived an EOS in the following form: $$p=4U_0(12.4r_0^{12}n^5-8.4r_0^6n^3)+\frac{1}{8}(6\pi^2)^{\frac{1}{3}}\hbar cn^{\frac{4}{3}}\,, \label{eq:lx09eos}$$ where $\hbar$ is the reduced Planck constant. The parameters in this expression, $U_0$ and $r_0$, are the depth of the potential and characteristic range of the interaction, respectively. The EOS is also dependent on the number of quarks in each cluster ($N_\mathrm{q}$) since it relates the energy density ($\epsilon$) and rest-mass density ($\rho$) to the number-density of quark clusters ($n$ in Eq. (\[eq:lx09eos\])). Similarly to the MIT bag-model EOS case, we use the rest-mass density parameter, which is $$\rho=m_\mathrm{u}\frac{N_\mathrm{q}}{3}n\,,$$ where $m_\mathrm{u}=931\mathrm{MeV}/c^2$ is the atomic mass unit. While several different choices of parameters are considered in Ref. [@lai2009], in our work we restrict our attention to $U_0=50\,\mathrm{MeV}$ and for $N_\mathrm{q}=18$. We also note that although it is not as obvious as for the MIT bag-model EOS, the LX EOS also has a nonzero surface density since expression Eq.  has a unique zero root when the number density is positive. ![The TOV solution sequences for MIT bag-model EOS (red solid line) and the LX EOS (blue dashed line) respectively. The left panel shows the mass-central density relationship for each model and the right panel is the mass-radius diagram. The bag constant we apply in this work for the MIT bag-model EOS satisfies the 2 solar mass constraint from observations.[]{data-label="fig:plot_tov"}](./TOV-crop.pdf){width="0.95\columnwidth"} Being a stiff EOS, the LX EOS is favored by the discovery of massive pulsars [@lai2011; @Demorest2010; @Antoniadis2013]. The rest-mass density and mass-radius relationships for spherical models can be seen in Fig. \[fig:plot\_tov\], and the characteristics of the maximum mass models are reported in Table \[tab:tovmax\]. The LX EOS has also been discussed in relation with the possibility of understanding some puzzling observations related to compact stars, such as the energy release during pulsar glitches [@zhou2014], the peculiar X-ray flares [@xu2009] and the optical/UV excess of X-ray-dim isolated NSs [@wang2017]. Particularly, a solid QS model has been suggested in order to understand those observations [@Xu2003]. However, as pointed out in [@zhou2014], the critical strain of such a star is very small. A starquake will be induced when the relative difference in ellipticity is $10^{-6}$, for most, between the actual configuration of the star and the configuration as if the star is a perfect fluid. This is consistent with the pulsar glitch observations on Vela. Therefore, we find it a good approximation to calculate the quasi-equilibrium configuration of such a star with perfect fluid assumption. Understanding the models and properties of the QS EOSs that we want to consider, we can modify the EOS part of the simulation code, which was originally designed for NS models, accordingly. As mentioned above, in the case of NSs, a piecewise-polytropic EOS is usually assumed to describe the EOS [@Read:2009a; @Rezzolla_book:2013]. In each piece, the pressure and rest-mass density are related as $$p_i = \kappa_i \rho^{\Gamma_i}= \kappa_i \rho^{1+1/n_i}\,, \qquad i=1,2,\ldots,N\,. \label{eq:polyEOS}$$ For QSs, due to the nonzero surface density and nonzero energy density integration constant, we will assume that the EOS is generally a polynomial $$p = \sum\limits_{i=1}^{N}\kappa_i \rho^{\Gamma_i}\,. \label{eq:qsEOS}$$ Given the relationship between $p$ and $\rho$, one can apply the first law of thermodynamics to obtain other quantities such as the energy density and the specific enthalpy. In the zero-temperature case, the first law of thermodynamics can be expressed as [@Rezzolla_book:2013] $$d\epsilon =\frac{\epsilon+p}{\rho} d\rho, \quad\mbox{or}\quad d\left(\frac{\epsilon}{\rho}\right)=\frac{p}{\rho^2}d\rho\,, \label{eq:1stlaw}$$ which can be integrated to obtain the total energy density. The integral constant is usually chosen to be 1, since when there is no internal energy, the energy density and the rest-mass density coincide (apart from the square of the speed of light). However, for QS EOSs (like in [@Guo2014] and the MIT bag-model EOS), the integral constant is different from unity and needs to be properly taken into account. [ccc]{}\ $C=-c^2$ & $\rho_s=1.4\rho_\mathrm{nuc}$ & $h_s=0.89697478c^2$\ $N=2$ & $\kappa_i$ & $\Gamma_i$\ $i=1$ & $2.7977907\times10^{15}$ & $\frac{4}{3}$\ $i=2$ & $-7.5279768\times10^{34}$ & $0$\ \ \ $C=0$ & $\rho_s=2.0\rho_\mathrm{nuc}$ & $h_s=0.96828675c^2$\ $N=3$ & $\kappa_i$ & $\Gamma_i$\ $i=1$ & $ 2.0824706\times10^{-39}$ & $5$\ $i=2$ & $ -6.1559375\times10^{-10}$ & $3$\ $i=3$ & $ 7.1307226\times10^{13}$ & $\frac{4}{3}$\ In this case, the energy density and specific enthalpy are related to the rest-mass density by $$\epsilon=\sum\limits_{i=1}^{N}\frac{\kappa_{i}}{\Gamma_{i}-1}\rho^{\Gamma_{i}}+\rho(1+C)\,, \label{eq:ene}$$ $$h=\frac{\epsilon+p}{\rho}=\sum\limits_{i=1}^{N}\frac{\Gamma_{i}\kappa_{i}}{\Gamma_{i}-1}\rho^{\Gamma_{i}-1}+1+C\,. \label{eq:enthalpy}$$ Here $C$ is the integral constant we mentioned above. It is usually taken to be zero for NS models. Here it is introduced again in order to accommodate stars that have different surface limit for the thermodynamic variables. The nonzero surface density of QSs requires a different boundary condition in our simulation. For typical NSs when we adjust the position of the surface, we are actually locating the points where the specific enthalpy is $1$. For QSs, the surface identification will be at values of the specific enthalpy different from unity and consistent with Eq. (\[eq:enthalpy\]). What we typically use as input parameter is the surface rest-mass density $\rho_s$, from which we then calculate $h_s$ using Eq. (\[eq:enthalpy\]). As an example, for MIT bag-model EOS the first law at zero temperature implies $$\begin{aligned} \epsilon & = & \frac{1}{1+\sigma}\left( \bar{C} \rho^{1+\sigma} + \sigma \epsilon_s \right) , \label{eq:mit_gen_eps} \\ p & = & \frac{\sigma}{1+\sigma}\left( \bar{C} \rho^{1+\sigma} -\epsilon_s \right) , \label{eq:mit_gen_p} \\ h & = & \bar{C} \rho^\sigma,\end{aligned}$$ where $\bar{C}$ is a constant of integration. The above EOS is of the form (\[eq:qsEOS\]) with $$\kappa_1 = \frac{\sigma \bar{C}}{1+\sigma},\quad \Gamma_1 = 1+\sigma,\quad \kappa_2 =-\frac{\sigma \epsilon_s}{1+\sigma},\quad \Gamma_2 = 0\,,$$ and $C=-1$. Having all thermodynamical variables in terms of the rest-mass density is convenient from the computational point of view since this is one of the fundamental variables used in the [<span style="font-variant:small-caps;">cocal</span>]{} code, therefore the modifications with respect to the EOS will be minimal. Given a fixed choice of $\sigma$ and $B$ for the MIT bag-model EOS, one can obtain a unique solution of the field equations under hydrostatic equilibrium (see Secs. \[sec:IWMformulation\] and \[sec:fluidformulation\]). In other words, the relationship between the gravitational mass versus the central energy density, the mass-radius relationship and the spacetime metric will not depend on the coefficient $\bar{C}$, as it is eliminated out from Eq.  (similar argument can be found in [@Li2017; @Bhattacharyya2016]). At the same time, it will indeed affect the rest mass hence the binding energy of the QS since it relates the rest-mass density and the number density of the components. Hence a reasonable choice for $\bar{C}$ will still be helpful although it will not affect anything that we are interested in for this work. Here we chose $\bar{C}$ such that the EOS corresponds to the $a_4 = 0.8$ model as in [@Alford2005] and we assume a rest mass of $931\,{\rm MeV}/c^2$ for each baryon number ($n_b=n_q/3$, where $n_q$ is the number density of quarks). Any other choices for $\bar{C}$ are in principle possible and they will not affect our solution except for the rest mass of the star. Actual values of these constants can be found in Table \[tab:EOSparfile\] in cgs units. In Eqs.  and we have employed the relationship which is quite similar to the explicit form of MIT bag model. By factoring out the rest mass, those two equations can be rewritten as a function of number density instead of rest-mass density. However, one clarification we want to discuss at this point is that, the choice of using Eq.  and is not essential. Moreover, Eqs.  and are related by the first law of thermodynamics \[see Eq. \], which is not essential as well. We can describe the MIT EOS Eq. (\[eq:mit\_gen\]), in a parametric form $(p(\rho),\epsilon(\rho))$ with an arbitrary parameter $\rho$, as long as the functions $p(\rho)$ and $\epsilon(\rho)$ satisfy Eq. (\[eq:mit\_gen\]). In doing so we choose to satisfy Eq. (\[eq:1stlaw\]) and therefore arrive at Eq. (\[eq:mit\_gen\_eps\]), (\[eq:mit\_gen\_p\]). As one can see from Eqs. (\[eq:rhoh\])–(\[eq:s\]), the fluid terms that appear are $q\rho=p$ and $h\rho=\epsilon+p$. Thus, for the MIT bag-model EOS, the only thermodynamic variable that appears in the field equations is $\epsilon$. Every model thus calculated will be uniquely defined by a deformation parameter and the central total energy density. The scaling constant which is analogous to scaling as $\kappa^{n/2}$ for NSs [@Rezzolla_book:2013], is here $\epsilon_s^{-1/2}$. EOS $(p/\rho)_c$ $\epsilon_c$ $\rho_c$ $M$ $\compa$ ----- -------------- ---------------------- ---------------------- ------- ---------- -- -- -- -- -- -- MIT 0.2940 $2.609\times10^{-3}$ $2.342\times10^{-3}$ 2.217 0.2706 LX 2.326 $2.451\times10^{-3}$ $1.744\times10^{-3}$ 3.325 0.3956 : Pressure, energy density, rest-mass density, gravitational mass, and compactness at the maximum mass of spherical solutions for the two EOSs in this work.[]{data-label="tab:tovmax"} Before concluding this discussion on the EOSs we should mention that although a polynomial-type EOS is implemented for the calculation of QSs with a finite surface density, the developments in the new version of [<span style="font-variant:small-caps;">cocal</span>]{} allow us to calculate any compact star with an EOS that can be described by a polynomial function, including NSs and hybrid stars. For instance, some phenomenological approaches suggested recently in Ref. [@Baym2017] also result in a polynomial-type EOS, which can be computed straightforwardly with the new code. Code tests {#sec:codetest} ========== The working properties of the [<span style="font-variant:small-caps;">cocal</span>]{} code for single rotating stars is presented in detail in previous works, [e.g., ]{}[@Huang08; @Uryu2016a][^2], so that here we will only mention the most important quantities that are used in our simulations. The method has its origins in the works of Ostriker and Marck [@Ostriker1968], who used it to compute Newtonian stars, and the works of Komatsu, Eriguchi, and Hachisu [@Komatsu89], who devised a stable numerical algorithm and obtained first axisymmetric general relativistic rotating stars. From this latest work the method is commonly referred as the KEH method and consists of an integral representation of the Poisson equation commonly referred as the representation formula. Since we have only one computational domain with trivial boundary conditions at infinity, the Green’s function is $G(x,x')=1/|x-x'|$ and is expanded as a series of the associated Legendre polynomials and trigonometric functions. The maximum number of the terms included in this expansion is given as $L$ in Table \[tab:grid\]. This approach is used to compute the gravitational fields $\{\alpha,\psi,\beta^i\}$ while hydrostatic equilibrium is achieved through Eq. (\[eq:firstint\]). At every step in the iteration to reach the solution at the desired accuracy, three constants need to be computed. The first one is the angular velocity of the star, $\Omega$, while the second one is the constant from the Euler integral, $\mathcal{E}$, in Eq. (\[eq:firstint\]), and, finally, the third constant is $R_0$, a normalization factor for the whole domain where the equations are solved.[^3] At every step during the iteration the nonlinear equation with respect to these three constants is solved typically by evaluating Eq. (\[eq:firstint\]) at three points in the star. For axisymmetric configurations, we use the center of the star and two points on the surface, one on the positive $x$-axis and one at the North pole of the stellar model. An axisymmetric equilibrium is achieved by setting the ratio between the polar axis over the equatorial radius along the $x$-axis. For triaxial configurations, on the other hand, the three points are the center of the star together with two points again on the surface, one at the positive $x$-axis, and one on the positive $y$-axis. Each triaxial solution has a fixed ratio of the radius on the $y$-axis over the radius of the $x$-axis. From a numerical point of view, [<span style="font-variant:small-caps;">cocal</span>]{} is a finite-difference code that uses spherical coordinates $(r,\theta,\phi)\in [0,r_b]\times [0,\pi]\times [0,2\pi]$[^4] and the basic parameters are summarized in Table \[tab:grid\]. In the angular directions $\theta,\phi$ the discretization is uniform, [i.e., ]{}$\Delta\theta=\pi/N_\theta$, and $\Delta_\phi=2\pi/N_\phi$. In the radial direction the grid is uniform until point $r_c$ with $\Delta r_i = r_c/\Nrm$, and in the interval $[r_c,r_b]$ the radial grid in non-uniform and follows a geometric series law [@Huang08]. While field variables are evaluated at the gridpoints, source terms under the integrals are evaluated at midpoints between two successive gridpoints since the corresponding integrals use the midpoint rule. For integrations in $r$ and $\phi$ we use a second-order midpoint rule. For integrations in $\theta$ we use a fourth-order midpoint rule. This was proven necessary to keep second order convergence at the region of maximum field strength [@Uryu:2012b]. Derivatives at midpoints are calculated using second-order rule for the angular variables $\theta,\phi$, and third-order order rule for the radial variable $r$ (again for keeping second-order convergence at same regions [@Uryu2012]). Derivatives evaluated at gridpoints always use a fourth-order formula in all variables. -------------- --- ------------------------------------------------------------------ $r_{a}$ : Radial coordinate where the grid $r_i$ starts. $r_{b}$ : Radial coordinate where the grid $r_i$ ends. $r_{c}$ : Radial coordinate between $r_{a}$ and $r_{b}$ where the grid changes from equidistant to non-equidistant. $N_{r}$ : Total number of intervals $\Dl r_i$ between $r_{a}$ and $r_{b}$. $\Nrm$ : Number of intervals $\Dl r_i$ in $[0,r_{c}]$. $\Nrf$ : Number of intervals $\Dl r_i$ in $[0,R(\theta,\phi)]$. $N_{\theta}$ : Total number of intervals $\Dl \theta_i$ for $\theta\in[0,\pi]$. $N_{\phi}$ : Total number of intervals $\Dl \phi_i$ for $\phi\in[0,2\pi]$. $L$ : Number of multipole in the Legendre expansion. -------------- --- ------------------------------------------------------------------ : Summary of parameters used for rotating star configurations.[]{data-label="tab:grid"} Type $r_a$ $r_b$ $r_c$ $N_r$ $\Nrm$ $\Nrf$ $N_\theta$ $N_\phi$ $L$ ------ ------- -------- ------- ------- -------- -------- ------------ ---------- ----- -- H2.0 0 $10^6$ 1.25 192 80 64 48 48 12 H2.5 0 $10^6$ 1.25 288 120 96 72 72 12 H3.0 0 $10^6$ 1.25 384 160 128 96 96 12 H3.5 0 $10^6$ 1.25 576 240 192 144 144 12 H4.0 0 $10^6$ 1.25 768 320 256 192 192 12 : Five different resolutions used for convergence tests. Parameters are shown in Table \[tab:grid\]. The number of points that covers the largest star radius is $\Nrf$.[]{data-label="tab:reso"} It is worth noting that, since for QSs the relationship between the specific enthalpy and the rest-mass density follows a general polynomial function \[[cf., ]{}Eq. \], a root-finding method needs to be employed when calculating the thermodynamical quantities from the enthalpy. The regular polynomial expression of the specific enthalpy with respect to rest-mass density allows us to use a Newton-Raphson method as the derivative can also be expressed easily. In view of this, the computational costs with a QS EOS are not significantly larger than those with NS EOS. However, in order to guarantee a solution of rest-mass density when the specific enthalpy is given, a bi-section root-finding method needs to be employed if the Newton-Raphson method does not converge sufficiently rapidly. In this case, the initial range of the bi-section method is set to be the specific enthalpy corresponding to the rest-mass densities at the stellar center and at the surface, respectively. In previous works [@Huang08; @Uryu:2012b; @Uryu2016a; @Tsokaros2016], the [<span style="font-variant:small-caps;">cocal</span>]{} code has been extensively tested, both with respect to its convergence properties, as well as with respect in actual evolutions with other well established codes [@Tsokaros2016]. In what follows we report the convergence tests we have performed in order to investigate the properties of the code under these new conditions. Before doing that, we note that special care is needed when using a root-finding method to calculate thermodynamical quantities for a given specific enthalpy in the case of rotating QSs. In particular, it is crucial to consider what is the accuracy set during the root-finding step. Of course, it is possible to require that the accuracy in the root-finding step is much higher than the other convergence criteria in the code to guarantee an accurate result. In this way, however, the computational costs will increase considerably, since the thermodynamical quantities need to be calculated at every gridpoint and at every iteration. We found that an accuracy of $10^{-10}$ for the thermodynamic variables solutions neither compromises the accuracy of the solutions, nor slows down the code significantly. Comparison with rotating NSs ---------------------------- Although the newly developed code presented here is intended for QS EOSs, it can be also used to produce rotating NSs if one restricts the EOS to a single polytrope. This can be accomplished by setting the polynomial terms to be only one, the surface rest-mass density $\rho_s=0$ and the energy integral constant $C=0$. In this case, Eq.  becomes $$p=\kappa\rho^\Gamma=\kappa\rho^{1+1/n}\,,$$ and the relationship between the energy density, the specific enthalpy and the rest-mass density \[Eqs. and \[eq:enthalpy\]\] will be exactly the same as that for a polytropic NS. We choose a stiff EOS with $n=0.3$ and produce axisymmetric solution sequences for small and high compactness $\compa:=M_{\rm ADM}/R=0.1, 0.2$, and $M_{\rm ADM}$ is the corresponding Arnowitt-Deser-Misner (ADM) mass for a nonrotating model, both with the original rotating-NS and the modified rotating-QS solver. The grid-structure parameters used are $N_r=240$, $\Nrm=80$, $\Nrf=64$, $N_\theta=96$, $N_\phi=192$, $L=12$, $r_b=10^4$, and $r_c=1.25$ (see Table \[tab:grid\]). Overall, we have found that the relative difference in all physical quantities is of the order of $10^{-6}$, which is what is expected since the criteria for convergence in [<span style="font-variant:small-caps;">cocal</span>]{} are that the relative difference in metric and fluid variables between two successive iterations is less than $10^{-6}$. Convergence test for rotating QSs --------------------------------- {width="0.95\columnwidth"} {width="0.95\columnwidth"} For the convergence analysis in this work we use the five resolutions shown in Table \[tab:reso\]. The outer boundary of the domain is placed at $r_b=10^6$, while the surface of the star is always inside the sphere $r=1$. The radius along the $x$-axis is exactly $r=1$ in the normalized variables. There are exactly $\Nrf$ intervals along the radii in the $x$, $y$, and $z$ directions. The number of Legendre terms used in the expansions is kept constant ($L=12$) in all resolutions since convergence with respect of those has been already investigated in [@Tsokaros2007]. When going from the low-resolution setup H2.0 to the high-resolution one H4.0, the spacings $\Delta r,\Delta\theta, \Delta\phi$ decrease as $2/3,\ 3/4,\ 2/3,\ 3/4$. As a result, if we denote as $f_{\mu}$, $f_{\nu}$ a quantity evaluated at two different resolutions, then $$f_{\mu}\,-\,f_{\nu} \, \approx \, A\left[ \left(\frac{\Delta_\mu}{\Delta_\nu}\right)^n-1 \right] \Delta^n_\nu\,, \label{eq:convtest}$$ where $A$ is a constant and $\Delta_\mu$ is the grid separation at resolution ${\rm H}\mu$. Choosing the combinations $f_{{\rm H}3.0}-f_{{\rm H}2.0}$, $f_{{\rm H}3.5}-f_{{\rm H}2.5}$, and $f_{{\rm H}4.0}-f_{{\rm H}3.0}$ so that we have $\Delta_{\mu} / \Delta_{\nu}=1/2$ and normalizing by $f_{{\rm H}4.0}$ we plot in Fig. \[fig:convtest\_RQS\] the relative error with respect to the grid spacing for $\Omega$, $\Madm$, $J$, $T/|W|$, $M_0$ and the eccentricity $e:=\sqrt{1-(\bR_z/\bR_x)^2}$, both for the LX EOS (left panel) and for the MIT bag-model EOS (right plot). The deformation is kept at $R_z/R_x=0.75$ for both EOSs, while the central densities are $\epsilon_c=1.301\times10^{-3}$, and $\epsilon_c = 7.361\times10^{-4}$ for LX and MIT bag-model EOS respectively. The dashed black line reports a reference first-order convergence, while the solid black line refers to second-order convergence. Note that quantities like the ADM mass, the angular velocity and the eccentricity converge to second order, while quantities like the angular momentum, the ratio $T/|W|$, and the rest-mass converge to an order that is closer to first. Furthermore, Fig. \[fig:convtest\_RQS\] shows that some quantities ([e.g., ]{}the ADM mass of the LX EOS) shows a convergence order that is larger than second, but this is an artefact of the specific deformation. In general, we found second-order convergence in $M_{\rm ADM},\ \Omega,\ e$ and at least first order for $J,\ T/|W|,\ M_0$. Note also that the two panels in Fig. \[fig:convtest\_RQS\] are very similar, even though the EOSs are quite different, with the MIT bag-model EOS being relatively soft ([i.e., ]{}$p\propto \rho^{4/3}$), while the LX EOS is comparatively stiff ([i.e., ]{}$p\propto \rho^5$). Hence, the overall larger error that is reported in Fig. \[fig:convtest\_RQS\] when compared to the corresponding Fig. 1 in Ref. [@Huang08], is mostly due to the finite rest-mass density at the stellar surface. In the original rotating-NS code, in fact, the surface was determined through a first-order interpolation scheme. This approach, however, is not sufficiently accurate for rotating QSs and would not lead to the desired convergence order unless the surface finder scheme was upgraded to second order. Triaxial solutions {#sec:triasolution} ================== The onset of a secular instability to triaxial solutions for the MIT bag-model EOS stars has been studied previously via a similar method in Ref. [@rosinska2003]. Surface-fitted coordinates have been used to accurately describe the discontinuous density at the surface of the star, and a set of equations similar to the one of the conformal flat approximation used here was solved. In order to find the secular bar-mode instability point, the authors of Ref. [@rosinska2003] performed a perturbation on the lapse function of an axisymmetric solution and build a series of triaxial quasi-equilibrium configurations to see whether this perturbation is damped or grows. Here, we build quasi-equilibrium sequences with constant rest mass (axisymmetric and triaxial) for both the MIT bag-model EOS and the LX EOS. We begin with the axisymmetric sequence in which we calculate a series of solutions with varying parameters, [i.e., ]{}the parameters that determine the compactness ([e.g., ]{}the central rest-mass density $\rho_c$) and the rotation ($R_z/R_x$). In doing so, we impose axisymmetry as a separate condition and manage to reach eccentricities as high as $e\simeq 0.96$ for $R_z/R_x=0.2656$ and compactness $\compa=0.1$. In order to access the triaxial branch of solutions, we recompute the above sequence of solutions but this time *without* imposing axisymmetry. As the rotation rate increases ($R_z/R_x$ decreases) the triaxial deformation ($R_y/R_x <1$) is *spontaneously* triggered, since at large rotation rate the triaxial configuration possesses lower total energy and is therefore favoured over the axisymmetric solution. This approach is different from the approach followed in Ref. [@rosinska2002], where the triaxial $m=2$ perturbation was triggered after a suitable modification of a metric potential. We keep decreasing $R_z/R_x$ to reach the mass shedding limit with the triaxial configuration. We can then move along the triaxial solution sequence by increasing $R_y/R_x$ which now acts as the new rotating parameter. The sequence is then terminated close to the axisymmetric sequence. The bifurcation point can be found by extrapolating this triaxial sequence towards the axisymmetric solutions. The largest triaxial deformation calculated in this work, for both the MIT bag-model EOS and the LX EOS, is $R_y/R_x=0.5078$ for the $\compa=0.1$ case (a three-dimensional of the surface for this solution is shown in Fig. \[fig:qs\_surf\]), $R_y/R_x=0.5234$ for $\compa=0.15$, and $R_y/R_x=0.6757$ for $\compa=0.2$. Similar with NSs [@Huang2007], the endpoint of the triaxial sequence happens in lower eccentricities as the compactness increases. ![Illustration of the three-dimensional surface of a QS solution with the largest triaxial deformation for the MIT bag-model with corresponding spherical compactness $\compa=0.2$. The axis ratio $R_y/R_x$ is 0.6757 and $R_z/R_x=0.4375$. The solid black lines on the surface corresponds to fixed values of the latitude angle and the fact that they are not parallel is a result of the triaxial deformation.[]{data-label="fig:qs_surf"}](./surf.pdf){width="0.95\columnwidth"} {width="0.95\columnwidth"} {width="0.95\columnwidth"} {width="0.95\columnwidth"} {width="0.95\columnwidth"} In Figs. \[fig:plot\_mit\] and \[fig:plot\_lx\], the relation between the $T/|W|$ ratio versus the eccentricity of the star has been plotted for three different compactnesses ($\compa=0.1,\ 0.15\ \mathrm{and}\ 0.2$) for both the MIT bag-model EOS and the LX EOS. Unlike in a Newtonian incompressible star, for which the bifurcation to triaxial deformation happens at $(T/|W|)_{\rm crit, Newt}\simeq 0.1375$ for any compactness, in general relativity the bifurcation point depends on the compactness. According to [@rosinska2002], $$\left(\frac{T}{|W|}\right)_\mathrm{\rm crit}=\left(\frac{T}{|W|}\right)_{\rm crit, Newt} +0.126 \, \compa\left(1+\compa\right) \,. \label{eq:twrelation}$$ This relation holds true not only for NSs but also for QSs with the MIT bag-model EOS (see Fig. 1 of [@rosinska2003]). The largest $T/|W|$ for the onset of secular instability is found to be $\simeq 0.17$ for rotating QSs in the configurations that we considered for both the MIT bag-model EOS and the LX EOS, and it will be even larger for higher compactnesses. Both the LX EOS and the MIT bag-model EOS in our calculations follow this relationship within a maximum error of $3\%$. This implies that the secular instability to a “Jacobi type” ellipsoidal figure in general relativity is not particularly affected by the stiffness of the EOS for quark matter. ![Plots of $\Omega M_{\rm ADM}$ versus eccentricity for MIT bag model sequences. Solid curves are axisymmetric solution sequences, and dashed curves are triaxial solution sequences, that correspond to $\compa=M/R=0.2$ (top green curve), $0.15$ (middle red curve) and $0.1$ (bottom blue curve) respectively. Note that $M$ is the spherical ADM mass. []{data-label="fig:plot_mit_omeM"}](./MIT.pdf){width="0.95\columnwidth"} ![The same as Fig. \[fig:plot\_mit\_omeM\] but for the LX EOS sequences. []{data-label="fig:plot_lx_omeM"}](./LX.pdf){width="0.95\columnwidth"} EOS $\compa$ $R_x$ $R_z/R_x$ $\epsilon_c$ $\Omega$ $\Madm$ $J$ $T/|W|$ $I$ $Z_{\rm p}$ --------- ---------- --------------- ----------------- -------------------------- ---------- --------- -------- --------- ------- ------------- -- -- MIT 0.1 7.021 (8.077) 0.5647 (0.5693) $ 6.200 \times 10^{-4} $ 0.02808 0.6515 0.4580 0.1520 16.31 0.1343 MIT 0.15 7.962 (9.922) 0.5565 (0.5640) $ 6.811 \times 10^{-4} $ 0.02985 1.169 1.293 0.1609 43.30 0.2231 MIT 0.2 8.415 (11.43) 0.5478 (0.5590) $ 7.696 \times 10^{-4} $ 0.03199 1.731 2.649 0.1706 82.83 0.3308 LX 0.1 5.698 (6.557) 0.5644 (0.5689) $9.144 \times 10^{-4} $ 0.03451 0.5312 0.3039 0.1518 8.805 0.1343 LX 0.15 6.515 (8.130) 0.5566 (0.5639) $9.542 \times 10^{-4} $ 0.03630 0.9686 0.8850 0.1607 24.38 0.2251 LX 0.2 6.972 (9.528) 0.5469 (0.5574) $9.977 \times 10^{-4} $ 0.03838 1.481 1.932 0.1715 50.34 0.3401 $n=0.3$ 0.1 6.624 (7.634) 0.5634(0.5693) $9.221 \times 10^{-4}$ 0.03180 0.5841 0.3708 0.1507 11.66 0.1328 $n=0.3$ 0.2 7.312 (9.979) 0.5394(0.5535) $1.243 \times 10^{-3}$ 0.03926 1.435 1.835 0.1688 46.72 0.3311 $n=0.5$ 0.1 10.30 (11.83) 0.5461(0.5536) $5.153 \times 10^{-4}$ 0.02197 0.8416 0.7644 0.1493 34.80 0.1281 It is worth noting that when compared with the rotating NSs calculated in Ref. [@Uryu2016a], rotating QSs have longer triaxial sequences. In another word, the triaxial sequence of rotating QSs terminates at larger eccentricity as well as larger triaxial deformation (in another word, smaller $R_y/R_x$ ratio). A rotating NS with $\Gamma=4$ and compactness $\compa=0.1$ bifurcates from axisymmetry at $e\simeq 0.825$ and can rotate as fast as to reach eccentricities $e< 0.9$ (see Fig. 6 in [@Uryu2016a]). For the QS models considered here and both EOSs, we have a bifurcation point at $e\simeq 0.825$ and the mass shedding limit at $e\simeq 0.93$. For more compact NSs with $\compa=0.2$, the bifurcation point happens at $e\simeq 0.835$ and the mass shedding limit at $e\simeq 0.88$. The corresponding compactness QS models bifurcate at $e\simeq 0.83$ and rotate as fast as $e\simeq 0.89$. A few remarks are useful to make at this point. First, we note that these values of eccentricity are strictly valid under the assumption of the conformal flatness approximation, which is however accurate for smaller compactnesses. These estimates are less accurate when the compactness increases and are slightly different when adopting more accurate formulations, such as the waveless approximation (see Fig. 6 in [@Uryu2016a]). Second, another difference between triaxial NSs and QSs is that for triaxial NSs, the ratio $T/|W|$ is essentially constant along the triaxial sequence, especially for higher compactnesses (for lower compactnesses there is an increase towards the mass shedding limit, but this is very slight). For rotating triaxial QSs, on the other hand, although this qualitative behaviour is still true, a greater curvature towards higher $T/|W|$ ratios can be seen. For example, for the $\compa=0.1$ models mentioned above, the difference between the critical value of $T/|W|$ and the one at mass-shedding limit is $(T/|W|)_{\rm ms}-(T/|W|)_{\rm crit}\simeq 0.0015$ for NSs while it is $0.0137$ for QSs. Third, in Ref. [@rosinska2003] it has been shown that at the bifurcation point the relation between the scaled angular frequency, $f/\bar{\epsilon}_s^{1/2}$ where $\bar{\epsilon}_s = \epsilon_s/(c^2\, 10^{14}\,\mathrm{g\,cm^{-3}}$), and the scaled gravitational mass $M_\mathrm{ADM}\bar{\epsilon}_s^{1/2}$, depends only very weakly on the bag constant. If we consider models with compactness $\compa=0.1$ model (see top line in Table \[tab:secular\]), $M_\mathrm{ADM}\bar{\epsilon}_s^{1/2} = 1.193\,M_\odot$ and $f/ \bar{\epsilon}_s^{1/2} = 495.9\,\mathrm{Hz}$, while the scaled bifurcation frequency for such a scaled mass model is roughly $492-494\,\mathrm{Hz}$ as deduced from Fig. 7 in [@rosinska2003]. Similarly, for the $\compa=0.15$ models, the renormalized ADM mass and frequency are $2.140\,M_\odot$ and $527.2\,\mathrm{Hz}$, while the corresponding range is $523-527\,\mathrm{Hz}$ in Ref. [@rosinska2003]; finally, for the $\compa=0.2$ case, the values are $3.168\,M_\odot$ and $565.0\,\mathrm{Hz}$, respectively, while the range $558-566\,\mathrm{Hz}$ is found in [@rosinska2003]. Overall, the comparison of these three values shows a very good agreement with the results presented in Fig. 7 of Ref. [@rosinska2003]. {width="0.95\columnwidth"} {width="0.95\columnwidth"} ![Estimate of the GW strain amplitude for the $\compa=0.2$ triaxial sequence for both the MIT bag-model EOS (blue solid curve) and the LX EOS (red dashed curve). The quantities are estimated according to the quadrupole formula. Shown is the GW strain for the $\ell=m=2$ mode normalized by the distance $D$ and the ADM mass $M_\mathrm{ADM}$ of the source.[]{data-label="fig:plot_gw_quad"}](./LX_rhm.pdf){width="0.95\columnwidth"} In order to understand the rotation properties of the triaxial solutions, we also report quantities such as dimensionless spin and dimensionless angular momentum in Fig. \[fig:plot\_mit\_omeM\]- \[fig:plot\_lx\_J\]. The dimensionless spin as a function of the eccentricity for MIT bag-model and LX model are shown in Fig. \[fig:plot\_mit\_omeM\] and \[fig:plot\_lx\_omeM\], respectively. The top panel of Fig. \[fig:plot\_lx\_J\] reports the spin angular momentum as a function of the eccentricity for the LX EOS. Similar as $T/|W|$, the angular momentum increases with the eccentricity. The main difference is that the relative positioning of the curves as a function of compactness is reversed when compared with the $T/|W|$ plots. In other words for a given eccentricity the greatest angular momentum is achieved for the smallest compactness, while the greatest $T/|W|$ for the largest one. This is true both for axisymmetric and triaxial solutions. Also as we can see from the bottom panel of Fig. \[fig:plot\_lx\_J\], more compact objects can reach greater rotational frequencies while less compact can reach larger angular momenta, which can exceed unity. According to Fig. \[fig:plot\_mit\_omeM\] and \[fig:plot\_lx\_omeM\] as well as the bottom panel of Fig. \[fig:plot\_lx\_J\], triaxial sequences lose angular velocity and gain spin angular momentum as one moves towards the mass shedding limit. An obviously interesting property of triaxially rotating compact stars is that they can act as strong sources of GWs. A full general-relativistic evolution needs to be employed in order to determine accurately the details of the GW emission from such triaxially rotating stars and this is beyond the scope of this paper (see however Ref. [@Tsokaros2017] for the case of NSs). At the same time, we can apply the quadrupole formula to make reasonable estimates using the quasi-equilibrium initial data we have computed. The relationship between the normalized GW strain and the eccentricity of the star is shown in the top panel of Fig. \[fig:plot\_gw\_quad\]. Compared with the results of triaxially rotating NSs calculated in [@Tsokaros2017], we find that the GW strain for QSs are several times larger for similar values of the compactness. For example, model G4C025 in [@Tsokaros2017] with $e=0.8685$ radiates GW with normalized strain 0.007357, while the corresponding amplitude for both the MIT bag-model EOS and the LX EOS is around 0.025 with same eccentricity (see Fig. \[fig:plot\_gw\_quad\]). Also shown in Fig. \[fig:plot\_gw\_quad\] for the two EOSs considered, are the relations between the strain and the eccentricity, which are very similar and both essentially linear. Triaxial supramassive solutions {#sec:triasupraQS} =============================== Besides the constant rest-mass sequences mentioned above, we have also built sequences with constant central rest-mass density for both the MIT bag-model EOS and the LX EOS. We recall that when fixing the central rest-mass density, the mass of the solutions will increase as the axis ratio $R_z/R_x$ decreases. Furthermore, since we do not impose axisymmetry, the triaxial deformation will be spontaneously triggered when $T/|W|$ is large enough. Therefore, with such calculations we can determine whether triaxial supramassive QSs, [i.e., ]{}triaxial solutions with ADM mass larger than the TOV maximum mass ($M_\mathrm{TOV}$) exist and the properties of such solutions[^5] (Note that all models shown in Figs. \[fig:plot\_mit\]–\[fig:plot\_lx\_J\] are not supramassive). According to [@Uryu2016b], triaxial supramassive NS does exist for the case with polytropic EOS in the range $\Gamma \agt 4$. Furthermore, for the case with a two segments piecewise polytropic EOS, sequences of triaxial supramassive NS solutions become longer, and hence the existence of supramassive triaxial NS becomes evident, when the EOS of the lower density region is stiff ($\Gamma=4$), and the higher density region is soft ($\Gamma=2.5$)[@Uryu2016b]. Therefore, it is likely that the triaxial supramassive QS also exists because the QS EOS used in this paper has an analogous property, namely, the effective $\Gamma$ is smaller (softer) in the higher density region, and larger (stiffer) in the lower density region (for the MIT model, see [@gourgoulhon1999]). Besides having an interest of its own, determining the existence of such solutions could be relevant to establish whether a BNS merger could lead to the formation of such an object. Based on current mass measurement constraints [@Demorest2010; @Antoniadis2013] and on known BNS systems, the mass of the post-merger product will very likely be larger than $M_\mathrm{TOV}$. In order to study this, we have fixed the central rest-mass density close to the value corresponding to $M_\mathrm{TOV}$ and built rotating solution sequences for both the MIT bag-model EOS and the LX EOS, respectively. In this way, we were indeed able to find triaxial supramassive solutions for both EOSs, reporting in Table.\[tab:triasupraqs\] the solutions with largest triaxial deformation, [i.e., ]{}the smallest ratio $R_y/R_x$. Finally, we note that although such models have large compactnesses and we are aware that the IWM formalism becomes increasingly inaccurate for large compactnesses ([i.e., ]{}with $\compa \gtrsim 0.3$), we also believe that the associated $\sim3\%$ errors will not change the qualitative result, namely, that triaxial supramassive QS models exist for the EOSs considered here. At the same time, we plan to re-investigate this point in the future, when more accurate methods, such as waveless formulation, will be employed to compute QS solutions. EOS $R_z/R_x$ $R_y/R_x$ $R_x$ $\epsilon_c$ $\Omega$ $\Madm$ $J$ $T/|W|$ $I$ $M_\mathrm{TOV}$ ----- ----------------- ----------------- --------------- -------------------------- ---------- --------- ------- --------- ------- ------------------ -- -- MIT 0.4375 (0.4713) 0.7657 (0.7938) 9.978 (16.32) $ 1.259 \times 10^{-3} $ 0.03870 2.862 6.847 0.1839 173.1 2.217 LX 0.4375 (0.4912) 0.7586 (0.8104) 7.660 (16.49) $1.348 \times 10^{-3} $ 0.05001 3.727 11.30 0.1948 222.1 3.325 Conclusion {#sec:disandconclu} ========== We have presented a new version of the [<span style="font-variant:small-caps;">cocal</span>]{} code to compute axisymmetric and triaxial solutions of uniformly rotating QSs in general relativity with two EOSs, [i.e., ]{}the MIT bag-model and the LX EOS. Comparisons have been made with NSs as well. Overall, three main properties are found when comparing solution sequences of QSs with of NSs. Firstly, QSs generally have a longer triaxial sequences of solutions than NSs. In another words, QSs can reach a larger triaxial deformation (or smaller $R_y/R_x$ ratio) before terminating the sequence at the mass-shedding limit; this is mostly due to the larger $T/|W|$ ratio that can be attained by QSs. Secondly, when considering similar triaxial configurations, QSs are (slightly) more efficient GW sources; this is mostly due to the finite surface rest-mass density and hence larger mass quadrupole for QSs. Thirdly, triaxial supramassive solutions can be found for QSs as well; this is due again to the fact that larger values of the $T/|W|$ ratio can be sustained before reaching the mass-shedding limit. Besides having an interest of its own within solutions of self-gravitating objects in general relativity, triaxially rotating compact stars are important sources for ground-based GW observatories. Our calculations have shown that for rotating QSs with different EOSs, the bifurcation point to triaxial sequence happens at a spin period of $\sim 1\,{\rm ms}$, so that the corresponding GW frequency is $\sim 2\,{\rm kHz}$ and hence within the band of GW observatories such as Advanced LIGO or Virgo. Indeed, exploiting the largest triaxial deformation solution obtained in our calculations, the GW strain amplitude can be as large as $10^{-23}$ at a distance of $\sim 30\,{\rm Mpc}$. Although this is an interesting prospect, it is still unclear whether such triaxial configurations can be produced in practice, since the radiation-reaction timescales needed for the triggering of the secular triaxial instability are still very uncertain, as are the other mechanisms that could contrast the instability. For example, if the triaxial deformation is induced in an isolated star, [e.g., ]{}a newly born fast rotating star, GW radiation may take away the excess angular momentum very rapidly so that the star would go back to the axisymmetric sequence again after the $T/|W|$ ratio drops below the critical value. Similarly, when considering stars in binary systems, there is the prospect that an accreting system, such as the one spinning up pulsars, could drive the accreting compact star to exceed the critical $T/|W|$ ratio, hence leading to a break of axisymmetry. In this process, which is also known as forced GW emission, the triaxial deformation can be maintained via the angular momentum supplied by the accreted matter. Notwithstanding the large uncertainties involved with the details of this picture, such as the presence or not of bifurcation point or the realistic degree of deformation attained by the unstable stars, the fact that these details depend sensitively on the EOS [@Uryu2016b], suggests that a detection of this type of signal could serve as an important probe for distinguishing the EOS of compact stars. Finally, we note that the triaxial configurations could also be invoked to explain the spin-up limit for rotating compact stars, which is far smaller than the mass-shedding limit. The results presented here and in Ref. [@Huang08] suggest that when triaxial deformations are taken into account, the rotational period of a compact star actually decreases as it gains angular momentum, [e.g., ]{}by accretion, along the triaxial sequence. As a result, the “spin-up” process provided by the accretion of matter onto the pulsar can actually spin down the pulsar if the bifurcation point is reached. In this case, no accreting pulsar could spin up faster than the period at the bifurcation point. Of course, depending on the microphysical properties of the QS ([e.g., ]{}the magnitude of the shear viscosity or of the breaking strain in the crust) it is also possible that other mechanisms of emission of gravitational waves, ([e.g., ]{}other dynamical instabilities such as the barmode instability [@Watts:2003nn; @Corvino:2010] or the $r$-mode instability [@Andersson1998; @Friedman1998; @Andersson99], or nonzero ellipticities) could intervene at lower spinning frequencies and therefore before the onset of instability to a triaxial deformation is reached [@Bildsten98; @Andersson:2009yt]. As a result, the search for fast spinning pulsars with more powerful radio telescopes, such as the Square Kilometre Array (SKA) and the Five-hundred-meter Aperture Spherical radio Telescope (FAST) [@SKAdoc; @Nan2011] could provide important clues on the properties of pulsars and test the validity of the solid QS assumption [@Xu2003]. It is a pleasure to thank J. G. Lu, J. Papenfort, Z. Younsi, and all the members of the Pulsar group in Peking University and the Relastro group in Frankfurt for useful discussions. We will thank Dr. L. Shao and Dr. Y. Hu for their help with GW experiments. E. Z. is grateful to the China Scholarship Council for supporting the joint PhD training in Frankfurt. This research is supported in part by the ERC synergy grant “BlackHoleCam: Imaging the Event Horizon of Black Holes” (Grant No. 610058), by “NewCompStar”, COST Action MP1304, by the LOEWE-Program in the Helmholtz International Center (HIC) for FAIR, by the European Union’s Horizon 2020 Research and Innovation Programme (Grant 671698) (call FETHPC-1-2014, project ExaHyPE). This work is supported by National Key R&D Program (No.2017YFA0402600) and NNSF (11673002,U1531243). A. T. is supported by NSF Grants PHY-1662211 and PHY-1602536, and NASA Grant 80NSSC17K0070. K. U. is supported by JSPS Grant-in-Aid for Scientific Research(C) 15K05085. [^1]: Such a quark-cluster star has also been named a [*strangeon*]{} star in Ref. [@lai2017]. [^2]: See [@Uryu2012; @Uryu:2012b; @Tsokaros2012; @Tsokaros2015] for the general binary case. [^3]: We recall that [<span style="font-variant:small-caps;">cocal</span>]{} uses normalized variables $\hat{x}^i:=x^i/R_0$ and the quantities listed in Table \[tab:reso\] refer to those and should be denoted by a hat. For simplicity, however, we have omitted these hats in the Table. [^4]: Note that the field equations for the shift vector Eq. (\[eq:MoC\_elip2\]) are expressed in Cartesian coordinates. [^5]: We recall that for NSs, a universal relation has been found between $M_\mathrm{TOV}$ and the maximum mass that can be sustained by axisymmetric solutions in uniform rotation, $M_{\rm max}$ (see also [@Weih2017] for the case of differentially rotating stars). More specifically, Breu and Rezzolla [@Breu2016] found that $M_{\rm max} \simeq (1.203 \pm 0.022) M_\mathrm{TOV}$ for a large class of EOSs; we expect a similar universal behaviour to be present also for QSs, although the scaling between $M_{\rm max}$ and $M_\mathrm{TOV}$ is likely to be different.

--- abstract: 'We investigate the evolution of closed strictly convex hypersurfaces in $\R^{n+1}$, $n=3$, for contracting normal velocities, including powers of the mean curvature, $H$, of the norm of the second fundamental form, $|A|$, and of the Gauss curvature, $K$. We prove convergence to a round point for $2$-pinched initial hypersurfaces. In $\R^{n+1}$, $n=2$, natural quantities exist for proving convergence to a round point for many normal velocities. Here we present their counterparts for arbitrary dimensions $n\in\N$.' address: - 'Martin Franzen, Universität Konstanz, Universitätsstrasse 10, D-78464 Konstanz, Germany' - 'Martin Westerholt-Raum, Max Planck Institute for Mathematics, Vivatsgasse 7, D-53111 Bonn, Germany' - 'Ferdinand Kuhl, Digital Competence, Annakirchstrasse 190, D-41063 Mönchengladbach, Germany' author: - Martin Franzen date: 'February 27, 2015.' title: Pinched hypersurfaces shrink to round points --- [^1] [^2] [^3] Overview ======== We consider the geometric flow equation $$\begin{aligned} \label{eq3:evol} \begin{cases} \frac{d}{dt} X=-F\nu,&\\ X(\cdot,0)=M_0 \end{cases}\end{aligned}$$ and ask whether closed strictly convex hypersurfaces $M_{0\leq t < T}$ in $\R^{n+1}$, $n=3$, shrink to round points. For the cubed mean curvature, $F=H^3$, the answer is affirmative if the initial hypersurface $M_0$ is $2$-pinched, the principal curvatures $\left(\lambda_i\right)_{1\leq i\leq 3}$ fulfill $$\begin{aligned} \frac{\lambda_i}{\lambda_j} \leq 2\end{aligned}$$ everywhere on $M_0$ for all $1\leq i,j\leq 3$. This is our main Theorem \[thm3:H3\]. Furthermore, we sketch the proof of similar results for the square of the norm of the second fundamental form, $F=|A|^2$, and the Gauss curvature, $F=K$. So far, strong pinching assumptions were needed to show convergence to a round point [@am:convex; @bc:deforming; @fs:convexity].\ **The paper is structured as follows**: *$\bullet$ Notation*: We give a quick introduction to differential geometric quantities used in this paper, the induced metric, the second fundamental form, and the principal curvatures. *$\bullet$ Linear operator $L$*: We calculate the linear operator $Lw:=\frac{d}{dt}w-F^{ij}w_{;ij}$ for a function $w$ of the principal curvatures $\lambda_i$, $i=1,2,3$, at a critical point of $w$. To improve readability, we also choose normal coordinates at that critical point, $g_{ij}=\delta_{ij}$, and $\left(h_{ij}\right)=\operatorname{diag}\left(\lambda_1,\lambda_2,\lambda_3\right)$. This lays the groundwork for subsequent calculations. *$\bullet$ Vanishing functions*: In $\R^{n+1}$, $n=2$, for many normal velocities $F$ the quantity $$\begin{aligned} \frac{\left(\lambda_1-\lambda_2\right)^2}{\left(\lambda_1\,\lambda_2\right)^2} F^2\end{aligned}$$ seems to be the natural choice when showing convergence to a round point. As in [@mf:when], we call this quantity a *vanishing function* for a normal velocity $F$. It is used by B. Andrews for the Gauss curvature flow [@ba:gauss], by F. Schulze and O. Schnürer for the $H^\sigma$-flow [@fs:convexity], by B. Andrews and X. Chen for the $|A|^\sigma$ and the $\operatorname{tr}A^\sigma$-flow [@ac:surfaces]. The quantity $$\begin{aligned} \sum_{i<j} \frac{\left(\lambda_i-\lambda_j\right)^2}{\left(\lambda_i\,\lambda_j\right)^2} F^2\end{aligned}$$ is the counterpart of a vanishing function for arbitrary dimensions $n\in\N$. In particular, we work with this quantity in $\R^{n+1}$, $n=3$. *$\bullet$ $H^3$-flow*: The proof of our main Theorem \[thm3:H3\] is based on investigating the quantities $$\begin{aligned} \phi_{H^3} =&\, \frac{(a-b)^2+(a-c)^2+(b-c)^2}{(a+b+c)^2}, \\ \text{and }\quad \psi_{H^3} =&\, \left( \frac{(a-b)^2}{(a\,b)^2} + \frac{(a-c)^2}{(a\,c)^2} + \frac{(b-c)^2}{(b\,c)^2} \right) \left(H^3\right)^2,\end{aligned}$$ which are homogeneous functions of the principal curvatures $a\equiv \lambda_1$, $b\equiv\lambda_2$, and $c\equiv\lambda_3$. First we show that the estimate $\phi_{H^3}\leq h := 1/8$ is preserved during the $H^3$-flow if the initial hypersurface $M_0$ is $2$-pinched. Next we prove that $\psi_{H^3}$ is bounded in time on the set where $\phi_{H^3}\leq h$. This involves the maximum-principle, the linear operator $L$ and our computer program $[\textsc{cp}]$. Finally, we show convergence to a round point combining the boundedness of $\psi_{H^3}$ and the proof of [@fs:convexity Theorem A.1.] by F. Schulze and O. Schnürer. *$\bullet$ $|A|^2$-flow and Gauss curvature flow*: We sketch the proof of results similar to our main Theorem \[thm3:H3\] for the $|A|^2$-flow, and for the Gauss curvature flow. *$\bullet$ Appendix*: Some of the Lemmas leading up to the proof our main Theorem \[thm3:H3\] rely on the computer program $[\textsc{cp}]$, where we use a Monte-Carlo method. For the convenience of the reader, we include the source code of $[\textsc{cp}]$ in three different programming languages, namely the computer algebra systems Mathematica, Sage, and Maple. Acknowledgments =============== We would like to thank O. Schnürer for suggesting the use of two monotone quantities instead of one. In particular, we thank O. Schnürer for proposing the quantity $\psi_{|A|^2}$, and M. Makowski for proposing the quantity $\phi_{|A|^2}$. We are also indebted to M. Westerholt-Raum and F. Kuhl for their help in translating the computer program $[\textsc{cp}]$ from Mathematica to Sage and to Maple. Notation ======== For a quick introduction of the standard notation we adopt the corresponding chapter from [@os:surfacesA2]. We use $X=X(x,\,t)$ to denote the embedding vector of an $n$-manifold $M_t$ into $\R^{n+1}$ and $\frac{d}{dt} X=\dot{X}$ for its total time derivative. It is convenient to identify $M_t$ and its embedding in $\R^{n+1}$. The normal velocity $F$ is a homogeneous symmetric function of the principal curvatures. We choose $\nu$ to be the outer unit normal vector to $M_t$. The embedding induces a metric $g_{ij} := \langle X_{,i},\, X_{,j} \rangle$ and the second fundamental form $h_{ij} := -\langle X_{,ij},\,\nu \rangle$ for all $i,\,j = 1,\ldots,n$. We write indices preceded by commas to indicate differentiation with respect to space components, $X_{,k} = \frac{\partial X}{\partial x_k}$ for all $k=1,\ldots,n$. We use the Einstein summation notation. When an index variable appears twice in a single term it implies summation of that term over all the values of the index. Indices are raised and lowered with respect to the metric or its inverse $\left(g^{ij}\right)$, $h_{ij} h^{ij} = h_{ij} g^{ik} h_{kl} g^{lj} = h^k_j h^j_k$. The principal curvatures $\lambda_i$, $i=1,\ldots,n$, are the eigenvalues of the second fundamental form $\left(h_{ij}\right)$ with respect to the induced metric $\left(g_{ij}\right)$. For $n=3$, we name the principle curvatures also $a\equiv \lambda_1$, $b\equiv \lambda_2$, and $c\equiv \lambda_3$. A surface is called strictly convex if all principal curvatures are strictly positive. We will assume this throughout the paper. Therefore, we may define the inverse of the second fundamental form denoted by $(\tilde h^{ij})$. Symmetric functions of the principal curvatures are well-defined, we will use the Gauss curvature $K=\frac{\det h_{ij}}{\det g_{ij}} = \prod^n_{i=1} \lambda_i$, the mean curvature $H=g^{ij} h_{ij} = \sum^n_{i=1} \lambda_i$, the square of the norm of the second fundamental form $|A|^2= h^{ij} h_{ij} = \sum^n_{i=1} \lambda_i^2$, and the trace of powers of the second fundamental form $\operatorname{tr}A^{\sigma} = \operatorname{tr}\left(h^i_j\right)^{\sigma} = \sum^n_{i=1} \lambda_i^\sigma$. We write indices preceded by semi-colons to indicate covariant differentiation with respect to the induced metric, e.g. $h_{ij;\,k} = h_{ij,k} - \Gamma^l_{ik} h_{lj} - \Gamma^l_{jk} h_{il}$, where $\Gamma^k_{ij} = \frac{1}{2} g^{kl} \left(g_{il,j} + g_{jl,i} - g_{ij,l}\right)$. It is often convenient to choose normal coordinates, coordinate systems such that at a point the metric tensor equals the Kronecker delta, $g_{ij}=\delta_{ij}$, in which $\left(h_{ij}\right)$ is diagonal, $(h_{ij})=\operatorname{diag}(\lambda_1,\ldots,\lambda_n)$. Whenever we use this notation, we will also assume that we have fixed such a coordinate system. We will only use a Euclidean metric for $\R^{n+1}$ so that the indices of $h_{ij;\,k}$ commute according to the Codazzi-Mainardi equations. A normal velocity $F$ can be considered as a function of principal curvatures $\lambda_i$, $i=1,\ldots,n$, or $(h_{ij},\,g_{ij})$. We set $F^{ij}=\frac{\partial F}{\partial h_{ij}}$, $F^{ij,\,kl}=\frac{\partial^2 F}{\partial h_{ij}\partial h_{kl}}$. Note that in coordinate systems with diagonal $h_{ij}$ and $g_{ij}=\delta_{ij}$ as mentioned above, $F^{ij}$ is diagonal. Linear operator $L$ =================== We begin this chapter with Definition \[def3:linear operator\] of the linear operator $Lw$ for a function $w$ of the principal curvatures $\lambda_i$, $i=1,\ldots,n$. Then we calculate the linear operator $Lw$ at a critical point of $w$ in $\R^{n+1}$, $n=3$. To improve readability, we also choose normal coordinates at that critical point, $g_{ij}=\delta_{ij}$, and $\left(h_{ij}\right)=\operatorname{diag}\left(\lambda_1,\lambda_2,\lambda_3\right)\equiv\operatorname{diag}\left(a,b,c\right)$. This is Lemma \[lem3:critical point\]. In Corollary \[cor3:critical point\] we will see that the linear operator $Lw$ has the form $$\begin{aligned} Lw = {\mathbf{C_w}} + {\mathbf{E_w}}\,{\mathbf{x_0}}^2 + {\mathbf{x_1}}^\top\, M^{\mathbf{R_w}}\, {\mathbf{x_1}} + {\mathbf{x_2}}^\top\, M^{\mathbf{S_w}}\, {\mathbf{x_2}} + {\mathbf{x_3}}^\top\, M^{\mathbf{T_w}}\, {\mathbf{x_3}},\end{aligned}$$ where ${\mathbf{C_w}}(a,b,c), {\mathbf{E_w}}(a,b,c)$ are functions in $\R$, and $M^{\mathbf{R_w}}(a,b,c)$, $M^{\mathbf{S_w}}(a,b,c)$, $M^{\mathbf{T_w}}(a,b,c)$ are functions in $\R^{2\times 2}$ with $$\begin{aligned} {\mathbf{x_0}} = h_{12;3},\; {\mathbf{x_1}} = \binom{h_{22;1}}{h_{33;1}},\; {\mathbf{x_2}} = \binom{h_{11;2}}{h_{33;2}},\; {\mathbf{x_3}} = \binom{h_{11;3}}{h_{22;3}}.\end{aligned}$$ In subsequent calculations we need the linear operator $Lw$ to be non-positive for some set $\Sc\subset\R^3_{+}$. We achieve this by checking the non-positivity of each of the functions ${\mathbf{C_w}}, {\mathbf{E_w}}$, $M^{\mathbf{R_w}}$, $M^{\mathbf{S_w}}$, and $M^{\mathbf{T_w}}$ on $\Sc\subset\R^3_{+}$. In Remark \[rem3:sufficient conditions\] we state the criterion we use in our computer program $[\textsc{cp}]$ to determine the negative semi-definiteness of $M^{\mathbf{R_w}}$, $M^{\mathbf{S_w}}$, and $M^{\mathbf{T_w}}$. \[def3:linear operator\] Let $w$ be a function of the principal curvatures. Then we define the linear operator $L$ by $$\begin{aligned} Lw=\frac{d}{dt}w-F^{ij}w_{;ij}, \end{aligned}$$ which is corresponding to the geometric flow equation . \[lem3:linear operator\] Let $w=w\big(h^j_i\big)$ be a function of the principal curvatures. Let $L$ be defined as in . Then we have $$\begin{aligned} \begin{split} Lw =&\, w^{ij}\left(h_{ij}F^{kl}h^m_kh_{lm}+h^m_ih_{jm}\left(F-F^{kl}h_{kl}\right)\right) \\ &\quad+\left(w^{ij}F^{kl,rs}-F^{ij}w^{kl,rs}\right)h_{kl;i}h_{rs;j}. \end{split} \end{aligned}$$ We refer to [@mf:when Lemma 4.5]. \[lem3:second derivatives\] Let $f$ be a normal velocity $F$ or a function $w$ of the principal curvatures. Then we have $$\begin{aligned} f^{ij,kl}\eta_{ij}\eta_{kl} = \sum_{i,j} \frac{\partial^2 f}{\partial \lambda_i\partial \lambda_j}\eta_{ii}\eta_{jj} +\sum_{i\neq j}\frac{\frac{\partial f}{\partial \lambda_i}-\frac{\partial f}{\partial \lambda_j}}{\lambda_i-\lambda_j}\eta_{ij}^2 \end{aligned}$$ for any symmetric matrix $(\eta_{ij})$ and $\lambda_i\neq\lambda_j$, or $\lambda_i=\lambda_j$ and the last term is interpreted as a limit. We refer to C. Gerhardt [@cg:curvature Lemma 2.1.14]. \[lem3:critical point\] Let $w=w\big(h^j_i\big)$ be a symmetric function of the principal curvatures $a$, $b$, and $c$. At a critical point of $w$, $w_{;i}=0$ for all $i=1,2,3$, we choose normal coordinates, $g_{ij}=\delta_{ij}$ and $\big(h_{ij}\big)=\operatorname{diag}(a,b,c)$. Then we have $$\begin{aligned} \begin{split} Lw =&\, {\mathbf{C_w}}(a,b,c) \\ & + {\mathbf{E_w}}(a,b,c)\,h_{12;3}^2 \\ & + {\mathbf{R_w}}(a,b,c,h_{11;1},h_{22;1},h_{33;1}) \\ & + {\mathbf{S_w}}(a,b,c,h_{11;2},h_{22;2},h_{33;2}) \\ & + {\mathbf{T_w}}(a,b,c,h_{11;3},h_{22;3},h_{33;3}) \end{split} \end{aligned}$$ The constant terms ${\mathbf{C_w}}$ are $$\begin{aligned} {\mathbf{C_w}}(a,b,c) =&\, a\,w_a\left(a^2\,F_a+b^2\,F_b+c^2\,F_c+a\left(F-a\,F_a-b\,F_b-c\,F_c\right)\right) \umbruch \\ & +b\,w_a\left(a^2\,F_a+b^2\,F_b+c^2\,F_c+b\left(F-a\,F_a-b\,F_b-c\,F_c\right)\right) \umbruch \\ & +c\,w_c\left(a^2\,F_a+b^2\,F_b+c^2\,F_c+c\left(F-a\,F_a-b\,F_b-c\,F_c\right)\right). \end{aligned}$$ The gradient terms ${\mathbf{E_w}}$ are $$\begin{aligned} {\mathbf{E_w}}(a,b,c)/2=&\, \Big( \left(w_c\left(F_a-F_b\right)-F_c\left(w_a-w_b\right)\right)/(a-b) \Big. \umbruch \\ & \Big. + \left(w_b\left(F_a-F_c\right)-F_b\left(w_a-w_c\right)\right)/(a-c) \Big. \umbruch \\ & \Big. + \left(w_a\left(F_b-F_c\right)-F_a\left(w_b-w_c\right)\right)/(b-c) \Big). \end{aligned}$$ The gradient terms ${\mathbf{R_w}}$ are $$\begin{aligned} {\mathbf{R_w}}(a,b,c,&\, h_{11;1},h_{22;1},h_{33;1}) \umbruch \\ =&\, w_a\left(\left(F_{aa}\,h_{11;1}^2+F_{bb}\,h_{22;1}^2+F_{cc}\,h_{33;1}^2\right) \right. \\ &\qquad\quad \left. +2\left(F_{ab}\,h_{11;1}\,h_{22;1} + F_{ac}\,h_{11;1}\,h_{33;1} + F_{bc}\,h_{22;1}\,h_{33;1} \right)\right) \umbruch \\ &\,+w_b\left(2\frac{F_a-F_b}{a-b} h_{22;1}^2 \right) \umbruch \\ &\,+w_c\left(2\frac{F_a-F_c}{a-c} h_{33;1}^2 \right) \umbruch \\ &\,-F_a\left(w_{aa}\,h_{11;1}^2+w_{bb}\,h_{22;1}^2 + w_{cc}\,h_{33;1}^2 \right. \\ &\qquad\quad \left.+2\left(w_{ab}\,h_{11;1}\,h_{22;1} + w_{ac}\,h_{11;1}\,h_{33;1} + w_{bc}\,h_{22;1}\,h_{33;1} \right)\right) \umbruch \\ &\,-F_b\left(2\frac{w_a-w_b}{a-b} h_{22;1}^2\right) \\ &\,-F_c\left(2\frac{w_a-w_c}{a-c} h_{33;1}^2\right). \end{aligned}$$ The gradient terms ${\mathbf{S_w}}$ are $$\begin{aligned} {\mathbf{S_w}}(a,b,c,&\,h_{11;2},h_{22;2},h_{33;2}) \\ =&\,w_a\left(2\frac{F_a-F_b}{a-b} h_{11;2}^2 \right) \\ &\,+w_b\left(F_{aa}\,h_{11;2}^2 + F_{bb}\,h_{22;2}^2 + F_{cc}\,h_{33;2}^2 \right. \\ &\qquad\quad \left.+2\left(F_{ab}\,h_{11;2}\,h_{22;2} + F_{ac}\,h_{11;2}\,h_{33;2} + F_{bc}\,h_{22;2}\,h_{33;2} \right)\right) \\ &\,+w_c\left(2\frac{F_b-F_c}{b-c} h_{33;2}^2 \right) \\ &\,-F_a\left(2\frac{w_a-w_b}{a-b} h_{11;2}^2 \right) \\ &\,-F_b\left(w_{aa}\,h_{11;2}^2+w_{bb}\,h_{22;2}^2 + w_{cc}\,h_{33;2}^2 \right. \\ &\qquad\quad \left. +2\left(w_{ab}\,h_{11;2}\,h_{22;2}+w_{ac}\,h_{11;2}\,h_{33;2}+w_{bc}\,h_{22;2}\,h_{33;2}\right)\right) \\ &\,-F_c\left(2\frac{w_b-w_c}{b-c} h_{33;2}^2\right). \end{aligned}$$ The gradient terms ${\mathbf{T_w}}$ are $$\begin{aligned} {\mathbf{T_w}}(a,b,c,&\,h_{11;3},h_{22;3},h_{33;3}) \umbruch \\ =&\,w_a\left(2\frac{F_a-F_c}{a-c} h_{11;3}^2 \right) \umbruch \\ &\,+w_b\left(2\frac{F_b-F_c}{b-c} h_{22;3}^2 \right) \umbruch \\ &\,+w_c\left(F_{aa}\,h_{11;3}^2+F_{bb}\,h_{22;3}^2+F_{cc}\,h_{33;3}^2 \right. \\ &\qquad\quad \left.+2\left(F_{ab}\,h_{11;3}\,h_{22;3} + F_{ac}\,h_{11;3}\,h_{33;3}+F_{bc}\,h_{22;3}\,h_{33;3}\right)\right) \umbruch \\ &\,-F_a\left(2\frac{w_a-w_c}{a-c} h_{11;3}^2 \right) \umbruch \\ &\,-F_b\left(2\frac{w_b-w_c}{b-c} h_{22;3}^2 \right) \umbruch \\ &\,-F_c\left(w_{aa}\,h_{11;3}^2 + w_{bb}\,h_{22;3}^2 + w_{cc}\,h_{33;3}^2 \right. \\ &\qquad\quad \left.+2\left(w_{ab}\,h_{11;3}\,h_{22;3} + w_{ac}\,h_{11;3}\,h_{33;3} + w_{bc}\,h_{22;3}\,h_{33;3} \right)\right). \end{aligned}$$ Furthermore, we have at a critical point of $w$ $$\begin{aligned} \label{id:critical point} \begin{split} h_{11;1} =&\, -\frac{1}{w_a}\left(w_b\,h_{22;1} + w_c\,h_{33;1} \right), \\ h_{22;2} =&\, -\frac{1}{w_b}\left(w_a\,h_{11;2} + w_c\,h_{33;2} \right), \\ h_{33;3} =&\, -\frac{1}{w_c}\left(w_a\,h_{11;3} + w_b\,h_{22;3} \right). \end{split} \end{aligned}$$ We use Lemma \[lem3:linear operator\], and Lemma \[lem3:second derivatives\] at a point, where we choose normal coordinates. This way we obtain the constant terms ${\mathbf{C_w}}$, and the four gradient terms ${\mathbf{E_w}}$, ${\mathbf{R_w}}$, ${\mathbf{S_w}}$, and ${\mathbf{T_w}}$.\ At a critical point of $w$, we have $w_i(a,b,c) = 0$ for $i=1,2,3$. This implies $$\begin{aligned} w_a\,h_{1l;i}\,g^{l1} + w_b\,h_{2l;i}\,g^{l2} + w_c\,h_{3l;i}\,g^{l3} = 0. \end{aligned}$$ Using normal coordinates we obtain $$\begin{aligned} w_a\,h_{11;i} + w_b\,h_{22;i} + w_c\,h_{33;i} = 0. \end{aligned}$$ Now we obtain for $i=1,2,3$ the identities $$\begin{aligned} h_{11;1} =&\, -\frac{1}{w_a}\left(w_b\,h_{22;1} + w_c\,h_{33;1} \right), \\ h_{22;2} =&\, -\frac{1}{w_b}\left(w_a\,h_{11;2} + w_c\,h_{33;2} \right), \\ h_{33;3} =&\, -\frac{1}{w_c}\left(w_a\,h_{11;3} + w_b\,h_{22;3} \right). \end{aligned}$$ This concludes the proof. \[cor3:critical point\] Let the gradient terms ${\mathbf{R_w}}$, ${\mathbf{S_w}}$, and ${\mathbf{T_w}}$ be defined as in Lemma \[lem3:critical point\].\ Then we have $$\begin{aligned} {\mathbf{R_w}}(a,b,c,h_{22;1},h_{33;1}) =&\, \binom{h_{22;1}}{h_{33;1}}^\top\, M^{\mathbf{R_w}}(a,b,c)\, \binom{h_{22;1}}{h_{33;1}}, \\ {\mathbf{S_w}}(a,b,c,h_{11;2},h_{33;2}) =&\, \binom{h_{11;2}}{h_{33;2}}^\top\, M^{\mathbf{S_w}}(a,b,c)\, \binom{h_{11;2}}{h_{33;2}}, \\ {\mathbf{T_w}}(a,b,c,h_{11;3},h_{22;3}) =&\, \binom{h_{11;3}}{h_{22;3}}^\top\, M^{\mathbf{T_w}}(a,b,c)\, \binom{h_{11;3}}{h_{22;3}}. \end{aligned}$$ The elements of the matrix $M^{\mathbf{R_w}}(a,b,c)$ are $$\begin{aligned} m_{11}^{\mathbf{R_w}}(a,b,c) =&\, 2\frac{F_a\,w_b-F_b\,w_a}{a-b} \\ &\, + F_{aa} \frac{w_b^2}{w_a} - 2 F_{ab}\,w_b + F_{bb}\,w_a \\ &\, - F_a \left( w_{aa} \frac{w_b^2}{w_a^2} - 2 w_{ab} \frac{w_b}{w_a} + w_{bb} \right), \umbruch \\ m_{12}^{\mathbf{R_w}}(a,b,c) =&\, F_{aa} \frac{w_b\,w_c}{w_a} - F_{ab}\,w_c - F_{ac}\,w_b + F_{bc}\,w_a \\ &\, -F_a \left( w_{aa} \frac{w_b\,w_c}{w_a^2} - w_{ab} \frac{w_c}{w_a} - w_{ac} \frac{w_b}{w_a} + w_{bc} \right), \umbruch \\ m_{22}^{\mathbf{R_w}}(a,b,c) =&\, 2\frac{F_a\,w_c - F_c\,w_a}{a-c} \\ &\, + F_{aa}\,\frac{w_c^2}{w_a} -2\,F_{ac}\,w_c +F_{cc}\,w_a \\ &\, -F_a\left(w_{aa} \frac{w_c^2}{w_a^2} -2 w_{ac}\,\frac{w_c}{w_a} + w_{cc} \right). \end{aligned}$$ The elements of the matrix $M^{\mathbf{S_w}}(a,b,c)$ are $$\begin{aligned} m_{11}^{\mathbf{S_w}}(a,b,c) =&\, 2\frac{F_a\,w_b-F_b\,w_a}{a-b} \\ &\, + F_{aa}\,w_b -2 F_{ab}\,w_a + F_{bb} \frac{w_a^2}{w_b} \\ &\, -F_b \left(w_{aa} - 2 w_{ab} \frac{w_a}{w_b} + w_{bb} \frac{w_a^2}{w_b^2} \right), \umbruch \\ m_{12}^{\mathbf{S_w}}(a,b,c) =&\, -F_{ab}\,w_c + F_{ac}\,w_b + F_{bb} \frac{w_a\,w_c}{w_b} - F_{bc}\,w_a \\ &\, -F_b \left(-w_{ab} \frac{w_c}{w_b} + w_{ac} + w_{bb} \frac{w_a\,w_c}{w_b^2} - w_{bc} \frac{w_a}{w_b} \right), \umbruch \\ m_{22}^{\mathbf{S_w}}(a,b,c) =&\, 2\frac{F_b\,w_c-F_c\,w_b}{b-c} \\ &\, +F_{bb} \frac{w_c^2}{w_b} - 2 F_{bc}\,w_c + F_{cc}\,w_b \\ &\, -F_b \left( w_{bb} \frac{w_c^2}{w_b^2} - 2 w_{bc} \frac{w_c}{w_b} + w_{cc} \right). \end{aligned}$$ The elements of the matrix $M^{\mathbf{T_w}}(a,b,c)$ are $$\begin{aligned} m_{11}^{\mathbf{T_w}}(a,b,c) =&\, 2 \frac{F_a\,w_c-F_c\,w_a}{a-c} \\ &\, +F_{aa}\,w_c - 2 F_{ac}\,w_a + F_{cc} \frac{w_a^2}{w_c} \\ &\, -F_c \left( w_{aa} - 2 w_{ac} \frac{w_a}{w_c} + w_{cc} \frac{w_a^2}{w_c^2} \right), \umbruch \\ m_{12}^{\mathbf{T_w}}(a,b,c) =&\, F_{ab}\,w_c - F_{ac}\,w_b - F_{bc}\,w_a + F_{cc} \frac{w_a\,w_b}{w_c} \\ &\, -F_c\left( w_{ab} - w_{ac} \frac{w_b}{w_c} - w_{bc} \frac{w_a}{w_c} + w_{cc} \frac{w_a\,w_b}{w_c^2} \right), \umbruch \\ m_{22}^{\mathbf{T_w}}(a,b,c) =&\, 2 \frac{F_b\,w_c - F_c\,w_b}{b-c} \\ &\, +F_{bb}\,w_c - 2 F_{bc}\,w_b + F_{cc}\,\frac{w_b^2}{w_c} \\ &\, -F_c\left(w_{bb} - 2 w_{bc} \frac{w_b}{w_c} + w_{cc} \frac{w_b^2}{w_c^2} \right). \end{aligned}$$ We use identities to replace $h_{11;1}$, $h_{22;2}$, and $h_{33;3}$ in ${\mathbf{R_w}}$, ${\mathbf{S_w}}$, and ${\mathbf{T_w}}$ from Lemma \[lem3:critical point\], respectively. Now we rewrite the quadratic forms ${\mathbf{R_w}}$, ${\mathbf{S_w}}$, and ${\mathbf{T_w}}$ as ${\mathbf{x}}^\top\,M\,{\mathbf{x}}$. This concludes the proof. \[rem3:sufficient conditions\] Under the assumptions of Lemma \[lem3:critical point\], the linear operator $Lw$ is non-positive at some critical point of $w$, if ${\mathbf{C_w}}$, ${\mathbf{E_w}}$ are non-positive, and $M^{\mathbf{R_w}}$, $M^{\mathbf{S_w}}$, and $M^{\mathbf{T_w}}$ are negative semi-definite there. This is a direct consequence of Lemma \[lem3:critical point\], and Corollary \[cor3:critical point\].\ Now let $M \in \R^{2\times 2}$ be a symmetric matrix. Then we have the equivalent conditions 1. $M=\begin{pmatrix} m_{11} & m_{12} \\ m_{12} & m_{22} \end{pmatrix}$ is negative semi-definite, 2. $\operatorname{tr}M=m_{11}+m_{22} \leq 0$, and $-\det M=m_{12}^2-m_{11}\,m_{22} \leq 0$. $\left. \right.$\ In $[\textsc{cp}]$, we check the non-positivity of the linear operator $Lw$ by checking the non-positivity of ${\mathbf{C_w}}$, ${\mathbf{E_w}}$, and by checking condition $(2)$ for the matrices $M^{\mathbf{R_w}}$, $M^{\mathbf{S_w}}$, and $M^{\mathbf{T_w}}$. Vanishing functions =================== In $\R^{n+1}$, $n=2$, for many normal velocities $F$ the quantity $$\begin{aligned} \frac{\left(\lambda_1-\lambda_2\right)^2}{\left(\lambda_1\,\lambda_2\right)^2} F^2\end{aligned}$$ seems to be the natural choice when showing convergence to a round point. As in [@mf:when], we call this quantity a *vanishing function* for a normal velocity $F$. It is used by B. Andrews for the Gauss curvature flow [@ba:gauss], by F. Schulze and O. Schnürer for the $H^\sigma$-flow [@fs:convexity], by B. Andrews and X. Chen for the $|A|^\sigma$ and the $\operatorname{tr}A^\sigma$-flow [@ac:surfaces]. First we give the Definition \[def3:vanishing function\] of a vanishing function in $\R^{n+1}$, $n=3$. In Remark \[rem3:vanishing function\] we then introduce $$\begin{aligned} \sum_{i<j} \frac{\left(\lambda_i-\lambda_j\right)^2}{\left(\lambda_i\,\lambda_j\right)^2} F^2\end{aligned}$$ as the counterpart of a vanishing function for arbitrary dimensions $n\in\N$. In this paper we work with the quantity in particular in $\R^{n+1}$, $n=3$. In Lemma \[lem3:vanishing function\] we deduce a simple but interesting estimate for vanishing functions for arbitrary dimensions $n\in\N$. We employ this Lemma \[lem3:vanishing function\] in the proof of our main Theorem \[thm3:H3\]. \[def3:vanishing function\] Let $v(a,b,c)\in C^2\left(\R^3_{+}\right)$ with $v\not\equiv 0$. Let ${\mathbf{C_w}}(a,b,c)$ be defined as in Lemma \[lem3:critical point\]. We call $v$ a *vanishing function* for a normal velocity $F$ if ${\mathbf{C_v}}(a,b,c) = 0$ for all $0<a,b,c$. \[exm3:vanishing function\] We have the following example of a vanishing function for a normal velocity $F$: $$\begin{aligned} \left( \frac{(a-b)^2}{(a\,b)^2} + \frac{(a-c)^2}{(a\,c)^2} + \frac{(b-c)^2}{(b\,c)^2} \right) F^2. \end{aligned}$$ \[rem3:vanishing function\] We can define a *vanishing function* for arbitrary dimensions $n\in\N$. Let $\lambda_i$, $i=1,\ldots,n$, denote the principal curvatures of a hypersurface in $\R^{n+1}$. Using Lemma \[lem3:linear operator\] we can define constant terms ${\mathbf{C_w}}(\lambda_1,\ldots,\lambda_n)$ as in Lemma \[lem3:critical point\] for an arbitrary $n\in\N$. We have the following example of a vanishing function: $$\begin{aligned} \label{id3:vanishing function} \sum_{i<j} \frac{\left(\lambda_i-\lambda_j\right)^2}{\left(\lambda_i\,\lambda_j\right)^2} F^2. \end{aligned}$$ Interestingly, we still obtain a vanishing function if we omit up to $n-1$ terms of the form $$\begin{aligned} \frac{\left(\lambda_i-\lambda_j\right)^2}{\left(\lambda_i\,\lambda_j\right)^2} F^2. \end{aligned}$$ This reminds us of [@hs:mean Theorem 1.5] by G. Huisken and C. Sinestrari. \[lem3:vanishing function\] Let $v$ be a vanishing function as defined in Remark \[rem3:vanishing function\]. Let $v\leq C^2$ on some set $\Sc\subset\R^n_{+}$, and for some constant $C>0$.\ Then we have $$\begin{aligned} 1 \leq \frac{\lambda_{\text{max}}}{\lambda_{\text{min}}} \leq 1 + C \frac{\lambda_{\text{max}}}{F} \qquad \text{on}\;\;\Sc. \end{aligned}$$ We assume $\lambda_{\text{min}} \equiv \lambda_1 \leq \ldots \leq \lambda_n \equiv \lambda_{\max}$ and obtain $$\begin{aligned} C^2 \geq \sum_{i<j} \frac{\left(\lambda_i-\lambda_j\right)^2}{\left(\lambda_i\,\lambda_j\right)^2} F^2 \geq \frac{\left(\lambda_n-\lambda_1\right)^2}{\left(\lambda_1\,\lambda_n\right)^2} F^2, \end{aligned}$$ which implies $$\begin{aligned} C \frac{\lambda_n}{F} \geq \frac{\lambda_n\left(\lambda_n-\lambda_1\right)}{\lambda_1\,\lambda_n} = \frac{\lambda_n}{\lambda_1}-1\geq 0. \end{aligned}$$ This concludes the proof. $H^3$-flow ========== The proof of our main Theorem \[thm3:H3\] is based on investigating $$\begin{aligned} \phi_{H^3} =&\, \frac{(a-b)^2+(a-c)^2+(b-c)^2}{(a+b+c)^2}, \\ \text{and }\quad \psi_{H^3} =&\, \left( \frac{(a-b)^2}{(a\,b)^2} + \frac{(a-c)^2}{(a\,c)^2} + \frac{(b-c)^2}{(b\,c)^2} \right) \left(H^3\right)^2.\end{aligned}$$ The quantity $\phi_{H^3}$ is inspired by the quantity used in [@gh:flow] by G. Huisken. The other quantity $\psi_{H^3}$ is a vanishing function. First we show that the estimate $\phi_{H^3}\leq h := 1/8$ is preserved during the $H^3$-flow if the initial hypersurface $M_0$ is $2$-pinched. This is Lemma \[lem3:phi\] and Corollary \[cor3:phi\]. Next we prove that $\psi_{H^3}$ is bounded in time on the set where $\phi_{H^3}\leq h$. This is Lemma \[lem3:psi\] and Corollary \[cor3:psi\]. The proofs of these Lemmas and Corollaries involve the maximum-principle, the linear operator $L$ and our computer program $[\textsc{cp}]$. In $[\textsc{cp}]$ we deal with computations of two kinds. One kind is the purely algebraic manipulation of terms, and could still be performed by pen and paper. The other kind of computations includes random numbers for a Monte-Carlo method, which appears to be very tedious to carry out with pen and paper. Finally, we show convergence to a round point combining the boundedness of $\psi_{H^3}$ and the proof of [@fs:convexity Theorem A.1.] by F. Schulze and O. Schnürer. This is our main Theorem \[thm3:H3\]. \[lem3:phi\] Let $\left(M_t\right)_{0\leq t < T}$ be a maximal solution of the $H^3$-flow, where $M_0$ is $2$-pinched. Then we have $$\begin{aligned} L\phi \leq 0 \end{aligned}$$ at a critical point of $\phi$, where $0<\phi \leq 1/8 =: h$. Let $\Sc_{C_2}$ be the $2$-pinched cone in the positive orthant. In $[\textsc{cp}]$, we compute $$\begin{aligned} \Sc_{C_2} :=&\, \{(\lambda_1,\lambda_2,\lambda_3) \in \R^3_{+} : \lambda_i/\lambda_j \leq 2 \text{ for all } 1\leq i,j\leq 3 \} \\ \Sc_h :=&\, \{(a,b,c) \in \R^3_{+} : 0 <\phi \leq 1/8 = h \}, \\ \Sc_{L\phi} :=&\, \{(a,b,c) \in \R^3_{+} : L\phi \leq 0 \} \end{aligned}$$ and show that $$\begin{aligned} \Sc_{C_2} \subset \Sc_h \subset \Sc_{L\phi} \end{aligned}$$ using a Monte-Carlo method. Here, we check the non-positivity of $L\phi$ as described in Remark \[rem3:sufficient conditions\].\ Since the functions $\lambda_i/\lambda_j$, $\phi$, and $L\phi$ are homogeneous in the principal curvatures, it suffices to compute the sets $\Sc_{C_2}$, $\Sc_h$, and $\Sc_{L\phi}$ in $[\textsc{cp}]$ for the radial projection $$\begin{aligned} \pi: \R^3_{+} \to \{a+b+c=1\},\quad(a,b,c) \mapsto (a,b,c)/(a+b+c). \end{aligned}$$ This concludes the computer-based proof. \[cor3:phi\] Let $\left(M_t\right)_{0\leq t < T}$ be a maximal solution of the $H^3$-flow, where $M_0$ is $2$-pinched. Then we have that $$\begin{aligned} \phi \leq h := 1/8 \end{aligned}$$ during the $H^3$-flow. This follows directly from Lemma \[lem3:phi\] using the maximum-principle. \[lem3:psi\] Let $\left(M_t\right)_{0\leq t < T}$ be a maximal solution of the $H^3$-flow, where $M_0$ is $2$-pinched. Then we have $$\begin{aligned} L\psi \leq 0 \end{aligned}$$ at a critical point of $\psi$, where $\psi>0$. By Corollary \[cor3:phi\], we have $$\begin{aligned} \phi \leq h \end{aligned}$$ during the $H^3$-flow. Let $\Sc_{C_2}$, $\Sc_h$ be defined as in Lemma \[lem3:phi\]. In $[\textsc{cp}]$, we also compute $$\begin{aligned} \Sc_{L\psi} := \{(a,b,c) \in \R^3_{+} : L\psi \leq 0 \} \end{aligned}$$ and show in particular the second inclusion of $$\begin{aligned} \Sc_{C_2} \subset \Sc_h \subset \Sc_{L\psi} \end{aligned}$$ using a Monte-Carlo method. By Lemma \[lem3:phi\], we have the first conclusion. This concludes the computer-based proof. \[cor3:psi\] Let $\left(M_t\right)_{0\leq t < T}$ be a maximal solution of the $H^3$-flow, where $M_0$ is $2$-pinched. Then we have that $$\begin{aligned} \max_{M_t}\,\psi \end{aligned}$$ is non-increasing during the $H^3$-flow. This follows directly from Lemma \[lem3:psi\] using the maximum-principle. \[thm3:H3\] Let $\left(M_t\right)_{0\leq t < T}$ be a maximal solution of the $H^3$-flow, where $M_0$ is $2$-pinched. Then $\left(M_t\right)_{0\leq t < T}$ converges to a round point. We closely follow proof of the corresponding [@fs:convexity Theorem A.1.] by F. Schulze and O. Schnürer.\ By [@fs:convexity Theorem 1.1] the surfaces $M_t$ become immediately strictly convex for $t>0$. Now choose a sufficiently small $0<\epsilon<T$ such that the $H^3$-flow is smooth and strictly convex on the interval $(\epsilon,T)$. Thus the quantity $\psi_{H^3}$ is well-defined on this interval, and bounded from above by Corollary \[cor3:psi\]. By Lemma \[lem3:vanishing function\] this implies $$\begin{aligned} 1 \leq \frac{\lambda_{\text{max}}}{\lambda_{\text{min}}} \leq 1 + \frac{C}{H^2} \end{aligned}$$ on $\left(M_t\right)_{(\epsilon<t<T)}$. Now the proof follows analogously to the proof of [@fs:convexity Theorem 1.2]. $|A|^2$-flow ============ A result similar to our main Theorem \[thm3:H3\] holds for the normal velocity $F=|A|^2$ and $3$-pinched hypersurfaces. For a proof consider $$\begin{aligned} \phi_{|A|^2} =&\, \frac{(a^2+b^2+c^2)(a\,b+a\,c+b\,c)^2}{(a\,b\,c)^2} \\ \text{and}\quad \psi_{|A|^2} =&\, \frac{(a+b+c)^2\left( (a-b)^2 + (a-c)^2 + (b-c)^2 \right)}{a\,b\,c},\end{aligned}$$ and O. Schnürer [@os:surfacesA2]. As in chapter on $H^3$-flow using $[\textsc{cp}]$ we obtain \[lem3:A2\] Let $\left(M_t\right)_{0\leq t < T}$ be a maximal solution of the $|A|^2$-flow, where $M_0$ is $3$-pinched. Then we have that $$\begin{aligned} \max_{M_t}\,\psi_{|A|^2} \end{aligned}$$ is non-increasing in time. Gauss curvature flow ==================== A result similar to our main Theorem \[thm3:H3\] holds for the normal velocity $F=K$ and $2$-pinched hypersurfaces. For a proof consider $$\begin{aligned} \phi_K =&\, \frac{(a-b)^2 + (a-c)^2 + (b-c)^2}{a^2+b^2+c^2} \umbruch \\ \text{and}\quad \psi_K =&\, \left( \frac{(a-b)^2}{(a\,b)^2} + \frac{(a-c)^2}{(a\,c)^2} + \frac{(b-c)^2}{(b\,c)^2} \right) \left(K\right)^2,\end{aligned}$$ and B. Chow [@bc:deforming]. As in chapter on $H^3$-flow using $[\textsc{cp}]$ we obtain \[lem3:K\] Let $\left(M_t\right)_{0\leq t < T}$ be a maximal solution of the Gauss curvature flow, where $M_0$ is $2$-pinched. Then we have that $$\begin{aligned} \max_{M_t}\,\psi_K \end{aligned}$$ is non-increasing in time. Outlook ======= Our aim is to show convergence to a round point without pinching requirements using vanishing functions in arbitrary dimensions. Instead of splitting the linear operator $L$ into constant terms and gradient terms we intend to work with integral estimates similar to G. Huisken [@gh:flow]. This way we seek to prove convergence to a round point for contracting normal velocities, including powers of the Gauss curvature, $K$, of the mean curvature, $H$, and of the norm of the second fundamental form, $|A|$. Appendix ======== Some of the Lemmas leading up to the proof our main Theorem \[thm3:H3\] rely on the computer program $[\textsc{cp}]$. First we compute the linear operator $L$ for the corresponding quantities $\phi$ and $\psi$. Next we use a Monte-Carlo method to compute the sets $\Sc_{C_2}$, $\Sc_h$, $\Sc_{L\phi}$, and $\Sc_{L\psi}$. Finally, we compute the two inclusions $$\begin{aligned} \label{id:inclusions} \begin{split} \Sc_{C_2} \subset&\, \Sc_h \subset \Sc_{L\phi}, \\ \Sc_{C_2} \subset&\, \Sc_h \subset \Sc_{L\psi}. \end{split}\end{aligned}$$ For the convenience of the reader, we include the source code of $[\textsc{cp}]$ in three different programming languages, namely for the computer algebra systems Mathematica, Sage, and Maple. The first part of the appendix is the Mathematica program, the second part is the Sage program, and the third part is the Maple program. In the first part we also visualize the two inclusions .\ At [www.arxiv.org](www.arxiv.org) we can only submit this article without the computer program $[\textsc{cp}]$. To download this article with the computer program $[\textsc{cp}]$ please go to [www.martinfranzen.de](www.martinfranzen.de). \#1 \[2\][ [\#2](http://www.ams.org/mathscinet-getitem?mr=#1) ]{} \[2\][\#2]{} [10]{} Ben Andrews, *Gauss curvature flow: the fate of the rolling stones*, Invent. math. **138** (1999), 151–-161. Ben Andrews, Xuzhong Chen, *Surfaces moving by powers of Gauss curvature*, [arXiv:1111.4616 \[math.DG\]]{}. Ben Andrews, James McCoy, *Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature*, Transactions of the American Mathematical Society (2012), no. 7, 3427–3447. Bennett Chow, *Deforming convex hypersurfaces by the nth root of the Gaussian curvature*, J. Differential Geom. **22** (1985), no. 1, 117–138. Martin Franzen, *When maximum-principle functions cease to exist*, [arXiv:1501.07259 \[math.DG\]]{}. Claus Gerhardt, *Curvature Problems*, Series in Geometry and Topology, vol. 39, International Press, Somerville, MA, 2006. Gerhard Huisken, *Flow by mean curvature of convex surfaces into spheres*, J. Differential Geom. **20** (1984), no. 1, 237-266. Gerhard Huisken, Carlos Sinestrari, *Mean curvature flow with surgeries of two-convex hypersurfaces*, Inventiones mathematicae **175** (2009), 137-221. Oliver C. Schnürer, *Surfaces contracting with speed $\vert A \vert^2$*, J. Differential Geom. **71** (2005), no. 3, 347-363. Felix Schulze, *Convexity estimates for flows by powers of the mean curvature, appendix with O. C. Schnürer*, Ann. Scuol Norm. Sup Pisa Cl. Sci. **(5)**, Vol. V (2006), 261-277. [^1]: We would like to thank F. Kuhl, B. Lambert, M. Langford, M. Makowski, O. Schnürer, [^2]: W. Stein, B. Stekeler, and M. Westerholt-Raum for discussions and support. [^3]: The author is a member of the DFG priority program SPP 1489.

--- abstract: 'We study the performance of holonomic quantum gates, driven by lasers, under the effect of a dissipative environment modeled as a thermal bath of oscillators. We show how to enhance the performance of the gates by suitable choice of the loop in the manifold of the controllable parameters of the laser. For a simplified, albeit realistic model, we find the surprising result that for a long time evolution the performance of the gate (properly estimated in terms of average fidelity) increases. On the basis of this result, we compare holonomic gates with the so-called Stimulated Raman adiabatic passage (STIRAP) gates.' author: - 'Daniele Parodi,$^{1,2}$ Maura Sassetti,$^{1,3}$ Paolo Solinas,$^{4}$ and Nino Zanghì$^{1,2}$' title: Environmental noise reduction for holonomic quantum gates --- Introduction ============ The major challenge for quantum computation is posed by the fact that generically quantum states are very delicate objects quite difficult to control with the required accuracy—typically, by means of external driving fields, e.g., a laser. The interaction with the many degrees of freedom of the environment causes decoherence; moreover, errors in processing the information may lead to a wrong output state. Among the approaches aiming at overcoming these difficulties are those for which the quantum gate depends very weakly on the details of the dynamics, in particular, the holonomic quantum computation (HQC) [@HQC] and the so-called Stimulated Raman adiabatic passage (STIRAP) [@Kis; @troiani-molinari; @roszak]. In the latter, the gate operator is obtained acting on the phase difference of the driving lasers during the evolution, while in the former the same goal is achieved by exploiting the non-commutative analogue of the Berry phase collected by a quantum state during a cyclic evolution. Concrete proposals have been put forward, for both Abelian [@Jones; @Falci] and non-Abelian holonomies [@HQC_proposal; @HQC_proposal1; @HQC_proposal2; @HQC_proposal3; @HQC_proposal4; @paper1-2]. The main advantage of the HQC is the robustness against noise deriving from a imperfect control of the driving fields [@par_noise; @par_noise1; @par_noise2; @par_noise3; @par_noise4; @par_noise5; @florio; @fuentes]. In a recent paper [@hqc_noise] we have shown that the disturbance of the environment on holonomic gates can be suppressed and the performance of the gate optimized for particular environments (purely superohmic thermal bath). In the present paper we consider a different sort of optimization, which is independent of the particular nature of the environment. By exploiting the full geometrical structure of HQC, we show how the performance of a holonomic gate can be enhanced by a suitable choice of the loop in the manifold of the parameters of the external driving field: by choosing the optimal loop which minimizes the “error” (properly estimated in terms of average fidelity loss). Our result is based on the observation that there are different loops in the parameter manifold producing the same gate and, since decoherence and dissipation crucially depend on the dynamics, it is possible to drive the system over trajectories which are less perturbed by the noise. For a simplified, albeit realistic model, we find the surprising result that the error decreases linearly as the gating time increases. Thus the disturbance of the environment can be drastically reduced. On the basis of this result, we compare holonomic gates with the STIRAP gates. In Sec. II the model is introduced and the explicit expression of the error is derived. In Sec. III we find the optimal loop, calculate the error, make a comparison with other approaches, and briefly sketch how to treat a different coupling with the environment. Model ===== The physical model is given by three degenerate (or quasidegenerate) states, $|+\rangle$, $|-\rangle$, and $|0\rangle$, optically connected to another state $|G\rangle$. The system is driven by lasers with different frequencies and polarizations, acting selectively on the degenerate states. This model describes various quantum systems interacting with a laser radiation, ranging from semiconductor quantum dots, such as excitons [@paper1-2] and spin-degenerate electron states [@troiani-molinari], to trapped ions [@HQC_proposal1] or neutral atoms [@HQC_proposal]. The (approximate) Hamiltonian modeling the effect of the laser on the system is (for simplicity, $\hbar=1$) [@HQC_proposal1; @paper1-2] $$\label{eq:system_hamiltonian} H_0(t)= \sum_{j=+,-,0} \big[ \epsilon |j\rangle \langle j| + (e^{-i \epsilon t} \Omega_j(t) |j\rangle \langle G| + H.c)\big] \mathbf{,}$$ where $\Omega_j(t)$ are the timedependent Rabi frequencies depending on controllable parameters, such as the phase and intensity of the lasers, and $\epsilon$ is the energy of the degenerate electron states. The Rabi frequencies are modulated within the adiabatic time $t_{ad}$, (which coincides with the gating time), to produce a loop in the parameter space and thereby realize the periodic condition $H_0(t_{ad})=H_0(0)$. The Hamiltonian (\[eq:system\_hamiltonian\]) has four time dependent eigenstates: two eigenstates $|E_{i}(t)\rangle$ , $i=1,2$, called bright states, and two eigenstates $|E_i(t)\rangle$, $i=3,4$, called dark states. The two dark states have degenerate eigenvalue $\epsilon$ and the two bright states have timedependent energies $\lambda_\pm(t) = [\epsilon \pm \sqrt{\epsilon^2 + 4 \Omega^2(t)}]/2$ with $\Omega^2(t)= \sum_{i=\pm,0} |\Omega_i(t)|^2$ [@dark-bright]. The evolution of the state is generated by $$\label{eq:U_t} U_t=T e^{-i\int^t_0 dt'H_0(t')},$$ where $T$ is the time-ordered operator. In the adiabatic approximation, the evolution of the state takes place in the degenerate subspace generated by $|+ \rangle$, $|- \rangle$, and $|0 \rangle$. This approximation allows to separate the dynamic contribution and the geometric contribution from the evolution operator. Expanding $U_t$ in the basis of instantaneous eigenstates of $H_0(t)$ (the bright and dark states), in the adiabatic approximation, we have $$U_t\cong \sum_j e^{-i\int^t_0 E_j(t')dt'}|E_j(t){\rangle}{\langle}E_j(t)|~\mathcal{U}_t, \label{eq:hol_operator}$$ where $$\mathcal{U}_t=Te^{\int_0^t d \tau V(\tau)}, \label{eq:hol_operator}$$ here $V$ is the operator with matrix elements $V_{ij}(t) = \langle E_i(t)| \partial_t | E_j(t) \rangle$. The unitary operator $\mathcal{U}_t$ plays the role of timedependent holonomic operator and is the fundamental ingredient for realizing complex geometric transformation whereas $ \sum_j e^{-i\int^t_0 E_j(t')dt'}|E_j(t){\rangle}{\langle}E_j(t)|$ is the dynamic contribution. Consider $\mathcal{U}_t$ for a closed loop, i.e., for $t=t_{ad}$, $$\label{eq:log_operator} \mathcal{U}=\mathcal{U}_{t_{ad}}.$$ If the initial state $|\psi_0{\rangle}$ is a superposition of $|+{\rangle}$ and $|-{\rangle}$, then $\mathcal{U}|\psi_0{\rangle}$ is still a superposition of the same vectors (in general, with different coefficients)[@paper1-2]. Thus the space spanned by $|+{\rangle}$ and $|-{\rangle}$ can be regarded as the “logical space” on which the “logical operator” $\mathcal{U}$ acts as a “quantum gate” operator. Note that for $t<t_{ad}$, $\mathcal{U}_t|\psi_0{\rangle}$ has, in general, also a component along $|0{\rangle}$. However, as it is easy to show [@dark-bright], at any instant $t<t_{ad}$, $\mathcal{U}_t|\psi_0{\rangle}$ can be expanded in the twodimensional space spanned by the dark states $|E_3(t){\rangle}$ and $|E_4(t){\rangle}$. It is important to observe that $\mathcal{U}$ depends only on global geometric features of the path in the parameter manifold and not on the details of the dynamical evolution [@HQC; @paper1-2]. To construct a complete set of holonomic quantum gates, it is sufficient to restrict the Rabi frequencies $\Omega_j (t)$ in such a way that the norm $\Omega$ of the vector $\vec{\Omega}=[\Omega_0 (t),\Omega_+ (t),\Omega_- (t)]$ is time independent and the vector lies on a real three dimensional sphere [@HQC_proposal1; @paper1-2]. We parametrize the evolution on this sphere as $\Omega_+(t)= \sin ~\theta(t) \cos ~\phi(t)$, $\Omega_-(t)= \sin ~\theta(t) \sin ~\phi(t)$ and $\Omega_0(t)= \cos ~\theta(t)$ with fixed initial (and final) point in $\theta(0)=0$, the north pole By straightforward calculation we obtain the analytical expression for $V(t)$ in eq. (\[eq:hol\_operator\]), $V(t)=i \sigma_y ~cos[\theta(t)]\dot{\phi}(t)$, where $\sigma_y$ is the usual Pauli matrix written in the basis of dark states. Thus, the operator (\[eq:hol\_operator\]) becomes $\mathcal{U}_t=\cos [a(t)] -i \sigma_y ~\sin [a(t)]$, here $a(t) = \int_0^{t} d\tau \dot{\phi}(\tau) \cos ~\theta(\tau)$. Accordingly, the logical operator $\mathcal{U}$ (\[eq:log\_operator\]) is $$\mathcal{U}=\cos~a -i \sigma_y \sin~a,$$ where $$a=a(t_{ad})=\int_0^{t_{ad}} d\tau \dot{\phi}(\tau) \cos ~\theta(\tau) \label{eq:solid_angle}$$ is the solid angle spanned on the sphere during the evolution. Note that the are many paths on the sphere which generate the same logical operator $\mathcal{U}$, and span the same solid angle $a$. In a previous work we have studied how interaction with the environment disturbs the logical operator $\mathcal{U}$ [@hqc_noise]. The goal of the present paper is to analyze whether and how such a disturbance can be minimized for a given $\mathcal{U}$. To this end, we model the environment as a thermal bath of harmonic oscillators with linear coupling between system and environment [@caldeira-leggett]. The total Hamiltonian is then $$\label{eq:bagno} H = H_0(t) + \sum_{\alpha=1}^N (\frac{p^2_{\alpha}}{2 m_{\alpha}} + \frac{1}{2} m_{\alpha} \omega_{\alpha}^2 x_{\alpha}^2 + c_{\alpha} x_{\alpha} A),$$ where $A$ is the system interaction operator called, from now on, noise operator. We now consider the time evolution of the reduced density matrix of the system, determined by the Hamiltonian (\[eq:bagno\]). We rely on the standard methods of the “master equation approach,” with the environment treated in the Born approximation and assumed to be at each time in its own thermal equilibrium state at temperature $T$. This allows to include the effect of the environment in the correlation function ($k_B=1$) $$g(\tau) = \int^\infty_0 J(\omega) \bigg[ \coth\bigg(\frac{\omega}{2 T}\bigg) \cos(\omega \tau) - i~ \sin(\omega \tau)\bigg] d\omega. \label{eq:autocorrelation}$$ Here the spectral density is $$J(\omega)=\frac{\pi}{2}\sum_{\alpha=1}^{N}\frac{c^2_{\alpha}}{m_{\alpha}\omega_{\alpha}} \delta(\omega-\omega_{\alpha}),$$ at the low frequencies regimes, is proportional to $\omega^s$, with $s\geq 0$, i.e., $s=1$ describes a Ohmic environments, typical of baths of conduction electrons, $s=3$ describes a super-Ohmic environment, typical of baths of phonons [@weiss; @hqc_noise]. The asymptotic decay of the real part of $g(\tau)$ defines the characteristic memory time of the environment. Denoting with $\tilde{\rho}(t)$ the time evolution of the reduced density matrix of the system in the interaction picture, e.g., $\tilde{\rho}(t) = U_t^{\dagger} \rho U_t$, one has [@weiss] [$$\begin{aligned} \tilde{\rho}(t_{ad})& =&\rho(0)+ \nonumber \\ &-& i\int_0^{t_{ad}} dt\int_0^t d\tau \{ g(\tau) [ \tilde{A} \tilde{A}^\prime \tilde{\rho}(t-\tau) - \tilde{A}^\prime \tilde{\rho}(t-\tau) \tilde{A} ] \nonumber \\ &+& g(-\tau) [ \tilde{\rho}(t-\tau) \tilde{A}^\prime \tilde{A} - \tilde{A} \tilde{\rho}(t-\tau) \tilde{A}^\prime ]. \label{eq:non_markov_{max}_eq}\end{aligned}$$ ]{} Here $\tilde{A}$ and $\tilde{A}^\prime$ stand for $\tilde{A}(t)$ and $\tilde{A}(t-\tau)$, with the tilde denoting the time evolution in the interaction picture. In quantum information the quality of a gate is usually evaluated by the fidelity $\mathcal{F}$, which measures the closeness between the unperturbed state and the final state, $$\label{eq:fidelity} \mathcal{F}={\langle}\psi_0(0)|\mathcal{U}^{\dag}\rho (t_{ad})\mathcal{U}|\psi_0(0){\rangle},$$ where $|\psi_0(0){\rangle}$ is the initial state, and $\rho(t_{ad})=\mathcal{U}\tilde{\rho}(t_{ad})\mathcal{U}^{\dag}$ is the reduced density matrix in the Schrödinger picture starting from the initial condition $\rho(0)=|\psi_0(0){\rangle}{\langle}\psi_0(0)|$. The average error is defined as the average fidelity loss, i.e., $$\label{eq:def_delta} \delta=<1- \mathcal{F}> = 1-<{\langle}\psi_0(0)|\tilde{\rho}(t_{ad})|\psi_0(0){\rangle}>,$$ where $<\cdots>$ denotes averaging with respect to the uniform distribution over the initial state $|\psi_0(0){\rangle}$. The right-handside of Eq. (\[eq:def\_delta\]) can be computed by the following steps: (1) solving Eq. (\[eq:non\_markov\_[max]{}\_eq\]) in strictly second order approximation; this approximation corresponds to replace $\tilde{\rho}(t-\tau)$ with $\rho(0)$; \(2) using the adiabatic approximation $U(t-\tau,t) \approx \exp(i \tau H_0(t))$; \(3) expanding the scalar product in Eq. (\[eq:def\_delta\]) with respect to a complete orthonormal basis $\{|\varphi_n(t){\rangle}\}$, $n=1,2,3$, orthogonal to $|\psi_0(t){\rangle}$. In this way, one obtains $$\delta = \Bigg< \sum_{n=1}^{3} \int_0^{t_{ad}} d t~ G(t) |\langle \psi_0(t) | A |\varphi_n(t) \rangle|^2 \Bigg>, \label{eq:error}$$ where $$G(t) = \int_0^t d \tau \big\{ \mbox{Re}[g(\tau)] \cos(\omega_{0n} t) + \mbox{Im}[g(\tau)] \sin(\omega_{0n} t))\big\}. \label{eq:G_t}$$ Here, $\omega_{0n}=\omega_0-\omega_n$ are the energy differences associated to the transition $\psi_0 \leftrightarrow \phi_n$, with $\omega_0=\epsilon$, $\omega_1=\lambda_+$, $\omega_2=\lambda_-$, and $\omega_3=\epsilon$. The interaction between system and environment is expressed by the noise operator $A$ in Eq. (\[eq:bagno\]). We shall now make the assumption that $A=\mbox{diag}\{0,0,0,1 \}$ in the $| G\rangle$, $| \pm \rangle$, and $| 0 \rangle$ basis. In this case the transition between degenerate states are forbidden, however the noise breaks their degeneracy, shifting one of them. In spite of its simple form, this $A$ is nevertheless a realistic noise operator for physical semiconductor systems [@roszak]. Minimizing the error ==================== The problem can be stated in the following way: given the noise operator $A$ and the logical operator $\mathcal{U}$, find a path on the parameter space (the surface of the sphere, described above) which minimizes the error $\delta$. The total error $\delta$, given by Eq. (\[eq:error\]), can be decomposed as $$\delta=\delta_{tr}+\delta_{pd},$$ where the transition error, $\delta_{tr}$, is the contribution to the sum of the nondegenerate states ($\omega_{0n}\neq0$) and the pure dephasing error $\delta_{pd}$ is the contribution of the degenerate states ($\omega_{0n}=0$). Thus $$\begin{aligned} \label{eq:errore_PD} \delta_{pd}&=&\frac{\pi}{8}\int_0^{t_{ad}}dt \int_0^{\infty} d\omega \frac{J(\omega)}{\omega} \coth\bigg(\frac{\omega}{2 T}\bigg) \nonumber \\ && \sin(\omega t) \bigg(1+\frac{1}{2}\sin^2 ~2a(t)\bigg)\sin^4~\theta(t) \end{aligned}$$ and $$\label{eq:errore_tr} \delta_{tr}=\sum_{n=+,-}\frac{1}{8\sqrt{1+[(\lambda_n-\epsilon)/\Omega}]^2}\Gamma_{0n}\int_0^{t_{ad}}\sin^2 ~2\theta(t) dt ,$$ where $$\Gamma_{0n} = J(|\omega_{0n}|) \bigg[\coth \left( \frac{|\omega_{0n}|}{2 T} \right) - \textrm{sgn}(\omega_{0n})\bigg]$$ correspond to the transition rates calculated by standard Fermi golden rules, supposing, as usual, $G(t)\approx G(\infty)$ for $g(\tau)$ strongly peaked around $\tau=0$. In the following we define for simplicity $$K=\sum_{n=+,-}\frac{1}{8\sqrt{1+[(\lambda_n-\epsilon)/\Omega}]^2}\Gamma_{0n}.$$ Since we are interested at long time evolution, we start discussing the transition error which dominates in this regime [@roszak; @alicki]. [![\[fig:d\_theta\] The error $\delta_{tr}$ versus $\theta_M$ for two different $a$ values: $a=\pi/2$ (dashed line) and $a=\pi/4$ (full line) correspond to NOT and Hadamard gate, respectively. ](figure1c.eps "fig:"){height="4.6cm"}]{} Transition rate --------------- As explained in Sec. \[model\], the holonomic paths are closed curves on the surface of the sphere which start from the north pole. It turns out that the curve minimizing $\delta_{tr}$ can be found among the loops which are composed by a simple sequence of three paths (see the Appendix): evolution along a meridian ($\phi=\mbox{const}$), evolution along a parallel ($\theta=\mbox{const}$) and a final evolution along a meridian to come back to the north pole. The error $\delta_{tr}$ in (\[eq:errore\_tr\]), depends on $a$ given by Eq. (\[eq:solid\_angle\]), $\theta_M$ (the maximum angle spanned during the evolution along the meridian), $\Delta \phi$ (the angle spanned along the parallel), and angular velocity $v$. We allow $\Delta \phi \geq 2 \pi$ which corresponds to cover more than one loop along the parallel. The velocity along the parallel is $v(t)=\dot{\phi}(t) \sin~\theta$ and that along the meridian is $v(t)= \dot{\theta}(t)$. In the following we assume that $v$ is constant, and it cannot exceed the maximal value of $v_{\mbox{max}}$, fixed by adiabatic condition $v_{\mbox{max}}\ll \Omega$. The parameters $a$, $\theta_M$, and $\Delta\phi$ are connected by the relation $a=\Delta \phi(1-\cos~\theta_M)$. The error $\delta_{tr}$ is then $$\delta_{tr}= \delta^M_{tr}+ \delta^P_{tr}, \label{eq:f}$$ where $$\delta^M_{tr}=\frac{K}{v}\bigg(\theta_M-\frac{1}{4}\sin 4\theta_M\bigg) \label{eq:fM}$$ is the contribution along the meridian and $$\delta^P_{tr}=K\frac{a}{v}\frac{\sin~\theta_M~\sin^2 ~2\theta_M}{1-\cos~\theta_M} \label{eq:fP}$$ is the contribution along the parallel. In Fig. \[fig:d\_theta\] $\delta_{tr}$ is plotted for $a=\pi/2$ and $a=\pi/4$ (corresponding to NOT and Hadamard gate, respectively) as a function of $\theta_M$. One can see that $\delta_{tr}$ has a local minimum for $\theta_M=\pi/2$ and a global minimum for $\theta_M=0$ where the error vanishes. This suggests that the best choice is to take $\theta_M$ as small as possible. It is interesting to consider the dependence of $\delta_{tr}$ also on the evolution time $t_{ad}$. For simplicity, we set the velocity $v=v_{\mbox{max}}$. In this case, changing $\theta_M$ (and then $\Delta \phi$) corresponds to a change in the evolution time. We obtain $$\theta_M=\arccos \bigg( 1-\frac{a}{2\pi m} \bigg), \label{eq:theta-t}$$ where $$m=\frac{1}{4\pi a}\big[(v_{\mbox{max}} t_{ad})^2+a^2\big]. \label{eq:m-t}$$ Using these relations, $\delta^M_{tr}$ and $\delta^P_{tr}$, given by (\[eq:fM\]) and (\[eq:fP\]) become functions of $t_{ad}$, $v_{\mbox{max}}$, and $a$. Note that $m$ measures the space covered along the parallel, in fact $\Delta\phi=2\pi m$. [![\[fig:d\_t\] The error $\delta_{tr}$ versus $v_{\mbox{max}} t_{ad}$ for two different $a$ values: $a=\pi/2$ (dashed line) and $a=\pi/4$ (full line) correspond to NOT and Hadamard gate, respectively. The dotted-dashed line shows the value of the error at $\theta=\pi/2$. The circles show the critical value of $v_{\mbox{max}} t_{ad}$ above which the best loop is the one with the minimal $\theta_M$. ](figure2d.eps "fig:"){height="4.6cm"}]{} In Fig. \[fig:d\_t\] we see the behavior of $\delta_{tr}$ as a function of $v_{\mbox{max}} t_{ad}$. The first minimum for both curves corresponds to $\theta_M=\pi/2$, then the curves for long $t_{ad}$ decrease asymptotically to zero corresponding to the region in which $\theta_M \rightarrow 0$. In this regime we have $\delta_{tr}\propto 1/t_{ad}$ which is drastically different from the results obtained with other methods where $\delta_{tr}\propto t_{ad}$, (see Refs [@alicki; @roszak] and below Sec. \[sec:comparison\]). It should be observed that this surprising results is a merit of holonomic approach which allows to choose the loop in the parameter space, without changing the logical operation as long as it subtends the same solid angle. Observe that small $\theta_M$ and long $t_{ad}$ mean large value of $m$, i.e., multiple loops around the north pole. Figure \[fig:d\_t\] shows that, for a given gate, there is a critical value $k_c$ of $v_{\mbox{max}} t_{ad}$ which discriminate between the choice of $\theta_M$ (e.g., $k= 6$ for the Hadamard gate and $k = 25$ for the NOT gate). For $v_{\mbox{max}}t_{ad}< k_c$ the best choice for the loop is $\theta_M = \pi/2$; For $v_{\mbox{max}}t_{ad}> k_c$ the best choice is the value of $\theta_M$ determined by eq. (\[eq:theta-t\]) and (\[eq:m-t\]). Note that the region $v_{\mbox{max}}t_{ad}> k_c$ is accessible with physical realistic parameters [@paper1-2]. For example, if we choose the laser intensity $\Omega=20$ meV and $v_{\mbox{max}}=\Omega/50$ (for which values the nonadiabatic transitions are forbidden), the critical parameter corresponds to the critical time of $15$ ps for the Hadamard gate and $42$ ps for the NOT gate. Pure Dephasing -------------- Until now we have ignored the pure dephasing effect because we have assumed that it is negligible in comparison with the transition error for long evolution time. Now, we check that the pure dephasing error contribution can indeed be neglected. We can write the pure dephasing error using Eq. (\[eq:errore\_PD\]) and splitting to parallel and meridian part as $$\begin{aligned} \label{eq:errore_pdP} &&\delta_{pd}^P=\int_0^{t_{ad}}dt\int_0^{\infty}d\omega\frac{J(\omega)}{\omega} \coth \bigg(\frac{\omega}{2 T}\bigg) \nonumber \\ && Q[a(t)]\sin~\omega t ~\sin^4~\theta_M\end{aligned}$$ and $$\begin{aligned} \label{eq:errore_pdL} &&\delta_{pd}^M=\int_{0}^{\frac{\theta_M}{v_{\mbox{max}}}}dt \int_0^{\infty}d\omega\frac{J(\omega)}{\omega}\coth \bigg(\frac{\omega}{2 T}\bigg) Q[a(t)]\nonumber \\ && \sin~\omega t\bigg\{\sin^4(v_{\mbox{max}} t)+\sin^4\bigg[\theta_M\bigg(1-\frac{v_{max} t}{\theta_M}\bigg)\bigg]\bigg\},\end{aligned}$$ where $Q[a(t)]= 1+ 1/2 ~\sin^2[2 a(t)]$. To estimate $\delta_{pd}$ we assume that $t_{ad}$ is longer with respect to the characteristic time of the bath. Remembering that $J(\omega)\propto \omega^s$, the pure dephasing error behavior along the parallel part at the temperature $T$ is $$\delta^P_{pd}\propto\left\{\begin{array}{lr} \left(\frac{\displaystyle 1}{\displaystyle t_{ad}}\right)^{s+3}, & T\ll1/t_{ad} \\ T \left(\frac{\displaystyle 1}{\displaystyle t_{ad}}\right)^{s+2}, & T\gg1/t_{ad} \end{array}\right.$$ while the along meridian is $$\delta_{pd}^M \propto \left(\frac{1}{t_{ad}}\right)^3.$$ Then, we can conclude that the pure dephasing can always be neglected for long time evolution because it decreases faster than the transition error. Comparison between HQC and STIRAP {#sec:comparison} --------------------------------- We make a comparison between holonomic quantum computation (HQC) and the STIRAP procedure which is an analogous approach to process quantum information. The STIRAP procedure ([@Kis; @roszak]) is, in its basic points, very similar to the holonomic information manipulation. The level spectrum, the information encoding, the evolution produced by adiabatic evolving laser are exactly the same. The fundamental difference is that in STIRAP the dynamical evolution is fixed (we must pass through a precise sequence of states) and then the corresponding loop in the parameter space is fixed. In particular, we go from the north pole to the south pole and back to the north pole along meridians. Since the loop, as in our model, is a sequence of meridian-parallel-meridian path, we can calculate the error and make a direct comparison. In this case, the transition error results proportional to $\delta_{tr}\propto t_{ad}$ and grows linearly in time while for HQC $\delta_{tr}\propto1/t_{ad}$. Therefore, the HQC is fundamentally the favorite for long application times with respect to the STIRAP ones. Moreover, we can show that the freedom in the choice of the loop allows us to construct HQC which perform better than the best STIRAP gates. In Ref. [@roszak] the minimum error (not depending on the evolution time) for STIRAP was obtained reaching a compromise between the necessity to minimize the transition, pure dephasing error and the constraint of adiabatic evolution. With realistic physical parameters [@hqc_noise] ($J(\omega)=k\omega^3 e^{(-\omega/\omega_c)^2}$, $\Omega=10$ meV, $\epsilon=1$eV, $v_{max}=\Omega/50$, $k=10^{-2} ($meV$)^{-2}, \omega_c=0.5$ meV and for low temperature), the total minimum error in Ref. [@roszak] is $\delta_{\mbox{stirap}}= 10^{-3}$. With the same parameters, we still have the possibility to increase the evolution time in order to reduce the environmental error. However, for evolution time $t_{ad}=50$ ps we obtain a total error $\delta= 1.5 \times 10^{-4}$ for the NOT gate and $\delta = 4 \times 10^{-5}$ for the Hadamard gate, respectively. As can be seen, the logical gate performance is greatly increased. More general noise ------------------ Until now we have discussed the possibility to minimize the environmental error by choosing a particular loop in the parameter sphere but the structure of the error functional clearly depends on the system-environment interaction. Then one might wonder if the same approach can be used for a different noise environment. For this reason, we now briefly analyze the case of noise matrix in the form $A=\mbox{diag}\{0,1,0,-1 \}$. Again, for long evolution we can neglect the contribution of the pure dephasing and focus on the transition error. In this case the interesting part of the error functional takes the form $$\delta_{tr}=K[(\frac{1}{2}\sin ~2\theta ~\cos ~2\theta)^2+(\sin~\theta~\sin ~2\phi)^2].$$ Even if the analysis in this case is much more complicated, it can be seen that $\delta_{tr}$ has an absolute minimum for $\theta_M=0$. The long time behavior is the same ($\delta_{tr} \propto 1/ t_{ad}$) such that the results are qualitatively analogous to the above ones: for small $\theta_M$ loops (or long evolution at fixed velocity) the holonomic quantum gate presents a decreasing error. Then even in this case it is possible to minimize the environmental error. Conclusions =========== In summary, we have analyzed the performance of holonomic quantum gates in the presence of environmental noise by focusing on the possibility to have small errors choosing different loops in the parameter manifold. Due to the geometric dependence, we can implement the same logical gate with different loops. Since different loops correspond to different dynamical evolutions, we have used this freedom to construct an evolution through “protected” or “weakly influenced” states leading to good holonomic quantum gates performances. This allows to select (once that the physical parameter are fixed) the best loop which minimizes the environmental effect. (Note that this optimization procedure is rather independent of the details of the simple model we have considered and arguably, it could be extended to more complicated systems without any substantial modification.) We have shown that for long time evolutions the noise decreases as $1/t_{ad}$ while in the other cases it increases linearly with adiabatic time. We also have shown that the same features can be found with different kinds of noise suggesting the possibility to find a way to minimize the environmental effect in the presence of any noise. These results open a new possibility for implementation of holonomic quantum gates to build quantum computation because they seem robust against both control error and environmental noise. Acknowledgment {#acknowledgment .unnumbered} ============== The autors thank E. De Vito for useful discussions. One of the authors (P. S.) acknowledges support from INFN. Financial support by the italian MIUR via PRIN05 and INFN is acknowledged. Minimizing theorem {#app:proof} ================== Let us consider the family $\mathcal{C}_n$ composed of the closed curves generated by a sequence of $n$ paths along a parallel ($\theta=const$) alternated with paths along a meridian ($\phi=const$). We call $C_n$ a generic curve in this family. For example, the family $\mathcal{C}_1$ contains all the closed curves composed by the sequence of path meridian-parallel-meridian while the family $\mathcal{C}_2$ contains the curves meridian-parallel-meridian-parallel-meridian. We argue that the closed curve minimizing the error in Eq. (\[eq:errore\_tr\]) can be found in the $\mathcal{C}_1$ family. First, we show that any closed curve in $\mathcal{C}_2$ spanning a solid angle $a$ on the sphere can be replaced by a closed curve in $\mathcal{C}_1$ spanning the same angle and producing a smaller error. In analogous way any closed curve in $\mathcal{C}_3$ can be replaced by a closed curve in $\mathcal{C}_2$ with smaller error and so on. By induction we obtain that any closed curve in $\mathcal{C}_n$ can be replaced by a curve in $\mathcal{C}_1$ spanning the same solid angle but producing smaller error. Since the curve belonging to $\mathcal{C}_n$ can approximate any closed curve on the sphere, the best curve can be found in $\mathcal{C}_1$. The crucial point is to show that any curve in $\mathcal{C}_2$ can be replaced by a curve in $\mathcal{C}_1$. Let us consider a generic curve $C_2$ in $\mathcal{C}_2$ spanning a solid angle $a$: composed by a segment of a meridian (with $\theta$ going from $0$ to $\theta_1$), a parallel (spanning a $\Delta \phi_1$ angle), meridian (with $\theta:\theta_1 \rightarrow \theta_2$), a parallel (spanning a $\Delta \phi_2$ angle), and finally a segment to the north pole along a meridian. Let us consider two closed curves $C_1^1$ and $C_1^2$ in $\mathcal{C}_1$ subtending the same solid angle $a$ with, respectively, $\theta_1$ and $\theta_2$ as maximum angle spanned during the evolution along the meridian. First we analyze (\[eq:f\]) along the meridian. Without losing generality, we can take $\theta_1 < \theta_2$; it is clear from Eq. (\[eq:fM\]) that the value of $\delta_{tr}$ along the meridian for $C_1^1$ is smaller that for $C_1^2$: $\delta^M_{C_1^1} < \delta^M_{C_1^2}$. We note from the Eq. (\[eq:fM\]), suitable extended to $C_2$, that the two paths along the meridians depends only on $\theta_2$ and then produce the same error of $C_1^2$, $$\delta^M_{C_1^1} < \delta^M_{C_1^2} = \delta^M_{C_2}. \label{app_eq:meridian_error}$$ The difference between the contribution along the parallel is $$\label{eq:dif1} \delta^P_{C_2}-\delta^{P}_{C_1^2}=\Delta\phi_1\bigg(\sin~\theta_1~\sin^2~2\theta_1-\frac{1-\cos~\theta_1}{1-\cos~\theta_2}\sin~\theta_2~\sin^2~2\theta_2\bigg)$$ and $$\label{eq:dif2} \delta^P_{C_2}-\delta^{P}_{C_1^1}=\Delta\phi_2 \bigg(\sin~\theta_2~\sin^2~2\theta_2-\frac{1-\cos~\theta_2}{1-\cos~\theta_1}\sin~\theta_1~\sin^2~2\theta_1\bigg).$$ Analysis of the positivity of the quantities given by Eqs. (\[eq:dif1\]) and (\[eq:dif2\]) shows that $\delta^P_{C_2}$ cannot be at the same time smaller than $\delta^{P}_{C_1^2}$ and $\delta^{P}_{C_2^2}$. In fact, there are two possibilities: If $\delta^P_{C_2} > \delta^P_{C_1^1}$, from Eq. (\[app\_eq:meridian\_error\]) and (\[eq:dif2\]), $$\delta_{C_2}= \delta^M_{C_2}+\delta^P_{C_2} > \delta^M_{C_1^1}+\delta^P_{C_1^1} = \delta_{C_1^1},$$ and the best closed curve is $C_1^1$. If $\delta^P_{C_2} > \delta^P_{C_1^2}$, from Eqs. (\[app\_eq:meridian\_error\]) and (\[eq:dif1\]), $$\delta_{C_2}= \delta^M_{C_2}+\delta^P_{C_2} > \delta^M_{C_1^2}+\delta^P_{C_1^2} = \delta_{C_1^2},$$ and the best closed curve is $C_1^2$. In the same way it can be shown that any closed curve in $\mathcal{C}_3$ can be replaced by a closed curve in $\mathcal{C}_2$ with smaller error. [99]{} P. Zanardi and M. Rasetti, Phys. Lett. A [**264**]{}, 94 (1999). Z. Kis and F. Renzoni, Phys. Rev. A [**65**]{}, 032318 (2002). F. Troiani, E. Molinari, and U. Hohenester, Phys. Rev. Lett. [**90**]{}, 206802 (2003). K. Roszak, A. Grodecka, P. Machnikowski, and T. Kuhn, Phys. Rev. B [**71**]{}, 195333 (2005). J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, Nature (London) [**403**]{}, 869 (2000). G. Falci, R. Fazio, G. M. Palma, J. Siewert, and V. Vedral, Nature (London) [**407**]{}, 355 (2000). R. G. Unanyan, B.W. Shore, and K. Bergmann, Phys. Rev. A [**59**]{}, 2910 (1999). L.-M. Duan, J.I. Cirac, and P. Zoller, Science [**292**]{}, 1695 (2001). L. Faoro, J. Siewert, and R. Fazio, Phys. Rev. Lett. [**90**]{}, 028301 (2003). I. Fuentes-Guridi, J. Pachos, S. Bose, V. Vedral, and S. Choi, Phys. Rev. A [**66**]{}, 022102 (2002). A. Recati, T. Calarco, P. Zanardi, J. I. Cirac, and P. Zoller, Phys. Rev. A [**66**]{}, 032309 (2002). P. Solinas, P. Zanardi, N. Zanghì, and F. Rossi, Phys. Rev. B [**67**]{}, 121307(R) (2003). A. Carollo, I. Fuentes-Guridi, M. F. Santos, and V. Vedral, Phys. Rev. Lett. [**90**]{}, 160402 (2003). G. De Chiara and G. M. Palma, Phys. Rev. Lett. [**91**]{}, 090404 (2003). A. Carollo, I. Fuentes-Guridi, M. F. Santos, and V. Vedral, Phys. Rev. Lett. [**92**]{}, 020402 (2004). V.I. Kuvshinov and A.V. Kuzmin, Phys. Lett. A, [**316**]{}, 391 (2003). P. Solinas, P. Zanardi, and N. Zanghì, Phys. Rev. A 70, 042316 (2004). S.-L. Zhu and P. Zanardi, Phys. Rev. A [**72**]{}, 020301(R) (2005). G. Florio, P. Facchi, R. Fazio, V. Giovannetti, and S. Pascazio, Phys. Rev. A [**73**]{}, 022327 (2006). I. Fuentes-Guridi, F. Girelli, and E. Livine, Phys. Rev. Lett. [**94**]{}, 020503 (2005) D. Parodi, M. Sassetti, P. Solinas, P. Zanardi, and N. Zanghì, Phys. Rev. A [**73**]{}, 052304 (2006). The explicit expression for the bright states is $|E_{1} \rangle = \frac{1}{\sqrt{2} \Omega} (\Omega |e\rangle + \sum_i \Omega_i |i \rangle)$ and $|E_{2} \rangle = \frac{1}{\sqrt{2} \Omega} (-\Omega |e\rangle + \sum_i \Omega_i |i \rangle)$; for the dark states is $|E_3\rangle = 1/(\Omega \sqrt{|\Omega_+|^2+ |\Omega_-|^2}) [ \Omega_0 (\Omega_+ |+ \rangle + \Omega_- |- \rangle) - (\Omega^2 - |\Omega_0|^2) |0 \rangle ]) $ and $|E_4 \rangle = 1 / \sqrt{|\Omega_+|^2 + |\Omega_-|^2} [\Omega_- |+ \rangle - \Omega_+ |- \rangle]$. A. O. Caldeira and A. J. Leggett, Phys. Rev. Lett. [**46**]{}, 211 (1981). U. Weiss, [*Quantum dissipative systems*]{} (World Scientific, Singapore, 1999). R. Alicki, M. Horodecki, P. Horodecki, R. Horodecki, L. Jacak, and P. Machnikowski, Phys. Rev. A [**70**]{}, 010501(R) (2004).

--- abstract: | We model the population characteristics of the sample of millisecond pulsars (MSPs) within a distance of $1.5$ kpc. We find that for a braking index $n=3$, the birth magnetic field distribution of the neutron stars as they switch on as radio-emitting MSPs can be represented by a Gaussian in the logarithm with mean $\log B(G)= 8.1$ and $\sigma_{\log B}=0.4$ and their birth spin period by a Gaussian with mean $P_0=4$ ms and $\sigma_{P_0}=1.3$ ms. We assume no field decay during the lifetime of MSPs. Our study, which takes into consideration acceleration effects on the observed spin-down rate, shows that most MSPs are born with periods that are close to the currently observed values and with average characteristic ages that are typically larger by a factor $\sim 1.5$ compared to the true age. The Galactic birth rate of the MSPs is deduced to be ${\mbox{$\, \stackrel{\scriptstyle >}{\scriptstyle \sim}\,$}}3.2 \times 10^{-6}$ yr$^{-1}$ near the upper end of previous estimates and larger than the semi-empirical birth rate $\sim 10^{-7}$ yr$^{-1}$ of the Low Mass X-ray Binaries (LMXBs), the currently favoured progenitors. The mean birth spin period deduced by us for the radio MSPs is a factor $\sim 2$ higher than the mean spin period observed for the accretion and nuclear powered X-ray pulsars, although this discrepancy can be resolved if we use a braking index $n=5$, the value appropriate to spin down caused by angular momentum losses by gravitational radiation or magnetic multipolar radiation. We discuss the arguments for and against the hypothesis that accretion induced collapse (AIC) may constitute the main route to the formation of the MSPs, pointing out that on the AIC scenario the low magnetic fields of the MSPs may simply reflect the field distribution in isolated magnetic white dwarfs which has recently been shown to be bi-modal with a dominant component that is likely to peak at fields below $10^3$ G which would scale to neutron star fields below $10^9$ G, under magnetic flux conservation. author: - | Lilia Ferrario and Dayal Wickramasinghe\ Department of Mathematics, The Australian National University, Canberra, ACT 0200, Australia date: 'Accepted. Received ; in original form' title: The birth properties of Galactic millisecond radio pulsars --- pulsars: general, stars: neutron, stars: magnetic fields, X-rays: binaries. Introduction ============ The properties of the MSPs and the “normal” radio pulsars place them in two nearly disjoint regions in the spin period ($P$) period derivative ($\dot P$) diagram. In the normal pulsars, $P$ and $\dot P$ are distributed about mean values of $\sim 0.6$ s and $\sim 10^{-15}$ s s$^{-1}$ respectively, with implied magnetic field strengths in the range $\sim 10^{11}-10^{13}$ G. In contrast, in the MSPs, $P$ and $\dot P$ are distributed about $\sim 5$ ms and $\sim 10^{-20}$ s s$^{-1}$ respectively, with field strengths in the range $\sim 10^8-10^9$  G. This bi-modality in the field distributions of the radio pulsars has been an enigma which has still to be fully resolved. There are also major differences in the population characteristics of these two groups of pulsars which provide important clues on their origin. Most ($\sim 85$%) of the MSPs are in binary systems (MSPs) on nearly circular orbits in contrast to the normal pulsars which tend to be isolated and when they are not, exhibit more eccentric orbits. Furthermore, proper motion studies have shown that while the average space velocity for normal pulsars is $\sim 400$ km s$^{-1}$ (Hobbs et al. 2005), the MSPs form a low-velocity population with typical transverse speeds of $\sim 85$ km s$^{-1}$ (Hobbs et al. 2005; Toscano et al. 1999). Differences in the incidence of binarity, eccentricity of orbits, and space motions are usually attributed to differences in the kick velocity imparted to the neutron stars at birth (Shklovskii 1970). The magnitude of the kick, and its effect on the binary system, will depend on the nature of the system, and whether the neutron star originates from the core collapse of a massive star with a supernova explosion, or from the accretion induced collapse (AIC) of a white dwarf. In the standard model, the MSPs are considered to be the end product of the evolution of low-mass and intermediate-mass X-ray binaries where it is assumed that the neutron star was formed by core collapse (CC) of a massive ($M> 8\Msun$) star, and is subsequently spun up to millisecond periods during an accretion disc phase (Bhattacharya & van den Heuvel 1991; Bisnovatyi-Kogan & Komberg 1974). We shall refer to this class of objects as the “Core-Collapsed LMXBs and IMXBs” or, more briefly, the LMXBs(CC)/IMXBs(CC). Accretion induced field decay is an integral part of this model which appears plausible from a theoretical view-point, particularly if the fields in neutron stars are of crustal origin (Konar & Bhattacharya 1997, but see Ruderman 2006 for an alternative model). Regardless of the origin of the low fields in the MSPs, a long standing problem with the LMXB/IMXB scenario has been the difficulty in reconciling their semi-empirical birth rates with those of the radio MSPs (e.g. Lorimer 1995; Cordes & Chernoff 1997). The problem with the birth rates has been confirmed by recent population synthesis calculations which have also highlighted the difficulties in explaining the observed orbital period distribution of MSPs on the LMXB(CC)/IMXB(CC) scenario (Pfhal et al. 2003). Another often discussed channel for the production of MSPs involves the AIC of an ONeMG white dwarf (Michel 1987). Here, during the course of mass transfer, a white dwarf reaches the Chandrasekhar limit, and collapses to form a neutron star (Bhattacharya & Van den Heuvel 1991). In the AIC scenario, we may expect the magnetic field distribution of the MSPs to reflect in some way the magnetic field distribution of their progenitor white dwarfs obviating the need for field decay. A low-mass or intermediate-mass X-ray binary phase may follow the collapse of the white dwarf and we shall refer to this class of objects as the “Accretion Induced Collapse LMXBs and IMXBs” or, more briefly, as the LMXBs(AIC)/IMXBs(AIC). Population synthesis calculations indicate that the expected birth rates from the AIC channel may be significantly higher than those from the LMXB(CC)/IMXB(CC) route (Hurley et al. 2002; Tout et al 2007; Hurley 2006, private communication). In this paper, we present an analysis of the 1.5 kpc sample of MSPs which is considered to be sufficiently sampled out (Lyne et al. 1998; Kramer et al. 1998) with the aim of establishing the MSP birth properties and constraining the different models that have been proposed for their origin. Our estimate of the Galactic birth rate of the MSPs is at the upper end of previous estimates (e.g. Cordes & Chernoff 1997) and again brings into question the LMXB(CC)/IMXB(CC) scenario as being the dominant route for the origin of the MSPs. The paper is arranged as follows. In section 2 we describe the data set and our model. Our results are presented and discussed in section 3 where we also present the case for and against LMXB(CC)/IMXB(CC) progenitors and AIC progenitors for the MSPs. Our conclusions are presented in section 4. The modelling of the radio properties of the MSPs ================================================= In 1998, Kramer et al. conducted a very detailed study aimed at comparing the radio emission properties of MSPs to those of normal pulsars. In their work, they restricted their comparative studies to objects within 1.5 kpc, on the grounds that the population of all radio-pulsars is sufficiently sampled out up to this distance, as first pointed out by Lyne et al. (1998). Recently, this assumption has gained further strength through the high-latitude survey of Burgay et al. (2006) in the region of the sky limited by $220\gradi<l<260\gradi$ and $|b|<60\gradi$ conducted with the 20-cm multi-beam receiver on the Parkes radio-telescope. If we restrict the pulsars in this survey to a distance of up to 1.5 Kpc, we find that all but one previously known radio-pulsars have been re-detected and 4 new objects, out of a total of 16, discovered. We therefore apply a correction factor of $1.25$ to obtain an estimate of the total number of MSPs in the $1.5$ kpc sample that we analyse, stressing the fact that this may only be a lower limit for the real number of objects up to this distance. We have restricted our analysis to binaries and isolated millisecond pulsars with spin periods shorter than 30 ms. In the restricted sample to 1.5 kpc, the Australia Telescope National Facility (ATNF) catalogue (Manchester et al. 2005) gives 24 MSPs in binaries and 11 isolated MSPs. We do not distinguish between these two groups since the isolated MSPs are likely to be an end product of binary systems in which the companion has been tidally disrupted or ablated (Radhakrishnan & Shukre 1986) with an otherwise similar evolutionary history to the binary MSPs (see, however, Michel 1987; Bailyn & Grindlay 1990 for alternative points of view). They exhibit very similar observational properties, except, perhaps, for the radio-luminosity which seems to be slightly lower in the isolated MSPs (Bailes et al. 1997). Of the $24$ binary MSPs, $14$ have periods above $10$ d and most have lower limits to the companion masses in the range $\sim 0.2 - 0.4 \Msun$ (Manchester et al. 2005) indicative of the end state of evolution of binary systems that evolve to longer periods (beyond the bifurcation period, see Podsiadlowski, Rappaport & Pfahl 2002) due to mass transfer from a low-mass giant (see sections 3.1 and 3.2) leading to He white dwarfs. The remaining shorter period systems appear to have either He or CO white dwarfs or very low-mass companions. The age of the millisecond pulsar is calculated according to $$\label{age} t=\frac{P}{(n-1)\dot P}\left[1-\left(\frac{P_0}{P}\right)^{n-1}\right]$$ where $n$ is the braking index, which is equal to 3 for the dipolar spin-down model, and $P$ and $P_0$ are the observed and the initial period of the MSP respectively. If $P_0<<P$ we obtain the characteristic age of the MSP: $$\label{char_age} \tau_c=\frac{P}{2\dot P}.$$ In the next sections we will show that $P_0$ is often too close to $P$ to be able to rely on $\tau_c$ for an estimate of the true age of a MSP. The very low spin-down rates of the MSPs have so far precluded any direct measurements of the braking index. An index of $n=3$ is indicated for old normal radio pulsars, but values $n\sim 1.5-2.8$ have been measured in younger pulsars (Lyne 1996; Hobbs et al. 2004). In this work, we have adopted $n=3$, but it is conceivable that a different value of $n$ may be appropriate for the MSPs. For instance, if spin down is by multi-polar radiation, the braking index will be somewhat larger than $3$, while if angular momentum is lost mainly by gravitational radiation, we expect $n=5$ (Camilo, Thorsett & Kulkarni 1994). In our modelling we adopt $n=3$ but we also discuss the implications of using a larger value of $n$. We synthesise the properties of the MSPs using essentially the method described in Ferrario & Wickramasinghe (2006, hereafter FW). However, in the present study there are two main differences. Firstly, we directly assume an initial magnetic field distribution for the MSPs without attempting to relate it back to the magnetic properties of the (main sequence) progenitors. We therefore have as our basic input the MSP birth magnetic field distribution, which we describe by a Gaussian in the logarithm, and the birth spin distribution also described by a Gaussian. We stress that here with “birth” characteristics of MSPs we refer to those characteristics that the MSPs have as they switch on as radio emitters, regardless of their previous history. Hence, the results of our calculations do not depend in any ways on the specific route(s) leading to the formation of the MSPs. Secondly, we take into consideration the three Doppler accelerations effects cited by Damour & Taylor (1991) which affect the observed spin-down rate of the MSPs, namely, (i) the Galactic differential rotation, (ii) the vertical acceleration $K_z$ in the Galactic potential and (iii) the intrinsic transverse velocity of the pulsar. Thus, the observed spin-down rate is given by (e.g. Toscano et al. 1999) $$\label{shk} \dot P_{\rm obs} =\dot P_i + \Delta \dot P$$ where $\dot P_i$ is the “intrinsic” spin-down rate and $\Delta\dot P$ is the term due to the aforementioned acceleration effects. Hence, when we compare our models to observations, we introduce these acceleration terms to our synthetic population to mimic the behaviour of the observed MSPs. We follow the motions of the stars we generate by integrating the equations of motion in the Galactic potential of Kuijken & Gilmore (1989) assuming that the neutron stars are born with a kick velocity given by a Gaussian distribution with velocity dispersion $\sigma_v$. To fit the observations, we also model the radio luminosity at 1400 MHZ and compare it to the members of our list with a measured value at this frequency. The studies of Kramer et al. (1998) and Kuz’min (2002) indicate that despite the large differences in periods and magnetic fields, normal pulsars and MSPs exhibit the same flux density spectra, therefore pointing towards the same emission mechanism, although the MSPs tend to be weaker sources on average (Kramer et al. 1998). Hence, similarly to many previous investigators (e.g. FW; Narayan & Ostriker 1990), we have assumed that the luminosity $L_{400}$ at 400 MHZ can be described by a mean luminosity of the form $$\label{lm} \log \langle L_{400}\rangle=\frac{1}{3}\log\left(\frac{\dot P}{P^3}\right)+\log L_0$$ Here the luminosities are in units of mJy kpc$^2$. We have modelled the spread around $L_{400}$ using the dithering function of Narayan & Ostriker (1990) to take into account the various intrinsic physical variations within the sources and also variations caused by different viewing geometries. This function is given by $$\label{gamfun} \rho_L(\lambda)=0.5\lambda^2\exp\left(-\lambda\right)\qquad\qquad (\lambda\ge 0)$$ where $$\label{dith} \lambda=b\left(\log\frac{L_{400}}{\langle L_{400}\rangle}+a\right)$$ and $a$ and $b$ are constants to be determined (Hartman et al. 1997). Kramer et al. (1998) find that by restricting their comparison analysis of normal radio-pulsars to MSPs to sources up to 1.5 kpc, the mean spectral indices of normal radio-pulsars and MSPs are essentially the same, i.e., $-1.6\pm 0.2$ (MSPs) and $-1.7\pm 0.1$ (normal pulsars). Hence our deduced radio luminosity at 400 MHZ is scaled to 1400 MHZ using a spectral index of $-1.7$ (as in FW). Once all the intrinsic properties of our model MSPs are determined, we check for pulsars detectability at 1400 MHZ by the Parkes multi-beam receiver (e.g. Manchester et al. 2001; Vranesevic et al. 2004). Furthermore, pulsars radio emission is anisotropic with pulsars at shorter periods exhibiting wider beams, hence we need to correct for this factor, since this will influence the birth rates of MSPs. For example, large beams would require smaller birth rates, since the MSPs would have a greater chance to be detected. However, there is as yet no agreement on the beaming fraction-period relationship, particularly for the MSPs. Rankin (1993), Gil et al. (1993) and Kramer et al. (1994) pointed out that observational evidence seems to suggest that the opening angles of normal radio-pulsars (that is, the last open dipolar field line) is proportional to $1/\sqrt{P}$. In the absence of a consensus on this issue, we use Kramer’s (1994) model at a frequency of 1.4 GHz for the opening half-angle $\theta$ (in degrees) of the pulsar beam: $$\label{op_angle} \theta=\frac{5.3\gradi}{P^{0.45}}.$$ These values of $\theta$ yield duty cycles of less than unity for periods down to about 1 ms. However, we would like to remark that our results are quite insensitive to slight modifications to the above $\theta-P$ relationship. Then, by assuming that the viewing angles of MSPs are randomly distributed, the fraction $f$ of the sky swept by the radiation beam is given by (Emmering & Chevalier 1989) $$\label{beam} f=\left(1-\cos\theta_r\right)+\left(\frac{\pi}{2}-\theta_r\right)\sin\theta_r$$ where $\theta_r$ is the half-opening angle now in radians. We will use this $f$ to compare our MSP synthetic population to the data sample under consideration. Results and discussion ====================== We have used as our observational constraints the 1-D projections of the data comprising the number distributions in period $P$, magnetic field $B$, period derivative $\dot P$, radio luminosity $L_{1400}$, $Z$-distribution and characteristic age $\tau_c$, as determined from equation \[shk\]. =0.6 Similarly to FW, the best fit model to the observations of the MSPs was determined “by eye” after conducting hundreds of trials. Our results are shown in Figure 1. The fit was obtained by setting $\sigma_{P_0}=1.3$ ms about a mean $P_0=4$ ms. The parameters for the luminosity model are ${\langle L_{400}\rangle}=5.4$, $a=1.5$ and $b=3.0$. Furthermore, the MSPs are imparted with a one-dimensional natal kick dispersion velocity of $50$ km s$^{-1}$, which yields an average transverse velocity of $83$ km s$^{-1}$. Our model reproduces the observed total number of MSPs with a local formation rate of ${\mbox{$\, \stackrel{\scriptstyle >}{\scriptstyle \sim}\,$}}4.5\times 10^{-9}$ yr$^{-1}~$kpc$^{-2}$ which translates into a Galactic birth rate of ${\mbox{$\, \stackrel{\scriptstyle >}{\scriptstyle \sim}\,$}}3.2\times 10^{-6}$ yr$^{-1}$. We emphasise that this should be seen as a lower limit for the MSP birth rate, since the 1.5 kpc sample is still likely to be somewhat incomplete even after the correction factor that we have applied. Our calculations show that the observed MSP magnetic field distribution can be modelled with a field which is initially (that is, at the onset of the MSP radio-emission phase) Gaussian in the logarithm. We find that, similarly to our modelling of normal isolated radio-pulsars (see FW), it is not necessary to assume any spontaneous field decay during the lifetime of the radio-emitting MSPs. Thus, the “high” magnetic field tail of MSPs arises from the dependence of the spin-down rate on the magnetic field, and not from a complex birth field distribution assigned to the parent population (e.g. as a result of accretion-induced field decay during a previous phase of mass accretion). In this context, we note that if the MSPs were born with higher field strengths than postulated by us and then decayed during their radio-emission lifetime towards weaker magnetic fields values, then we would expect a continuous field distribution filling up the gap between the normal radio-pulsars and the MSPs. This was first noted by Camilo et al. (1994), who also point out that there is no indication that old globular cluster MSPs have magnetic fields which are lower than those of MSPs in the Galactic disc. Another observational peculiarity of MSPs is that some of these objects appear to be older than the Galactic disc (10 Gyr) if one uses equation \[char\_age\] to assign them an age. Toscano et al. (1999) found that by correcting their spin-down rates for the transverse velocity (Shklovskii) effect they exacerbated this paradox, since the correction resulted in a decrease in the spin-down rate $\dot P$ (and thus of the derived magnetic field strength) and an increase in characteristic age. As a consequence, nearly half of their corrected sample exhibited characteristic ages comparable to or greater than the age of the Galactic disc. Our theoretical results agree with their findings and are presented in Figure 2 where we have plotted the observed $\dot P_{obs}$ and intrinsic $\dot P_i$ spin-down rates of our synthetic population. This clearly supports the view that transverse velocity effects do play an important role in the observations of MSPs. =0.75 Camilo et al. (1994) proposed three possibilities to solve this paradox. The first is that the magnetic field structure of MSPs is multipolar and thus the braking index that appears in equation \[age\] may be greater than 3. Alternatively, due to gravitational radiation, $n=5$. The second is that the magnetic field decays, so that high values of $\tau_c$ could be attained by a certain choice for the decay time-scale, although they discarded this possibility as outlined earlier. Finally, the third possibility that Camilo et al. (1994) proposed is that at least some MSPs may have been born with $P_0\sim P$. Our modelling has shown that if we start with an initial period distribution that is Gaussian with mean at 4 ms, we obtain a current day period distribution that is close to what is observed. In particular, we can reproduce the sharp rise in the observed period distribution near $\sim 3$ ms. In our model, most pulsars have initial periods that are quite close to the observed periods. This yields an average characteristic age which is larger than the average true age (as given in equation \[age\]) by nearly 50%. Further observational support in favour of birth periods being close to observed periods in MSPs also comes from studies of individual systems. For instance, for PSR J0437-4715, Johnston et al. (1993) and Bell et al. (1995) find $P=5.757$ ms, $P_{\rm orb}=5.741$ d and $\tau_c=4.4-4.91$. Sarna, Ergma, Gerskevits-Antipova (2000) derive the mass of the companion of PSR J0437-4715 to be $0.21\pm 0.01$ M$_\odot$ with a cooling age of $1.26-2.25$ Gyr. This implies that PSR J0437-4715 is much younger than inferred through its characteristic age and was born with a period close to the current period. The lack of sub-millisecond pulsars is apparent in the distribution of the MSPs and has also been noted in the millisecond X-ray pulsars (Chakrabarty 2005). Our study indicates that this is consistent with our assumption of a Gaussian distribution of initial birth periods that peaks at $4$ ms and rules out the possibility that most millisecond pulsars are born at sub-millisecond periods. Here, selection effects may be playing a role, however, Camilo et al. (2000) estimated a loss of sensitivity of only 20% below about 2 ms, which is far too low to explain the sudden drop in the number of MSPs below this period. This may suggest that neutron stars can never achieve break-up spin periods ($\sim 0.4-0.7$ ms depending on equation of state, Cook et al. 1994). Hence, loss of angular momentum caused by gravitational radiation may limit the neutron star spin rate (Wagoner 1984). Current estimates to the lower limits for spin periods are about 1.4 ms (Levin & Ushomirsky 2001), which is close to the spin of the recently discovered MSP in the globular cluster Terzan 5 (Hessels et al. 2006). The LMXB(CC)/IMXB(CC) scenario and its relation to birth properties ------------------------------------------------------------------- In the LMXB(CC) route, matter is transfered from low-mass main-sequence donors and binary evolution occurs towards shorter periods driven by magnetic braking and gravitational radiation. Mass transfer continues past the period minimum over a Hubble time or until the companion is evaporated. Binary MSPs that result from this route are expected to have very low-mass companions with ultra short orbital periods. In contrast, in the IMXB(CC) route the donor stars are of intermediate mass ${\mbox{$\, \stackrel{\scriptstyle >}{\scriptstyle \sim}\,$}}2\Msun$, and binary evolution is driven by nuclear evolution past the bifurcation period towards longer periods. For the lower mass donor stars, mass transfer phase ends when the helium core of the donor star is exposed as a low-mass ($\sim 0.2- 0.4 \Msun $) helium white dwarf. For the more massive donor stars, mass transfer can terminate when a CO or an ONeMg white dwarf core is exposed. Although the observed sample of binary MSPs consists of systems with all of the above companions, recent population synthesis calculations have not been successful in modelling the observed orbital period distributions (Pfahl et al. 2003). Thus, Pfahl et al. (2003) find that the LMXB(CC) route leads to a significant population of low period binary MSPs peaking at $P_{\rm orb}\sim 0.03$ d, but this population is not represented in the observed sample of MSPs. Indeed, the shortest observed binary period for the MSPs is $P_{\rm orb} = 0.1$ d for PSR J2051-0827 (Stappers et al. 2001). On the other hand, the binary periods of the ultra-compact X-ray binaries are generally significantly shorter with a few of them exhibiting orbital periods of 0.03 days. Hence, this may be an indication that either (i) as their LXMB(CC) evolution continues, they will end up ablating their companion and thus appearing as isolated MSPs, or (ii) the ultra-compact LMXBs(CC) and the MSPs are not evolutionarly linked. In contrast, the IMXB(CC) route predicts a population of binaries that peaks roughly at the observed periods $P_{\rm orb} \sim 6-30$ d, but with a width that falls short by a factor of $\sim 10-100$ (depending on assumptions on the common envelope parameter) in comparison to the observations of binary MSPs. Indeed, the majority of the binary MSPs in the ATNF sample (Manchester et al. 2005) do not have orbital periods that fall in the most probable region (Pfahl et al. 2003) predicted for either LMXB(CC) or IMXB(CC) evolution. There is also the problem with the birth rates mentioned in section 1. Attempts at reconciling the LMXB(CC)/IMXB(CC) rates with the birth rates of MSPs have not been successful (Pfahl et al. 2003) and the present results go in the direction of making this discrepancy larger. It has been suggested that the above discrepancies may disappear when more realistic models are constructed that allow for limit cycles that may arise from X-ray irradiation of the donor stars, and for the intricacies in common envelope evolution. This remains a possibility. However, it should be noted that the semi-empirical birth rates are based almost entirely on the observed LMXBs, which, given the arguments above, cannot be the dominant progenitors of the binary MSPs. The same comment also applies to our discussion of the AIC that undergo a phase of mass transfer after collapse (see section 3.2). In the evolution that leads up to LMXBs(CC) and IMXBs(CC), one of the stellar components (usually, the initially more massive) evolves into a neutron star through core collapse with a field distribution peaking near $\log B(G)=12.5$ (as observed in the isolated radio-pulsars). The observed field distribution of the MSPs, on the other hand, peaks at $\log B(G) = 8.4$. This discrepancy is often explained by accretion-induced field decay or evolution. The presence of MSPs in old systems suggests that if the low value of the magnetic field is due to field decay, it must do so mainly during the accretion phase prior to the neutron star becoming a radio MSP. The manner in which the field is expected to decay in accreting neutron stars depends on the origin of the magnetic fields, and here there is no consensus. There is no clear evidence for field decay in ordinary pulsars on a time scale of $10^7 - 10^8$ yr. However, accretion can enhance field decay, particularly if the fields are of crustal origin. Two competing effects have been considered. Accretion raises the temperature and reduces the conductivity in regions of the crust that carry the current, thereby enhancing field decay. Accretion also pushes the current forming region inward towards regions of higher density and conductivity where the field can be frozen. Konar & Bhattacharya (1997) have shown that these two effects, when taken together, could lead to an asymptotic “frozen” field strength that is a factor of $10^{-1}-10^{-4}$ below the initial field strength. The asymptotic value depends on the accretion rate and the total mass accreted. On the other hand, Wijers (1997) presented strong evidence against accretion-induced field decay which is proportional to the accreted mass onto the neutron star. Thus, the standard model does not explain the observed characteristics of the MSP birth field distribution as they switch-on as radio-emitters. For instance, if we consider the route that contributes to the majority of binary MSPs, namely those having orbital periods $P_{\rm orb} \ge 10 -1000$ d with low mass He WD companions arising from the evolution of intermediate mass donors, we may expect the accretion history, and therefore the birth field distribution, to depend on the orbital period. It is therefore not immediately apparent why this field should have a nearly Gaussian distribution with such a narrow width. The problem becomes even more severe when more than one channel is considered (see discussion in Tout et al. 2007). The detection of coherent X-ray pulsations with millisecond periods in a handful of LMXBs (Lamb & Yu 2005) is often used in support of the idea of accretion induced field decay (Wijnands & van der Kliss 1998). However, whether this is evidence simply for field submersion and spin up during an accretion disc phase, or for field decay and spin up, remains to be established. Cumming et al. (2001) have argued that the majority of the LMXBs do not show coherent pulsations because they may have fields significantly less than $10^8$ G due to field submersion which, at face value, is inconsistent with the fields seen in the radio-MSPs, but their calculations also indicate that the field will re-emerge on a time scale of $\sim 1000$ yr although it is unclear to what value. Indeed, for the LXMB(CC)/IMXB(CC) standard scenario to be viable, the field would be required to re-emerge to values that are similar to those observed in the radio MSPs. If we adopt the contentious viewpoint that magnetic fields do not decay due to accretion, but are simply temporarily submerged, and re-emerge to their original values of a few $\times 10^{12}$ G at the end of the LMXB(CC)/IMXB(CC) phase, then we may expect a population of high field MSPs. The objects in such a population would have a birth rate that is $10^{-4}$ times the birth rate of normal radio-pulsars and would therefore be unlikely to be represented in the current sample of radio-pulsars. Furthermore, since they would spin down very rapidly to much longer periods (with characteristic time scales of only a few hundred years), they would have an even smaller chance to be detected as high field radio-MSPs. Finally, we note that on the LMXB(CC)/IMXB(CC) hypothesis, we expect the birth spin period distribution of the MSPs as they become radio-emitters, to be similar to the observed spin period distribution of the LMXBs(CC). However, observations of accretion and nuclear powered LMXBs show that their spin periods peak near 2 ms (Lamb & Yu 2005). This could indicate either a different origin for the radio MSPs (see section 3.2), or that the braking index is significantly larger than adopted by us. We have carried out calculations for different braking indices and find that a braking index of $n= 5$ (appropriate to angular momentum loss by gravitational radiation or magnetic multipolar radiation) will bring the two distributions into closer agreement. The AIC scenario and its relation to birth properties ----------------------------------------------------- According to current models, accretion induced collapse leads to the formation of a rapidly spinning (a few milliseconds) neutron star when an ONeMg white dwarf accretes matter in a binary system and reaches the Chandrasekhar limit. Recent calculations have re-affirmed that an AIC is the expected outcome of thermal time scale mass transfer in such systems with orbital periods of the order of a few days (Ivanova & Taam 2004). The magnetic fluxes in these cores may thus reflect the magnetic fluxes seen in the isolated white dwarfs. Until recently, it was believed that the magnetic fields of the isolated white dwarfs could be described by a single distribution. However, it is now evident that the distribution is bi-modal, comprising of a high and a low field component. The high field component ($10-15$% of all white dwarfs) has a distribution that peaks at $\log B(G) \sim 7.5 $ with a half width $\sigma\log B = 7.3$ (e.g. Wickramasinghe & Ferrario 2005). This distribution declines towards lower fields with very few stars detected in the field range $10^5 - 10^6$ G (magnetic field gap). The incidence of magnetism rises again towards lower magnetic fields with some $15 - 25$% of white dwarfs being magnetic at the kilo-Gauss level (Jordan et al. 2006). Since the new detections of Jordan et al. (2006) are at the limit of the sensitivity of current spectropolarimetric surveys, it appears likely that *all* white dwarfs will be found to be magnetic, with the majority ($\sim 85$%) belonging to the low field group (${\mbox{$\, \stackrel{\scriptstyle <}{\scriptstyle \sim}\,$}}1,000$ G). A field of a kilo-Gauss scales under magnetic flux conservation to a neutron star field of $10^9$ G. Although the peak of the low field distribution has still to be established observationally, it is conceivable that the fields will be distributed in a Gaussian manner about a peak that will map on to the observed field distribution of the radio MSPs. Given the high mass transfer rates required for AIC, the Ohmic diffusion time scale will be much larger than the accretion time scale (Cumming 2002), so we expect the white dwarf field to be submerged during the build up of the white dwarf mass prior to collapse. For the above scenario to be viable, we need to postulate that the submerged field will re-emerge without decay to its flux conserved value at the birth of the neutron star. In this context, we note that there is no evidence of accretion-induced field decay in the AM Herculis-type Cataclysmic Variables, where a highly magnetic white dwarf has been accreting mass over billion years from a companion. In fact, their well studied field configurations are very similar to those observed and modelled in the isolated high field magnetic white dwarfs (e.g. Wickramasinghe & Ferrario 2000 and references therein). We expect the white dwarf to be spun up to near break up velocity prior to collapse (e.g. like the white dwarfs observed in dwarf novae). However, angular momentum (and mass) must necessarily be lost during the subsequent collapse to a neutron star (Bailyn & Grindlay 1990) so that detailed models are required to establish the expected birth spin and mass distributions of the resulting neutron star. Dessart et al. (2006) have conducted 2.5-dimensional radiation-hydrodynamics simulations of the AIC of white dwarfs to neutron stars. Their calculations show that these lead to the formation of neutron stars with rotational periods of a few (2.2-6.3) milliseconds. Hence, even if the binary system were to be disrupted following a kick during the AIC, these “runaway” newly born neutron stars would appear as isolated radio-MSPs of the type currently observed. A proportion of the white dwarfs that could be subjected to AIC will inevitably belong to the high field group ($\sim 10^6-10^9$ G), and will result in rapidly rotating (millisecond) pulsars with fields in the range $10^{10}-10^{14}$ G on collapse, if we assume magnetic flux conservation (see Figure 1 in FW). This proportion could be as high as $50$% because highly magnetic white dwarfs tend to be more massive than their non-magnetic (or weakly magnetic) counterparts (mean mass of $0.92\Msun$, Wickramasinghe & Ferrario 2005). Assuming that the kicks are not field dependent and thus preferentially disrupt these systems, we may also expect a group of MSPs with high fields. However, given the much higher spin-down rates of these objects, and their low birth rates as compared to normal radio-pulsars, we expect them to make a small contribution which would be dominated by the lowest field objects in the distribution which would have the longest lifetimes as radio-pulsars. A possible candidate could be the binary radio-pulsar PSR B0655+64 which has a relatively high magnetic field ($B=1.17^{10}$ G), short orbital period ($P_{\rm orb}=1.03$ d) and is on a nearly circular orbit (Damashek, Taylor& Hulse 1978; Edwards & Bailes 2001). The population synthesis calculations of Hurley et al. (2002) yielded an AIC rate that is two orders of magnitude higher than the LMXB(CC) rate that results from the evolution of binary systems with primaries that are less massive than $2\Msun$. A more detailed investigation of the AIC and core collapse rates and orbital period distributions expected from such calculations has been presented by Tout et al. (2007). Here it is shown that as with the core collapse route, *the AIC route also generates binary MSPs of all of the observed types*. We note in particular that a class of long period ($P_{\rm orb} \ge 10$ d) binary MSPs with He white dwarf companions is predicted, and this closely follows the observed $P_{\rm orb}-M_{WD}$ relationship (Van Kerkwijk et al. 2005). We conclude this section by noting that neutron stars that are formed via AIC may also go through a mass transfer phase prior to their switching-on as radio MSP. We therefore expect that the known sample of LMXBs/IMXBs will have a contribution from both neutron stars that have resulted from the core collapse of massive stars and from the AICs. According to current estimates of the AIC rates, the LMXBs(AIC)/IMXBs(AIC) may dominate over the LMXBs(CC)/IMXBs(CC). However, because of field submersion, it may be difficult at present to distinguish between these two possibilities. Conclusions =========== We have presented an analysis of the properties of the MSPs and placed constraints on the magnetic field and the spin period distributions of the MSPs at the time they turn on as radio emitters. We find that if we assume a braking index $n=3$, the field distribution can be represented by a Gaussian in the logarithm with mean $\log B({\rm G})=8.1$ and $\sigma_{\log B}=0.4$ and the birth spin period by a Gaussian with mean $P_0=4$ ms and $\sigma_P=1.3$ ms. Our study, which allows for acceleration effects on the observed spin-down rate, shows that (i) most MSPs are born with periods that are close to the currently observed values (ii) the characteristic ages of MSPs are typically much larger than their true age, and (iii) sub-millisecond pulsars are rare or do not exist. We also find a Galactic birth rate for the MSPs of ${\mbox{$\, \stackrel{\scriptstyle >}{\scriptstyle \sim}\,$}}3.2\times 10^{-6}$ yr $^-1$. We have used our results to discuss the relative merits of the LMXB(CC)/IMXB(CC) and AIC scenarios that have been proposed for explaining the origin of the MSPs. Our main conclusions can be summarised as follows. - The birth rate that we deduce for the MSPs is significantly larger (by a factor $\sim 100$) than the semi-empirical birth rates quoted for the LMXBs/IMXBs and is more in accord with the expected birth rates from the AIC route from population synthesis calculations. - The AIC scenario relates the MSP neutron star field distribution to their white dwarf ancestry without invoking any kind of field decay, and finds some support from the recently discovered bi-modality of the magnetic field distribution of the white dwarfs. The low fields of the MSPs may simply arise from the low field component of the white dwarf magnetic field distribution that is expected to peak below $10^3$ G and to scale to fields below $10^9$ G under magnetic flux conservation. The nearly Gaussian distribution that we deduce for the neutron star birth magnetic fields in the MSPs may thus have a more ready explanation on the AIC scenario. - On the standard LMXB(CC)/IMXB(CC) picture, the majority of the MSPs which have intermediate or long orbital periods, come from the IMXB(CC) route. However, the predictions of the expected orbital period distributions from population synthesis calculations are not in good agreement with the observations of the MSPs. It remains to be seen if better agreement with the observed period distribution can be obtained with predictions from the AIC route which is expected to also produce systems with the same range of orbital periods and companion masses at the end of mass transfer prior to the radio MSP phase. - Population synthesis calculations predict that the standard LMXB(CC) route results in a significant class of LMXBs with periods less than $1$ hr, and these are observed as ultra compact LMXBs. However, there is not as yet a single MSP with such a low period. A likely possibility is that the mass of the companion has been reduced to negligible values by mass transfer, or that the companion has been fully ablated by the time the pulsar turns on. These systems could result in isolated MSPs. However, the AIC route also leads to similar systems and therefore end products. - The peak in the spin period distribution of accretion and nuclear powered X-ray pulsars occurs at $\sim 2$ ms, shorter that the $\sim 4$ ms that we have deduced for the birth spin period of MSPs assuming a braking index of $n=3$. We find that the two distributions can be brought into closer agreement if we assume a braking index $n=5$ which may suggest that the spin down in the MSPs is dominated by angular momentum losses by gravitational radiation or by magnetic multipolar radiation. Alternatively, this may be an indicator that the two groups are not associated, and have neutron stars with intrinsically different properties (e.g. mean mass) which is reflected in their birth spin periods. We differ a more detailed discussion of the expected outcomes from the AIC route to a subsequent paper (Tout et al. 2007), where we present a detailed comparison of the birth rates and orbital period distributions of the different types of radio MSPs that result from the usual LMXB(CC)/IMXB(CC) route and the AIC route. We conclude by noting that if the AICs provide the dominant route leading to the MSPs, one has to make the proposition that neutron star fields do not decay, even in accreting stars, which remains contentious at the present time. This issue is likely to be resolved by detailed observations, as has been done in the case of white dwarfs where the general consensus appears to be that there is no evidence for field decay either in single magnetic white dwarfs, or in accreting magnetic white dwarfs in binaries. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Chris Tout and Jarrod Hurley for helpful discussions and the anonymous Referee for a careful reading of our manuscript and for numerous useful comments. Bailes M., et al. 1997, ApJ, 481, 386 Bailyn C.D., Grindlay J.E. 1990, ApJ, 353, 159 Bell J.F., Bailes M., Manchester R.N., Weisberg J.M., Lyne A.G., 1995, ApJ, 316, L81 Bhattacharya D., van den Heuvel E. P. J. 1991, Phys. Rev., 203, 1 Bisnovatyi-Kogan G.S ., Komberg B.V. 1974, Sov. Astron., 18, 217 Burgay M., Joshi B.C., D’Amico N., Possenti A., Lyne A.G., Manchester R.N., McLaughlin M.A., Kramer M., Camilo F., Freire P.C.C., MNRAS, 2006, 368, 283 Camilo F., Thorsett S.E., Kulkarni S.R. 1994, ApJ, 421, L15 Camilo F., Lorimer D.R., Freire P., Lyne A.G., Manchester R.N., 2000, ApJ, 535, 975 Chakrabarty D., 2005, in “Binary Radio Pulsars”, 2005, Ed: F.A. Rasio & I.H. Stairs, San Francisco: ASP Conf. Ser., 328, 279 Cook G.B., Shapiro S.L., Teukolsky S.A., 1994, ApJ, 424, 823 Cordes J.M., Chernoff D.F., ApJ, 1997, 482, 971 Cumming, A., 2002, MNRAS, 333, 589 Cumming A., Zweibel E. Bildsten L., 2001, ApJ, 557, 958 Damashek M., Taylor J.H., Hulse R.A., 1978, ApJ, 225, L31 Damour T., Taylor J.H., 1991, ApJ, 366, 501 Dessart L., Burrows A., Ott C.D., Livne E., Yoon S.-C., Langer N., 2006, ApJ, 644, 1063 Edwards R.T., Bailes M., 2001, ApJ, 553, 801 Emmering R. T., Chevalier R.A. 1989, ApJ, 345, 931 Ferrario L., Wickramasinghe D.T. 2006, MNRAS, 356, 1576 (FW) Gil J.A., Kija J., Seiradakis J.H. 1993, A&A, 272, 268 Hartman J.W., Bhattacharya D., Wijers R., Verbunt F., 1997, A&A, 322, 477 Hessels J.W.T., Ransom, S.M., Stairs, I.H., Freire P.C.C., Kaspi V.M., Camilo, F., 2006, Science, 311, 1901 Hobbs G., Lyne A.G., Kramer M., Martin C.E., Jordan C., 2004, MNRAS, 353, 1311 Hobbs G., Lorimer D.R., Lyne A.G., Kramer M., 2005, MNRAS, 360, 974 Hurley J.H, Tout C.A., Pols O.R., 2002, MNRAS, 329, 897 Ivanova N, Taam R.E., 2004, ApJ, 601, 1058 Johnston S. et al., 1993, Nature, 316, 613 Jordan S., Aznar Cuadrado R., Napiwotzki R., Schmid H.M., Solanki S.K., 2006, astro-ph/0610875 Konar S., Bhattacharya D., 1997, MNRAS, 284, 311 Kramer, M., Wielebinski, R., Jessner, A., Gil, J. A., Seiradakis, J. H. 1994, A&AS, 107, 515 Kramer M., Xilouris K.M., Lorimer D.R., Doroshenko O., Jessner A., Wielebinski R., Wolszczan A., Camilo F., 1998, ApJ, 501, 270 Kuijken K., Gilmore G., 1989, MNRAS, 239, 571 Kuz’min A.D., 2002, Astronomy Reports, Volume 46, Issue 6, 451 Lamb F., Yu W., 2005, in “Binary Radio Pulsars”, Ed: F.A. Rasio & I.H. Stairs, San Francisco: ASP Conf. Ser., 328, 299 Levin Y., Ushomirsky G., 2001, MNRAS, 324, 917 Lorimer D.R., 1995, MNRAS, 274, 300 Lorimer D.R., Lyne A.G., Festin L., Nicastro L., 1995, Nature, 316, 393 Lyne A.G., 1996, in “Pulsars: Problems and Progress”, San Francisco: ASP Conf. Ser., 105, 73 Lyne A.G., Manchester R.N., Lorimer D.R., Bailes M., D’Amico N., Tauris T. M., Johnston S., Bell J.F., Nicastro L. 1998, MNRAS, 295 743 Manchester R.N., et al., 2001, MNRAS, 328, 17 Manchester R.N., Hobbs G.B., Teoh A., Hobbs M. AJ, 2005, 129, 1993 Michel F.C. 1987, Nature, 329, 310 Narayan R., Ostriker J.P., 1990, ApJ, 352, 222 Pfahl E., Rappaport S., Podsiadlowski Ph., 2003, ApJ, 597, 1036 Radhakrishnan V., Shukre C.S., 1986, Ap&SS., 118, 329 Rankin J.M., 1993, ApJ, 405, 285 Podsiadlowski Ph., Rappaport S., Pfahl E., 2002, ApJ, 565, 1107 Ruderman M., 2006, in “The Electromagnetic Spectrum of Neutron Stars” , Eds. A. Baykal, S.K. Yerli, M. Gilfanov, S. Grebenev, Dordrecht: Springer, 210, 47 Sarna M.J., Ergma E., Gerskevits-Antipova J, 2000, MNRAS, 316, 84 Shklovskii I.S., 1970, Soviet Astron., 13, 562 Stappers B.W., van Kerkwijk M.H., Bell, J.F., Kulkarni, S.R., 2001, ApJ, 548, 183 Toscano M., Sandhu J.S., Bailes M., Manchester R.N., Britton M.C., Kulkarni S.R., Anderson S.B., Stappers B.W., 1999, MNRAS, 307, 925 Tout C.A., et al., 2007, MNRAS, to be submitted Van Kerkwijk M.H., Bassa C.G., Jacoby B.A., Jonker P.G., 2005, in “Binary Radio Pulsars”, 2005, Ed: F.A. Rasio & I.H. Stairs, San Francisco: ASP Conf. Ser., 328, 357 Vranesevic N., et al., 2004, ApJL, 617, L139 Wagoner, R.V., 1984, ApJ 278, 345 Wijers R.A.M.J., 1997, MNRAS, 287, 607 Wijnands R, van der Klis M., 1998, Nature, 394, 344. Wickramasinghe D.T., Ferrario L., 2000, PASP, 112, 873 Wickramasinghe D.T., Ferrario L., 2005, MNRAS, 356, 1576

--- abstract: | We reconsider the tree level color-singlet contribution for the inclusive $J/\psi$ production in $\Upsilon$ decay with the $\alpha_{s}^{5}$ order QCD process $\Upsilon\to J/\psi+c\bar{c}+g$ and $\alpha^{2}\alpha_s^{2}$ order QED processes $\Upsilon\to\gamma^{\ast}\to J/\psi+c\bar{c}$ and $\Upsilon\to J/\psi+gg$. It is found that the contribution of the QED process is compatible with that of the QCD process, and the numerical results for the QCD process alone is an order of magnitude smaller than the previous theoretical predictions, and our theoretical prediction in total is about an order of magnitude smaller than the recent CLEO measurement on the branching fraction $\mathcal{B}(\Upsilon\to J/\psi+X)$. It indicates that the $J/\psi$ production mechanism in $\Upsilon$ decay is not well understood, and further theoretical work and experimental analysis are still necessary. author: - 'Zhi-Guo He' - 'Jian-Xiong Wang' title: 'Inclusive $J/\psi$ Production In $\Upsilon$ Decay Via Color-singlet Mechanism' --- Introduction ============ Since the discovery of $c\bar{c}$ state $J/\psi$ and $b\bar{b}$ state $\Upsilon$ more than three decades ago, heavy-quarkonium system has served as a good laboratory for testing QCD from both perturbative and non-perturbative aspects. With the accumulation of new experimental data and the development of interesting theory, considerable attention has been attracted to study heavy-quarkonium spectrum, decay and production (for a review see [@Brambilla:2004wf]). On the theoretical side, the non-relativistic QCD(NRQCD)[@Bodwin:1994jh] effective field theory was introduced, based on which the production and decay of heavy quarkonium can be calculated with a rigorous factorization formalism. This formalism separates the physics on the energy scale larger than the quark mass $m_Q$, related to the annihilation or production of $Q\bar{Q}$ pair, from the physics on the scale of $m_{Q}v^2$ order, relevant to the formation of the bound state. Consequently, the inclusive production and decay rates of heavy quarkonium are factorized into the product of short-distance coefficients, which could be calculated perturbatively as the expansion of $\alpha_{s}$, and the corresponding long-distance matrix elements, which are determined by some non-perturbative methods. The long-distance matrix elements are weighted by the powers of $v$, the velocity of heavy quark in the rest frame of the bound state. One important feature of NRQCD is that it allows the contribution of $Q\bar{Q}$ pair in color-octet configuration in short distance, and the color-octet state will subsequently evaluate into physics state through the emission of soft gluons. The introduction of NRQCD has greatly improved our understanding of the production mechanism of heavy quarkonium. One remarkable success of NRQCD is that the transverse momentum ($p_t$) distributions of $J/\psi$ and $\psi^{\prime}$ production at Fermilab Tevatron[@Abe:1992ww] could be well described by the color-octet mechanism[@Braaten:1994vv]. However, this mechanism could not correctly explain the CDF measurements of $J/\psi$ polarization[@Affolder:2000nn]. Just about one or two years ago, the next-to-leading order (NLO) QCD corrections to both the color-singlet and color-octet processes have been obtained. For the color-octet process[@Gong:2008ft], it is found that the leading order (LO) results are little changed when the NLO QCD corrections are taken into account. In the color-singlet case, the theoretical predictions at QCD NLO are significantly changed from the LO results on the $p_t$ distribution and polarization of $J/\psi$[@Campbell:2007ws]. Although this still could not resolve the puzzle. The large impact of the color-singlet NLO QCD corrections on the LO results indicates that the contribution of the color-octet mechanism may not as important as we expected before. Furthermore, the theoretical predictions[@Artoisenet:2008fc] for the $p_t$ distribution of $\Upsilon$ can compatible with the data of $\Upsilon$ production at Tevatron[@Acosta:2001gv] within the theoretical uncertainty when considering some of the important next-to-next-to-leading-order (NNLO) $\alpha_{s}^{5}$ contribution. However, it still cannot explain the recent polarization measurement by D0 Collaboration [@:2008za] In the case of $J/\psi$ production in $e^{+}e^{-}$ annihilation, the existence of color-octet mechanism also faces to a challenge. The NRQCD approach predicts that the $J/\psi$ production in $e^{+}e^{-}$ annihilation at LO in $\alpha_{s}$ is dominated by $e^{+}e^{-}\to J/\psi+gg$, and $e^{+}e^{-}\to J/\psi+c\bar{c}$ and $e^{+}e^{-}\to J/\psi+g$, in which the first two are color-singlet subprocesses and the last one is color-octet subprocess. The color-octet contribution[@Braaten:1995ez] predicts there is a peak in $J/\psi$ momentum spectrum near the kinematic end point. Unfortunately, The peak was not found in the experimental observation of BABAR[@Aubert:2001pd] and Belle[@Abe:2001za]. By using the soft-collinear effective theory (SCET), the color-octet predictions[@Fleming:2003gt] could be softened, but it depends on a unknown non-perturbative shape function. Belle also extended their analysis by deriving associated $J/\psi$ production with $c\bar{c}$ pair from inclusive $J/\psi$ production production[@Abe:2002rb]. The NLO QCD calculations shown that both $\sigma[e^{+}e^{-}\to J/\psi+c\bar{c}+X]$[@Zhang:2005cha; @Gong:2009ng] and $\sigma[e^{+}e^{-}\to J/\psi+X_{\mathrm{non-}c\bar{c}}]$[@Ma:2008gq; @Gong:2009kp] may be explained by considering only the contribution of color-singlet process. However, it point out in Ref. [@Gong:2009ng] that the color-octet contribution is still not yet completely ruled out due to the incomplete measurement in the experimental analysis. To improve our understanding of $J/\psi$ production mechanism, it was proposed[@Cheung:1996mh; @Napsuciale:1997bz] that the $\Upsilon$ decay may provide an alternate probe of $J/\psi$ production in rich gluon environment. Experimentally, the branching ratio of $\Upsilon \to J/\psi+X$ has already been reported to be $(1.1\pm0.4\pm0.2)\times10^{-3}$ by CLEO based on about 20 events in Ref.[@Fulton:1988ug]. The ARGUS Collaboration obtained an upper limit of $0.68\times10^{-3}$[@Albrecht:1992ap] at $90\%$ confidence level. With about 35 times larger data sample than previous work, an improved measurement of $J/\psi$ branching ratio and momentum spectrum have been obtained recently by CLEO Collaboration with $\mathcal{B}(\Upsilon\to J/\psi+X)=(6.4\pm0.4\pm0.6)\times10^{-4}$[@Briere:2004ug]. Theoretically, the color-octet prediction is $\mathcal{B}(\Upsilon\to J/\psi+X)\simeq6.2\times10^{-4}$[@Napsuciale:1997bz] with $10\%$ contribution from $\psi(2S)$ feed-down and another $10\%$ from $\chi_{cJ}$[@Trottier:1993ze]. However, it was found that the branching ratio of color-singlet process $\Upsilon\to J/\psi+c\bar{c}g$ is about $5.9\times10^{-4}$[@Li:1999ar], which is also in agreement with experimental measurement. Although both the color-singlet and color-octet decay modes may explain the total decay rate independently, their predictions on the $J/\psi$ momentum spectrum are significantly different. The maximum value of $J/\psi$ momentum in the color-singlet and color-octet process are 3.7 GeV and 4.5 GeV respectively. The CLEO collaboration found that the experimental result of $J/\psi$ momentum spectrum is much softer than color-octet predictions and somewhat softer than color-singlet predictions. The process $\Upsilon\to J/\psi+X$ also was studied in color evaporation model[@Fritzsch:1978ey] more than thirty years ago, but this model can not give systematic predictions of $J/\psi$ production. Another early theoretical work on the process $\Upsilon\to J/\psi+X$ could be found in Ref.[@Bigi:1978tj]. There is a very well agreement between the LO color-singlet predictions[@Li:1999ar] and experimental measurements[@Briere:2004ug]. But it seems difficult to understand the situation in comparison with the case of the $J/\psi$ production at B factories, where there are huge discrepancies between the LO theoretical predictions and the experimental measurements. Therefore, we re-calculate the branching ratio of color-singlet process $\Upsilon \to J/\psi+c\bar{c}+g$ in this paper. And the results show that it is an order of magnitude smaller than the previous theoretical prediction  [@Li:1999ar]. Therefore, there is an order of magnitude discrepancy between the LO theoretical prediction and experimental measurement for $\Upsilon\to J/\psi+X$ now. To further clarify the situation, we also estimate the leading-order contribution of the QED processes $\Upsilon\to\gamma^{\ast}\to J/\psi+c\bar{c}$ and $\Upsilon\to J/\psi+ gg$ at $\alpha^{2}\alpha_{s}^{2}$ order, in which the process $\Upsilon\to J/\psi+gg$ includes two gauge invariant subsets, $\Upsilon\to\gamma^{\ast}\to J/\psi+gg$ and $\Upsilon\to\gamma^{\ast}gg$ followed by $\gamma^{\ast}\to J/\psi$. The final results show that the contribution from the QED processes are compatible with that from the QCD process. The rest of paper is organized as follows: In section II, the basic formula and method used in the calculation are presented. In section III, we describe the calculation on the branching ratio of the QCD process $\Upsilon\to J/\psi+c\bar{c}+g$ and $J/\psi$ momentum spectrum. In section IV, we estimate the contribution of the two QED processes $\Upsilon\to J/\psi+c\bar{c}$ and $\Upsilon\to J/\psi+gg$. The final results and summary are given in the last section. Description of Our basic calculation formula ============================================ At leading order in $v_{Q}$, for S-wave heavy-quarkonium production and decay, the color-singlet model predictions are equal to that based on NRQCD effective theory. Then we express $d\Gamma(\Upsilon\to J/\psi+X)$ as: $$d\Gamma(\Upsilon\to J/\psi+X)=d\hat{\Gamma}(b\bar{b}[^3S_1,\underline{1}]\to c\bar{c}[^3S_1,\underline{1}]+X) \langle\Upsilon|\mathcal{O}_1(^3S_1)|\Upsilon\rangle \langle\mathcal{O}^{\psi}_1(^3S_1)\rangle,$$ where $d\Gamma(b\bar{b}[^3S_1,\underline{1}]\to c\bar{c}[^3S_1,\underline{1}]+X)$ represents color-singlet $b\bar{b}$ pair in spin-triplet state decay into color-singlet $c\bar{c}$ pair in spin-triplet state with anything, which is calculated perturbatively, and $\langle\Upsilon|\mathcal{O}_1(^3S_1)|\Upsilon\rangle$ and $\langle\mathcal{O}^{\psi}_1(^3S_1)\rangle$ are the long-distance matrix elements, which can be related to the nonrelativistic wave functions as: $$\langle\Upsilon|\mathcal{O}_1(^3S_1)|\Upsilon\rangle\simeq \frac{3}{2\pi}|R_{\Upsilon}(0)|^{2}, \langle\mathcal{O}^{\psi}_1(^3S_1)\rangle = \frac{9}{2\pi}|R_{\psi}(0)|^{2}.$$ We employ spinor projection method[@Kuhn:1979bb] to calculate the short-distance part $d\hat{\Gamma}$. In the nonrelativistic limit, the amplitude of $b\bar{b}[^3S_1,\underline{1}]\to c\bar{c}[^3S_1,\underline{1}]+X$ could be written as[@Cho:1995vh]: $$\begin{aligned} &&\mathcal{M}(b\bar{b}[^3S_1,\underline{1}](p_0)\to c\bar{c}[^{3}S,\underline{1}](p_{1})+X)=\sum_{s_1,s_2} \sum_{i,l}\sum_{s_3,s_4}\sum_{k,l}\nonumber\\ &\times&\langle s_1;s_2\mid 1 S_z\rangle \langle 3i;\bar{3}j\mid 1\rangle\times\langle s_3;s_4\mid 1 S_z\rangle\langle 3k;\bar{3}l\mid 1\rangle\nonumber\\ &\times& {\cal M}(b_i(\frac{p_{0}}{2},s_1)\bar{b}_j(\frac{p_{0}}{2},s_2)\to c_k(\frac{p_{1}}{2},s_3)\bar{c}_l(\frac{p_{1}}{2},s_4)+X)\end{aligned}$$ where $\langle 3i;\bar{3}j\mid 1\rangle=\delta_{ij}/\sqrt{N_c}$, $\langle 3k;\bar{3}l\mid 1\rangle=\delta_{kl}/\sqrt{N_c}$, $\langle s_1;s_2\mid 1 S_z\rangle$, and $\langle s_3;s_4\mid 1 S_{z}\rangle$ are the SU(3)-color, SU(2)-spin and angular momentum Clebsch-Gordan (C-G) coefficients for $Q\bar{Q}$ projecting on certain appropriate configurations at short distance. At leading order in $v_Q(Q=b,c)$, the projection of spinors $u(\frac{p_{0}}{2},s_1)\bar{v}(\frac{p_{0}}{2},s_2)$ and $v(\frac{p_{1}}{2},s_3)\bar{u}(\frac{p_{1}}{2},s_4)$ could be expressed as: $$\Pi_b=\sum_{s_1,s_2}\langle s_1;s_2\mid 1 S_z\rangle u(\frac{p_{0}}{2},s_1)\bar{v}(\frac{p_{0}}{2},s_2) =\frac{1}{2\sqrt{2}}\slashed{\epsilon}(S_z) (\slashed{p}_0-2m_b),$$ $$\Pi_c=\sum_{s_1,s_2}\langle s_1;s_2\mid 1S_z\rangle v(\frac{p_{1}}{2},s_3)\bar{u}(\frac{p_{1}}{2},s_4) =\frac{1}{2\sqrt{2}}\slashed{\epsilon}(S_z) (\slashed{p}_1+2m_c),$$ where $\epsilon(S_z)$ is the polarization vector of the heavy quarkonium. For a spin=1 state with momentum $p$, the sum over its all possible states $S_z$ is $$\sum_{S_z}\epsilon_{\alpha}(S_z)\epsilon^{\ast}_{\beta}(S_z)= (-g_{\alpha\beta}+\frac{p_\alpha p_\beta}{p^2})$$ According to the spinor projection method, the relation between $d\hat{\Gamma}$ and $|\mathcal{M}|^2$ for the $b\bar{b}[^3S_1,\underline{1}]\to c\bar{c}[^3S_1,\underline{1}]+X$ is $$d\hat{\Gamma}(b\bar{b}[^3S_1,\underline{1}]\to c\bar{c}[^3S_1,\underline{1}]+c\bar{c}+g)=\frac{1}{3} \frac{1}{4m_b}\frac{\sum|\mathcal{M}|^2}{3m_bm_c(2N_c)^2}d\Phi_{n}$$ where $\sum$ means to sum over all possible polarization states of the particles in this process and $\Phi_{n}$ is the n-body phase space. The factor $(1/2N_c)^2$ with $N_c=3$ comes from the normalization factor of the NRQCD 4-Fermion operator. Since our calculation gives different results from the previous theoretical prediction [@Li:1999ar], we further checked our results by using two different way to do all the calculations. One is to apply the above formula to write a piece of program to do the calculations for each process described in the following two section. Another is just using the Feynman Diagram Calculation (FDC) Package [@FDC] to generate all the needed Fortran source and then do the numerical calculation. We obtained exactly the same results by using these two methods. Moreover, to check gauge invariance, in the expression of FDC version source, the gluon polarization vector is explicit kept and then is replaced by its 4-momentum in the final numerical calculation. Definitely the result must be zero and our results confirm it. The QCD process $\Upsilon\to J/\psi+c\bar{c}+g$ =============================================== Now we proceed to calculate the total decay rate of $\Upsilon\to J/\psi+c\bar{c}+g$ and its contribution to the $J/\psi$ momentum spectrum. At leading order in $\alpha_{s}$, there are six Feynman diagrams which are shown in Fig. 1. The amplitude $\mathcal{M}$ could be factorized as: $$\begin{aligned} \mathcal{M}(b\bar{b}[^3S_1,\underline{1}](p_{_0})\to c\bar{c}[^3S_1,\underline{1}](p_{_1})+c(p_{_2})\bar{c}(p_{_3})+g(p_{_4})) =\nonumber\\ \mathcal{M}_b(b\bar{b}[^3S_1,\underline{1}]\to g^{\ast}g^{\ast}g) \times\mathcal{M}_c(g^{\ast}g^{\ast}\to c\bar{c}[^3S_1,\underline{1}]+c\bar{c}),\end{aligned}$$ in which the later one is universal for all the six diagrams and it is $$\mathcal{M}_c=\frac{g_s^2}{(p_2+p_1/2)^2(p_3+p_1/2)^2} \bar{u}(p_2)\gamma^{\mu}\Pi_{c}\gamma^{\nu}v(p_3).$$ The amplitude of $\mathcal{M}_b(b\bar{b}[^3S_1,\underline{1}]\to g^{\ast}g^{\ast}g)$, for example for the first diagram, is $$\begin{aligned} \mathcal{M}^{1}_{b}=g_s^3C_{1} \mathrm{Tr}[\Pi_b\gamma^{\mu}\frac{\frac{-\slashed{p}_0}{2}+\frac{\slashed{p}_1}{2}+\slashed{p}_3+m_b} {(-p_0/2+p_1/2+p_3)^2-m_b^2}\gamma^{\nu} \frac{\frac{\slashed{p}_0}{2}-\slashed{p}_4+m_b} {(p_0/2-p_4)^2-m_b^2}\slashed{e}_3]\end{aligned}$$ where $C_{1}$ is the corresponding color coefficient and $\slashed{e}_3$ is the polarization vector of the real gluon. The amplitude $\mathcal{M}^{i}_{b}$ for the other five diagrams could be obtained in a similar way. An analytical expression of $\sum|\mathcal{M}|^2$ is obtained in the calculation, but is too lengthy to be presented here. ![The six Feynman diagrams for the short-distance process: $b\bar{b}[^3S_1,1]\to c\bar{c}[^3S_1,1]+c\bar{c}+g$. ](Feynman1.eps) The four-body phase space $\Phi_{4}$ for $b\bar{b}[^3S_1,\underline{1}]\to c\bar{c}[^3S_1,\underline{1}]+c\bar{c}+g$ is defined as $$d\Phi_{4}(p_{_0}\to p_{_1}+p_{_2}+p_{_3}+p_{_4})=\prod_{k=1}^{4}\frac{d^3\vec{p}_{_k}}{(2\pi)^32E_k}(2\pi)^4 \delta^4(p_0-\sum_{k=1}^{4}p_{_k})$$ There are many ways to perform the four-body phase-space integration. Here we briefly introduce our methods. Using the two following identical equation $$\int\frac{d^4p_{_{234}}}{(2\pi)^4}(2\pi)^4\delta^{4}(p_{_{234}}-p_{_2}-p_{_3}-p_{_4})\equiv1, \int\frac{d^4p_{_{34}}}{(2\pi)^4}(2\pi)^4\delta^{4}(p_{_{34}}-p_{_3}-p_{_4})\equiv1,$$ we transform the four-body space into the combination of three two-body phase spaces, which is given by $$\begin{aligned} &&d\Phi_{4}(p_{_0}\to p_{_1}+p_{_2}+p_{_3}+p_{_4})=\frac{ds_{_{234}}}{2\pi}\frac{ds_{_{34}}}{2\pi} \nonumber\\&&d\Phi_2(p_{_0}\to p_{_1}+p_{_{234}}) d\Phi_2(p_{_{234}}\to p_{_2}+p_{_{34}})d\Phi_{2}(p_{_{34}}\to p_{_3}+p_{_4})\end{aligned}$$ where $s_{_{234}}=p_{_{234}}^2,s_{_{34}}=p_{_{34}}^2$. The three two-body phase spaces integration are described by the three-momenta $\vec{p}_{_1},\vec{p}_{_2}^{\ast},\vec{p}_{_3}^{\ast\ast}$ and their solid angle element $d\Omega_{_0},d\Omega_{_{234}}^{\ast},d\Omega_{_{34}}^{\ast\ast}$ in the rest frames of $p_{_0}$, $p_{_{234}}$, and $p_{_{34}}$ respectively. Then the expression of four-body phase space becomes $$d\Phi_{4}=\int \frac{ds_{_{234}}}{2\pi} \int \frac{|\vec{p}_{1}|}{8(2\pi)^2m_b}d\Omega_{_0} \int\frac{ds_{_{34}}}{2\pi} \int d\Omega_{_{234}}^{\ast}\frac{|\vec{p}_{_2}^{\ast}|}{4(2\pi)^2\sqrt{s_{_{234}}}} \int\frac{|\vec{p}_{_3}^{\ast\ast}|}{4(2\pi)^2\sqrt{s_{_{34}}}} d\Omega_{_{34}}^{\ast\ast}.$$ where $|\vec{p}_{1}|$, $|\vec{p}_{_2}^{\ast}|$ and $|\vec{p}_{_3}^{\ast\ast}|$ are given in the equations below in the rest frame of $p_{0}$, $p_{_{234}}$ and $p_{_{34}}$ respectively $$|\vec{p}_{1}|=\frac{\sqrt{16m_b^4+(-4m_c^2+s_{_{234}})^2-8m_b^2(4m_c^2+s_{_{234}})}} {4m_b}$$ $$|\vec{p}_{_2}^{\ast}|=\frac{\sqrt{(s_{_{234}}-(m_c-\sqrt{s_{_{34}}})^2) (s_{_{234}}-(m_c+\sqrt{s_{_{34}}})^2)}} {2\sqrt{s_{_{234}}}}$$ $$|\vec{p}_{_3}^{\ast\ast}|=\frac{s_{_{34}}-m_c^2}{2\sqrt{s_{_{34}}}}$$ The integration ranges of $s_{_{234}}$ and $s_{_{34}}$ are $$4m_c^2<s_{_{234}}<(2m_b-2m_c)^2,m_c^2<s_{_{34}}<(\sqrt{s_{_{234}}}-m_c)^2.$$ For space-symmetry, $d\Omega_{_0}$ and $d\phi_{_{234}}^{\ast}$ could be integrated out directly then $|\mathcal{M}|^2$ only dependent on five variables $s_{_{234}}$, $s_{_{34}}$, $\theta_{_{234}}^{\ast}$, $\theta_{_{34}}^{\ast\ast}$, and $\phi_{_{34}}^{\ast\ast}$. To get the total decay rate, the non-trivial integral with these five variables is performed by three steps. First, we do the integration $d\Omega_{_{34}}^{\ast\ast}$ in the rest frame of $p_{_{34}}$, then we integrate out $s_{_{34}}$ and $\theta_{_{234}}^{\ast}$ in the rest frame of $p_{_{234}}$, the last variable $s_{_{234}}$ is integrated out in $\Upsilon$ rest frame. Since $|\vec{p}_{1}|$ only depend on $s_{_{234}}$, the $J/\psi$ momentum spectrum could be easily obtained by replacing $d s_{_{234}}$ with $\frac{d s_{_{234}}}{d|\vec{p}_{1}|}d|\vec{p}_{1}|$. The phase space integrations for the total rate and $J/\psi$ momentum spectrum are calculated numerically.   r     0.275     0.296     0.317     0.327     0.338     0.361     0.381   --------- ----------- ----------- ----------- ----------- ----------- ----------- -----------   f(r)  0.904 0.567 0.345 0.269 0.202 0.105 0.055 : The values of $f(r)$ for different $r={m_c}/{m_b}$ By dimension analysis, it is easy to represent the decay width and differential decay width of $\Upsilon\to J/\psi+c\bar{c}+g$ as $$\Gamma(\Upsilon\to J/\psi+c\bar{c}+g)=\frac{\alpha_s^5}{m_b^{5}}f(r) \frac{\langle\Upsilon|\mathcal{O}_1(^3S_1)|\Upsilon\rangle}{2N_c} \frac{\langle\mathcal{O}^{\psi}_1(^3S_1)\rangle}{3\times2N_c}.$$ $$\frac{d\Gamma}{d|\vec{p}_1|}(\Upsilon\to J/\psi+c\bar{c}+g) =\frac{\alpha^s_5}{m_b^{6}}g(r,|\vec{p}_1|/m_b) \frac{\langle\Upsilon|\mathcal{O}_1(^3S_1)|\Upsilon\rangle}{2N_c} \frac{\langle\mathcal{O}^{\psi}_1(^3S_1)\rangle}{3\times2N_c}.$$ where $r={m_c}/{m_b}$ and $f(r)$ are dimensionless, and $f(r)$ function is same as $h(r)$ in Ref.[@Li:1999ar]. To ensure the validity of our calculations, we use two different kinds of computer codes for cross check and obtain exactly the same results for $f(r)$ and $g(r,|\vec{p}_1|/m_b)$. When $r=0.327$, the decay width is $$\Gamma(\Upsilon\to J/\psi+c\bar{c}+g)= \frac{\alpha_{s}^5}{m_b^{5}}\frac{\langle\Upsilon|\mathcal{O}_1(^3S_1)|\Upsilon\rangle}{2N_c} \frac{\langle\mathcal{O}^{\psi}_1(^3S_1)\rangle}{3\times2N_c}\times0.269.$$ To compare our results with those in Ref. [@Li:1999ar], the numerical results of $f(r)$ in the range of $0.275\leq r\leq0.381$ are listed in Tab.\[I\]. It is easy to see that the results of $f(r)$ are about an order of magnitude smaller than that given in Ref.[@Li:1999ar] and $f(r)$ changes a little sharper than that when $r$ goes from 0.275 to 0.381. Besides $f(r)$, the decay width $\Gamma(\Upsilon\to J/\psi+c\bar{c}+g)$ is also dependent on the choice of the values of the two long-distance matrix elements $\langle\Upsilon|\mathcal{O}_1(^3S_1)|\Upsilon\rangle, \langle\mathcal{O}^{\psi}_1(^3S_1)\rangle$, the coupling constant $\alpha_{s}$ and the mass of b-quark. To reduce the uncertainty of theoretical predictions, we normalize it by the decay width of $\Upsilon\to \mathrm{light\; hadron}$, which includes two dominate decay modes $\Upsilon\to ggg$ and $\Upsilon\to\gamma^{\ast}\to q\bar{q}\;\mathrm(q=u,d,s,c)$. At leading order in $\alpha_{s}$ and $v_b$, we have $$\Gamma(\Upsilon\to ggg)= \frac{20\alpha_s^{3}(\pi^2-9)}{243m_b^2})\langle\Upsilon|\mathcal{O}_1(^3S_1)|\Upsilon\rangle,$$ $$\Gamma(\Upsilon\to q\bar{q})= \frac{2\pi N_ce_q^2 e_b^2\alpha^2}{m_b^2} \langle\Upsilon|\mathcal{O}_1(^3S_1)|\Upsilon\rangle.$$ Then the normalized width $\Gamma^{c\bar{c}g}_{\mathrm{Nor}}$ is given by $$\Gamma^{c\bar{c}g}_{\mathrm{Nor}}= \frac{f(r)\alpha_s^{5}\langle\mathcal{O}^{\psi}_1(^3S_1)\rangle} {3(2N_c)^2(\frac{20}{243}\alpha_s^3(\pi^2-9)+\sum_{q}2\pi N_c e_q^2 e_b^2\alpha^2)m_b^3}$$ and the branching ratio turns to be $$\mathcal{B}(\Upsilon\to J/\psi+c\bar{c}+g)= \Gamma^{c\bar{c}g}_{\mathrm{Nor}}\times\mathcal{B}(\Upsilon\to \mathrm{light\;hadron}).$$ Since the process $\Upsilon\to J/\psi+c\bar{c}+g$ can be viewed as $\Upsilon\to g g^{\ast}g^{\ast}$ followed by $g^{\ast}g^{\ast}\to J/\psi+c\bar{c}$, as suggested in Ref.[@Cheung:1996mh], it is reasonable to chose $\alpha_{s}(2m_c)=0.259$. Using $e_u=\frac{2}{3}$, $e_d=-\frac{1}{3}$,$e_s=-\frac{1}{3}$,$e_c=\frac{2}{3}$, $e_b=\frac{1}{3}$, $\alpha=\frac{1}{128}$, $r=\frac{1.548}{4.73}\simeq0.327$, $m_b=4.73\mathrm{GeV}$, $|R_\psi(0)|^2=0.81\mathrm{GeV^{3}}$ being calculated in potential model[@Eichten:1995ch] and $\mathcal{B}(\Upsilon\to\mathrm{light\;hadron})=92\%$[@Amsler:2008zzb], we predict $$\mathcal{B}(\Upsilon\to J/\psi+c\bar{c}+g)=2.12\times10^{-5}$$ The normalized $J/\psi$ momentum spectrum $d\Gamma_{Nor}/{d|\vec{p}_1|}$ is shown in Fig. 3. It is easy to see that the shape of the $J/\psi$ momentum spectrum is similar with that in Ref.[@Li:1999ar], although the prediction for the total decay width is an order of magnitude smaller than the experimental data. The QED Process $\Upsilon\to J/\psi+X$ ======================================= There are two QED processes $\Upsilon\to J/\psi+c\bar{c}$ and $\Upsilon\to J/\psi+gg$ at the leading order in $\alpha_{s}$ and $\alpha$. Both of them are considered in this work. We will present a few simple steps and analytic results for them in the following. ![The typical Feynman diagrams for the QED processes of inclusive $J/\psi$ production: (a) $b\bar{b}[^3S_1,1]\to \gamma^{\ast}\to\bar{c}[^3S_1,1]+c\bar{c}$, (b) $b\bar{b}[^3S_1,1]\to \gamma^{\ast}\to\bar{c}[^3S_1,1]+gg$, (c) $b\bar{b}[^3S_1,1]\to c\bar{c}[^3S_1,1]+gg$. ](Feynman2.eps) $\Upsilon\to\gamma^{\ast}\to J/\psi+c\bar{c}$ --------------------------------------------- At the leading order, there are four Feynman diagrams for $\Upsilon(p_0)\to\gamma^{\ast}\to J/\psi(p_1)+c(p_2)\bar{c}(p_3)$, two of which are shown in Fig. 2a. The calculation procedure for this process is very similar to that for the $J/\psi$ production in association with $c\bar{c}$ pair in $e^{+}e^{-}$ annihilation. The differential decay width is given by $$\begin{aligned} &&\frac{d\Gamma}{d|\vec{p}_{_1}|}(\Upsilon\to\gamma^{\ast}\to J/\psi+c\bar{c}) =\frac{2\pi C_AC_F^2e_b^2e_c^2\alpha^2\alpha_s^2 \langle\Upsilon|\mathcal{O}_1(^3S_1)|\Upsilon\rangle \langle\mathcal{O}^{\psi}_1(^3S_1)\rangle \sqrt{x_1^2-4r^2}} {9(2N_c)^2m_b^6 r\,{x_1}^4\,{( \kappa - x_1 ) }^3\, {( -2 + x_1 ) }^2\,{( \kappa + x_1 ) }^3}\nonumber\\&& ( 2\,\kappa x_1( -2{\kappa}^6( 1 + 2r^2 ) {x_1}^2 + {\kappa}^4( 6r^6( -4 + 3{x_1}^2 ) + 2{x_1}^2( -4 + {x_1}^2( -2 + 9x_1 ) ) -\nonumber\\&& 4r^4( 16 + x_1( -16 + x_1( -8 + 9x_1 ) ) ) + r^2( -2 + x_1 ) ( 16 + x_1( -24 + x_1( -14 + 39x_1 ) ) ) ) +\nonumber\\&& 2{\kappa}^2{x_1}^2( 8{x_1}^2 + 7{x_1}^4 - 18{x_1}^5 - 4r^6( -8 + x_1( 8 + x_1 ) ) + 4r^4( 20 + x_1( -40 + x_1( 13 + 4x_1 ) ) ) +\nonumber\\&& r^2( 32 + x_1( -96 + x_1( 60 + ( 76 - 37x_1 ) x_1 ) ) ) ) + {x_1}^4( 6r^6( 4 + {x_1}^2 ) + 2{x_1}^2( -4 + {x_1}^2( -4 + 9x_1 ) ) +\nonumber\\&& 4r^4( 8 + x_1( 32 + ( -26 + x_1 ) x_1 ) ) + r^2( -32 + x_1( 128 + x_1( -124 - 60x_1 + 39{x_1}^2 ) ) ) ) ) +\nonumber\\&& {( \kappa - x_1 ) }^3{( \kappa + x_1 ) }^3 ( -6r^6( 4 + {x_1}^2 ) + 2{x_1}^2( 4 + {x_1}^2( -13 + 8x_1 ) ) 4r^4( -16 + x_1+ \nonumber\\&&( 16 + x_1( -4 + 5x_1 ) ) ) + r^2( -32 + x_1( 64 + x_1( 4 + ( 4 - 7x_1 ) x_1 ) ) ) ) \log \frac{x_1-\kappa }{x_1+\kappa } ),\end{aligned}$$ where $C_A=3$ and $C_F=\frac{4}{3}$ are the color factors, and there are $x_1=\sqrt{|\vec{p}_{_1}|^2+4m_c^2}/m_b$ and $\kappa=\sqrt{(x_1+2r)(x_1-2r)(1+r^2-x_1)(1-x_1)}/{(1+r^2-x_1)}$. Integrating $|\vec{p}_{_1}|$ numerically and normalizing $\Gamma(\Upsilon\to\gamma^{\ast}\to J/\psi+c\bar{c})$ by $\Gamma(\Upsilon\to\mathrm{light\; hadron})$, we obtain $$\Gamma^{c\bar{c}}_{\mathrm{Normal}}=\frac{\Gamma(\Upsilon\to\gamma^{\ast}\to J/\psi+c\bar{c})}{\Gamma(\Upsilon\to\mathrm{light\; hadron})}= \frac{3.85\alpha^2\alpha_s^2\langle\mathcal{O}^{\psi}_1(^3S_1)\rangle}{{6N_c(\frac{20}{243}\alpha_s^3(\pi^2-9) +\sum_{q}2\pi e_q^2 e_b^2\alpha^2)m_b^3}}.$$ By choosing the same numerical values for $r$, $m_b$, $e_q$, $\alpha$ $\langle\mathcal{O}^{\psi}_1(^3S_1)\rangle$ and $\mathcal{B}(\Upsilon\to\mathrm{light\; hadron})$ as those in Sec.III, the numerical result is $$\mathcal{B}(\Upsilon\to\gamma^{\ast}\to J/\psi+c\bar{c})=1.06\times10^{-6},$$ and the normalized $J/\psi$ momentum spectrum is shown in Fig. 3. $\Upsilon\to J/\psi+gg$ ----------------------- The process $\Upsilon(p_{_0})\to J/\psi(p_{_1})+g(p_{_2})g(p_{_3})$ includes two parts, $\Upsilon\to\gamma^{\ast}\to J/\psi+gg$ and $\Upsilon\to gg\gamma^{\ast}$ and $\gamma^{\ast}\to J/\psi$. There are six Feynman diagrams for each part at the leading order with the typical ones shown in Fig. 2b and 2c. To calculate the contribution of the two parts together, the differential decay width is represented as $$\begin{aligned} &&\frac{d\Gamma}{d|\vec{p}_{_1}|}(\Upsilon\to J/\psi+gg)= \frac{ 32\pi C_A C_F e_c^2e_b^2\alpha^2\alpha_s^2 \langle\Upsilon|\mathcal{O}_1(^3S_1)|\Upsilon\rangle \langle\mathcal{O}^{\psi}_1(^3S_1)\rangle \sqrt{x_1^2-4r^2}} {9(2N_c)^2 m_b^6\,r^3\,x_1\,\left( -1 + r^2 \right) \,{\left( 2\,r^2 - x_1 \right) }^3\,{\left( -2 + x_1 \right) }^3} \nonumber\\&& ( ( -1 + r) \,( 1 + r ) \,( 2\,r^2 - x_1 ) \,( -2 + x_1 ) \,{\sqrt{-4\,r^2 + {x_1}^2}}\, ( 8 + 8\,r^8 - 4\,r^6\,( -4 + 3\,x_1 ) + \nonumber\\&& r^4\,( -2 + x_1 ) \,( -16 + 7\,x_1 ) + x_1\,( -12 + ( 7 - 2\,x_1 ) \,x_1 ) - 2\,r^2\,( -1 + x_1 ) \,( 8 + ( -7 + x_1 ) \,x_1 ) ) + \nonumber\\&& 2\,( 1 + r^2 - x_1 ) \,( -( ( 2\,r^2 - x_1 ) \, ( 8 + 2\,r^8 + x_1\,( -12 + 5\,x_1 ) + r^6\,( 40 + x_1\,( -32 + 5\,x_1 ) ) +\nonumber\\&& r^4\,( 6 - ( -2 + x_1 ) \,x_1\,( -19 + 6\,x_1 ) ) + r^2\,x_1\,( -6 + x_1\,( 13 + 2\,( -5 + x_1 ) \,x_1 ) ) ) \nonumber\\&& \log (\frac{-2 + x_1 - {\sqrt{-4\,r^2 + {x_1}^2}}}{-2 + x_1 + {\sqrt{-4\,r^2 + {x_1}^2}}}) ) + r^2\,( -2 + x_1 ) \,( 8\,r^{10} - 12\,r^8\,x_1 + {x_1}^2\,( 5 + 2\,( -3 + x_1 ) \,x_1 ) + \nonumber\\&& r^6\,( 6 + x_1\,( -6 + 5\,x_1 ) ) + r^4\,( 40 + x_1\,( -38 + 13\,x_1 ) ) + r^2\,( 2 + x_1\,( -32 + ( 31 - 10\,x_1 ) \,x_1 ) ) ) \nonumber\\&& \log (\frac{-2\,r^2 + x_1 + {\sqrt{-4\,r^2 + {x_1}^2}}}{-2\,r^2 + x_1 - {\sqrt{-4\,r^2 + {x_1}^2}}}) ) ), \label{eqn:jpsigg}\end{aligned}$$ Where there is $x_1=\sqrt{|\vec{p}_{_1}|^2+4m_c^2}/m_b$. And the normalized decay width becomes $$\Gamma^{gg}_{\mathrm{Normal}}=\frac{\Gamma(\Upsilon\to J/\psi+gg)}{\Gamma(\Upsilon\to\mathrm{light\; hadron})}= \frac{60.8\alpha^2\alpha_s^2\langle\mathcal{O}^{\psi}_1(^3S_1)\rangle}{{6N_c(\frac{20}{243}\alpha_s^3(\pi^2-9)+\sum_{q}2\pi e_q^2 e_b^2\alpha^2)m_b^3}}.$$ By using the same parameters as above. We obtain $$\mathcal{B}(\Upsilon\to J/\psi+gg)=1.67\times10^{-5}$$ and the normalized $J/\psi$ momentum spectrum is plotted in Fig. 3. In the numerical result, about $85.2\%$ contribution comes from the $\Upsilon\to gg\gamma^{\ast}(J/\psi)$ part, $18.2\%$ from the $\Upsilon\to\gamma^{\ast}\to J/\psi gg$ part and $-3.4\%$ from the interference part. Summary And Discussion ====================== To sum up all the contributions of the color-singlet QED and QCD processes considered above, the branching ratio of direct $J/\psi$ production in $\Upsilon$ decay is $$\mathcal{B_\mathrm{Direct}}(\Upsilon\to J/\psi+X)=3.9\times10^{-5},$$ and the corresponding normalized $J/\psi$ momentum distribution is given by the solid line in Fig. 3. It can be seen in Fig. 3 that the contribution of the QCD process is dominated in small $p_\psi$ region, while the effect of the QED process $J/\psi+gg$ is more important in large $p_\psi$ region. In Eq. (\[eqn:jpsigg\]) and the dot-dashed line in Fig. 3, the logarithmic divergence at the kinematic end point is obvious shown for the QED process $J/\psi+gg$. It was pointed out in Ref [@Fleming:2003gt; @Lin:2004eu; @Ma:2008gq] that both the $\alpha_{s}$ and $v_{b}$ expansion failed near the kinematic end point region in the similar processes $e^{+}e^{-}\to J/\psi+X$ and $\Upsilon\to \gamma+X$ because of the large perturbative and non-perturbative corrections, and the logarithmic divergent behavior can be soften by applying the resummation in the SCET. Whatever it can improve the $J/\psi$ momentum spectrum largely near the kinematic end point, but the corrections to the total decay width is small. Therefore we omit the resummation effect here.  Our calculations show that at the leading order in $\alpha_{s}$, $v_{b}$ and $v_c$, the QCD process $\Upsilon\to J/\psi+c\bar{c}+g$ only accounts for $54.4\%$ of the LO theoretical prediction for total branching ratio, in spite of a enhancement factor $\alpha_s^3/\alpha^2$ that is associated with the QCD and QED coupling constants when compared to the QED processes. The main reason lies on the fact that the virtuality of the two virtual gluons are both of $m_b^{2}$ order in the QCD process while the virtuality of the photon is fixed to $4m_c^2$ in the QED processes dominated by $\Upsilon\to gg\gamma^{\ast}(J/\psi)$, and moreover the four-body phase space of the QCD process is also less than the three-body one of the QED processes. On the experimental side, the CLEO collaboration find[@Briere:2004ug] that the feed-down of $\chi_{cJ}$ to $J/\psi$ are $<8.2,11,10$ percent for $J=0,1,2$ respectively and the feed-down of $\psi(2S)$ is about $24$ percent in $\Upsilon \rightarrow J/\psi+X$. Therefore it indicates that the experimental result of direct $J/\psi$ production would be $$\mathcal{B_\mathrm{Direct}}(\Upsilon\to J/\psi+X)=3.52\times10^{-4}$$ which is about 9 times larger than the presented theoretical results based on the color-singlet calculations. This means that unlike the conclusion before[@Li:1999ar] the branching ratio of $\Upsilon\to J/\psi+X$ can not be explained by color-singlet model at the leading order. From the theoretical point of view, the color-octet mechanism can account for most $J/\psi$ production, but its predictions for the $J/\psi$ momentum spectrum is not agree with the experimental data. The color-singlet predictions on the shape of the $J/\psi$ momentum spectrum is more closer to the experimental result, but the discrepancy of the branching ratio between them is large. For all the numerical results, we used the theoretically normalized decay width to estimate the branching ratio. Alternatively, by using $\langle\Upsilon|O_1(^3S_1)|\Upsilon\rangle=2.9\mathrm{GeV^3}$[@Cheung:1996mh] to calculate the partial decay width and choosing the total decay width of $\Upsilon$ $51.4$ keV from the experimental measurement[@Amsler:2008zzb], the branching ratio will be enhanced by a factor of about 3, which still can not explain the experimental results. Therefore, it means that the NLO QCD correction is important, just like in the known cases, the NLO QCD corrections for $J/\psi$ production in $e^{+}e^{-}$ annihilation show that the $K$-factor are about $1.97$ and $1.2$ for $e^{+}e^{-}\to\gamma^{\ast}\to J/\psi+c\bar{c}$ and $e^{+}e^{-}\to\gamma^{\ast}\to J/\psi+gg$ processes respectively; the NLO QCD correction in $J/\psi$ related $\Upsilon$ exclusive decays are also found quite important [@Hao:2006nf]. In addition, the contribution of $\mathcal{O}(\alpha_s^6)$ processes $b\bar{b}(^3S_1,1)\to c\bar{c}(^3S_1,1)+gg$ and $b\bar{b}(^3S_1,1)\to c\bar{c}(^3S_1,1)+gggg$ to the branching ratio has been estimated to be of $10^{-4}$ order[@Trottier:1993ze]. So that the next important step is to give an explicit and complete calculations of them, which will be very helpful to understand the conflict between the theory and experiment. Furthermore, to obtain the full QCD correction for the inclusive $J/\psi$ production in $\Upsilon$ decay would be a very interesting and challenge work for explaining the experimental data. But it will involve very complicated work at the QCD NLO and is beyond the scope of this work. {#section .unnumbered} We thank Dr. S. Y. Li for helpful discussions. This work was supported by the National Natural Science Foundation of China (No. 10775141) and Chinese Academy of Sciences under Project No. KJCX3-SYW-N2. Zhiguo He is currently supported by the CPAN08-PD14 contract of the CSD2007-00042 Consolider-Ingenio 2010 program, and by the FPA2007-66665-C02-01/ project (Spain). [99]{} N. Brambilla [*et al.*]{} \[Quarkonium Working Group\], \[arXiv:hep-ph/0412158\]. G. T. Bodwin, E. Braaten and G. P. Lepage, Phys. Rev.  D [**51**]{}, 1125 (1995) \[Erratum-ibid.  D [**55**]{}, 5853 (1997)\] \[arXiv:hep-ph/9407339\]. F. Abe [*et al.*]{} \[CDF Collaboration\], Phys. Rev. Lett.  [**69**]{}, 3704 (1992). E. Braaten and S. Fleming, Phys. Rev. Lett.  [**74**]{}, 3327 (1995) \[arXiv:hep-ph/9411365\]. A. A. Affolder [*et al.*]{} \[CDF Collaboration\], Phys. Rev. Lett.  [**85**]{}, 2886 (2000). B. Gong, X. Q. Li and J. X. Wang, Phys. Lett.  B [**673**]{}, 197 (2009). J. M. Campbell, F. Maltoni and F. Tramontano, Phys. Rev. Lett.  [**98**]{}, 252002 (2007) \[arXiv:hep-ph/0703113\]; B. Gong and J. X. Wang, Phys. Rev. Lett.  [**100**]{}, 232001 (2008) \[arXiv:0802.3727 \[hep-ph\]\]; B. Gong and J. X. Wang, Phys. Rev.  D [**78**]{}, 074011 (2008) \[arXiv:0805.2469 \[hep-ph\]\]. P. Artoisenet, J. M. Campbell, J. P. Lansberg, F. Maltoni and F. Tramontano, Phys. Rev. Lett.  [**101**]{}, 152001 (2008) \[arXiv:0806.3282 \[hep-ph\]\]. D. E. Acosta [*et al.*]{} \[CDF Collaboration\], Phys. Rev. Lett.  [**88**]{}, 161802 (2002). V. M. Abazov [*et al.*]{} \[D0 Collaboration\], Phys. Rev. Lett.  [**101**]{}, 182004 (2008) \[arXiv:0804.2799 \[hep-ex\]\]. E. Braaten and Y. Q. Chen, Phys. Rev. Lett.  [**76**]{}, 730 (1996) \[arXiv:hep-ph/9508373\]; F. Yuan, C. F. Qiao and K. T. Chao, Phys. Rev.  D [**56**]{}, 321 (1997) \[arXiv:hep-ph/9703438\]. B. Aubert [*et al.*]{} \[BABAR Collaboration\], Phys. Rev. Lett.  [**87**]{}, 162002 (2001) \[arXiv:hep-ex/0106044\]. K. Abe [*et al.*]{} \[BELLE Collaboration\], Phys. Rev. Lett.  [**88**]{}, 052001 (2002) \[arXiv:hep-ex/0110012\]. S. Fleming, A. K. Leibovich and T. Mehen, Phys. Rev.  D [**68**]{}, 094011 (2003) \[arXiv:hep-ph/0306139\]. K. Abe [*et al.*]{} \[Belle Collaboration\], Phys. Rev. Lett.  [**89**]{}, 142001 (2002) \[arXiv:hep-ex/0205104\]. P. Pakhlov [*et al.*]{} \[Belle Collaboration\], arXiv:0901.2775 \[hep-ex\]. Y. J. Zhang, Y. j. Gao and K. T. Chao, Phys. Rev. Lett.  [**96**]{}, 092001 (2006) \[arXiv:hep-ph/0506076\]. B. Gong and J. X. Wang, Phys. Rev.  D [**80**]{}, 054015 (2009) arXiv:0904.1103 \[hep-ph\]. Y. Q. Ma, Y. J. Zhang and K. T. Chao, Phys. Rev. Lett.  [**102**]{}, 162002 (2009) \[arXiv:0812.5106 \[hep-ph\]\]. B. Gong and J. X. Wang, Phys. Rev. Lett.  [**102**]{}, 162003 (2009) \[arXiv:0901.0117 \[hep-ph\]\]; K. M. Cheung, W. Y. Keung and T. C. Yuan, Phys. Rev.  D [**54**]{}, 929 (1996) \[arXiv:hep-ph/9602423\]. M. Napsuciale, Phys. Rev.  D [**57**]{}, 5711 (1998) \[arXiv:hep-ph/9710488\]. R. Fulton [*et al.*]{} \[CLEO Collaboration\], Phys. Lett.  B [**224**]{}, 445 (1989). H. Albrecht [*et al.*]{} \[ARGUS Collaboration\], Z. Phys.  C [**55**]{}, 25 (1992). R. A. Briere [*et al.*]{} \[CLEO Collaboration\], Phys. Rev.  D [**70**]{}, 072001 (2004) \[arXiv:hep-ex/0407030\]. H. D. Trottier, Phys. Lett.  B [**320**]{}, 145 (1994) \[arXiv:hep-ph/9307315\]. S. Y. Li, Q. B. Xie and Q. Wang, Phys. Lett.  B [**482**]{}, 65 (2000) \[arXiv:hep-ph/9912328\]; W. Han and S. Y. Li, Phys. Rev.  D [**74**]{}, 117502 (2006) \[arXiv:hep-ph/0607251\]. H. Fritzsch and K. H. Streng, Phys. Lett.  B [**77**]{}, 299 (1978). I. I. Y. Bigi and S. Nussinov, Phys. Lett.  B [**82**]{}, 281 (1979). J. H. Kuhn, J. Kaplan and E. G. O. Safiani, Nucl. Phys.  B [**157**]{}, 125 (1979). B. Guberina, J. H. Kuhn, R. D. Peccei and R. Ruckl, Nucl. Phys.  B [**174**]{}, 317 (1980). P. L. Cho and A. K. Leibovich, Phys. Rev.  D [**53**]{}, 150 (1996) \[arXiv:hep-ph/9505329\]; P. L. Cho and A. K. Leibovich, Phys. Rev.  D [**53**]{}, 6203 (1996) \[arXiv:hep-ph/9511315\]; P. Ko, J. Lee and H. S. Song, Phys. Rev.  D [**54**]{}, 4312 (1996) \[Erratum-ibid.  D [**60**]{}, 119902 (1999)\] \[arXiv:hep-ph/9602223\]. J.-X. Wang, Nucl. Instrum. Meth. A [**534**]{}, 241 (2004). E. J. Eichten and C. Quigg, Phys. Rev.  D [**52**]{}, 1726 (1995) \[arXiv:hep-ph/9503356\]. Z. H. Lin and G. h. Zhu, Phys. Lett.  B [**597**]{}, 382 (2004) \[arXiv:hep-ph/0406121\]. C. Amsler [*et al.*]{} \[Particle Data Group\], Phys. Lett.  B [**667**]{}, 1 (2008). G. Hao, Y. Jia, C. F. Qiao and P. Sun, JHEP [**0702**]{}, 057 (2007) \[arXiv:hep-ph/0612173\]. B. Gong, Y. Jia and J. X. Wang, Phys. Lett.  B [**670**]{}, 350 (2009) \[arXiv:0808.1034 \[hep-ph\]\].

--- abstract: 'We study gravitational perturbations in the Randall–Sundrum two-brane background with scalar-curvature terms in the action for the branes, allowing for positive as well as negative bulk gravitational constant. In the zero-mode approximation, we derive the linearized gravitational equations, which have the same form as in the original Randall–Sundrum model but with different expressions for the effective physical constants. We develop a generic method for finding tachyonic modes in the theory, which, in the model under consideration, may exist only if the bulk gravitational constant is negative. In this case, if both brane gravitational constants are nonzero, the theory contains one or two tachyonic mass eigenvalues in the gravitational sector. If one of the brane gravitational constants is set to zero, then either a single tachyonic mass eigenvalue is present or tachyonic modes are totally absent depending on the relation between the nonzero brane gravitational constant and brane separation. In the case of negative bulk gravitational constant, the massive gravitational modes have ghost-like character, while the massless gravitational mode is not a ghost in the case where tachyons are absent.' address: | $^a$Bogolyubov Institute for Theoretical Physics, Kiev 03143, Ukraine\ $^b$Department of Physics, Taras Shevchenko National University, Kiev 03022, Ukraine author: - 'Yuri Shtanov$^a$[^1] and Alexander Viznyuk$^{a,\,b}$[^2]' title: 'Linearized gravity on the Randall–Sundrum two-brane background with curvature terms in the action for the branes' --- Introduction ============ Linear gravitational perturbations of the flat braneworld models were studied beginning from the seminal papers by Randall & Sundrum [@RS1; @RS2], where their spectrum was shown to contain, besides the zero mode, also an infinite tower of Kaluza–Klein massive modes. Since then, perturbations in various types of braneworld scenarios and in various approximations were considered in the flat case as well as on the cosmological background (which is a much more complicated issue still far from being well understood; see [@cosmology] and references therein). In this paper, we study certain aspects of linear perturbations on a particular simple background, which, to our knowledge, have not been previously investigated. We consider the Randall–Sundrum two-brane model (the so-called RS1 model [@RS1]) supplemented by scalar-curvature terms in the action for both branes. Historically, the induced scalar-curvature term for the brane was introduced in [@DGP] (see also [@CHS]) as a method of making gravity on the brane effectively four-dimensional even in the flat infinite bulk space. The corresponding cosmological models were initiated in [@CHS; @DDG]. The effects of these scalar-curvature terms in the two-brane setup under consideration were recently studied in [@effects; @CD; @Padilla; @Smolyakov; @mirror]. Perturbations in the one-brane (RS2 [@RS2]) counterpart of this model were previously investigated in [@KTT; @Tanaka]. The study of the braneworld models is usually confined to the case of positive bulk and brane gravitational constants. (The signs of the gravitational constants in this case are defined relative to the signs of the conventional matter Lagrangians on the branes.) This can be explained by the fact that only positive values of the bulk gravitational constant are allowed in the Randall–Sundrum model, since its sign coincides with the sign of the effective Newton’s constant in that model [@RS1; @RS2]. When one adds curvature terms in the action for the branes, this assumption might be relaxed, and, in this paper, we allow for positive as well as negative bulk gravitational constant while keeping gravitational constants on the branes positive. More generally, one could consider various relations between the signs of the four-dimensional gravitational constants in the action for the branes, bulk gravitational constant, and conventional matter Lagrangians [@CD; @Padilla; @Smolyakov; @mirror]. Although we are not aware of any fundamental multidimensional theory that can produce different signs of the gravitational couplings for the bulk and branes, we consider this possibility from the viewpoint of the effective action regardless of the unknown underlying theory. The study of both signs of the bulk gravitational constant is partially motivated by the existence of braneworld models with interesting behaviour that require negative brane tension. One of them is the model of [*disappearing dark energy*]{} (DDE) [@SS; @AS]. The DDE model is a braneworld model of expanding universe which, after the current period of acceleration, re-enters the matter-dominated regime continuing indefinitely in the future. The merit of this model of dark energy is the absence of the cosmological event horizon owing to the fact that the universe becomes flat, rather than De Sitter, in the asymptotic future. The DDE model is based on the generic braneworld action with the bulk cosmological constant and brane tension satisfying the Randall–Sundrum constraint, and also including the curvature term in the action for the brane. For the consistency of this model with the current cosmological observations (specifically, for the condition $\Omega_{\rm m} < 1$ on the dark-matter cosmological parameter), the brane tension has to be [*negative*]{} [@SS]. Negative brane tension is required also for the existence of unusual ‘quiescent’ singularities [@SS1] in the AdS-embedded braneworld models, which occur during the universe expansion and are characterized by [*finiteness*]{} of the scale factor, Hubble parameter, and matter density. The bulk gravitational constant enters the homogeneous cosmological equations on the brane in even power; therefore, its sign does not matter on the level of the homogeneous cosmology on the brane [@SS]. However, the relation between the sign of the bulk gravitational constant and the sign of the brane tension is of importance for the small-scale gravitational physics in the braneworld, in particular, for the behaviour of cosmological perturbations. This can be seen already from the fact that the property of ‘localization’ of five-dimensional gravity in the neighbourhood of the brane (that the warp factor locally decreases as one moves away from the brane) requires the brane tension and the bulk gravitational constant to be of the same sign. This localization property may turn out to be important for a consistent braneworld theory and, therefore, for negative-tension branes, it may require negative gravitational constant in the bulk. Thus, it seems important to keep open this possibility when generalizing the Randall–Sundrum model by including the curvature terms in the action for the branes. The issue of ghosts in a theory with positive bulk gravitational constant but arbitrary signs of the [*brane*]{} gravitational constants was recently under investigation in [@Padilla], and certain regions of parameters were ruled out. In this paper, we consider somewhat complimentary situation where the brane gravitational constants are always positive with respect to the conventional matter Lagrangian, while the bulk gravitational constant can be of any sign.[^3] We will be mainly concerned with the issue of tachyons in the theory of this kind, which, to our knowledge, was not discussed in the literature before. Our original method of finding tachyonic modes in the two-brane background is quite general and can be applied to other braneworld models leading to the same system of equations for the bulk gravitational modes, in particular, to the model with the Gauss–Bonnet action in the bulk considered in [@CD]. It will be shown below that, unlike in the pure Randall–Sundrum case, the presence of the curvature terms in the action for the branes leads to a possibility of unwanted tachyonic modes if the bulk gravitational constant is negative. We demonstrate that, in this case, there can be only one or two tachyonic mass eigenvalues in the theory under consideration and determine the range of parameters for which tachyonic modes do not exist. We will also see that the massive gravitational modes have ghost-like character in the case of negative bulk gravitational constant, while the massless gravitational mode is not a ghost in the case where tachyons are absent. These results are in agreement with those of [@Padilla]. Thus, it is mainly the presence of ghosts in the Kaluza–Klein massive spectrum of gravity that makes models with negative bulk gravitational constant problematic. This paper is organized in the following manner. After describing the model, we review a suitable theory of linear gravitational perturbations on the Randall–Sundrum two-brane background. Then, in the zero-mode approximation, we derive the linearized system of gravitational equations with matter confined to the visible and/or hidden brane. These equations have the same form as in the original RS1 model but with different physical constants. After that, we specially investigate the case of negative bulk gravitational constant and show that the linearized theory can contain tachyonic gravitational modes. In this case, one or two tachyonic mass eigenvalues are observed if both brane gravitational constants are nonzero. If one of the brane gravitational constants is zero, then either a single tachyonic mass eigenvalue is present or tachyonic modes are totally absent; which of these two possibility is realized depends upon the values of the nonzero brane gravitational constant and brane separation. We determine the range of parameters for which tachyonic modes are present or absent. Following [@Padilla], we also consider the ghost modes for the radion and graviton in our theory. In the case of negative bulk gravitational constant, the range of parameters where the radion is ghost free is rather narrow, and all massive gravitational modes have ghost-like nature. We also calculate the effective gravitational potentials of static matter sources on the visible and hidden brane in the generic case. The model ========= The action of the model in the neighbourhood of one brane has the form $$\label{action} S = M^3 \left[\int_{\rm bulk}\!\! \left( {\cal R} - 2 \Lambda \right) - 2 \int_{\rm brane}\!\! K \right] + \int_{\rm brane}\!\! \left( m^2 R - 2 \sigma \right) + \int_{\rm brane}\!\! L \left( h_{ab}, \phi \right) \, ,$$ where the first part, proportional to the cube of the bulk Planck mass $M^3$, describes the bulk bounded by the brane, and the remaining integrals are taken over the brane. Here, ${\cal R}$ is the curvature scalar in the bulk, $R$ is the curvature scalar of the induced metric $h_{ab}$ on the brane, $K$ is the trace of the tensor of the intrinsic curvature $K_{ab}$ of the brane with respect to the inner normal, $L \left(h_{ab}, \phi \right)$ is the Lagrangian of the matter fields $\phi$ on the brane, and integration in (\[action\]) implies natural volume elements in the bulk and on the brane. The action is similar in the neighbourhood of the other brane. In principle, two branes in our model may have different Planck masses $m$, and, to allow for solutions with flat vacuum branes, their tensions $\sigma$ must have opposite signs and satisfy the well-known constraint [@RS1; @RS2] $$\label{lambda-rs} \Lambda_{\rm RS} \equiv {\Lambda \over 2} + {\sigma^2 \over 3 M^6} = 0 \, .$$ Note that, in this paper, we allow for positive as well as negative signs of the bulk Planck mass parameter $M$, the consequences of which will become clear later. Action (\[action\]) leads to the bulk described by the usual Einstein equation with cosmological constant: $$\label{bulk} {\cal G}_{ab} + \Lambda g_{ab} = 0 \, ,$$ while the field equation on the brane is $$\label{brane} m^2 G_{ab} + \sigma h_{ab} = \tau_{ab} + M^3 \left(K_{ab} - h_{ab} K \right) \, ,$$ where $\tau_{ab}$ is the stress–energy tensor on the brane stemming from the last term in action (\[action\]). By contracting the Gauss identity $$\label{Gauss} R_{abc}{}^d = h_a{}^f h_b{}^g h_c{}^k h^d{}_j {\cal R}_{fgk}{}^j + K_{ac} K_b{}^d - K_{bc} K_a{}^d$$ on the brane and using Eq. (\[bulk\]), one obtains the ‘constraint’ equation $$\label{constraint} R - 2 \Lambda + K_{ab} K^{ab} - K^2 = 0\, ,$$ which, together with (\[brane\]), implies the following closed scalar equation on the brane: $$\label{closed} M^6 \left( R - 2 \Lambda \right) + \left( m^2 G_{ab} + \sigma h_{ab} - \tau_{ab} \right) \left( m^2 G^{ab} + \sigma h^{ab} - \tau^{ab} \right) - {1 \over 3} \left( m^2 R - 4 \sigma + \tau \right)^2 = 0\, ,$$ where $\tau = h^{ab} \tau_{ab}$. In the case of a vacuum brane ($\tau_{ab} = 0$), Eq. (\[closed\]) takes the form $$\label{vacuum} \left(M^6 + \frac23 \sigma m^2 \right) R + m^4 \left( R_{ab} R^{ab} - \frac13 R^2 \right) - 4 M^6 \Lambda_{\rm RS} = 0 \, ,$$ where $\Lambda_{\rm RS}$ is given by Eq. (\[lambda-rs\]). It should be noted that the second term in Eq. (\[vacuum\]) has [*precisely*]{} the form of one of the terms in the expression for the conformal anomaly, which describes the vacuum polarization at the one-loop level in curved space-time (see, e.g., [@BD]).[^4] Another useful relation is the Codazzi identity $$\label{Codazzi} D_a \left(K^a{}_b - h^a{}_b K \right) = 0 \, ,$$ which is valid at any timelike hypersurface in the bulk, in particular, on the branes, due to Eq. (\[bulk\]). Here, $D_a$ denotes the unique covariant derivative on the timelike hypersurface associated with the induced metric $h_{ab}$. The gravitational equations in the bulk can be integrated by using Gaussian normal coordinates, as described, e.g., in [@Shtanov]. Specifically, in the Gaussian normal coordinates $(x, y)$, where $x = \{x^\alpha\}$ are the coordinates on the brane and $y$ is the fifth coordinate in the bulk, the metric is written as $$\label{bulk-metric} d s^2 = dy^2 + h_{\alpha\beta} (x, y) dx^\alpha dx^\beta \, .$$ Introducing also the tensor of extrinsic curvature $K_{ab}$ of every hypersurface $y = {\rm const}$, one can obtain the following system of differential equations for the components $h_{\alpha\beta}$ and $K^\alpha{}_\beta$: $$\begin{aligned} \label{s} {\partial K^\alpha{}_\beta \over \partial y} &=& R^\alpha{}_\beta - K K^\alpha{}_\beta - \frac16 \delta^\alpha{}_\beta \left( R + 2 \Lambda + K^\mu{}_\nu K^\nu{}_\mu - K^2 \right) \nonumber \\ &=& R^\alpha{}_\beta - K K^\alpha{}_\beta - \frac23 \delta^\alpha{}_\beta \Lambda \, ,\end{aligned}$$ $$\label{metric} {\partial h_{\alpha\beta} \over \partial y} = 2 h_{\alpha\gamma} K^\gamma{}_\beta \, ,$$ where $R^\alpha{}_\beta$ are the components of the Ricci tensor of the metric $h_{\alpha\beta}$ induced on the hypersurface $y = {\rm const}$, $R = R^\alpha{}_\alpha$ is its scalar curvature, and $K = K^\alpha{}_\alpha$ is the trace of the tensor of extrinsic curvature. The second equality in (\[s\]) is true by virtue of the ‘constraint’ equation (\[constraint\]). Equations (\[s\]) and (\[metric\]) together with the ‘constraint’ equation (\[constraint\]) represent the $4\!+\!1$ splitting of the Einstein equations in Gaussian normal coordinates. The initial conditions for these equations are defined on the brane through Eq. (\[brane\]). Linear perturbations ==================== Linear perturbations of the RS1 model are well studied (see, e.g., [@linear; @SV] and references therein). Here, we would like to see the modifications arising from the presence of the scalar-curvature terms in the action for the branes (nonzero values of the masses $m$ and $m_*$). Several aspects of this setup were also studied in [@CD; @Padilla; @Smolyakov; @mirror]. Our treatment in this and in the subsequent section is similar to that of [@SV]. The perturbed metric of our solution in Gaussian normal coordinates has the form (\[bulk-metric\]) with $$\label{induced} h_{\alpha\beta} (x, y) = a^2 (y) \Bigl[ \eta_{\alpha\beta} + \gamma_{\alpha\beta} (x, y) \Bigr]\, ,$$ where $$\label{scale} \quad a(y) = \exp(-ky) \, , \quad k = {\sigma \over 3 M^3} \, ,$$ and we emphasize that $k$ can be positive as well as negative depending on the signs of $M$ and $\sigma$. The perturbations of the tensor of extrinsic curvature and of the Einstein tensor have the form $$\label{perturb} \delta K^\alpha{}_\beta = \frac12 {\partial \gamma^\alpha{}_\beta \over \partial y} \, , \quad G_{\alpha\beta} = - \frac12 \Box \bar \gamma_{\alpha\beta} + \partial^\gamma \partial_{(\alpha} \bar \gamma_{\beta)\gamma} - \frac12 \eta_{\alpha\beta} \partial^\gamma \partial^\delta \bar \gamma_{\gamma\delta}\, ,$$ where $\bar \gamma_{\alpha\beta} = \gamma_{\alpha\beta} - \frac12 \eta_{\alpha\beta} \gamma$, $\gamma = \gamma^\alpha{}_\alpha$, and $\Box = \partial^\alpha\partial_\alpha$. Here and below, the indices of $\partial_\alpha$ and $\gamma_{\alpha\beta}$ are raised and lowered with respect to the flat metric $\eta_{\alpha\beta}$. Using the freedom of choice of the coordinates $x^\alpha$ on the brane, one can choose the harmonic gauge in which $\partial^\alpha \bar \gamma_{\alpha\beta} = 0$ on one of the branes. In this gauge, we have $$\label{harmoncor} G_{\alpha\beta} = - \frac12 \Box \bar \gamma_{\alpha\beta}$$ on that brane. In the unperturbed solution, the first (visible) brane is assumed to be at $y = 0$, and the second (hidden) brane is at $y = \rho$. First, we consider the situation where the hidden brane does not have matter on it (stress–energy tensor equal to zero). Then, when studying perturbations, it is convenient to choose Gaussian normal coordinates with respect to the hidden brane. Thus, the hidden brane remains at $y = \rho$, while the position of the visible brane is linearly perturbed to become $y = \phi(x)$, which is the so-called radion degree of freedom. Let $m$ and $\sigma$ denote the Planck mass and tension of the visible brane, and let those of the hidden brane be $m_*$ and $\sigma_* = - \sigma$, respectively. The linearly perturbed boundary equation (\[brane\]) on the second (hidden) brane becomes $$- m_*^2 G^\alpha{}_\beta = M^3 \delta S^\alpha{}_\beta \, ,$$ where $S^\alpha{}_\beta = K^\alpha{}_\beta - \delta^\alpha{}_\beta K$, and we have taken into account that the extrinsic curvature is calculated with respect to the normal in the positive direction of $y$. Choosing harmonic coordinates on the hidden brane, for which (\[harmoncor\]) is satisfied, we have $$\label{hidden} {m_*^2 \over a_*^2} \Box \bar \gamma^\alpha{}_\beta = M^3 \left( {\partial \bar \gamma^\alpha{}_\beta \over \partial y} + \frac12 \delta^\alpha{}_\beta {\partial \bar \gamma \over \partial y} \right) \, ,$$ where $a_* = a(\rho) = e^{-k\rho}$ and $\bar \gamma = \bar \gamma^\alpha{}_\alpha$. Linearization of the vacuum constraint equation (\[vacuum\]) implies the condition $\Box \bar \gamma = 0$ on the hidden brane if $$\label{nonsin} M^6 + \frac23 \sigma_* m_*^2 \ne 0 \, ,$$ which we assume to be the case. This condition and Eq. (\[hidden\]) implies the condition $\partial \bar \gamma / \partial y = 0$ at the hidden brane. Then the Codazzi relation (\[Codazzi\]) implies the condition $\partial \left(\partial^\alpha \bar \gamma_{\alpha\beta}\right) / \partial y = 0$ at the same brane. Now we turn to Eqs. (\[s\]) and (\[metric\]). Using (\[perturb\]), we can write the second-order differential equations for perturbations $\bar \gamma_{\alpha\beta}$ in the bulk. First, we verify that the Gaussian normal coordinates $x^\alpha$ remain harmonic in the bulk. We introduce the quantity $$v^\alpha = \partial_\beta \bar \gamma^{\beta\alpha} \, ,$$ which is an indicator of the harmonicity of the coordinates $x^\alpha$ on the hypersurface $y = {\rm const}$. Then we can write the system of differential equations for $v^\alpha$ and $\bar \gamma$ that stems from system (\[s\]), (\[metric\]): $$\label{harmon} {\partial^2 v^\alpha \over \partial y^2} = 4k {\partial v^\alpha \over \partial y} + k {\partial \left(\partial^\alpha \bar \gamma \right) \over \partial y} \, , \qquad {\partial^2 \bar \gamma \over \partial y^2} = - \frac{1}{a^2} \left( 2 \partial_\alpha v^\alpha + \Box \bar \gamma \right) + 8k {\partial \bar \gamma \over \partial y} \, ,$$ with the following boundary conditions at the hidden brane ($y = \rho$): $$\label{init-harmon} v^\alpha = 0 \, , \quad {\partial v^\alpha \over \partial y} = 0 \, , \quad \Box \bar \gamma = 0 \, , \quad {\partial \bar \gamma \over \partial y} = 0 \, .$$ The unique solution of system (\[harmon\]) in the bulk with the boundary conditions (\[init-harmon\]) is $$\label{harmonrel} v^\alpha (x, y) \equiv 0 \, , \quad \bar \gamma (x, y) \equiv \bar \gamma (x) \, , \quad \Box \bar \gamma (x) = 0 \, .$$ In particular, this means that the Gaussian normal coordinates which we are using remain harmonic with respect to $x$ all over the bulk. Taking into account relations (\[harmonrel\]), from (\[s\]), (\[metric\]) one obtains the system of equations for perturbations in the bulk: $$\label{eq-bulk} {\partial^2 \bar \gamma_{\alpha\beta} \over \partial y^2} - 4k {\partial \bar \gamma_{\alpha\beta} \over \partial y} + {1 \over a^2} \Box \bar \gamma_{\alpha\beta} = 0$$ with the boundary condition at $y = \rho$ which stems from (\[hidden\]): $$\label{b-hidden} {\partial \bar \gamma_{\alpha\beta} \over \partial y} = {m_*^2 \over M^3 a_*^2} \Box \bar \gamma_{\alpha\beta} \, .$$ To obtain the boundary equations on the visible brane, one must take into account its ‘bending’ in the bulk: $y = \phi(x)$. The induced metric on the visible brane in the linear approximation becomes $$\label{indmet} h^{\rm vis}_{\alpha\beta} = (1 - 2 k \phi) \eta_{\alpha\beta} + \gamma_{\alpha\beta} \, ,$$ so that its perturbation is $$\quad \gamma^{\rm vis}_{\alpha\beta} = \gamma_{\alpha\beta} - 2 k \phi \eta_{\alpha\beta} \, , \quad \bar \gamma^{\rm vis}_{\alpha\beta} = \bar \gamma_{\alpha\beta} + 2 k \phi \eta_{\alpha\beta} \, .$$ Substituting it to the boundary condition at the visible brane $$m^2 G^\alpha{}_\beta = M^3 \delta S^\alpha{}_\beta + \tau^\alpha{}_\beta \, ,$$ we obtain the boundary condition at $y = 0$: $$\label{g} m^2 G_{\alpha\beta} \equiv -{m^2 \over 2} \Box \bar \gamma_{\alpha\beta} + 2 m^2 k \left( \partial_\alpha \partial_\beta - \eta_{\alpha\beta} \Box \right) \phi = \tau_{\alpha\beta} + M^3 \left( \eta_{\alpha\beta} \Box - \partial_\alpha \partial_\beta \right) \phi + \frac12 M^3 {\partial \bar \gamma_{\alpha\beta} \over \partial y} \, .$$ Taking trace of this equation, we obtain the equation for the radion field $\phi$: $$\label{radion} - 3 A \Box \phi = \tau \, ,$$ where $\tau \equiv \eta^{\alpha\beta} \tau_{\alpha\beta}$ is the trace of the stress–energy tensor, and $A = M^3 + 2 k m^2$. Thus, the radion field is coupled to the trace of the stress–energy tensor, as is the case in the Randall–Sundrum model [@linear], but with different coupling constant. Using Eq. (\[radion\]), from (\[g\]) we obtain $$\label{b-visible} -{m^2 \over 2} \Box \bar \gamma_{\alpha\beta} = \tau_{\alpha\beta} - \frac13 \eta_{\alpha\beta} \tau - A \partial_\alpha \partial_\beta \phi + \frac12 M^3 {\partial \bar \gamma_{\alpha\beta} \over \partial y} \, .$$ Now we have to solve the bulk equations (\[eq-bulk\]) with the boundary conditions (\[radion\]), (\[b-hidden\]) and (\[b-visible\]). Proceeding to the Fourier transform with momenta $p_\alpha$ in the coordinates $x^\alpha$ and omitting the tensor indices, we have for the Fourier image $\psi (q, y)$ of $\bar \gamma_{\alpha\beta} (x, y)$: $$\label{eq-psi} \psi'' - 4 k \psi' + q^2 e^{2 k y} \psi = 0 \, ,$$ where the prime denotes the derivative with respect to $y$, and $q = \sqrt{- p^2}$ (here we assume $p^2 \equiv p^\alpha p_\alpha \le 0$; the tachyonic case will be studied in Sec. \[tachyon\]). After the standard change of variable and function $$z (y) = {q e^{ky} \over k} \, , \qquad \psi (z) = z^2 \chi (z) \, ,$$ we get the equation $$\label{eq-chi} z^2 \chi'' + z \chi' + \left(z^2 - 4\right) \chi = 0 \, ,$$ in which the prime denotes the derivative with respect to $z$. Note that $z (y)$ is a monotonic function of $y$ for both signs of $k$, but the sign of $z$ coincides with the sign of the constant $k$. The boundary conditions follow from (\[b-visible\]) and (\[b-hidden\]): $$\begin{aligned} \label{b1-vis} - m^2 k^2 z_0^2 \chi (z_0) &=& {2 T \over z_0^2} + k M^3 \Bigl[ z_0 \chi' (z_0) + 2 \chi (z_0) \Bigr] \, , \quad z_0 = z(0) = \frac{q}{k} \, , \\ \label{b1-hid} m_*^2 k^2 z_*^2 \chi (z_*) &=& k M^3 \Bigl[ z_* \chi' (z_*) + 2 \chi (z_*) \Bigr] \, , \quad z_* = z(\rho) = \frac{q}{k} e^{k\rho} \, ,\end{aligned}$$ where $T$ stands for the Fourier transform of the expression $$\label{t} T_{\alpha\beta} = \tau_{\alpha\beta} - \frac13 \eta_{\alpha\beta} \tau - A \partial_\alpha \partial_\beta \phi$$ with tensor indices omitted. The general solution of the Bessel equation (\[eq-chi\]) is given by $$\label{chi} \chi (z) = P J_2 \Bigl(|z|\Bigr) + Q Y_2 \Bigl(|z|\Bigr) \, ,$$ where $J_2$ and $Y_2$ are the Bessel functions, $P$ and $Q$ are constants, and the modulus of $z$ reflects the fact that the domain of $z$ is positive or negative depending on the sign of $k$. Using the recurrence relations $$\label{recurrence} z J'_2 (z) + 2 J_2 (z) = z J_1 (z) \, , \quad z Y'_2 (z) + 2 Y_2 (z) = z Y_1 (z) \, ,$$ we obtain from (\[b1-vis\]) and (\[b1-hid\]): $$\begin{aligned} \label{b2-vis} - m^2 k^2 z_0^2 \left[ P J_2 \Bigl(|z_0|\Bigr) + Q Y_2 \Bigl(|z_0|\Bigr) \right] &=& {2 T \over z_0^2} + k M^3 |z_0| \left[ P J_1 \Bigl(|z_0|\Bigr) + Q Y_1 \Bigl(|z_0|\Bigr) \right] \, , \\ \label{b2-hid} m_*^2 k^2 z_*^2 \left[ P J_2 \Bigl(|z_*|\Bigr) + Q Y_2 \Bigl(|z_*|\Bigr) \right] &=& k M^3 |z_*| \left[ P J_1 \Bigl(|z_*|\Bigr) + Q Y_1 \Bigl(|z_*|\Bigr) \right] \, ,\end{aligned}$$ solving which, one finds the constants $P$ and $Q$ and obtains the solution for $\psi (z)$: $$\label{psi} \psi (z) = \left({6 T z^2 \over \sigma |z_0|^3}\right) {C_Y^* J_2 \Bigl( |z| \Bigr) - C_J^* Y_2 \Bigl( |z| \Bigr) \over C_Y^0 C_J^* - C_J^0 C_Y^*} \, ,$$ where the constants are given by $$\label{cs} \begin{array}{l} C_Y^0 = Y_1 \Bigl( |z_0| \Bigr) + \displaystyle {m^2 \over M^3} k |z_0| Y_2 \Bigl( |z_0| \Bigr) \, , \quad C_J^0 = J_1 \Bigl( |z_0| \Bigr) + {m^2 \over M^3} k |z_0| J_2 \Bigl( |z_0| \Bigr) \, , \medskip \\ C_Y^* = Y_1 \Bigl( |z_*| \Bigr) - \displaystyle {m_*^2 \over M^3} k |z_*| Y_2 \Bigl( |z_*| \Bigr) \, , \quad C_J^* = J_1 \Bigl( |z_*| \Bigr) - {m_*^2 \over M^3} k |z_*| J_2 \Bigl( |z_*| \Bigr) \, . \end{array}$$ These results differ from the similar results [@linear; @SV] of the RS1 model by the presence of the terms containing the brane Planck masses $m$ and $m_*$ in Eqs. (\[cs\]). The spectrum of the model is determined by the equality of the denominator of (\[psi\]) to zero. Introducing the dimensionless variable $s = q/|k|$ and parameters $\mu = k m^2 / M^3$, $\mu_* = k m_*^2 / M^3$, and $\alpha = e^{k\rho}$, we obtain the following equation for the spectrum: $$\label{spec} F_1(s) + \mu s F(s) + \alpha \mu_* s F_*(s) + \alpha \mu \mu_* s^2 F_2 (s) = 0 \, ,$$ where $$\begin{array}{l}\label{fs} F_1(s) = J_1 (s) Y_1 (\alpha s) - J_1 (\alpha s) Y_1 (s) \, , \\ F(s) = J_2 (s) Y_1 (\alpha s) - J_1 (\alpha s) Y_2 (s) \, , \\ F_*(s) = J_2 (\alpha s) Y_1 (s) - J_1 (s) Y_2 (\alpha s) \, , \\ F_2(s) = J_2 (\alpha s) Y_2 (s) - J_2 (s) Y_2 (\alpha s) \, . \end{array}$$ If both masses $m$ and $m_*$ are nonzero, then the ultraviolet asymptotics of the spectrum for the Kaluza–Klein modes is determined by the zeros of the last term in (\[spec\]), so that $$s_n \sim {\pi n \over \alpha - 1} \, , \quad n \gg 1 \, ,$$ which coincides with the asymptotics of the spectrum in the Randall–Sundrum model, determined by the zeros of the first term in (\[spec\]). If $m \ne 0$, $m_* = 0$, then the asymptotics of the spectrum is determined by the second term in (\[spec\]): $$s_n \sim {\pi n - \frac{\pi}{2}\over \alpha - 1} \, , \quad n \gg 1 \, .$$ If $m = 0$, $m_* \ne 0$, then it is determined by the third term in (\[spec\]): $$s_n \sim {\pi n + \frac{\pi}{2}\over \alpha - 1} \, , \quad n \gg 1 \, .$$ In these last two cases, the spectrum is somewhat shifted. Linearized equations in the zero-mode approximation =================================================== In the zero-mode approximation [@linear; @SV], one considers the limit as $q \to 0$. In this limit, using Eqs. (\[g\]), (\[radion\]), and (\[psi\]) and expanding all functions of $q = \sqrt{- p^2}$ in powers of $q$, we obtain the linearized gravity equation in the case of matter present only on the visible brane: $$\label{g-vis} G_{\alpha\beta} = {2 k \over A - B e^{- 2 k\rho}} \left[ \tau_{\alpha\beta} - {B e^{-2 k\rho} \over 3 A} \left( \eta_{\alpha\beta} - {p_\alpha p_\beta \over p^2} \right) \tau \right] + {\cal O} \left(p^2\right) \, ,$$ where it should be stressed that $G_{\alpha\beta}$ is the Einstein tensor of the induced metric (\[indmet\]) on the brane, and the constants $A$ and $B$ are given by[^5] $$\label{AB} A = M^3 + 2 k m^2 \, , \quad B = M^3 - 2 k m_*^2 \, .$$ The effective Newton’s constant $G_{\rm N}$ is given by the relation $$\label{newton} 8 \pi G_{\rm N} = {2 k \over A - B e^{- 2 k\rho}} \, ,$$ and one should note the extra contribution from the radion in (\[g-vis\]), which involves the trace of the stress–energy tensor.[^6] If $k > 0$, this contribution is exponentially suppressed for large separations between the branes, $k\rho \gg 1$. If matter is present only on the hidden brane, then it still induces curvature on the visible brane [@linear; @SV]. In our theory, we obtain the result $$\label{g-hid} G_{\alpha\beta} = {2 k \over A e^{2 k\rho} - B } \left[ \tau^*_{\alpha\beta} - {e^{- 2 k\rho} \over 3} \left( \eta_{\alpha\beta} - {p_\alpha p_\beta \over p^2} \right) \tau^* \right] + {\cal O} \left(p^2\right) \, ,$$ where $\tau^*_{\alpha\beta}$ is the stress–energy tensor on the hidden brane, and $\tau^*$ is its trace. If both branes contain matter, then the results on the right-hand sides of (\[g-vis\]) and (\[g-hid\]) simply add together. A few comments are in order about the obtained results. First of all, in the limit of zero Planck masses for the branes, $m = m_* = 0$, we have $A = B = M^3$, and they turn to the results previously obtained for the Randall–Sundrum two-brane model [@linear; @SV]. The presence of two new mass parameters $m$ and $m_*$ extends the freedom of the model. Thus, if the constant $B$ turns out to be sufficiently small, then the scalar contribution to the right-hand side of (\[g-vis\]), proportional to the trace of the stress–energy tensor, may become negligibly small. Note, however, that it is not possible to set either the constant $A$ or the constant $B$ exactly to zero in our expressions since, in this case, the theory becomes singular. This can be seen, e.g., from Eq. (\[vacuum\]), in which the first (linear in curvature) term is proportional to $A$ \[the same property is observed in the general equation (\[closed\])\]. In particular, the nonzero value of the constant $B$ was already assumed in the linearization scheme \[see Eq. (\[nonsin\])\]. The special cases where either $A$ or $B$ is equal to zero must be studied separately. Some results in this direction were recently reported in [@Smolyakov], where it was pointed out that the linearized theory possesses some additional symmetry in this case. To see how this degeneracy arises in some more detail, we turn to the Gauss identity (\[Gauss\]) again and, following the procedure first employed in [@SMS], contract it once on the brane using equations (\[bulk\]) and (\[brane\]). We obtain the effective equation on the brane that generalizes the result of [@SMS] to the presence of the brane curvature term: $$\label{effective} G_{ab} + \Lambda_{\rm RS} {M^3 \over A} h_{ab} = {2 \sigma \over 3 M^3 A} \tau_{ab} + {1 \over M^3 A} Q_{ab} - {M^3 \over A} W_{ab} \, ,$$ where $\Lambda_{\rm RS}$ is given by (\[lambda-rs\]), $$Q_{ab} = \frac13 E E_{ab} - E_{ac} E^{c}{}_b + \frac12 \left(E_{cd} E^{cd} - \frac13 E^2 \right) h_{ab}$$ is the quadratic expression with respect to the ‘bare’ Einstein equation $E_{ab} \equiv m^2 G_{ab} - \tau_{ab}$ on the brane, $E = h^{ab} E_{ab}$, and $W_{ab} \equiv h^c{}_a h^e{}_b W_{cdef} n^d n^f$ is a projection of the bulk Weyl tensor $W_{abcd}$ to the brane. One can see that [*all*]{} the couplings in (\[effective\]), including the effective cosmological and gravitational constants, are inversely proportional to the constant $A$, which indicates that the theory becomes degenerate in the case $A = 0$. In the absence of the curvature term on the brane ($m = 0$), we have $A = M^3$, which brings us to the original result of [@SMS]. Our second remark is that, unlike in the original Randall–Sundrum model ($m = m_* = 0$), in our theory the sign of the constant $M$ is not fixed by the zero-mode approximation: apart from the scalar contribution described by the trace of the stress–energy tensor, matching with the general-relativity limit fixes only the sign of the overall constant in (\[g-vis\]) and (\[g-hid\]). In particular, for a sufficiently small absolute value of $M$, namely, $|M^3 / 2 k| \ll m^2,\, m_*^2\,$, the sign of $M$ does not matter. In the formal limit $M \to 0$ with $k > 0$ (hence, $k \to + \infty$), the equation for the visible brane (\[g-vis\]) turns to the usual linearized Einstein equation. In the formal simultaneous limit $M \to 0$ and $\sigma \to 0$ so that $k = \sigma / M^3$ is fixed, expressions (\[g-vis\]) and (\[g-hid\]) become $$\label{lim-vis} G_{\alpha\beta} = {1 \over m^2 + m_*^2 e^{- 2 k\rho}} \left[ \tau_{\alpha\beta} + {m_*^2 e^{-2 k\rho} \over 3 m^2} \left( \eta_{\alpha\beta} - {p_\alpha p_\beta \over p^2} \right) \tau \right] + {\cal O} \left(p^2\right)$$ and $$\label{lim-hid} G_{\alpha\beta} = {1 \over m^2 e^{2 k\rho} + m_*^2 } \left[ \tau^*_{\alpha\beta} - {e^{- 2 k\rho} \over 3} \left( \eta_{\alpha\beta} - {p_\alpha p_\beta \over p^2} \right) \tau^* \right] + {\cal O} \left(p^2\right) \, ,$$ respectively. However, in the following section we will see that the massive gravitational modes in the theory with negative value of $M$ have ghost-like nature. Finally, we note that our result does not explicitly contain the constant $\sigma$ but contains it only in the combination $k = \sigma / 3 M^3$. Therefore, for one and the same effective law of gravity (\[g-vis\]) and (\[g-hid\]), the visible brane can have either positive or negative brane tension, depending on the sign of $M$. In particular, the zero-mode graviton is ‘localized’ around the visible brane ($k > 0$) even if its tension is negative, provided $M$ is also negative. If $k > 0$, then, in the limit $\rho \to \infty$, we pass to the one-brane model in Eq. (\[g-vis\]), which has the form of the corresponding equation of general relativity. Tachyonic modes and ghosts {#tachyon} ========================== In the original Randall–Sundrum model, negative values of the bulk Planck mass $M$ are nonphysical because this leads to negative effective Newton’s constant. This can be seen by setting $m = m_* = 0$ in Eq. (\[newton\]), thus having $A = B = M^3$ in it. If $M < 0$, then the effective Newton’s constant is negative for any sign of $k$, which means that the massless graviton becomes a ghost. The presence of the curvature terms in the action for the brane relaxes the situation with the massless gravitational modes and thus relaxes the necessity of dealing only with positive values of $M$. Negative values of $M$ are of interest in view of some of the braneworld cosmological models with [*negative*]{} brane tension, in particular, the model of disappearing dark energy [@SS; @AS], as discussed in the introduction. However, unlike in the pure Randall–Sundrum case, the presence of the curvature terms in the action for the branes leads to a possibility of unwanted tachyonic modes and ghost-like character of the massive modes in the gravitational sector of the theory if $M < 0$. In this section, we demonstrate that there can be only one or two tachyonic mass eigenvalues in the theory under consideration and determine the range of parameters for which tachyonic modes do not exist. We also show that the massive gravitational modes have ghost-like character in the case of negative $M$. In looking for tachyonic modes, one needs to solve Eq. (\[eq-psi\]) for $q^2 = - p^2 < 0$, i.e., $$\label{eqt-psi} \psi'' - 4 k \psi' - p^2 e^{2 k y} \psi = 0 \, , \quad p^2 > 0 \, ,$$ where the prime denotes the derivative with respect to $y$. After the standard change of variable and function $$z (y) = p e^{ky} / k \, , \qquad \psi (z) = z^2 \chi (z) \, ,$$ we get the equation for the new function $\chi(z)$: $$\label{eqt-chi} z^2 \chi'' + z \chi' - \left(z^2 + 4\right) \chi = 0 \, ,$$ in which the prime denotes the derivative with respect to $z$. With matter present on the visible brane only, the boundary conditions are similar to (\[b1-vis\]) and (\[b1-hid\]): $$\begin{aligned} \label{bt-vis} m^2 k^2 z_0^2 \chi (z_0) &=& {2 T \over z_0^2} + k M^3 \Bigl[ z_0 \chi' (z_0) + 2 \chi (z_0) \Bigr] \, , \quad z_0 = z(0) = \frac{p}{k} \, , \\ \label{bt-hid} - m_*^2 k^2 z_*^2 \chi (z_*) &=& k M^3 \Bigl[ z_* \chi' (z_*) + 2 \chi (z_*) \Bigr] \, , \quad z_* = z(\rho) = \frac{p}{k} e^{k\rho} \, ,\end{aligned}$$ where $T$ is the Fourier transform of expression (\[t\]), with tensor indices omitted. Solution of (\[eqt-chi\]) is now given by the modified Bessel functions $I_2$ and $K_2$: $$\label{t-chi} \chi (z) = P I_2 \Bigl(|z|\Bigr) + Q K_2 \Bigl(|z|\Bigr) \, ,$$ where $P$ and $Q$ are constants, and the modulus of $z$ again reflects the fact that the domain of $z$ is positive or negative depending on the sign of $k$. The recurrence relations of type (\[recurrence\]) are valid also for the modified Bessel functions: $$\label{recur-mod} z I'_2 (z) + 2 I_2 (z) = z I_1 (z) \, , \quad z K'_2 (z) + 2 K_2 (z) = - z K_1 (z) \, ,$$ and we can use them in deriving the solution similar to (\[psi\]): $$\label{psi-t} \psi (z) = \left({6 T z^2 \over \sigma |z_0|^3}\right) {C_K^* I_2 \Bigl( |z| \Bigr) + C_I^* K_2 \Bigl( |z| \Bigr) \over C_I^* C_K^0 - C_I^0 C_K^*} \, ,$$ where $$\label{Cs}\begin{array}{l} C_I^0 = I_1 \Bigl( |z_0| \Bigr) - \displaystyle {m^2 \over M^3} k |z_0| I_2 \Bigl( |z_0| \Bigr) \, , \quad C_K^0 = K_1 \Bigl( |z_0| \Bigr) + {m^2 \over M^3} k |z_0| K_2 \Bigl( |z_0| \Bigr) \medskip \\ C_I^* = I_1 \Bigl( |z_*| \Bigr) + \displaystyle {m_*^2 \over M^3} k |z_*| I_2 \Bigl( |z_*| \Bigr) \, , \quad C_K^* = K_1 \Bigl( |z_*| \Bigr) - {m_*^2 \over M^3} k |z_*| K_2 \Bigl( |z_*| \Bigr) \, . \end{array}$$ At this point, we note that the restriction to the brane at $y = 0$ and the limit of brane separation $\rho \to \infty$ brings expression (\[psi-t\]) to the form obtained for the one-brane case in [@Tanaka]. Our result generalizes it to the case of two branes with arbitrary sign of the bulk gravitational constant. Tachyonic modes correspond to those values of $p$ for which the denominator of (\[psi-t\]) turns to zero: $$C_I^* C_K^0 - C_I^0 C_K^* = 0 \, .$$ Since the transformation $p \to e^{- k\rho}\!p$ followed by $k \rightarrow - k$ and $m \leftrightarrow m_*$ does not change the spectrum of the theory, it is sufficient to study only the case $k > 0$. Using the dimensionless variable $s = p/k$ and parameters $\mu = k m^2 /M^3$, $\mu_* = k m_*^2 /M^3$, and $\alpha = e^{k\rho}$, we obtain the equation for tachyonic modes: $$\label{E} E(s) \equiv D_1(s) + \mu s D(s) + \alpha \mu_* s D_*(s) + \alpha \mu \mu_* s^2 D_2 (s) = 0 \, ,$$ where $$\begin{array}{l}\label{ds} D_1(s) = I_1 (\alpha s) K_1 (s) - I_1 (s) K_1 (\alpha s) \, , \\ D(s) = I_2 (s) K_1 (\alpha s) + I_1 (\alpha s) K_2 (s) \, , \\ D_*(s) = I_1 (s) K_2 (\alpha s) + I_2 (\alpha s) K_1 (s) \, , \\ D_2(s) = I_2 (\alpha s) K_2 (s) - I_2 (s) K_2 (\alpha s) \, . \end{array}$$ Since $\alpha > 1$ for $k > 0$, all functions in (\[ds\]) are strictly positive for positive $s$. This implies that tachyonic modes are absent in the case $\mu,\,\mu_* > 0$, or, equivalently, $M > 0$. However, tachyonic modes may be present in the opposite case $M < 0$. It is possible to indicate the corresponding range of parameters where tachyonic modes are present or absent. To do this, it is convenient to introduce the following auxiliary function of two variables $s$ and $\bar s$: $$\label{E2} \bar E(s, \bar s) \equiv D_1(s) + \mu \bar s D(s) + \alpha \mu_* \bar s D_*(s) + \alpha \mu \mu_* \bar s^2 D_2 (s)$$ \[to be compared with (\[E\])\]. By construction, $\bar E (s, s) \equiv E(s)$. First, we consider the case where both $\mu$ and $\mu_*$ are nonzero (in the present case, they are then both negative). Then the equation $$\label{bareq} \bar E (s, \bar s) = 0$$ gives the two branches of solutions with respect to $\bar s$: $$\label{bars} \bar s_\pm (s) = {\Bigl|\mu D(s) + \alpha \mu_* D_*(s) \Bigr| \pm \sqrt{\Bigl(\mu D(s) + \alpha \mu_* D_*(s)\Bigr)^2 - 4 \alpha \mu \mu_* D_1(s) D_2(s) } \over 2 \alpha \mu \mu_* D_2 (s) } \, ,$$ and solving the original equation (\[E\]) is equivalent to solving one of the equations $$\label{eqs} \bar s_\pm (s) = s \, .$$ It is easy to verify that the expression under the square root of (\[bars\]) is strictly positive for positive $s$ so that the two solutions $\bar s_\pm (s)$ exist for all $s >0$ and are positive. The asymptotic behavior of these solutions for small and large $s$ can easily be found: $$\bar s_+ (s) \sim {4 \Bigl| \alpha^2 \mu + \mu_* \Bigr| \over \left(\alpha^4 - 1\right) \mu \mu_* s } \, , \quad \bar s_-(s) \sim {\left(\alpha^2 - 1\right) s \over 2 \Bigl| \alpha^2 \mu + \mu_* \Bigr| } \, , \qquad s \to 0 \, ,$$ $$\bar s_\pm (s) \to {\Bigl| \mu + \alpha \mu_* \Bigr| \pm \Bigl| \mu - \alpha \mu_* \Bigr| \over 2 \alpha \mu \mu_*} = {\rm const} \, , \qquad s \to \infty \, .$$ From these expressions it is clear that the graph of $\bar s_+(s)$ crosses the graph of $f(s) = s$ at least once for any values of parameters in the range $\mu, \mu_* < 0$ under consideration. Thus, Eq. (\[eqs\]) has a solution, and at least one tachyonic mass eigenvalue is present in this range of parameters. Numerical computation indicates that there is exactly one solution connected with the branch $\bar s_+(s)$ in this case. It is also clear that the graph of $\bar s_-(s)$ definitely crosses the graph of $f(s) = s$ in the case where $\bar s'_-(0) > 1$, or $$\label{s_} {\left(\alpha^2 - 1\right) \over 2 \Bigl| \alpha^2 \mu + \mu_* \Bigr| } > 1 \, .$$ Again, numerical computation indicates that there is only one tachyonic solution connected with the branch $\bar s_-(s)$ in this case. They also indicate that tachyonic modes connected with the branch $\bar s_-(s)$ are absent in the case of the opposite inequality in (\[s\_\]). Thus, tachyonic modes exist for all values of parameters in the range $\mu, \mu_* < 0$. In the case under consideration, $M < 0$, one can expect tachyonic modes to be absent only if one of the brane Planck masses $m$ or $m_*$ is zero.[^7] We show that this is indeed the case and determine the range of masses and brane separations for which tachyonic modes are absent. In the case $\mu < 0$, $\mu_* = 0$, the function $\bar E(s, \bar s)$ given by Eq. (\[E2\]) takes the simple form $$\bar E(s, \bar s) \equiv D_1(s) + \mu \bar s D(s) \, ,$$ and Eq. (\[bareq\]) has a single solution with respect to $\bar s$: $$\bar s (s) = - {D_1(s) \over \mu D(s)} = {D_1 (s) \over |\mu| D(s)} \, .$$ It can be verified that the function $D_1(s)/D(s)$ is convex upwards, so that the equation $\bar s (s) = s$ has exactly one solution or no solutions in the range $s > 0$ depending on the value of the derivative $\bar s'(0)$. Specifically, a solution exists if $\bar s'(0) > 1$, and there are no solutions in the opposite case $\bar s'(0) \le 1$. Calculating the derivative $\bar s'(0)$, we obtain that exactly one tachyonic mass eigenvalue is present in the theory if $$|\mu| < \frac12 \left(1 - \alpha^{-2} \right) \, ,$$ and tachyonic modes are absent if the value of the Planck mass $m$ is sufficiently large, namely, if $$\label{mass} |\mu| \ge \frac12 \left(1 - \alpha^{-2} \right) \, .$$ In the limit of infinite separation between branes, $\alpha = e^{k\rho} \to \infty$, the condition of absence of tachyonic modes becomes $|\mu| \ge 1/2$, which coincides with the condition $A \ge 0$, where $A$ is given by (\[AB\]). Interestingly, this is also the condition of positivity of the effective Newton’s constant in the zero-mode approximation (\[g-vis\]), (\[g-hid\]) in the same limit. The case $\mu = 0$, $\mu_* < 0$ is analyzed in quite a similar way. Now the function $\bar E(s, \bar s)$ given by Eq. (\[E2\]) takes the form $$\bar E(s, \bar s) \equiv D_1(s) + \alpha \mu_* \bar s D_*(s) \, ,$$ and Eq. (\[bareq\]) has one solution with respect to $\bar s$: $$\bar s (s) = - {D_1(s) \over \alpha \mu_* D_*(s)} = {D_1 (s) \over \alpha |\mu_*| D_*(s)} \, .$$ Again, it can be verified that the equation $\bar s (s) = s$ has exactly one solution in the range $s > 0$ if $\bar s'(0) > 1$, and there are no solutions in the opposite case $\bar s' (0) \le 1$. Calculating the derivative $\bar s' (0)$, we obtain that exactly one tachyonic mass eigenvalue is present in the theory if $$|\mu_*| < \frac12 \left(\alpha^2 - 1 \right) \, ,$$ and tachyonic modes are absent if the value of the Planck mass $m_*$ is sufficiently large, namely, if $$\label{masstar} |\mu_*| \ge \frac12 \left(\alpha^2 - 1 \right) \, .$$ We note that our method of finding the range of parameters where tachyonic terms are present or absent is not restricted to the model under investigation and can be used whenever the equation for tachyonic modes has the form (\[E\]), as is the case, e.g., in the theory with arbitrary signs of the brane gravitational constants [@CD; @Padilla; @Smolyakov; @mirror] and/or with the Gauss–Bonnet action in the bulk [@CD]. The issue of ghosts in the gravitational sector of the complementary theory with positive value of $M$ but arbitrary signs of the brane gravitational constants was considered in [@Padilla], and we apply the results obtained therein to our case. First, we start with the radion. The radion degree of freedom in our formalism is connected with the possibility of brane bending in the bulk. After identifying the physical degrees of freedom for the radion, one can obtain the conditions for its ghost-free character in our model using the results of [@Padilla]: $$\label{ghost-free} M^3 \left({1 \over 1 - 2 \mu_*} - {e^{- 2 k \rho} \over 1 + 2 \mu} \right) \geq 0 \, ,$$ which we expressed in terms of our parameters $\mu = k m^2 / M^3$ and $\mu_* = k m_*^2 / M^3$ restricting ourselves to the case $k > 0$ and taking into account that $M$ can be of any sign. Then the conditions of absence of both tachyons and radion ghosts in the case $M < 0$ following from (\[mass\]), (\[masstar\]), and (\[ghost-free\]) are $$\label{tachyghost} \mu_* = 0 \, , \ \ \frac12 \left(1 - e^{- 2 k \rho} \right) \le |\mu| < \frac12 \, , \quad \mbox{and} \quad \mu = 0 \, , \ \ |\mu_*| \ge \frac12 \left(e^{2 k \rho} - 1 \right) \, .$$ These conditions on the constants of the theory can be seen to be rather restrictive. Following [@Padilla], we can also show that the massive gravitational modes in the theory under consideration have ghost-like nature. For free metric perturbations in the form (\[induced\]), (\[scale\]) described by the transverse traceless modes $\gamma_{\alpha\beta} (x, y)$ with the boundary conditions (\[b-hidden\]) and (\[b-visible\]) in which we set $\phi = 0$ and $\tau_{\alpha\beta} = 0$, one obtains the gravitational part of action (\[action\]) to quadratic order in the form $$\begin{aligned} S &=& {M^3 \over 2} \int_0^\rho dy e^{- 2ky} \int dx \left( \gamma^{\alpha\beta} \Box \gamma_{\alpha\beta} - e^{- 2ky} \partial_y \gamma^{\alpha\beta} \partial_y \gamma_{\alpha\beta} \right) \nonumber \\ &+& {m^2 \over 2} \int\limits_{y = 0} dx \gamma^{\alpha\beta} \Box \gamma_{\alpha\beta} + {m_*^2 \over 2} e^{- 2 k\rho} \int\limits_{y = \rho} dx \gamma^{\alpha\beta} \Box \gamma_{\alpha\beta} \, . \label{secord}\end{aligned}$$ Expanding the perturbation in the modes $\psi(q,y)$ that are solutions of Eq. (\[eq-psi\]) with the corresponding boundary conditions, $$\gamma_{\alpha\beta} (x, y) = \sum_q \chi_{\alpha\beta} (q, x) \psi (q, y) \, ,$$ substituting this expansion into action (\[secord\]), and using the orthogonality condition $$\displaystyle M^3 \int_0^\rho dy e^{-2ky} \psi (q_1,y) \psi (q_2, y) + m^2 \psi (q_1,0) \psi (q_2, 0) + m_*^2 e^{-2k\rho} \psi (q_1, \rho) \psi (q_2, \rho) = 0$$ for $q_1 \ne q_2$, we arrive at the following quadratic effective action (cf. with [@Padilla]): $$S = \frac12 \sum_q C_q \int dx \chi^{\alpha\beta} (q, x) \left( \Box - q^2 \right) \chi_{\alpha\beta} (q, x) \, ,$$ where $$C_q = M^3 \int_0^\rho dy e^{-2ky} [\psi (q, y)]^2 + m^2 [\psi (q, 0)]^2 + m_*^2 e^{-2k\rho} [\psi (q, \rho)]^2 \, .$$ For the massless mode ($q = 0$), we have $\psi(0, y) \equiv {\rm const}$, and the constant $C_0$ is given by $$C_0 = {M^3 \over 2k} [\psi (0, 0)]^2 \left[ 1 + 2 \mu - \left(1 - 2 \mu_* \right) e^{-2k\rho} \right]$$ and is positive in all cases in which tachyonic modes are absent in the theory, as can be seen from conditions (\[mass\]) and (\[masstar\]). Thus, the massless graviton is not a ghost. However, for the massive modes ($q \ne 0$), using Eq. (\[eq-psi\]), one can obtain the expression $$C_q = {M^3 \over q^2} \int_0^\rho dy e^{-4ky} [\psi' (q, y)]^2 \, , \quad q \ne 0 \, ,$$ which shows that the massive modes have positive norm in the case $M > 0$, and have ghost-like nature in the case $M < 0$. Corrections to Newton’s law {#corr} =========================== In this section, we compute the gravitational potential $V(r)$ on the visible brane induced by a static point source located on the visible or hidden brane and determine corrections to the Newton’s law in the physically reasonable case $k > 0$, i.e., where the zero-mode graviton is localized around the visible brane. Our starting formula is $$h^{\rm (vis)}_{00} = - (1 + 2 V) \, ,$$ where $h^{\rm (vis)}_{\alpha\beta}$ is the induced metric on the visible brane. Matter source on the visible brane ---------------------------------- If matter source is on the visible brane only, then the induced metric $h^{\rm (vis)}_{\alpha\beta}$ is given by Eq. (\[indmet\]). The stress–energy tensor of a static point source of mass ${\cal M}$ is $\tau_{00} = {\cal M} \delta \left( {\vec r\,} \right)$ with other components being zero, and $\tau = - \tau_{00}$. Its Fourier image is $\tau_{00} (p_\alpha) = 2 \pi {\cal M} \delta \left(p_0\right)$, containing only tachyonic modes. Hence, we use the formulas of Sec. \[tachyon\]. We take into account (\[radion\]) and (\[psi-t\]) for the Fourier transform of the solution for the induced metric on the first brane with the source on the same brane. Collecting all expressions together, we obtain the Fourier image of the gravitational potential $$\label{vp} V (p_\alpha) = {2 \pi {\cal M} \delta \left( p_0 \right) \over 3} \left[ {k \over A {\vec p\,}^2} - {2 \over M^3 |{\vec p\,}|} \cdot f \Bigl( |{\vec p\,}| \Bigr) \right] \, ,$$ where the function $f (p)$ denotes the second fraction in expression (\[psi-t\]) taken at the position of the visible brane ($z = z_0$): $$f(p) \equiv {C_K^* I_2 \Bigl( |z_0| \Bigr) + C_I^* K_2 \Bigl( |z_0| \Bigr) \over C_I^* C_K^0 - C_I^0 C_K^*} \, .$$ The potential $V(r)$ is obtained by taking the inverse Fourier transform of (\[vp\]): $$V(r) = - {k {\cal M} \over 3 \pi^2 M^3 r} \left[ I (r) - {\pi M^3 \over 4 A} \right] \, ,$$ where $$\label{I} I(r) = \int\limits_0^\infty d s \sin ( k r s )\, \Psi (s) \, , \qquad \Psi(s) = {D (s) + \mu_* \alpha s D_2 (s) \over E(s) } \, ,$$ and $E(s)$ is given by Eq. (\[E\]). The integral in (\[I\]) cannot be evaluated exactly, but it can be approximated in different regions of $r$, as it is done, e.g., in [@KTT; @JKP]. [**1.**]{} On very large spatial scales $kr \gg \alpha \equiv e^{k\rho}$, we need the asymptotics of the function $\Psi (s)$ in the region $\alpha s \ll krs \sim 1$. It is given by the expression $$\begin{aligned} \Psi(s) &\sim& {2 \over 1 + 2 \mu + \alpha^{-2} \left(2 \mu_* - 1 \right) } \cdot \frac{1}{s} \nonumber \\ &+& { \left( \alpha^2 - 1 \right) (2 \mu_* - 1) \left[ 3 \alpha^2 - 1 + 2 \mu_* \left( 1 - \alpha^2 \right) \right] + 4 \alpha^2 \ln\alpha \over 4 \left[ \alpha^2 (1 + 2 \mu) + 2 \mu_* - 1 \right]^2} \cdot s \, ,\end{aligned}$$ and the integral in (\[I\]) is approximated by using the regularization [@KTT; @JKP] $$\int\limits_0^\infty s \sin (krs) ds \quad \rightarrow \quad \lim_{\epsilon \to 0} \int\limits_0^\infty s \sin (krs) e^{- \epsilon s} ds = 0$$ with the result $$I (r) \approx {\pi \over 1 + 2 \mu + \alpha^{-2} \left(2 \mu_* - 1 \right) } = {\rm const} \, .$$ The potential in this region has Newtonian form $$V(r) = - {G {\cal M} \over r} \, , \qquad G = G_{\rm N} \left(1 + {B \over 3 A \alpha^2 } \right) \, ,$$ where $G_{\rm N}$ given by (\[newton\]), which is in complete agreement with the zero-mode approximation (\[g-vis\]). The theory has continuous Newtonian limit as $\alpha \to \infty$. [**2.**]{} On the scales $1 \ll kr \ll \alpha$, it is the region of integration $1/\alpha \ll s \ll 1$ that substantially contributes to the integral in (\[I\]). In this region, we have $$\Psi (s) \approx {1 \over 1 + 2 \mu} \cdot \frac1s - {\log (s/2) \over (1 + 2 \mu)^2 } \cdot s \, ,$$ substituting which to (\[I\]), we obtain $$I (r) = {\pi \over 1 + 2 \mu} + {\pi \over 2 (1 + 2 \mu )^2} \cdot {1 \over (kr)^2}$$ and $$V(r) = - {G {\cal M} \over r} \left( 1 + {2 M^3 \over 3 A (kr)^2} \right) \, , \qquad G = G_{\rm N} \left( 1 - {B \over A \alpha^2} \right) \, .$$ We observe corrections to the Newtonian potential similar to those of the Randall–Sundrum model [@RS2] but with somewhat different relative constant. [**3.**]{} In the case $kr \ll 1$, we can use the asymptotics for the function $\Psi (s)$ at infinity: $$\Psi (s) \sim {1 \over \mu s + 1 + 15 \mu / 8} \left[1 + {\cal O} \left( s^{-1} \right) \right] \, , \quad \mu \ne 0 \, .$$ This case is further partitioned into two asymptotic regions, depending on the magnitude of $\mu$. 3a. $kr \ll |\mu|$. Here, if $\mu$ is negative, then it cannot be small by absolute value since, in this latter case, the theory contains tachyons. We obtain $$I (r) \approx {\pi \over 2 \mu} + \frac1\mu \left({15 \over 8} + \frac1\mu \right) kr \log \left[\left({15 \over 8} + \frac1\mu \right) kr \right]$$ and $$V (r) = - {G {\cal M} \over r} - {k {\cal M} \over 3 \pi^2 m^2} \left({15 \over 8} + \frac1\mu \right) \log \left[\left({15 \over 8} + \frac1\mu \right) kr \right] \, , \qquad G = {1 \over 8 \pi m^2 } \cdot {\mu + 2/3 \over \mu + 1/2} \, .$$ The logarithmic corrections in these expressions assume that the expression $15/8 + 1/ \mu $ is not very small by absolute value. 3b. $\mu \ll kr \ll 1$. Here we must consider only positive $\mu$. In this case, $$I (r) \approx {1 \over kr} \, ,$$ and the gravitational law is five-dimensional: $$V (r) = - {{\cal M} \over 3 \pi^2 M^3 r^2} \left( 1 - \frac\pi4 kr \right) \, .$$ The expressions for the gravitational potentials obtained in cases 3a and 3b are analogous to those of [@DGP]. The same results would be obtained in the linear approximation in the one-brane case considered in [@Tanaka], as is clear from the remark made after Eq. (\[Cs\]), which identifies the corresponding propagators. Mater source on the hidden brane -------------------------------- In a similar way one can consider the case where the stationary matter resides on the hidden brane with mass ${\cal M}_*$ defined as $\tau^*_{00} = {\cal M}_* \delta ({\vec r\,})$. In this case, we obtain the following expression for the gravitational potential on the visible brane: $$V(r) = - {k {\cal M}_* \over 3 \pi^2 M^3 \alpha r } I(r) \, ,$$ where $$\label{I2} I(r) = \int\limits_0^\infty ds \sin (krs) \Psi (s) \, , \qquad \Psi(s) = {I_1 (s) K_2 (s) + I_2 (s) K_1 (s) \over E(s) } \, ,$$ and $E(s)$ is given by Eq. (\[E\]). The asymptotic expressions for $\Psi (s)$ can be found in various regions: $$\Psi (s) \approx {2 \alpha \over 2 \mu_* - 1 + \alpha^2 ( 2 \mu + 1) } \cdot \frac1s + {\cal O} (s) \, , \quad s \ll \alpha^{-1} \, ,$$ $$\Psi(s) \approx {\sqrt{2 \pi} e^{- \alpha s} \over (1 + 2 \mu) \mu_* \sqrt{\alpha s}} \left[1 + \left( \frac{15}{8} - \frac1{\mu_*} \right) {1 \over \alpha s} + {\cal O} (s^2) \right] \, , \quad \alpha^{-1} \ll s \ll 1 \, , \quad \mu_* \ne 0 \, ,$$ $$\Psi(s) \approx {\sqrt{2 \pi \alpha s}\, e^{- \alpha s} \over (1 + 2 \mu) } \left[1 + \frac{3}{8\alpha s} + {\cal O} (s^2) \right] \, , \quad \alpha^{-1} \ll s \ll 1 \, , \quad \mu_* = 0 \, ,$$ $$\Psi (s) \approx {2 \sqrt\alpha e^{- \alpha s} \over \mu \mu_* \alpha s^2 + \Bigl( \mu + \mu_* \alpha + 15 \mu \mu_* \alpha / 8 \Bigr) s} \, , \quad s \gg 1 \, , \quad \mu^2 + \mu_*^2 \ne 0 \, ,$$ $$\Psi (s) \approx 2 \sqrt\alpha\, e^{- \alpha s} \, , \quad s \gg 1 \, , \quad \mu = \mu_* = 0 \, .$$ Using these expression, it is not difficult to obtain the estimates for the gravitational potential $V(r)$ caused by the presence of the static source on the hidden brane in various regions. We have $$V(r) \approx - {4 G_{\rm N} {\cal M}_* \over 3 \alpha^2 r} \, , \quad kr \gg \alpha \, ,$$ where $G_{\rm N}$ is given by (\[newton\]). Again, in this distance range, the result can be obtained by using the zero-mode approximation (\[g-hid\]). In the case $kr \ll \alpha$, the result crucially depends on whether $\mu_*$ is zero or not: $$\label{mu0} V(r) \approx - {c_1 k^2 {\cal M}_* \over \pi^2 A \alpha^3 } \left[ 1 - c_2 \left( {kr \over \alpha } \right)^2 \right] \, , \quad kr \ll \alpha \, , \quad \mu_* = 0 \, ,$$ $$\label{mu} V (r) \approx - {2 k^2 {\cal M}_* \over 3 \pi^2 A \alpha^3 } \left[ \left( 1 + {c_3 \over \mu_* } - {c_4 \over \mu_*^2} \right) - \left( c_5 + {c_6 \over \mu_* } - {c_7 \over \mu_*^2} \right) \cdot \left( { kr \over \alpha } \right)^2 \right] \, , \quad kr \ll \alpha \, , \quad \mu_* \ne 0 \, ,$$ Here the constants $c_n$ take the following approximate values: $$c_1 \approx 1.77 \, , \quad c_2 \approx 1.02 \, , \quad c_3 \approx 1.3 \, , \quad c_4 \approx 0.35 \, , \quad c_5 \approx 0.06 \, , \quad c_6 \approx 1.12 \, , \quad c_7 \approx 0.24 \, .$$ Expression (\[mu\]) is not valid for sufficiently small $\mu_*$ since we know that, in the limit of $\mu_* \to 0$, the asymptotics changes to (\[mu0\]). In fact, comparison with the exact numerical integration of (\[I2\]) shows that our approximate result (\[mu\]) is only good for $|\mu_*| \gsim 0.5$. Discussion ========== It is known that the braneworld model becomes rather rich in its cosmological manifestations if curvature term is present in the action for the brane (see [@DGP; @CHS; @DDG; @KTT; @SS; @AS; @SS1]). In this paper, we studied the model with scalar-curvature terms for the branes on the original Randall–Sundrum two-brane background. The linearized gravitational equations (\[g-vis\]) and (\[g-hid\]) in this case have the same structure as in the original Randall–Sundrum model but with different physical constants. In the limit of vanishing brane Planck masses $m$ and $m_*$, they tend to the known results [@KTT; @SV; @JKP], which are physically reasonable only for the bulk Planck mass $M > 0$. In the opposite limit of $M \to 0$ while $k = \sigma/M^3$ is fixed, they produce reasonable results (\[lim-vis\]) and (\[lim-hid\]) independently of the sign of $M$. In this paper, we developed a general method for detecting tachyonic modes in the braneworld theory, which can be generalized to theories giving rise to equations of the similar kind, e.g., the theory with Gauss–Bonnet term in the bulk action [@CD]. In our case, for negative values of $M$, the linearized theory typically contains tachyonic modes in the gravitational sector. If both brane Planck masses are nonzero, then we have one or two tachyonic mass eigenvalues depending on the constants of the theory, the conditions of which were determined in Sec. \[tachyon\]. However, in the case where one of the brane Planck masses is zero, tachyonic modes are absent if the other brane has sufficiently high Planck mass \[given by Eq. (\[mass\]) for the visible brane, and by Eq. (\[masstar\]) for the hidden brane\]. In the case of negative bulk Planck mass $M$, the zero-mode graviton is ‘localized’ around the brane with [*negative*]{} tension and is not a ghost in all cases where tachyonic modes are absent in the theory. However, in all cases with negative $M$, the massive gravitational modes of the theory under consideration have ghost-like character. The conditions of absence of both radion ghosts and gravitational tachyons are expressed by (\[tachyghost\]). Exploring both signs of the bulk gravitational constant may be interesting in connection with some braneworld cosmological models requiring [*negative*]{} brane tension, such as the model of [*disappearing dark energy*]{} (DDE) recently discussed in [@SS; @AS] or the braneworld models with ‘quiescent’ cosmological singularities during expansion [@SS1]. The DDE model [@SS; @AS] represents a cosmological braneworld with the Randall–Sundrum constraint (\[lambda-rs\]), negative brane tension, and the condition $|\mu| \ge 1/2$, which is required for physical consistency and which implies inequality (\[mass\]). The ‘quiescent’ singularities in the AdS-embedded braneworld models occur during the universe expansion and are characterized by [*finiteness*]{} of the scale factor, Hubble parameter, and matter density. The braneworld cosmological equations involve the bulk gravitational constant only in even power; therefore, their behaviour is independent of its sign on the homogeneous and isotropic level [@SS]. However, as noted in the introduction, the sign of the bulk gravitational constant is of importance for the small-scale gravitational physics in a braneworld universe, in particular, for the behaviour of cosmological perturbations. The results of the present paper indicate that models with negative bulk gravitational constant can be free from tachyons, although they are plagued with massive ghosts in the gravitational sector. Perhaps, the unwanted situation with ghosts can be remedied by modifications of the bulk action. It should be emphasized that these results do not relate to the braneworld cosmological models with negative brane tension but positive bulk and brane gravitational constants, which require future investigation. Acknowledgments {#acknowledgments .unnumbered} =============== The authors are grateful to Varun Sahni for valuable comments and suggestions. Yu. S. acknowledges warm hospitality of the Inter-University Centre for Astronomy and Astrophysics (IUCAA) in Pune, India. [99]{} Randall L and Sundrum R 1999 [*Phys. Rev. Lett.*]{} [**83**]{} 3370 ([*Preprint*]{} hep-ph/9905221) Randall L and Sundrum R 1999 [*Phys. Rev. Lett.*]{} [**83**]{} 4690 ([*Preprint*]{} hep-th/9906064) Deruelle N 2003 [*Astrophys. Space Sci.*]{} [**283**]{} 619 ([*Preprint*]{} gr-qc/0301035)\ Maartens R 2003 [*Prog. Theor. Phys. Suppl.*]{} [**148**]{} 213 ([*Preprint*]{} gr-qc/0304089) Dvali G, Gabadadze G and Porrati M 2000 [*Phys. Lett.*]{} B [**485**]{} 208 ([*Preprint*]{} hep-th/0005016)\ Dvali G and Gabadadze G 2001 [*Phys. Rev.*]{} D [**63**]{} 065007 ([*Preprint*]{} hep-th/0008054) Collins H and Holdom B 2000 [*Phys. Rev.*]{} D [**62**]{} 105009 ([*Preprint*]{} hep-ph/0003173)\ Collins H and Holdom B 2000 [*Phys. Rev.*]{} D [**62**]{} 124008 ([*Preprint*]{} hep-th/0006158)\ Shtanov Yu V 2000 [*On brane-world cosmology*]{}, ([*Preprint*]{} hep-th/0005193) Deffayet C 2001 [*Phys. Lett.*]{} B [**502**]{} 199 ([*Preprint*]{} hep-th/0010186)\ Deffayet C, Dvali G and Gabadadze G 2002 [*Phys. Rev.*]{} D [**65**]{} 044023 ([*Preprint*]{} astro-ph/0105068) Davoudiasl H, Hewett J L and Rizzo T G 2003 [*JHEP*]{} [**0308**]{} 034 ([*Preprint*]{} hep-ph/0305086)\ Dubovsky S L and Libanov M V 2003 [*JHEP*]{} [**0311**]{} 038 ([*Preprint*]{} hep-th/0309131) Charmousis C and Dufaux J-F 2004 [*Phys. Rev.*]{} D [**70**]{} 106002 ([*Preprint*]{} hep-th/0311267) Padilla A 2004 [*Class. Quantum Grav.*]{} [**21**]{} 2899 ([*Preprint*]{} hep-th/0402079) Smolyakov M N 2004 [*Nucl. Phys.*]{} B [**695**]{} 301 ([*Preprint*]{} hep-th/0403034) Shtanov Yu and Viznyuk A 2004 [*Mirror branes*]{}, ([*Preprint*]{} hep-th/0404077) Kiritsis E, Tetradis N and Tomaras T N 2002 [*JHEP*]{} [**0203**]{} 019 ([*Preprint*]{} hep-th/0202037) Tanaka T 2004 [*Phys. Rev.*]{} D [**69**]{} 024001 ([*Preprint*]{} gr-qc/0305031) Sahni V and Shtanov Yu 2003 [*JCAP*]{} [**0311**]{} 014 ([*Preprint*]{} astro-ph/0202346)\ Sahni V and Shtanov Yu 2002 [*Int. J. Mod. Phys.*]{} D [**11**]{} 1515 ([*Preprint*]{} gr-qc/0205111) Alam U and Sahni V 2002 [*Supernova constraints on braneworld dark energy*]{}, ([*Preprint*]{} astro-ph/0209443) Shtanov Yu and Sahni V 2002 [*Class. Quantum Grav.*]{} [**19**]{} L101 ([*Preprint*]{} gr-qc/0204040) Birrell N D and Davies P C W 1982 [*Quantum Fields in Curved Space*]{} (Cambridge: Cambridge University Press) Shtanov Yu V 2002 [*Phys. Lett.*]{} B [**543**]{} 121 ([*Preprint*]{} hep-th/0108211) Garriga J and Tanaka T 2000 [*Phys. Rev. Lett.*]{} [**84**]{} 2778 ([*Preprint*]{} hep-th/9911055)\ Charmousis C, Gregory R and Rubakov V A 2000 [*Phys. Rev.*]{} D [**62**]{} 067505 ([*Preprint*]{} hep-th/9912160)\ Barvinsky A O, Kamenshchik A Yu, Rathke A and Kiefer C 2003 [*Phys. Rev.*]{} D [**67**]{} 023513 ([*Preprint*]{} hep-th/0206188) Smolyakov M N and Volobuev I P 2002 [*Linearized gravity, Newtonian limit and light deflection in RS1 model*]{}, ([*Preprint*]{} NPI MSU 2002-18/702, hep-th/0208025) Shiromizu T, Maeda K and Sasaki M 2001 [*Phys. Rev.*]{} D [**62**]{} 024012 ([*Preprint*]{} hep-th/9910076) Jung E, Kim S H and Park D K 2003 [*Nucl. Phys.*]{} B [**669**]{} 306 ([*Preprint*]{} hep-th/0305156) [^1]: E-mail: shtanov@bitp.kiev.ua [^2]: E-mail: viznyuk@bitp.kiev.ua [^3]: The results obtained in one of such theories are usually easy to apply to another theory by using the change of the overall sign of the action. [^4]: It is interesting that, while the conformal anomaly term $R_{ab} R^{ab} - \frac13R^2$ cannot be obtained by the variation of a local four-dimensional Lagrangian, the very same term is obtained via the variation of a local Lagrangian in the five-dimensional braneworld theory under investigation [@SS]. Also note that this term is absent in the original Randall–Sundrum model which has $m = 0$. [^5]: The constant $A$ is the same as in Eq. (\[radion\]). [^6]: In Eq. (\[g-vis\]) as well as in similar equations of this section, we formally express the radion field through the trace of the stress–energy tensor using Eq. (\[radion\]), similarly to how it is done, e.g., in [@SV]. [^7]: Tachyonic modes are obviously absent in the Randall–Sundrum model ($m = m_* = 0$) even in the case of negative bulk gravitational constant ($M < 0$), but this model is already excluded as resulting in negative effective Newton’s constant on the brane (see the beginning of this section).

--- abstract: 'We study the relationship between quasihomotopy and path homotopy for Sobolev maps between manifolds. We employ singular integrals on manifolds to show that, in the critical exponent case, path homotopy implies quasihomotopy – and observe the rather surprising fact that $n$-quasihomotopic maps need not be path homotopic. We also study the case where the target is an aspherical manifold, e.g. a manifold with nonpositive sectional curvature, and the contrasting case of the target being a sphere.' address: 'ul. Śniadeckich 8, 00-656 Warszawa' author: - Elefterios Soultanis bibliography: - 'abib.bib' title: Path and quasihomotopy for Sobolev maps between manifolds --- Introduction ============ Let $M$ and $N$ be compact Riemannian manifolds with $n=\dim M\ge 2$. The study of harmonic and $p$-harmonic maps between $M$ and $N$ naturally leads to questions about homotopies between finite energy Sobolev maps [@eel64; @eel78; @eel88; @ver12; @pig09]. However classical homotopy is incompatible with Sobolev maps: on one hand Sobolev maps need not be continuous, and on the other classical homototopy classes are not stable under convergence in the Sobolev norm. Indeed, an easy example by B. White [@whi86] showed that the identity map $S^3\to S^3$ is homotopic to maps of arbitrarily small energy, whilst not being homotopic to a constant map. F. Burstall, in [@bur84], studied energy minimization within classes of maps with prescribed 1-homotopy class, and White [@whi86] introduced the notion of *$d$-homotopy* for an integer $d\le n=\dim M$. *Two maps $u,v\in {W^{1,p}(M;N)}$ are $d$-homotopic, $d<p$, if the restrictions of $u$ and $v$ to a $d$-skeleton of a generic triangulation of $M$ (which are continuous by the Sobolev embedding theorem) are classically homotopic.* White proved [@whi86; @whi88] that Sobolev maps $u\in {W^{1,p}(M;N)}$ ($p\le n$) have a well defined $(\lfloor p\rfloor-1)$-homotopy type (i.e. the homotopy class of the restriction of $u$ does not depend on the generic $(\lfloor p\rfloor-1)$-dimensional skeleton) that is stable under weak convergence in ${W^{1,p}(M;N)}$, and therefore well suited for variational minimization problems. Connections of of $d$-homotopy with the topology of the Sobolev space ${W^{1,p}(M;N)}$ are already visible in [@whi86]. The notion of *path homotopy*, introduced by H. Brezis and Y. Li in [@bre01] utilizes this idea. *Two maps $u,v\in {W^{1,p}(M;N)}$ ($1<p<\infty$) are path homotopic if there exists a continuous path $h\in C([0,1]; {W^{1,p}(M;N)})$ joining $u$ and $v$.* They proved [@bre01 Theorem 0.2] that ${W^{1,p}(M;N)}$ is always path connected when $1<p<2$, while a deep result of Hang and Lin [@han03] states that, for $1<p<n$, *two maps $u,v\in {W^{1,p}(M;N)}$ are path homotopic if and only if they are $(\lfloor p\rfloor-1)$-homotopic.* When $p=n$ this equivalence does not remain valid. Instead, Sobolev maps $u\in {W^{1,n}(M;N)}$ have well-defined homotopy classes (due to the density of Sobolev maps [@uhl83] and a result of White [@whi86], see Theorem \[white\].) When $p>n$ the Sobolev embedding implies that Sobolev maps are continuous and indeed by results in Appendix A in [@bre01] path homotopy is equivalent to classical homotopy. With the emergence of analysis on metric spaces (see [@haj96; @hei98; @hei01; @sha00] and the monographs [@bjo11; @HKST07]) the study of energy minimization problems between more general spaces has become viable. The first steps in this direction were taken by N. Korevaar and R. Schoen [@kor93] – who studied the existence of minimizers of 2-energy in homotopy classes of maps from a manifold to a nonpositively curved metric space (see [@bri99]) – and J. Jost [@jost94; @jost95; @jost96; @jost97] who studied the related problem of minimizing 2-energy in equivariance classes of maps from $(1,2)$-Poincaré space spaces to nonpositively curved metric spaces. In the more general setting both $d$-homotopy and path homotopy become problematic. The lack of triangulations in metric spaces on the one hand, and the fact that the topology of Newton Sobolev spaces ${N^{1,p}(X;Y)}$ depends on the embedding of $Y$ into a Banach space (see [@haj07]) on the other, make both notions of homotopy difficult to work with. In [@teri1], for the purpose of studying minimizers of $p$-energy in homotopy classes of maps from a $(1,p)$-Poincaré space to a nonpositively curved metric space a third notion, called *$p$-quasihomotopy*, was introduced. Here we state the definition for manifolds. It is based on the known fact that Sobolev maps $u\in {W^{1,p}(M;N)}$ have $p$-quasicontinuous representatives, i.e. *for every $\varepsilon>0$ there is an open set $E\subset M$ with ${\operatorname{Cap}}_p(E)<\varepsilon$ so that $u|_{M\setminus E}$ is continuous.* Quasicontinuity may be seen as a refinement of the almost continuity of measurable maps. *Two quasicontinuous representatives $u,v\in {W^{1,p}(M;N)}$ ($1<p<\infty$) are $p$-quasihomotopic if there is a map $H:M\times [0,1]\to N$ with the following property: for any $\varepsilon>0$ there is an open set $E\subset M$ with ${\operatorname{Cap}}_p(E)<\varepsilon$ so that $H|_{M\setminus E\times [0,1]}$ is a (continuous) homotopy between $u|_{M\setminus E}$ and $v|_{M\setminus E}$.* Capacity is a much finer measure of smallness than the Lebesgue measure; a set $E\subset M$ of zero $p$-capacity has Hausdorff dimension at most $n-p$, and sets of small $p$-capacity have small Hausdorff content, $${\operatorname{Cap}}_p(E)\le c(n,p,q){\mathcal{H}}^{n-q}_\infty(E)\textrm{ for any } 1<q<p$$ (Theorem 5.3 in [@mal03].) Thus, while quasihomotopy allows for discontinuities, it does so in a sense a minimal amount, preserving *some* amount of topology. For example, a set of zero $p$-capacity, $p>1$, does not separate a space, whereas a set of measure zero may. There is also a $p$-quasicontinuous counterpart to the fact that if the preimage of a point of a continuous function (from a connected space) is nonempty and open, then the function must be constant (see Lemma 5.3 in [@teri1]). As such, $p$-quasihomotopy is a natural relaxation of classical homotopy to encompass Sobolev maps. Indeed, under the additional assumption that the target space has hyperbolic universal cover there always exists minimizers of $p$-energy in quasihomotopy classes in the metric setting, see Theorem 1.1. in [@teri2]. When $p>n$ the fact any nonempty set has $p$-capacity $\ge \varepsilon_0$ for some small number $\varepsilon_0$ implies that $p$-quasihomotopy coincides with classical homotopy, and thus with path homotopy. However when $1<p<n$ the notion of $p$-quasihomotopy turns out to differ from the other two. Theorem 1.4 in [@teri1] states that when $1<p<n$, if $u,v\in {W^{1,p}(M;N)}$ are $p$-quasihomotopic then they are path homotopic. The proof in fact yields more: *if $1<p\le n$ and $u,v\in {W^{1,p}(M;N)}$ are $p$-quasihomotopic then $u$ and $v$ are $d$-homotopic, where $d=\lceil p\rceil-1$ is the largest integer $<p$.* Since $\lfloor p\rfloor-1<\lceil p\rceil-1$ unless $p$ is an integer it is expected that path homotopic maps need not be quasihomotopic. Indeed the constant map and $$x\mapsto \frac{x}{|x|}\in {W^{1,p}(B^2;S^1)},\ 1<p<2$$ are path homotopic but not $p$-quasihomotopic (see Section 4.2 in [@teri1]). The first main theorem in this paper considers the remaining case $p=n$. \[main\] Let $M$ and $N$ be smooth compact Riemannian manifolds, with $n=\dim M$. If two maps $f,g\in W^{1,n}(M;N)$ are path homotopic then they are $n$-quasihomotopic. The relationships between path-, quasi-, and $d$-homotopy are summarized in the table below. ------------------ ------------------------------------- -------------------- --------------------------------- ${W^{1,p}(M;N)}$ \[1ex\] $1<p<n$ $p$-quasihomotopy $\Rightarrow$ $([p]-1)$-homotopy $\Leftrightarrow$ path homotopy \[1ex\] $p=n$ path homotopy $\Rightarrow$ $p$-quasihomotopy $\Rightarrow$ $(n-1)$-homotopy \[1ex\] $p>n$ $p$-quasihomotopy $\Leftrightarrow$ homotopy $\Leftrightarrow$ path homotopy \[1ex\] ------------------ ------------------------------------- -------------------- --------------------------------- Surprisingly, the converse of Theorem \[main\] fails. Namely it can happen that two maps $f,g\in {W^{1,n}(M;N)}$ are $n$-homotopic but not path homotopic. An example to this effect is given in Corollary \[sn\]. It is noteworthy that in the example the target has the rational homology type of a sphere (in this case it is in fact a sphere) in light of the discussion in [@gol12] (see in particular Theorems 1.4 and 1.5 there). An $n$-manifold $M$ is a rational homology sphere if $$\begin{aligned} H^k_{dR}(M)=\left\{ \begin{array}{ll} 0 &, k\ne 0,n \\ {\mathbb{Z}}&, k=0,\ k=n, \end{array} \right.\end{aligned}$$ where $H^k_{dR}(M)$ denotes the *de Rham cohomology* of $M$. For generic manifolds $M,N$, particularly rational homology sphere targets, the implications between path- and quasihomotopy depend on $p$. In contrast, for *aspherical* target manifolds the situation is simpler. An $m$-manifold $N$ is apsherical if the homotopy groups $\pi_k(N)$ vanish for all $k\ge 2$. Using Whiteheads theorem (Theorem 4.5 in [@hat02]) aspherical manifolds may be characterized as those with contractible universal cover. Aspherical manifolds include, as an important subclass, manifolds of nonpositive sectional curvature. For general $p\in (1,\infty)$ we have the following theorem. \[pqpath\] Suppose $M$ and $N$ are compact smooth Riemannian manifolds, $N$ aspherical and $1<p<\infty$. If two maps $u,v\in {W^{1,p}(M;N)}$ are $p$-quasihomotopic then they are path homotopic. When $p\ge 2$ we can say more. \[main2\] Let $2\le p <\infty$, $M,N$ be smooth compact Riemannian manifolds, $N$ being aspherical. Then two maps $f,g\in {W^{1,p}(M;N)}$ are path homotopic if and only they are $p$-quasihomotopic. The restriction $p\ge 2$ is essential. Indeed by Theorem 0.2 in [@bre01] the space ${W^{1,p}(M;N)}$ is always path connected when $1<p<2$, while there may exists distinct $p$-quasihomotopy classes (see the example above). ### Outline {#outline .unnumbered} The proof of Theorem \[main\] is based on approximating a given Sobolev map with suitable mollified maps and showing the convergence is quasiuniform (Theorem \[molli\]). The second section is devoted to mollification and the use of singular integrals to accomplish this. Section 3 deals with the aspherical case. For nonpositively curved targets Proposition \[pqpath\] follows directly from Theorem 1.1 and Proposition 1.5 in [@teri1] but the more general case of aspherical targets requires somewhat different arguments and the use of Theorem \[molli\]. Theorem \[main2\] is an immediate consequence of Theorem \[main3\], presented in this Section. The last Section is devoted to proving that ${W^{1,p}(M;S^k)}$ is $p$-quasiconnected, i.e. any two maps in ${W^{1,p}(M;S^k)}$ are $p$-quasihomotopic, when $p\le k$ (Proposition \[mn\]). Some of the auxiliary results (e.g. Proposition \[cap\]) may be interesting in themselves. Proposition \[mn\] serves as an example showing that sometimes – though not in general – path homotopy - and $p$-quasihomotopyclasses coincide. The paper is closed by remarking that ${W^{1,p}(B^{k+1};S^k)}$, while path connected when $p<k+1$, is not $p$-quasiconnected for $k<p<k+1$. Critical exponent case ====================== The proof strategy of Theorem \[main\] utilizes Brian White’s result. \[white\] Two Lipschitz maps in ${W^{1,n}(M;N)}$ are path homotopic if and only if they are homotopic. Moreover for each $u\in {W^{1,n}(M;N)}$ there is a number $\varepsilon >0$ so that if $\|u-v\|_{1,n}<\varepsilon$ then $u$ and $v$ are path homotopic. Coupled with the fact, due to Schoen-Uhlenbeck [@uhl83], that $Lip(M;N)$ is dense in ${W^{1,n}(M;N)}$ the question, whether path homotopy implies $n$-quasihomotopy, is reduced to the following statement. *For every $u\in {W^{1,n}(M;N)}$ and $\varepsilon>0$ there is a Lipschitz map $u_\varepsilon$ with $\|u-u_\varepsilon\|_{1,n}<\varepsilon$ such that $u_\varepsilon$ is $n$-quasihomotopic to $u$.* We will construct such functions by means of mollifying the original function. Mollifiers ---------- Suppose $\psi:[0,\infty)\to [0,1]$ is a Lipschitz cut-off function with ${\operatorname{spt}}\psi\subset [0,1)$. Given $r>0$ define $\psi_r:M\to {\mathbb{R}}$ by $$\psi_r(p)=\int_M\psi\left(\frac{|p-z|}{r}\right){\mathrm {d}}z.$$ Given $u\in L^p(M;{\mathbb{R}}^\nu)$ and $r>0$ set $$\psi_r\ast u(p)=\frac{1}{\psi_r(p)}\int_M\psi\left(\frac{|p-z|}{r}\right)u(z){\mathrm {d}}z, \ p\in M$$ \[lipmol\] For each $u\in L^1_{loc}(M;{\mathbb{R}}^\nu)$ and $r>0$ the map $\psi_r\ast u:M\to {\mathbb{R}}^\nu$ is Lipschitz continuous. Moreover $$\psi_r\ast u (x)\le C r{\mathcal{M}}|u| (x)$$ for almost every $x\in M$, with $C$, depending only on $\psi,\ M$ and $\nu$. For $g\in L^1_{loc}(M)$ and arbitrary $x,y\in M$ we have $$\begin{aligned} &\left|\int_M\psi\left(\frac{|x-z|}{r}\right)g(z){\mathrm {d}}z-\int_M\psi\left(\frac{|y-z|}{r}\right)g(z){\mathrm {d}}z\right|\nonumber \\ \le & {\operatorname{Lip}}(\psi) \int_{B(x,r+d(x,y))}\left|\frac{|x-z|-|y-z|}{r}\right||g(z)|{\mathrm {d}}z\nonumber\\ \le & {\operatorname{Lip}}(\psi)\frac{d(x,y)}{r}\int_{B(x,r+d(x,y)) }|g|{\mathrm {d}}z.\label{apu}\end{aligned}$$ The lipschitz continuity of $\psi_r\ast u$ follows from this by expressing the difference $\psi_r\ast u(x)-\psi_r\ast u(y)$ ,where $d(x,y)<r$, as $$\begin{aligned} \frac{\psi_r(y)-\psi_r(x)}{\psi_r(x)\psi_r(y)}&\int_M\psi\left(\frac{|x-z|}{r}\right)u(z){\mathrm {d}}z \\ +\frac{1}{\psi_r(y)}&\left[\int_M\psi\left(\frac{|x-z|}{r}\right)u(z){\mathrm {d}}z-\int_M\psi\left(\frac{|y-z|}{r}\right)u(z){\mathrm {d}}z \right]\end{aligned}$$ and applying (\[apu\]) and the doubling property of the measure. The estimate in the claim follows by a standard decomposition of the integral into annular regions, see [@HKST07; @hei01]. \[0\](Schoen-Uhlenbeck) Let $u\in {W^{1,p}(M;N)}$. For $r>0$ we have $${\operatorname{dist}}(N,\varphi_r\ast u(p))\lesssim \left(\int_{B_r(p)}|Du|^n{\mathrm {d}}z\right)^{1/n}$$ for all $p\in M$. Consequently for each $u\in {W^{1,n}(M;N)}$ there is $r_0>0$ so that $$\sup_{p\in M}{\operatorname{dist}}(N,\varphi_r\ast u(p))<\varepsilon_0$$ whenever $r<r_0$. Let $p\in M$. For a.e. $z\in B_r(p)$ $${\operatorname{dist}}(N,\varphi_r\ast u(p))\le \|u(z)-\varphi_r\ast u(p)\|.$$ Taking an average integral over $B_r(p)$ we obtain $${\operatorname{dist}}(N,\varphi_r\ast u(p))\le \dashint_{B_r(p)}\|u(z)-\varphi_r\ast u(p)\|{\mathrm {d}}z.$$ By the $(1,n)$-Poincare inequality (which every manifold of dimension $n$ supports) $$\begin{aligned} \dashint_{B_r(p)}\|u(z)-\varphi_r\ast u(p)\|{\mathrm {d}}z&\le \frac{1}{\varphi_r(p)}\int_{B_r(p)}\dashint_{B_r(p)}\varphi\left(\frac{|p-w|}{r}\right)\|u(z)-u(w)\|{\mathrm {d}}z{\mathrm {d}}w\\ &\lesssim \dashint_{B_r(p)}\dashint_{B_r(p)}\|u(z)-u(w)\|{\mathrm {d}}z{\mathrm {d}}w\\ &\lesssim r\left(\dashint_{B_r(p)}|Du|^n{\mathrm {d}}z\right)^{1/n}\simeq \left(\int_{B_r(p)}|Du|^n{\mathrm {d}}z\right)^{1/n}. \end{aligned}$$ The implied constants in the estimates depend only on the data of $M$ and on $N$. The second assertion follows directly from the absolute continuity of the measure $|Du|{\mathrm {d}}z$. Singular integrals ------------------ Let us set some notation. Let $\varphi:{\mathbb{R}}\to {\mathbb{R}}$ be a smooth cutoff function and define the *kernel* $k_r:(0,\infty)\to {\mathbb{R}}$, $$k_r(t)=\frac{\varphi(t/r)}{t^{n-1}}.$$ We abuse notation by writing $$k_r(p,q)=k_r(|p-q|),\ p,q\in M$$ and finally, given $g\in L^p(M)$ ($1<p<\infty$), we define the convolution $$k_r\ast g (x)=\int_Mk_r(x,z)g(z){\mathrm {d}}z, \ x\in M.$$ By the compactness of $M$ there exists $r_1$ so that $$\exp_x:B^n(r_1)\to B(x,r_1)$$ is a 2-bilipschitz diffeomorphism for all $x\in M$. Thus, when $r<r_1$ we may use a change of variables given by the exponential map and write the integral above $$k_r\ast g (x)=\int_{B^n(r)}k_r(|\xi|) g(\exp_x\xi)J\exp_x\xi {\mathrm {d}}\xi.$$ Let $1<p<\infty$. Given $g\in L^p(M)$ the function $k_r\ast g$ has distributional gradient $$\nabla _x(k_r\ast g)v= -PV \int_M k_r'(|x-z|)\langle \nabla_xd_z,v\rangle g(z){\mathrm {d}}z,\ v\in T_xM.$$ We refer to [@see59; @cal56] for the existence and basic properties of singular integrals on manifolds (see in particular Chapter IV in [@koh57] and the example in [@see59 D].) The distributional derivative is determined by the condition $$\int_M\langle \nabla(k_r\ast g),V\rangle {\mathrm {d}}x=-\int_M(k_r\ast g)div V{\mathrm {d}}x$$ for all smooth vector fields $V$ on $M$. We may write $$\begin{aligned} &\int_M PV \int_M k'_r(|x-z|)\langle \nabla_xd_z,V_x\rangle g(z){\mathrm {d}}z{\mathrm {d}}x\nonumber \\ =&\lim_{\delta\to 0}\int_M \int_{M\setminus B_\delta(x)} k'_r(|x-z|)\langle \nabla_xd_z,V_x\rangle g(z){\mathrm {d}}z{\mathrm {d}}x\nonumber \\ =&\lim_{\delta\to 0}\int_Mg(z)\int_{M\setminus B_\delta(z)}k'_r(|x-z|)\langle \nabla_xd_z,V_x\rangle {\mathrm {d}}x{\mathrm {d}}z.\label{pv} \end{aligned}$$ Note that when $x\ne z$ the vector $\nabla_xd_z$ is the unit vector normal to $\partial B_\delta (z)$ at $x$. Thus $-\nabla_xd_z$ is the unit normal to $\partial(M\setminus B_\delta(z))$ at $x$. The divergence theorem gives $$\begin{aligned} &\int_{M\setminus B_\delta(z)}k'_r(|x-z|)\langle \nabla_xd_z,V_x\rangle {\mathrm {d}}x \nonumber\\ =&-\int_{M\setminus B_\delta(z)}k_r(|x-z|)div V_x{\mathrm {d}}x+ \int_{\partial B_\delta(z)}k_r(|y-z|)\langle \nabla_yd_z,V_y\rangle {\mathrm {d}}\sigma(y)\nonumber \\ =&-\int_{M\setminus B_\delta(z)}k_r(|x-z|)div V_x{\mathrm {d}}x+O(\delta)\label{o}. \end{aligned}$$ The second term is $O(\delta)$ since it may be estimated using again the divergence theorem: $$\left|\int_{\partial B_\delta(z)}k_r(|y-z|)\langle \nabla_yd_z,V_y\rangle {\mathrm {d}}\sigma(y)\right|=\left|k_r(\delta)\int_{B_\delta(z)}div V_y{\mathrm {d}}y \right| \lesssim \delta^{1-n} \delta^n.$$ Plugging (\[o\]) in (\[pv\]) we obtain $$\begin{aligned} &\int_M PV \int_M k'_r(|x-z|)\langle \nabla_xd_z,V_x\rangle g(z){\mathrm {d}}z{\mathrm {d}}x\nonumber\\ =& -\lim_{\delta\to 0}\int_M g(z)\int_{M\setminus B_\delta(z)}k_r(|x-z|)div V_x{\mathrm {d}}x {\mathrm {d}}z + \lim_{\delta\to 0}\int_MO(\delta){\mathrm {d}}z\nonumber\\ =&- \lim_{\delta\to 0}\int_M \int_{M\setminus B_\delta(x)}k_r(|x-z|)g(z)div V_x{\mathrm {d}}z {\mathrm {d}}x = -\int_M(k_r\ast g)divV{\mathrm {d}}x. \end{aligned}$$ Thus we are done. \[unibound\] The operators $g\mapsto k_r\ast g$ ($r>0$) are uniformly bounded $$L^p(M)\to {{W^{1,p}(M)}},$$ i.e. $$\label{bound} \int_M|k_r\ast g(x)|^p{\mathrm {d}}x+\int_M|\nabla_x(k_r\ast g)|^p{\mathrm {d}}x\le C\int_M|g|^p{\mathrm {d}}x,$$ $g\in L^p(M)$, for all $0<r<r_1$. For a.e. $x\in M$ we have, $v\in T_xM$ and $r>0$ $$\begin{aligned} &|\nabla_x(k_r\ast g)v|=\left|PV \int_Mk'_r(|x-z|)\langle\nabla_xd_z,v\rangle g(z){\mathrm {d}}z\right| \\ \le & |v|\int_M\frac{|\varphi'(|x-z|/r)}{r|x-z|^{n-1}}|g(z)|{\mathrm {d}}z+ \left|PV \int_M\frac{\varphi(|x-z|/r)}{|x-z|^n}\langle \nabla_xd_z,v\rangle g(z){\mathrm {d}}z\right|. \end{aligned}$$ Using this and the estimate in Lemma \[lipmol\] we obtain the estimate $$\begin{aligned} |k_r\ast g(x)|^p+|\nabla_x(k_r\ast g)|^p \le & C(r^p+1){\mathcal{M}}g(x)^p\nonumber \\ &+\sup_{|v|=1} \left|PV \int_M\frac{\varphi(|x-z|/r)}{|x-z|^n}\langle \nabla_xd_z,v\rangle g(z){\mathrm {d}}z\right|^p\label{yet} \end{aligned}$$ In light of (\[yet\]) it suffices to demonstrate the (uniform) boundedness of $T_j:L^p(M)\to L^p(M)$ given by $$T_jg(x)= PV \int_M\frac{\varphi(|x-z|/r)}{|x-z|^n}\langle \nabla_xd_z,\partial_j\rangle g(z){\mathrm {d}}z$$ for each $j=1,\ldots,\dim M$, when $r<r_1$. The operator $T_j:L^p(M)\to L^p(M)$ is bounded with norm independent of $r\in (0,r_1)$. Since $r<r_1$ and the integrand in $T_j$ vanishes outside $B(x,r)$ which is bilipschitz diffeomorphic to $B^n(r)$ through the exponential map $\exp_x:B^n(r)\to B(x,r)$, the operator $T_j$ may be written $$T_jg(x)=PV\int_{B^n(r)}\varphi(|\xi|/r)\frac{\xi_j}{|\xi|^{n+1}}g(\exp_x\xi)J\exp_x(\xi){\mathrm {d}}\xi.$$ By Definition 4 in [@see59 B] it is sufficient to prove the boundedness, uniformly in $r$, for the Euclidean operator $\widetilde T_j:L^p({\mathbb{R}}^n)\to L^p({\mathbb{R}}^n)$ given by the same kernel: $$\widetilde T_jh(x)=PV\int_{{\mathbb{R}}^n}\varphi(|\xi|/r)\frac{\xi_j}{|\xi|^{n+1}}h(x-\xi){\mathrm {d}}\xi.$$ By Theorem 5.4.1 in [@gra14] (cf. Chapter 5, Theorem 5.1 in [@duo01]) this is implied by the following two conditions. Denote $$\displaystyle K_r(y)=\varphi(|y|/r)\frac{y_j}{|y|^{n+1}}.$$ - $\|\widehat{K_r}\|_{\infty}\le A$, and - $|\nabla K_r(y)|\le \frac{B}{|y|^{n+1}}$. A change of variables implies $\widehat{K_r}(\xi)=\widehat{K_1}(r\xi)$ so that $$\|\widehat{K_r}\|_{\infty}\le \|\widehat{K_1}\|_{\infty}:=A.$$ We may estimate $$|\nabla K_r(y)|\le \chi_{B^n(r)}(y)\left[\frac{\|\varphi'\|_\infty}{r|y|^n}+\|\varphi\|_\infty|\nabla(y_j/|y|^{n+1})|\right]\le \frac{C(n,\varphi)}{|y|^{n+1}}.$$ Consequently both (1) and (2) are satisfied with constants independent of $r$. This completes the proof of the sublemma. Having a bound $\|T_j\|_{L^p(M)\to L^p(M)}\le C$ where $C$ is independent of $r$ we obtain the estimate (\[bound\]) with constant $C$ independent of $r$. This proves Lemma \[unibound\]. The proof of Theorem \[main\] ----------------------------- Using Lemma \[0\] we define a net of approximating maps with values in $N$. \[mollidef\] Let $u\in {W^{1,n}(M;N)}$, and let $r_0$ be the constant in Lemma \[0\]. For $0<r\le r_0$ set $$u_r(p)=\pi(\varphi_r\ast u(p)),\ p\in M.$$ Additionally, we set $$u_0=u.$$ For each $r>0$ the maps $u_r:M\to N$ are clearly Lipschitz. *The resulting map $M\times [0,r_0]\ni (p,t)\mapsto u_t(p)$ is a key component in the proof of Theorem \[main\].* \[easy\] Let $r>0$. Then $u_s\to u_r$ uniformly as $s\to r$. \[hard\] The maps $u_r$ converge $n$-quasiuniformly to $u$, i.e. for each $\varepsilon >0$ there exists an open set $U$ with ${\operatorname{Cap}}_n(U)<\varepsilon$ such that $(u_r)|_{M\setminus U}\to u|_{M\setminus U}$ uniformly as $r\to 0$. We will estimate the difference $\|u_r(p)-u_s(p)\|$ by splitting it into two parts. Let $b$ be any vector in ${\mathbb{R}}^\nu$. We will later choose it appropriately. $$\begin{aligned} \|u_r(p)-u_s(p)\|\le & \|\varphi_r\ast u(p)-\varphi_s\ast u(p)\|=\|\varphi_r\ast [u-b](p)-\varphi_s\ast [u-b](p)\|\nonumber\\ \label{1}\le & \left|\frac{1}{\varphi_r(p)}-\frac{1}{\varphi_s(p)} \right|\int_{B_{r}(p)}\varphi\left(\frac{|p-z|}{r}\right)\|u(z)-b\|{\mathrm {d}}z\\ \label{2} &+ \frac{1}{\varphi_s(p)}\int_{B_{s\vee r}(p)}\left|\varphi\left(\frac{|p-z|}{r}\right)-\varphi\left(\frac{|p-z|}{s}\right) \right|\|u(z)-b\|{\mathrm {d}}z\end{aligned}$$ Let us estimate the two terms (\[1\]) and (\[2\]) separately, starting with the latter. Throughout we assume that $|r-s|<r$, which implies that $\varphi_{r\vee s}(p)\lesssim \varphi_s(p)$ with constant depending only on $M$. $$\begin{aligned} (\ref{2})&\le \frac{1}{\varphi_s(p)}\int_{B_{s\vee r}(p)}{\operatorname{Lip}}(\varphi)|p-z|\left|\frac{1}{r}-\frac{1}{s} \right|\|u(z)-b\|{\mathrm {d}}z\\ &\lesssim |r/s-1|\vee |s/r-1|\dashint_{B_{r\vee s}(p)}\|u-b\|{\mathrm {d}}z.\end{aligned}$$ A similar computation yields the same bound for (\[1\]). Thus we arrive at $$\|u_r(p)-u_s(p)\|\lesssim |r/s-1|\vee |s/r-1|\dashint_{B_{r\vee s}(p)}\|u-b\|{\mathrm {d}}z.$$ Now we choose $b=u_{B_{s\vee r}(p)}$ and use the $(1,n)$-Poincare inequality to estimate $$\dashint_{B_{r\vee s}(p)}\|u-b\|{\mathrm {d}}z\lesssim \left(\int_{B_{s\vee r}(p)}|Du|^n{\mathrm {d}}z\right)^{1/n}\lesssim \|Du\|_{L^n(M)}.$$ Combining these we arrive at $$\begin{aligned} \|u_r(p)-u_s(p)\|\lesssim (|r/s-1|\vee |s/r-1|)\|Du\|_{L^n(M)}\end{aligned}$$ for all $p$. Thus $u_s\to u_r$ uniformly as $s\to r$, as long as $r\ne 0$. Proposition \[hard\] requires more work. We begin by estimating the difference of $u$ and $u_r$ by an expression which we study in more detail \[est\] Let $u\in {W^{1,p}(M;N)}$. For $p$-q.e. $x\in M$ we have $$\|u(x)-u_r(x)\|\le \int_M\frac{\varphi(|z-x|/r)}{|z-x|^{n-1}}|Du|(z){\mathrm {d}}z.$$ The proof is similar to [@hei01 p. 28, (4.5)]. \[quasiuni\] Let $1<p<\infty$ and let $(f_k)\subset {N^{1,p}(M)}$ be a bounded secuence with $0\le f_{k+1}\le f_k$ pointwise and $\|f_k\|_{L^p}\to 0$ as $k\to \infty$. Then $f_k\to 0$ $p$-quasiuniformly. Since ${N^{1,p}(M)}$ is reflexive we may pass to a subsequence converging weakly to 0, and by the Mazur lemma a sequence of convex combinations converges to 0 in norm. Passing to another subsequence if needed, we may assume that the sequence of convex combinations, $$h_m=\lambda_1^mf_{k_1}+\cdots +\lambda_{N_m}^mf_{k_{N_m}},\ (k_1<\cdots<k_{N_m}),$$ converges to zero $p$-quasiuniformly. The monotonicity now imples $$0\le f_{k_{N_m}}\le h_m$$ so that a subsequence of $(f_k)$ converges $p$-quasiuniformly to zero. Since the sequence is pointwise nonincreasing the whole sequence converges to zero $p$-quasiuniformly. These auxiliary results yield Proposition \[hard\]. By Lemma \[est\] we have $$\|u(x)-u_r(x)\|\lesssim k_r\ast |Du| (x)$$for $p$-quasievery $x\in M$. Choosing $\varphi$ nonincreasing we get that $$k_r\ast |Du|\le k_s\ast |Du|$$ pointwise whenever $r<s$, and further, $$k_r\ast |Du|\stackrel{L^n}{\longrightarrow} 0$$as $r\to 0$. By lemma \[unibound\] the functions $k_r\ast |Du|$ have uniformly bounded $W^{1,n}$-norms (in $r$) so by Lemma \[quasiuni\] we have that $k_r\ast |Du|\to 0$ $n$-quasiuniformly. Consequently $u_r\to u$ $n$-quasiuniformly. \[molli\] Let $u\in {W^{1,n}(M;N)}$. The map $M\times [0,r_0]\to N$ given by $$(p,r)\mapsto u_r(p)$$ in \[mollidef\] defines an $n$-quasihomotopy $u\simeq u_{r_0}$. Denote $H(p,r)=u_r(p)$ and suppose $\varepsilon>0$ is given. Let $U$ be the open set satisfying the claim of Proposition \[hard\]. We claim that $H|_{M\setminus U\times [0,r_0]}$ is continuous. For this it suffices to show that $(u_s)|_{M\setminus U}\to (u_r)|_{M\setminus U}$ uniformly as $s\to r$. This, however, follows immediately from \[easy\] and \[hard\]. We close this Section with the proof of Theorem \[main\]. Suppose $u,v\in {W^{1,n}(M;N)}$ are path homotopic. For small enough $\varepsilon$ we have, by Theorems \[molli\] and \[white\], that $u_\varepsilon$ is both $n$-quasihomotopic and path homotopic to $u$. The same holds for $v$ and $v_\varepsilon$. It follows that $u_\varepsilon$ and $v_\varepsilon$ are path homotopic and since they are Lipschitz, homotopic (Theorem \[white\]). Thus $u_\varepsilon$ and $v_\varepsilon$ are $n$-quasihomotopic. Consequently $u$ and $v$ are $n$-quasihomotopic. Aspherical targets ================== A topological space $X$ is called *aspherical* if $\pi_i(X)=0$ for every $i>1$. It is well known that for smooth Riemannian manifolds the vanishing of higher homotopy groups is equivalent to having contractible universal cover. In particular manifolds with nonpositive sectional curvature are aspherical. The equivalence stated in Theorem \[main2\] can be seen as a Sobolev version of Whiteheads theorem [@hat02]. Before turning our attention to Theorem \[main2\] let us present a proof of Theorem \[pqpath\]. Suppose $N$ is aspherical and let $f,g\in {W^{1,p}(M;N)}$ be $p$-quasihomotopic. We devide the proof into three cases: - By Theorem 1.4 in [@teri1] $f$ and $g$ are path homotopic. - In this case path homotopy and $p$-quasihomotopy coincide, see the discussion in the introduction. - This is the only case that requires some work. By Theorem \[molli\] $f,g$ are $n$-quasihomotopic to Lipschitz maps $f_0,g_0$ so we may assume that $f$ and $g$ are themselves Lipschitz. Since $N$ is aspherical it is path representable [@teri2 Proposition 3.4] and thus by [@teri2 Theorem 1.2] $(f,g)\in {N^{1,n}(M;N)}\cap {\operatorname{Lip}}(M;N)$ has a lift $h\in {N^{1,n}(M;\widehat{N}_{diag})}$ where $\widehat{N}_{diag}$ is the diagonal cover of $N$ (see [@teri2 Subsection 2.4]). Since $g_h=g_{(f,g)}\le {\operatorname{LIP}}(f)+{\operatorname{LIP}}(g)$ almost everywhere (Lemma 4.3 in [@teri2]) it follows that $h$ is in fact Lipschitz. Thus the continuous map $(f,g):M\to N\times N$ admits a (continuous) lift $h:M\to \widehat{N}_{diag}$. By Proposition 3.2 in [@teri2] $f$ and $g$ are homotopic, hence path homotopic in ${W^{1,n}(M;N)}$. When $p\ge 2$, a Sobolev map $f\in {W^{1,p}(M;N)}$ induces a homorphism$u_\ast:\pi(M,x_0)\to \pi(N,f(x_0))$ [@sch79] (see also [@whi88; @nak93]). For almost every $x_0\in M$ an induced homomorphism satisfies, for all $[\gamma]\in \pi(M,x_0)$: - $u_\ast[\gamma]=[u\circ\gamma]$ if $\gamma$ is such that $u\circ\gamma$ is continuous - $u_\ast[\gamma]=[u\circ \gamma']$ for some $\gamma'\sim \gamma$. It is known that no such induced homomorphism need exist for a Sobolev map $f\in {W^{1,p}(M;N)}$ when $1<p<2$. To connect induced homomorphisms to $p$-quasihomotopies we recall the notion of a *fundamental system of loops* from [@teri2]. Given a $p$-quasicontinuous representative $u\in {W^{1,p}(M;N)}$, an upper gradient $g\in L^p(M)$ and an exceptional path family $\Gamma_0$ of curves in $M$, such that $g$ is an upper gradient of $u$ along any curve $\gamma\notin \Gamma_0$, and a *basepoint* $x_0\in M$ with ${\mathcal{M}}g^g(x_0)<\infty$, the collection of loops $${\mathcal{F}}_{x_0}(g,\Gamma_0)=\{\alpha\beta^{-1}: \Gamma_{x_0x}\setminus\Gamma_0,\ {\mathcal{M}}g^p(x)<\infty \}$$ is called the fundamental system of loops. Recall the definition of ${\operatorname{spt}}_p\Gamma_0$ of a negligible path family: $${\operatorname{spt}}_p\Gamma_0=\bigcap\{{\mathcal{M}}\rho^p=\infty\}$$ where the intersection is taken over all admissible metrics $\rho\in L^p(M)$ for which $$\int_\gamma\rho=\infty\textrm{ for all }\gamma\in \Gamma_0.$$ \[disse\] There is a constant $C$ with the following property. If $\Gamma_0$ is a path family and $g\in L^p(M)$ a nonnegative Borel function with $$\int_{\gamma}g=\infty,\ \gamma\in \Gamma_0,$$ then for any $x,y\notin \{{\mathcal{M}}g^p=\infty\}$ there exists a curve $\gamma\notin \Gamma_0$ joining $x$ and $y$ with $$\ell(\gamma)\le Cd(x,y).$$ By Lemma 4.5 in [@teri2] and Theorem 2 (4) in [@kei03] we have $$d(x,y)^{1-p}\le C{\operatorname{Mod}}_p(\Gamma_{xy}\setminus \Gamma_g; \mu_{xy}),$$ where $$\mu_{xy}(A)=\int_A\left[\frac{d(x,z)}{\mu(B(x,d(x,z)))}+\frac{d(y,z)}{\mu(B(y,d(y,z)))} \right]{\mathrm {d}}\mu(z), \ A\subset X.$$ In particular $\Gamma_{xy}\setminus \Gamma_g$ is nonempty. Note that $\Gamma_0\subset \Gamma_g$. If $\ell(\gamma)\ge Dd(x,y)$ for all $\gamma\in \Gamma_{xy}\setminus \Gamma_g\subset \Gamma_{xy}\setminus \Gamma_0$ then $\rho=1/(Dd(x,y))$ is admissible for $\Gamma_{xy}\setminus \Gamma_g$ and thus $${\operatorname{Mod}}_p(\Gamma_{xy}\setminus \Gamma_g; \mu_{xy})\le CD^{-p}d(x,y)^{1-p}.$$Combining the two inequalitites yields the required bound on $D$. \[loop\] Let $p\ge 2$, and $u\in {W^{1,p}(M;N)}$ be a quasicontinuous representative. Given an upper gradient $g$ of $u$, a path family $\Gamma_0$ of zero $p$-modulus, and a point $x_0\notin {\operatorname{spt}}_p\Gamma_0$ with ${\mathcal{M}}g^p(x_0)<\infty$, we have $$u_\ast\pi(M,x_0)=u_\sharp {\mathcal{F}}_{x_0}(g,\Gamma_0).$$ Let $u\in {W^{1,p}(M;N)}$ and let $g,\Gamma_0$ be as in the claim. Set $$E=\{x_0: {\mathcal{M}}g^p(x_0)=\infty \}\cup {\operatorname{spt}}\Gamma_0$$ and choose and arbitrary point $x_0\notin E$. For any $\gamma\in {\mathcal{F}}_{x_0}(g,\Gamma_0)$ clearly $[u\circ \gamma]\in u_\ast\pi(M,x_0)$. Thus we only need to prove the other inclusion. To this end, fix a loop $\gamma$ based on $x_0$. Take a tubular neighbourhood $T$ of $\gamma$ so that any loop in $T$ is homotopic with $\gamma$. Take a finite chain of open balls $x_0\in B_0,B_1,\ldots,B_k$ of radii $r>0$ such that $2C\overline B_j\subset T$, and $B_j\cap B_{j+1}\ne \varnothing$, where $C$ is the constant in Lemma \[disse\]. Since $|E|=0$ there exists, for each $j$, points $y_j\in (B_j \cap B_{j+1})\setminus E$ (with the convention that $y_0=x_0$ and $y_k\in (B_0\cap B_k)\setminus E$.) By Lemma \[disse\] there exists a curve $\gamma_j\notin \Gamma_0$ joining $y_j$ and $y_{j+1}$ with $\ell(\gamma_j)\le Cd(y_j,y_{j+1})$ (here $y_{k+1}=x_0$). Hence $|\gamma_j|\subset T$. The loop $\gamma'=\gamma_0\cdots\gamma_{k+1}$ belongs to ${\mathcal{F}}_{x_0}(g,\Gamma_0)$ and is contained in $T$, and therefore homotopic with $\gamma$. It follows that $[u\circ \gamma']=u_\ast[\gamma']=u_\ast[\gamma]$ and since $\gamma$ was arbitrary we obtain $u_\ast\pi(M,x_0)\le u_\sharp{\mathcal{F}}_{x_0}(g,\Gamma_0)$. The proof is complete. \[pconj\] Let $p\ge 2$. Two maps, $u,v\in {W^{1,p}(M;N)}$, are $p$-quasihomotopic if and only if $ u_\sharp\pi(M)$ and $v_\sharp\pi(M)$ are conjugated subgroups of $\pi(N)$. By [@teri2 Theorem 1.2 and 1,3] the maps $u,v$ are $p$-quasihomotopic if and only if $$\label{eq z} (u,v)_\sharp{\mathcal{F}}_{x_0}(g,\Gamma_0)\le p_\ast\pi(\widehat{N}_{diag},[\alpha])$$ for some $[\alpha]\in p^{-1}(u(x_0),v(x_0))$, and some $x_0\in M$. Here $(p,\widehat{N}_{diag})$ is the *diagonal cover* of $N$ which consists of homotopy classes of all paths in $N$ (see [@teri2] for the precise construction). A modification of the proof of [@teri2 Lemma 2.18] yields $$p_\ast\pi(\widehat{N}_{diag},[\alpha])=\{([\gamma],[\alpha^{-1}\gamma \alpha]): [\gamma]\in \pi(N,u(x_0)) \}\le \pi(N,u(x_0))\times\pi(N,v(x_0)).$$ On the other hand by Lemma \[loop\] $$(u,v)_\sharp{\mathcal{F}}_{x_0}(g,\Gamma_0)= (u,v)_\ast\pi(M,x_0)=\{ (u_\ast[\gamma],v_\ast[\gamma]): [\gamma]\in \pi(M,x_0)\}.$$ By these two identities (\[eq z\]) is equivalent to $$u_\ast[\gamma]=[\alpha]^{-1}v_\ast[\gamma][\alpha]$$ for all $[\gamma]\in \pi(M,x_0)$. Hence we are done. \[pathconj\] If $u,v\in {W^{1,p}(M;N)}$ are path homotopic ($p\ge 2$) then for almost every $x_0\in M$ $u_\ast\pi(M, x_0)$ and $v_\ast\pi(M, x_0)$ are conjugated. Suppose first that $p<n$. Then by [@han03 Theorem 1.1] $u$ and $v$ are $[p-1]$-homotopic and, since $p\ge 2$, in particular $1$-homotopic. Fix a $1$-skeleton $K$ of $M$ containing a point $x_0\in \{{\mathcal{M}}(|Du|^p+|Dv|^p)<\infty \}$, and such that $u|_{K}$ and $v|_{K}$ are (continuous and) homotopic by a homotopy $h:K\times [0,1]\to N$. To prove that the image subgroups of the homomorphisms are conjugated, take a loop $\gamma$ with basepoint $x_0$. By [@hat02 Section 4.1, Theorem 4.8] $\gamma$ is homotopic to a loop $\gamma'$ which lies in $K$. Thus the image loops $u\circ\gamma'$ and $v\circ\gamma'$ are conjugated by $$H(s,t)=h(\gamma(s),t),\ t,s\in [0,1]^2.$$ Denoting by $\alpha$ the path $t\mapsto h(x_0,t)$ we thus have $$[u\circ\gamma']=[\alpha^{-1}(v\circ\gamma')\alpha].$$ Consequently $$u_\ast([\gamma])=u_\ast([\gamma'])=(v_\ast([\gamma']))^{[\alpha]}=(v_\ast([\gamma]))^{[\alpha]}, \ [\gamma]\in \pi(M,x_0).$$ This proves the claim in the case $p<n$. In case $p\ge n$ it follows from Theorem \[main\] and Theorem \[homotopy\] that $u$ and $v$ are $p$-quasihomotopic. The claim now follows from Lemma \[pconj\] above. Combining Proposition \[pqpath\] and Lemmata \[pconj\] and \[pathconj\] we obtain the following theorem, which directly implies Theorem \[main2\]. \[main3\] Let $p\ge 2$, and $N$ aspherical. Then two maps $u,v\in {W^{1,p}(M;N)}$ are path homotopic if and only if the subgroups $u_\ast\pi(M)$ and $v_\ast\pi(M)$ are conjugated. Suppose $u,v$ are path homotopic. Then Lemma \[pathconj\] implies the claim. If, conversely, $u_\ast\pi(M)$ and $v_\ast\pi(M)$ are conjugated, Lemma \[pconj\] implies that $u$ and $v$ are $p$-quasihomotopic. By Proposition \[pqpath\] $u$ and $v$ are path homotopic. Quasiconnectedness of ${W^{1,p}(M;S^k)}$ ======================================== In this section the following result is proven. \[mn\] Suppose $M$ is a smooth compact riemannian manifold, possibly with boundary, and $1<p\le k$. Then $u\in {W^{1,p}(M;S^k)}$ is $p$-quasiconnected, i.e. every map is $p$-quasihomotopic to a constant. We single out the following corollary. \[sn\] Suppose $2 \le k$ and $1<p\le k$. Then any two maps in ${W^{1,p}(S^k;S^k)}$ are $p$-quasihomotopic. The proof of Theorem \[mn\] is based on the example given in [@bre03] after Theorem 3. We begin by observing that that in a suitable range of $p$’s points have small preimages under Sobolev maps. \[cap\] Let $f\in {W^{1,p}(M;N)}$ be a $p$-quasicontinuous representative, $1<p\le \dim N$. Then for almost every $y\in N$ we have $${\operatorname{Cap}}_p(f^{-1}(y))=0.$$ For $y\in N$, consider the function $u_k\in {{W^{1,p}(M)}}$ given by $$u_k(x)=w_k\circ f,$$ where $w_k:N\to {\mathbb{R}}$ is defined by $$\begin{aligned} w_k(z)=\left\{ \begin{array}{ll} 1 &, z\in B(y,1/k^2)\\ (\log k)^{-1}\log\left(\frac{1/k}{|z-y|}\right) &, z\in A(y,1/k^2,1/k)\\ 0 &, z\notin B(y,1/k) \end{array} \right. \end{aligned}$$ Then $u_k|_{f^{-1}(y)}\equiv 1$ $p$-quasieverywhere and therefore $${\operatorname{Cap}}_p(f^{-1}(y))\le \liminf_{k\to\infty}\|u_k\|_{1,p}^p.$$ We have the pointwise estimates $$\begin{aligned} &0\le u_k(x)\le \chi_{B(y,1/k)}(f(x)),\\ &|\nabla u_k|(x)\le |\nabla w_k|(f(x))|\nabla f|(x)\le (\log k)^{-1}\frac{\chi_{A(y,1/k^2,1/k)}(f(x))}{|f(x)-y|}|\nabla f|(x) \end{aligned}$$ almost everywhere. Thus $$\begin{aligned} {\operatorname{Cap}}_p(f^{-1}(y))\le & \liminf_{k\to\infty}\left[\int_M\chi_{B(y,1/k)}\circ f {\mathrm {d}}x\right. \\ &\left. +(\log k)^{-p}\int_M\frac{\chi_{A(y,1/k^2,1/k)}(f(x))}{|f(x)-y|^p} |\nabla f|^p{\mathrm {d}}x\right]. \end{aligned}$$ Integrating over $y\in N$ and using Fatou and Fubini we obtain $$\begin{aligned} \label{cap1} &\int_N{\operatorname{Cap}}_p(f^{-1}(y)){\mathrm {d}}y\\ \le & \liminf_{k\to\infty}\int_M\int_N\left[\chi_{B(y,1/k)}(f(x))+(\log k)^{-p}|\nabla f|^p(x)\frac{\chi_{A(y,1/k^2,1/k)}(f(x))}{|f(x)-y|^p}\right]{\mathrm {d}}y{\mathrm {d}}x \nonumber \end{aligned}$$ Since $$\int_M\int_N\chi_{B(y,1/k)}(f(x)){\mathrm {d}}y{\mathrm {d}}x =\int_M\left(\int_N\chi_{B(f(x),1/k)}(y){\mathrm {d}}y\right){\mathrm {d}}x \le C/k^{\dim N}$$ inequality (\[cap1\]) becomes $$\begin{aligned} \label{cap2} &\int_N{\operatorname{Cap}}_p(f^{-1}(y)){\mathrm {d}}y\nonumber \\ \le & \liminf_{k\to\infty}\int_M\int_N(\log k)^{-p}|\nabla f|^p(x)\frac{\chi_{A(y,1/k^2,1/k)}(f(x))}{|f(x)-y|^p}{\mathrm {d}}y{\mathrm {d}}x. \end{aligned}$$ The righthand integral in turn may be written as $$\begin{aligned} (\log k)^{-p}\int_M|\nabla f|^p(x)\left(\int_N\frac{\chi_{A(f(x),1/k^2,1/k)}(y)}{|f(x)-y|^p}{\mathrm {d}}y\right){\mathrm {d}}x. \end{aligned}$$ For sufficiently large $k\ge 1$ one may estimate $$\begin{aligned} \int_N\frac{\chi_{A(f(x),1/k^2,1/k)}(y)}{|f(x)-y|^p}{\mathrm {d}}y \lesssim C\int_{{\mathbb{R}}^{\dim N}}\chi_{A(0,1/k^2,1/k)}(y)\frac{{\mathrm {d}}y}{|y|^p}\simeq \int_{1/k^2}^{1/k} t^{\dim N-1-p}{\mathrm {d}}t. \end{aligned}$$ Since $p\le \dim N$ we obtain $$\int_{1/k^2}^{1/k} t^{\dim N-1-p}{\mathrm {d}}t\le \int_{1/k^2}^{1/k} t^{-1}{\mathrm {d}}t=\log k.$$ Plugging all these inequalities into (\[cap2\]) we obtain $$\begin{aligned} \int_N{\operatorname{Cap}}_p(f^{-1}(y)){\mathrm {d}}y\le C \liminf_{k\to\infty}\int_M(\log k)^{1-p}|\nabla f|^p(x){\mathrm {d}}x=0, \end{aligned}$$ thus completing the proof. \[capcor\] Let $2\le k$ and $1<p\le k$. For a $p$-quasicontinuous representative $f\in {W^{1,p}(M;S^k)}$ the following holds for almost every $y\in S^k$. $$\lim_{r\to 0}{\operatorname{Cap}}_p(f^{-1} B(y,r))=0.$$ Let $\varepsilon>0$ be arbitrary and let $U\subset M$ be open with ${\operatorname{Cap}}_p(U)<\varepsilon$ and $f|_{M\setminus U}$ continuous. We may estimate $${\operatorname{Cap}}_p(f^{-1} B(y,r))\le {\operatorname{Cap}}_p((f|_{M\setminus U})^{-1}( \overline B(y,r)))+{\operatorname{Cap}}_p(U).$$ The sets $(f|_{M\setminus U})^{-1}( \overline B(y,r))$ are compact and decrease to $(f|_{M\setminus U})^{-1}(y)$ as $r>0$ decreases. By the monotonicity of capacity for compact sets therefore $$\limsup_{r\to 0}{\operatorname{Cap}}_p((f|_{M\setminus U})^{-1}( \overline B(y,r)))={\operatorname{Cap}}_p((f|_{M\setminus U})^{-1}(y)).$$ The latter quantity is zero for almost every $y\in S^k$ by Lemma \[cap\] above. Thus we obtain $${\operatorname{Cap}}_p(f^{-1} B(y,r))\le 0+{\operatorname{Cap}}_p(U)<\varepsilon.$$ Since $\varepsilon >0$ was arbitrary the claim follows. Suppose $f\in {W^{1,p}(M;S^k)}$. Choose $y_0\in S^k$ so that the claim of Corollary \[capcor\] holds for $y=y_0$. Define $h:S^k\times [0,\infty]\to S^k$ by $$\begin{aligned} h(x,t)=\left\{\begin{array}{ll} \frac{x-ty_0}{|x-ty_0|},&\ 0\le t<\infty\\ -y_0,&\ t=\infty \end{array}\right. \end{aligned}$$ Note that $h|_{S^k\setminus\{x_0\}\times [0,\infty]}$ is continuous. We claim that $$H(x,t)=h(f(x),t),\ (x,t)\in M\times [0,\infty]$$ is a $p$-quasihomotopy $f\simeq -y_0$. Given $\varepsilon>0$ let $U$ be an open set with ${\operatorname{Cap}}_p(U)<\varepsilon/2$ and $f|_{M\setminus U}$ continuous. Further let $r>0$ be small enough so that ${\operatorname{Cap}}_p(f^{-1}B(y_0,r))<\varepsilon/2$. Set $E=U\cup [(f^{-1}B(x_0,r))\setminus U]$. Then $E$ is open, ${\operatorname{Cap}}_p(E)<\varepsilon$ and $H|_{M\setminus E\times [0,\infty]}$ is continuous, $$H(x,0)=\frac{f(x)}{|f(x)|}=f(x),\ H(x,\infty)=-x_0,\ x\in M\setminus E.$$ A similar procedure yields a continuous path in ${W^{1,p}(S^n;S^n)}$ between $f$ and a constant map when $p<n$ (see [@bre03]), but not when $p=n$. Indeed, in the latter case it is not possible to connect every map to a constant path by a continuous path ([@bre03 Lemma 1”]) and so we see that the converse of Theorem \[main\] is not true. In closing we remark that ${W^{1,p}(B^{k+1};S^k)}$, $k<p<k+1$ provides another example where path and $p$-quasihomotopy differ. Consider the map $g:(0,1]\times S^k\to B^{k+1}$ given by $$g(t,y)=ty.$$ This is a $p$-quasihomotopy equivalence ($p<k+1$) since the map $h(x)=(|x|,x/|x|)$ is $p$-quasicontinuous and $g\circ h=id_{B^{k+1}}$, $h\circ g=id_{(0,1]\times S^k}$ $p$-quasieverywhere. Thus, postcomposition with $g$ defines a continuous map $$G:{W^{1,p}(B^{k+1};S^k)}\to {W^{1,p}((0,1]\times S^k;S^k)},\ Gf=f\circ g,$$ which preserves $p$-quasihomotopy classes and is bijective (the map $f\mapsto f\circ h$ is an inverse to $G$). It is known ([@bre01], Proposition 0.2) that ${W^{1,p}((0,1]\times S^k;S^k)}$ is path connected when $p<k+1$. However, when $k<p<k+1$, the Sobolev space ${W^{1,p}((0,1]\times S^k;S^k)}$ and consequently ${W^{1,p}(B^{k+1};S^k)}$ is not $p$-quasiconnected. (This easily seen by noting that the map $f(t,y)=y$, $(t,y)\in (0,1]\times S^k$, is not $p$-quasihomotopic to a constant map.) ### Acknowledgements {#acknowledgements .unnumbered} I would like to thank Pekka Pankka for reading the manuscript and making many valuable comments. I also thank Pawel Goldstein for useful discussions.

--- abstract: 'Satellite-based quantum terminals are a feasible way to extend the reach of quantum communication protocols such as quantum key distribution (QKD) to the global scale. To that end, prior demonstrations have shown QKD transmissions from airborne platforms to receivers on ground, but none have shown QKD transmissions from ground to a moving aircraft, the latter scenario having simplicity and flexibility advantages for a hypothetical satellite. Here we demonstrate QKD from a ground transmitter to a receiver prototype mounted on an airplane in flight. We have specifically designed our receiver prototype to consist of many components that are compatible with the environment and resource constraints of a satellite. Coupled with our relocatable ground station system, optical links with distances of were maintained and quantum signals transmitted while traversing angular rates similar to those observed of low-Earth-orbit satellites. For some passes of the aircraft over the ground station, links were established within of position data transmission, and with link times of a few minutes and received quantum bit error rates typically , we generated secure keys up to in length. By successfully generating secure keys over several different pass configurations, we demonstrate the viability of technology that constitutes a quantum receiver satellite payload and provide a blueprint for future satellite missions to build upon.' address: - '$^1$ Institute for Quantum Computing, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada' - '$^2$ Department of Physics and Astronomy, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada' - '$^3$ Department of Electrical and Computer Engineering, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada' author: - 'Christopher J. Pugh$^{1,2}$, Sarah Kaiser$^{1,2}$[^1], Jean-Philippe Bourgoin$^{1,2}$, Jeongwan Jin$^{1,2}$, Nigar Sultana$^{1,3}$, Sascha Agne$^{1,2}$, Elena Anisimova$^{1,2}$, Vadim Makarov$^{2,1,3}$, Eric Choi$^{1}$[^2], Brendon L. Higgins$^{1,2}$ and Thomas Jennewein$^{1,2}$' title: Airborne demonstration of a quantum key distribution receiver payload --- [*Keywords*]{}: quantum key distribution, quantum communication, quantum cryptography, quantum optics, photon detection, quantum satellite payload Introduction ============ Quantum key distribution (QKD) [@Bennet84; @SBC09] establishes cryptographic keys between two distant parties in a way that is cryptanalytically unbreakable. Ground based implementations of QKD using optical fiber links are limited to distances of a few hundred kilometers due to absorption losses, which scale exponentially with distance, leading to insufficient signal-to-noise [@SWV09; @LCW10; @KLH15]. Alternatively, free space links have been demonstrated over ground with varying distances, both in stationary [@BHK98; @HND02; @UTSComm07; @SWF07; @VDS14] and moving [@Nauerth13; @WYLZ13; @BHG15] configurations. But despite losses due to geometric effects scaling quadratically with distance, the addition of atmospheric absorption and turbulence, and the necessity of having clear line of sight, limit terrestrial free-space transmissions to, also, a few hundred kilometers. Much greater distances could be spanned in free-space transmissions outside Earth’s atmosphere. Utilizing orbiting satellites therefore has potential to allow the establishment of global QKD networks, with “quantum” satellites acting as intermediaries. Such satellites could operate as untrusted nodes linking two ground stations simultaneously [@HBK00; @TCS14], or trusted nodes connecting any two ground stations on Earth at different times [@RTG02; @UJK09; @VJT08; @EAW11; @MYM11; @YCL13; @VDT16]. The majority of such analyses propose a quantum downlink, where photons are generated at the satellite and transmitted to receivers on the ground. Since 2010, the Canadian Space Agency (CSA) has studied the proposed Quantum Encryption and Science Satellite (QEYSSat) [@JBH14], where a mission concept was developed in partnership with COM DEV (now Honeywell Aerospace). This concept, in contrast to many other missions, proposes a quantum uplink, placing the receiver on the satellite while keeping the quantum source at the ground station. Under similar conditions, the uplink configuration has a lower key generation rate than the downlink, owing to atmospheric turbulence affecting the beam path earlier in the propagation. Nevertheless, comprehensive theoretical comparative study of QKD under uplink and downlink conditions—which included the effects of atmospheric turbulence, absorption, beam propagation, optical component losses, detector characteristics, noise contributions, and representative pointing and collection capabilities at a hypothetical satellite—concluded that an uplink approach is viable, with the reduction in generated key bits (compared to downlink) being less than one order of magnitude [@BMH13]. Importantly, an uplink also possesses a number of advantages over a downlink, including relative simplicity of the satellite design, not requiring high-rate true random number generators, relaxed requirements on data processing and storage (only the photon reception events need be considered, which are many orders of magnitude fewer than the source events), and the flexibility of being able to incorporate and explore various different quantum source types with the same receiver apparatus (which would have major associated costs were the source located on the satellite, as for downlink). Recently, China launched a quantum science satellite which aims to perform many quantum experiments with optical links between space and ground [@G16; @C16]. However its exact capabilities are unverified as no details or results have been published at this time. Demonstrations of QKD with moving and airborne platforms take important steps to verifying the readiness of quantum technology, and the supporting classical technology, for deployment within a satellite payload. To date, however, reported demonstrations of QKD with aircraft have operated exclusively in the downlink configuration [@Nauerth13; @WYLZ13], where the quantum states are generated and transmitted from the aircraft to a receiver at a stationary ground location. Here we demonstrate QKD uplink to a receiver on a moving aircraft. Our apparatuses incorporate coarse- and fine-pointing systems necessary to establish and maintain optical link, quantum source and measurement components that conduct polarization-encoded QKD, and suitable post-processing algorithms to extract secure key. The results show good performance at the same angular rates exhibited by low-Earth-orbit (LEO) satellites. Our QKD receiver makes extensive use of components custom-designed according to the mass, volume, power, thermal, and vacuum operating environment requirements of systems to be embedded in a satellite payload—many components are already space suitable, and others have a clear path to flight. In a recent study, conducted with the University of Toronto Institute for Aerospace Studies Space Flight Laboratory (UTIAS SFL), a realistic satellite concept was developed, incorporating the space-ready receiver apparatus demonstrated here, integrated into the flight-proven NEMO-150 [@SFL] micro-satellite bus (see below). This, together with our airborne operational demonstration, illustrates the technological advancements made towards the development of a space-suitable QKD receiver, and highlights the feasibility and technological readiness of an uplink QKD satellite. Apparatuses and Methods ======================= Concept ------- ![Flight paths for the arc and line, followed from left to right. The star indicates the location of the ground station at Smith Falls–Montague Airport. The inner portions represent where the quantum link was active. Photo produced using GPSVisualizer.com, map data 2016 Google, imagery 2016 Cnes/Spot Image, DigitalGlobe, Landsat, New York GIS, USDA Farm Service Agency.[]{data-label="fig:Flight7km"}](FlightPath7kmBoth.jpg){width="0.8\linewidth"} The apparatuses for our demonstration consist of a QKD source and transmitter, located at a ground station near the airstrip of Smiths Falls–Montague Airport, and a QKD receiver, located on a Twin Otter research aircraft from the National Research Council of Canada. Optical links were only attempted at night, to limit optical noise. One systems-test day-time flight was conducted (where the optical links were not attempted), followed by night-time flights. Two night-time flights were conducted, each two hours duration and consisting of several passes of varying trajectories. Optical links were established using tracking feedback to two-axis motors, guided by strong beacon lasers (at a wavelength different from the quantum signal) and an imaging camera, at each of the two sites. The QKD signals produced by the source were guided through a telescope on the ground and pointed to the receiver on the aircraft. There the QKD signal polarizations and times-of-arrival were recorded for later correlation and processing to complete the QKD protocol and extract the key. We focused on two path types: arcs with (approximately) constant radius around the ground station, and straight lines past the ground station. For straight line paths, the distance we quote is the minimum. Over the two nights we performed 14 passes with nominal distances of , , , and , in both line and arc configurations, at an altitude of above sea level—see, for example, Fig. \[fig:Flight7km\] for the flight path of an arc at radius. For this flight mission concept, a sequence of GPS coordinates was calculated for each flight, with the start angle relative to the ground station and the distance were used as input. These coordinates were transferred to the flight software of the aircraft by the pilots. We developed a decision tree such that, based on the observed performance of each pass, we could immediately select an appropriate course of action (e.g. to perform troubleshooting or collect data under different conditions). That a mission concept such as this is viable shows that a similar mission concept, appropriate for an orbiting satellite receiver, can realistically be achieved. Source and Transmitter ---------------------- [0.475]{} ![Left, schematic diagram of the quantum source and transmitter apparatus. AWG, arbitrary waveform generator; WDM, wavelength division multiplexer; PBS, polarizing beam splitter; OA, optical attenuator; F, band-pass filter; PT, polarization tomography; TT, time tagger. Other acronyms and details given in the text. The red border indicates components that are mounted on the motors. Right, ground station located at Smiths Falls–Montague airport, showing (right-to-left) the trailer where the source is located, motor mount with transmitter telescope attached, Wi-Fi antenna, and calibration telescope.[]{data-label="fig:Transmitter"}](TransmitterSchematic.pdf "fig:"){width="\linewidth"} [0.475]{} ![Left, schematic diagram of the quantum source and transmitter apparatus. AWG, arbitrary waveform generator; WDM, wavelength division multiplexer; PBS, polarizing beam splitter; OA, optical attenuator; F, band-pass filter; PT, polarization tomography; TT, time tagger. Other acronyms and details given in the text. The red border indicates components that are mounted on the motors. Right, ground station located at Smiths Falls–Montague airport, showing (right-to-left) the trailer where the source is located, motor mount with transmitter telescope attached, Wi-Fi antenna, and calibration telescope.[]{data-label="fig:Transmitter"}](Transmitter.jpg "fig:"){width="\linewidth"} Our QKD source is a significantly improved version of a previous-generation apparatus [@YMB13], implementing BB84 with decoy states [@LMC05] at . Weak coherent pulses at wavelength are generated by combining a narrow-band continuous-wave (CW) laser (L1) with triggered-pulsing laser (L2) through sum frequency generation in a periodically poled magnesium oxide (PPMgO) waveguide (see Fig. \[fig:Transmitter\]). For each pulse, one of three intensity levels is chosen: signal, decoy, or vacuum, with probabilities of , , and , respectively. Signal and decoy levels are generated using a fast electro-optical intensity modulator (IM) calibrated to emit $\mu \approx 0.5$ and $\nu \approx 0.1$ mean photon number at the entrance of the transmitter telescope, respectively. The vacuum state is made by suppressing the laser trigger. Each of the four BB84 polarizations—horizontal (H), vertical (V), diagonal (D), and anti-diagonal (A)—are imposed using two electro-optical phase modulators (PMs), each in one arm of a balanced Mach-Zehnder polarization interferometer. With a balanced input (D), the PMs can address any point on the circle through D, right-circular (R), A, and left-circular (L). A subsequent unitary rotation takes these to D, H, A, and V, respectively. The intensity and polarization states are generated according to a randomized sequence that repeats every 1000 pulses. Although this is insecure, it is sufficient for our demonstration, while upgrading to a fully random sequence (e.g., given by a quantum random number generator) is straightforward and a suitable system has been identified. Pulse intensities are measured locally through the weak output of an optical fiber splitter (90:10) connected to a silicon avalanche photodiode (Si-APD) operating in Geiger mode with active quenching. The bulk of the pulse power is guided from the source to the transmitter through single-mode optical fiber. The beam passes through a band-pass ( bandwidth) filter (to impede Trojan-horse attacks [@JAK14]) and then a 75:25 beam splitter. We employ a polarization correction system to undo the unknown unitary rotation applied by the single-mode fiber—the reflected of pulses undergo characterization, while the remaining of pulses pass through a triplet of wave plates (WPs) in motorized rotation stages that apply a compensation operation to the states, and are finally transmitted through a diameter Sky-Watcher BK 1206AZ3 refractive telescope. The polarization characterization subsystem consists of two beam paths, where each path passes through a port of a rotating chopper wheel that contains linear polarizers. The linear polarizers are each calibrated to project onto the H, V, D, or A state—however, one of the two beam paths contains a quarter-wave plate just prior to the chopper wheel, thereby facilitating projections onto a tomographically complete set of three polarization bases: H/V, D/A, and R/L. The actual state any given photon is projected to depends on which blade of the wheel is open at the time the photon passes through (the rotation of the wheel is also recorded). The two beams are each coupled into fiber and directed to Si-APDs. With near real-time analysis of source and detection data (performed on per-second integrated counts), we obtain tomographic reconstructions, for each of the generated polarization states, of the states at the transmitter after the rotation applied by the fiber. We then optimize the compensating wave plate triplet (a sequence of quarter-, half-, and quarter-wave plate) to maximize the fidelity of the states expected after compensation with the nominally generated states. The optimal positions are given to the motorized stages, applying the (rotated) wave plates to pulses that are then transmitted through the telescope towards the receiver. During our airborne trials, the QKD source optics and electronics, as well as computers for data recording and pointing feedback, were located inside of a trailer to maintain thermal and humidity stability. The transmitter pointing stages, polarization characterization optics, and telescope were located just outside the trailer, with cabling running through a small window. Equipped with an electric generator, our ground station is relocatable and self-sufficient. Receiver -------- [0.475]{} ![Left, schematic diagram of the receiver apparatus. F, band-pass filters; WB, wide-field beacon (produced by the IRL). Other acronyms and details given in the text. The red border indicates components that are mounted on the motors. Right, receiver apparatus facing out the port-side door of the NRC Twin Otter research aircraft, showing (clockwise) the telescope, beacon assembly, motor mount, IRL, Wi-Fi antenna, and FPC (behind). Other components not visible (primarily electronics) are mounted in front of the seats seen at the left of the picture.[]{data-label="fig:Receiver"}](ReceiverSchematic.pdf "fig:"){width="\linewidth"} [0.475]{} ![Left, schematic diagram of the receiver apparatus. F, band-pass filters; WB, wide-field beacon (produced by the IRL). Other acronyms and details given in the text. The red border indicates components that are mounted on the motors. Right, receiver apparatus facing out the port-side door of the NRC Twin Otter research aircraft, showing (clockwise) the telescope, beacon assembly, motor mount, IRL, Wi-Fi antenna, and FPC (behind). Other components not visible (primarily electronics) are mounted in front of the seats seen at the left of the picture.[]{data-label="fig:Receiver"}](Receiver.jpg "fig:"){width="\linewidth"} At the receiver (Fig. \[fig:Receiver\]), the signal is collected by a Tele Vue NP101is refractive telescope with a aperture, and coupled into a sequence of custom components developed under contract with the Canadian Space Agency [@CSA_QKDR]. First of these is a fine-pointing unit (FPU), developed with Institut National d’Optique (INO) and Neptec Design Group, which guides both the quantum and beacon signals with a fast-steering mirror (FSM). Inside the FPU, a dichroic mirror separates the quantum and beacon signals—the beacon is reflected towards a quad-cell photo-sensor (QS), providing position feedback to a fine-pointing controller (FPC) that guides the fast-steering mirror in a closed loop [@PKB17]. The FPU, measuring and , has a field of view. The collected quantum beam is guided through a pinhole, acting as a spatial-mode filter [@SCB15], followed by a pair of ( bandwidth) spectral filters. It then passes into a custom integrated optical assembly (IOA) developed with INO, containing a passive-basis-choice polarization analysis module with a 50:50 beam splitter and polarizing beam splitters. The IOA, measuring and , produces four beams coupled into multimode fibers, corresponding to the four BB84 measurement states (H, V, D, and A) with state contrasts between 532:1 and 2577:1. The four IOA output fibers are guided to a detector module (DM) containing four Excelitas Technologies SLiK Si-APD detectors operating in Geiger mode with passive quenching. The DM measures and , and operates at steady state (including thermoelectric cooling of detector active areas to ) to give a detection efficiency of , biased above breakdown. The detectors trigger low-voltage differential signalling pulses which are measured at a control and data processing unit (CDPU) based on Xiphos Systems Corporation’s Q7 processor card (recently flown on GHGSat [@GHG]) with a custom daughterboard. The CDPU utilizes an ARM Cortex-A9 processor and measures , , drawing while operating. A field-programmable gate array embedded in the CDPU is programmed to implement time-tagging of detection pulses with a resolution of , while data storage, communication, and processing software running in the Linux operating system implement the receiver-side QKD protocol. The receiver telescope was mounted facing out the cabin door on the port side of the aircraft, and flown with the door removed. The electronics and computers were located six feet forward in the aircraft cabin, and optical fibers and cables conducted signals between the electronics and the receiver telescope and pointing equipment. Acquisition and Calibration --------------------------- The transmitter and receiver each have a beacon laser assembly (BLA) consisting of three fiber launchers with fixed divergence angles of and individual tip/tilt control. These are mounted on each telescope and fed strong () laser light from fiber-coupled beacon laser source (BLS) arrays located away from the telescopes. A beacon camera (BC)—a 50 frame-per-second, 2 megapixel imaging camera with an band-pass filter ( bandwidth)—is also mounted to each telescope. Each telescope is attached to a commercial two-axis motor system (transmitter: ASA DDM85 Standard, receiver: FLIR PTU-D300E), providing first-stage “coarse” pointing. When light at the beacon wavelength is visible as a bright spot on the camera image, our custom pointing software (running on PCs at each site) controls the angular speeds of the motors to minimize the deviation of the spot’s position from a calibrated reference position. The control feedback loop incorporates the estimated angular speed of the spot (based on position differences between recent images, taking into account previous motor motions), a factor proportional to the spot’s current deviation, and a factor proportional to the accumulated (integrated) spot deviations. The pointing software operates as a state machine, and also includes a “coasting” state to handle short drop-outs of the beacon signal, and “acquiring” and “searching” states to support the initial acquisition of the beacon. To achieve initial acquisition, we employ inertial navigation modules (INMs), containing GPS receivers and attitude sensors, mounted to the telescopes. Each site transmits their GPS location to the other site via a classical RF (Wi-Fi) link, and then calculates the other site’s orientation relative to its own based on its local attitude data. During initial testing, the INMs exhibited an attitude uncertainty of about —significantly larger than the divergence of the beacon lasers. To mitigate this, we turn on a bright infra-red light-emitting diode array (IRL) at the receiver with much greater divergence (of order ), allowing the transmitter to find, and point towards, the receiver. Once the receiver sees the transmitter’s beacon spot in its camera image and has moved to position, the IRL is switched off, and two-way beacon tracking continues for the remainder of the pass. A necessary practical feature of our transmitter and receiver apparatuses is that they can be independently calibrated, as they would not be co-located prior to establishing a link (much like for a satellite mission). To align each of the beacon lasers with the quantum signal beam path, we first inject alignment laser power into each telescope, and point the telescope towards a separate larger-diameter () telescope, located away, equipped with a camera imaging the far field. We then observe the position of the beacon beams on the camera image, and adjust the tip and tilt of each beacon fiber launcher to center its output over the signal spot. To calibrate the reference position of the beacon camera at the transmitter and the collimation of the transmitted quantum beam, we optimize the power received (using the alignment laser injected into the transmitter telescope) at another telescope located at a sufficient distance down the runway. The receiver beacon camera, which has greater tolerance due to the receiver’s fine-pointing unit, is calibrated using a corner cube located away in the NRC hangar. These alignments were done prior to each flight. These independent calibrations allowed link acquisition to begin immediately upon the arrival of the airplane in the vicinity of the ground station. Results ======= In total, seven of the 14 airplane passes over the ground station successfully established a quantum signal link. Issues, including minor equipment failures (e.g., a loose beacon camera lens) and accidental controller misconfigurations, particularly hampered link establishment during the first night—two of the seven attempts were successful. These issues were addressed during the intervening day, and the second night had considerably better link establishment rate—five of seven attempts. (We attribute the two failures on the second night—both attempted straight-line paths—to the fixed orientation of the Wi-Fi transceiver at the aircraft being poor for this geometry, particularly at the beginning of a pass.) Secret key was extracted out of six of the seven successful passes. From data collected during these passes, we observe the performance of the system at various distances and with angular speeds. Circular-arc passes allowed us to demonstrate longer duration of key exchange, compared to straight-line passes, as the receiver telescope held a relatively constant position during the pass, making link establishment and pointing easier. Straight line passes, however, are much more representative of a satellite passing over a ground station, as they simulate the change in angular speed that would be experienced during such a pass. The maximum angular rate is reached when the airplane is closest to the ground station for that pass—the greatest maximum angular rate we measured for our passes was at a distance of (arc). This angular rate is consistent with overflying LEO spacecraft such as for a orbit, as baselined for QEYSSat, or for the International Space Station (ISS). ---------------------------------------------- ------------ ------------ ------------ ------------ ------------ ------------ ------------ arc 1 line arc 2 line arc arc arc 2016-09-21 2016-09-21 2016-09-22 2016-09-22 2016-09-22 2016-09-22 2016-09-22 Parameter 2:57:45 3:30:45 1:15:23 2:19:33 2:24:45 2:42:16 2:57:42 Classical link duration \[\] 288 172 352 34 170 210 289 Quantum link duration \[\] 235 158 250 33 158 206 269 Mean speed \[\] 208 200 198 236 216 259 212 Maximum angular speed \[\] 0.76 0.45 0.75 1.0 1.28 0.60 0.37 Transmitter pointing error ($10^{-3}$)\[\] 22.0 4.85 1.33 3.42 2.91 1.58 2.82 Receiver pointing error ($10^{-3}$)\[\] 125 126 63.0 86.5 89.8 78.6 87.2 Receiver fine-pointing error ($10^{-3}$)\[\] 2.73 9.98 No data 2.62 2.39 3.01 12.7 Source QBER \[\] 5.08 3.58 3.32 2.66 4.37 2.80 3.39 Signal QBER \[\] 13.13 5.24 3.42 2.96 5.20 2.96 3.30 Decoy QBER \[\] 19.54 11.1 6.13 6.35 7.93 5.97 8.46 Theoretical loss \[\] 52.1 41.6–44.8 28.1 33.3–35.1 30.9 32.1 39.9 Mean measured loss \[\] 48.0 51.1 34.5 39.5 34.4 39.4 42.6 Error correction efficiency 1.4 1.16 1.33 1.4 1.18 1.46 1.27 Signal-to-noise threshold 0 1500 2000 1000 1000 2000 2500 Sifted key length \[bits\] 152508 95710 5212446 853066 5102122 2348086 1175317 Secure key length \[bits\] None 9566\* 867771 71648 44244 200297 70947 ---------------------------------------------- ------------ ------------ ------------ ------------ ------------ ------------ ------------ : Summary of data from passes where a quantum link was established. All times are UTC. Except where indicated (\*), secure key lengths incorporate finite-size effects. \[tab:Flights\] Table \[tab:Flights\] summarizes the seven passes where quantum signal was successfully transmitted to the receiver aboard the aircraft. Passes typically lasted a few minutes, with the aircraft travelling at . To quantify pointing performance, we define the typical pointing error as the measured distance of the beacon spot from the calibrated reference point on the camera image, discarding times when the motors had just begun tracking. The mean typical pointing error at the transmitter varied from over the passes; at the receiver, it was . The receiver’s fine-pointing unit measured pointing errors similar to the pointing error of the transmitter, between , where the deviation was measured from the centre of the quad-cell sensor. (These values are used in the link analysis model, below.) [0.475]{} ![Results for the arc pass (left) and the straight-line pass (right). a) and e) show the speeds of the azimuthal (coarse) motor at the transmitter. The insets, corresponding to the shaded portions, show the motor speed during initial acquisition, with times $t_1$ through $t_4$ identifying establishment of the Wi-Fi link, identification of the beacon spot, lock to the beacon spot, and first counts received, respectively. The oscillation prior to this in a) is from a spiralling search state of the pointing software. b) and f) show coarse- and fine-pointing performance at the receiver. Where there are no coarse pointing data (e.g., at the beginning of a pass), no beacon spot was found in the camera image. This corresponds with large fluctuations in the fine-pointing deviation—in the absence of beacon light, the unit operates on electrical noise generated at the quad cell. c) and g) show the estimated time of flight of the photons from the transmitter to the receiver (used in event time-correlation), calculated from per-second GPS coordinates at each site. The smooth curve in g) is particularly characteristic of the straight-line pass, with a similar shape to that of a satellite pass. d) and e) show the total detection rates at the receiver and the QBER of the signal.[]{data-label="fig:Pointing"}](Pointing7kmArc.pdf "fig:"){width="\linewidth"} [0.475]{} ![Results for the arc pass (left) and the straight-line pass (right). a) and e) show the speeds of the azimuthal (coarse) motor at the transmitter. The insets, corresponding to the shaded portions, show the motor speed during initial acquisition, with times $t_1$ through $t_4$ identifying establishment of the Wi-Fi link, identification of the beacon spot, lock to the beacon spot, and first counts received, respectively. The oscillation prior to this in a) is from a spiralling search state of the pointing software. b) and f) show coarse- and fine-pointing performance at the receiver. Where there are no coarse pointing data (e.g., at the beginning of a pass), no beacon spot was found in the camera image. This corresponds with large fluctuations in the fine-pointing deviation—in the absence of beacon light, the unit operates on electrical noise generated at the quad cell. c) and g) show the estimated time of flight of the photons from the transmitter to the receiver (used in event time-correlation), calculated from per-second GPS coordinates at each site. The smooth curve in g) is particularly characteristic of the straight-line pass, with a similar shape to that of a satellite pass. d) and e) show the total detection rates at the receiver and the QBER of the signal.[]{data-label="fig:Pointing"}](Pointing7kmLine.pdf "fig:"){width="\linewidth"} Figure \[fig:Pointing\] shows observed results for two representative passes, including the motor speed of the transmitter in the horizontal axis and link acquisition stages, the coarse- and fine-pointing errors at the receiver, the calculated time of flight of the quantum signal from the transmitter to the receiver, the rate of detections of all four DM channels combined, and the quantum bit error rate (QBER) of the signal. There, the maximal angular speed was about during beacon pointing lock, after initial acquisition. The mean measured loss of the quantum link during the flights varied from . Our theoretical loss model [@BMH13] assumes a mid-latitude, rural atmospheric model in summer with the ground station located above sea level and visibility. Other model parameters include detector efficiency and receiver optical transmittance of (determined from the measured properties of the receiver prototype). We simulate the effect of atmospheric turbulence at our location using Hufnagel–Valley parameterization of atmospheric conditions [@H74; @H98], with a sea-level turbulence strength of and high-altitude wind-speed of . The measured pointing accuracy, aircraft altitude, and ground distance for each pass was also used. The divergence angle of the quantum beam could not be measured during the flight campaign. For the model we assume diffraction-limited divergence, resulting in lower bound theoretical loss estimates. Indeed, in the experiment, a number of passes were conducted with the transmitter intentionally slightly defocussed so as to avoid saturating the detectors. Consequently, the experimental losses we observed are generally higher than the theoretical losses. The difference between the theoretical loss of an arc pass and the minimum theoretical loss of a line pass at the same nominal distance is due to varying pointing accuracy experienced for each pass, as well as the actual ground distance and altitude deviating from nominal. For QKD analysis we utilize a signal-to-noise (SNR) filter [@EHM12], which assesses the total counts in each frame of data and discards any frame with counts less than a threshold, prior to distilling key bits. We choose thresholds between 1000 and 2500, depending on the pass. Background detection rates at the beginning and end of the pass are sufficiently low that those frames are discarded by the SNR filter. Some drop-outs can be seen in Fig. \[fig:Pointing\]h)—these frames are also discarded by the SNR filter. The source’s intrinsic QBER, as predicted by the polarization correction system, varied between for each pass. At the receiver, the FPU, IOA, and fibers leading to the detectors were shielded with black cloth to minimize stray light entering the detectors, leading to typical total background detection rates of . The QBER measured at the receiver drops to a few percent upon optical link lock, and rests at due to the random noise of background detections at all other times. For passes where secure key was generated, the QBER measured at the receiver, after the SNR filter, varied from . The received QBER during the first night flight was observed to be higher than for the second night, possibly due to an issue with the wave plate motorized stage controller. We generate secure key bits from the data collected during each pass using algorithms tailored for the asymmetric processing resources that would be available with a satellite platform [@BGH15]. These algorithms consist of source and receiver event time-correlation (performed at the ground station), error correction utilizing low-density parity check codes, and privacy amplification via reduced-Toeplitz-matrix two-universal hashes. To ensure security, the uncertainty due to the finite number of samples used to estimate link parameters must be taken into account. Of the six passes from which key could be extracted, five yielded secure key including these finite-size effects (where we use the common ten-standard-deviation heuristic to bound parameter estimates [@SLL09]). The remaining pass had too few counts and could only generate secure key assuming no finite-size effects. Discussion and Conclusion ========================= We have successfully demonstrated quantum key distribution to a satellite receiver payload prototype on an aircraft moving at up to . Our pointing and tracking system was able to establish and maintain an optical link with milli-degree precision over distances while BB84 decoy-state signals were sent across the channel to the aircraft moving at the angular speeds of a LEO satellite. Our custom fine-pointing system, IOA, DM, and CDPU, along with the other commercial components, all performed in concert on the aircraft to generate secure keys, of tens to hundreds of kilobits in length, in various flight scenarios, including the straight-line paths approximating the apparent trajectory a LEO satellite. With source intrinsic QBER typically and post-processing algorithms representative of what would be achievable with a satellite platform, we extracted finite-size secure key for many of the tested passes. The details of path-to-flight modifications necessary to construct space-suitable versions of our receiver components varies. Some elements present on the CDPU daughterboard, for example, will need to be replaced with radiation-hard equivalent versions. Or, for the IOA, glues designed for low out-gassing must be used. Sensitivity of the Si-APDs in the DM to proton radiation in orbit is of particular note, as such radiation can significantly increase dark counts. However, strategies including cooling and thermal annealing [@AHB17], as well as laser annealing [@LAH17], are capable of mitigating these effects, and a space suitable prototype DM implementing these strategies is being developed. [0.4]{} ![The UTIAS SFL NEMO-150 micro-satellite bus housing the quantum receiver payload. Left, external view showing extended telescope and baffle. Right, cross-section showing the possible placement of the main receiver payload components demonstrated during this airborne QKD campaign. Images provided by UTIAS SFL.[]{data-label="fig:Sat"}](QEYSSOutside.jpg "fig:"){width="\linewidth"} [0.25]{} ![The UTIAS SFL NEMO-150 micro-satellite bus housing the quantum receiver payload. Left, external view showing extended telescope and baffle. Right, cross-section showing the possible placement of the main receiver payload components demonstrated during this airborne QKD campaign. Images provided by UTIAS SFL.[]{data-label="fig:Sat"}](QEYSSInside.pdf "fig:"){width="\linewidth"} For pointing to a satellite from the ground, initial acquisition will likely not have a real-time classical communication link to exchange position data. In this case, however, predictions of the satellite position at the time when a link is to be established can be used, as the orbital trajectory of a satellite is predictable with far greater accuracy than the flight path of an airplane. In this context, point-ahead may be necessary (depending on the transmitter’s divergence) to ensure that the quantum beam is coincident with the satellite when it arrives, owing to the satellite’s motion during the time of flight of the optical signals. A fine-pointing system would likely also be required to achieve sufficient accuracy over the significantly larger transmission distance. For the aircraft, this was not necessary. One advantage of the uplink approach is source flexibility. While we have demonstrated only operation with a weak coherent pulse source here, we fully expect that QKD using entangled photon pairs generated at the appropriate wavelength by, for example, spontaneous parametric down-conversion will produce equivalent results under a BBM92-style protocol [@BBM92], with one photon of each pair measured on the ground. To support this, no aspect of the receiver prototype need be modified. Our system demonstrates the viability of an uplink QKD satellite mission. The core quantum components of a QKD satellite receiver have been demonstrated and have clear path to inclusion in space-faring system. In particular, see Fig. \[fig:Sat\] from a recent study conducted with UTIAS SFL, which shows our receiver hardware—FPU, FPC, IOA, DM, and CDPU—with minor modifications, cohesively integrated onto the flight-proven NEMO-150 micro-satellite bus. With the feasibility of performing uplink QKD with moving platforms well supported with satellite-ready hardware, QKD at the global scale utilizing satellite uplinks is within reach. Acknowledgements ================ The authors acknowledge funding from the Canadian Space Agency Flights and Fieldwork for the Advancement of Science and Technology (FAST) program as well as the Space Technology Development Program (STDP), Ontario Research Fund, the National Sciences and Engineering Research Council, the Canadian Institute for Advanced Research, and the Canada Foundation for Innovation. CJP acknowledges support from the Natural Sciences and Engineering Research Council Canadian Graduate Scholarship–Doctoral Program and the Ontario Government Ontario Graduate Scholarship Program. JJ acknowledges support from the Korean Institute for Science and Technology. SK acknowledges support from the Mike and Ophelia Lazaridis Fellowship Program. The authors thank Jeremy Dillon and the Flight Research Laboratory team at the National Research Council of Canada for their expertise in integrating and flying scientific aircraft payloads, training, and assistance during the flights. We thank the members of the Smiths Falls Flying Club, especially Peter Campbell, for access to their airfield and facilities, Phil Kaye for providing hangar space and assistance, and Ramy Tannous for assistance at the ground station. We thank Ian D’Souza and Jeff Kehoe for assistance with preliminary equipment test flights, and Rolf Horn for allowing us to set up a temporary ground station on his property. We also thank Dotfast-Consulting, Excelitas Technologies, Institut National d’Optique, Neptec Design Group, and Xiphos Systems Corporation for the development and support of the custom components. CJP, EC, and TJ managed the project, and planned and executed logistics. SK, CJP, and TJ conducted feasibility and aircraft flight-path studies, with link analysis conducted by CJP and JPB. CJP, SK, JPB, BLH, and TJ designed and tested system components, with industry partners. SK, EA, VM, and TJ designed and built the receiver detector module. CJP and SK designed and assembled the receiver payload. BLH developed the coarse pointing system, data acquisition, data processing, and polarization compensation system software, supervised by TJ. JPB, NS, and TJ designed and built the quantum source. CJP, SK, JPB, JJ, SA, BLH, and TJ conducted outdoor full-system calibration and tests. CJP and JJ integrated the receiver payload into the aircraft, assisted by BLH. CJP, JPB, BLH and TJ developed and managed flight operations and mission tasking. CJP operated the receiver in flight, assisted by JJ. BLH conducted data acquisition and pointing at the ground station. JPB operated the quantum source, assisted by BLH. NS, SA, and TJ supported ground station operations. CJP analyzed the data, supervised by BLH and TJ. TJ conceived and supervised the project. CJP and BLH wrote the manuscript, with contributions from all authors. References ========== [10]{} C H Bennett and G Brassard. Quantum cryptography: Public key distribution and coin tossing. In [*Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing*]{}, pages 175–179, New York, 1984. IEEE Press. Valerio Scarani, Helle Bechmann-Pasquinucci, Nicolas J Cerf, Miloslav Du šek, Norbert Lütkenhaus, and Momtchil Peev. The security of practical quantum key distribution. , 81:1301–1350, Sep 2009. D Stucki, N Walenta, F Vannel, R T Thew, N Gisin, H Zbinden, S Gray, C R Towery, and S Ten. High rate, long-distance quantum key distribution over 250 km of ultra low loss fibres. , 11(7):075003, 2009. Yang Liu, *et al.* Decoy-state quantum key distribution with polarized photons over 200 km. , 18(8):8587–8594, Apr 2010. Boris Korzh, Charles Ci Wen Lim, Raphael Houlmann, Nicolas Gisin, Ming Jun Li, Daniel Nolan, Bruno Sanguinetti, Rob Thew, and Hugo Zbinden. Provably secure and practical quantum key distribution over 307 km of optical fibre. , 9:163–168, 2015. W T Buttler, R J Hughes, P G Kwiat, S K Lamoreaux, G G Luther, G L Morgan, J E Nordholt, C G Peterson, and C M Simmons. Practical free-space quantum key distribution over 1 km. , 81:3283–3286, Oct 1998. Richard J Hughes, Jane E Nordholt, Derek Derkacs, and Charles G Peterson. Practical free-space quantum key distribution over 10 km in daylight and at night. , 4(1):43, 2002. R Ursin, *et al.* Entanglement-based quantum communication over 144 km. , 3(7):481–486, Jul 2007. Tobias Schmitt-Manderbach, *et al.* Experimental demonstration of free-space decoy-state quantum key distribution over 144 km. , 98:010504, Jan 2007. Giuseppe Vallone, Vincenzo D’Ambrosio, Anna Sponselli, Sergei Slussarenko, Lorenzo Marrucci, Fabio Sciarrino, and Paolo Villoresi. Free-space quantum key distribution by rotation-invariant twisted photons. , 113:060503, Aug 2014. S Nauerth, F Moll, M Rau, C Fuchs, J Horwath, S Frick, and H Weinfurter. Air-to-ground quantum communication. , 7:382–386, 2013. Jian-Yu Wang, *et al.* Direct and full-scale experimental verifications towards ground-satellite quantum key distribution. , 7(5):387–393, May 2013. Jean-Philippe Bourgoin, Brendon L Higgins, Nikolay Gigov, Catherine Holloway, Christopher J Pugh, Sarah Kaiser, Miles Cranmer, and Thomas Jennewein. Free-space quantum key distribution to a moving receiver. , 23(26):33437–33447, Dec 2015. R J Hughes, W T Buttler, P G Kwiat, S K Lamoreuax, G L Morgan, J E Nordholt, and C G Peterson. Quantum cryptography for secure satellite communications. In [*2000 IEEE Aerospace Conference. Proceedings (Cat. No.00TH8484)*]{}, volume 1, pages 191–200, 2000. Z Tang, R Chandrasekara, Y Y Sean, C Cheng, C Wildfeuer, and A Ling. Near-space flight of a correlated photon system. , 4:6366, 2014. J G Rarity, P R Tapster, P M Gorman, and P Knight. Ground to satellite secure key exchange using quantum cryptography. , 4(1):82, 2002. R Ursin, *et al.* Space-quest, experiments with quantum entanglement in space. , 40(3):26–29, 2009. P Villoresi, *et al.* Experimental verification of the feasibility of a quantum channel between space and [E]{}arth. , 10(3):033038, 2008. R Etengu, F M Abbou, H Y Wong, A Abid, N Nortiza, and A Setharaman. Performance comparison of [BB84]{} and [B92]{} satellite-based free space quantum optical communication systems in the presence of channel effects. , 32:37–47, April 2011. Evan Meyer-Scott, Zhizhong Yan, Allison MacDonald, Jean-Philippe Bourgoin, Hannes Hübel, and Thomas Jennewein. How to implement decoy-state quantum key distribution for a satellite uplink with [50-dB]{} channel loss. , 84:062326, Dec 2011. Juan Yin, *et al.* Experimental quasi-single-photon transmission from satellite to earth. , 21(17):20032–20040, Aug 2013. Giuseppe Vallone, Daniele Dequal, M Tomasin, M Schiavon, F Vedovato, Davide Bacco, Simone Gaiarin, Giuseppe Bianco, Vincenza Luceri, and Paolo Villoresi. Satellite quantum communication towards [GEO]{} distances. , 9900:99000J–99000J–8, 2016. T Jennewein, *et al.* : a mission proposal for a quantum receiver in space. , 8997:89970A–89970A–7, 2014. J-P Bourgoin, *et al.* A comprehensive design and performance analysis of low [E]{}arth orbit satellite quantum communication. , 15(2):023006, 2013. E Gibney. Chinese satellite is one giant step for the quantum internet. , 535:478–479, 2016. Quantum optics lifts off. , 10:689, 2016. . Products & services: Satellite platforms. <http://utias-sfl.net/?page_id=89>. Zhizhong Yan, Evan Meyer-Scott, Jean-Philippe Bourgoin, Brendon L Higgins, Nikolay Gigov, Allison MacDonald, Hannes Hübel, and Thomas Jennewein. Novel high-speed polarization source for decoy-state [BB84]{} quantum key distribution over free space and satellite links. , 31(9):1399–1408, May 2013. Hoi-Kwong Lo, Xiongfeng Ma, and Kai Chen. Decoy state quantum key distribution. , 94:230504, Jun 2005. Nitin Jain, Elena Anisimova, Imran Khan, Vadim Makarov, Christoph Marquardt, and Gerd Leuchs. Trojan-horse attacks threaten the security of practical quantum cryptography. , 16(12):123030, 2014. . Quantum key distribution receiver. contract no. 9F063-120711/002/MTB. Christopher J Pugh, *et al.* A fine pointing system suitable for quantum communications on a satellite. In preparation. Shihan Sajeed, Poompong Chaiwongkhot, Jean-Philippe Bourgoin, Thomas Jennewein, Norbert Lütkenhaus, and Vadim Makarov. Security loophole in free-space quantum key distribution due to spatial-mode detector-efficiency mismatch. , 91:062301, Jun 2015. . <http://www.ghgsat.com/>. . [E]{}. Hufnagel. Variations of [A]{}tomspheric [T]{}urbulence. In [*Digest of [T]{}opical [M]{}eeting on [O]{}ptical [P]{}ropagation Through [T]{}urbulence, OSA Technical Digest Series (Optical Society of America, Washinton, D.C.)*]{}, pages [W]{}[A]{}1–1–[W]{}[A]{}1–4, 1974. http://www.ast.cam.ac.uk/sites/default/files/Hufnagel\_Var\_251109.pdf J. W. Hardy. . ([*Oxford series in optical and imaging sciences.*]{}) (Oxford: Oxford University Press), 1998. C Erven, B Heim, E Meyer-Scott, J P Bourgoin, R Laflamme, G Weihs, and T Jennewein. Studying free-space transmission statistics and improving free-space quantum key distribution in the turbulent atmosphere. , 14(12):123018, 2012. Jean-Philippe Bourgoin, Nikolay Gigov, Brendon L Higgins, Zhizhong Yan, Evan Meyer-Scott, Amir K Khandani, Norbert Lütkenhaus, and Thomas Jennewein. Experimental quantum key distribution with simulated ground-to-satellite photon losses and processing limitations. , 92:052339, Nov 2015. Shi-Hai Sun, Lin-Mei Liang, and Cheng-Zu Li. Decoy state quantum key distribution with finite resources. , 373(30):2533 – 2536, 2009. Elena Anisimova, Brendon L Higgins, Jean-Philippe Bourgoin, Miles Cranmer, Eric Choi, Danya Hudson, Louis P. Piche, Alan Scott, Vadim Makarov, and Thomas Jennewein. Mitigation of radiation damage of single photon detectors for space applications. . in press (arXiv:1702.01186). Jin Gyu Lim, Elena Anisimova, Brendon L Higgins, Jean-Philippe Bourgoin, Thomas Jennewein, and Vadim Makarov. Laser annealing heals radiation damage in avalanche photodiodes. . in press (arXiv:1701.08907). C H Bennet, G Brassard, and N D Mermin. Quantum cryptography without [B]{}ell’s theorem. , 68:557–559, 1992. [^1]: Present Address: Department of Physics and Astronomy, Macquarie University, Balaclava Road, North Ryde, NSW, 2109, Australia [^2]: Present Address: Magellan Aerospace, 3701 Carling Avenue, Ottawa, Ontario K2H 8S2, Canada

--- author: - 'Hiromasa Watanabe,' - 'Georg Bergner,' - 'Norbert Bodendorfer,' - 'Shotaro Shiba Funai,' - 'Masanori Hanada,' - 'Enrico Rinaldi,' - 'Andreas Schäfer,' - and Pavlos Vranas bibliography: - 'partial-deconfinement-test.bib' title: Partial Deconfinement at Strong Coupling on the Lattice --- Introduction {#sec:introduction} ============ Review of partial deconfinement {#sec:review} =============================== Partial deconfinement: the Gaussian matrix model {#sec:Gaussian-matrix-model} ================================================ Partial deconfinement: the Yang-Mills matrix model {#sec:Yang-Mills-MM} ================================================== Conclusion and discussion {#sec:conclusion} ========================= M. H. thanks Pavel Buividovich, Raghav Jha, David Schaich, Hidehiko Shimada and Masaki Tezuka for discussions and comments. G. B. acknowledges support from the Deutsche Forschungs-gemeinschaft (DFG) Grant No. BE 5942/2-1. N. B. was supported by an International Junior Research Group grant of the Elite Network of Bavaria. The work of M. H. was partially supported by the STFC Ernest Rutherford Grant ST/R003599/1 and JSPS KAKENHI Grants17K1428. P. V. was supported by DOE LLNL Contract No. [DE-AC52-07NA27344]{}. The numerical simulations were performed on ATHENE, the HPC cluster of the Regensburg University Compute Centre; the HPC cluster ARA of the University of Jena; and ‘pochi’ at the University of Tsukuba. Computing support for this work came also from the Lawrence Livermore National Laboratory (LLNL) Institutional Computing Grand Challenge program.

--- abstract: 'We propose a novel approach to the problem of polarizabilities and dissociation in electric fields from the static limit of the Vignale-Kohn (VK) functional. We consider the response to the purely scalar part of the VK response potential. This potential has ground-state properties that notably improve over the full VK response density and over usual (semi-)local functionals. The correct qualitative behavior of our potentials means that it is expected to work well for polarizabilities in cases such as the H$_2$ chain, and it will also correctly dissociate open-shell fragments in a field.' author: - 'Neepa T. Maitra' - Meta van Faassen title: 'An improved exchange-correlation potential for polarizability and dissociation in DFT' --- Density functional theory (DFT) [@HK64; @KS65; @K99b] has become the most popular electronic structure method in a wide range of problems in quantum chemistry, achieving an unprecedented balance between accuracy and efficiency. But in some applications there are fundamental problems with standard exchange-correlation (xc) approximations such as LDA and GGAs. The response to electric fields is severely overestimated for long-range molecules and for molecular chains [@CPGB98; @GSGB99]. Even for the simple H$_2$ molecule dissociating in an electric field, LDA and GGAs yield unphysical fractional charges on each atom [@GB96; @GSGB99]. This is a problem in many topical applications, including molecular electronics [@TFSB05; @SZVV05] and nonlinear optics devices [@KKP04; @KRM94]. It is now well known that the functionals in these cases must depend in an ultranonlocal way on the density [@GSGB99; @MWY03; @FBLB02]. There have been two approaches: The first is exact exchange (EXX): this has implicit non-local density-dependence through orbital-dependence [@TS76]. The second is Vignale-Kohn theory (VK) [@VK96], where a linear response calculation is performed within time-dependent current-DFT (TDCDFT), a natural extension of time-dependent-DFT [@RG84]. VK uses functionals of the current-density: the (time-integrated) current through a small volume in space contains information about the density response far from that volume, so functionals that are local in the current-density are ultranonlocal in the density. Both EXX and VK have field-counteracting terms that decrease the response compared with LDA/GGA, and so yield improved static polarizabilities for many polymer chains. An exception is the hydrogen chain, for which VK performs almost as poorly as LDA [@FBLB02]. Another problem that needs non-local functionals is the dissociation of simple diatomics in electric fields. As the molecule is pulled apart, a field-counteracting step develops in the exact Kohn-Sham (KS) potential midway between the atoms. In the limit of large separation, the step size approaches a constant that realigns the highest occupied molecular orbitals of the two atoms in the case of two open-shell fragments, and vanishes in the case of two closed-shell fragments. EXX captures the step in the latter, but until now, no density functional approximation has captured the step in the former case. VK has never before been applied to this problem. The calculations in [@FBLB02] utilise the zero-frequency limit of VK, which contains additional “dynamical” xc fields, on top of LDA [@UVb98]. The VK xc vector potential has both longitudinal as well as transverse components. Static response calculations are technically outside the realm of validity of VK [@VK96], but its success in many such cases suggest such dynamical terms are also present in the true static functional and contain essential physics [@FBLB02; @GV05]. The fact that a transverse xc field persists in the static limit of VK means that, in contrast to usual response methods, caution must be used when interpreting the zero-frequency limit as a ground-state perturbation. Satisfaction of the adiabatic theorem requires that the response in the static limit is representable by a scalar KS potential [@GDP96]: For a perturbation turned on slowly enough, the system remains arbitrarily close to the instantaneous ground-state. Functionals in any time-dependent theory should reduce to ground-state DFT ones, where xc effects are contained in a scalar potential. However, a transverse field, with its non-zero curl, represents a non-conservative force, and so cannot correspond to a scalar potential underlying a conserved energy. In this paper we reconsider the VK response to electric fields from a new perspective. We first generate the self-consistent xc vector potential from a full VK calculation in the static limit, but then discard its transverse component. Thereby we eliminate the non-conservative part of the force. We gauge-transform the longitudinal part into a scalar potential which we view as a [*ground-state xc response potential*]{}. This approach is quite distinct from the previous use of VK [@FBLB02], where the key player is the [*density response*]{} of the VK potential, $n_1^{VK}$. This is not the ground-state density response of the scalar potential above, because $n_1^{VK}$ is the full response to both the longitudinal and transverse fields of VK. Here, we consider the true ground-state response to just the scalar part of the VK potential. We show that this ground-state VK potential has desirable features arising from global field-counteracting terms. The dissociation limit of the electron-pair bond is correctly obtained, in contrast to the notorious fractional charges that result from all previous density functional approximations, including LDA, GGA and EXX [@GB96; @GSGB99; @B01]. For cases where the VK response density has not performed well, eg. hydrogen chains, features of this scalar VK potential suggest that our approach will perform very well. Dynamical terms that proved crucial in the usual VK approach to polarizabilities, play an even more fundamental role here. Consider a system initially in its field-free ground-state, of density $n_0(\br)$. In VK response theory, xc-contributions to a perturbative time-dependent field are contained in a vector potential that is a local functional of the induced current-density $\bj(\br t)$ [@VK96], ${\bf a}\xc[{\bf j}](\br t)$: in the frequency domain, i(,) = v\^[ALDA]{}() - (, )/n\_0() \[eq:axc\] where, in the static limit, $\omega\to 0$, \_[ij]{} = -n\_0()\^2f\^[dyn]{}()( + -\_[ij]{}), \[eq:sigma\] with $\bu(\br) = \bj(\br)/n_0(\br)$. Here $f\xc^{\rm dyn}(\br)$ is a dynamical correction to the static scalar xc kernel of the homogeneous electron gas $f\xc^{\rm hom}$ in the long-wavelength limit, evaluated at the field-free density at $\br$, i.e. f\^[dyn]{}() = \_[0]{}f\^[hom]{}(n\_0()) - .\_[n\_0()]{} The [*static*]{} xc response of the homogeneous electron gas is given by the second-derivative of the xc energy density $e\xc^{\rm hom}$. This differs from the zero frequency limit of the scalar kernel by $f\xc^{\rm dyn} = 4\mu\xc^{\rm dyn}/3n^2$, where $\mu\xc^{\rm dyn}$ is the elastic shear modulus at $\omega\to 0$ [@QV02]. Thus VK contains dynamical corrections to ALDA that persist all the way through $\omega\to 0$. The response kernels are defined in VK by taking the wavevector $q$ of the perturbing field to zero before taking the static limit, and these limits do not commute [@GV05; @UVb98; @QV02; @VK96]. We now define a ground-state potential from the static limit of the VK response: Taking $\omega \to 0$, the longitudinal component of Eq. (\[eq:axc\]) is gauge-transformed to a scalar potential, $v\xc^{(1)}[n](\br)$, a non-local functional of the density: $i\omega{\bf a}_{\sss XC,L}[{\bf j}](\br \omega) = \nabla v\xc^{(1)}[n](\br)$. We restrict now to cylindrically symmetric linear systems, and [*approximate*]{} this potential in the following way. We consider Eqs. (\[eq:axc\]) and (\[eq:sigma\]) for a purely one-dimensional inhomogeneity [@UVb98; @SZVV05], so we effectively average over the transverse directions in our cases. We then obtain an approximate potential [@UVb98; @SZVV05] for linear systems: v\^[(1)]{}(z)= f\^[hom]{}(z)n\_1(z)-f\^[dyn]{}(z)\_[-]{}\^[z]{} dz’ n\_1(z’)\ - \_z\^dz’f\^[dyn]{}(z’)n\_1(z’)\ + \_z\^dz’()\^2 f\^[dyn]{}(z’)\_[-]{}\^[z’]{}dz”n\_1(z”) \[eq:vk1d\] Here $n_0'(z) = dn_0(\br)/dz$ (where $\br = (0,0,z)$ is along the bond-axis) and $n_1(\br)=\nabla\cdot\bj/(i\omega)$ is the system’s density response, which is taken in the zero-frequency limit. This potential consists of four terms: the first is the LDA response, local in the density response. The second term is directly proportional to the local current-density response. The third and fourth display global behavior across the molecule, and are the key terms for the purposes of this paper. It follows from the structure of the fourth term that any polarization of the density, be it local or of charge-transfer nature, yields field-counteracting behavior. The third term tends to align along the field, but is generally smaller than the fourth. The first step is to run a zero-frequency VK response calculation on the chosen system in a weak external field, $Ez$, placed along the bond-axis, $z$. We utilize the ADF program package [@ADF] with the TDCDFT extension; see Refs. [@FBLB02] for implementation details. In the second step, the resulting density response is inserted in Eq. (\[eq:vk1d\]) to define our ground-state potential. We study three classes of systems that are challenging for usual semi-local density functionals: (i) dissociation of the electron-pair bond (ii) a dimer composed of two closed-shell fragments at large but finite separation, and (iii) a molecular chain. Atomic units are used throughout. [*(i) Dissociation of the electron-pair bond: H$_2$-like systems*]{} The step that forms in the exact KS potential when a molecule composed of open-shell units dissociates in an electric field prevents dissociation to fractionally charged species, and is consistent with the physical picture of two locally polarized species [@GSGB99]. An analogous step occurs in field-free dissociation of a heteroatomic molecule composed of open-shell units [@PPLB82; @P85b; @PL97c; @AB85; @GB96]. Its origin is static correlation, and it is particularly difficult for approximations to capture, eluding not only LDA/GGAs but also EXX. Ref. [@B01] has a density-matrix solution for this. The lower left panel of Figure \[fig:H2\] shows the field-free density (scaled by 0.01), the exact density response within bound-state perturbation theory, and the VK density response $n_1^{\rm VK}$ for H$_2$ at bond-length 10 au in a field of 0.001 au. The exact response demonstrates local polarization with no charge-transfer, as expected. However, this is not true of the VK (or LDA) response density, which yields fractionally charged atoms. Correspondingly, the VK polarizability is grossly overestimated, as it is in the LDA. The top panel of this same figure shows the exact xc response potential $v\xc^{(1)}$ (within bound-state perturbation theory) and the VK and LDA xc response potentials (i.e. subtracting the field-free potential). The exact was obtained by numerically inverting the KS equation, for the exact KS bonding-orbital composed of polarized atomic orbitals. The salient feature of the exact $v\xc^{(1)}$ is the field-counteracting step which compensates for the difference in potential at the two separated atoms, and thus re-aligns them when the total potential is considered. This step is missed by the LDA which has an along-field component, strengthening the applied field. The step is also missing in EXX (not shown), as the potential there is simply minus half the Hartree. The VK potential does capture the step. In the right panel we see that the fourth term in Eq. (\[eq:vk1d\]) is responsible for this. The VK potential falls a little short of the exact step, but once the Hartree response potential (field-counteracting [@GSGB99]) is added to the VK xc potential, we expect that the total step completely compensates the change in potential created by the external field. The VK potential re-aligns the two atoms, and so the ground-state of this system would certainly not involve any charge transfer across the system, like in the exact case, in contrast to the VK density response (see Fig. 1). Indeed, as the bond-length $R$ increases, one finds that the VK density globally polarizes even more, while the VK step size increases as $ER$, maintaining the atoms at the same level. As the molecule dissociates, stretched H$_2$ has the metallic-like feature that its HOMO-LUMO gap vanishes. This may underlie the reason why VK exactly captures the step, since VK is based on the response of a metallic system, the weakly inhomogeneous electron gas. It is important that the VK potential be evaluated on the “wrong” density response to the full VK fields: if evaluated on the exact response density, the field-counteracting term is reduced by a factor of about 20. [*(ii) Dimer of two closed-shell units*]{} Field-counteracting behavior also arises from the exchange interaction between closed shells, as explained in Refs. [@GSGB99; @GB01]. In contrast to the step of (i), EXX methods can retrieve this step [@GSGB99]. In Figure \[fig:2H2\] we see that the step is also nicely reproduced by the VK potential, shown in the top panel. Also shown there is the exact xc response potential, calculated from highly accurate wavefunction techniques in Ref. [@GSGB99]. Although details of the exact potential are missing in the VK response potential, the step is clearly captured, with the correct magnitude of drop in the potential between the up- and down-field molecules. In contrast, the LDA or any GGA has no net drop. The right panel shows that the net step in the VK potential is a result of competition between the third and fourth terms. Note that in this case there is largely local polarization in the molecules, with small charge transfer (hardly discernable in the lower left panel). The VK density response and polarizability are again very close to the LDA ones, both overestimating the exact polarizability. However, the ground-state response using the VK potential shown, is expected to yield polarizabilities much closer to the exact one, because of the field-counteracting behavior. The magnitude of the exact step between two closed-shell units decreases as the intermolecular separation increases [@GSGB99]. This is [*not*]{} the case for the VK step asymptotically. Essentially, the VK step arises from integrals over the current-density (fourth term in Eq. (\[eq:vk1d\])); this integral persists at large separation, due to the local polarization. [*(iii) Long-chain molecules*]{} Polarizabilities and hyperpolarizabilities of long-chain polymers are notoriously overestimated in LDA/GGA calculations. Recent work has shed much light on the role of ultranonlocal density dependence in these systems. It has been demonstrated that orbital functionals such as EXX within OEP can significantly improve the polarizabilities [@KKP04; @GSGB99; @MWY03], due to field-counteracting behavior in the exchange potential. Approximate exchange methods such as KLI [@KLI92] also give corrections over LDA, although not as strongly as full exchange [@KKP04]. For example, for the hydrogen chain in Figure \[fig:H2chain\], the LDA, KLI, and OEP polarizabilities are 114.6, 90.6, and 84.2 a.u, respectively. The VK functional was also applied [@FBLB02] and gave 110.20 a.u, hardly an improvement over LDA; in contrast to its dramatic improvement for many other molecular chains. Again, the scalar VK [*potential*]{} has field-counteracting behavior, as shown in Figure \[fig:H2chain\]. This potential will give improved results (comparable to KLI) when used in a computation of ground-state response and polarizabilities. In Ref. [@KKP04], the importance of intermolecular barriers in the [*field-free*]{} ground-state potential was stressed: these are lacking in LDA, underestimated in KLI, but captured well in OEP. Although formal arguments point to LDA as the correct field-free potential to be used in conjunction with the VK response potential, it will be interesting to compare polarizabilities using the VK response potential on top of a field-free OEP potential. In summary, we have shown that the scalar part of the VK response potential contains crucial field-counteracting terms making it a good candidate for response properties of effectively one-dimensional systems in electric fields. It is the first approximation that captures the step in a molecule composed of open-shell fragments dissociating in an electric field. It captures field-counteracting terms in systems of two or more closed-shell units, and is promising for long-chains, being numerically less intensive than EXX methods which also have been successful for these problems. The scalar VK potential works well even when the full VK response does not! In all cases studied, $n_1^{VK}$ is very close to the LDA response, suggesting that the effects of the transverse VK and dynamical-longitudinal fields somewhat cancel. Satisfaction of the adiabatic theorem means that the transverse component of the response field should vanish in the static limit, suggesting that VK should be corrected by dropping its transverse part. This supports our results, where the ground-state response to the longitudinal component was considered. An alternative choice is to discard the transverse field from the start, and consider the self-consistent response to the purely longitudinal field. The results are not as good: the field-counteracting nature of Eq. (\[eq:vk1d\]) is most effective when evaluated on the full (wrong) density-response $n_1^{VK}$ rather than that obtained self-consistently. The non-self-consistent aspect of our present approach is somewhat dissatisfying from a rigorous viewpoint; on the other hand, there are many other situations in DFT where such post-self-consistent approaches have success eg. Ref. [@PKZB99]. Finally, we note that the VK response potential smears over some local details of the true response potential (see e.g. Fig. \[fig:2H2\]). Work is underway to investigate if this has a significant effect on the global polarizability. We thank S. Kümmel for his OEP and KLI data on the H$_2$ chain, and K. Burke and G. Vignale for fruitful discussions. This work is financially supported by the American Chemical Society’s Petroleum Research Fund and National Science Foundation Career CHE-0547913. [99]{} P. Hohenberg and W. Kohn, Phys. Rev. [**136**]{}, B 864 (1964). W. Kohn and L.J. Sham, Phys. Rev. [**140**]{}, A 1133 (1965). W. Kohn, Rev. Mod. Phys. [**71**]{}, 1253 (1999). B. Champagne [*et. al.*]{}, J. Chem. Phys. [**109**]{}, 10489 (1998). S.J.A. van Gisbergen [*et. al.*]{}, Phys. Rev. Lett. [**83**]{}, 694 (1999); Phys. Rev. A [**62**]{} 012507 (2000). O.V. Gritsenko and E.J. Baerends, Phys.Rev. A [**54**]{}, 1957 (1996). C. Toher et al., Phys. Rev. Lett. [**95**]{}, 146402 (2005). N. Sai et al., Phys. Rev. Lett.[**94**]{}, 186810 (2005). An effectively one-dimensional approach was also adopted here. S. Kümmel, L. Kronik, and J. P. Perdew, Phys. Rev. Lett. [**93**]{}, 213002 (2004). D.R. Kanis, M.A. Ratner, and T.J. Marks, Chem. Rev. [**94**]{}, 195 (1994). M. van Faassen et al., Phys. Rev. Lett. [**88**]{}, 186401 (2002); J. Chem. Phys. [**118**]{}, 1044 (2003). P. Mori-Sanchez, Q. Wu, and W. Yang, J. Chem. Phys. [**119**]{}, 11001 (2003). J.D. Talman and W.G. Shadwick, Phys. Rev. A [**14**]{}, 36 (1976). G. Vignale and W. Kohn, Phys. Rev. Lett. [**77**]{}, 2037 (1996); G. Vignale, C.A. Ullrich, and S. Conti, Phys. Rev. Lett. [**79**]{}, 4878 (1997). E. Runge and E.K.U. Gross, Phys. Rev. Lett. [**52**]{}, 997 (1984); C. A. Ullrich and G. Vignale, Phys. Rev. B, [**58**]{}, 15756 (1998); ibid [**65**]{}, 245102 (2002). G. F. Giuliani and G. Vignale, [*Quantum Theory of the Electron Liquid*]{}, Cambridge, 2005. E.J. Baerends, Phys.Rev. Lett. [**87**]{}, 133004 (2001); M.A. Buijse and E.J. Baerends, Mol. Phys. [**100**]{}, 401 (2002). Z. Qian and G. Vignale, Phys. Rev. B. [**65**]{}, 235121 (2002). E.K.U. Gross, J.F. Dobson, and M. Petersilka, in [*Topics in Current Chemistry*]{}, [**181**]{}, 81 (1996). ADF 2003.01, available from http://www.scm.com J.P. Perdew et al, Phys. Rev. Lett. [**49**]{}, 1691 (1982). J.P. Perdew and M. Levy, Phys. Rev. B [**56**]{}, 16021 (1997). J.P. Perdew, in [*Density Functional Methods in Physics*]{}, ed. R.M. Dreizler and J. da Providencia (Plenum, NY, 1985), p. 265. C.O. Almbladh and U.von Barth, Phys. Rev. B. [**31**]{}, 3231, (1985). O.V. Gritsenko and E.J. Baerends, Phys. Rev. A [**64**]{}, 042506 (2001). J.B. Krieger, Y. Li, and G. J. Iafrate, Phys. Rev. A [**46**]{}, 5453 (1992). J. P. Perdew et al., Phys. Rev. Lett.[**82**]{} 2544 (1999).

--- abstract: 'We consider Fisher-KPP equation with advection: $u_t=u_{xx}-\beta u_x+f(u)$ for $x\in (g(t),h(t))$, where $g(t)$ and $h(t)$ are two free boundaries satisfying Stefan conditions. This equation is used to describe the population dynamics in advective environments. We study the influence of the advection coefficient $-\beta$ on the long time behavior of the solutions. We find two parameters $c_0$ and $\beta^*$ with $\beta^*>c_0>0$ which play key roles in the dynamics, here $c_0$ is the minimal speed of the traveling waves of Fisher-KPP equation. More precisely, by studying a family of the initial data $\{ \sigma \phi \}_{\sigma >0}$ (where $\phi$ is some compactly supported positive function), we show that, (1) in case $\beta\in (0,c_0)$, there exists $\sigma^*\geqslant0$ such that spreading happens when $\sigma>\sigma^*$ (i.e., $u(t,\cdot;\sigma\phi)\to 1$ locally uniformly in ${\mathbb{R}}$) and vanishing happens when $\sigma \in (0,\sigma^*]$ (i.e., $[g(t),h(t)]$ remains bounded and $u(t,\cdot;\sigma\phi)\to 0$ uniformly in $[g(t),h(t)]$); (2) in case $\beta\in (c_0,\beta^*)$, there exists $\sigma^*>0$ such that virtual spreading happens when $\sigma>\sigma^*$ (i.e., $u(t,\cdot;\sigma \phi)\to 0$ locally uniformly in $[g(t),\infty)$ and $u(t,\cdot + ct;\sigma \phi )\to 1$ locally uniformly in ${\mathbb{R}}$ for some $c>\beta -c_0$), vanishing happens when $\sigma\in (0,\sigma^*)$, and in the transition case $\sigma=\sigma^*$, $u(t, \cdot+o(t);\sigma \phi)\to V^*(\cdot-(\beta-c_0)t )$ uniformly, the latter is a traveling wave with a “big head“ near the free boundary $x=(\beta-c_0)t$ and with an infinite long tail” on the left; (3) in case $\beta = c_0$, there exists $\sigma^*>0$ such that virtual spreading happens when $\sigma > \sigma^*$ and $u(t,\cdot;\sigma \phi)\to 0$ uniformly in $[g(t),h(t)]$ when $\sigma \in (0,\sigma^*]$; (4) in case $\beta\geqslant \beta^*$, vanishing happens for any solution.' author: - 'Hong Gu$^\dag$, Bendong Lou$^\dag$ and Maolin Zhou$^\ddag$' title: 'Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries$^\S$' --- [^1] [^2] [^3] [^4] Introduction ============ In this paper, we consider the following problem $$\label{p} \left\{ \begin{array}{ll} u_t =u_{xx}- \beta u_{x}+f(u), & g(t)< x<h(t),\; t>0,\\ u(t,g(t))=0,\ \ g'(t)=-\mu u_x(t, g(t)), & t>0,\\ u(t,h(t))=0,\ \ h'(t)=-\mu u_x (t, h(t)) , & t>0,\\ -g(0)=h(0)= h_0,\ \ u(0,x) =u_0 (x),& -h_0\leqslant x \leqslant h_0, \end{array} \right. \tag{$P$}$$ where $\mu$ and $\beta$ are positive constants, $h_0>0$ and $u_0$ is a nonnegative $C^2$ function with support in $[-h_0,h_0]$, $f:[0,\infty)\to {\mathbb{R}}$ is a $C^1$ function satisfying $$\label{f} \left\{ \begin{array}{ll} f(0)=f(1)=0,\quad (1-u)f(u)>0 \mbox{ for } u>0 \mbox{ and } u\not= 1,\\ f'(0)>0,\ f'(1)<0 \mbox{ and } f(u)\leqslant f'(0)u \mbox{ for } u\geqslant 0. \end{array} \right. \tag{$F$}$$ The problem is used to model the spreading of a new or invasive species, under the influence of diffusion and advection. The unknown $u(t,x)$ denotes the population density over a one dimensional habitat and the free boundaries $x=g(t)$ and $x=h(t)$ represent the expanding fronts of the species. We assume that the free boundaries move according to one-phase Stefan condition, which is a kind of free boundary conditions widely used in the study of melting of ice [@R], wound healing [@CF], and population dynamics [@BDK; @DuLin; @DuLou]. The derivation of one-phase or two-phase Stefan conditions in population models as singular limits of competition-diffusion systems can be found in [@HIMN; @HMS] etc. When $\beta=0$ (i.e., there is no advection in the environment), the qualitative properties of the problem was studied by Du and Lin [@DuLin] for logistic nonlinearity $f(u)=u(1-u)$. Among others, they proved that, when $h_0\geqslant \frac{\pi}{2}$, any solution of with $\beta=0$ grows up and converges to $1$ (which is called [*spreading*]{} phenomena); when $h_0<\frac{\pi}{2}$, spreading happens if $\mu$ is large and [*vanishing*]{} happens if $\mu$ is small (i.e., the solution converges to $0$). The vanishing phenomena is a remarkable result since it shows that the presence of free boundaries may avoid the so-called [*hair-trigger effect*]{}, which is a phenomena shown in [@AW]: spreading always happens for a solution of the Cauchy problem for $u_t =u_{xx}+f(u)$, no matter how small the positive initial data is. Recently, Du and Lou [@DuLou] extended the results in [@DuLin] to the problem with general monostable, bistable and combustion types of $f$, and gave a rather complete description on the long time behavior of the solutions. In addition, Kaneko and Yamada [@KY], Liu and Lou [@LL1; @LL2] studied the problem with $\beta=0$ and with a fixed boundary $g(t)\equiv 0$, Du and Guo [@DuGuo; @DuGuo2], Du, Matano and Wang [@DMW], Zhou and Xiao [@ZX], Wang [@Wang] studied the problem (without advection) in higher dimension spaces and/or in spatial heterogeneous environments. Besides the qualitative properties, another interesting problem is the asymptotic spreading speeds of the free boundaries when spreading happens. Du and Lin [@DuLin], Du and Lou [@DuLou] proved that, when spreading happens for a solution $(u,g,h)$ of the problem with $\beta=0$, $$\label{speed-000} c^*:= \lim_{t\to\infty}\frac{h(t)}{t} = \lim_{t\to\infty}\frac{-g(t)}{t} >0.$$ Recently, Du, Matsuzawa and Zhou [@DMZ] improved this result to better ones: $$\label{speed-001} \lim_{t\to \infty} h'(t) = \lim_{t\to \infty} [-g'(t)] =c^*,\quad \lim_{t\to \infty} [h(t)- c^* t ] =H_\infty,\quad \lim_{t\to \infty} [g(t)+ c^* t ] =G_\infty,$$ for some $H_\infty,\ G_\infty \in {\mathbb{R}}$. In this paper we consider the problem with $\beta>0$, which means that the spreading of a species is affected by advection. In the field of ecology, organisms can often sense and respond to local environmental cues by moving towards favorable habitats, and these movement usually depend upon a combination of local biotic and abiotic factors such as stream, climate, food and predators. For example, some diseases spread along the wind direction. In 2009, Maidana and Yang [@Maiyang] studied the propagation of West Nile Virus from New York City to California state. It was observed that West Nile Virus appeared for the first time in New York City in the summer of 1999. In the second year the wave front travels 187km to the north and 1100km to the south. Therefore, they took account of the advection movement and showed that bird advection becomes an important factor for lower mosquito biting rates. Another example is that Averill [@Ave] considered the effect of intermediate advection on the dynamics of two-species competition system, and provided a concrete range of advection strength for the coexistence of two competing species. Moreover, three different kinds of transitions from small advection to large advection were illustrated theoretically and numerically. Many other examples involving advection can also be found in the field of ecology (cf. [@BS; @BC; @BP; @PL; @RSB; @SO; @SG; @VL] etc.). From a mathematical point of view, to involve the influence of advection, one of the simplest but probably still realistic approaches is to assume that species can move up along the gradient of the density, as considered in [@BH; @HL; @HAES; @RSB; @SO; @SG; @VL] etc. Gu, Lin and Lou [@GLL1; @GLL2] studied the problem with small advection. They proved a spreading-vanishing dichotomy result on the long time behavior of positive solutions of , which is similar as the conclusions in [@DuLin; @DuLou] for equations without advection. They also proved that, when spreading happens for a solution of with small advection, its rightward spreading speed is bigger than the leftward one: $$\label{speed-002} \lim_{t\to\infty}\frac{h(t)}{t} > \lim_{t\to\infty}\frac{-g(t)}{t} >0.$$ Recently, Kaneko and Matsuzawa [@KM] improved this result to some conclusions like . Our main purpose in this paper is to study the influence of the advection term $-\beta u_x$ on the long time behavior of solutions of . As we will see below, our study improves the results in [@GLL1; @GLL2; @KM] since we will study the problem for all $\beta>0$, not only for small $\beta$. Especially, when $\beta$ is large, the phenomena is much more complicated and more interesting than the case where $\beta$ is small. We point that the problem for the equations with bistable type of nonlinearity, or the problems (with monostable or bistable type of nonlinearity) in the interval $[0,h(t)]$ with $x=h(t)$ a free boundary and $x=0$ a fixed boundary where $u$ satisfies a general Robin boundary condition can be considered similarly. In fact, in our forthcoming papers [@G; @GLiu; @GL] we study these problems and obtain similar results as in this paper. To sketch the influence of $\beta$, we introduce two important traveling waves. First, consider the following problem $$\label{c0} \left\{ \begin{array}{ll} q''(z) - c q' (z) +f(q)=0,\quad z\in {\mathbb{R}},\\ q(-\infty)=0,\ q(+\infty)=1,\ q(0)=1/2,\quad q' (z)>0 \mbox{ for } z \in {\mathbb{R}}. \end{array} \right.$$ It is well known that this problem has a solution $q(z;c)$ if and only if $c\geqslant c_0 $, where $$c_0:= 2\sqrt{f'(0)}$$ is called the minimal speed of the traveling waves of Fisher-KPP equation. Denote $Q(z) := q(z;c_0)$, then $u(t,x) = Q (x - (\beta -c_0)t)$ is a traveling wave of $u_t = u_{xx} - \beta u_x +f(u)$. It travels leftward (resp. rightward) if and only if $\beta <c_0$ (resp. $\beta >c_0$). Next we consider the following problem $$\label{c*} \left\{ \begin{array}{ll} q''(z) + (c-\beta ) q'(z) + f(q)=0,\quad z \in (-\infty,0),\\ q(0)=0,\ q(-\infty)=1,\ -\mu q'(0)=c,\quad q' (z)<0\mbox{ for } z\in (-\infty, 0]. \end{array} \right.$$ As is shown in Lemma \[lem:semi-wave\] (see also [@DuLin; @DuLou; @GLL2]), for any $\beta>0$, this problem has a unique solution $(c,q)=(c^*,U^*(z))$. So $u(t,x) = U^* (x-c^* t)$ is a solution of $u_t = u_{xx} -\beta u_x +f(u)$, with $u(t,c^* t)=0,\ c^* = - \mu u_x(t, c^* t)$. It is called a [*traveling semi-wave*]{} in [@DuLou] since it is only defined for $x\leqslant c^* t$. We also write $c^*$ as $c^*(\beta)$ to emphasize the dependence of $c^*$ on $\beta$, then we will show in Lemma \[lem:semi-wave\] that the equation $\beta - c_0 = c^*(\beta)$ has a unique root $\beta^* >c_0$: $$\label{def:beta*} \beta^* - c_0 = c^*(\beta^*).$$ We will see below that the traveling wave $Q(x -(\beta -c_0) t)$ and the traveling semi-wave $U^* (x-c^* t)$ are of special importance in the study of spreading solutions. To explain their roles intuitively, we consider the problem with initial data $u_0 (x)$ which is even and $$u_0(x) = \left\{ \begin{array}{ll} 1, & x\in [0, h_0-1],\\ \mbox{smooth and decreasing}, & x\in [h_0 -1, h_0], \end{array} \right. \qquad \mbox{ with } h_0 \gg 1.$$ It is easily seen by the maximum principle that $u(t,\cdot)$ has exactly one maximum point. As usual, we call the sharp decreasing part in the graph of $u(t,\cdot)$ the [*front*]{}, and call the sharp increasing part on the left side the [*back*]{}. Now we sketch the influence of the advection $-\beta u_x$. [*Case 1*]{}. When $\beta \in (0, c_0)$, the advection influence is not strong, the solution has enough space between the back and the front to grow up and to converge to 1. Its front approaches a profile like $U^*(\cdot)$ and moves rightward at a speed $\approx c^*$. Its back approaches a profile like $U^*(-\; \cdot)$ and moves leftward at a speed smaller than $ c_0 -\beta$ (see details in Theorem \[thm:profile of spreading sol\] below). This case is similar as the spreading phenomena in [@DuLin; @DuLou] for the equation with $\beta=0$. [*Case 2*]{}. When $\beta \in (c_0, \beta^*)$ with $\beta^*$ being the unique root of , the traveling wave $Q(x - (\beta-c_0)t)$ travels rightward at a speed $\beta -c_0>0$. Hence the back of the solution $u$, with a shape like $Q(x-(\beta-c_0)t)$, is pushed by $Q(x - (\beta-c_0)t)$ to move rightward at a speed $\approx \beta -c_0$, and so $u\to 0$ locally uniformly. But, when the initial domain is wide enough, the solution still have enough space to grow up between the back and the front since the front moves rightward (at a speed $\approx c^*$) faster than the back. In this paper we call such a phenomena as [*virtual spreading*]{} (see Theorem \[thm:middle beta\] and Lemma \[lem:condition for virtual spreading\] below). [*Case 3*]{}. When $\beta = c_0$, the traveling wave $Q(x - (\beta-c_0)t) = Q(x)$ is indeed a stationary solution of $_1$. However, the back of the solution $u$ still moves rightward at a speed $O(t^{-1})$ since it starts from a compactly supported initial data $u_0$ (cf. [@HNRR] and see details below). Hence virtual spreading still happens when $h_0$ is sufficiently large, since the front moves rightward at speed $c^*$, faster than the back. [*Case 4*]{}. When $\beta > \beta^*$, the back moves rightward (at a speed $\approx \beta-c_0$) faster than the front (which moves rightward at speed $\approx c^*< \beta -c_0$). So the solution is suppressed by its back, and then $u\to 0$ uniformly. In summary, the long time behavior of the solutions is quite different for $\beta\in (0,c_0),\; \beta =c_0,\; \beta\in (c_0, \beta^*)$ and $\beta> \beta^*$. This paper is organized as the following. In section 2 we present our main results. In section 3 we give some preliminaries including the comparison principles, stationary solutions, several types of traveling waves, zero number arguments and some upper bound estimates. In section 4 we study the influence of the advection on the long time behavior of the solutions. In section 5, we revisit the virtual spreading phenomena and give a uniform convergence for such solutions. Main Results ============ Throughout this paper we choose initial data $u_0$ from the following set: $$\label{def:X} \mathscr {X}(h_0):= \Big\{ \phi \in C^2 ([-h_0,h_0]) \mid \phi(-h_0)= \phi (h_0)=0,\; \phi(x) \geqslant ,\not\equiv 0 \ \mbox{in } (-h_0,h_0).\Big\}$$ where $h_0 >0$ is any given real number. By a similar argument as in [@DuLin; @DuLou], one can show that, for any initial data $u_0\in\mathscr{X}(h_0)$, the problem has a time-global solution $(u,g,h)$, with $u\in C^{1+\nu/2,2+\nu}((0,\infty)\times[g(t),h(t)])$ and $g,h\in C^{1+\nu/2}((0,\infty))$ for any $\nu\in(0,1)$. Moreover, it follows from the maximum principle that, when $t>0$, the solution $u$ is positive in $(g(t),h(t))$, $u_x(t,g(t))>0$ and $u_x(t,h(t))<0$. Hence $g'(t)<0$, $h'(t)>0$. Denote $$g_{\infty}:=\lim_{t\to\infty}g(t),\quad h_{\infty}:= \lim_{t\to\infty}h(t),\quad I(t):= [g(t),h(t)] \quad \mbox{and} \quad I_{\infty}:=(g_{\infty},h_{\infty}).$$ In what follows, we mainly consider the solution of with initial data $u_0=\sigma\phi$ for some given $\phi\in\mathscr {X}(h_0)$ and $\sigma\geqslant0$. We also use $(u(t,x;\sigma\phi),g(t;\sigma\phi),h(t;\sigma\phi))$ to denote such a solution. Now we list some possible situations for the solutions of . - [*spreading*]{} : $I_\infty ={\mathbb{R}}$ and $$\label{eq spreading} \lim_{t\to\infty}u(t,\cdot)=1 \mbox{ locally uniformly in ${\mathbb{R}}$};$$ - [*vanishing*]{} : $I_\infty$ is a bounded interval and $$\label{eq vanishing} \lim_{t\to\infty}\max_{g(t)\leqslant x\leqslant h(t)} u(t,x)=0;$$ - [*virtual spreading*]{} : $g_\infty>-\infty,\ h_\infty=+\infty$, $$\label{eq v s} \lim_{t\to\infty}u(t,\cdot)=0 \mbox{ locally uniformly in } I_\infty$$ and $$\label{eq v s 2} \lim_{t\to\infty}u(t,\cdot+ct)=1 \mbox{ locally uniformly in } {\mathbb{R}}, \quad \mbox{ for some } c >0;$$ - [*virtual vanishing*]{} : $g_\infty>-\infty,\ h_\infty=+\infty$ and holds. When the advection is small, we have the following conclusion on the long time behavior of the solutions. \[thm:small beta\] Assume $0<\beta<c_0$ and $(u,g,h)$ is a time-global solution of with initial data $u_0 = \sigma \phi$ for some $\phi\in \mathscr {X}(h_0)$. Then there exists $\sigma^* = \sigma^* (h_0, \phi) \in [0,\infty]$ such that [(i)]{} [vanishing]{} happens when $\sigma \in [0,\sigma^*]$, with $|I_\infty| = h_\infty -g_\infty \leqslant\frac{2\pi}{\sqrt{c_0 ^2-\beta^2}}$; [(ii)]{} [spreading]{} happens when $\sigma> \sigma^*$. From this theorem we see that the long time behavior of the solutions of with small advection: $\beta \in (0,c_0)$ is similar as the case without advection: $\beta =0$ (cf. [@DuLin; @DuLou; @GLL1]). The main reason is that in both cases the problem has exactly two stationary solutions: $0$ and $1$ in ${\mathbb{R}}$. The proof of this theorem, which is given in subsection 4.2, is also similar as that for $\beta=0$. Next we consider the case where the advection is not small: $\beta \geqslant c_0$. The most interesting phenomena appears in the problem with medium-sized advection: $\beta\in [c_0, \beta^*)$, where $\beta^*$ is the unique root of . \[thm:middle beta\] Assume $c_0<\beta<\beta^*$ and $(u,g,h)$ is a time-global solution of with initial data $u_0 =\sigma \phi$ for some $\phi\in \mathscr{X}(h_0)$. Then there exists $\sigma^* = \sigma^* (h_0, \phi) \in (0,\infty]$ such that [(i)]{} [virtual spreading]{} happens when $\sigma >\sigma^*$, and $$\lim_{t\to\infty}u(t,\cdot+ct)=1 \mbox{ locally uniformly in ${\mathbb{R}}$}, \quad \mbox{ for any } c\in (\beta -c_0, c^*),$$ where $c^* =c^*(\beta)$ is the speed of the traveling semi-wave in ; [(ii)]{} [vanishing]{} happens when $0<\sigma <\sigma^*$; [(iii)]{} in the [transition]{} case $\sigma =\sigma^*$: $g_\infty>-\infty,\ h_\infty=+\infty$, $$\lim_{t\to\infty}h'(t)=\beta-c_0 \quad \mbox{and} \quad h(t)= (\beta -c_0)t+\varrho (t)$$ with $\varrho(t) = o(t)$ and $\varrho(t)\to\infty\ (t\to \infty)$. In addition, $$\label{u to V*} \lim\limits_{t\to\infty} \| u(t, \cdot ) - V^*(\cdot -(\beta-c_0)t - \varrho(t)) \|_{L^\infty (I(t))} =0,$$ where $V^*(z)$ is the unique solution of $$\label{vv} \left\{ \begin{array}{ll} q''(z)-c_0 q'(z)+f(q)=0 \quad \mbox{for } z\in (-\infty,0),\\ q(0)=0,\ q(-\infty)=0,\ q(z)>0 \mbox{ for } z\in (-\infty,0),\ -\mu q'(0)=\beta-c_0. \end{array} \right.$$ In the next section we will see that $V^*$ has a [*tadpole-like*]{} shape: it has a big head“ and a boundary on the right side and an infinite long tail” on the left side. So we call $V^*(x-(\beta -c_0)t)$ a [*tadpole-like traveling wave*]{} with speed $\beta-c_0$, which exists if and only if $\beta\in(c_0,\beta^*)$ (see Lemma \[lem:tadpole tw bata&lt;beta\*\] below). Theorem \[thm:middle beta\] (iii) implies that, roughly, $u(t,x)$ converges to this traveling wave. In Aronson and Weinberger [@AW], it was shown that any positive solution of the Cauchy problem for Fisher-KPP equation converges to 1 (i.e., [*hair-trigger effect*]{}). In [@DuLin; @DuLou], by introducing the free boundaries, the authors proved a spreading-vanishing dichotomy on the long time behavior of the solutions of Fisher-KPP equation. In particular, vanishing may happen for some solutions. Now our Theorem \[thm:middle beta\] gives the third possibility besides the virtual spreading and vanishing, that is, with a medium-sized advection in the equation, there may exist a transition state: the solution converges to a [*tadpole-like traveling wave*]{}. This interesting phenomena is new comparing with the results for Cauchy problems and for free boundary problems without advection. \[thm:beta=c0\] Assume $\beta=c_0$ and $(u,g,h)$ is a time-global solution of with initial data $u_0 =\sigma \phi$ for some $\phi\in \mathscr{X}(h_0)$. Then there exists $\sigma_*,\; \sigma^* \in (0,\infty]$ with $\sigma_* \leqslant \sigma^*$ such that [(i)]{} [virtual spreading]{} happens when $\sigma>\sigma^*$, and $$\lim_{t\to\infty}u(t, \cdot + c t)=1 \mbox{ locally uniformly in ${\mathbb{R}}$}, \quad \mbox{ for any } c\in (0,c^*),$$ where $c^*=c^*(\beta)$ is the speed of the traveling semi-wave in ; [(ii)]{} [vanishing]{} happens when $0<\sigma<\sigma_*$; [(iii)]{} [virtual vanishing]{} happens when $\sigma\in [\sigma_*, \sigma^*]$. The transition cases in Theorem \[thm:middle beta\] and Theorem \[thm:beta=c0\] are different. In case $c_0<\beta<\beta^*$, a solution $u(t,x;\sigma\phi)$ is a transition one only if the initial value is taken the sharp threshold value $\sigma^*\phi$. However, in case $\beta=c_0$, we obtain transition solutions whose initial data are taken from $\{\sigma\phi\mid \sigma\in [\sigma_*, \sigma^*]\}$. Whether or not this domain is a singleton: $\sigma_*=\sigma^*$ is still open now. The difficulty in studying this problem is that virtual vanishing solutions have no shapes", so it is not easy to compare one to another. The conclusions for the problem with large advection: $\beta \geqslant \beta^*$ is rather simple. \[thm:large beta\] Assume $\beta\geqslant \beta^*$ and $(u,g,h)$ is a time-global solution of with initial data $u_0\in\mathscr {X}(h_0)$. Then vanishing happens. Besides the convergence/dichotomy/trichotomy results on the long time behavior of the solutions as stated in the previous theorems, we can say more about the solutions when (virtual) spreading happens. It turns out that, when $\beta\in [c_0, \beta^*)$, the virtual spreading solution can be characterized by the rightward traveling semi-wave $U^*(x-c^* t)$ and the traveling wave $Q(x -(\beta-c_0)t)$; when $\beta\in (0,c_0)$, the spreading solution can be characterized by $U^*(x-c^* t)$ and the leftward traveling semi-wave $U^*_l (x - c^*_l t)$. Here $(c^*_l, U^*_l)$ (with $c_l^*<0$) is the unique solution of the following problem with $\beta\in(0,c_0)$ (see details in subsection 3.3) $$\label{q*l} \left\{ \begin{array}{ll} q''(z) + (c-\beta) q'(z) + f(q)=0,\quad z \in (0,\infty),\\ q(0)=0,\ q(\infty)=1,\ -\mu q'(0)=c,\ q' (z)>0\mbox{ for }z\in (0, \infty). \end{array} \right.$$ Using these traveling waves we can give the asymptotic profiles for (virtual) spreading solutions. \[thm:profile of spreading sol\] Assume spreading or virtual spreading happens for a solution of as in Theorems \[thm:small beta\], \[thm:middle beta\] or \[thm:beta=c0\]. Let $(c^*,U^*)$ be the unique solution of with $c^*>0$. - When $\beta\in(0,c_0)$, let $(c_l^*,U_l^*)$ be the unique solution of with $0< -c_l^* <c^*$. Then there exist $H_\infty$, $G_\infty\in{\mathbb{R}}$ such that $$\label{right spreading speed} \lim\limits_{t\to\infty}[h(t)-c^*t] = H_\infty ,\quad \lim\limits_{t\to\infty}h'(t)=c^*,$$ $$\label{left spreading speed} \lim\limits_{t\to\infty}[g(t)-c_l^*t] = G_\infty ,\quad \lim\limits_{t\to\infty}g'(t)=c_l^*,$$ and, if we extend $U^*$, $U_l^*$ to be zero outside their supports we have $$\label{profile convergence 1} \lim\limits_{t\to\infty} \left\| u(t,\cdot) - U^*(\cdot -c^*t - H_\infty) \cdot U^*_l \big(\cdot - c^*_l t - G_\infty \big)\right\|_{L^\infty (I(t))} =0.$$ - When $\beta\in [c_0,\beta^*)$, holds for some $H_\infty\in{\mathbb{R}}$. Moreover, if we extend $U^*$ to be zero outside its support, then $$\label{profile convergence 2} \lim\limits_{t\to\infty} \left\| u(t,\cdot)- U^*(\cdot -c^* t- H_\infty) \cdot Q \big(\cdot - (\beta-c_0)t-\theta(t) \big)\right\|_{L^\infty (I(t))}=0$$ for some function $\theta(t)$ satisfying $\theta(t)=o(t)$ and $\theta(t)\to \infty\ (t\to \infty)$. Assume $\beta \in [0,c_0)$ and spreading happens for a solution $(u,g,h)$ of . The asymptotic spreading speed $\lim_{t\to \infty} \frac{h(t)}{t}=c^*$ was obtained in [@DuLin; @DuLou] for the case $\beta =0$, and in [@GLL2] for the case $\beta\in (0,c_0)$. Recently, Du, Matsuzawa and Zhou [@DMZ], Kaneko and Matsuzawa [@KM] improved them to analogues of , and . Note that our theorem includes both the case $\beta\in (0,c_0)$ and the case $\beta\in [c_0, \beta^*)$. The proof of will be given in the last section, based on the fact that $Q$ is steeper than any other entire solution (see section 5 below and [@DGM]). Preliminaries ============= In this section we first give some comparison principles and then present all the bounded stationary solutions and traveling wave solutions of $_1$ which will be used for comparison. In the fourth subsection we give some results on the zero numbers of the solutions of linear equations which will play key roles in our approach. In the last subsection we give some precise upper bound estimates for the solutions. The comparison principle ------------------------ In this subsection we give two types of comparison principles which will be used frequently in this paper. Similar as [@DuLin; @DuLou], we have \[lem:comp1\] Assume $T\in(0,\infty)$, $\overline g(t),\ \overline h(t)\in C^1([0,T])$, $\overline u(t,x)\in C(\overline D_T)\cap C^{1,2}(D_T)$ with $D_T=\{(t,x)\in{\mathbb{R}}^2\mid0<t\leqslant T, \overline g(t)<x<\overline h(t)\}$, and $$\begin{aligned} \left\{ \begin{array}{lll} \overline u_{t} \geqslant \overline u_{xx} -\beta \overline u_x+f(\overline u),\; & 0<t \leqslant T,\ \overline g(t)<x<\overline h(t), \\ \overline u= 0,\quad \overline g'(t)\leqslant -\mu \overline u_x,\quad & 0<t \leqslant T, \ x=\overline g(t),\\ \overline u= 0,\quad \overline h'(t)\geqslant -\mu \overline u_x,\quad &0<t \leqslant T, \ x=\overline h(t). \end{array} \right.\end{aligned}$$ If $$\mbox{$[-h_0, h_0]\subset [\overline g(0), \overline h(0)]$ \quad and \quad $u_0(x)\leqslant \overline u(0,x)$ for $x\in[-h_0,h_0]$,}$$ and $(u,g, h)$ is a solution of , then $$\mbox{ $g(t)\geqslant \overline g(t),\; h(t)\leqslant\overline h(t)$\quad for $t\in(0, T]$,}$$ $$\mbox{$u(t,x)\leqslant \overline u(t,x)$ \quad for $t\in (0, T]$ and $x\in (g(t), h(t))$.}$$ \[lem:comp2\] Assume $T\in(0,\infty)$, $l(t),\, k(t)\in C^1([0,T])$, $w(t,x)\in C(\overline D_T)\cap C^{1,2}(D_T)$ with $D_T=\{(t,x)\in{\mathbb{R}}^2\mid0<t\leqslant T, l(t)<x<k(t)\}$, and $$\begin{aligned} \left\{ \begin{array}{lll} w_{t}\geqslant w_{xx}-\beta w_x+ f(w),\; &0<t \leqslant T,\ l(t)<x<k(t), \\ w\geqslant u, &0<t \leqslant T, \ x= l(t),\\ w= 0,\quad k'(t)\geqslant -\mu w_x,\quad &0<t \leqslant T, \ x=k(t), \end{array} \right.\end{aligned}$$ with $$\label{com prin 2} \mbox{$g(t)\leqslant l(t)\leqslant h(t) $ for $t\in[0,T]$,\quad $h_0\leqslant k(0),$\quad $u_0(x)\leqslant w(0,x)$ for $x\in[l(0),h_0]$,}$$ where $(u,g, h)$ is a solution of . Then $$\mbox{ $h(t)\leqslant k(t)$ for $t\in(0, T]$,\quad $u(t,x)\leqslant w(t,x)$ for $t\in (0, T]$\ and $ l(t)<x< h(t)$.}$$ The proof of Lemma \[lem:comp1\] is identical to that of Lemma 5.7 in [@DuLin], and a minor modification of this proof yields Lemma \[lem:comp2\]. The function $\overline u$, or the triple $(\overline u,\overline g,\overline h)$ in Lemmas \[lem:comp1\] and the function $w$, or the triple $(w,l,k)$ in Lemma \[lem:comp2\] are often called the upper solutions of . There is a symmetric version of Lemma \[lem:comp2\], where the conditions on the left and right boundaries are interchanged. The lower solutions can be defined analogously by reversing all the inequalities except for $g(t)\leqslant l(t)\leqslant h(t)$ in . We also have corresponding comparison results for lower solutions in each case. Phase plane analysis and stationary solutions {#subsec:phase plane} --------------------------------------------- We first use the phase plane analysis to study the following equation $$\label{phase sol} q''(z) + \gamma q'(z)+f(q)=0,\quad q(z) \geqslant 0 \quad \mbox{ for } z\in J,$$ where $J$ is some interval in ${\mathbb{R}}$. Note that a nonnegative stationary solution $u$ of $_1$ solves with $\gamma = - \beta$, a nonnegative traveling wave $u(t,x)= q(x-ct)$ of $_1$ solves with $\gamma = c-\beta$ and $J={\mathbb{R}}$. The equation is equivalent to the system $$\label{q-p} \left\{ \begin{array}{l} q'(z)=p,\\ p'(z)= - \gamma p- f(q). \end{array} \right.$$ A solution $(q(z), p(z))$ of this system traces out a trajectory in the $q$-$p$ phase plane (cf. [@AW; @DuLou; @Pet]). Such a trajectory has slope $$\label{Pq} \frac{\mathrm{d}p}{\mathrm{d}q}= - \gamma -\frac{f(q)}{p}$$ at any point where $p\neq0$. It is easily seen that $(0,0)$ and $(1,0)$ are two singular points on the phase plane. We are only interested in the case $\gamma < c_0 := 2\sqrt{f'(0)}$. For such a $\gamma$, the eigenvalues of the corresponding linearizations at the singular points are $$\lambda_{0}^{\pm}=\frac{-\gamma \pm\sqrt{\gamma^2 -4f'(0)}}{2}\ \ (\mbox{at } (0,0))\quad \mbox{and} \quad \lambda_{1}^{\pm}=\frac{-\gamma \pm\sqrt{\gamma^2 -4 f'(1)}}{2}\ \ (\mbox{at } (1,0)),$$ respectively. Since $f'(0)>0$ and $f'(1)<0$, $(1,0)$ is a saddle point, $(0,0)$ is a center when $\gamma=0$, or a focus when $0<|\gamma|<c_0$, or a node when $\gamma \leqslant - c_0$. By the phase plane analysis (cf. [@AW; @DuLou; @Pet]), it is not difficult to give all kinds of bounded, nonnegative solutions of for $\gamma <c_0$ (see Figure 1). **Constant solutions:** $q\equiv 0$ and $q\equiv 1$. **Strictly decreasing solutions on the half-line in case $\gamma<c_0$:** $q(\cdot)=U(\cdot-z_0;\gamma)$ for any $z_0\in{\mathbb{R}}$, where $U\in C^2((-\infty,0])$ is the unique solution of in $(-\infty,0)$, with $U(0;\gamma)=0$, $U(-\infty;\gamma)=1$ and $U'(\cdot;\gamma)<0$ in $(-\infty,0]$ [(]{}see $\Gamma_1$ and $\Gamma_5$ in Figure 1[)]{}. Denote $$P(\gamma) := -\mu U'(0;\gamma).$$ Using the comparison principle for the ordinary differential equation we have $P'(\gamma)<0$ for $\gamma\in (-\infty,c_0)$, $P(c_0 -0) =0$ and $P(-\infty)= +\infty$ (see Figure 2 (a)). **Strictly increasing solutions on the half-line in case $\gamma\in(-c_0,c_0)$:** $q(\cdot)=U_l (\cdot-z_0;\gamma)$ for any $z_0\in{\mathbb{R}}$, where $U_l \in C^2([0,\infty))$ is the unique solution of in $(0,\infty)$, with $U_l (0;\gamma)=0$, $U_l (\infty;\gamma)=1$ and $U'_l (\cdot;\gamma)>0$ in $[0,+\infty)$ [(]{}see $\Gamma_4$ in Figure 1 (a)[)]{}. **Solutions with compact supports in case $\gamma\in(-c_0,c_0)$:** $q(\cdot)= W(\cdot - z_0;b,\gamma)$ for any $z_0\in{\mathbb{R}}$, where for each $b \in (0,P(\gamma))$, there exists a unique $L(b,\gamma) >0$ such that $W \in C^2([-L(b,\gamma), 0])$ is the unique solution of in $(-L(b,\gamma),0)$ with $W(-L(b,\gamma);b,\gamma)=W(0;b,\gamma)=0$ and $b = -\mu W'(0;b,\gamma)$ [(]{}see $\Gamma_2$ and $\Gamma_3$ in Figure 1 (a)[)]{}. Each point $(\gamma, b)$ in the set $S_1:= \{(\gamma,b) \mid 0<b<P(\gamma),\; -c_0 <\gamma <c_0\}$ in Figure 2 (a) corresponds to such a compactly supported solution $W(z;b,\gamma)$.       **Tadpole-like solutions in case $\gamma \leqslant -c_0$:** $q(\cdot)=V(\cdot-z_0;b,\gamma)$ for any $z_0\in{\mathbb{R}}$, where for each $b \in (0,P(\gamma))$, $V\in C^2((-\infty,0])$ is the unique solution of in $(-\infty,0)$ with $V (0;b,\gamma)=0$, $V(-\infty;b,\gamma)=0$ and $b =-\mu V'(0;b,\gamma)$ [(]{}see $\Gamma_6$ and $\Gamma_7$ in Figure 1 (b)[)]{}. Each point $(\gamma, b)$ in the set $S_2:= \{(\gamma,b) \mid 0<b<P(\gamma),\; \gamma \leqslant -c_0\}$ in Figure 2 (a) corresponds to such a tadpole-like solution $V(z;b,\gamma)$. We call $V$ a [*tadpole-like solution*]{} since its graph has a big head“ and a boundary on the right side, and an infinite long tail” on the left side. Similarly, when we construct a traveling wave with the form $V(x-ct;b,\gamma)$, we call it a [*tadpole-like traveling wave*]{}. **Strictly increasing solutions in ${\mathbb{R}}$ in case $\gamma\leqslant -c_0$:** $q(\cdot)=Q(\cdot-z_0;\gamma)$ for any $z_0\in{\mathbb{R}}$, where $Q\in C^2({\mathbb{R}})$ is the unique solution of in ${\mathbb{R}}$ with $Q (-\infty;\gamma)=0$, $Q (\infty ;\gamma)=1$, $Q(0;\gamma)=1/2$ and $Q'(z;\gamma) >0$ in ${\mathbb{R}}$ [(]{}see $\Gamma_8$ in Figure 1 (b)[)]{}. Each nonnegative stationary solution of $_1$ is a solution of the problem with $\gamma =-\beta$. Using the above results we see that, when $\beta\in (0,c_0)$, a bounded, nonnegative stationary solution of $_1$ is either $0$, or $1$, or a strictly decreasing solutions $U(\cdot-z_0;-\beta)\ (z_0\in {\mathbb{R}})$ defined on $(-\infty, z_0]$, or a compactly supported solution $W(\cdot-z_0;b,-\beta)\ (z_0\in {\mathbb{R}})$ for some $b\in (0,P(-\beta))=(0,-\mu U'(0;-\beta))$, or a strictly increasing solution $U_l (\cdot -z_0; -\beta)\ (z_0\in {\mathbb{R}})$ defined on $[z_0, \infty)$. When $\beta\geqslant c_0$, a bounded, nonnegative stationary solution of $_1$ is either $0$, or $1$, or a strictly decreasing solutions $U(\cdot-z_0;-\beta)\ (z_0\in {\mathbb{R}})$ defined on $(-\infty, z_0]$, or a tadpole-like function $V(\cdot-z_0;b,-\beta)\ (z_0\in {\mathbb{R}})$ for some $b\in (0,P(-\beta))=(0,-\mu U'(0;-\beta))$, or a strictly increasing solution $Q(\cdot-z_0; -\beta)\ (z_0\in {\mathbb{R}})$ defined in ${\mathbb{R}}$. Traveling waves --------------- If $u(t,x)=q(x-ct)$ is a traveling wave of $u_t=u_{xx} -\beta u_x +f(u)$, then $(c,q)$ solves with $\gamma =c- \beta$, that is, $$\label{eq tw} q''(z) + (c-\beta)q'(z) +f(q)=0.$$ In this paper we will use several types of traveling waves which are specified now. [**(I) Traveling wave**]{} $Q(x-ct;c-\beta)$ for any $c\leqslant \beta-c_0$, where $q(z)=Q(z;c-\beta)$ satisfies and $$\label{tw R} q(-\infty )=0,\ q(\infty)=1,\ q(0)=\frac{1}{2},\ q'(z)>0 \mbox{ for } z\in {\mathbb{R}}.$$ The existence of such solutions has been given in the previous subsection. $U^* (x-c^*t)$ with $c^* \in(0,c_0+\beta)$, where $q(z)= U^*(z) := U(z;c^*-\beta)$ satisfies with $c=c^*$ and $$\label{traveling semi-wave} q(0)=0,\ q(-\infty)=1,\ -\mu q'(0) = c^*,\ q'(z)<0 \mbox{ for } z\in (-\infty, 0],$$ that is, $(c^*, U^* (z) )$ is a solution of (cf. point A in Figure 2 (b) and in Figure 3). $U^*(x-c^*t)$ is called a traveling semi-wave as in [@DuLou] since $U^*(z)$ is defined only on the half-line $(-\infty,0]$. \[lem:semi-wave\] Assume $\beta>0$. Then - there exists a unique $c^*=c^*(\beta) \in (0,c_0+\beta)$ such that the problem and with $c=c^*$ has a solution, which is unique and denoted by $U^*(z)$; - $0<\frac{\rm d}{{\rm d}\beta} c^*(\beta)<1$ for $\beta>0$; - there exists a unique $\beta^* > c_0$ such that $$\label{def beta *} c^*(\beta) -\beta +c_0>0 \ \ {\rm (} \mbox{resp. } =0,\ <0 {\rm )}\ \mbox{ when } \ \beta <\beta^* \ \ {\rm (} \mbox{resp. } \beta =\beta^*,\ \beta >\beta^* {\rm )}.$$ \(i) For any $c< c_0 +\beta$, the problem with $\gamma= c-\beta <c_0$ has a unique strictly decreasing solution $q(\cdot)=U(\cdot;c-\beta)$ in $(-\infty,0]$, satisfying $U(0;c-\beta )=0$, $U(-\infty;c-\beta )=1$ and $U'(\cdot;c-\beta)<0$ in $(-\infty,0]$. Denote $P(c-\beta) := -\mu U'(0;c-\beta)$ as above, then $P(c-\beta)$ is strictly decreasing in $c\in (-\infty ,c_0+\beta)$, $$(P(c-\beta) - c) \big|_{c=0} = P(-\beta)>0\quad \mbox{and} \quad (P(c-\beta) - c) \big|_{c=c_0+\beta-0} = -c_0 -\beta <0$$ (see Figure 2 (b)). Hence the equation $P(c-\beta)=c$ has a unique root $c=c^*(\beta)\in (0,c_0+\beta)$, that is, $$\label{eq P c} c^*(\beta) = P( c^*(\beta) -\beta)= -\mu U'(0; c^*(\beta) -\beta).$$ \(ii) Differentiating $P( c^*(\beta) -\beta) = c^*(\beta)$ in $\beta$ and using the fact $P'(\gamma)<0$ for $\gamma<c_0$ we have $$\frac{{\rm d} c^*(\beta)}{{\rm d}\beta} = \frac{-P'( c^*(\beta) -\beta)} {1 - P'(c^*(\beta) -\beta ) } \in (0,1).$$ \(iii) Set $\beta^*:= P(-c_0)+c_0 >c_0$. Then $c=\beta^* -c_0$ is a root of $P(c-\beta^*) =c$ in $(0,c_0+\beta^*)$. By the definition of $c^*(\beta)$ and by its uniqueness we have $\beta^*-c_0 =c^*(\beta^*)$. Moreover, the inequalities in (ii) shows that the function $c^*(\beta)-\beta +c_0$ is strictly decreasing in $\beta>0$ and so it has a unique zero $\beta^*$. This proves . [**(III) Leftward traveling semi-wave**]{} $U^*_l (x - c^*_lt)$ in case $\beta\in (0,c_0)$, where $c_l^*=c^*_l (\beta) \in (\beta -c_0, 0)$, $q(z)= U^*_l (z) := U_l (z;c^*_l-\beta)$ satisfies with $c=c^*_l$ and $$\label{traveling semi-wave-left} q(0)=0,\ q(\infty)=1,\ -\mu q'(0) = c^*_l,\ q'(z)>0 \mbox{ for } z\in [0, \infty).$$ For any given $\beta\in (0,c_0)$, the existence and uniqueness of such a solution can be proved as in Lemma \[lem:semi-wave\] (i). [**(IV) Tadpole-like traveling wave**]{} $V(x-ct; b, c-\beta)$ in case $\beta > c_0$. For any $c\in (0,\beta-c_0]$ and any $b\in (0,P(c-\beta))$, $V(x-ct;b,c-\beta)$ is a tadpole-like traveling wave if the function $q(z):=V(z;b,c-\beta)$ satisfies and $$\label{tadpole tw0} q(0)= q(-\infty)=0,\ q(z)>0 \mbox{ for } z\in (-\infty, 0) \ \mbox{and} \ -\mu q'(0) = b$$ (cf. points $B,\ C,\ E$ in Figure 3 (a)). In particular, when $b=c$, the function $V(z;c,c-\beta)$ is a solution of and $$\label{tadpole tw} q(0)= q(-\infty)=0,\ q(z)>0 \mbox{ for } z\in (-\infty, 0) \ \mbox{and} \ -\mu q'(0) = c$$ (cf. points $B,\ C$ in Figure 3 (a)). On the existence of such solutions we have the following results. \[lem:tadpole tw bata&lt;beta\*\] Let $\beta^*$ be the constant given in Lemma \[lem:semi-wave\]. Assume $c_0<\beta<\beta^*$. Then - for any $b\in (0,P(-c_0))$, and with $c=\beta-c_0$ has a unique tadpole-like solution $V(z;b,-c_0)$ (cf. points $B,\ E$ in Figure 3 (a)). Moreover, there exists $z_b <0$ such that $$\label{V to Q} V(\cdot + z_b; b,-c_0)\to Q(\cdot) \mbox{ locally uniformly in } {\mathbb{R}},\quad \mbox{as } b\to P(-c_0);$$ - $q(z) =V^*(z):= V(z; \beta -c_0,-c_0)$ is the unique tadpole-like solution of and with $c=\beta-c_0$, that is, the unique solution of ; - for any $\delta\in (0,\beta-c_0)$, $q(z)=V_{\delta} (z):= V(z; \beta-c_0-\delta, -c_0-\delta)$ is a tadpole-like solution of and with $c=\beta-c_0-\delta$. Moreover, $V_\delta(z)\to V^*(z)$ locally uniformly in $(-\infty, 0]$ as $\delta \to 0$ (cf. point $C$ in Figure 3 (a)). \(i) Since $c_0<\beta<\beta^*$, we have $0<\beta-c_0 < c^*$ by Lemma \[lem:semi-wave\]. On the $c$-$b$ plane (see Figure 3 (a)), any point $E(\beta -c_0,b)$ with $b\in (0,P(-c_0))$ corresponds to a tadpole-like solution $V(z;b,-c_0)$ of and with $c=\beta -c_0$ (cf. trajectories $\Gamma_6$ or $\Gamma_7$ in Figure 1 (b). Note that such a solution does not necessarily satisfy Stefan condition since $b$ may be not equal to $c$). As $b\to P(\beta -c_0)$ (i.e., point $E$ moves up to $F$ in Figure 3 (a)), the trajectory of $V(z; b,-c_0)$ approaches the union of the trajectories of $Q$ and $U^*$ (i.e., $\Gamma_6 \to \Gamma_5\cup \Gamma_8$). Denote $z_b:=\min \{z<0 \mid V(z;b,-c_0)=\frac12\}$. Then the trajectory of $V(\cdot+z_b; b,-c_0)$ approaches that of $Q(\cdot)$ (since $Q(0)=\frac12$), and so we obtain by continuity. \(ii) On the $c$-$b$ plane, the line $\{b=c\}$ passes through the domain $S_2(\beta):= \{(c,b)\mid 0<b<P(c-\beta),0<c<\beta-c_0\}$ and leaves it at a point $B(\beta-c_0, \beta-c_0)$. This point corresponds to the desired tadpole-like solution $V^*(z):=V(z; \beta-c_0 , -c_0)$. \(iii) For any small $\delta >0$, we consider the point $C(\beta-c_0-\delta, \beta-c_0-\delta)$ on the $c$-$b$ plane. Since $C\in S_2(\beta)\cap \{b=c\}$, it corresponds to a tadpole-like solution $V_{\delta} (z):= V(z; \beta-c_0-\delta, -c_0-\delta)$ of and with $c=\beta-c_0-\delta$. In particular, $V_\delta(z)$ satisfies the following initial value problem $$\left\{ \begin{array}{l} q''(z) -(c_0 +\delta) q'(z) +f(q)=0,\ \ z<0,\\ q(0)=0,\ \ -\mu q'(0)=\beta - c_0 -\delta. \end{array} \right.$$ Since $V^*$ satisfies this problem with $\delta =0$ and since $V_\delta$ depends on $\delta$ continuously, we have $V_\delta(\cdot)\to V^*(\cdot)$ as $\delta\to 0$, uniformly in $[-M, 0]$ for any $M>0$. This proves the lemma. In a similar way, one can prove the following lemma (cf. Figure 3 (b)). \[lem:tadpole tw beta=beta\*\] Assume $\beta=\beta^*$. Then for any small $\delta>0$, $q(z)=V^*_{\delta} (z):= V(z; \beta^*-c_0-\delta, -c_0-\delta)$ is the unique tadpole-like solution of and with $\beta =\beta^*$, $c=\beta^*-c_0-\delta$. Moreover, $V^*_\delta(\cdot)\to U^*(\cdot)$ locally uniformly in $(-\infty, 0]$ as $\delta \to 0$, where $U^*(z)$ is the unique solution of and with $\beta=\beta^*$, $c^*=c^*(\beta^*) = \beta^* -c_0$. [**(V) Compactly supported traveling wave**]{} $W(x-ct;c,c-\beta)$ in case $\beta\in [c_0,\beta^*)$, where for any $c\in(\beta-c_0,c^*(\beta))$, $q(z)=W(z;c, c-\beta)$ satisfies and $$\label{compact tw} q(0)= q(-L(c,c-\beta))=0,\quad q(z)>0 \mbox{ in } (-L(c,c-\beta), 0) \quad \mbox{and} \quad -\mu q'(0) = c.$$ \[lem:compact tw\] Assume $c_0 \leqslant \beta < \beta^*$. For any $\delta\in(0,c^*(\beta)-\beta+c_0)$, $q(z)=W_{\delta}(z):= W(z;\beta -c_0 +\delta, -c_0 +\delta )$ is the unique solution of the problem and with $c=\beta -c_0 +\delta$. Moreover, $L_\delta\to\infty$ and $D_\delta\to1$ as $\delta\to c^*(\beta)-\beta+c_0$, where $L_\delta:=L(\beta-c_0+\delta,-c_0+\delta)$ denotes the width of the support of $W_\delta(\cdot)$, and $D_\delta$ denotes its height. We only prove the case $\beta\in(c_0,\beta^*)$, the proof for the case $\beta=c_0$ is similar. When $c_0<\beta<\beta^*$, we have $c^*(\beta)-\beta+c_0>0$ by Lemma \[lem:semi-wave\]. On the $c$-$b$ plane (see Figure 3 (a)), the line $b=c$ leaves the domain $S_2(\beta)$ and enters $S_1(\beta):=\{(c,b)\mid 0<b<P(c-\beta),\ \beta-c_0<c<\beta+c_0\}$ at a point $B(\beta-c_0,\beta-c_0)$, then it passes through $S_1(\beta)$ and leaves it finally at $A(c^*(\beta),c^*(\beta))$. For any $\delta\in(0,c^*(\beta)-\beta+c_0)$, the point $(c,b)=(\beta-c_0+\delta,\beta-c_0+\delta)$ is on the line segment $AB$ (cf. point $D$ in Figure 3 (a)), it corresponds to a trajectory like $\Gamma_2$ on the $q$-$p$ phase plane, and so it defines a compactly supported function $W_\delta(z):=W(z;\beta-c_0+\delta,-c_0+\delta)$. As $\delta\to c^*(\beta)-\beta+c_0$, the point $(\beta-c_0+\delta,\beta-c_0+\delta)$ approaches point $A$ in Figure 3 (a), this implies that its corresponding trajectory approaches the union of the trajectories of $U^*(\cdot)$ and $U_l(\cdot;c^*(\beta)-\beta)$ (i.e., $\Gamma_2\to\Gamma_1\cup\Gamma_4$ in Figure 1 (a)). Therefore, the corresponding function $W_\delta$ satisfies $\max\limits_{-L_\delta\leqslant z\leqslant0} W_\delta(z)=D_\delta\to1$, and the width $L_\delta$ of its support tends to $\infty$ as $\delta\to c^*(\beta)-\beta+c_0$. Zero number arguments --------------------- In what follows, we use $\mathcal{Z}_I[w(\cdot)]$ to denote the number of zeros of a continuous function $w(\cdot)$ defined in $I\subset{\mathbb{R}}$. The following lemma is an easy consequence of the proofs of Theorems C and D in Angenent [@A]. \[angenent\] Let $u:[0,T]\times [0, 1]\to {\mathbb{R}}$ be a bounded classical solution of $$\label{linear} u_t=a(t,x)u_{xx}+b(t,x)u_x+c(t,x)u$$ with boundary conditions $$u(t,0)=l_0(t), \;u(t, 1)=l_1(t),$$ where $l_0, l_1\in C^1([0,T])$, and each function is either identically zero or never zero for $t\in [0,T]$. In the special case where $l_0(t),\ l_1(t)\equiv 0$ we assume further that $u(t,\cdot) \not\equiv 0$ for each $t\in [0,T]$. Suppose also that $$a, 1/a, a_t, a_x, a_{xx}, b, b_t, b_x, c \in L^\infty, \mbox{ and } u(0,\cdot)\not\equiv 0 \mbox{ when $l_0=l_1\equiv 0$}.$$ Then for each $t\in (0, T]$, $\mathcal{Z}_{[0,1]} [u(t,\cdot)]<\infty$. Moreover, $\mathcal{Z}_{[0,1]} [u(t,\cdot)]$ is nonincreasing in $t$ for $t\in (0, T]$, and if for some $t_0\in(0, T]$ the function $u(t_0,\cdot)$ has a degenerate zero $x_0\in [0,1]$, then $\mathcal{Z}_{[0,1]} [u(t_1, \cdot)] >\mathcal{Z}_{[0,1]} [u(t_2, \cdot)] $ for all $t_1, t_2\in (0,T]$ satisfying $t_1<t_0<t_2$. For convenience of applications in this paper we give a variant of Lemma \[angenent\]. \[zero-number\] Let $\xi_1(t)<\xi_2(t)$ be two continuous functions for $t\in (t_0, t_1)$. If $u(t,x)$ is a continuous function for $t\in (t_0, t_1)$ and $x\in J(t):= [\xi_1(t),\xi_2(t)]$, and satisfies in the classical sense for such $(t,x)$, with $$u(t,\xi_1(t))\not=0,\; u(t, \xi_2(t))\not=0 \mbox{ for } t\in (t_0, t_1),$$ then for each $t\in (t_0, t_1)$, $\mathcal{Z}_{J(t)} [u(t,\cdot)] <\infty$. Moreover $\mathcal{Z}_{J(t)}[u(t,\cdot)]$ is nonincreasing in $t$ for $t\in (t_0, t_1)$, and if for some $s\in (t_0, t_1)$ the function $u(s,\cdot)$ has a degenerate zero $x_0\in J(s)$, then $\mathcal{Z}_{J(s_1)} [u(s_1,\cdot)]> \mathcal{Z}_{J(s_2)} [u(s_2,\cdot)]$ for all $s_1, s_2$ satisfying $t_0<s_1<s<s_2<t_1$. For any given $t^*\in (t_0, t_1)$, we can find $\epsilon>0$ and $\delta>0$ small such that $u(t,x)\not=0$ for $t\in T_{t^*}:=[t^*-\delta, t^*+\delta]\subset (t_0, t_1)$ and $x\in [\xi_1(t),\xi_1(t^*)+\epsilon]\cup [ \xi_2(t^*)-\epsilon, \xi_2(t)]$. Hence we may apply Lemma \[angenent\] with $[0,T]\times [0,1]$ replaced by $T_{t^*}\times [\xi_1(t^*)+\epsilon,\xi_2(t^*)-\epsilon]$ to see that the conclusions for $\mathcal{Z}_{J(t)} [u(t,\cdot)]$ hold for $t\in T_{t^*}$. Since any compact subinterval of $(t_0, t_1)$ can be covered by finitely many such $T_{t^*}$, we see that $\mathcal{Z}_{J(t)}[u(t,\cdot)]$ has the required properties over any compact subinterval of $(t_0, t_1)$. It follows that $\mathcal{Z}_{J(t)}[u(t,\cdot)]$ has the required properties for $t\in (t_0, t_1)$. In our approach we will compare the solution $u$ of with traveling semi-wave $U^*$ or tadpole-like traveling wave $V^*$ or compactly supported traveling wave $W_\delta$ by studying the number of their intersection points. Now we give some preliminary results. We use $\Psi(x-ct-C)$ (for some $c>0$ and some $C\in{\mathbb{R}}$) to represent one of the traveling waves $U^*(x-c^*t-C)$, $V^*(x-(\beta-c_0)t-C)$ and $W_\delta(x-(\beta-c_0+\delta)t-C)$. Denote the support of $\Psi(x-ct-C)$ by $[k_1(t),k_2(t)]$, where $k_2(t)=ct+C$, $k_1(t)=-\infty$ in case $\Psi=U^*$ or $\Psi=V^*$, and $k_1(t)=k_2(t)-L_\delta$ in case $\Psi=W_\delta$. Denote $$r(t):=\min\{h(t),k_2(t)\},\quad R(t):=\max\{h(t),k_2(t)\},\quad l(t):=\max\{g(t),k_1(t)\}$$ and $$\eta(t,x):=u(t,x)-\Psi(x-ct-C),\quad x\in J(t):=[l(t),r(t)],\ t\in(t_1,t_2)$$ (here we only consider the case where $J(t)\neq\emptyset $ for each $t\in(t_1,t_2)$, otherwise, $u$ and $\Psi$ has no common domain and so there is no need to compare them). We notice that $\eta$ satisfies $$\eta_t=\eta_{xx}-\beta\eta_x+c(t,x)\eta\quad \mbox{for }x\in(l(t),r(t)),\ t\in(t_1,t_2)$$ with $c(t,x):=[f(u(t,x))-f(\Psi(x-ct-C))]/\eta(t,x)$ when $\eta(t,x)\neq0$, and $c(t,x)=0$ otherwise. Using Lemmas \[angenent\] and \[zero-number\] one can obtain the following result on the number of zeros of $\eta(t,\cdot)$. \[lem:zeros between u and Psi\] For any given $C\in{\mathbb{R}}$, let $r(t),\ R(t),\ l(t)$ and $\eta$ be defined as above. Then - $\mathcal{Z}_{J(t)}[\eta(t,\cdot)]$ is finite and nonincreasing in $t\in(t_1,t_2)$; - if $t_0\in(t_1,t_2)$ such that $r(t_0)=R(t_0)$, or $\eta(t_0,\cdot)$ has a degenerate zero in the interior of $J(t_0)$, then $\mathcal{Z}_{J(\tau_1)}[\eta(\tau_1,\cdot)]>\mathcal{Z}_{J(\tau_2)}[\eta(\tau_2,\cdot)]$ for any $t_1<\tau_1<t_0<\tau_2<t_2$. [*Sketch of the proof*]{}. The proof is essentially identical to that of Lemma 2.3 in [@DLZ]. We give a sketch here for the readers’ convenience. Note that when $h(t)=k_2(t) = r(t)=R(t)$, $x=r(t)$ becomes a degenerate zero of $\eta$ on the boundary since both $u$ and $\Psi$ satisfy Stefan condition on their right boundaries. We claim that $h(t)\equiv k_2(t)$ in an interval $(t_3,t_4)\subset(t_1,t_2)$ is impossible. For, otherwise, we can consider $\zeta=\eta e^{-\frac{\beta}{2} x}$ instead of $\eta$, which satisfies an equation without advection and so can be extended outside $r(t)$ as an odd function with respect to $x=r(t)$. For this extended function $x=r(t)$ is an interior degenerate zero in time interval $(t_3,t_4)$, this contradicts Lemma \[angenent\]. On the other hand, $g(t)=k_1(t)$ at most once, and only possible in case $\Psi=W_\delta$. Therefore, the set of the times when $r(t)=R(t)$ or $g(t)=k_1(t)$ is a nowhere dense set, and for other times we have $\mathcal{Z}_{J(t)}[\eta(t,\cdot)]<\infty$ by Lemma \[angenent\]. Assume $r(t_0)=R(t_0)$ and $r(t)<R(t)$ for $t\in[t_0-\epsilon,t_0)$. Assume further that $\eta(t,\cdot)$ has nondegenerate zeros $\{z_i(t)\}_{i=1}^m$ with $$l(t)<z_1(t)<z_2(t)<\cdots<z_m(t)<r(t),\quad t\in[t_0-\epsilon,t_0).$$ Then by [@F Theorem 2] one can prove that $\lim\limits_{t\to t_0}z_i(t)$ exist (denoted by $\bar{z}_i$) and $\{\bar{z}_i\}_{i=1}^m$ are the only zeros of $\eta(t_0,\cdot)$. Moreover, by the maximum principle we have $\bar{z}_m=r(t_0)$, that is, the largest zero $z_m(t)$ tends to the right boundary. Then using the maximum principle again in the domain $\{(t,x)\mid t_0<t<t_0+\epsilon_1,r(t)-\epsilon_1<z<r(t)\}$ for some small $\epsilon_1$, we can show that the boundary zero $r(t_0)$ disappear immediately after time $t_0$. In summary, $$\mathcal{Z}_{J(\tau_1)}[\eta(\tau_1,\cdot)]=m\geqslant\mathcal{Z}_{J(t_0)}[\eta(t_0,\cdot)]> \mathcal{Z}_{J(\tau_2)}[\eta(\tau_2,\cdot)]$$ for $t_0-\epsilon< \tau_1 <t_0 < \tau_2 <t_0+\epsilon_1$. [$\Box$]{} As can be expected, the presence of the advection makes the maximum points prefer to move rightward. Indeed we can show that the local maximum points concentrate near the right boundary under certain conditions. Using zero number properties Lemma \[zero-number\] to $u_x$, we see that $u_x(t,\cdot)$ has only nondegenerate zeros for all large $t$. Hence, $u(t,\cdot)$ has fixed number of (nondegenerate) local maximum points for large $t$. \[lem:max at right\] Assume, for some $T\geqslant 0$, $u(t,\cdot)$ has exactly $N$ ($N$ is a positive integer) local maximum points $\{\xi_i(t)\}_{i=1}^{N}$ for all $t\geqslant T$, with $$g(t)<\xi_1(t)<\xi_2(t)<\cdots<\xi_N(t)<h(t).$$ If $N\geqslant 2$, then $$\label{xi1(t)>=eta(t)} \xi_1(t)\geqslant \beta\cdot(t-T)+C\quad \mbox{for }T\leqslant t<T_\infty,$$ for some $C\in{\mathbb{R}}$, where $$\label{monotonicity outside} T_\infty= \left\{ \begin{array}{ll} \inf \mathcal{T},\ \ & \mbox{ if } \mathcal{T}:=\{t\geqslant T\mid\beta\cdot(t-T)+C=h(t)\}\neq\emptyset,\\ \infty, \ \ & \mbox{ if } \mathcal{T}=\emptyset. \end{array} \right.$$ Choose $C=\frac{1}{2}[g(T)+\xi_1(T)]$ and define $\rho(t):=\beta(t-T)+C$, then $\rho(T)=C<\xi_1(T)$. Hence $T_1:=\inf\{s\geqslant T\mid\rho(s)=\xi_1(s)\}>T$. If $T_1=T_\infty$, then holds. Now we assume $T<T_1<T_\infty$. By the definitions of $\xi_1(t)$ and $T_1$ we have $$\label{monotonicity outside for [T0,T1)} u_x(t,x)>0\quad \mbox{for }x\in[g(t),\rho(t)],\ T\leqslant t<T_1,$$ and $$\label{u_x(T1)=0} u_x(T_1,\rho(T_1))=u_x(T_1,\xi_1(T_1))=0.$$ Define $$\zeta(t,x):=u(t,x)-u(t,2\rho(t)-x)\quad \mbox{for }x\in[l(t),\rho(t)],\ t\geqslant T,$$ with $l(t):=\max\{g(t),2\rho(t)-\xi_N(t)\}$. A direct calculation shows that $$\zeta_t=\zeta_{xx}-\beta\zeta_x+c\zeta\quad \mbox{for }x\in[l(t),\rho(t)],\ t\in[T,T_1),$$ where $c$ is a bounded function. Since $$\left\{ \begin{array}{ll} \zeta(T,x)<0 &\mbox{ for }x\in[l(T),\rho(T)],\\ \zeta(t,\rho(t))=0 &\mbox{ for }t\in[T,T_1],\\ \zeta(t,l(t))<0&\mbox{ for } t\in[T,T_1]. \end{array} \right.$$ The last inequality follows from the following analysis. By the monotonicity of $u_x$ and the fact that $\xi_N(t)$ is the rightmost local maximum point, we see that, if $\zeta(t_1, l(t_1))=0$ for some $t_1\in(T,T_1]$ (denote $t_1$ the first of such times), then $\zeta(t_1, l(t_1)+\varepsilon )>0$ for some small $\varepsilon >0$. This, however, is impossible since $\zeta(t_1, x)<0$ for all $x\in (l(t_1), \rho(t_1))$ by the maximum principle and by the fact that $\zeta(t,l(t))<0$ for $t\in[T,t_1)$. Now we use the Hopf lemma for $\zeta$ in the domain $\{(t,x)\mid l(t)\leqslant x\leqslant \rho(t),\ T\leqslant t\leqslant T_1\}$ to derive $$\zeta(T_1,\rho(T_1))=0\quad \mbox{and}\quad \zeta_x(T_1,\rho (T_1))>0.$$ The latter, however, contradicts . This proves $T_1=T_\infty$. Upper bound estimate -------------------- In order to study the convergence of $u$ we give some precise upper bound estimates for the solutions. In this paper we always write $$\label{def A} A:= 2 \max\big\{ 1, \|u_0\|_{L^\infty ([-h_0, h_0])}\big\}.$$ 1\. [*Bound of $u$ near the free boundary $x=h(t)$*]{}. For any $\delta \in (0,-f'(1))$, set $$\bar{g}(t):= g(t),\quad \bar{h}(t):= c^* t - M e^{-\delta t} +H \ \mbox{ for some } M,\ H>0,$$ and $$\bar{u}(t,x) := (1+Ae^{-\delta t}) U^* (x-\bar{h}(t) ) \ \mbox{ for } x\leqslant \bar{h}(t),\ t>0,$$ where $U^*$ is the rightward traveling semi-wave (the solution of ). A direct calculation as in the proof of [@DMZ Lemma 3.2] shows that $(\bar{u},\bar{g},\bar{h})$ is an upper solution of provided $M,H>0$ are large. Hence we have $$\label{right bound} u(t,x) \leqslant \bar{u}(t, x) \mbox{ for } x\in [g(t), h(t)] \mbox{ and } t>0,\quad h(t) \leqslant \bar{h}(t) \mbox{ for } t>0.$$ 2\. [*Bound of $u$ in case $\beta \geqslant c_0$*]{}. We define a function $f_A(s)\in C^2([0,\infty),{\mathbb{R}})$ such that $$\label{fA} f_A (s) \left\{ \begin{array}{ll} = f'(0) s, & 0\leqslant s \leqslant 1,\\ >0 , & 1<s<A,\\ <0 , & s> A, \end{array} \right. \quad f'_A (A)<0,\quad f(s)\leqslant f_A (s)\leqslant f'(0) s\ \mbox{for}\ s\geqslant 0.$$ Denote by $Q_A(z)$ the unique solution of with $c=c_0$, with $f$ replaced by $f_A$ and $q(+\infty)=1$ replaced by $q(+\infty)=A$. Then by the comparison principle we have $$\label{left bound beta large} u(t, x) \leqslant Q_A (x- (\beta -c_0)t + x_0)\quad \mbox{for } x\in [g(t),h(t)],\; t>0,$$ provided $x_0 >0$ is large enough. Using $f_A$ we consider the Cauchy problem: $$\left\{ \begin{array}{l} (u_1)_t = (u_1)_{xx} - \beta (u_1)_x + f_A ( u_1), \quad x\in {\mathbb{R}},\; t>0,\\ u_1 (0,x) = \tilde{u}_0 (x) := \left\{ \begin{array}{ll} u_0(x), & x\in [-h_0, h_0],\\ 0, & |x|>h_0. \end{array} \right. \end{array} \right.$$ Since $f_A \geqslant f$, by the comparison principle we have $$\label{u < u1} u(t,x)\leqslant u_1(t,x),\quad x\in [g(t), h(t)],\; t>0.$$ Set $y:= x- \beta t$ and $u_2 (t,y):= u_1 (t,x)= u_1 (t,y+\beta t )$. Then $u_2 $ is a solution of $$\left\{ \begin{array}{l} (u_2)_t = (u_2)_{yy} + f_A ( u_2), \quad y\in {\mathbb{R}},\; t>0,\\ u_2 (0,y) = \tilde{u}_0 (y)\in [0,A], \quad y\in {\mathbb{R}}. \end{array} \right.$$ Set $\tilde{u}(t,y) := \frac{1}{A}u_2(t,-y) $, then $\tilde{u}(t,y)$ is the solution of $$\left\{ \begin{array}{l} \tilde{u}_t = \tilde{u}_{yy}+ \frac{1}{A} f_A \big(A \tilde{u} \big), \quad y\in {\mathbb{R}}, \; t>0,\\ \tilde{u}(0,y) = \frac{1}{A}\tilde{u}_0 (-y) \in [0,1], \quad y\in {\mathbb{R}}. \end{array} \right.$$ By and by the definitions of $u_2$ and $\tilde{u}$ we have $$\label{u < tildeu} u(t,x) \leqslant A \tilde{u}(t, \beta t -x),\quad x\in [g(t),h(t)],\; t>0.$$ On the other hand, by Proposition 2.3 in [@HNRR] and its proof, there exist $C_1,C_2,t_0$ depending on $u_0$ such that $$\tilde{u} \Big(t,c_0 t - \frac{3}{c_0} \ln \big(1+\frac{t}{t_0}\big) +y \Big) \leqslant C_1 Z (t,y ), \quad y\geqslant h_0,\; t>0,$$ where $$\label{def Z} Z(t,y):= \frac{1}{\sqrt{t_0}}y e^{- \frac{c_0}{2} y} \big[ C_2e^{\frac{-y^2}{4(t+t_0)}} +h(t,y) \big],\quad y\in {\mathbb{R}}, t>0,$$ with $h(t,y)$ satisfying $$\limsup\limits_{t\to \infty} \sup\limits_{0\leqslant y \leqslant \sqrt{t+1}} |h(t,y)| \leqslant \frac{C_2}{2}.$$ In particular, there exists $C_3 >0$ such that $$\tilde{u} \Big(t,c_0 t - \frac{3}{c_0} \ln \big(1+\frac{t}{t_0}\big) +y \Big) \leqslant C_3 e^{- \frac{5c_0}{12} y},\quad y\in [0, \sqrt{t+1}].$$ Combining with we have $$\label{left 1} u (t,x) \leqslant C_4 e^{-\frac{5c_0}{12} (Y(t)-x)}\quad \mbox{for } Y(t)-\sqrt{t+1} \leqslant x\leqslant \min \big\{Y(t), h(t)\big\},\; t\gg1,$$ where $C_4>0$ is a constant and $$\label{def Y} Y(t):= (\beta-c_0) t + \frac{3}{c_0} \ln \big(1+\frac{t}{t_0}\big),\quad t>0.$$ \[cor:-h0 h(t)\] [(i)]{} Assume $\beta =c_0$. Then there exists $C$ depending on $u_0,c_0,h_0$ such that $$u(t, -h_0)\leqslant C t^{-\frac{5}{4}}\quad \mbox{for } t>0 \mbox{ large};$$ [(ii)]{} Assume $\beta\in(c_0,\beta^*]$ and $h(t) =(\beta-c_0) t +O(1)$. Then there exists $C$ depending on $u_0, c_0, h_0$ such that $$u (t,x ) \leqslant C t^{-\frac{5}{4}}\quad \mbox{for } x\in \Big[h(t)-\frac{\pi}{2}, h(t)\Big] \mbox{ and } t>0 \mbox{ large}.$$ We only prove (ii) since (i) can be proved similarly. In Case (ii), $$Y(t)=(\beta-c_0) t + \frac{3}{c_0} \ln \big(1+\frac{t}{t_0}\big),$$ and so, for any $x\in [h(t)-\frac{\pi}{2}, h(t) ]$, we have $$Y(t) -x = \frac{3}{c_0} \ln \big(1+\frac{t}{t_0}\big) + O(1) \in [0, \sqrt{t+1}],\quad \mbox{when } t\gg 1.$$ Using we have $$\label{u h0<} u (t, x)\leqslant C t^{-\frac{5}{4}}\quad \mbox{when } t\gg 1,$$ where $C>0$ are some constants depending on $u_0,c_0,h_0$. For any given $m\in (0,1)$, denote $$\chi(t):= \min\{x\in [g(t),h(t)] \mid u(t,x)=m\} \mbox{ and } \widetilde{\chi}(t):= \min\{y\in {\mathbb{R}}\mid \tilde{u}(t,y)=m/A\}.$$ Then by [@HNRR Theorem 1.1] we have $$c_0 t -\frac{3}{c_0} \ln t -C \leqslant \widetilde{\chi}(t) \leqslant c_0 t -\frac{3}{c_0} \ln t +C ,\quad t\gg1,$$ for some $C>0$. Hence by we have $$\label{chi(t)>=} \chi(t)\geqslant \beta t -\widetilde{\chi}(t) \geqslant (\beta -c_0)t +\frac{3}{c_0} \ln t -C,\quad t\gg1.$$ Influence of $\beta$ on the long time behavior of solutions =========================================================== In this section we consider the influence of $\beta$ on the long time behavior of the solutions. In subsection 1 we give a locally uniformly convergence result. In subsection 2 we consider the small advection $\beta \in (0,c_0)$ and prove Theorem \[thm:small beta\]. In subsection 3 we first prove the boundedness of $g_\infty$ for $\beta \geqslant c_0$, the boundedness of $h_\infty$ for $\beta \geqslant \beta^*$, and then prove Theorem \[thm:large beta\] for large advection $\beta \geqslant \beta^*$. In subsection 4, we consider with medium-sized advection $\beta\in [c_0, \beta^*)$ and prove Theorems \[thm:middle beta\] and \[thm:beta=c0\]. The argument for the case $\beta\in [c_0, \beta^*)$ are longer and much more complicated than the cases with small or large advection. Convergence result ------------------ First we give a locally uniformly convergence result for $\beta \geqslant c_0$. \[lem:beta&gt;c0 u to 0\] Assume $\beta\geqslant c_0$. Then $u(t,\cdot)$ converges as $t\to\infty$ to $0$ locally uniformly in $I_\infty$. When $\beta >c_0$, the conclusion follows easily from since the upper solution $Q_A(x-(\beta -c_0)t +x_0)$ is a rightward traveling wave with positive speed $\beta-c_0$ and $Q_A(z)\to 0$ as $z\to -\infty$. Assume $\beta =c_0$. Then for any $[a,b]\subset I_\infty$, when $t$ is sufficiently large we have $$\frac{3}{c_0} \ln \big( 1+\frac{t}{t_0} \big) -\sqrt{t+1} < x< \frac{3}{c_0} \ln \big( 1+\frac{t}{t_0} \big) \quad \mbox{ for any } x\in [a,b],$$ and so by , for any $x\in [a,b]$, $$u(t,x) \leqslant C_4 e^{-\frac{5c_0}{12} \big[ \frac{3}{c_0} \ln \big( 1+\frac{t}{t_0} \big) -b\big] } \to 0,\quad \mbox{as } t\to \infty.$$ This proved the lemma. \[thm:convergence\] Let $(u,g, h)$ be a time-global solution of . Then as $t\to\infty$, $u(t,\cdot)$ converges to $0$ or to $1$ locally uniformly in $I_\infty$ when $\beta\in(0,c_0)$; $u(t,\cdot)$ converges to $0$ locally uniformly in $I_\infty$ when $\beta\geqslant c_0$. Moreover, $\lim_{t\to\infty}\|u(t,\cdot)\|_{L^{\infty}([g(t),h(t)])}=0$ if $I_{\infty}$ is a bounded interval. Using a similar argument as proving [@DM Theorem 1.1], [@DuLou Theorem 1.1], [@LL1 Theorem 1.1], one can show that $u(t,\cdot)$ converges, as $t\to\infty$, to a stationary solution, that is, a solution $v$ of $v_{xx}-\beta v_x+f(v)=0$, locally uniformly in $x\in I_\infty$. Moreover, one can show by Hopf lemma that $v=0$ when $g_\infty>-\infty$ or $h_\infty <\infty$. In other word, the limit $v$ can not be a non-trivial solution with endpoint. Therefore, when $\beta\in (0,c_0)$, the only possible choice for the $\omega$-limit of $u$ in the topology of $L^\infty_{\rm loc} (I_\infty)$ is $0$ or $1$; when $\beta\geqslant c_0$, the conclusion follows from Lemma \[lem:beta&gt;c0 u to 0\]. Finally, when $I_\infty$ is bounded, the uniform convergence for $u$ is also proved in the same way as that in [@DuLou; @LL1]. Problem with small advection: $0<\beta<c_0$ ------------------------------------------- In a similar way as proving [@GLL1 Lemma 2.2] and [@DuLou Theorem 3.2, Corollary 4.5] one can show the following conditions for spreading and for vanishing. \[lem:&lt;c0\] Assume $\beta\in(0,c_0)$. Let $(u,g,h)$ be a solution of . [(i)]{} If $h_0<H^*:= \frac{\pi}{\sqrt{{c^2_0}-\beta^2}}$ and if $\|u_0\|_{L^\infty([-h_0,h_0])}$ is sufficiently small, then vanishing happens; [(ii)]{} if $h_0\geqslant H^*$, then spreading happens. This lemma implies that, when $\beta\in (0,c_0)$, $2H^*$ is a critical width of the interval $I(t)$. Spreading happens if and only if $|I_\infty| >2H^*$. This extends the results in [@DuLin; @DuLou] for $\beta =0$, where it was shown that the critical width $2H^*= \frac{2\pi}{c_0}=\frac{\pi}{\sqrt{f'(0)}}$. [**Proof of Theorem \[thm:small beta\]:**]{} By Lemma \[lem:&lt;c0\] (ii) we see that spreading happens if $|I_\infty|>2H^*$. By the definition of spreading, this implies that $I_\infty={\mathbb{R}}$. Hence both the case $g_\infty>-\infty$, $h_\infty=\infty$ and the case $g_\infty=-\infty$, $h_\infty<\infty$ are impossible. $I_\infty$ is either a bounded interval with width $|I_\infty|\leqslant2H^*$ or the whole line ${\mathbb{R}}$. Using Theorem \[thm:convergence\] and Lemma \[lem:&lt;c0\] again, we can get the spreading-vanishing dichotomy result for the long-time behavior of the solutions of . Then the sharp threshold of the initial data $\sigma\phi$ can be proved in a similar way as in [@DuLou Theorem 5.2]. [$\Box$]{} Boundedness of $g_\infty$ and $h_\infty$, the proof of Theorem \[thm:large beta\] --------------------------------------------------------------------------------- Whether $g_\infty$ and $h_\infty$ are bounded or not is also a part of the conclusions in the long time behavior of $(u,g,h)$. In this subsection we show that $g_\infty>-\infty$ if $\beta \geqslant c_0$, and $h_\infty<\infty$ if $\beta\geqslant\beta^*$. We will prove these conclusions by using Corollary \[cor:-h0 h(t)\]. For this purpose we need the monotonicity of $u_x$. When $\beta =0$, Du and Lou [@DuLou] proved the monotonicity of $u_x$ in $[h_0, h(t)]$ and in $[g(t),-h_0]$. When $\beta>0$ we find that this is true only on the left side: $x\in[g(t),-h_0]$. \[lem:center\] Assume $(u,g, h)$ is a solution of . Then $$\label{g+h} g(t)+h(t)>-2h_0 \quad\mbox{for all } t>0,$$ $$\label{rough-symmetry} u_x(t,x)>0\quad \mbox{for all }x\in [g(t), -h_0],\ t>0.$$ It is easily seen by the continuity that, when $t>0$ is sufficiently small, $g(t)+h(t)>-2h_0$ and $u_x(t,x)>0$ for $g(t)\leqslant x\leqslant -h_0$. Define $$T_1:=\sup\big\{s\mid g(t)+h(t)>-2h_0\ \mbox{for all}\ t\in(0,s)\big\},$$ $$T_2:=\sup\big\{s\mid u_x(t,x)>0\ \mbox{for any}\ x\in[g(t),-h_0],\ t\in(0,s)\big\}.$$ We prove that $T_1=T_2=+\infty$. Otherwise, either $T_1<T_2\leqslant +\infty$, or $T_2\leqslant T_1\leqslant +\infty$ and $T_2<+\infty$. 1\. If $T_1<T_2\leqslant +\infty$, then $$g(t)+h(t)>-2h_0 \mbox{ for } t\in[0,T_1)\quad \mbox{and}\quad g(T_1)+h(T_1)=-2h_0.$$ Hence $$\label{T1} g'(T_1)+h'(T_1) \leqslant 0.$$ Set $G_{T_1}:=\{(t,x)|t\in(0,T_1],x\in(g(t),-h_0)\}$ and $$w(t,x):=u(t,x)-u(t,-2h_0-x) \mbox{ in } \overline{G}_{T_1}.$$ Since $-h_0\leqslant -2h_0-x\leqslant -2h_0-g(t)\leqslant h(t)$ when $(t,x)\in\overline{G}_{T_1}$, $w$ is well-defined over $\overline{G}_{T_1}$ and it satisfies $$\begin{aligned} w_t-w_{xx}-\beta w_x -c(t,x)w=-2\beta u_x(t,x) \leqslant 0 \mbox{ for } (t,x)\in G_{T_1},\end{aligned}$$ where $c$ is a bounded function, and $$w(t,-h_0)=0,\ w(t,g(t))\leqslant0\ \mbox{for}\ t\in(0,T_1].$$ Moreover, $$w(T_1,g(T_1))=u(T_1,g(T_1))-u(T_1,-2h_0-g(T_1))=u(T_1,g(T_1))-u(T_1,h(T_1))=0.$$ Then by the strong maximum principle and the Hopf lemma, we have $$w(t,x)<0\ \mbox{for}\ (t,x)\in G_{T_1},\ \mbox{and}\ w_x(T_1,g(T_1))<0.$$ Thus $$g'(T_1)+h'(T_1)=-\mu[u_x(T_1,g(T_1))+u_x(T_1,h(T_1))]=-\mu w_x(T_1,g(T_1))>0.$$ This contradicts . 2\. If $T_2\leqslant T_1\leqslant \infty$ and $T_2<+\infty$, then $$u_x(t,x)>0,\ t\in(0,T_2),\ x\in[g(t),-h_0].$$ By the definition of $T_2$, there exists $y\in (g(T_2),-h_0]$ such that $u_x(T_2,y)= 0$. Denote $x_0$ the minimum of such $y$. By the continuity and the monotonicity of $g(t)$, there exists $T_0\in[0,T_2)$ such that $x_0=g(T_0)$. Let $$G_{T_2}:=\{(t,x)|t\in(T_0,T_2],\ x\in(g(t),x_0)\},$$ $$z(t,x):=u(t,x)-u(t,2x_0-x)\ \mbox{for}\ (t,x)\in\overline{G}_{T_2}.$$ Using the maximum principle for $z(t,x)$ in $G_{T_2}$ as above we conclude that $z_x (T_2, x_0)>0$. This contradicts the definition of $x_0$. Combining the above two steps we obtain $T_1=T_2=+\infty$. A direct consequence of Lemma \[lem:center\] is $g_{\infty}+h_{\infty}\geqslant -2h_0$. So we have \[3case\] There are only three possible situations for $I_\infty = (g_\infty, h_\infty)$: [(i)]{} $I_\infty ={\mathbb{R}}$; [(ii)]{} $I_\infty$ is a finite interval; and [(iii)]{} $I_\infty =(g_\infty, \infty)$ with $g_\infty>-\infty$. Indeed, (i) and (ii) are possible when $\beta\in (0,c_0)$ (see Theorem \[thm:small beta\]), (ii) and (iii) are possible when $\beta \geqslant c_0$ (see Theorems \[thm:middle beta\], \[thm:beta=c0\] and \[thm:large beta\]). By the monotonicity of $u(t,\cdot)$ in $[g(t),-h_0]$, we can prove the boundedness of $g_\infty$. \[prop:g\_infty&gt;-infty\] Assume $\beta\geqslant c_0$ and $(u,g,h)$ is a solution of . Then $g_\infty>-\infty$. First we consider the case $\beta>c_0$. Let $f_A(s)$ be defined as in and let $Q_A(z)$ be the unique solution of with $c=c_0$, with $f$ replaced by $f_A$ and $q(+\infty)=1$ replaced by $q(+\infty)=A$, as in . Since $$\label{QA(-infty)} Q_A(z)\sim-Cze^{\frac{c_0}{2}z}\mbox{ as }z\to-\infty$$ for some $C>0$ (cf. [@AW; @HNRR]), there exist $T_1>0$, $C_1>0$ such that, when $t\geqslant T_1$, $$\begin{aligned} Q_A(-h_0-(\beta-c_0)t+x_0)&\leqslant & -2C(-h_0-(\beta-c_0)t+x_0)e^{\frac{c_0}{2}(-h_0-(\beta-c_0)t+x_0)} \\ &\leqslant &C_1 te^{-\frac{c_0}{2}(\beta-c_0)t}\leqslant C_1 e^{-\frac{c_0}{4}(\beta-c_0)t},\end{aligned}$$ where $x_0>0$ is large such that holds. By Lemma \[lem:center\] and we have $$\label{u(t,x)<=C1e^{-t/4}} u(t,x)\leqslant u(t,-h_0)\leqslant Q_A(-h_0-(\beta-c_0)t+x_0)\leqslant C_1e^{-\frac{c_0}{4}(\beta-c_0)t}$$ for $x\in[g(t),-h_0]$ and $t\geqslant T_1$. Set $\delta:=\min\{1,\frac{c_0}{4}(\beta-c_0)\}$, $\epsilon_1:=C_1e^{-\frac{\beta\pi+c_0(\beta-c_0)T_1}{4}}$, $$k(t):=-g(T_1)+\frac{\pi}{2}+\frac{\mu\epsilon_1}{\delta}(1-e^{-\delta t})\mbox{ for }t\geqslant 0$$ and $$w(t,x):=\epsilon_1 e^{-\delta t}e^{\frac{\beta}{2}(x+k(t))}\sin(x+k(t))\quad \mbox{ for }-k(t)\leqslant x\leqslant -k(t)+\frac{\pi}{2},\ t\geqslant 0.$$ A direct calculation shows that $$\begin{aligned} w_t-w_{xx}+\beta w_x-f(w)&=&w\Big[1-\delta+\frac{\beta}{2}k'(t)+\frac{\beta^2}{4}\Big]+k'(t)\hat{w}-f(w) \\ &\geqslant& w\Big[1-\delta+\frac{\beta^2}{4}-f'(0)\Big]\geqslant(1-\delta)w\geqslant0,\end{aligned}$$ for $-k(t)\leqslant x\leqslant -k(t)+\frac{\pi}{2},\ t>0$, where $\hat{w}=\epsilon_1 e^{-\delta t}e^{\frac{\beta}{2}(x+k(t))}\cos(x+k(t))$, $$-k'(t)=-\epsilon_1\mu e^{-\delta t}=-\mu w_x(t,-k(t)),\quad t>0,$$ and $$w\Big(t,-k(t)+\frac{\pi}{2}\Big)=\epsilon_1 e^{\frac{\beta\pi}{4}}e^{-\delta t} \geqslant C_1 e^{-\frac{c_0(\beta-c_0)}{4}(t+T_1)}\geqslant u(t+T_1,x)$$ for $g(t+T_1)\leqslant x\leqslant -h_0$, $t>0$. Hence for $t\geqslant 0$, either $g(t+T_1)\geqslant -k(t)+\frac{\pi}{2}\geqslant g(T_1)-\frac{\mu\epsilon_1}{\delta}$, or $u(t+T_1,\cdot)$ and $w(t,\cdot)$ have common domain. In the latter case, by comparing them on their common domain we have $$g(t+T_1)\geqslant -k(t)\geqslant g(T_1)-\frac{\pi}{2}-\frac{\mu\epsilon_1}{\delta}>-\infty.$$ This proves $g_\infty>-\infty$. Next we consider the case $\beta=c_0$. By Corollary \[cor:-h0 h(t)\] and Lemma \[lem:center\], there exist $T_2>\frac{5}{4}$ and $C>0$ such that $$u(t,x)\leqslant u(t,-h_0)\leqslant Ct^{-\frac{5}{4}}\mbox{ for }g(t)\leqslant x\leqslant -h_0,\ t\geqslant T_2.$$ Set $\epsilon_2:=Ce^{-\frac{\beta\pi}{4}}$ and define $$k_2(t):=-g(T_2)+\frac{\pi}{2}+4\mu\epsilon_2[T_2^{-\frac{1}{4}}-(t+T_2)^{-\frac{1}{4}}]\mbox{ for }t>0,$$ $$w_2(t,x):=\epsilon_2(t+T_2)^{-\frac{5}{4}}e^{\frac{\beta}{2}(x+k_2(t))}\sin(x+k_2(t)) \mbox{ for }-k_2(t)\leqslant x\leqslant -k_2(t)+\frac{\pi}{2},\ t\geqslant 0.$$ A similar discussion as above shows that $(w_2,-k_2,-k_2+\frac{\pi}{2})$ is an upper solution, and so $$g(t+T_2)\geqslant -k_2(t)\geqslant g(T_2)-\frac{\pi}{2}-4\mu\epsilon_2 T_2^{-\frac{1}{4}}>-\infty.$$ This proves the proposition. Next we prove the boundedness of $h_\infty$ when $\beta \geqslant \beta^*$. \[&gt;=b\^\*-h\_infty\] Assume $\beta\geqslant \beta^*$ and $(u,g,h)$ is a solution of . Then $h_\infty<\infty$. 1\. First we consider the case $\beta>\beta^*$. In this case we have $c^*(\beta)<\beta-c_0$. Denote $\nu:=\beta-c_0-c^*(\beta)>0$. By , and , there exist $T_1>0$, $C_1>0$ such that, for $x\in(g(t),h(t))$ and $t\geqslant T_1$, we have $$\begin{aligned} u(t,x)&\leqslant & Q_A(x-(\beta-c_0)t+x_0)\leqslant Q_A(h(t)-(\beta-c_0)t+x_0) \\ &\leqslant & Q_A(-\nu t+H+x_0)\leqslant -2C(-\nu t+ H +x_0)e^{\frac{c_0}{2}(-\nu t+ H +x_0)} \\ &\leqslant & C_1e^{-\frac{c_0\nu}{4}t}.\end{aligned}$$ Set $\delta:=\frac{1}{2}\min\{1,\frac{c_0\nu}{4}\}$ and choose $T_2>T_1$ such that $$\epsilon_3 :=C_1 e^{\frac{\beta\pi-c_0\nu T_2}{4}} < \frac{2}{\beta \mu}.$$ Define $$k_3(t):=h(T_2)+\frac{\pi}{2}+\frac{\mu\epsilon_3}{\delta}(1-e^{-\delta t})\mbox{ for } t\geqslant 0$$ and $$w_3(t,x):=\epsilon_3 e^{-\delta t}e^{\frac{\beta}{2}(x-k_3(t))}\cos \Big( x-k_3(t)+\frac{\pi}{2} \Big) \mbox{ for }k_3(t)-\frac{\pi}{2}\leqslant x\leqslant k_3(t),\ t\geqslant 0.$$ A direct calculation as in the proof of Proposition \[prop:g\_infty&gt;-infty\] shows that $(w_3,k_3(t)-\frac{\pi}{2},k_3(t))$ is an upper solution and $$h(t+T_2)\leqslant k_3(t)\leqslant h(T_2)+\frac{\pi}{2}+\frac{\mu\epsilon_3}{\delta}<\infty.$$ 2\. Next we consider the case $\beta=\beta^*$. We first show that, for some large $T_3$, $$\label{u(t,x)<U*(x-c*t+h0)} u(t,x)<U^*(x-c^*(\beta^*)t+h_0)\mbox{ for }x\in[g(t),h(t)],\ t\geqslant T_3,$$ where $U^*(x-c^*(\beta^*)t+h_0)$ is the rightward traveling semi-wave with endpoint at $c^*(\beta^*) t-h_0$. At time $t=0$, $u(t,x)$ and $U^*(x+h_0)$ intersect at $x=-h_0$. Then for small time $t>0$, they intersect at exact one point. We claim that the case $c^*(\beta^*) t-h_0<h(t)$ for all $t\geqslant 0$ is impossible. Otherwise, combining with we have $h(t)=c^* (\beta^*) t+O(1)$, and so by Corollary \[cor:-h0 h(t)\] there exist $T_4>0$ and $C>0$ such that $$u(t,x)\leqslant Ct^{-\frac{5}{4}}\mbox{ for }h(t)-\frac{\pi}{2}\leqslant x\leqslant h(t),\ t\geqslant T_4.$$ Set $\epsilon_4:=Ce^{\frac{\beta\pi}{4}}$, $T_5:=\max\{1,T_4, \frac{5}{4}+\frac{\beta\mu\epsilon_4}{2}\}$ and define $$k_4(t):=h(T_5)+\frac{\pi}{2}+4\mu\epsilon_4[T_5^{-\frac{1}{4}}-(t+T_5)^{-\frac{1}{4}}],\quad t\geqslant 0,$$ $$w_4(t,x):=\epsilon_4(t+T_5)^{-\frac{5}{4}}e^{\frac{\beta}{2}(x-k_4(t))}\cos\Big(x-k_4(t)+\frac{\pi}{2}\Big), \quad k_4(t)-\frac{\pi}{2}\leqslant x\leqslant k_4(t),\ t\geqslant 0.$$ A direct calculation shows that $(w_4,k_4(t)-\frac{\pi}{2},k_4(t))$ is an upper solution, and so $$h(t+T_5)\leqslant k_4(t)\leqslant h(T_5)+\frac{\pi}{2}+4\mu\epsilon_4 T_5^{-\frac{1}{4}}<\infty,$$ contradicts our assumption $c^*(\beta^*) t-h_0<h(t)$ for all $t$. Therefore, there exists $T_6>0$ such that $h(T_6)=c^*(\beta^*) T_6-h_0$ and by Lemma \[lem:zeros between u and Psi\], the unique intersection point between $u$ and $U^*$ disappears after $T_6$. This implies holds when $x\in[g(t),h(t)]$ and $t>T_3$ for any $T_3 >T_6$. On the other hand, by Lemma \[lem:tadpole tw beta=beta\*\], $V^*_{\delta_1}(z):=V(z;\beta^*-c_0-\delta_1,-c_0-\delta_1)$ approaches $U^*(z)$ locally uniformly in $(-\infty,0]$ as $\delta_1\to0$. Hence there exists $\delta_1>0$ sufficiently small such that $V^*_{\delta_1} (z)$ is close to $U^*(z)$ and so $$u(T_3,x)<V^*_{\delta_1}(x-c^*(\beta^*)T_3+h_0)\mbox{ for }x\in[g(t),h(t)].$$ By comparison $u(t+T_3,x)\leqslant V^*_{\delta_1}(x-(\beta^*-c_0-\delta_1)t-c^*(\beta^*)T_3+h_0)$ and so $h(t+T_3)$ is blocked by the right endpoint $(\beta^* -c_0-\delta_1)t+c^*(\beta^*)T_3-h_0$ of $V^*_{\delta_1}$: $$h(t+T_3)\leqslant (\beta^*-c_0-\delta_1)t+c^*(\beta^*)T_3-h_0,\quad t\geqslant0.$$ Using we see that, for any $x\in[g(t),h(t)]$ and sufficiently large $t$, $$u(t,x)\leqslant Q_A(x-(\beta^*-c_0)t+x_0)\leqslant Q_A(-\delta_1t+x_1)\leqslant C_2 e^{-\frac{c_0\delta_1}{4}t},$$ for some $x_1\in{\mathbb{R}}$ and $C_2>0$. The rest proof is similar as that in Step 1. This proves the proposition. [**Proof of Theorem \[thm:large beta\]:**]{} The conclusions follow from Proposition \[prop:g\_infty&gt;-infty\], \[&gt;=b\^\*-h\_infty\] and Theorem \[thm:convergence\] immediately. [$\Box$]{} Problem with medium-sized advection: $c_0\leqslant \beta<\beta^*$ ----------------------------------------------------------------- In this subsection we consider the case $\beta\in [c_0,\beta^*)$. In this case, the long time behavior of the solutions is complicated and more interesting. Besides vanishing, we find some new phnomena: virtual spreading, virtual vanishing and convergence to the tadpole-like traveling wave. In the first part, we give some sufficient conditions for vanishing; in the second part we give a necessary and sufficient condition for virtual spreading; in the third part we study the limits of $h'(t)$ and $u(t,x)$ when vanishing and virtual spreading do not happen; in the last part we finish the proofs of Theorems \[thm:middle beta\] and \[thm:beta=c0\]. ### Vanishing phenomena When $\beta\geqslant c_0$, we have $g_\infty > -\infty$ by Proposition \[prop:g\_infty&gt;-infty\], which implies that $u\to 0$ locally uniformly. We now show that the convergence can be a uniform one when the initial data $u_0$ is sufficiently small. \[lem:condition for vanishing\] Assume $\beta\geqslant c_0$ and $(u,g,h)$ is the solution of . If $\|u_0\|_{L^{\infty}([-h_0,h_0])}$ is sufficiently small, then vanishing happens. Choose $\delta>0$ small such that $$\frac{\pi^2}{h_0^2(1+\delta)^2}\geqslant 4\delta+\beta h_0\delta^2.$$ Set $k(t):=h_0(1+\delta-\frac{\delta}{2}e^{-\delta t})$, $\epsilon:= \frac{h_0^2\delta^2}{\pi\mu}(1+\frac{\delta}{2})$ and $$\label{vanishing-w} w(t,x):=\epsilon e^{-\delta t}e^{\frac{\beta}{2}(x-k(t))}\cos \frac{\pi x}{2k(t)}\ \ \mbox{ for }-k(t)\leqslant x\leqslant k(t),\ t>0.$$ A direct calculation shows that, for $x\in(-k(t),k(t))$ and $t>0$, $$w_t-w_{xx}+\beta w_x-f(w)\geqslant \frac{1}{4}\Big( \frac{\pi^2}{h_0^2(1+\delta)^2}- 4\delta-\beta h_0\delta^2 \Big) w\geqslant 0.$$ On the other hand, for any $t>0$, $$\mu w_x(t,-k(t))\leqslant -\mu w_x(t,k(t))=\frac{\pi\mu \epsilon}{2k(t)}e^{-\delta t} \leqslant \frac{\pi\mu \epsilon}{2h_0(1+\frac{\delta}{2})}e^{-\delta t} =\frac{\delta^2}{2}h_0e^{-\delta t}=k'(t).$$ Hence $(w,-k,k)$ is an upper solution of . Clearly $k(0)=h_0(1+\frac{\delta}{2})>h_0$ and $w(t, \pm k(t))=0$ for $t\geqslant 0$. If $\|u_0\|_{L^\infty([-h_0,h_0])}$ is small such that $$\|u_0\|_{L^\infty([-h_0,h_0])}\leqslant \epsilon e^{-\frac{\beta}{2}h_0(2+\frac{\delta}{2})}\cos \frac{\pi}{2+\delta}=w(0,-h_0),$$ then $u_0(x)\leqslant w(0,x)$ for $x\in[-h_0,h_0]$. By the comparison principle, we have $$-h_0(1+\delta)\leqslant -k(t)\leqslant g(t)<h(t)\leqslant k(t)\leqslant h_0(1+\delta),$$ $$\|u(t,\cdot)\|_{L^\infty([g(t),h(t)])} \leqslant \|w(t,\cdot)\|_{L^\infty([-k(t),k(t)])} \leqslant \epsilon e^{-\delta t} \to 0\quad \mbox{as } t\to \infty.$$ This proves the lemma. For any given $h_0>0$ and $\phi\in\mathscr {X}(h_0)$, we write the solution $(u,g,h)$ also as $(u(t,x;\sigma\phi)$, $g(t;\sigma\phi)$,$h(t;\sigma\phi))$ to emphasize the dependence on the initial data $u_0=\sigma\phi$. Set $$\label{def E0} E_0:=\{\sigma\geqslant 0 \mid \mbox{vanishing happens for } u(\cdot,\cdot;\sigma\phi)\},\quad \sigma_*:=\sup E_0.$$ Lemma \[lem:condition for vanishing\] implies that $\sigma\in E_0$ for all small $\sigma>0$. By the comparison principle we have $[0,\sigma_*)\subset E_0$. In case $\sigma_*=\infty$ (this happens in particular when $\liminf_{s\to \infty} \frac{-f(s)}{s} \gg 1$ and $\beta =0$, see [@DuLou Proposition 5.4]), there is nothing left to prove. Hence we only consider the case $\sigma_*\in(0,\infty)$. \[thm:sigma\_\*\] Assume $c_0\leqslant \beta<\beta^*$. For any $\phi\in\mathscr {X}(h_0)$, let $E_0$ and $\sigma_*$ be defined as in . If $\sigma_*\in(0,\infty)$, then $E_0 = [0,\sigma_*)$. If $\sigma\geqslant \sigma_*$, then $g(\infty;\sigma\phi)>-\infty$, $h(\infty;\sigma\phi)=\infty$, and $u(t,\cdot;\sigma\phi)\to 0$ locally uniformly in $(g(\infty;\sigma\phi), \infty)$. For any positive $\sigma_0\in E_0$, since $u(t,\cdot;\sigma_0\phi)\to 0$ uniformly, we can find a large $T_0>0$ such that $u(T_0,x;\sigma_0\phi)< w(0,x)$, where $w(t,x)$ is defined as in , with $h_0$ replaced by $H:= \max\{h(\infty;\sigma_0\phi), -g(\infty;\sigma_0\phi)\}<\infty$. By continuity, there exists $\epsilon>0$ such that $u(T_0,x;\sigma\phi)< w(0,x)$ for every $\sigma\in[\sigma_0,\sigma_0+\epsilon)$. As in the proof of Lemma \[lem:condition for vanishing\], we conclude that vanishing happens for $u(t,x;\sigma\phi)$, that is, $\sigma \in E_0$. Therefore, $E_0\backslash \{0\}$ is an open set, and so $E_0=[0,\sigma_*)$. The rest of the conclusions follow from Theorem \[thm:convergence\] and Proposition \[prop:g\_infty&gt;-infty\]. We finish this part by giving another sufficient condition for vanishing. \[lem:condition 2 for vanishing\] Assume $c_0 < \beta<\beta^*$. Let $(u,g,h)$ be a solution of . Then vanishing happens if there exist $t_1\geqslant 0$, $x_1\in{\mathbb{R}}$ such that $$h(t_1)\leqslant x_1,\ u(t_1,x)\leqslant V^*(x-x_1)\mbox{ for }x\in[g(t_1),h(t_1)],$$ where $V^*(z)$ is the tadpole-like solution as in Lemma \[lem:tadpole tw bata&lt;beta\*\] (ii). Since $V^*(x-(\beta-c_0)t-x_1)$ is a solution of $_1$ satisfying Stefan free boundary condition at $x=k(t):=(\beta-c_0)t+x_1$: $$k'(t)=\beta-c_0=-\mu(V^*)'(0)$$ by Lemma \[lem:tadpole tw bata&lt;beta\*\] (ii). By the comparison principle we have $$h(t_1+1)<k(1),\ u(t_1+1,x)<V^*(x-\beta+c_0-x_1)\mbox{ for }x\in[g(t_1+1),h(t_1+1)].$$ By Lemma \[lem:tadpole tw bata&lt;beta\*\] (iii), $V_\delta(z):=V(z;\beta-c_0-\delta,-c_0-\delta)\to V^*(z)$ locally uniformly in $(-\infty,0]$ as $\delta\to0$. Hence for sufficiently small $\delta>0$, we have $$u(t_1+1,x)<V_\delta(x-\beta+c_0-x_1)\mbox{ for }x\in[g(t_1+1),h(t_1+1)].$$ Since $V_\delta(x-(\beta-c_0-\delta)t-\beta+c_0-x_1)$ is a solution of $_1$ satisfying Stefan free boundary condition at $x=k_\delta(t):=(\beta-c_0-\delta)t+\beta-c_0+x_1$, by comparison we have $$h(t+t_1+1)\leqslant k_\delta(t)=(\beta-c_0-\delta)t+\beta-c_0+x_1.$$ A similar argument as in step 1 of the proof of Proposition \[&gt;=b\^\*-h\_infty\] shows that $h_\infty<\infty$, this implies that vanishing happens by Theorem \[thm:convergence\]. ### A necessary and sufficient condition for virtual spreading \[lem:condition for virtual spreading\] Assume $c_0\leqslant\beta<\beta^*$. Let $(u,g,h)$ be a solution of . Then virtual spreading happens if and only if, for any $\delta\in(0,c^*(\beta)-\beta+c_0)$, there exist $t_1$ and $x_1$ such that $$\label{condition for v s} u(t_1,x)\geqslant W_\delta(x-x_1)\mbox{ for }x\in[x_1-L_\delta,x_1],$$ where $W_\delta(z)$, $L_\delta:=L(\beta-c_0+\delta,-c_0+\delta)$ are the notation in Lemma \[lem:compact tw\]. The inequality follows from the definition of virtual spreading immediately. We only need to show that is a sufficient condition for virtual spreading. Since $W_\delta(x-(\beta-c_0+\delta)t-x_1)$ satisfies $_1$ and Stefan free boundary condition at $x=r(t):=(\beta-c_0+\delta)t+x_1$. Comparing $u$ and $W_\delta$ we have $$\label{u(t+t2,x)>W_delta} u(t+t_1,x)>W_\delta(x-(\beta-c_0+\delta)t-x_1)\mbox{ for }x\in[r(t)-L_\delta,r(t)].$$ In particular, this is true at $t=1$. Since $W_\delta(z)$ depends on $\delta$ continuously, we have $$u(t_1+1,x)>W_{\delta+\epsilon}(x-(\beta-c_0+\delta+\epsilon)-x_1) \mbox{ for }x\in[r(1)+\epsilon-L_{\delta+\epsilon},r(1)+\epsilon],$$ for any $\epsilon\in(0,\epsilon_0]$ provided $\epsilon_0>0$ is small. Using the comparison principle again between $u(t+t_1+1,x)$ and $W_{\delta+\epsilon}(x-(\beta-c_0+\delta+\epsilon)(t+1)-x_1)$, we have $$h(t+t_1+1)\geqslant (\beta-c_0+\delta+\epsilon)(t+1)+x_1,\ t>0.$$ This implies that $$\label{H} H(t):=h(t+t_1+1)-(\beta-c_0+\delta)t\geqslant \epsilon(t+1)+x_1\to\infty\mbox{ as }t\to\infty.$$ Set $$\label{G} G(t):=g(t+t_1+1)-(\beta-c_0+\delta)t$$ and $$\label{w-G-H} w(t,x):=u(t+t_1+1,x+(\beta-c_0+\delta)t)\mbox{ for }G(t)\leqslant x\leqslant H(t),\ t\geqslant 0.$$ Then $G(t)\to-\infty$ as $t\to\infty$ by Proposition \[prop:g\_infty&gt;-infty\], $w$ satisfies $$\label{w(t,x)>W_delta} w(t,x)>W_\delta(x-(\beta-c_0+\delta)-x_1)\mbox{ for }x\in[x_1+\beta-c_0+\delta-L_\delta,x_1+\beta-c_0+\delta],\ t\geqslant0$$ by and $$\label{shift-w} \left\{ \begin{array}{ll} w_t=w_{xx}-(c_0-\delta)w_{x}+f(w),\quad t>0,\ G(t)< x<H(t),\\ w(t,G(t))=0,\ \ G'(t)=-\mu w_x(t, G(t))-(\beta-c_0+\delta),\quad t>0,\\ w(t,H(t))=0,\ \ H'(t)=-\mu w_x (t, H(t))-(\beta-c_0+\delta) ,\quad t>0,\\ G(0)=g(t_1+1),\ H(0)= h(t_1+1),\ \ w(0,x) =u(t_1+1,x) \mbox{ for } G(0)\leqslant x \leqslant H(0). \end{array} \right.$$ In a similar way as proving Theorem \[thm:convergence\] (cf. the proof of [@DuLou Theorem 1.1]), one can show that $w(t,\cdot)$ converges to a stationary solution of $_1$ locally uniformly in ${\mathbb{R}}$. By , such a stationary solution must be 1. This means spreading happens for $w$ and so virtual spreading happens for $u$. This proves the lemma. ### The limits of $h$ and $u$ when vanishing and virtual spreading do not happen In this part we always assume $c_0\leqslant \beta<\beta^*$ and $\sigma_*\in (0,\infty)$ for given $\phi\in \mathscr{X}(h_0)$, where $\sigma_*$ is defined by . We consider the limits of $h(t;\sigma\phi)$, $h'(t;\sigma\phi)$ and $u(t,\cdot+h(t);\sigma\phi)$ when vanishing does not happen, that is, when $\sigma\geqslant\sigma_*$. \[lem:h(t)-(beta-c0)t=+infty\] Assume $c_0\leqslant \beta<\beta^*$. If vanishing does not happen for a solution $u$ of , then $$\label{h(t)-(beta-c0)t=+infty} \lim\limits_{t\to\infty}[h(t)-(\beta-c_0)t]=+\infty.$$ When $\beta=c_0$, we have $g_\infty>-\infty$ by Proposition \[prop:g\_infty&gt;-infty\]. If $h_\infty<\infty$, then vanishing happens for $u$ by Theorem \[thm:convergence\], contradicts our assumption. Therefore, holds when $\beta=c_0$. We now consider the case $c_0<\beta<\beta^*$. First we prove that $$h(t)>(\beta-c_0)t-h_0\mbox{ for any }t>0.$$ Set $$\eta_1(t,x):=u(t,x)-V^*(x-(\beta-c_0)t+h_0)\mbox{ for }x\in J_1 (t),\ t>0,$$ where $$J_1(t):=[g(t),\min\{h(t),(\beta-c_0)t-h_0\}]\mbox{ for }t>0.$$ It is easily seen that, for $0<t\ll1$, $$\label{Z_J(t)[eta1]} (\beta-c_0)t-h_0<h(t)\mbox{ and }\mathcal{Z}_{J_1(t)}[\eta_1(t,\cdot)]=1.$$ We claim that this is true for all $t>0$. Otherwise, there exists $T_1>0$ such that $(\beta-c_0)t-h_0<h(t)$ for $0<t<T_1$ and $(\beta-c_0)T_1-h_0=h(T_1)$. By Lemma \[lem:zeros between u and Psi\] we have $\mathcal{Z}_{J_1(t)}[\eta_1(t,\cdot)]=1$ for $0<t<T_1$ and $\mathcal{Z}_{J_1(t)}[\eta_1(t,\cdot)]=0$ for $t>T_1$. Therefore, $$u(t,x)<V^*(x-(\beta-c_0)t+h_0)\mbox{ for }x\in I(t),\ T_1<t\ll T_1+1.$$ This implies that vanishing happens for $u$ by Lemma \[lem:condition 2 for vanishing\], contradicts our assumption. Next we prove that, for any large $M>0$, $h(t)>(\beta-c_0)t+M$ when $t$ is large. Without loss of generality we assume $$u_0'(-h_0)>0,\ u_0'(h_0)<0\mbox{ and }u_0(x)>0\mbox{ for }x\in(-h_0,h_0).$$ (Otherwise one can replace $u_0(x)$ by $u(1,x)$ to proceed the following analysis.) So there exists $X>h_0$ large such that $u_0(x)$ intersects $V^*(x-M)$ at exactly two points for any $M\geqslant X$. Set $$\eta_2(t,x):=u(t,x)-V^*(x-(\beta-c_0)t-M)\mbox{ for }x\in J_2(t),\ t>0,$$ where $$J_2(t):=[g(t),\min\{h(t),(\beta-c_0)t+M\}].$$ Then $\mathcal{Z}_{J_2(t)}[\eta_2(t,\cdot)]=2$ for $0<t\ll1$. Denote by $\xi_1(t)$ and $\xi_2(t)$ with $\xi_1(t)<\xi_2(t)$ the two zeros of $\eta_2(t,\cdot)$. Then we have the following situations about the relations among $\xi_1(t)$, $\xi_2(t)$, $h(t)$ and $(\beta -c_0)t +M $. *Case 1*. $h(t)<(\beta-c_0)t+M$ for all $t>0$. In this case, combining with we have $h(t)=(\beta-c_0)t+O(1)$. Using a similar argument as in step 2 of the proof of Proposition \[&gt;=b\^\*-h\_infty\] we can derive $h_\infty<\infty$. This implies that vanishing happens for $u$, contradicts our assumption. *Case 2*. There exists $T_2>0$ such that $h(t)<(\beta-c_0)t+M$ for $0<t<T_2$ and $h(T_2)=(\beta-c_0)T_2+M$. This includes several subcases. *Subcase 2-1*. $\xi_1(t)$ meets $\xi_2(t)$ at time $t=T_3<T_2$. In this case, $\xi_1(T_3)=\xi_2(T_3)$ is a degenerate zero of $\eta_2(T_3,\cdot)$ and so $\mathcal{Z}_{I(t)}[\eta_2(t,\cdot)]=0$ for $T_3<t\ll T_3+1$. This indicates that $$\label{u(t+1,x)<V*} u(t,x)<V^*(x-(\beta-c_0)t-M),\quad x\in[g(t),h(t)],\ T_3<t\ll T_3+1,$$ and so vanishing happens by Lemma \[lem:condition 2 for vanishing\], contradicts our assumption. *Subcase 2-2*. $\xi_1(t)<\xi_2(t)<h(t)$ for $0<t\leqslant T_2$. This means a new intersection point $(h(T_2),0)$ between $u$ and $V^*$ emerges on the boundary. This is impossible by Lemma \[lem:zeros between u and Psi\]. *Subcase 2-3*. $\xi_1(t)<\xi_2(t)<h(t)$ for $0<t<T_2$ and $\xi_1(T_2)=\xi_2(T_2)=h(T_2)$. This means the two intersection points between $u$ and $V^*$ move rightward to $(h(t),0)$ at time $T_2$. By Lemma \[lem:zeros between u and Psi\], this is the unique zero of $\eta_2(T_2,\cdot)$ and it will disappear after time $T_2$. Hence holds for $t>T_2$. Then vanishing happens, a contradiction. *Subcase 2-4*. $\xi_1(t)<\xi_2(t)<h(t)$ for $0<t<T_2$ and $\xi_1(T_2)<\xi_2(T_2)=h(T_2)=(\beta-c_0)T_2+M$. By Lemma \[lem:zeros between u and Psi\], $\mathcal{Z}_{J_2(t)}[\eta_2(t,\cdot)]=1<2$ for $T_2<t\ll T_2+1$, where $J_2(t):=[g(t),(\beta-c_0)t+M]$. Using the maximum principle for $\eta_2(t,x)$ in the domain $$\Omega:=\{(t,x)\mid\xi_1(t)<x<\xi_2(t),\ 0<t\leqslant T_2\}$$ and using Hopf lemma at $(t,x)=(T_2,h(T_2))=(T_2,\xi_2(T_2))$ we have $(\eta_2)_x(T_2,h(T_2))<0$, that is, $$\label{u_x(T2,h(T2))<(V*)'(0)} u_x(T_2,h(T_2))<(V^*)'(0)=-\frac{\beta-c_0}{\mu},$$ and so $$\label{h'(T2)>beta-c0} h'(T_2)=-\mu u_x(T_2,h(T_2))>\beta-c_0.$$ We claim that $$\label{(beta-c0)t+M<h(t)} (\beta-c_0)t+M<h(t)\mbox{ for all }t>T_2$$ and so $\mathcal{Z}_{J_2(t)}[\eta_2(t,\cdot)]=1$ for all $t>T_2$. Indeed, if $(\beta-c_0)t+M$ catches up $h(t)$ again at time $t=T_4$, then the unique intersection point $(\xi_1(t),u(t,\xi_1(t)))$ (for $t\in[T_2,T_4)$) moves to $(h(t),0)$ at time $T_4$ and then it disappear after time $T_4$ by Lemma \[lem:zeros between u and Psi\]. This implies that holds for $t>T_4$ and so vanishing happens, a contradiction. is true for any $M>0$ and so holds. \[lem:h’(t)=beta-c0\] Assume $c_0\leqslant \beta<\beta^*$. If vanishing and virtual spreading do not happen for the solution $u$ of , then $$\label{h'(t)=beta-c0} \lim\limits_{t\to\infty}h'(t)=\beta-c_0.$$ We divide the proof into several steps. *Step 1*. We first prove $h'(t)>\beta-c_0$ for all large $t$. This is clear when $\beta=c_0$. We now assume $c_0<\beta<\beta^*$. For readers’ convenience, we first sketch the idea of our proof. We put a tadpole-like traveling wave $V^*(x-(\beta -c_0)t-C)$ whose right endpoint $r(t):= (\beta -c_0)t+C$ lies right to $h_0$. As $t$ increasing, both $h(t)$ and $r(t)$ move rightward, but $h(t)$ moves faster by Lemma \[lem:h(t)-(beta-c0)t=+infty\]. Hence $h(t)$ catches up $r(t)$ at some time $T$. We will show that at this moment $u>V^*$ near $x=h(T)$ and so $h'(T)\geqslant \beta -c_0$ (in fact, strict inequality holds by Hopf lemma). Since the shift $C$ of $V^*$ can be chosen continuously we indeed obtain $h'(t)>\beta -c_0$ for all large time $t$. Now we give the details of the proof. As in the proof of the previous lemma, there exists $X>h_0$ such that $u_0(x)$ intersects $V^*(x-M)$ at exactly two points for any $M\geqslant X$. By , there exists $T_X>0$ such that $h(t)-(\beta-c_0)t>X$ for all $t\geqslant T_X$. For any $a>h(T_X)$ denote $T_a$ the unique time such that $h(T_a)=a$. Set $X_a:=h(T_a)-(\beta-c_0)T_a\ (>X)$. We study the intersection points between $u(t,\cdot)$ and $V^*(x-(\beta-c_0)t-X_a)$. As in the proof of the previous lemma, only subcase 2-4 is possible: there exists $T^*>0$ such that $$\xi_1(t)<\xi_2(t)<h(t)\mbox{ for }0<t<T^*\quad \mbox{and}\quad \xi_1(T^*)<\xi_2(T^*)=h(T^*)=(\beta-c_0)T^*+X_a,$$ and as proving we have $$(\beta-c_0)t+X_a<h(t)\quad \mbox{for all }t>T^*.$$ Therefore $T^*$ is nothing but $T_a$. By we have $$h'(T_a)=-\mu u_x(T_a,h(T_a))=-\mu u_x(T_a,a)>\beta-c_0.$$ Since $a>h(T_X)$ is arbitrary, $T_a$ is continuous and strictly increasing in $a$, we indeed have $$h'(t)>\beta-c_0\quad \mbox{for all }t>T_X.$$ *Step 2*. We prove $$\label{h(t)-(beta-c_0+delta)t=-infty} \lim\limits_{t\to\infty}[h(t)-(\beta-c_0+\delta)t]=-\infty\mbox{ for all }\delta\in(0,c^*-\beta+c_0).$$ For any $\delta\in(0,c^*-\beta+c_0)$, we choose $\delta_1\in(0,\delta)$ and consider the compactly supported traveling wave $W_{\delta_1}(x-c_1t-M)$, where $c_1=\beta-c_0+\delta_1$, $M>0$ is a large real number such that $u_0(x)$ has no intersection point with $W_{\delta_1}(x-M)$. Clearly is proved if we have $h(t)<c_1t+M$ for all $t>0$. If, otherwise, there exists some $T_1>0$ such that $$h(t)<c_1t+M\mbox{ for }t\in[0,T_1),\ h(T_1)=c_1T_1+M,$$ then there exists $T_2\in(0,T_1)$ such that $h(t)$ catches up the left boundary $l_1(t):=c_1t+M-L_{\delta_1}$ of the support of $W_\delta(x-c_1t-M)$ at time $T_2$ and never lags behind it again. So in the time interval $(T_2,T_1)$. $$\mathcal{Z}_{J_1(t)}[\zeta_1(t,\cdot)]=1\mbox{ for }t\in[T_2,T_1],$$ where $J_1(t):=[l_1(t),h(t)]$ and $$\zeta_1(t,x):=u(t,x)-W_{\delta_1}(x-c_1t-M)\mbox{ for } x\in J_1(t),\ t\in[T_2,T_1].$$ By Lemma \[lem:zeros between u and Psi\], the unique zero $\zeta_1(t,\cdot)$ moves to $(h(t),0)$ at time $t=T_1$ and it disappears after $T_1$. Hence $$u(T_1,x)\geqslant W_{\delta_1}(x-c_1T_1-M)\mbox{ for }x\in[l_1(T_1),c_1T_1+M]=[l(T_1),h(T_1)].$$ This implies that virtual spreading happens for $u$ by Lemma \[lem:condition for virtual spreading\], contradicts our assumption. *Step 3*. Based on Step 2 we prove $$\label{h'(t)<beta-c_0+delta} h'(t)<\beta-c_0+\delta\mbox{ for large }t,$$ for any $\delta\in(0,c^*-\beta+c_0)$. Fix such a $\delta$, we consider $u(t,x)$ and $W_\delta(x-(\beta-c_0+\delta)t+h_0)$. It is easily seen that these two functions intersect at exactly one point in their common domain $J_2(t):=[g(t),r(t)]$ for small $t>0$, where $r(t):=(\beta-c_0+\delta)t-h_0$. By Step 2, there exists $T_3>0$ such that $$r(t)<h(t)\mbox{ for }t\in[0,T_3),\ r(T_3)=h(T_3).$$ If the left boundary $l_2(t):=r(t)-L_\delta$ of the support of $W_\delta(x-r(t))$ lags behind $g(t)$ till $t=T_3$: $l_2(t)<g(t)$ for $t\in[0,T_3)$, then $$u(T_3,x)\leqslant W_\delta(x-r(T_3))\mbox{ for }x\in[g(T_3),h(T_3)].$$ Using Hopf lemma at $h(T_3)$ we have $$\label{h'(T3)<beta-c0+delta} h'(T_3)=-\mu u_x(T_3,h(T_3))<-\mu W_\delta'(0)=\beta-c_0+\delta.$$ If there exists $T_4\in(0,T_3)$ such that $$l_2(t)<g(t)\mbox{ for }t\in[0,T_4),\ l_2(T_4)=g(T_4).$$ Then either $W_\delta(x-r(T_4))\leqslant u(T_4,x)$ in $[l_2(T_4),r(T_4)]$ or $\mathcal{Z}_{J_2(T_4)}[u(T_4,\cdot)-W_\delta(\cdot-r(T_4))]=2$ by the zero number arguments. In the former case, virtual spreading happens for $u$ by Lemma \[lem:condition for virtual spreading\], contradicts our assumption. In the latter case, we have $$\mathcal{Z}_{[l_2(t),r(t)]}[u(t,\cdot)-W_\delta(\cdot-r(t))]=2\mbox{ for }T_4\leqslant t\ll T_4+1.$$ In a similar way as in the proof of the previous lemma we see that the only possibility is that $r(t)$ catches up $h(t)$ at $t=T_3$, and the other intersection point between $u(T_3,\cdot)$ and $W_\delta(\cdot-r(T_3))$ stays on the left. Hence we have again at time $t=T_3$. Using a similar idea as in step 1 of the current proof, we obtain for all large time $t$. *Step 4*. Combining Step 1 with Step 3 we have $$\beta-c_0<h'(t)<\beta-c_0+\delta\mbox{ for large }t.$$ Since $\delta>0$ can be arbitrarily small, we proves . \[lem:max at right beta\] Under the assumption of Lemma \[lem:h’(t)=beta-c0\], $u(t,\cdot)$ has exactly one local maximum point for large $t$. Using zero number argument Lemma \[zero-number\] to $u_x(t,\cdot)$ we see that $u(t,\cdot)$ has exactly $N$ local maximum points for large $t$, where $N$ is a positive integer. If $N\geqslant2$, then by Lemma \[lem:max at right\] the leftmost maximum point $\xi_1(t)$ moves right at a speed not less than $\beta$. On the other hand, indicates $h(t)$ moves right at a speed $\beta-c_0$. Therefore, after some time, $\xi_1(t)$ reaches $h(t)$, this is a contradiction. \[thm:V-convergence\] Assume that vanishing and virtual spreading do not happen for the solution $u$ of . - If $c_0<\beta<\beta^*$, then $$\label{u converges to V} \lim\limits_{t\to\infty}\left\|u(t,\cdot)-V^*(\cdot-h(t))\right\|_{L^\infty(I(t))}=0;$$ - If $\beta=c_0$, then $$\label{u converges to 0} \lim\limits_{t\to\infty}\left\|u(t,\cdot)\right\|_{L^\infty(I(t))}=0;$$ 1\. We first prove the locally uniform convergence near $h(t)$. Set $w(t,x):=u(t,x+h(t))$ and $G(t):=g(t)-h(t)$ for $t\geqslant 0$. Then $$\label{w-G-H-problem} \left\{ \begin{array}{ll} w_t = w_{xx}-(\beta-h'(t))w_{x} +f(w), & t>0,\ G(t)< x<0,\\ w(t,G(t))=0,\ G'(t)=-\mu w_x(t,G(t))+\mu w_x(t,0) , & t>0,\\ w(t,0)=0,\ h'(t)=-\mu w_x(t,0), & t>0,\\ G(0)=-2h_0,\ \ w(0,x) =u_0 (x+h_0),& -2h_0\leqslant x \leqslant 0. \end{array} \right.$$ It is easy to know that $G_{\infty}:=\lim_{t\to\infty}G(t)=-\infty$. Since $w\in C^{1+\nu/2,2+\nu}([1,\infty)\times[G(t),0])$, $h\in C^{1+\nu/2}([1,\infty))$ for any $\nu\in(0,1)$ and $h'(t)\to \beta -c_0$ by Lemma \[lem:h’(t)=beta-c0\], there exists a sequence $\{t_n\}_{n=1}^\infty$ satisfying $t_n\to\infty$ as $n\to\infty$ such that $$w(t+t_n,x)\to v(t,x)\mbox{ as }n\to\infty\mbox{ locally uniformly in }(t,x)\in{\mathbb{R}}\times(-\infty,0],$$ and $v$ is a solution of $$\left\{ \begin{array}{ll} v_t = v_{xx}- c_0v_{x} +f(v), & t\in{\mathbb{R}},\ x<0,\\ v(t,0)=0,\ v_x(t,0)=-\frac{\beta-c_0}{\mu}, & t\in{\mathbb{R}}. \end{array} \right.$$ In case $\beta\in(c_0,\beta^*)$, we show that $v(t,x)\equiv V^*(x)$ for all $t\in{\mathbb{R}}$. If this is not true, then there exists $(t_0,x_0)\in{\mathbb{R}}\times(-\infty,0)$ such that $v(t_0,x_0)\neq V^*(x_0)$. Then for sufficiently small $\epsilon>0$, when $t\in(0,\epsilon)$ we have $v(t_0+t,x_0)\neq V^*(x_0)$. Using zero number result Lemma \[angenent\] for $\eta(t,x):=v(t_0+t,x)-V^*(x)$ in $(t,x)\in[0,\epsilon]\times[x_0,0]$, we see that $\mathcal{Z}_{[x_0,0]}[\eta(t,\cdot)]<\infty$ for $t\in(0,\epsilon)$, and it decreases strictly once it has a degenerate point in $[x_0,0]$. This contradicts the fact that $x=0$ is a degenerate zero of $\eta(t,\cdot)$ for all $t\in (0,\epsilon)$. Therefore, $v(t,x)\equiv V^*(x)$, and so $w(t+t_n,x)\to V^*(x)$ as $n\to \infty$ locally uniformly in $(t,x)\in {\mathbb{R}}\times (-\infty, 0]$. By the uniqueness of $V^*(x)$ we actually proves $u(t,\cdot+h(t))=w(t,\cdot) \to V^*(\cdot)$ as $t\to \infty$ uniformly in $[-M, 0]$ for any $M>0$. In case $\beta =c_0$, a similar discussion as above shows that $v(t,x)\equiv 0$ and so $u(t,\cdot+h(t)) \to 0$ as $t\to \infty$ uniformly in $[-M, 0]$ for any $M>0$. 2\. We prove the uniform convergence in $I(t)$ in case $c_0<\beta<\beta^*$. For any small $\epsilon>0$, there exists a large $M>0$ such that $$V^*(x)\leqslant V^*(-M)\leqslant \frac{\epsilon}{3}\mbox{ for }x\leqslant -M.$$ Taking $T>0$ sufficiently large, by Step 1 we have $$\label{u(+h(t))-V*<epsilon/3} G(t)<-M,\quad \|u(t,\cdot+h(t))-V^*(\cdot)\|_{L^\infty([-M,0])}<\frac{\epsilon}{3}\mbox{ for }t\geqslant T.$$ Hence, the function $u(t,\cdot+h(t))$ has a maximum point in $[-M,0]$. It is the unique maximum point by Lemma \[lem:max at right beta\]. Hence $u(t,\cdot+h(t))$ is increasing in $ [G(t),-M]$, and so $$0\leqslant u(t,x+h(t))\leqslant u(t,h(t)-M)\leqslant V^*(-M)+\frac{\epsilon}{3}\leqslant \frac{2\epsilon}{3} \quad \mbox{for } x\in [G(t),-M],\ t\geqslant T.$$ This implies that $$\|u(t,\cdot+h(t))-V^*(\cdot)\|_{L^\infty([G(t),-M])}\leqslant \epsilon\mbox{ for }t\geqslant T.$$ Combining with we proves . 3\. We now prove in case $\beta=c_0$. By Lemma \[lem:max at right beta\], $u(t,\cdot)$ has exactly one maximum point $\xi(t)$ when $t$ is large, say, when $t\geqslant T$ for some $T>0$. There are three cases: [*Case 1*]{}. $u(t,\xi(t))\to 0$ as $t\to \infty$; [*Case 2*]{}. $u(t,\xi(t))\to 1$ as $t\to \infty$; [*Case 3*]{}. There exist $d\in (0,1)$ and a sequence $\{t_n\}_{n=1}^\infty \subset [T, \infty)$ with $t_n \to \infty$ such that $u(t_n,\xi(t_n))=d$ for $n=1,2,\cdots$. The limit in follows from Case 1 immediately. We now derive contradictions for Case 2 and Case 3. [*Case 2*]{}. By Lemma \[lem:compact tw\], there exists $\delta_1 \in (0, c^*(\beta))$ such that the equation in has a compactly supported traveling wave $W_{\delta_1} (x-{\delta_1} t)$ with $$\label{com tw} W_{\delta_1} (0)= W_{\delta_1} (-L_{\delta_1}) =0,\quad D_{\delta_1} := \max\limits_{-L_{\delta_1} \leqslant z\leqslant 0} W_{\delta_1} (z) =\frac12 \ \ \mbox{ and }\ \ {\delta_1} =-\mu W'_{\delta_1} (0).$$ By Lemmas \[lem:beta&gt;c0 u to 0\] and \[lem:center\], $u(t,\cdot)\to 0$ as $t\to \infty$ uniformly in $[g(t), 2L_{\delta_1}]$, by the result in step 1 above, $u(t,\cdot)\to 0$ as $t\to \infty$ uniformly in $[h(t)-2L_{\delta_1}, h(t)]$. Hence we may assume that, for some $T_1 >T$, $$2L_{\delta_1} <\xi(t)<h(t)-2L_{\delta_1} \mbox{ and } u(t,\xi(t))>D_{\delta_1} =\frac12 \mbox{ for all } t\geqslant T_1.$$ Now we consider the traveling wave $w_1(t,x) :=W_{\delta_1} (x-{\delta_1} t +{\delta_1} T_1 -g_\infty)$. Clearly, when $t=T_1$ it has no contact point with $u(T_1, x)$. Since it moves rightward with speed ${\delta_1} >0$ and since $h'(t)\to 0$, the right endpoint $r_1(t):= {\delta_1} t -{\delta_1} T_1 +g_\infty$ of $w_1$ reaches $x=h(t)$ after some time. Before that, $r_1(t)$ first meets $g(t)$ at time $T_2 >T_1$, and then its left endpoint $l_1(t) := r_1 (t)-L_{\delta_1}$ meets $g(t)$ at time $T_3 >T_2$. By the zero number argument, for $t\in [T_2, T_3)$ we have $\mathcal{Z}_{[g(t),r_1(t)]} [w_1 (t,\cdot) - u(t,\cdot)] =1$, and for $T_3 <t\ll T_3 +1$, either $$\label{w1 < u} w_1 (t,x) < u(t,x)\quad \mbox{for } x\in [l_1 (t), r_1(t)],$$ or, $\mathcal{Z}_{[l_1(t) , r_1(t)]} [w_1 (t,\cdot) - u(t,\cdot)] =2$. In the latter case, the two contact points between $w_1$ and $u$ can not remain and move across $x=\xi(t)$ where $u(t,\xi(t))>\frac12 \geqslant w_1(t,\xi(t))$. Therefore, before $w_1(t,x)$ moves into the interval $[h(t)-L_{\delta_1}, h(t)]$, the two contact points disappear at some time $T_4 >T_3$, and so holds for $t=T_4$. Once holds at some time, it holds for all larger time since $w_1 $ is a lower solution of . This leads to virtual spreading for $u$ by Lemma \[lem:condition for virtual spreading\], a contradiction. [*Case 3*]{}. As above we select a compactly supported traveling wave $W_{{\delta_2}} (x-{\delta_2} t)$ for some ${\delta_2}\in (0,c^*(\beta))$ such that $$\label{com tw2} W_{{\delta_2}} (0)= W_{{\delta_2}} (-L_{{\delta_2}}) =0,\quad D_{{\delta_2}} := \max\limits_{-L_{{\delta_2}} \leqslant z\leqslant 0} W_{{\delta_2}} (z)= W_{{\delta_2}} ( -\tilde{z} )= d \ \ \mbox{ and }\ \ {\delta_2} =-\mu W'_{{\delta_2}} (0),$$ where $-\tilde{z}\in (-L_{{\delta_2}}, 0)$ is the maximum point of $W_{{\delta_2}}(z)$. By the locally uniform convergence in the above step 1 and in Lemma \[lem:beta&gt;c0 u to 0\], there exists $n_0 $ such that $$\label{case 3} 2L_{{\delta_2}} <\xi(t_n)< h(t_n)-2L_{{\delta_2}} \mbox{ for all } n\geqslant n_0.$$ Since $\xi(t_n) -{\delta_2} t_n < h(t_n) -{\delta_2}t_n \to -\infty $ as $n\to \infty$, there exists $n_1 >n_0$ such that $$C:= \xi(t_{n_1}) -{\delta_2} t_{n_1} + {\delta_2} t_{n_0} +\tilde{z} \leqslant g_\infty.$$ Now we consider the traveling wave $w_2(t,x):=W_{{\delta_2}} (x-{\delta_2} t +{\delta_2} t_{n_0} -C)$ for $t\geqslant t_{n_0}$. Since $w_2(t_{n_0},x) =W_{{\delta_2}} (x-C)$, $w_2 (t_{n_0},\cdot)$ has no contact point with $u(t_{n_0},x)$. Since $w_2$ moves rightward with speed ${\delta_2} >0$ and since $h'(t)\to 0$, the right endpoint $r_2 (t):= {\delta_2} t - {\delta_2} t_{n_0} +C$ of $w_2$ reaches $x=h(t)$ after some time. Before that, $r_2 (t)$ first meets $g(t)$ at some time $T_5 > t_{n_0}$, and then the left endpoint $l_2(t):= r_2 (t)- L_{{\delta_2}}$ of $w_2$ meets $g(t)$ at some time $T_6 >T_5$. We remark that $T_6 <t_{n_1}$. In fact, by we have $$r_2 (t_{n_1}) ={\delta_2} t_{n_1} -{\delta_2} t_{n_0} +C =\xi(t_{n_1}) +\tilde{z} >2L_{{\delta_2}} > g(T_6) +L_{{\delta_2}} = r_2 (T_6).$$ Now, for $t\in [T_5, T_6)$, using the zero number argument we have $\mathcal{Z}_{[g(t),r_2 (t)]} [w_2 (t,\cdot) - u(t,\cdot)] =1$. For $T_6 <t\ll T_6 +1$, we have either $$\label{w2 < u} w_2 (t,x) \leqslant u(t,x)\quad \mbox{for } x\in [l_2 (t), r_2 (t)],$$ or, $\mathcal{Z}_{[l_2 (t), r_2 (t)]} [w_2 (t,\cdot) - u(t,\cdot)] =2$. can not be true, since it implies virtual spreading for $u$ by Lemma \[lem:condition for virtual spreading\]. In case $w_2(t,\cdot)-u(t,\cdot)$ has two zeros for $T_6 <t\ll T_6 +1$, by the zero number argument, the two zeros unite to be one degenerate zero $\xi(t_{n_1})$ at time $t_{n_1}$ (note that $\xi (t_{n_1})$ is the maximum point of both $w_2(t_{n_1}, \cdot)$ and $u(t_{n_1}, \cdot)$). So after $t_{n_1}$, $w_2$ and $u$ have no contact points. This implies that $w_2 (t,x) <u(t,x)$ ($w_2 > u$ is impossible since the support of $u$ is wider than that of $w_2$). This again leads to virtual spreading for $u$ by Lemma \[lem:condition for virtual spreading\], a contradiction. This proves Theorem \[thm:V-convergence\]. \[rem:h to beta-c0\] By Lemma \[lem:h’(t)=beta-c0\] we have $h(t)=(\beta-c_0)t+\varrho(t)$ for some $\varrho(t)=o(t)$. Hence the uniform convergence in can be rewritten as . ### Proofs of Theorems \[thm:middle beta\] and \[thm:beta=c0\] In the last of this subsection we prove Theorem \[thm:middle beta\] and Theorem \[thm:beta=c0\]. Remember we use $(u(t,x;\sigma\phi), g(t;\sigma\phi), h(t;\sigma\phi))$ to denote the solution of with initial data $u_0 =\sigma \phi$ for some given $\phi\in \mathscr{X}(h_0)$. Define $E_0$ and $\sigma_*$ as in , and when $c_0 \leqslant \beta <\beta^*$, denote $$E_1:=\{\sigma> 0 \mid \mbox{virtual spreading happens for } (u, g, h)\},\quad \sigma^*:=\inf E_1.$$ By the comparison principle we have $[\sigma,\infty)\subset E_1$ if $\sigma\in E_1$. Thus $(\sigma^*,\infty)\subset E_1$. [**Proof of Theorem \[thm:middle beta\]:**]{} If $\sigma_*=\infty$, then there is nothing left to prove. We assume $\sigma_*\in(0,\infty)$ in the following. We first prove $\sigma_*=\sigma^*$. Otherwise, $\sigma_*<\sigma^*$, and so there exist $\sigma_1,\ \sigma_2\in(\sigma_*,\sigma^*)$ with $\sigma_1<\sigma_2$. By the strong comparison principle we have $$g(t;\sigma_1 \phi)>g(t;\sigma_2 \phi),\quad h(t;\sigma_1 \phi)<h(t;\sigma_2 \phi)$$ and $$u(t,x;\sigma_1\phi)<u(t,x;\sigma_2\phi) \ \mbox{ for } x\in I^{\sigma_1} (t):=[g(t;\sigma_1\phi),h(t;\sigma_1\phi)],\ t>0.$$ Since these inequalities are strict at $t=1$, there exists $\epsilon>0$ small such that $$u(1,x;\sigma_1\phi)<u(1,x-\epsilon;\sigma_2\phi)\mbox{ for }x\in I^{\sigma_1}(1).$$ By the comparison principle again we have $$u(t,x;\sigma_1\phi)<u(t,x-\epsilon;\sigma_2\phi) \mbox{ for }x\in I^{\sigma_1}(t),\ t\geqslant 1.$$ And so $$\label{h1 < h2} u(t,x+h(t;\sigma_1\phi);\sigma_1\phi)<u(t,x+h(t;\sigma_1\phi)-\epsilon;\sigma_2\phi) \mbox{ for }x\in[g(t;\sigma_1\phi)-h(t;\sigma_1\phi),0],\ t\geqslant 1.$$ By Theorem \[thm:V-convergence\] (i), both $u(t,x+h(t;\sigma_1\phi);\sigma_1\phi)$ and $u(t,x+h(t;\sigma_2\phi);\sigma_2\phi)$ converge to the tadpole-like function $V^*(x)$ uniformly. Taking limits as $t\to \infty$ in we deduce a contradiction by $h(t;\sigma_1\phi)-\epsilon-h(t;\sigma_2\phi)\leqslant -\epsilon$. This proves $\sigma_*=\sigma^*$. It is easily shown as in the proof of Theorem \[thm:sigma\_\*\] that $E_0\setminus \{0\}$ is open, and $E_1$ is open by Lemma \[lem:condition for virtual spreading\], so neither vanishing nor virtual spreading happens for $(u(t,x;\sigma\phi)$, $g(t;\sigma\phi)$,$h(t;;\sigma\phi))$ with $\sigma=\sigma^*$. Thus $u(t,x; \sigma^*\phi)$ is a transition solution and it converges to $V^*$ as in Theorem \[thm:V-convergence\] and Remark \[rem:h to beta-c0\]. Other conclusions in Theorem \[thm:middle beta\] follow from the previous lemmas and theorems. [$\Box$]{} [**Proof of Theorem \[thm:beta=c0\]:**]{} If $\sigma_*=\infty$, then there is nothing left to prove. If $\sigma_*\in(0,\infty)$ and $\sigma^*=\infty$, then vanishing happens for $u(t,x;\sigma\phi)$ with $\sigma<\sigma_*$, and virtual vanishing happens for $u(t,x;\sigma\phi)$ with $\sigma\geqslant \sigma_*$. Finally we consider the case $0<\sigma_*\leqslant \sigma^*<\infty$. We show that $E_1$ is an open set. Indeed, if $\sigma_1\in E_1$, then for any $\delta\in(0,c^*(\beta))$ there exists $T_1>0$, $x_1\in{\mathbb{R}}$ such that $$u(T_1,x;\sigma_1\phi)>W_\delta(x-x_1)\mbox{ for }x\in[x_1-L_\delta,x_1],$$ since $u(T_1,\cdot;\sigma_1\phi)$ depends on $\sigma_1$ continuously, there exists $\epsilon>0$ such that $$u(T_1,x;\sigma\phi)>W_\delta(x-x_1)\mbox{ for }x\in[x_1-L_\delta,x_1],$$ for any $\sigma\in[\sigma_1-\epsilon,\sigma_1+\epsilon]$. By Lemma \[lem:condition for virtual spreading\], virtual spreading happens for $(u(t,x;\sigma\phi)$, $g(t;\sigma\phi)$,$h(t;\sigma\phi))$. Hence $E_1$ is an open set, and so $E_1=(\sigma^*,\infty)$. This proves the theorem. [$\Box$]{} Uniform convergence when (virtual) spreading happens ==================================================== In the main results Theorems \[thm:small beta\], \[thm:middle beta\] and \[thm:beta=c0\], we observe (virtual) spreading phenomena, which is the case where the solution converges to $1$ locally uniformly in a fixed or moving coordinate frame. In this section we consider the asymptotic profiles for such solutions in the whole domain. Throughout this section we assume $0<\beta<\beta^*$. Locally uniform convergence of the front ---------------------------------------- We first describe the asymptotic profile near the front $x=h(t)$. In a similar way as [@DMZ; @KM; @LL2] one can show that \[prop:asymptotic profile near x=h(t)\] Assume $0<\beta<\beta^*$. If (virtual) spreading happens for a solution of , then there exists $H_\infty\in{\mathbb{R}}$ such that $$\label{h(t)-c*t converges to H} \lim\limits_{t\to\infty}[h(t)-c^*t]=H_\infty,\ \lim\limits_{t\to\infty}h'(t)=c^*,$$ $$\label{u(t,x+h(t)) converges to U*(x)} \lim\limits_{t\to\infty}u(t,\cdot+h(t))=U^*(\cdot)\mbox{ locally uniformly in }(-\infty,0].$$ For small advection: $0<\beta<c_0$, one can give a uniform convergence for the solution $(u,g,h)$ of as in [@DMZ; @KM; @LL2]. \[prop:profile when beta is small\] Assume $0<\beta<c_0$. If spreading happens for a solution $(u,g,h)$ of , then there exist $G_\infty,\ H_\infty\in{\mathbb{R}}$ such that holds and $$\lim\limits_{t\to\infty}[g(t)-c_l^*t] = G_\infty ,\ \lim\limits_{t\to\infty}g'(t)=c_l^*,$$ $$\lim\limits_{t\to\infty}\|u(t,\cdot)-U^*(\cdot-c^*t-H_\infty)\cdot U_l^*(\cdot-c_l^*t-G_\infty)\|_{L^{\infty}([g(t),h(t)])}=0,$$ if we extend $U^*$ and $U^*_l$ to be zero outside their supports. Locally uniform convergence of the back --------------------------------------- In this subsection we show that, when $c_0 \leqslant \beta<\beta^*$, the back of a virtual spreading solution $u$ converges to a traveling wave $Q$ locally uniformly. We will use the following definition: \[def:steeper\] Let $u_1$, $u_2$ be two entire solutions of $u_t=u_{xx}-\beta u_x+f(u)$ satisfying $u_{1x}(t,x)>0$ and $u_{2x}(t,x)>0$ for all $x\in{\mathbb{R}}, t\in{\mathbb{R}}$. We say that $u_1$ is **steeper than** $u_2$ if for any $t_1$, $t_2$ and $x_1$ in ${\mathbb{R}}$ such that $u_1(t_1,x_1)=u_2(t_2, x_1)$, we have either $$u_1(\cdot+t_1,\cdot)\equiv u_2(\cdot+t_2,\cdot)\mbox{ or } (u_1)_x(t_1,x_1)>(u_2)_x(t_2,x_1).$$ As above, $u_1$ and $u_2$ are called entire solutions since they are defined for all $t\in{\mathbb{R}}$. The above property implies that the graph of the solution $u_1$ (at any chosen time moment $t_1$) and that of the solution $u_2$ (at any chosen time moment $t_2$) can intersect at most once unless they are identical, and that if they intersect at a single point, then $u_1 -u_2$ is negative on the left-hand side of the intersection point, while positive on the right-hand side. \[thm:left limit\] Assume $\beta \in [c_0, \beta^*)$. If virtual spreading happens for a solution $(u,g,h)$ of , then there exists a continuous function $\theta(t)$ with $\theta(t)=o(t)$ and $\theta(t) \to\infty\ (t\to \infty)$ such that for any $M>0$, $$\label{left limit} \lim\limits_{t\to\infty}\|u(t,\cdot)-Q(\cdot-(\beta-c_0)t -\theta(t))\|_{L^\infty([g(t),(\beta-c_0)t +\theta(t)+M])}=0.$$ The proof is long and is divided into several steps. We will use $C$ and $T$ to denote positive constants which may be different case by case. *Step 1*. [*A rough estimate for the speed of the back*]{}. For any $\delta_1 \in (0,c^*(\beta)-\beta +c_0)$, we consider the compactly supported traveling wave $W_{\delta_1}(x-c_1 t)$ with $c_1=\beta-c_0+\delta_1 $, where $W_{\delta_1}(z):=W(z;c_1,-c_0+\delta_1 )$ is the solution of and with $c=c_1$, whose support is $[-L_{\delta_1},0]=[-L(c_1,-c_0+\delta_1),0]$. As in Lemma \[lem:compact tw\] we denote $D_{\delta_1}:=\max\limits_{-L_{\delta_1}\leqslant z\leqslant0} W_{\delta_1}(z)$. Write $$\label{def:m} m:= \frac12 \inf\{D_{\delta} \mid 0< \delta <c^*(\beta) -\beta +c_0\}.$$ Then $m \in (0,1)$ by the phase plane analysis. By our assumption, virtual spreading happens: $u(t,\cdot+c t)\to1$ locally uniformly in ${\mathbb{R}}$ for some $c>0$. Hence for any given $\delta_1 \in (0,c^*-\beta +c_0)$ there exist a large $T_0$ and $r\in{\mathbb{R}}$ such that $$u(T_0, x)\geqslant W_{\delta_1}(x-r)\quad \mbox{for }x\in[r-L_{\delta_1},r].$$ By comparison we have $$u(t+T_0, x)> W_{\delta_1}(x-c_1t-r)\quad \mbox{for }x\in[r+c_1t-L_{\delta_1},r+c_1t],\ t>0.$$ Therefore $\chi(t):=\min \{x\in[g(t),h(t)]\big|u(t,x)=m \}$ satisfies $$\label{chi(t)<=c1t+r+chi0} \chi(t+T_0) < c_1 t +r\quad \mbox{for }t>0.$$ Combining with we have $$\label{<=chi(t)<=} (\beta-c_0)t+\frac{3}{c_0}\ln t-C\leqslant\chi(t)\leqslant(\beta-c_0+\delta_1)t+C,\quad t\gg1.$$ *Step 2*. [*Truncation of the solution*]{}. Instead of $u$ we will consider its truncation on $[g(t),\xi(t)]$ for some $\xi(t)\in(g(t),h(t))$. Let $\epsilon \in (0,\frac12 (1-m))$ be any given small constant. We define $\xi(t)$ as a position near $h(t)$ where $u$ takes value $1-\epsilon$. More precisely, by the definition of the rightward traveling semi-wave $U^*(x-c^*(\beta)t)$, there exists $M_1 = M_1(\epsilon)>0$ sufficiently large such that $U^*(-M_1)>1-\frac{\epsilon}{2}$. By , $u(t,h(t)-M_1)> 1- \epsilon$ for sufficiently large $t$, and so there exists $\xi(t)\in[h(t)-M_1, h(t)]$ such that $$\label{u(t,xi(t))=1-epsilon} u(t,\xi(t))=1-\epsilon,\quad \mbox{for large } t.$$ By and we have $$\label{xi-chi to infty} \xi(t)-\chi(t)\geqslant h(t) -M_1 -\chi(t) \geqslant (c^* -\beta +c_0 -\delta_1) t +O(1)\to\infty ,\quad \mbox{as } t\to\infty.$$ Since the convergence in holds in fact in $C^2_{\rm loc}((-\infty,0])$ topology by parabolic estimate, it follows from $U^*_x(x)<0$ that $$\label{ux<0} u_x(t,x)<0 \mbox{ for } x\in [h(t)-2M_1, h(t)] \mbox{ and } t\gg 1.$$ So the leftmost local maximum point $\xi_1(t)$ of $u(t,\cdot)$ satisfies $\xi_1(t)< h(t) -2M_1 < \xi(t)$ for $t\gg 1$. We now show that $$\label{u>1-epsilon} u(t,x)\geqslant 1-\epsilon \mbox{ for } x\in [\xi_1(t),\xi(t)],\ t\gg 1.$$ In case $u(t,\cdot)$ has exactly one local maximum point $\xi_1(t)$ for large $t$, holds since $u(t,\cdot)$ is decreasing in $[\xi_1(t),\xi(t)]$ and $u(t,\xi(t))=1-\epsilon$. We now consider the case that $u(t,\cdot)$ has exactly $N\ (\geqslant2)$ local maximum points $\{\xi_i(t)\}_{i=1}^N$ with $g(t)<\xi_1(t)<\cdots <\xi_N(t)<h(t)$ for large $t$. We remark that this case is possible only if $\beta<c^*(\beta)$. In fact, when $\beta\geqslant c^*(\beta)$, by and we have $$0<h(t)-\xi_1(t)<(c^*(\beta)-\beta)t+C\leqslant C,\quad t\gg1.$$ This contradicts the locally uniform convergence and the fact that $U^*$ is a strictly decreasing function. So, in the following, we assume that $$\label{beta<c*} \beta<c^*(\beta)\quad \mbox{and}\quad \xi_1(t)\geqslant \beta t -C \mbox{ for some } C>0.$$ Choose a small $\delta \in (0, c_0)$ and consider the solution $q(z):= W(z;b,-\delta)$ of with $\gamma=-\delta$, where $b\in(0,P(-\delta))$. This solution corresponds to a point $G\in S_1$ as in Figure 2 (a), and its trajectory is a curve like $\Gamma_2$ in Figure 1 (a). When $b\to P(-\delta)$, the trajectory $\Gamma_2\to\Gamma_1$ in Figure 1 (a). As in subsection 3.2, for the above given $\epsilon>0$ and $\delta\in(0,c_0)$, there exists $b=b(\epsilon,\delta)\in(0,P(-\delta))$ such that $\mathscr{W}(z):=W(z;b(\epsilon,\delta),-\delta)$ and $\mathcal{L}=L(b(\epsilon,\delta),-\delta)$ satisfy $$\mathscr{W}(0)=\mathscr{W}(-\mathcal{L})=0,\quad \mathscr{W}(z)>0\mbox{ for }z\in(-\mathcal{L},0),$$ $$\mathscr{W}(\hat{z})=\max\limits_{-\mathcal{L}\leqslant z\leqslant0} \mathscr{W}(z)= 1- \epsilon \quad\mbox{for some }\hat{z}\in(-\mathcal{L},0),$$ We prove by contradiction. By we only need to prove for $x\in [\xi_1(t), h(t)-2M_1]$. Assume that there exist a time sequence $\{t_n\}_{n=1}^\infty$ and a sequence $\{y_n\}_{n=1}^\infty$ with $t_n\to \infty$ and $y_n \in [\xi_1(t_n), h(t_n)-2M_1]$ such that $$\label{un<1-epsilon} u(t_n, y_n) <1-\epsilon \mbox{ for all } n.$$ For each $n$, we define a continuous function of $\tau$ by $$\psi_n (\tau) := h(\tau) + (\beta -\delta) (t_n -\tau) - y_n -M_1 -\hat{z}.$$ It is easily seen that $$\psi_n (t_n) \geqslant h(t_n) -(h(t_n)-2M_1) -M_1 - \hat{z}>0.$$ For $\rho \in (0,1)$, by $y_n \geqslant \xi_1(t_n) \geqslant \beta t_n -C$ we have $$\begin{aligned} \psi_n (\rho t_n) & = & h(\rho t_n) + (\beta -\delta)(1-\rho) t_n - y_n +O(1)\\ \ & \leqslant & [c^* \rho - (\beta-\delta) \rho -\delta] t_n +O(1) \to -\infty\mbox{ as } n\to \infty,\end{aligned}$$ provided $\rho >0$ is sufficiently small. Hence for such a $\rho $ and for any large $n$, there exists $\tau_n \in (\rho t_n, t_n)$ such that $\psi_n (\tau_n ) =0$. For any large $n$, by we have $$u(\tau_n, x) \geqslant U^* (x- h(\tau_n)) -\frac{\epsilon}{2} \geqslant \mathscr{W}(x-h(\tau_n)+M_1),\quad x\in [h(\tau_n) -M_1 -\mathcal{L}, h(\tau_n)-M_1].$$ Set $r_n(t):= (\beta-\delta)t + h(\tau_n)-M_1$. Since $\mathscr{W}(x-r_n(t))$ is a compactly supported traveling wave of $_1$, and its right endpoint $$r_n(t) = (\beta -\delta)t + c^* \tau_n +H_\infty -M_1 + o(1) < h(t+\tau_n) =c^* t +c^* \tau_n + H_\infty +o(1)$$ by and , provided $n$ is sufficiently large. Hence $\mathscr{W}(x- r_n(t)) $ is a lower solution of and by the comparison principle we have $$u(t+\tau_n, x) \geqslant \mathscr{W} (x-r_n(t) ) \ \mbox{ for } x\in [r_n(t)- \mathcal{L} , r_n(t)],\ t>0.$$ In particular, at $t= t_n -\tau_n >0$ and $x=y_n$, by $\psi_n (\tau_n)=0$ we have $$1-\epsilon > u(t_n,y_n) \geqslant \mathscr{W} (-\hat{z}) = 1-\epsilon,$$ a contradiction. This proves . In what follows, we write $\hat{u}(t,x):=u(t,x)\big|_{x\in[g(t),\xi(t)]}$ as a truncation of $u$. *Step 3*. [*Truncation of tadpole-like traveling waves*]{}. For any $b\in(0,P(-c_0))$ recall that $V(z;b,-c_0)$ is a tadpole-like solution of (cf. point $H$ in Figure 2 (a)). We choose $b=b(\epsilon)$ near $P(-c_0)$ such that $$\max\limits_{z\leqslant 0}V(z;b(\epsilon),-c_0) =V(\bar{z};b(\epsilon),-c_0)=1-2\epsilon,$$ for some $\bar{z}<0$. In a similar way as above, we write $$\widehat{V}(x-(\beta-c_0)t):=V(x-(\beta-c_0)t;b(\epsilon),-c_0)\big|_{x\in(-\infty,\bar{z}+(\beta-c_0)t]}$$ as a truncation $V$. *Step 4*. [*Comparison between $\hat{u}$ and $\widehat{V}$*]{}. To study the asymptotic profile of $\hat{u}$, we compare $\hat{u}$ with a family of the shifts of $\widehat{V}$. Without loss of generality, we may assume $u(t,x)$ satisfies all the properties in steps 1-3 from time $t=0$. Since $\hat{u}_x(0,g(0))>0$, one can choose $X>0$ large such that $\hat{u}(0,x)$ and $\widehat{V}(x-\hat{x})$ (for any $\hat{x}\geqslant X$) intersect at exactly one point $\hat{y}$, and $$\hat{u}(0,x)<\widehat{V}(x-\hat{x})\mbox{ for }x\in[g(0),\hat{y}),\ \hat{u}(0,x)>\widehat{V}(x-\hat{x})\mbox{ for }x\in(\hat{y},\min\{\xi(0),\bar{z}+\hat{x}\}).$$ Since the back of $u(t,\cdot)$ moves rightward faster than $(\beta-c_0)t+\frac{3}{c_0}\ln t-C$ by , it will exceed $\widehat{V} (x-(\beta-c_0)t-\hat{x})$ at some time $\hat{T}>0$, that is, their intersection point $(y(t),\hat{u}(t,y(t)))$ starting from $(\hat{y}, \hat{u}(0,\hat{y}))$ exists only in time interval $[0,\hat{T}]$. For each $\hat{x}\geqslant X$, both $\hat{u}(t,x)$ and $\widehat{V}(x-(\beta-c_0)t-\hat{x})$ are solutions of $_1$. We now compare them and show that for $t\in [0,\hat{T})$, $$\label{hat{u} < and > widehat{V}} \left\{ \begin{array}{l} \mbox{there exists }y(t)\in(g(t),\xi(t))\cap(-\infty,\eta(t)]\mbox{ such that }\\ \hat{u}(t,x)<\widehat{V}(x-(\beta-c_0)t-\hat{x})\mbox{ for }x\in[g(t),y(t)), \\ \hat{u}(t,x)>\widehat{V}(x-(\beta-c_0)t-\hat{x})\mbox{ for }x\in(y(t),\min\{\xi(t),\eta(t)\}\big]. \end{array}\right.$$ where $\eta(t):= (\beta -c_0)t +\hat{x} +\bar{z}$. By the comparison principle, this is true provide we exclude the following two possibilities: \(A) the right endpoint $(\xi(t),1-\epsilon)$ of $\hat{u}(t,\cdot)$ touches $\widehat{V}$ at some time $t\in (0,\hat{T})$; \(B) the right endpoint $(\eta(t), 1- 2\epsilon )$ of $\widehat{V} (x-(\beta-c_0)t-\hat{x})$ touches $\hat{u}$ at some time $t\in (0,\hat{T})$. \(A) of course is impossible because $\hat{u}(t,\cdot)$ takes value $1-\epsilon$ at $x=\xi(t)$, bigger than $\max \widehat{V}$. (B) is impossible when $\eta(t) \in [\xi_1(t), \xi(t)]$ since in this case $\hat{u}(t,\eta(t))\geqslant 1-\epsilon >\max \widehat{V}$ by . When $\eta(t)<\xi_1(t)$, $\widehat{V}_x(x-(\beta-c_0)t-\hat{x})\big|_{x=\eta(t)}=\widehat{V}_x(\bar{z})=0$ and $\hat{u}_x(t,\eta(t))>0$. Hence $(\eta(t),1-2\epsilon )$ can not be a new emerging intersection point between $\hat{u}$ and $\widehat{V}$. This excludes the possibility of (B). *Step 5*. [*Slope of the back of $u(t,\cdot)$*]{}. By and by the Hopf lemma, at the unique intersection point $(y(t),\hat{u}(t,y(t)))$ between $\hat{u}$ and $\widehat{V}$ we have $$\hat{u}(t,y(t))=\widehat{V} (y(t)-(\beta-c_0)t-\hat{x})\quad\mbox{and}\quad\hat{u}_x(t,y(t))>\widehat{V} _x(y(t)-(\beta-c_0)t-\hat{x}).$$ Denote $y^0$ the unique root of $\widehat{V}(z)=m$ in $(-\infty,\bar{z})$. For any given large $t$, we take $\hat{x}= \chi(t)-(\beta -c_0)t -y^0$, then the function $\hat{u}(t,x)$ and $\widehat{V}(x-(\beta-c_0)t-\hat{x})$ intersect exactly at $x=\chi(t)$: $$\hat{u}(t,\chi (t))=m= \widehat{V}(y^0) = \widehat{V}(\chi (t)-(\beta-c_0)t-\hat{x}).$$ By the Hopf lemma we have $$\label{u steeper than V} \hat{u}_x(t, \chi (t))>\widehat{V}_x(y^0).$$ *Step 6*. [*Convergence of the back of $u$ and the slope of the limit function*]{}. For any increasing sequence $\{t_n\}_{n=0}^{\infty}$ with $t_n\to\infty \ (n\to \infty)$, we set $x_n:= \chi (t_n)$ and define $$\hat{u}_n(t,x):=\hat{u}(t+t_n,x+x_n)\mbox{ for }g(t+t_n)-x_n\leqslant x\leqslant \xi(t+t_n)-x_n,\ t>-t_n.$$ Clearly, $\hat{u}_n(0,0)=\hat{u}(t_n,x_n)=m$ for $n\in\mathbb{N}$. For any given $t\in{\mathbb{R}}$, $g(t+t_n)-x_n\to-\infty$ as $n\to\infty$ and by and we have $$\begin{aligned} \xi(t+t_n)-x_n & = &\xi(t+t_n)-\xi(t_n)+ [\xi(t_n)-\chi(t_n)]\\ & \geqslant & h(t+t_n) - M_1 -h(t_n) + [\xi(t_n)-\chi(t_n)]\\ & = & c^* t +O(1) + [\xi(t_n)-\chi(t_n)] \to\infty\mbox{ as }n\to\infty.\end{aligned}$$ Since $\hat{u}_n(t,x)$ is bounded in $L^\infty$ norm, by parabolic estimate, it is also bounded in $C^{1+\nu/2,2+\nu}$ $([-M,M]\times[-M,M])$ norm for any $M>0$ and any $\nu\in(0,1)$. By Cantor’s diagonal argument, there exists a subsequence $\{n_j\}$ of $\{n\}$ such that $$\lim\limits_{j\to\infty}\hat{u}_{n_j}(t,x)=w(t,x)\mbox{ in }C^{1,2}_{\rm loc}({\mathbb{R}}^2)\mbox{ topology},$$ where $w\in C^{1,2}({\mathbb{R}}^2)$ is an entire solution of $_1$ with $w(0,0)=m$. By we have $$\label{w steeper than V} w_x(0,0)= \lim\limits_{j\to\infty}(\hat{u}_{n_j})_x (0,0) = \lim\limits_{j\to\infty} \hat{u}_x (t_{n_j}, x_{n_j}) =\lim\limits_{j\to\infty} \hat{u}_x (t_{n_j}, \chi (t_{n_j})) \geqslant \widehat{V}_x(y^0).$$ For the solution $Q(z)$ of - with $c=\beta-c_0$, there exists a unique $y^*\in{\mathbb{R}}$ such that $Q(y^*)= m$. By the phase plane analysis (Lemma \[lem:tadpole tw bata&lt;beta\*\] (i)), $V(\cdot+y^0 ;b(\epsilon),-c_0)= \widehat{V}(\cdot+y^0) \to Q(\cdot+y^*)$ in $C_{\rm loc}^2({\mathbb{R}})$ topology, as $\epsilon\to0$, or equivalently, as $b(\epsilon)\to P(-c_0)$. Taking limit as $\epsilon\to0$ in we have $$\label{w steeper than Q} w_x(0,0)\geqslant Q'(y^*).$$ On the other hand, both $u_1(t,x):= Q(x-(\beta-c_0)t+y^*)$ and $u_2(t,x):= w(t,x)$ are entire solutions of $_1$. By [@DGM Lemma 2.8], $u_1$ is steeper than $u_2$ in the sense of Definition \[def:steeper\]. In particular, taking $t_1 =t_2 =x_1 =0$ in Definition \[def:steeper\] we have $u_1(0,0)=Q(y^*)=m=w(0,0)=u_2(0,0)$. Hence $$w(t,x)\equiv Q(x-(\beta-c_0)t+y^*)\quad\mbox{for all }t,\ x\in{\mathbb{R}}$$ by Definition \[def:steeper\] and the inequality . Therefore, $\lim\limits_{j\to\infty}\hat{u}_{n_j}(t,x)= Q(x-(\beta-c_0)t+y^*)$ in $C^{1,2}_{\rm loc}({\mathbb{R}}^2)$ topology. By the uniqueness of the limit function $Q$ we have $$\lim\limits_{n\to\infty}u_n(t,x)=\lim\limits_{n\to\infty}u(t+t_n,x+\chi(t_n)) =Q(x-(\beta-c_0)t+y^*)\mbox{ in }C^{1,2}_{\rm loc}({\mathbb{R}}^2)\mbox{ topology}.$$ Since $\{t_n\}$ is an arbitrarily chosen sequence we obtain $$\lim\limits_{\tau\to\infty}u(t+\tau,x+\chi(\tau))=Q(x-(\beta-c_0)t +y^*)\mbox{ in }C^{1,2}_{\rm loc}({\mathbb{R}}^2)\mbox{ topology}.$$ Taking $t=0$ we have $$\label{u to Q} \lim\limits_{\tau\to\infty}u(\tau,x+\chi(\tau))=Q(x+y^*)\mbox{ in }C^2_{\rm loc}({\mathbb{R}})\mbox{ topology}.$$ Define $$\theta(\tau):= \chi(\tau) - (\beta-c_0) \tau -y^*,$$ which is a continuous function of $\tau$. Then by we have $$\frac{3}{c_0}\ln \tau -C \leqslant \theta (\tau) \leqslant \delta_1 \tau +C,\quad \tau \gg 1.$$ Since this is true for any small $\delta_1 >0$ (see Step 1) we have $\theta(\tau)= o(\tau)\ (\tau\to \infty)$. Thus by we have $$\label{u to Q 2} \lim\limits_{\tau\to\infty} [u(\tau,x) - Q(x-(\beta-c_0)\tau-\theta(\tau))]=0$$ uniformly in $[(\beta-c_0)\tau +\theta(\tau)-M, (\beta-c_0)\tau + \theta(\tau)+M]$ for any $M>0$. . For any $\varepsilon_0>0$, there exists $M>0$ such that $Q(z)\leqslant Q(-M)\leqslant \varepsilon_0$ for $z<-M$. For this $M$, we choose $T>0$ large such that when $t>T$ we have $$u(t, (\beta-c_0)t +\theta(t)-M )\leqslant 2 Q(-M ) \leqslant 2\varepsilon_0$$ by . $u(t,\cdot)$ is increasing in $[g(t), (\beta -c_0)t+\theta(t)-M]$ by Lemma \[lem:max at right\], hence, when $t>T$ and $x\in [g(t), (\beta -c_0)t+\theta(t)-M]$ we have $$|u(t,x)-Q(x-(\beta-c_0)t-\theta(t))| \leqslant u(t,(\beta-c_0)t +\theta(t)-M )+ Q(-M) \leqslant 3\varepsilon_0.$$ Combining with we obtain the conclusion . Uniform convergence ------------------- In this subsection, we complete the proof of Theorem \[thm:profile of spreading sol\]. [**Proof of Theorem \[thm:profile of spreading sol\]:**]{} , and are proved in Propositions \[prop:asymptotic profile near x=h(t)\] and \[prop:profile when beta is small\]. We only need to prove for $c_0\leqslant \beta<\beta^*$. For any given small $\varepsilon >0$, we will prove $$\label{convergence 2 0} |u(t,x)-U^*(x-c^* t -H_\infty) \cdot Q(x-(\beta -c_0)t-\theta(t))| < C\varepsilon \quad \mbox{ for } x\in I(t) \mbox{ and large } t,$$ where $C>0$ is a constant independent of $t$ and $x$. By Lemma \[lem:compact tw\], for any $\delta \in (0, c^*(\beta)-\beta +c_0)$ the problem has a compactly supported traveling wave $W_\delta (x-(\beta-c_0+\delta)t)$, where $W_\delta (z)$ (with support $[-L_\delta, 0]$) is the unique solution of the problem and , whose maximum and maximum point are denoted by $D_\delta$ and $-z_\delta$, respectively. Moreover, for the above given $\varepsilon >0$, Lemma \[lem:compact tw\] also indicates that, there exists $\delta_\varepsilon \in (0, c^*(\beta)-\beta +c_0)$ such that $$D_\delta = W_\delta (-z_\delta) \in (1-\varepsilon, 1) \mbox{ when } \delta\in (\delta_\varepsilon, c^*(\beta) -\beta +c_0).$$ We select $\delta_1, \delta_0, \delta_2 \in (\delta_\varepsilon, c^*(\beta)-\beta +c_0 )$ with $\delta_1 >\delta_0 >\delta_2$ and fix them. For $i=1$ and $2$, denote $\kappa_i = (1-D_{\delta_i})/(3\varepsilon)$, then $\kappa_i \in (0, \frac13)$. By the definitions of $U^*(z)$ and $Q(z)$, there exists $M(\varepsilon) >0$ such that when $M>M (\varepsilon)$, $$\label{U* Q =1} 1-\varepsilon \leqslant U^*(z) \leqslant 1\mbox{ for } z\leqslant -M,\qquad 1-\varepsilon \leqslant Q(z) \leqslant 1 \mbox{ for } z\geqslant M$$ and there exists $M(\delta_1, \delta_2)>M(\varepsilon)$ such that when $M>M(\delta_1, \delta_2)$, $$\label{UQ > Ddelta} Q(z) > D_{\delta_1} +\kappa_1 \varepsilon \mbox{ for } z\in [M-L_{\delta_1}, M],\qquad U^*(z) > D_{\delta_2} +\kappa_2 \varepsilon \mbox{ for } z\in [-2M, -2M+L_{\delta_2}].$$ In what follows we fix an $M>M(\delta_1, \delta_2)>M(\varepsilon)$. Since the solution $\eta (t)$ of the problem $$\eta_t = f(\eta),\quad \eta(0)=1+\|u_0\|_{L^\infty}$$ is an upper solution of and since $\eta(t)\to 1$ as $t\to \infty$, there exists a time $T_1 =T_1 (\varepsilon) >0$ such that $$\label{u<1+ep} u(t,x) < 1+\varepsilon \quad \mbox{ for } x\in I(t),\ t>T_1.$$ By Theorem \[thm:left limit\], there exists $T_2 >T_1$ such that when $t>T_2$ we have $$\label{u-Q small} |u(t,x)-Q(x-(\beta-c_0)t-\theta(t))| < \kappa_1 \varepsilon \mbox{ for } x\in I_l (t) := [g(t),(\beta-c_0)t+\theta(t)+M],$$ where $\theta(t)$ is a continuous positive function with $\theta(t)=o(t)$ and $\theta(t)\to \infty\ (t\to \infty)$. By Proposition \[prop:asymptotic profile near x=h(t)\], there exists $T_3 >T_2$ such that when $t>T_3$ we have $$\label{u-U small} |u(t,x)-U^*(x-c^*t-H_\infty)|<\kappa_2 \varepsilon \mbox{ for } x\in I_r (t):=[h(t)-2M,h(t)],$$ (we extend $U^* (z)$ to be zero for $z>0$ if necessary). We now prove $$\label{u not small middle} u(t,x) \geqslant 1- \varepsilon\mbox{ for } x\in I_c (t):= [(\beta-c_0)t + \theta(t)+M, h(t)-2M] \mbox{ and large } t.$$ Once this is proved, combining it with the above results we obtain with $C=3$. In the following we prove by contradiction. Assume that there exist a time sequence $\{t_n\}_{n=1}^\infty$ with $t_n \to \infty$ and a sequence $\{y_n\}$ with $y_n\in I_c (t_n)$ for each $n$ such that $$\label{u(t,yn)<1-2ep} u(t_n,y_n) < 1-\varepsilon\quad\mbox{for all } n.$$ We divide the interval $I_c(t)$ into $I^1_c (t)$ and $I^2_c (t)$, where $$I^1_c (t):= [(\beta-c_0)t+\theta(t)+M, (\beta-c_0+\delta_0)t],\quad I^2_c (t):= [(\beta-c_0+\delta_0)t, h(t)-2M].$$ Our idea to derive contradictions is the following. We put a compactly supported traveling wave $W_{\delta_1}(x-(\beta-c_0+\delta_1)t+C_1)$ (resp. $W_{\delta_2}(x-(\beta-c_0+\delta_2)t+C_2)$) under $u(t+\tau, x)$ at time $t=0$ in the interval $I_l(\tau)$ (resp. $I_r(\tau)$), and then as $t$ increases to $t_n -\tau$, its maximum point exactly reaches $y_n\in I^1_c (t_n)$ (resp. $y_n\in I^2_c (t_n)$), this leads to a contradiction. First we consider the case that $\{y_n\}$ has a subsequence (denoted it again by $\{y_n\}$) such that $y_n\in I^1_c (t_n)$ for each $n$. Define a continuous function of $\tau$: $$\psi_n^{(1)}(\tau) := \delta_1 \tau -\theta (\tau) + y_n -(\beta-c_0 +\delta_1)t_n - M +z_{\delta_1} \mbox{ for } \tau\leqslant t_n.$$ Since $y_n \in I^1_c (t_n)$, it is easily seen that $$\psi_n^{(1)}(t_n) = y_n -(\beta -c_0)t_n -\theta (t_n) -M +z_{\delta_1} \geqslant z_{\delta_1} >0,$$ and for $\rho_1 = \frac12 (1-\frac{\delta_0}{\delta_1})\in (0,1)$ we have $$\psi_n^{(1)}(\rho_1 t_n) \leqslant ( \rho_1 \delta_1 +\delta_0 -\delta_1 ) t_n + O(1) \to -\infty\mbox{ as } n\to \infty.$$ Hence, when $n$ is sufficiently large, there exists $\tau_n \in (\rho_1 t_n, t_n)$ such that $\psi_n^{(1)} (\tau_n ) =0$. For such a large $n$, by and we have $$u(\tau_n, x) \geqslant Q(x-(\beta-c_0)\tau_n -\theta(\tau_n)) -\kappa_1 \varepsilon \geqslant D_{\delta_1} \geqslant W_{\delta_1} (x-X) \quad \mbox{ for } x\in [X-L_{\delta_1}, X],$$ where $X:= (\beta -c_0)\tau_n +\theta(\tau_n) +M$. Using comparison principle we have $$u(t+\tau_n, x) \geqslant W_{\delta_1} (x-(\beta -c_0 +\delta_1)t -X) \mbox{ for } x\in J_1(t),\ t>0,$$ where $J_1(t) := [(\beta-c_0+\delta_1)t +X-L_{\delta_1}, (\beta-c_0+\delta_1)t +X]$. By $\psi_n^{(1)}(\tau_n)=0$ we have $$y_n =(\beta -c_0 +\delta_1) (t_n -\tau_n) +X -z_{\delta_1} \in J_1 (t_n -\tau_n).$$ Hence by taking $t=t_n -\tau_n$ and $x=y_n$ we have $$1-\varepsilon > u(t_n,y_n) \geqslant W_{\delta_1} (y_n -(\beta-c_0+\delta_1)(t_n -\tau_n) -X) =W_{\delta_1} (-z_{\delta_1}) = D_{\delta_1} > 1-\varepsilon,$$ a contradiction. Next we consider the case that $\{y_n\}$ has a subsequence (denoted it again by $\{y_n\}$) such that $y_n\in I^2_c (t_n)$ for each $n$. The proof is similar as above. Define a continuous function $$\psi_n^{(2)}(\tau) := h(\tau) - (\beta -c_0 +\delta_2) \tau - 2M + L_{\delta_2} -y_n +(\beta -c_0+\delta_2) t_n -z_{\delta_2} \mbox{ for } \tau\leqslant t_n.$$ Since $y_n \in I^2_c (t_n)$, it is easily seen that $$\psi_n^{(2)}(t_n) = h(t_n)- 2M +L_{\delta_2} - y_n - z_{\delta_2} \geqslant L_{\delta_2} - z_{\delta_2} >0,$$ and for $\rho_2 \in (0,1)$ we have $$\begin{aligned} \psi_n^{(2)}(\rho_2 t_n) & \leqslant & h(\rho_2 t_n) -(\beta -c_0 +\delta_2) \rho_2 t_n -y_n +(\beta -c_0+\delta_2) t_n +O(1) \\ & = & c^*(\beta) \rho_2 t_n + (\beta -c_0 +\delta_2) (1-\rho_2) t_n -y_n +O(1) \\ & \leqslant & [-\rho_2 (\beta -c_0 +\delta_2) +c^*(\beta) \rho_2 - \delta_0 +\delta_2] t_n + O(1) .\end{aligned}$$ Since $\delta_0 > \delta_2$, the coefficient of $t_n$ in the last line is negative when $\rho_2 >0$ is sufficiently small. Hence for such a $\rho_2$, $\psi_n^{(2)}(\rho_2 t_n)\to -\infty$ as $n\to \infty$. Consequently, for any large $n$, there exists $\tau'_n \in (\rho_2 t_n, t_n)$ such that $\psi_n^{(2)} (\tau'_n) =0$. By and we have $$u(\tau'_n, x) \geqslant U^* (x-c^* \tau'_n -H_\infty) -\kappa_2 \varepsilon > D_{\delta_2} \geqslant W_{\delta_2} (x-X') \quad \mbox{ for } x\in [X'-L_{\delta_2}, X'],$$ where $X':= h(\tau'_n) -2M +L_{\delta_2}$. By the comparison principle we have $$u(t+\tau'_n, x) \geqslant W_{\delta_2} (x-(\beta -c_0 +\delta_2)t -X') \mbox{ for } x\in J_2(t),\ t>0,$$ where $J_2(t) := [(\beta-c_0+\delta_2)t +X'-L_{\delta_2}, (\beta-c_0+\delta_2)t +X']$. By $\psi_n^{(2)}(\tau'_n)=0$ we have $$y_n =(\beta -c_0 +\delta_2) (t_n -\tau'_n) +X' -z_{\delta_2} \in J_2 (t_n -\tau'_n).$$ Hence at $t=t_n -\tau'_n$ and $x=y_n$ we have $$1-\varepsilon > u(t_n,y_n) \geqslant W_{\delta_2} (y_n -(\beta-c_0+\delta_2)(t_n -\tau'_n) -X') =W_{\delta_2} (-z_{\delta_2}) = D_{\delta_2} > 1-\varepsilon,$$ a contradiction. This completes the proof of Theorem \[thm:profile of spreading sol\]. [$\Box$]{} [123456]{} S.B. Angenent, [*The zero set of a solution of a parabolic equation*]{}, J. Reine Angew. Math., 390 (1988), 79-96. D.G. Aronson and H.F. Weinberger, [*Multidimensional nonlinear diffusions arising in population genetics*]{}, Adv. Math., 30 (1978), 33-76. I.E. Averill, [*The Effect of Intermediate Advection on Two Competing Species*]{}, Doctor of Philosophy, Ohio State University, Mathematics, 2012. M. Ballyk and H. Smith, [*A model of microbial growth in a plug flow reactor with wall attachment*]{}, Math. Biosci., 158 (1999), 95-126. F. Belgacem and C. Cosner, [*The effect of dispersal along environmental gradients on the dynamics of populations in heterogeneous environment*]{}, Can. Appl. Math. Quart., 3 (1995), 379-397. H. Berestycki and F. Hamel, [*Front propagation in periodic excitable media*]{}, Comm. Pure Appl. Math., 55 (2002), no.8, 949-1032. G. Bunting, Y. Du, and K. Krakowski, [*Spreading speed revisited: analysis of a free boundary model*]{}, Netw. Heterog. Media 7(4) (2012), 583-603. J. Byers and J. Pringle, [*Going against the flow: retention, range limits and invasions in advective environments*]{}, Mar. Ecol. Prog. Ser., 313 (2006), 27-41. X. Chen and A. Friedman, [*A free boundary problem arising in a model of wound healing*]{}, SIAM J. Math. Anal., 32 (2001), 778-800. Y. Du and Z.M. Guo, [*The Stefan problem for the Fisher-KPP equation*]{}, J. Diff. Eqns., 253 (2012), 996-1035. Y. Du and Z.M. Guo, [*Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary II*]{}, J. Diff. Eqns., 250 (2011), 4336-4366. Y. Du and Z. Lin, [*Spreading-vanishing dichtomy in the diffusive logistic model with a free boundary*]{}, SIAM J. Math. Anal., 42 (2010), 377–405. Y. Du and B. Lou, [*Spreading and vanishing in nonlinear diffusion problems with free boundaries*]{}, J. Eur. Math. Soc., to appear. (arXiv:1301.5373) Y. Du, B. Lou and M. Zhou, [*Nonlinear diffusion problems with free boundaries: convergence and transition speed*]{}, preprint. Y. Du and H. Matano, [*Convergence and sharp thresholds for propagation in nonlinear diffusion problems*]{}, J. Eur. Math. Soc., 12 (2010), 279-312. Y. Du, H. Matano and K. Wang, [*Regularity and asymptotic behavior of nonlinear Stefan problems*]{}, Arch. Rational Mech. Anal., 212 (2014), 957-1010. Y. Du, H. Matsuzawa and M. Zhou, [*Sharp estimate of the spreading speed determined by nonlinear free boundary problems*]{}, SIAM J. Math. Anal., 46 (2014), 375-396. A. Ducrot, T. Giletti and H. Matano, [*Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations*]{}, Trans. Amer. Math. Soc., 366 (2014), 5541-5566. F.J. Fernandez, Unique continuation for parabolic operators. II, [*Comm. Part. Diff. Eqns.*]{} 28 (2003), 1597-1604. H. Gu, [*On a non-monotone traveling semi-wave of the Fisher-KPP equation with advection*]{}, preprint. H. Gu and X. Liu, [*Long time behavior of solutions of a reaction-advection-diffusion equation with free and Robin boundary conditions*]{}, in preparation. H. Gu and B. Lou, [*On the Allen-Cahn equation with advection and free boundaries*]{}, in preparation. H. Gu, Z. Lin and B. Lou, [*Long time behavior of solutions of a diffusion-advection logistic model with free boundaries*]{}, Appl. Math. Letters, 37 (2014), 49-53. H. Gu, Z. Lin and B. Lou, [*Different asymptotic spreading speeds induced by advection in a diffusion problem with free boundaries*]{}, Proc. Amer. Math. Soc., to appear. (arXiv:1302.6345) F. Hamel, J. Nolen, J. Roquejoffre, and L. Ryzhik, [*A short proof of the logarithmic Bramson correction in Fisher-KPP equations*]{}, Netw. Heterog. Media, 8 (2013), 275-289. D. Hilhorst, M. Iida, M. Mimura and H. Ninomiya, [*A competition-diffusion system approximation to the classical two-phase stefan problem*]{}, Japan J. Indust. Appl. Math., 18 (2001), 161-180. D. Hilhorst, M. Mimura and R. Schätzle, [*Vanishing latent heat limit in a Stefan-like problem arising in biology*]{}, Nonlinear Anal. RWA, 4 (2003), 261-285 S.B. Hsu and Y. Lou, [*Single phytoplankton species growth with light and advection in a water column*]{}, SIMA J. Appl. Math. 70 (2010), 2942-2974. J. Huisman, M. Arrayas, U. Ebert, and B. Sommeijer, [*How do sinking phytoplankton species manage to persist?*]{}, Amer. Naturalist, 159 (2002), 245-254. Y. Kanako and H. Matsuzawa, [*Spreading speed and sharp asymptotic profiles of solutions in free boundary problems for reaction-advection-diffusion equations* ]{}, preprint. Y. Kaneko and Y. Yamada, [*A free boundary problem for a reaction-diffusion equation appearing in ecology*]{}, Adv. Math. Sci. Appl. 21 (2011) 467-492. X. Liu and B. Lou, [*Asymptotic behavior of solutions to diffusion problems with Robin and free boundary conditions*]{}, Math. Model. Nat. Phenom., 8 (2013), 18-32. X. Liu and B. Lou, [*Long time behavior of solutions of a reaction-diffusion equation with Robin and free boundary conditions*]{}, preprint. N. A. Maidana, H. Yang, [*Spatial spreading of West Nile Virus described by traveling waves*]{}, J. Theoretical Bio., 258 (2009), 403-417. I.G. Petrovski, [*Ordinary Differential Equations*]{}, Prentice-Hall, Englewood Cliffs, New Jersey, 1966, Dover, New York, 1973. A. Potapov and M. Lewis, [*Climate and competition: the effect of moving range boundaries on habitat invasibility*]{}, Bull. Math. Biol., 66(5) (2004), 975-1008. G.A. Riley, H. Stommel, and D.F. Bumpus, [*Quantitative ecology of the plankton of the western North Atlantic*]{}, Bulletin of the Bingham Oceanographic Collection Yale University 12 (1949), 1-169. L.I. Rubinstein, [*The Stefan Problem*]{}, American Mathematical Society, Providence, RI, 1971. N. Shigesada and A. Okubo, [*Analysis of the self-shading effect on algal vertical distribution in natural waters*]{}, J. Math. Biol., 12 (1981), 311-326. D.C. Speirs and W.S.C. Gurney, [*Population persistence in rivers and estuaries*]{}, Ecology 8(5) (2001), 1219-1237. O. Vasilyeva and F. Lutscher, [*Population dynamics in rivers: analysis of steady states*]{}, Can. Appl. Math. Q., 18(4) (2010), 439-469. M.X. Wang, [*The diffusive logistic equation with a free boundary and sign-changing coefficient*]{}, J. Diff. Eqns., (258) 2015, 1252-1266. P. Zhou and D.M. Xiao, [*The diffusive logistic model with a free boundary in heterogeneous environment*]{}, J. Diff. Eqns., 256 (2014), 1927-1954. [^1]: $\S$ This research was partly supported by NSFC (No. 11271285). [^2]: $\dag$ Department of Mathematics, Tongji University, Shanghai 200092, China. [^3]: $\ddag$ Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo 153-8914, Japan. [^4]: [**Emails:**]{} [honggu87@126.com]{} (H. Gu), [blou@tongji.edu.cn]{} (B. Lou), [zhouml@ms.u-tokyo.ac.jp]{} (M. Zhou)

--- author: - | \ Department of Physics,\ Columbia University,\ New York, NY   10027\ E-mail: - For the RBC and UKQCD Collaborations title: NLO and NNLO chiral fits for 2+1 flavor DWF ensembles --- Introduction ============ The RBC and UKQCD Collaborations have been generating 2+1 flavor domain wall fermion (DWF) QCD ensembles over the last few years. Extensive results have been published from the first ensemble, which had two dynamical quark masses, a variety of valence quark masses, $1/a = 1.72(2)$ GeV and a spatial volume of $(2.75 \; {\rm fm})^3$ [@24cubed; @24cubed-bk; @24cubed-kl3]. The largest sources of systematic errors in these results are the $O(a^2)$ errors and errors from extrapolating from our simulation light quark masses to the physical light quark masses. We estimated both of these to be about 4% for $f_\pi$, for example. We now have a second ensemble at a smaller lattice spacing, which allows us to extrapolate to $a = 0 $, assuming that our data is in the region where the errors are of $O(a^2)$. The second ensemble also has lighter dynamical quarks and allows us to probe the reliability of the chiral extrapolation for our data. This report details our analysis of the combined data from both ensembles, including chiral and continuum extrapolations. The details of our ensembles are given in Table \[ensembles\]. For the results from our $1/a = 1.72$ GeV ensembles given in [@24cubed], measurements were made on ensembles of length 3600 MD time units (after thermalization). We have generated more lattices, so that for the $1/a = 1.72$ GeV ensemble with light quark mass $m_l = 0.005$, our results come from 8080 MD time units and for the heavier light quark mass, $m_l = 0.01$, 7180 time units. This more than doubles the measurements from our earlier work. For the $1/a = 2.32 $ GeV ensemble, we have 6100 MD time units, after thermalization, for $m_l = 0.004$, 6220 for $m_l = 0.006$, and 5020 for $m_l = 0.008$. The analysis we present here uses our measurements of light-light pseudoscalar masses (pions), strange-light pseudoscalar masses (kaons) and the mass of the $\Omega$ baryon, $m_\Omega$ to set the lattice scale and determine the physical light and strange quark masses (we assume $m_u = m_d$ throughout). As inputs, we take the known values for $m_\pi$, $m_K$ and $m_\Omega$. We have also measured the light-light and light-strange pseudoscalar decay constants, and predictions for these are an output of our analysis and a check on our systematic errors. A major focus of our analysis is how well full NLO and NNLO chiral perturbation theory formulae fit our data and whether the apparent convergence of the series and estimates of the size of neglected terms are consistent with general theoretical estimates and known values for $f_\pi$ and $f_K$. [ccccc]{} Ensemble & $(m_l^{\rm lat}, \widetilde{m}_l)$ & $m_\pi$ (MeV) & $(m_{\rm val}^{\rm lat}, \widetilde{m}_{\rm val})$ & $m_\pi^{\rm val}$ (MeV)\ ---------------------------------- $1/a = 1.72(2)$ GeV $24^3 \times 64 \times 16$ $(2.75 \; {\rm fm})^3$ $m_h^{\rm lat}= 0.04$ $\widetilde{m}_h= 116.5$ MeV $m_{\rm res}^{\rm lat}= 0.00315$ $m_{\rm res} = 8.5$ MeV ---------------------------------- : \[ensembles\]Run parameters for the simulations presented here. Quark and pion masses are given in MeV. Quark masses are reported in $\overline{\rm MS}(2\; {\rm GeV})$, which uses the lattice spacings we determine from our analysis and a separate NPR measurement of the quark mass renormalization factors. The notation is as in [@24cubed] – in particular, quark masses with a tilde are total quark masses. & ------------------- (0.005, 22.5 MeV) (0.01, 35.5 MeV) ------------------- : \[ensembles\]Run parameters for the simulations presented here. Quark and pion masses are given in MeV. Quark masses are reported in $\overline{\rm MS}(2\; {\rm GeV})$, which uses the lattice spacings we determine from our analysis and a separate NPR measurement of the quark mass renormalization factors. The notation is as in [@24cubed] – in particular, quark masses with a tilde are total quark masses. & ----- 328 417 ----- : \[ensembles\]Run parameters for the simulations presented here. Quark and pion masses are given in MeV. Quark masses are reported in $\overline{\rm MS}(2\; {\rm GeV})$, which uses the lattice spacings we determine from our analysis and a separate NPR measurement of the quark mass renormalization factors. The notation is as in [@24cubed] – in particular, quark masses with a tilde are total quark masses. & -------------------- (0.001, 11.2 MeV) (0.005, 22.0 MeV) (0.010, 35.5 MeV) (0.020, 62.5 MeV) (0.030, 89.5 MeV) (0.040, 116.5 MeV) -------------------- : \[ensembles\]Run parameters for the simulations presented here. Quark and pion masses are given in MeV. Quark masses are reported in $\overline{\rm MS}(2\; {\rm GeV})$, which uses the lattice spacings we determine from our analysis and a separate NPR measurement of the quark mass renormalization factors. The notation is as in [@24cubed] – in particular, quark masses with a tilde are total quark masses. & [c]{} 239\ 328\ 417\ \ \ \ ----------------------------------- $1/a = 2.32(3)$ GeV $32^3 \times 64 \times 16$ $(2.72 \; {\rm fm})^3$ $m_h^{\rm lat}= 0.03$ $\widetilde{m}_h= 113.1$ MeV $m_{\rm res}^{\rm lat}= 0.000664$ $m_{\rm res} = 2.45$ MeV ----------------------------------- : \[ensembles\]Run parameters for the simulations presented here. Quark and pion masses are given in MeV. Quark masses are reported in $\overline{\rm MS}(2\; {\rm GeV})$, which uses the lattice spacings we determine from our analysis and a separate NPR measurement of the quark mass renormalization factors. The notation is as in [@24cubed] – in particular, quark masses with a tilde are total quark masses. & -------------------- (0.004, 17.2 MeV ) (0.006, 24.6 MeV ) (0.008, 32.0 MeV ) -------------------- : \[ensembles\]Run parameters for the simulations presented here. Quark and pion masses are given in MeV. Quark masses are reported in $\overline{\rm MS}(2\; {\rm GeV})$, which uses the lattice spacings we determine from our analysis and a separate NPR measurement of the quark mass renormalization factors. The notation is as in [@24cubed] – in particular, quark masses with a tilde are total quark masses. & ----- 295 350 397 ----- : \[ensembles\]Run parameters for the simulations presented here. Quark and pion masses are given in MeV. Quark masses are reported in $\overline{\rm MS}(2\; {\rm GeV})$, which uses the lattice spacings we determine from our analysis and a separate NPR measurement of the quark mass renormalization factors. The notation is as in [@24cubed] – in particular, quark masses with a tilde are total quark masses. & -------------------- (0.002, 9.83 MeV) (0.004, 17.2 MeV) (0.006, 24.6 MeV) (0.008, 32.0 MeV) (0.025, 94.7 MeV) (0.030, 113.1 MeV) -------------------- : \[ensembles\]Run parameters for the simulations presented here. Quark and pion masses are given in MeV. Quark masses are reported in $\overline{\rm MS}(2\; {\rm GeV})$, which uses the lattice spacings we determine from our analysis and a separate NPR measurement of the quark mass renormalization factors. The notation is as in [@24cubed] – in particular, quark masses with a tilde are total quark masses. & [c]{} 226\ 295\ 350\ 397\ \ \ Observables, reweighting and global fits ======================================== For light-light and heavy-light pseudoscalars, we use Coulomb gauge fixed wall source propagators. We fit the propagators starting at 10 lattice spacings from the wall source and find no dependence on this choice of fit range. The plateaus are very long for these states - we fit over 44 lattice spacings for both ensembles. Coulomb gauge fixed box sources are used for the $\Omega$, with a box size of $16^3$ for the $1/a = 1.72$ GeV ensemble and $20^3$ for the 2.32 GeV ensemble. These sources also give very good signals, and we fit over a range of 7 lattice spacings. We find statistical errors on the pseudoscalar masses and decay constants in the range of 0.2-0.5%, and errors on $m_\Omega$ of 0.2-0.7%. With these measurements, we then fit our data to SU(2) chiral perturbation theory (ChPT) formulae. With SU(2), we do not assume that $m_s$ (or alternatively $m_K$) is small. We do need $m_l \ll m_s$. For the light-strange sector, we use SU(2) for kaons, as we did in [@24cubed]. In our SU(2) fits, we include $O(a^2)$ corrections to leading order LEC’s, and neglect any $a^2$ dependence for NLO, or higher, LEC’s. The residual mass effects of DWF are taken into account by including the contribution of $m_{\rm res}$ in the total quark mass. Of course, there are many ways to write down SU(2) ChPT formula at NLO, since any rearrangement of the series that only changes it at NNLO is equally valid. Since we have no [*a priori*]{} reason to know which particular ordering is the most convergent for a particular quantity, and it may seem unlikely that any one reordering will be optimal for all quantities, we use the series as an expansion in $f$, the chiral limit value for the pion decay constant. The pseudoscalar masses that enter at NLO and higher are just $m^2 = 2 B m_q$, the leading order expressions. This view of the series also readily extends to the full, continuum NNLO forms, as given in [@bijnens]. Since the total lattice quark mass enters into the ChPT expressions and we are working at two different lattice spacings, we need a renormalization factor to relate bare quarks at one lattice spacing to the other. We have gotten this ratio three different ways: 1) from NPR calculations at the two different lattice spacings, 2) from matching the two lattice spacings at unphysical quark masses [@kelly] and 3) from a global fit where the ratio is a free parameter and it is fit for in an overall $\chi^2$ minimization step. All three methods agree within their errors. A last issue in our global fits is the difference between the dynamical heavy quark mass in the simulations, $m_h$, and the physical strange quark mass, $m_s$. When the simulations are run, the value of $m_h$ to use is not known. Only after a complete analysis of the data does one gain knowledge about the correct value of $m_h$. In SU(2) ChPT, unlike SU(3), all LEC’s are implicit functions of $m_h$ and cannot be extrapolated to the physical value. (For valence heavy quark mass dependence, it is easy to interpolate between the values for a hadron mass that are measured with different valence quark masses to a self-consistently determined strange quark mass.) By reweighting our observables from the simulated $m_h$ to the desired $m_s$, we can remove any systematic error from the (generally mild) dependence on $m_h$. The left graph in Figure \[fig:fpi\_rw\] shows the unitary values for $f_\pi$ as a function of the reweighted dynamical heavy quark mass for three of our ensembles. The right graph shows the ratio of the reweighted value of $f_\pi$ to the unreweighted value for the $1/a = 2.32 $ GeV, 0.004/0.03 ensemble. Four stochastic estimators per mass step were used here and studying the reweighting dependence versus the number of stochastic hits indicates that four are sufficient. We see a clear signal for a small dependence on the strange quark mass. More details of this procedure are also given in [@chulwoo]. Three different global fits, using the above procedure have been performed by members of our collaboration and the agreement for physical quantities is very good. Two types of fits use results from matching the lattices at unphysical quark masses to constrain the ratio of quark renormalization factors and/or lattice spacings. A third fit self-consistently fits the data, determining the ratio of lattice spacings, quark mass renormalizations and LECs that give the best $\chi^2$. We do uncorrelated fits to our data, since we find our covariance matrices are very singular, due to the data being strongly correlated. As shown in [@dawson] in such a case an uncorrelated fit gives the correct answer, but the quoted $\chi^2$ is not a reliable goodness-of-fit indicator. The left graph of Figure \[fig:chi-sq\] shows a histogram of the uncorrelated $\chi^2$ for our fits, which involve 125 data points and about 20 parameters, depending on precisely which fit was done. All of our partially quenched data for light-light and heavy-light pseudoscalars and the $\Omega$ are fit simultaneously, under an outer jackknife loop, which produces the errors. The fit is to NLO order in ChPT, including finite volume effects. The physical values for $m_\pi$, $m_K$ and $m_\Omega$ are used to determine the lattice scale and quark masses. From NPR [@yaoki], we use a value of $Z_m = 1.590$ for the $1/a = 2.32$ GeV ensemble to convert lattice quark masses into continuum masses renormalized in $\overline{\rm MS}(2\; {\rm GeV})$. The histogram shows that the fits are in good agreement with the error bars on each data point. The right graph of Figure \[fig:chi-sq\] shows the comparison between the NLO ChPT fit results and the data for pions made of degenerate quarks. Curvature consistent with the expected chiral logarithms is seen. Figure \[fig:fpi\_unitary\] shows our results for the unitary light-light pseudoscalar decay constant and our fits. We find $f_\pi = 122.2 \pm 3.4_{\rm stat} \; {\rm MeV}$. The right graph shows the LO and NLO contribution to the fit and one sees that in the region where we have data, the NLO contribution is a 20-30% correction to the LO result. From this one would expect an NNLO error of order $0.2^2$ to $0.3^2$ or $O(4-9\%)$. One sees that our prediction for $f_\pi$ is low by roughly this amount. We also estimate a similar ChPT systematic effect by looking at simple, analytic fits to our data [@kelly]. We quote a preliminary ChPT systematic error given by the square of the ratio of NLO to LO in the lightest quark mass region where we have data. [*A priori*]{}, one has little information about the size of NNLO ChPT contributions. If the series is reasonably convergent when the next order is added, the NNLO terms should be roughly the square of the NLO contributions. Of course ChPT differs from a renormalizable field theory in that many new LEC’s enter at the next order, as well as contributions from LEC’s at the current order times logarithms. For DWF, where we have continuum chiral symmetries at finite lattice spacing, we can fit our data to the continuum NNLO SU(2) formulae, to help address these questions. ![\[fig:fpi\_rw\] The left graph shows the reweighted value for $f_\pi$ for some of our ensembles. The right graph is the ratio of $f_\pi$, reweighted to the quark mass given on the horizontal axis, to the unreweighted $f_\pi$.](plot_reweighted_fpi.pdf "fig:"){width="40.00000%"} ![\[fig:fpi\_rw\] The left graph shows the reweighted value for $f_\pi$ for some of our ensembles. The right graph is the ratio of $f_\pi$, reweighted to the quark mass given on the horizontal axis, to the unreweighted $f_\pi$.](plot_reweighted_fpi_ratio_correct.pdf "fig:"){width="40.00000%"} ![\[fig:chi-sq\] The left graph is a histogram of the deviations of fit values from measured values, in units of the $\sigma$ for the data points. The right graph shows results from the partially quenched, NLO ChPT fits for $m_\pi^2$.](histogram_deviation_nlo_fv_no_chisq_dof.pdf "fig:"){width="40.00000%"} ![\[fig:chi-sq\] The left graph is a histogram of the deviations of fit values from measured values, in units of the $\sigma$ for the data points. The right graph shows results from the partially quenched, NLO ChPT fits for $m_\pi^2$.](plot_mpi2_deg_over_mq.pdf "fig:"){width="40.00000%"} ![\[fig:fpi\_unitary\] The left graph shows the unitary, light-light pseudoscalar decay constant versus quark mass, from our NLO fit. The right graph shows the contribution of various terms to the fit.](plot_fpi_unitary.pdf "fig:"){width="40.00000%"} ![\[fig:fpi\_unitary\] The left graph shows the unitary, light-light pseudoscalar decay constant versus quark mass, from our NLO fit. The right graph shows the contribution of various terms to the fit.](plot_fpi_unitary_terms.pdf "fig:"){width="40.00000%"} Full NNLO fit ============= We have also fit our data to the full, continuum NNLO ChPT results for SU(2) given by Bijnens and Lahde [@bijnens]. This adds 13 new parameters to our fits, five $L_i$ and eight, linearly-independent combinations of the $K_i$. For now, we have done NNLO fits keeping the lattice spacing and ratio of quark mass renormalization factors fixed to the values returned from NLO fits. The left graph in Figure \[fig:nnlo\] shows the results for $f_\pi$ from a full NNLO fit to all of our partially quenched, light-light masses and decay constants. Both lattice spacings are fit simultaneously using a standard, least-squares approach and the uncorrelated $\chi^2 = 21.8$ with 125 data points and 33 parameters. Statistical errors come from a jackknife analysis. The NNLO fit predicts $f_\pi = 133 \pm 13_{\rm stat}$ MeV, $f = 130.4 \pm 20.0_{\rm stat}$ and gives values for $m_{ud}$ and $m_s$ the same, within statistical errors, as the NLO fits. The blue band in the figure gives the one $\sigma$ error for the LO contribution, the green shows the error for the LO + NLO contribution and the black the error for the LO + NLO + NNLO contribution. The large size of the green error band means that, for the majority of our fits under the jackknife loop, the size of the NNLO contributions is very large and the series is not convergent. This observation, along with the increase in the statistical error for $f_\pi$ from the extra degrees of freedom in the fit, means that we cannot get an accurate extrapolation to physical light quark masses by fitting our data, which has $m_\pi = 220$ to 420 MeV, to NNLO order in ChPT. The right graph in Figure \[fig:nnlo\] shows the results for $f_\pi$ from a full NNLO fit, with the constraint that the SU(2) chiral limit decay constant, $f = 122$ MeV. (This value of $f$ comes from the phenomenological value for $\bar{l}_4$ and has an uncertainty of about 1 MeV.) The total, uncorrelated $\chi^2$ for these fits is about 25 and gives $f_\pi = 127.9 \pm 1.8_{\rm stat}$ MeV. (The smaller statistical error for this fit comes from the strong constraint in the chiral limit.) However, the large error on the LO + NLO contribution, means that, in a $\sim 25$% of our fits, we have very large NNLO contributions. Thus we cannot demonstrate a reasonably convergent series at this point, even with a constraint. ![\[fig:nnlo\] The left graph is $f_\pi$ from a full NNLO fit to our data. The right graph is also for a full NNLO fit, but with the constraint that the SU(2) chiral limit decay constant, $f = 122.0$ MeV.](plot_fpi_unitary_terms_fit_error_unconstrain0j.pdf "fig:"){width="40.00000%"} ![\[fig:nnlo\] The left graph is $f_\pi$ from a full NNLO fit to our data. The right graph is also for a full NNLO fit, but with the constraint that the SU(2) chiral limit decay constant, $f = 122.0$ MeV.](plot_fpi_unitary_terms_fit_error_constrin12j.pdf "fig:"){width="40.00000%"} Summary and Conclusions ======================= We have found that our 2+1 flavor DWF QCD data from two lattice spacings, with partially quenched $m_\pi$ from 220 to 420 MeV, can be fit with either NLO or full NNLO ChPT formulae. Standard, least-squares fits yield small, uncorrelated values for $\chi^2$, so the fit formulae well represent our data. For unconstrained NNLO fits, we find $f_\pi = 133 \pm 13_{\rm stat}$ MeV and the series is not convergent. Constraining the chiral limit value for $f$ to be 122.0 MeV gives $f_\pi = 127.9 \pm 1.8_{\rm stat}$ and the series appears less poorly convergent. With the current data set, NNLO fits do not provide a reliable extrapolation to physical light quark masses. Turning to NLO fits, we find the NLO terms for $f_\pi$ are 20-30% the size of the LO term, at quark masses where we have data. From this, we estimate a 4-9% correction due to NNLO terms and our value for $f_\pi$ deviates from the physical value by about this much. We take the square of the fractional size of the NLO correction as an estimate of our systematic ChPT error. We also have used NPR to determine the quark mass renormalization, allowing us to determine values for $m_{ud}$ and $m_s$. Our preliminary results are: -------------------------------------------------------------------------- ------------------------------------------------------------------- $ m_{ud}^{\overline{\rm MS}}(2 \; {\rm GeV}) = 3.47 \pm 0.10_{\rm stat} \pm 0.17_{\rm NPR} \; {\rm MeV } $ $ m_{s}^{\overline{\rm MS}}(2 \; {\rm $m_{s}/m_{ud} = 27.19 \pm 0.35_{\rm stat}$ GeV}) = 94.3 \pm 3.4_{\rm stat} \pm 4.5_{\rm NPR} \; {\rm MeV}$ $f_\pi = 122.2 \pm 3.4_{\rm stat} \pm 7.3_{\rm ChPT} \; {\rm MeV} $ $f_K = 149.7 \pm 3.8_{\rm stat} \pm 2.0_{\rm ChPT} \; {\rm MeV} $ $f_K/f_\pi = 1.225 \pm 0.012_{\rm stat} \pm 0.014_{\rm ChPT}$ $f = 113.0 \pm 3.8_{\rm stat} \pm 6.8_{\rm ChPT} \; {\rm MeV}$ $f_K^{(0)} = 144.8 \pm 4.2_{\rm stat} \pm 2.0_{\rm ChPT} \; {\rm MeV}$ $f_K^{(0)}/f = 1.282 \pm 0.015_{\rm stat} \pm 0.017_{\rm ChPT}$ -------------------------------------------------------------------------- ------------------------------------------------------------------- This work and the author were supported in part by US DOE grant \#DE-FG02-92ER40699. We thank Johans Bijnens for his Fortran code for the evaluation of NNLO SU(2) ChPT. The RBC and UKQCD collaborations receive additional support from the US DOE, the RIKEN-BNL Research Center, and PPARC in the UK. Part of this work used computer time granted by the USQCD Collaboration. We thank RIKEN, BNL, the US DOE and PPARC in the UK for providing facilities essential to this work. Producing the computationally demanding $1/a = 2.32$ GeV ensembles has only been possible through the transformative scale of resources made available from the Argonne Leadership Class Facility. [99]{} C. Allton, [*et. al.*]{}, Phys. Rev. D78:114509, 2008. D. J. Antonio, [*et. al.*]{}, Phys. Rev. Lett. 100:032001, 2008. P. A. Boyle, [*et. al.*]{}, Phys. Rev. Lett. 100:141601,2008. J. Bijnens and T. A. Lahde, Phys. Rev. D72:074502, 2005. Y. Aoki, Pos(LAT2009)012. C. Kelly, P. Boyle and C. T. Sachrajda \[RBC and UKQCD Collaborations\] PoS(LAT2009)087. C. Jung, Pos(LAT2009)002. C. Dawson, Pos(LAT2009)072.

--- abstract: 'Photoproduction of vector mesons are computed in dipole model in proton-proton ultraperipheral collisions(UPCs) at the CERN Larger Hadron Collider (LHC). The dipole model framework is employed in the calculations of cross sections of diffractive processes. Parameters of the bCGC model are refitted with the latest experimental data. The bCGC model and Boosted Gaussian wave functions are employed in the calculations. We obtain predictions of rapidity distributions of $J/\psi$ and $\psi(2s)$ mesons in proton-proton ultraperipheral collisions. The predictions give a good descriptions to the experimental data of LHCb. Predictions of $\phi$ and $\omega$ mesons are also calculated in this paper.' author: - 'Ya-Ping Xie' - Xurong Chen title: 'Photoproduction of vector mesons in proton-proton ultraperipheral collisions at the Larger Hadron Collider' --- introduction ============ Diffractive photoproduction of vector mesons in hadron-hadron and electron-proton collisions can help us study the QCD dynamics and gluon saturation effect at high energy level [@Bertulani:2005ru; @Baltz:2007kq]. The H1 and ZEUS collaborations have measured the cross sections of $J/\psi$ in diffractive process at HERA [@Chekanov:2002xi; @Chekanov:2004mw; @Aktas:2005xu; @Alexa:2013xxa]. The LHCb collaborations have measured the rapidity distributions of $J/\psi$ and $\psi(2s)$ in proton-proton and nucleus-nucleus ultraperipheral collisions (UPCs) at the LHC[@Aaij:2013jxj; @Aaij:2014iea; @LHCb:2016oce; @Abbas:2013oua; @Abelev:2012ba; @TheALICE:2014dwa; @Adam:2015gsa; @Adam:2015sia]. Various theoretical approaches can be found to compute the production of vector mesons in UPCs and diffractive processes [@Klein:1999qj; @Frankfurt:2002sv; @Goncalves:2005yr; @Ryskin:1992ui; @Toll:2012mb; @Adeluyi:2012ph; @Xie:2016ino; @Xie:2017mil].\ In hadron-hadron UPCs, the direct hadronic interaction is suppressed. The photon-induced interaction is dominant in hadron-hadron UPCs. Vector mesons can be produced in photon-induced process. The dipole model is a phenomenological model in small-x physics [@Forshaw:2003ki]. In the dipole model, the interaction between virtual photon and proton can be viewed as three steps. Firstly, the virtual photon splits into quark and antiquark. Therefore, the quark-antiquark interacts with proton by exchange gluons. Finally, the quark-antiquark recombine into other particles, for example, vector mesons or real photon. The important aspect of dipole model is the cross section of a pair of quark-antiquark scattering off a proton via gluons exchange. Dipole amplitude is the imaginary part of total photon-proton cross section. It is important in the diffractive process to calculate the production of vector mesons since the vector meson can be viewed as a probe of the interaction between the dipole and the proton. The Golec-Biernat-Wusthoff (GBW) model was firstly introduced to describe the dipole cross section in saturation physics [@GolecBiernat:1998js]. The Bartel-Golec-Biermat-Kowalski (BGBK) model are extensive model of the GBW model considering the gluon density evolution according to DGLAP equation [@Bartels:2002cj]. The Color-Glass-Condensate (CGC) model was introduced based on Balitsky-Kovchegov (BK) evolution equation [@Iancu:2003ge; @Soyez:2007kg; @Ahmady:2016ujw]. The bSat and bCGC models are impact parameter dependent dipole models based on the BGBK and CGC models [@Kowalski:2003hm; @Kowalski:2006hc; @Rezaeian:2012ji; @Watt:2007nr; @Rezaeian:2013tka]. These models all contains free parameters which are determined by fit on cross sections of the inclusive production in deep inelastic scattering.\ In the photoproduction of vector meson in diffractive process, the light-cone wave functions of photon and vector meson are employed in the amplitude. The light-cone wave function of photon can be analytically computed, but the light-cone function of the vector meson can’t be computed analytically. The phenomenological models are used for the vector mesons. The Boosted Gaussian model is a successful model for $J/\psi$ and excited states. The production of $J/\psi$ and $\psi(2s)$ can be used to check the validity of the Boosted Gaussian model. Using the dipole amplitude and light-cone functions of photon and vector meson, the cross section in diffractive process can be evaluated as a function of Bjorken x.\ On other side, the cross sections of heavy vector mesons in diffractive process is investigated in perturbative QCD approach [@Jones:2013pga; @Jones:2013eda; @Jones:2016icr]. The vector meson amplitude is proportional to the gluon density. The leptonic decay width of the heavy vector meson is included in the amplitude. Rapidity gap survival factor is introduced in this paper [@Khoze:2013dha].\ In this paper, the bCGC model is employed to perform the production of vector mesons in diffractive process. Then multiplying the photon flux and rapidity gap survival factor, we obtain the rapidity distributions of the vector mesons in proton-proton UPCs. Similar works can be found in Ref. [@Ducati:2013tva]. The aim of this paper is to update the prediction of exclusive production of $J/\psi$ and $\psi(2s)$ mesons and compute the rapidity distributions of $\phi$ and $\omega$ are performed in bCGC model using the Boosted Gaussian wave functions in proton-proton UPCs. We obtain new parameters of bCGC model in this paper and we consider the contribution of rapidity gap survival factors in this paper too. In Section II, the theoretical framework is reviewed. In Section III, the parameters of the bCGC model are fitted using the latest experimental data. In section IV, the numerical results are presented and some discussions are also listed. The conclusions are in section IV. vector meson production in the dipole model =========================================== In this paper, we focus on the production of heavy mesons in proton-proton UPCs. The rapidity distributions of heavy meson production in UPCs is the product of cross sections of $\gamma+\mathrm{p}\to V+\mathrm{p}$, the photon flux factor and rapidity gap survival factor. The rapidity distributions of heavy mesons in proton-proton UPCs is given as follows[@Jones:2013pga; @Ducati:2013tva] $$\begin{aligned} \frac{d\sigma}{dy}=S^2(W^+)k^+\frac{dn}{dk^+}(k^+)\sigma^{\gamma p\to Vp}(W^+)+S^2(W^-)k^-\frac{dn}{dk^-}(k^-)\sigma^{\gamma p\to Vp}(W^-). \label{dsdy}\end{aligned}$$ In above equation, $k$ is momentum of the radiated photon from proton. $\mathrm{y}$ is the rapidity of the vector meson. $k^{\pm}=M_V/2\exp(\pm |\mathrm{y}|)$. $\mathrm{W}^{\pm}$ is the center mass energy in diffractive process In UPCs, $W^{\pm}=(2k^\pm\sqrt{s})^{1/2}$ with $\sqrt{s}$ center-energy. $S^2(W)$ is rapidity gap survival factor in Good-Walker model[@Jones:2013pga; @Khoze:2002dc], and $dn/dk$ is photon flux[@Bertulani:2005ru]. It is given by $$\begin{aligned} \frac{dn}{dk}(k)=\frac{\alpha_{em}}{2\pi k}\Big[1+\Big(1-\frac{2k}{\sqrt{s}}\Big)^2\Big]\Big( \ln \Omega-\frac{11}{6}+\frac{3}{\Omega}-\frac{3}{2\Omega^2}+\frac{1}{3\Omega^3}\Big),\end{aligned}$$ where $\Omega=1+0.71/Q^2_{min}$, with $Q^2_{min}=k^2/\gamma^2_L$, $\gamma_L$ is the lorentz boost factor with $\gamma_L=\sqrt{s}/2m_p$. The cross sections of $\sigma^{\gamma p\to Vp}(W)$ is integrated by $|t|$ as. $$\sigma^{\gamma p\to Vp}(W)=\int dt\frac{d\sigma^{\gamma p\to Vp}}{dt}.$$ Then, the differential cross section of $\gamma+p\to V+p$ is given as [@Kowalski:2003hm; @Kowalski:2006hc] $$\begin{aligned} \frac{d\sigma^{\gamma p\to Vp}(x)}{dt}=\frac{R_g^2(1+\beta^2)}{16\pi^2} |\mathcal{A}^{\gamma p\to Vp}(x,Q^2,\Delta)|^2, \label{dsigma1}\end{aligned}$$ with $x=\frac{M_V}{\sqrt{s}}\exp(\mp| \mathrm{y|})$ or $x=M_V^2/W^2$. The amplitude $\mathcal{A}^{\gamma p\to Vp}(x,Q^2,\Delta)$ in Eq. (\[dsigma1\]) is written as $$\begin{aligned} \mathcal{A}^{\gamma p\to Vp}(x, Q^2,\Delta)= i\int d^2r\int_0^1\frac{dz}{4\pi} \int d^2b(\psi_V^*\psi_{\gamma})_{T}(z,r,Q^2)e^{-i(\bm b-(1-z)\bm r)\cdot\bm \Delta }\mathcal{N}(x,\bm r,\bm b),\notag\\ \label{amp}\end{aligned}$$ where T denotes the transverse overlap function of photon and vector meson functions with $Q^2=0$, since the photon is real one in UPCs. And $\beta$ is ratio of the imaginary part to the real part amplitude. $$\beta=\tan (\frac{\pi}{2}\delta), \quad\text{with}\quad \delta=\frac{\partial \ln (\mathrm{Im}\mathcal{A}(x))}{\partial \ln(1/x)}.$$ The factor $R_g^2$ reflects the skewedness [@Shuvaev:1999ce], it gives $$R_g=\frac{2^{2\delta+3}}{\sqrt{\pi}}\frac{\Gamma(\delta+5/2)}{\Gamma(\delta+4)}.$$ In the bCGC model, the dipole amplitude is given as [@Iancu:2003ge; @Rezaeian:2013tka] $$\begin{aligned} \mathcal{N}(x,\bm r,\bm b)=2\times\begin{cases} \mathcal{N}_0(\frac{rQs}{2})^{2(\gamma_s+(1/\kappa\lambda Y)\ln(2/rQs))},\quad\! rQs\leqslant 2,\\ 1-\exp\big(-A\ln^2(B rQs)\big),\quad\quad\!\! rQs>2, \end{cases} \end{aligned}$$ where $Qs(x,\bm b)=(x/x_0)^{\lambda/2}\exp(-\frac{\bm b^2}{4\gamma_sB_p})$, $\kappa=9.9$, and $Y=\ln(1/x)$. $A$ and $B$ are given as $$\begin{split} & A=-\frac{\mathcal{N}^2_0\gamma_s^2}{(1-\mathcal{N}_0)^2\ln(1-\mathcal{N}_0)},\\ &B=\frac{1}{2}(1-\mathcal{N}_0)^{-(1-\mathcal{N}_0)/(2\mathcal{N}_0\gamma_s)}. \end{split}$$ In the bCGC model, $B_p$, $x_0$, $\gamma_s$, $\mathcal{N}_0$ and $\lambda$ are free parameters and they are fitted from the experimental data.\ The overlap of photon and vector meson in Eq. (\[amp\]) we use are given as follows $$\begin{aligned} (\Psi_V^*\Psi_{\gamma})_T(r,z,Q^2)=e_fe\frac{N_c}{\pi z(1-z)}\lbrace m_f^2 K_0(\epsilon r)\phi_T(r,z)-(z^2+(1-z)^2)\epsilon K_1(\epsilon r)\partial_r \phi_T(r,z)\rbrace,\notag\\ \end{aligned}$$ where $e_f$ is effective charge for mesons, $\epsilon=\sqrt{z(1-z)Q^2+m_f^2}$ and $\phi_T(r,z)$ is the scalar functions, $K_0(x)$ and $K_1(x)$ are second kind Bessel functions. There is no analytic expression for the scalar functions of the vector mesons. There are some successful models for the scalar functions. The Boosted Gaussian model is a phenomenological model. The scalar function of $J/\psi$ in Boosted Gaussian model is written as $$\begin{aligned} \phi^{1s}_T(r,z)=N_Tz(1-z)\exp\big(-\frac{m_f^2\mathcal{R}_{1s}^2}{8z(1-z)}- \frac{2z(1-z)r^2}{\mathcal{R}^2_{1s}}+\frac{m_f^2\mathcal{R}^2_{1s}}{2}\big). \end{aligned}$$ The scalar function for $\psi(2s)$ meson in Boosted Gaussian model is given as [@Armesto:2014sma] $$\begin{aligned} \phi^{2s}_T(r,z)&=&N_Tz(1-z)\exp\big(-\frac{m_f^2\mathcal{R}^2_{2s}}{8z(1-z)}- \frac{2z(1-z)r^2}{\mathcal{R}^2_{2s}}+\frac{m_f^2\mathcal{R}^2_{2s}}{2}\big)\notag\\ &\times&\Big[1+\alpha_{2s}\Big(2+\frac{m_f^2\mathcal{R}^2_{2s}}{8z(1-z)}- \frac{4z(1-z)r^2}{\mathcal{R}_{2s}^2}-m_f^2\mathcal{R}_{2s}^2\Big)\Big]. \end{aligned}$$ There are several free parameters of the Boosted Gaussian wave functions. They are presented in Table. \[wave\]. The parameters of $\omega$ meson are obtained in this work. The parameters are determined by the normalization condition and the lepton decay width. meson $e_f$ mass $f_V$ $m_f$ $N_T$ $\mathcal{R}^2$ $\alpha_{2s}$ ------------ --------------- ------- -------- ------- ------- ----------------- --------------- GeV GeV GeV $\text{GeV}^2 $ $\omega$ $1/3\sqrt{2}$ 0.782 0.0458 0.14 0.895 15.78 $\phi$ $1/3$ 1.020 0.076 0.14 0.919 11.2 $J/\psi$ $2/3$ 3.097 0.274 1.27 0.596 2.45 $J/\psi$ $2/3$ 3.097 0.274 1.40 0.57 2.45 $\psi(2s)$ $2/3$ 3.686 0.198 1.27 0.70 3.72 -0.61 $\psi(2s)$ $2/3$ 3.686 0.198 1.40 0.67 3.72 -0.61 : Parameters of the scalar functions of the Boosted Gaussian model for $\omega$, $\phi$, $J/\psi$ and $\psi(2s)$ mesons, the parameters of $\rho$, $J/\psi$ and $\psi(2s)$ are taken from [@Kowalski:2006hc; @Armesto:2014sma].[]{data-label="wave"} parameters fit for the bCGC model ================================= In the bCGC model, there are several free parameters need to be fitted from the experimental data. In dipole model, the cross section of the virtual photon and the proton in Deep Inelastic Scattering (DIS) are written as $$\begin{aligned} \sigma_{T,L}^{\gamma^*p}(x,Q^2)=&&\sum_{f=u,d,s}\int d^2 \bm r\int \frac{dz}{4\pi}(\psi^*\psi)_{T,L}^f(z,\bm r, Q^2) \sigma_{q\bar{q}}(x,\bm r)\notag\\ &&+\sum_{f=c}\int d^2 r\int \frac{dz}{4\pi}(\psi^*\psi)_{T,L}^f(z,\bm r,Q^2) \sigma_{q\bar{q}}(\hat{x},\bm r). \label{dipolecross} \end{aligned}$$ where $x=x_{Bjorken}$ and $\hat{x}=x_{Bjorken}(1+4m^2_c/Q^2)$. The dipole cross section $\sigma_{q\bar{q}}(x,\bm r)$ is integrated as $$\begin{aligned} \sigma_{q\bar{q}}(x,\bm r)=\int d^2\bm b\mathcal{N}(x,\bm r,\bm b). \end{aligned}$$ The square of the wave functions of the virtual photons are given by $$\begin{aligned} (\psi^*\psi)_T(z,\bm r,Q^2)&=&\frac{2N_c}{\pi}\alpha_{em}e_f^2\{[z^2+(1-z)^2]\epsilon^2K^2_1(\epsilon r) +m_f^2K^2_0(\epsilon r)\}; \\ (\psi^*\psi)_L(z,\bm r, Q^2)&=&\frac{8N_c}{\pi}\alpha_{em}e_f^2Q^2z^2(1-z)^2K_0^2(\epsilon r). \end{aligned}$$ The proton structure functions $F_2(x,Q^2)$ and $F_L(x,Q^2)$ are written as $$\begin{aligned} F_2(x,Q^2)&=&\frac{Q^2}{4\pi^2\alpha_{em}}[\sigma_T^{\gamma^*p}(x,Q^2)+\sigma_L^{\gamma^*p}(x,Q^2)];\\ F_L(x,Q^2)&=&\frac{Q^2}{4\pi^2\alpha_{em}}\sigma^{\gamma^*p}_L(x,Q^2). \end{aligned}$$ The reduce cross section in DIS is given by $$\begin{aligned} \sigma_r(x,\text{y},Q^2)=F_2(x,Q^2)-\frac{\text{y}^2}{1+(1-\text{y})^2}F_L(x,Q^2).\end{aligned}$$ where $\text{y}=Q^2/(xs)$. In 2015, H1 and ZEUS released the latest combined reduce cross sections [@Abramowicz:2015mha].\ In this paper, we refit the free parameters of the bCGC model using the reduce cross sections released in 2015. The experimental data are selected from $x<0.01$ and $0.40\;\text{GeV}^2\leqslant Q^2\leqslant 45\;\text{GeV}^2$. The parameters fitted in this paper are presented in Table \[IPP2\] with two fits. $m_{u,d,s}$/GeV $m_{c}$/GeV $B_p$/ $\mathrm{GeV}^2$ $\gamma_s$ $N_0$ $x_0$ $\lambda$ $\chi^2$/d.o.f ------- ----------------- ------------- ------------------------- ------------ -------- ---------- ----------- ---------------- Fit 1 0.14 1.27 5.746 0.6924 0.3159 0.001849 0.2039 607/467=1.300 Fit 2 0.14 1.4 5.852 0.6932 0.3144 0.001978 0.2012 629/467=1.347 : Parameters for bCGC model fitted from the reduce cross sections with $x<0.01$ and $0.40\; \text{GeV}^2\leqslant Q^2\leqslant45\;\text{GeV}^2$ released in 2015 [@Abramowicz:2015mha].[]{data-label="IPP2"} ![(Color online) Proton structure functions $F_2(x,Q^2)$ calculated in bCGC model with parameters presented in Table \[IPP2\] and compared with the experimental data from H1 and ZEUS collaboration [@Aaron:2009aa].[]{data-label="F2"}](bccF2.eps){width="5in"} ![(Color online) Proton charmed structure functions $F^{c\bar{c}}_2(x,Q^2)$ calculated in bCGC model with parameters presented in Table \[IPP2\] and compared with charmed structure functions $F^{c\bar{c}}_2\approx \sigma_r^{c\bar{c}}$ from H1 and ZEUS collaboration [@Abramowicz:1900rp][]{data-label="FCC"}](bccFCC.eps){width="4in"} Using the parameters of the bCGC model, the proton structure function $F_2(x,Q^2)$ can be evaluated in the bCGC model and compared with experimental data. The proton structure functions for proton is shown in Fig. \[F2\]. It can be seen that the bCGC model give a good description to structure function $F_2(x,Q^2)$ using two fits parameters. The charmed proton structure function $F^{c\bar{c}}_2(x,Q^2)$ is shown In Fig. \[FCC\], it can be seen that the two fits parameters give different predictions for the charmed structure function. numerical results and discussions ================================= Firstly, we compute the cross sections of $J/\psi$ in diffractive process and compare the predictions with experimental data. The amplitude of $\gamma+ p\to J/\psi +p$ are performed in bCGC model with the Boosted Gaussian wave functions. The parameters with $m_c=1.27$ GeV and $m_c=1.4$ GeV in Table. \[IPP2\] are used in the calculations. Predictions of cross section of $J/\psi$ meson in diffractive process are shown in Fig. \[sigma\]. The upper band of bCGC are using parameters with $m_c=1.27$ GeV and the lower band of bCGC are using parameters with $m_c=1.4$ GeV as presented in Table \[IPP2\]. It can be seen that the predictions using parameters $m_c$=1.27 GeV give a better description than the fit with $m_c$=1.4 GeV. It can be concluded that the dipole model is sensitive to the quark mass. The cross sections labeled H1 and ZEUS are measured directly by H1 and ZEUS collaboration. The cross section labeled ALICE and LHCb are also not measured directly. They are extracted from p-Pb and proton-proton UPCs. The cross sections of LHCb are divided by the rapidity gap survival factor and photon flux as presented Eq. (\[dsdy\]). Therefore, we need add the rapidity gap survival factor contribution as Eq. (\[dsdy\]). The rapidity gap survival factor we use are from Refs. [@LHCb:2016oce; @Jones:2016icr].\ ![(Color online) Predictions of cross sections of diffractive process as a function of W calculated in the bSat and bCGC models with the Boosted Gaussian wave functions compared with the experimental data from H1[@Aktas:2005xu; @Alexa:2013xxa], ZEUS [@Chekanov:2002xi; @Chekanov:2004mw], ALICE [@TheALICE:2014dwa] and LHCb [@Aaij:2014iea], The upper band of bCGC are using parameters with $m_c=1.27$ GeV and the lower band of bCGC are using parameters with $m_c=1.4$ GeV.[]{data-label="sigma"}](epv.eps){width="4in"} Secondly, we compute the rapidity distributions of $J/\psi$ and $\psi(2s)$ mesons as Eq. (\[dsdy\]). The rapidity gap survival factors and photon flux are included. The parameters in Table. \[IPP2\] of bCGC model are used in the calculations. The rapidity distributions of $J/\psi$ and $\psi(2s)$ mesons computed in two fits parameters are shown in Fig. \[jpsi\] and Fig. \[psi2s\] . The experimental data of LHCb are also presented in the same figures. The upper band of bCGC are using parameters with $m_c=1.27$ GeV and the lower band of bCGC are using parameters with $m_c=1.4$ GeV. It can be seen that our predictions give a good descriptions to the experimental data. In Ref. [@Ducati:2013tva], the rapidity distributions of $J/\psi$ and $\psi(2s)$ mesons had been computed in CGC model using the Boosted Gaussian wave functions, but the parameters of the Boosted Gaussian functions were not presented in Ref. [@Ducati:2013tva]. The predictions of this paper are close to the results in Ref. [@Ducati:2013tva; @Fiore:2014oha]. We use the bCGC models with parameters fitted from combined H1 and ZEUS data and we present the detail parameters for the Boosted Gaussian wave functions and rapidity gap survival factors. In Ref. [@Goncalves:2016sqy], the rapidity distributions $J/\psi$ are obtained in the bCGC model, but the rapidity gap survival factor is unity. In our calculation, the rapidity gap survival factor is about $0.6\sim 0.9$. And we find that the rapidity gap survival factor is important in the final results of rapidity distributions.\ ![(Color online) Predictions of rapidity distributions of $J/\psi$ meson in proton-proton ultraperipheral collisions at the LHC compared with the experimental data of the LHCb collaboration[@Aaij:2014iea; @LHCb:2016oce]. The upper band of bCGC are using parameters with $m_c=1.27$ GeV and the lower band of bCGC are using parameters with $m_c=1.4$ GeV.[]{data-label="jpsi"}](jpsi01.eps "fig:"){width="3in"} ![(Color online) Predictions of rapidity distributions of $J/\psi$ meson in proton-proton ultraperipheral collisions at the LHC compared with the experimental data of the LHCb collaboration[@Aaij:2014iea; @LHCb:2016oce]. The upper band of bCGC are using parameters with $m_c=1.27$ GeV and the lower band of bCGC are using parameters with $m_c=1.4$ GeV.[]{data-label="jpsi"}](jpsi02.eps "fig:"){width="3in"} ![(Color online) Predictions of rapidity distributions of $\psi(2s)$ meson in proton-proton ultraperipheral collisions at the LHC compared with the experimental data of the LHCb collaboration[@Aaij:2014iea; @LHCb:2016oce]. The upper band of bCGC are using parameters with $m_c=1.27$ GeV and the lower band of bCGC are using parameters with $m_c=1.4$ GeV.[]{data-label="psi2s"}](psi2s01.eps "fig:"){width="3in"} ![(Color online) Predictions of rapidity distributions of $\psi(2s)$ meson in proton-proton ultraperipheral collisions at the LHC compared with the experimental data of the LHCb collaboration[@Aaij:2014iea; @LHCb:2016oce]. The upper band of bCGC are using parameters with $m_c=1.27$ GeV and the lower band of bCGC are using parameters with $m_c=1.4$ GeV.[]{data-label="psi2s"}](psi2s02.eps "fig:"){width="3in"} ![(Color online) Predictions of rapidity distributions of $\phi$ meson computed in bCGC model using the Boosted Gaussian wave function in proton-proton ultraperipheral collisions at the LHC. The upper band of bCGC are using parameters of Fit 2 and the lower band of bCGC are using parameters of Fit 1.[]{data-label="phi"}](phi01.eps "fig:"){width="3in"} ![(Color online) Predictions of rapidity distributions of $\phi$ meson computed in bCGC model using the Boosted Gaussian wave function in proton-proton ultraperipheral collisions at the LHC. The upper band of bCGC are using parameters of Fit 2 and the lower band of bCGC are using parameters of Fit 1.[]{data-label="phi"}](phi02.eps "fig:"){width="3in"} ![(Color online)Predictions of rapidity distributions of $\omega$ meson computed in bCGC model using the Boosted Gaussian wave function in proton-proton ultraperipheral collisions at the LHC. The upper band of bCGC are using parameters of Fit 2 and the lower band of bCGC are using parameters of Fit 1. []{data-label="omega"}](omega01.eps "fig:"){width="3in"} ![(Color online)Predictions of rapidity distributions of $\omega$ meson computed in bCGC model using the Boosted Gaussian wave function in proton-proton ultraperipheral collisions at the LHC. The upper band of bCGC are using parameters of Fit 2 and the lower band of bCGC are using parameters of Fit 1. []{data-label="omega"}](omega02.eps "fig:"){width="3in"} Finally, the rapidity distributions of $\phi$ and $\omega$ mesons are also computed in the bCGC model with the Boosted Gaussian wave function in this paper. The predictions are shown in Fig. \[phi\] and \[omega\]. The quark mass is $m_q=0.14$ GeV in the calculations and the upper band of bCGC are using parameters of Fit 2 and the lower band of bCGC are using parameters of Fit 1. Since there is no information of the rapidity gap survival factors for these two mesons now, the rapidity gap survival factors are taken as unity for $\phi$ and $\omega$ in the calculations. In Ref.[@Cisek:2010jk], the authors presented the exclusive $\phi$ production in proton-proton UPCs at the LHC. The rapidity distributions of $\phi$ at LHC are smaller than the results in the paper since the different approaches are employed in the two paper. There is no experimental data for the $\phi$ and $\omega$ mesons in proton-proton UPCs at the LHC. We hope that the experimental data will be measured in the future. We can compare the theoretical prediction with the experimental data.\ conclusion ========== In this paper, we have studied the exclusive photoproduction of $J/\psi$, $\psi (2s)$, $\phi$ and $\omega$ in proton-proton UPCs at the LHC. The bCGC model and the Boosted Gaussian wave functions are employed in the calculation. The parameters of the bCGC model are refitted with the experimental data released in 2015. The theoretical predictions of $J/\psi$ and $\psi(2s) $ mesons rapidity distributions are evaluated in bCGC model and compared with the experimental data measured by the LHCb collaboration. It can be seen that the predictions of bCGC model give a good description to the experimental data. It is concluded that the bCGC are successful phenomenological model for the small-x physics and the Boosted Gaussian wave functions are good candidates for the $J/\psi$ and $\psi(2s)$ mesons. The rapidity gap survival factor is important in the calculations multiplied together with the photon flux. The quark mass is sensitive to the exclusive vector mesons photoproduction in proton-proton UPCs. The rapidity distributions of $\phi$ and $\omega$ mesons are also performed in this paper. The predictions of the $\phi$ and $\omega$ mesons can be employed in future experiments. Acknowledgements ================ We thank the useful discussions with M. V. T. Machado and V. P. Gonçalves. This work is supported in part by Key Research Program of Frontier Sciences,CAS (Grant No QYZDY-SSW-SLH006) and the National 973 project in China (No: 2014CB845406). [99]{} C. A. Bertulani, S. R. Klein and J. Nystrand, Ann. Rev. Nucl. Part. Sci.,  [**55**]{}: 271 (2005) \[nucl-ex/0502005\]. A. J. Baltz, G. Baur, D. d’Enterria, L. Frankfurt [*et al.*]{}, Phys. Rept.,  [**458**]{}:1 (2008) \[arXiv:0706.3356 \[nucl-ex\]\]. S. Chekanov [*et al.*]{} \[ZEUS Collaboration\], Eur. Phys. J. C, [**24**]{}: 345 (2002) \[hep-ex/0201043\]. S. Chekanov [*et al.*]{} \[ZEUS Collaboration\], Nucl. Phys. B, [**695**]{}: 3 (2004) \[hep-ex/0404008\]. A. Aktas [*et al.*]{} \[H1 Collaboration\], Eur. Phys. J. C, [**46**]{}: 585 (2006) \[hep-ex/0510016\]. C. Alexa [*et al.*]{} \[H1 Collaboration\], Eur. Phys. J. C, [**73**]{}(6): 2466 (2013) \[arXiv:1304.5162 \[hep-ex\]\]. R. Aaij [*et al.*]{} \[LHCb Collaboration\], J. Phys. G, [**40**]{}:045001 (2013) \[arXiv:1301.7084 \[hep-ex\]\]. R. Aaij [*et al.*]{} \[LHCb Collaboration\], J. Phys. G, [**41**]{}: 055002 (2014) \[arXiv:1401.3288 \[hep-ex\]\]. The LHCb Collaboration \[LHCb Collaboration\], LHCb-CONF-2016-007, CERN-LHCb-CONF-2016-007, oai:cds.cern.ch:2209532. B. Abelev [*et al.*]{} \[ALICE Collaboration\], Phys. Lett. B, [**718**]{}: 1273 (2013) \[arXiv:1209.3715 \[nucl-ex\]\]. E. Abbas [*et al.*]{} \[ALICE Collaboration\], Eur. Phys. J. C, [**73**]{}(11): 2617 (2013) \[arXiv:1305.1467 \[nucl-ex\]\]. B. B. Abelev [*et al.*]{} \[ALICE Collaboration\], Phys. Rev. Lett.,  [**113**]{},(23): 232504 (2014) \[arXiv:1406.7819 \[nucl-ex\]\]. J. Adam [*et al.*]{} \[ALICE Collaboration\], Phys. Lett. B, [**751**]{}: 358 (2015) \[arXiv:1508.05076 \[nucl-ex\]\]. J. Adam [*et al.*]{} \[ALICE Collaboration\], JHEP, [**1509**]{}: 095 (2015) \[arXiv:1503.09177 \[nucl-ex\]\]. S. Klein and J. Nystrand, Phys. Rev. C, [**60**]{}: 014903 (1999) \[hep-ph/9902259\]. S. R. Klein, J. Nystrand and R. Vogt, Eur. Phys. J. C, [**21**]{}: 563 (2001) \[hep-ph/0005157\]. S. R. Klein, J. Nystrand and R. Vogt, Phys. Rev. C, [**66**]{}: 044906 (2002) \[hep-ph/0206220\]. L. Frankfurt, M. Strikman and M. Zhalov, Phys. Rev. C, [**67**]{}: 034901 (2003) \[hep-ph/0210303\]. L. Frankfurt, V. Guzey, M. Strikman and M. Zhalov, Phys. Lett. B, [**752**]{}: 51 (2016) \[arXiv:1506.07150 \[hep-ph\]\]. V. P. Goncalves and M. V. T. Machado, Eur. Phys. J. C, [**40**]{}: 519 (2005) \[hep-ph/0501099\]. V. P. Goncalves and M. V. T. Machado, Phys. Rev. C, [**80**]{}: 054901 (2009) \[arXiv:0907.4123 \[hep-ph\]\]. V. P. Goncalves and M. V. T. Machado, Phys. Rev. C, [**84**]{}: 011902 (2011) \[arXiv:1106.3036 \[hep-ph\]\]. V. P. Gonçalves, B. D. Moreira and F. S. Navarra, Phys. Lett. B, [**742**]{}: 172 (2015) \[arXiv:1408.1344 \[hep-ph\]\]. M. G. Ryskin, Z. Phys. C, [**57**]{}: 89 (1993). doi:10.1007/BF01555742 A. D. Martin, C. Nockles, M. G. Ryskin and T. Teubner, Phys. Lett. B, [**662**]{}: 252 (2008) \[arXiv:0709.4406 \[hep-ph\]\]. V. Guzey and M. Zhalov, JHEP, [**1310**]{}: 207 (2013) \[arXiv:1307.4526 \[hep-ph\]\]. A. Adeluyi and C. A. Bertulani, Phys. Rev. C, [**85**]{}: 044904 (2012) \[arXiv:1201.0146 \[nucl-th\]\]. A. Adeluyi and T. Nguyen, Phys. Rev. C, [**87**]{}(2): 027901 (2013) \[arXiv:1302.4288 \[nucl-th\]\]. T. Toll and T. Ullrich, Phys. Rev. C, [**87**]{}(2): 024913 (2013) \[arXiv:1211.3048 \[hep-ph\]\]. E. Andrade-II, I. González, A. Deppman and C. A. Bertulani, Phys. Rev. C, [**92**]{}: 064903 (2015) \[arXiv:1509.08701 \[hep-ph\]\]. T. Lappi and H. Mantysaari, Phys. Rev. C, [**87**]{}(3): 032201 (2013) \[arXiv:1301.4095 \[hep-ph\]\]. T. Lappi and H. Mantysaari, Phys. Rev. C, [**83**]{}: 065202 (2011) \[arXiv:1011.1988 \[hep-ph\]\]. Y. P. Xie and X. Chen, Eur. Phys. J. C, [**76**]{}(6): 316 (2016) \[arXiv:1602.00937 \[hep-ph\]\]. Y. P. Xie and X. Chen, Nucl. Phys. A, [**957**]{}: 477 (2017) \[arXiv:1512.08105 \[hep-ph\]\]. Y. P. Xie and X. Chen, Nucl. Phys. A, [**959**]{}: 56 (2017). Y. P. Xie and X. Chen, Nucl. Phys. A [**970**]{}, 316 (2018). doi:10.1016/j.nuclphysa.2017.12.003 J. R. Forshaw, R. Sandapen and G. Shaw, Phys. Rev. D, [**69**]{}: 094013 (2004) \[hep-ph/0312172\]. K. J. Golec-Biernat and M. Wusthoff, Phys. Rev. D, [**59**]{}: 014017 (1998) \[hep-ph/9807513\]. K. J. Golec-Biernat and M. Wusthoff, Phys. Rev. D, [**60**]{}: 114023 (1999) \[hep-ph/9903358\]. J. Bartels, K. J. Golec-Biernat and H. Kowalski, Phys. Rev. D, [**66**]{}: 014001 (2002) \[hep-ph/0203258\]. E. Iancu, K. Itakura and S. Munier, Phys. Lett. B, [**590**]{}: 199 (2004) \[hep-ph/0310338\]. G. Soyez, Phys. Lett. B, [**655**]{}: 32 (2007) \[arXiv:0705.3672 \[hep-ph\]\]. M. Ahmady, R. Sandapen and N. Sharma, Phys. Rev. D, [**94**]{}(7): 074018 (2016) \[arXiv:1605.07665 \[hep-ph\]\]. H. Kowalski and D. Teaney, Phys. Rev. D, [**68**]{}: 114005 (2003) \[hep-ph/0304189\]. H. Kowalski, L. Motyka and G. Watt, Phys. Rev. D, [**74**]{}: 074016 (2006) \[hep-ph/0606272\]. A. H. Rezaeian, M. Siddikov, M. Van de Klundert and R. Venugopalan, Phys. Rev. D, [**87**]{}(3): 034002 (2013) \[arXiv:1212.2974\]. G. Watt and H. Kowalski, Phys. Rev. D, [**78**]{}: 014016 (2008) \[arXiv:0712.2670 \[hep-ph\]\]. A. H. Rezaeian and I. Schmidt, Phys. Rev. D, [**88**]{}: 074016 (2013) \[arXiv:1307.0825 \[hep-ph\]\]. S. P. Jones, A. D. Martin, M. G. Ryskin and T. Teubner, JHEP, [**1311**]{}: 085 (2013) \[arXiv:1307.7099 \[hep-ph\]\]. S. P. Jones, A. D. Martin, M. G. Ryskin and T. Teubner, J. Phys. G, [**41**]{}: 055009 (2014) \[arXiv:1312.6795 \[hep-ph\]\]. S. P. Jones, A. D. Martin, M. G. Ryskin and T. Teubner, J. Phys. G, [**44**]{}(3): 03LT01 (2017) \[arXiv:1611.03711 \[hep-ph\]\]. V. A. Khoze, A. D. Martin and M. G. Ryskin, Eur. Phys. J. C, [**73**]{}: 2503 (2013) \[arXiv:1306.2149 \[hep-ph\]\]. M. B. Gay Ducati, M. T. Griep and M. V. T. Machado, Phys. Rev. D, [**88**]{}: 017504 (2013) \[arXiv:1305.4611 \[hep-ph\]\]. V. A. Khoze, A. D. Martin and M. G. Ryskin, Eur. Phys. J. C, [**24**]{}: 459 (2002) \[hep-ph/0201301\]. A. G. Shuvaev, K. J. Golec-Biernat, A. D. Martin [*et al*]{}, Phys. Rev. D, [**60**]{}: 014015 (1999) \[hep-ph/9902410\]. N. Armesto and A. H. Rezaeian, Phys. Rev. D, [**90**]{}(5): 054003 (2014) \[arXiv:1402.4831 \[hep-ph\]\]. H. Abramowicz [*et al.*]{} \[H1 and ZEUS Collaborations\], Eur. Phys. J. C, [**75**]{}(12): 580 (2015) \[arXiv:1506.06042 \[hep-ex\]\]. F. D. Aaron [*et al.*]{} \[H1 and ZEUS Collaborations\], JHEP, [**1001**]{}: 109 (2010) \[arXiv:0911.0884 \[hep-ex\]\]. H. Abramowicz [*et al.*]{} \[H1 and ZEUS Collaborations\], Eur. Phys. J. C, [**73**]{}(2): 2311 (2013) \[arXiv:1211.1182 \[hep-ex\]\]. R. Fiore, L. Jenkovszky, V. Libov and M. Machado, Theor. Math. Phys. ,  [**182**]{}(1): 141 (2015) \[Teor. Mat. Fiz., [**182**]{}(1): 171 (2014)\] \[arXiv:1408.0530 \[hep-ph\]\]. V. P. Goncalves, B. D. Moreira and F. S. Navarra, Phys. Rev. D, [**95**]{} (5): 054011 (2017) \[arXiv:1612.06254 \[hep-ph\]\]. A. Cisek, W. Schafer and A. Szczurek, Phys. Lett. B, [**690**]{}: 168 (2010) \[arXiv:1004.0070 \[hep-ph\]\].

--- abstract: 'Understanding the evolution of a set of genes or species is a fundamental problem in evolutionary biology. The problem we study here takes as input a set of trees describing [possibly discordant]{} evolutionary scenarios for a given set of genes or species, and aims at finding a single tree that minimizes the leaf-removal distance to the input trees. This problem is a specific instance of the general consensus/supertree problem, widely used to combine or summarize discordant evolutionary trees. The problem we introduce is specifically tailored to address the case of discrepancies between the input trees due to the misplacement of individual taxa. Most supertree or consensus tree problems are computationally intractable, and we show that the problem we introduce is also NP-hard. We provide tractability results in form of a 2-approximation algorithm and a parameterized algorithm with respect to the number of removed leaves. We also introduce a variant that minimizes the maximum number $d$ of leaves that are removed from any input tree, and provide a parameterized algorithm for this problem with parameter $d$.' author: - Cedric Chauve - Mark Jones - Manuel Lafond - Céline Scornavacca - Mathias Weller bibliography: - 'mast.bib' title: 'Constructing a Consensus Phylogeny from a Leaf-Removal Distance' --- Introduction {#sec:introduction} ============ In the present paper, we consider a very generic computational biology problem: given a collection of trees representing, possibly discordant, evolutionary scenarios for a set of biological entities (genes or species – also called *taxa* in the following), we want to compute a single tree that agrees as much as possible with the input trees. Several questions in computational biology can be phrased in this generic framework. For example, for a given set of homologous gene sequences that have been aligned, one can sample *evolutionary trees* for this gene family according to a well defined posterior distribution and then ask how this collection of trees can be combined into a single gene tree, a problem known as *tree amalgamation* [@DBLP:journals/bioinformatics/ScornavaccaJS15]. In phylogenomics, one aims at *inferring a species tree* from a collection of input trees obtained from whole-genome sequence data. A first approach considers gene families and proceeds by computing individual *gene trees* from a large set of gene families, and then combining this collection of gene trees into a unique species tree for the given set of taxa; this requires handling the discordant signal observed in the gene trees due to evolutionary processes such as gene duplication and loss [@10.1073/pnas.1412770112], lateral gene transfer [@10.1073/pnas.1202997109], or incomplete lineage sorting [@10.1093/sysbio/syw082]. Another approach concatenates the sequence data into a single large multiple sequence alignment, that is then partitioned into overlapping subsets of taxa for which partial evolutionary trees are computed, and a unique species tree is then inferred by combining the resulting collection of partial trees [@10.1126/science.1253451]. For example, the Maximum Agreement Subtree (MAST) problem considers a collection of input trees[^1], all having the same leaf labels and looks for a tree of maximum size (number of leaves), which agrees with each of the input trees. This problem is tractable for trees with bounded degree but NP-hard generally [@Amir1997]. The MAST problem is a *consensus problem*, because the input trees share the same leaf labels set, and the output tree is called a *consensus* tree. In the *supertree framework*, the input trees might not all have identical label sets, but the output is a tree on the whole label set, called a *supertree*. For example, in the Robinson-Foulds (RF) supertree problem, the goal is to find a supertree that minimizes the sum of the RF-distances to the individual input trees [@10.1093/bioinformatics/btw600]. One way to compute consensus trees and supertrees that is closely related to our work is to modify the collection of input trees minimally in such a way that the resulting modified trees all agree. For example, in the MAST problem, modifications of the input trees consist in removing a minimum number of taxa from the whole label set, while in the Agreement Supertree by Edge Contraction (AST-EC) problem, one is asked to contract a minimum number of edges of the input trees such that the resulting (possibly non-binary) trees all agree with at least one supertree [@DBLP:journals/siamcomp/Fernandez-BacaG15]; in the case where the input trees are all triplets (rooted trees on three leaves), this supertree problem is known as the Minimum Rooted Triplets Inconsistency problem [@journals/dam/byrka2010]. The SPR Supertree problem considers a similar problem where the input trees can be modified with the Subtree-Prune-and-Regraft (SPR) operator [@whidden2014supertrees]. =-1 In the present work, we introduce a new consensus problem, called [`LR-Consensus`]{}. Given a collection of input trees having the same leaf labels set, we want to remove a minimum number of leaves – an operation called a Leaf-Removal (LR) – from the input trees such that the resulting pruned trees all agree. Alternatively, this can be stated as finding a consensus tree that minimizes the cumulated *leaf-removal distance* to the collection of input trees. This problem also applies to tree amalgamation and to species tree inference from one-to-one orthologous gene families, where the LR operation aims at correcting the misplacement of a single taxon in an input tree. In the next section, we formally define the problems we consider, and how they relate to other supertree problems. Next we show that the [`LR-Consensus`]{} problem is NP-hard and that in some instances, a large number of leaves need to be removed to lead to a consensus tree. We then provide a 2-approximation algorithm, and show that the problem is fixed-parameter tractable (FPT) when parameterized by the total number of LR. However, these FPT algorithms have impractical time complexity, and thus, to answer the need for practical algorithms, we introduce a variant of the [`LR-Consensus`]{} problem, where we ask if a consensus tree can be obtained by removing at most $d$ leaves from each input tree, and describe an FPT algorithm with parameter $d$. Preliminary notions and problems statement ========================================== #### Trees. All trees in the rest of the document are assumed to be rooted and binary. If $T$ is a tree, we denote its root by $r(T)$ and its leaf set by ${\mathcal{L}}(T)$. Each leaf is labeled by a distinct element from a *label set* ${\mathcal{X}}$, and we denote by ${\mathcal{X}}(T)$ the set of labels of the leaves of $T$. We may sometimes use ${\mathcal{L}}(T)$ and ${\mathcal{X}}(T)$ interchangeably. For some $X \subseteq {\mathcal{X}}$, we denote by $lca_T(X)$ the *least common ancestor* of $X$ in $T$. The subtree rooted at a node $u \in V(T)$ is denoted $T_u$ and we may write ${\mathcal{L}}_T(u)$ for ${\mathcal{L}}(T_u)$. If $T_1$ and $T_2$ are two trees and $e$ is an edge of $T_1$, grafting $T_2$ on $e$ consists in subdividing $e$ and letting the resulting degree $2$ node become the parent of $r(T_2)$. Grafting $T_2$ above $T_1$ consists in creating a new node $r$, then letting $r$ become the parent of $r(T_1)$ and $r(T_2)$. Grafting $T_2$ on $T_1$ means grafting $T_2$ either on an edge of $T_1$ or above $T_1$. #### The leaf removal operation. =-1 For a subset $L \subseteq {\mathcal{X}}$, we denote by $T - L$ the tree obtained from $T$ by removing every leaf labeled by $L$, contracting the resulting non-root vertices of degree two, and repeatedly deleting the resulting root vertex while it has degree one. The *restriction* $T|_L$ of $T$ to $L$ is the tree $T - ({\mathcal{X}}\setminus L)$, *i.e.* the tree obtained by removing every leaf *not* in $L$. A *triplet* is a rooted tree on $3$ leaves. We denote a triplet ${R}$ with leaf set $\{a,b,c\}$ by $ab|c$ if $c$ is the leaf that is a direct child of the root (the parent of $a$ and $b$ being its other child). We say ${R}= ab|c$ is a triplet of a tree $T$ if $T|_{\{a,b,c\}} = {R}$. We denote $tr(T) = \{ab|c : ab|c$ is a triplet of $T\}$. We define a *distance function* $d_{LR}$ between two trees $T_1$ and $T_2$ on the same label set ${\mathcal{X}}$ consisting in the minimum number of labels to remove from ${\mathcal{X}}$ so that the two trees are equal. That is, $$d_{LR}(T_1, T_2) = \min \{ |X| : X \subseteq {\mathcal{X}}\mbox{ and } T_1 - X = T_2 - X \}$$ Note that $d_{LR}$ is closely related to the Maximum Agreement Subtree (MAST) between two trees on the same label set ${\mathcal{X}}$, which consists in a subset $X' \subseteq {\mathcal{X}}$ of maximum size such that $T_1|_{X'} = T_2|_{X'}$: $d_{LR}(T_1, T_2) = |{\mathcal{X}}| - |X'|$. The MAST of two binary trees on the same label set can be computed in time $O(n \log n)$, where $n = |{\mathcal{X}}|$ [@DBLP:journals/siamcomp/ColeFHPT00], and so $d_{LR}$ can be found within the [same]{} time complexity. #### Problem statements. In this paper, we are interested in finding a tree $T$ on ${\mathcal{X}}$ minimizing the sum of $d_{LR}$ distances to a given set of input trees. [`LR-Consensus`]{}\ [**Given**]{}: a set of trees ${\mathcal{T}}= \{T_1, \ldots, T_t\}$ with ${\mathcal{X}}(T_1) = \ldots = {\mathcal{X}}(T_t) = {\mathcal{X}}$.\ [**Find**]{}: a tree $T$ on label set ${\mathcal{X}}$ that minimizes $\sum_{T_i \in {\mathcal{T}}} d_{LR}(T, T_i)$.\ We can reformulate the [`LR-Consensus`]{} problem as the problem of removing a minimum number of leaves from the input trees so that they are *compatible*. Although the equivalence between both formulations is obvious, the later formulation will often be more convenient. We need to introduce more definitions in order to establish this equivalence. A set of trees ${\mathcal{T}}= \{T_1, \ldots, T_t\}$ is called *compatible* if there is a tree $T$ such that ${\mathcal{X}}(T) = \bigcup_{T_i \in {\mathcal{T}}}{\mathcal{X}}(T_i)$ and $T|_{{\mathcal{X}}(T_i)} = T_i$ for every $i \in [t]$. In this case, we say that $T$ *displays* ${\mathcal{T}}$. A list ${\mathcal{C}}= ({\mathcal{X}}_1, \ldots, {\mathcal{X}}_t)$ of subsets of ${\mathcal{X}}$ is a *leaf-disagreement* for ${\mathcal{T}}$ if $\{T_1 - {\mathcal{X}}_1, \ldots, T_t - {\mathcal{X}}_t\}$ is compatible. The *size* of ${\mathcal{C}}$ is $\sum_{i \in [t]}|{\mathcal{X}}_i|$. We denote by ${AST_{LR}({\mathcal{T}})}$ the minimum size of a leaf-disagreement for ${\mathcal{T}}$, and may sometimes write ${AST_{LR}(T_1, \ldots, T_t)}$ instead of ${AST_{LR}({\mathcal{T}})}$. A subset ${\mathcal{X}}' \subseteq {\mathcal{X}}$ of labels is a *label-disagreement* for ${\mathcal{T}}$ if $\{T_1 - {\mathcal{X}}', \ldots, T_t - {\mathcal{X}}'\}$ is compatible. Note that, if ${\mathcal{T}}= \{T_1, T_2\}$, then the minimum size of a label-disagreement for ${\mathcal{T}}$ is $d_{LR}(T_1, T_2)$. We may now define the [`AST-LR`]{} problem [(see Figure \[fig:example\] for an example)]{}. `Agreement Subtrees by Leaf-Removals` ([`AST-LR`]{})\ [**Given**]{}: a set of trees ${\mathcal{T}}= \{T_1, \ldots, T_t\}$ with ${\mathcal{X}}(T_1) = \ldots = {\mathcal{X}}(T_t) = {\mathcal{X}}$.\ [**Find**]{}: a leaf-disagreement ${\mathcal{C}}$ for ${\mathcal{T}}$ of minimum size.\ {width="\textwidth"} \[lem:equiv-problems\] Let ${\mathcal{T}}= \{T_1, \ldots, T_t\}$ be a set of trees on the same label set [${\mathcal{X}}$]{}, with $n = |{\mathcal{X}}|$. Given a supertree $T$ such that $v := \sum_{T_i \in {\mathcal{T}}}d_{LR}(T, T_i)$, one can compute in time $O(t n \log(n))$ a leaf-disagreement ${\mathcal{C}}$ of size at most $v$. Conversely, given a leaf-disagreement ${\mathcal{C}}$ for ${\mathcal{T}}$ of size $v$, one can compute in time $O(t n \log^2 (tn))$ a supertree $T$ such that $\sum_{T_i \in {\mathcal{T}}}d_{LR}(T, T_i) \leq v$. From Lemma \[lem:equiv-problems\][^2] both problems share the same optimality value, the NP-hardness of one implies the hardness of the other and approximating one problem within a factor $c$ implies that the other problem can be approximated within a factor $c$. We conclude this subsection with the introduction of a parameterized variant of the [`AST-LR`]{} problem. **<span style="font-variant:small-caps;">[`AST-LR-d`]{}</span>**\ [**Input**]{}: a set of trees ${\mathcal{T}}= \{T_1, \ldots, T_t\}$ with ${\mathcal{L}}(T_1) = \ldots = {\mathcal{L}}(T_t) = {\mathcal{X}}$, and an integer $d$.\ [**Question**]{}: Are there ${\mathcal{X}}_1, \ldots, {\mathcal{X}}_t \subseteq {\mathcal{X}}$ such that $|{\mathcal{X}}_i| \leq d$ for each $i \in [t]$, and $\{T_1 - {\mathcal{X}}_1, \ldots, T_t - {\mathcal{X}}_t\}$ is compatible?\ =-1 We call a tree $T^*$ a *solution* to the ${\texttt{AST-LR-d}}{}$ instance if $d_{LR}(T_i, T^*) \leq d$ for each $i \in [t]$. #### Relation to other supertree/consensus tree problems. =-1 The most widely studied supertree problem based on modifying the input trees is the SPR Supertree problem, where arbitrarily large subtrees can be moved in the input trees to make them all agree (see [@whidden2014supertrees] and references there). The interest of this problem is that the SPR operation is very general, modelling lateral gene transfer and introgression. The LR operation we introduce is a limited SPR, where the displaced subtree is composed of a single leaf. An alternative to the SPR operation to move subtrees within a tree is the Edge Contraction (EC) operation, that contracts an edge of an input tree, thus increasing the degree of the parent node. This operation allows correcting the local misplacement of a full subtree. [`AST-EC`]{} is NP-complete but can be solved in $O((2t)^ptn^2)$ time where $p$ is the number of required EC operations [@DBLP:journals/siamcomp/Fernandez-BacaG15]. Compared to the two problems described above, an LR models a very specific type of error in evolutionary trees, that is the misplacement of a single taxon (a single leaf) in one of the input trees. This error occurs frequently in reconstructing evolutionary trees, and can be caused for example by some evolutionary process specific to the corresponding input tree (recent incomplete lineage sorting, or recent lateral transfer for example). Conversely, it is not well adapted to model errors, due for example to ancient evolutionary events that impacts large subtrees. However, an attractive feature of the LR operation is that computing the LR distance is equivalent to computing the [`MAST`]{} cost and is thus tractable, unlike the SPR distance which is hard to compute. This suggests that the [`LR-Consensus`]{} problem might be easier to solve than the SPR Supertree problem, and we provide indeed several tractability results. Compared to the [`AST-EC`]{} problem, the [`AST-LR`]{} problem is naturally more adapted to correct single taxa misplacements as the EC operation is very local and the number of EC required to correct a taxon misplacement is linear in the length of the path to its correct location, while the LR cost of correcting this is unitary. Last, [`LR-Consensus`]{} is more flexible than the [`MAST`]{} problem as it relies on modifications of the input trees, while with the way [`MAST`]{} corrects a misplaced leaf requires to remove this leaf from all input trees. This shows that the problems [`AST-LR`]{} and [`AST-LR-d`]{} complement well the existing corpus of gene trees correction models. Hardness and approximability of [`AST-LR`]{} {#sec:hardness} ============================================ In this section, we show that the ${\texttt{AST-LR}}{}$ problem is NP-hard, from which the [`LR-Consensus`]{} hardness follows. We then describe a simple factor $2$ approximation algorithm. The algorithm turns out to be useful for analyzing the worst case scenario for ${\texttt{AST-LR}}{}$ in terms of the required number of leaves to remove, as we show that there are ${\texttt{AST-LR}}{}$ instances that require removing about $n - \sqrt{n}$ leaves in each input tree. NP-hardness of [`AST-LR`]{} {#np-hardness-of-ast-lr .unnumbered} --------------------------- We assume here that we are considering the decision version of ${\texttt{AST-LR}}{}$, *i.e.* deciding whether there is a leaf-disagreement of size at most ${\ell}$ for a given ${\ell}$. We use a reduction from the [`MinRTI`]{} problem: given a set ${\mathcal{R}}$ of rooted triplets, find a subset ${\mathcal{R}}' \subset {\mathcal{R}}$ of minimum cardinality such that ${\mathcal{R}}\setminus {\mathcal{R}}'$ is compatible. The [`MinRTI`]{} problem is NP-Hard [@journals/dam/byrka2010] (even $W[2]$-hard and hard to approximate within a $O(\log n)$ factor). Denote by ${MINRTI({\mathcal{R}})}$ the minimum number of triplets to remove from ${\mathcal{R}}$ to attain compatibility. We describe the reduction here. Let ${\mathcal{R}}= \{R_1, \ldots, R_t\}$ be an instance of ${\texttt{MinRTI}}$, with the label set $L := \bigcup_{i = 1}^t {\mathcal{X}}(R_i)$. For a given integer $m$, we construct an [`AST-LR`]{} instance ${\mathcal{T}}= \{T_1, \ldots, T_t\}$ which is such that $MINRTI({\mathcal{R}}) \leq m$ if and only if ${AST_{LR}({\mathcal{T}})} \leq t(|L| - 3) + m$. We first construct a tree $Z$ with additional labels which will serve as our main gadget. Let $\{L_i\}_{1 \leq i \leq t}$ be a collection of $t$ new label sets, each of size $(|L|t)^{10}$, all disjoint from each other and all disjoint from $L$. Each tree in our ${\texttt{AST-LR}}{}$ instance will be on label set ${\mathcal{X}}= L \cup L_1 \cup \ldots \cup L_t$. For each $i \in [t]$, let $X_i$ be any tree with label set $L_i$. Obtain $Z$ by taking any tree on $t$ leaves $l_1, \ldots, l_t$, then replacing each leaf $l_i$ by the $X_i$ tree (*i.e.* $l_i$ is replaced by $r(X_i)$). Denote by $r_Z(X_i)$ the root of the $X_i$ subtree in $Z$. Then for each $i \in [t]$, we construct $T_i$ from $R_i$ as follows. Let $L' = L \setminus {\mathcal{X}}(R_i)$ be the set of labels not appearing in $R_i$, noting that $|L'| = |L| - 3$. Let $T_{L'}$ be any tree with label set $L'$, and obtain the tree $Z_i$ by grafting $T_{L'}$ on the edge between $r_Z(X_i)$ and its parent. Finally, $T_i$ is obtained by grafting $R_i$ above $Z_i$. [See Figure \[fig:hardness\] for an example.]{} Note that each tree $T_i$ has label set ${\mathcal{X}}$ as desired. Also, it is not difficult to see that this reduction can be carried out in polynomial time. This construction can now be used to show the following. {width="35.00000%"} \[thm:np-hard\] The [`AST-LR`]{} and [`LR-Consensus`]{} problems are NP-hard. [The idea of the proof is to show that in the constructed [`AST-LR`]{} instance, we are “forced” to solve the corresponding [`MinRTI`]{} instance. In more detail, we show that $MINRTI({\mathcal{R}}) \leq m$ if and only if ${AST_{LR}({\mathcal{T}})} \leq t(|L| - 3) + m$. In one direction, given a set ${\mathcal{R}}'$ of size $m$ such that ${\mathcal{R}}\setminus{\mathcal{R}}'$ is compatible, one can show that the following leaf removals from ${\mathcal{T}}$ make it compatible: remove, from each $T_i$, the leaves $L' = L \setminus {\mathcal{X}}(R_i)$ that were inserted into the $Z$ subtree, then for each $R_i \in {\mathcal{R}}'$, remove a single leaf in ${\mathcal{X}}(R_i)$ from $T_i$. This sums up to $t(|L| - 3) + m$ leaf removals. Conversely, it can be shown that there always exists an optimal solution for ${\mathcal{T}}$ that removes, for each $T_i$, all the leaves $L' = L \setminus {\mathcal{X}}(R_i)$ inserted in the $Z$ subtree, plus an additional single leaf $l$ from $m$ trees $T_{i_1}, \dots, T_{i_m}$ such that $l \in L$. The corresponding triplets $R_{i_1},\dots, R_{i_m}$ can be removed from ${\mathcal{R}}$ so that it becomes compatible.]{} Approximating [`AST-LR`]{} and bounding worst-case scenarios {#approximating-ast-lr-and-bounding-worst-case-scenarios .unnumbered} ------------------------------------------------------------ Given the above result, it is natural to turn to approximation algorithms in order to solve [`AST-LR`]{} or [`LR-Consensus`]{} instances. It turns out that there is a simple factor $2$ approximation for [`LR-Consensus`]{} which is achieved by interpreting the problem as finding a median in a metric space. Indeed, it is not hard to see that $d_{LR}$ is a metric (over the space of trees on the same label set ${\mathcal{X}}$). A direct consequence, using an argument akin to the one in [@books/gusfield1997 p.351], is the following. \[lem:2-approx\] The following is a factor $2$ approximation algorithm for [`LR-Consensus`]{}: return the tree $T \in {\mathcal{T}}$ that minimizes $\sum_{T_i \in {\mathcal{T}}}d_{LR}(T, T_i)$. Theorem \[lem:2-approx\] can be used to lower-bound the ‘worst’ possible instance of [`AST-LR`]{}. We show that in some cases, we can only keep about $\sqrt{|{\mathcal{X}}|}$ leaves per tree. That is, there are instances for which ${AST_{LR}({\mathcal{T}})} = \Omega(t(n - \sqrt{n}))$, where $t$ is the number of trees and $n = |{\mathcal{X}}|$. The argument is based on a probabilistic argument, for which we will make use of the following result [@bryant2003size Theorem 4.3.iv]. \[thm:expectedmast\] For any constant $c > e/\sqrt{2}$, there is some $n_0$ such that for all $n \geq n_0$, the following holds: if $T_1$ and $T_2$ are two binary trees on $n$ leaves chosen randomly, uniformly and independently, then $\mathbb{E}[d_{LR}(T_1, T_2)] \geq n - c\sqrt{n}$. \[thm:worst-case\] There are instances of [`AST-LR`]{} in which $\Omega(t(n - \sqrt{n}))$ leaves need to be deleted. =-1 The above is shown by demonstrating that, by picking a set ${\mathcal{T}}$ of $t$ random trees, the expected optimal sum of distances $\min_T \sum_{T_i \in {\mathcal{T}}} d_{LR}(T, T_i)$ is $\Omega(t(n - \sqrt{n})$. This is not direct though, since the tree $T^*$ that minimizes this sum is not itself random, and so we cannot apply Theorem \[thm:expectedmast\] directly on $T^*$. We can however, show that the tree $T' \in {\mathcal{T}}$ obtained using the 2-approximation, which is random, has expected sum of distances $\Omega(t(n - \sqrt{n}))$. Since $T^*$ requires, at best, half the leaf deletions of $T'$, the result follows. Note that finding a non-trivial upper bound on ${AST_{LR}({\mathcal{T}})}$ is open. Fixed-parameter tractability of [`AST-LR`]{} and [`AST-LR-d`]{}. {#sec:fpt} ================================================================ An alternative way to deal with computational hardness is parameterized complexity. In this section, we first show that [`AST-LR`]{} is fixed-parameter-tractable with respect to $q := {AST_{LR}({\mathcal{T}})}$. More precisely, we show that [`AST-LR`]{} can be solved in $O(12^q n^3)$ time, where $n := |{\mathcal{X}}|$. We then consider an alternative parameter $d$, and show that finding a tree $T^*$, if it exists, such that $d_{LR}(T_i, T^*) \leq d$ for every input tree $T_i$, can be done in $O(c^d d^{3d}(n^3 + tn \log n))$ time for some constant $c$. Parameterization by $q$ ----------------------- The principle of the algorithm is the following. It is known that a set of trees ${\mathcal{T}}= \{T_1, \ldots, T_t\}$ is compatible if and only if the union of their triplet decomposition $tr({\mathcal{T}}) = \bigcup_{T_i \in {\mathcal{T}}} tr(T_i)$ is compatible [@bryant1997building]. In a step-by-step fashion, we identify a conflicting set of triplets in $tr({\mathcal{T}})$, each time branching into the (bounded) possible leaf-removals that can resolve the conflict. We stop when either $tr({\mathcal{T}})$ is compatible after the performed leaf-removals, or when more than $q$ leaves were deleted. We employ a two phase strategy. In the first phase, we eliminate direct conflicts in $tr({\mathcal{T}})$, *i.e.* if at least two of $ab|c, ac|b$ and $bc|a$ appear in $tr({\mathcal{T}})$, then we recursively branch into the three ways of choosing one of the $3$ triplets, and remove one leaf in each $T_i$ disagreeing with the chosen triplet (we branch into the three possible choices, either removing $a$, $b$ or $c$). The chosen triplet is locked in $tr({\mathcal{T}})$ and cannot be changed later. When the first phase is completed, there are no direct conflicts and $tr({\mathcal{T}})$ consists in a *full set of triplets* on ${\mathcal{X}}$. That is, for each distinct $a,b,c \in {\mathcal{X}}$, $tr({\mathcal{T}})$ contains exactly one triplet on label set $\{a,b,c\}$. Now, a full set of triplets is not necessarily compatible, and so in the second phase we modify $tr({\mathcal{T}})$, again deleting leaves, in order to make it compatible. Only the triplets that have not been locked previously can be modified. This second phase is analogous to the FPT algorithm for *dense* [`MinRTI`]{} presented in [@guillemot2010kernel]. The dense [`MinRTI`]{} is a variant of the [`MinRTI`]{} problem, introduced in Section \[sec:hardness\], in which the input is a full set of triplets and one has to decide whether $p$ triplets can be deleted to attain compatibility. A full set of triplets ${\mathcal{R}}$ is compatible if and only if for any set of four labels $\{a,b,c,d\}$, ${\mathcal{R}}$ does not contain the subset $\{ab|c, cd|b, bd|a\}$ nor the subset $\{ab|c, cd|b, ad|b\}$. One can check, through an exhaustive enumeration of the possibilities, that there are only four ways to correct a conflicting set of triplets ${R}_1, {R}_2, {R}_3$ where ${R}_1 = ab|c, {R}_2 = cd|b, {R}_3 \in \{bd|a, ad|b\}$. We can: (1) transform ${R}_1$ to $bc|a$; (2) transform ${R}_1$ to $ac|b$; (3) transform ${R}_2$ to $bd|c$; (4) transform ${R}_3$ to $ab|d$. This leads to a $O(4^p n^3)$ algorithm for solving dense [`MinRTI`]{}: find a conflicting set of four labels, and branch on the four possibilities, locking the transformed triplet each time. =-1 For the second phase of [`AST-LR`]{}, we propose a slight variation of this algorithm. Each time a triplet ${R}$ is chosen and locked, say ${R}= ab|c$, the trees containing $ac|b$ or $bc|a$ must loose $a,b$ or $c$. We branch into these three possibilities. Thus for each conflicting $4$-set, there are four ways of choosing a triplet, then for each such choice, three possible leaves to delete from a tree. This gives $12$ choices to branch into recursively. Algorithm \[alg:fpt2\] is described in detail in the Appendix and its analysis yields the following. \[thm:fpt-in-q\] [`AST-LR`]{} can be solved in time $O(12^q t n^3)$. Although Theorem \[thm:fpt-in-q\] is theoretically interesting as it shows that [`AST-LR`]{} is in FPT with respect to $q$, the $12^q$ factor might be too high for practical purposes, motivating the alternative approach below. Parameterization by maximum distance $d$ ---------------------------------------- We now describe an algorithm for the [`AST-LR-d`]{} problem, running in time $O(c^d d^{3d}(n^3 + tn \log n))$ that, if it exists, finds a solution (where here $c$ is a constant not depending on $d$ nor $n$). We employ the following branch-and-bound strategy, keeping a candidate solution at each step. Initially, the candidate solution is the input tree $T_1$ and, if $T_1$ is indeed a solution, we return it. Otherwise (in particular if $d_{LR}(T_1,T_i) > d$ for some input tree $T_i$), we branch on a set of “leaf-prune-and-regraft” operations on $T_1$. In such an operation, we prune one leaf from $T_1$ and regraft it somewhere else. If we have not produced a solution after $d$ such operations, then we halt this branch of the algorithm (as any solution must be reachable from $T_1$ by at most $d$ operations). The resulting search tree has depth at most $d$. In order to bound the running time of the algorithm, we need to bound the number of “leaf-prune-and-regraft” operations to try at each branching step. There are two steps to this: first, we bound the set of candidate leaves to prune, second, given a leaf, we bound the number of places where to regraft it. To bound the candidate set of leaves to prune, let us call a leaf $x$ *interesting* if there is a solution $T^*$, and minimal sets $X_1,X_i \subseteq {\mathcal{X}}$ of size at most $d$, such that $T_1 - X_1 = T^* - X_1$, $T_i - X_i = T^* - X_i$, and $x \in X_1 \setminus X_i$, where $T_i$ is an arbitrary input tree for which $d_{LR}(T_1,T_i) > d$. It can be shown that an interesting leaf $x$ must exist if there is a solution. Moreover, though we cannot identify $x$ before we know $T^*$, we can nevertheless construct a set $S$ of size $O(d^2)$ containing all interesting leaves. Thus, in our branching step, it suffices to consider leaves in $S$. [Assuming we have chosen the correct $x$, we then bound the number of places to try regrafting $x$. Because of the way we chose $x$, we may assume there is a solution $T^*$ and $X_i \subseteq {\mathcal{X}}$ such that $|X_i| \leq d$, $T_i - X_i = T^* - X_i$ and $x \notin X_i$. Thus we may treat $T_i$ as a “guide” on where to regraft $x$. Due to the differences between $T_1$, $T_i$ and $T^*$, this guide does not give us an exact location in $T_1$ to regraft $x$. Nevertheless, we can show that the number of candidate locations to regraft $x$ can be bounded by $O(d)$. Thus, in total we have $O(d^3)$ branches at each step in our search tree of depth $d$, and therefore have to consider $O((O(3^d))^{d}) = O(c^dd^{3d})$ subproblems.]{} \[thm:fpt-in-d\] [`AST-LR-d`]{} can be solved in time $O(c^d d^{3d}(n^3 + tn \log n))$, where $c$ is a constant not depending on $d$ or $n$. Conclusion ========== To conclude, we introduced a new supertree/consensus problem, based on a simple combinatorial operator acting on trees, the Leaf-Removal. We showed that, although this supertree problem is NP-hard, it admits interesting tractability results, that compare well with existing algorithms. Future research should explore if various simple combinatorial operators, that individually define relatively tractable supertree problems (for example LR and EC) can be combined into a unified supertree problem while maintaining approximability and fixed-parameter tractability. [MJ was partially supported by Labex NUMEV (ANR-10-LABX-20) and Vidi grant 639.072.602 from The Netherlands Organization for Scientific Research (NWO). CC was supported by NSERC Discovery Grant 249834. CS was partially supported by the French Agence Nationale de la Recherche Investissements d’Avenir/Bioinformatique (ANR-10-BINF-01-01, ANR-10- BINF-01-02, Ancestrome). ML was supported by NSERC PDF Grant. ]{} Omitted proofs ============== [Here we give proofs for several results whose proofs were omitted in the main paper. Note that the proof of Theorem \[thm:fpt-in-d\] is deferred to its own section.]{} **Lemma \[lem:equiv-problems\]** *(restated). Let ${\mathcal{T}}= \{T_1, \ldots, T_t\}$ be a set of trees on the same label set ${\mathcal{X}}$. Then, given a supertree $T$ such that $v := \sum_{T_i \in {\mathcal{T}}}d_{LR}(T, T_i)$, one can compute in time $O(t n \log n)$ a leaf-disagreement ${\mathcal{C}}$ of size at most $v$, where $n = |{\mathcal{X}}|$. Conversely, given a leaf-disagreement ${\mathcal{C}}$ for ${\mathcal{T}}$ of size $v$, one can compute in time $O(t n \log^2 (tn))$ a supertree $T$ such that $\sum_{T_i \in {\mathcal{T}}}d_{LR}(T, T_i) \leq v$.* In the first direction, for each $T_i \in {\mathcal{T}}$, there is a set $X_i \subseteq {\mathcal{X}}$ of size $d_{LR}(T, T_i)$ such that $T_i - X_i = T - X_i$. Moreover, $X_i$ can be found in time $O(n \log n)$. Thus $(X_1, \ldots, X_t)$ is a leaf-disagreement of the desired size and can be found in time $O(t n \log n)$. Conversely, let ${\mathcal{C}}= (X_1, \ldots, X_t)$ be a leaf-disagreement of size $v$. As ${\mathcal{T}}' = \{T_1 - X_1, \ldots, T_t - X_t\}$ is compatible, there is a tree $T$ that displays ${\mathcal{T}}'$, and it is easy to see that the sum of distances between $T$ and ${\mathcal{T}}'$ is at most the size of ${\mathcal{C}}$. As for the complexity, it is shown in [@deng_et_al:LIPIcs:2016:6088] how to compute in time $O(tn \log^2 (tn))$, given a set of trees ${\mathcal{T}}'$, a tree $T$ displaying ${\mathcal{T}}'$ if one exists. We next consider the case where ${\mathcal{T}}$ consists only of two trees. \[lem:two-trees\] Let $T_1, T_2$ be two trees on the same label set ${\mathcal{X}}$. Then ${AST_{LR}(T_1, T_2)} = d_{LR}(T_1, T_2)$. Moreover, every optimal leaf-disagreement ${\mathcal{C}}= ({\mathcal{X}}_1', {\mathcal{X}}_2')$ for $T_1$ and $T_2$ can be obtained in the following manner: for every label-disagreement ${\mathcal{X}}'$ of size $d_{LR}(T_1, T_2)$, partition ${\mathcal{X}}'$ into ${\mathcal{X}}'_1, {\mathcal{X}}'_2$. Let ${\mathcal{X}}' \subset {\mathcal{X}}$ such that $|{\mathcal{X}}'| = d_{LR}(T_1, T_2)$ and $T_1 - {\mathcal{X}}' = T_2 - {\mathcal{X}}'$. Then clearly, for any bipartition $({\mathcal{X}}'_1, {\mathcal{X}}'_2)$ of ${\mathcal{X}}'$, $T_1' := T_1 - {\mathcal{X}}'_1$ and $T_2' := T_2 - {\mathcal{X}}'_2$ are compatible, since the leaves that $T_1'$ and $T_2'$ have in common yield the same subtree, and leaves that appear in only one tree cannot create incompatibility. In particular, ${AST_{LR}(T_1, T_2)} \leq d_{LR}(T_1, T_2)$. Conversely, let ${\mathcal{C}}= ({\mathcal{X}}'_1, {\mathcal{X}}'_2)$ be a minimum leaf-disagreement. We have ${\mathcal{X}}'_1 \cap {\mathcal{X}}'_2 = \emptyset$, for if there is some ${\ell}\in {\mathcal{X}}'_1 \cap {\mathcal{X}}'_2$, then ${\ell}$ could be reinserted into one of the two trees without creating incompatibility. Thus ${\mathcal{C}}$ is a bipartition of ${\mathcal{X}}' = {\mathcal{X}}'_1 \cup {\mathcal{X}}'_2$. Moreover, we must have $T_1 - {\mathcal{X}}' = T_2 - {\mathcal{X}}'$, implying $|{\mathcal{X}}'| \geq d_{LR}(T_1, T_2)$. Combined with the above inequality, $|{\mathcal{X}}'| = d_{LR}(T_1, T_2)$, and the Lemma follows. It follows from Lemma \[lem:two-trees\] that any optimal label-disagreement ${\mathcal{X}}'$ can be turned into an optimal leaf-disagreement, which is convenient as ${\mathcal{X}}'$ can be found in polynomial time. We will make heavy use of this property later on. Note that the same type of equivalence does not hold when $3$ or more trees are given, *i.e.* computing a MAST of three trees does not necessarily yield a leaf-disagreement of minimum size. Consider for example the instance ${\mathcal{T}}= \{T_1,T_2,T_3\}$ in Figure \[fig:example\]. An optimal leaf-disagreement for ${\mathcal{T}}$ has size $2$ and consists of any pair of distinct leaves. On the other hand, an optimal leaf-disagreement for ${\mathcal{T}}$ has size $3$, and moreover each leaf corresponds to a different label. **Theorem \[thm:np-hard\]** *(restated). The [`AST-LR`]{} and [`LR-Consensus`]{} problems are NP-hard.* We begin by restating the reduction from ${\texttt{MinRTI}}$ to [`AST-LR`]{}. Let ${\mathcal{R}}= \{R_1, \ldots, R_t\}$ be an instance of ${\texttt{MinRTI}}$, with the label set $L := \bigcup_{i = 1}^t {\mathcal{X}}(R_i)$. For a given integer $m$, we construct an [`AST-LR`]{} instance ${\mathcal{T}}= \{T_1, \ldots, T_t\}$ which is such that $MINRTI({\mathcal{R}}) \leq m$ if and only if ${AST_{LR}({\mathcal{T}})} \leq t(|L| - 3) + m$. We first construct a tree $Z$ with additional labels which will serve as our main gadget. Let $\{L_i\}_{1 \leq i \leq t}$ be a collection of $t$ new label sets, each of size $(|L|t)^{10}$, all disjoint from each other and all disjoint from $L$. Each tree in our ${\texttt{AST-LR}}{}$ instance will be on label set ${\mathcal{X}}= L \cup L_1 \cup \ldots \cup L_t$. For each $i \in [t]$, let $X_i$ be any tree with label set $L_i$. Obtain $Z$ by taking any tree on $t$ leaves $l_1, \ldots, l_t$, then replacing each leaf $l_i$ by the $X_i$ tree (*i.e.* $l_i$ is replaced by $r(X_i)$). Denote by $r_Z(X_i)$ the root of the $X_i$ subtree in $Z$. Then for each $i \in [t]$, we construct $T_i$ from $R_i$ as follows. Let $L' = L \setminus {\mathcal{X}}(R_i)$ be the set of labels not appearing in $R_i$, noting that $|L'| = |L| - 3$. Let $T_{L'}$ be any tree with label set $L'$, and obtain the tree $Z_i$ by grafting $T_{L'}$ on the edge between $r_Z(X_i)$ and its parent. Finally, $T_i$ is obtained by grafting $R_i$ above $Z_i$. [See Figure \[fig:hardness\] for an example.]{} Note that each tree $T_i$ has label set ${\mathcal{X}}$ as desired. Also, it is not difficult to see that this reduction can be carried out in polynomial time. We now show that $MINRTI({\mathcal{R}}) \leq m$ if and only if ${AST_{LR}({\mathcal{T}})} \leq t(|L| - 3) + m$. ($\Rightarrow$) Let ${\mathcal{R}}' \subset {\mathcal{R}}$ such that $|{\mathcal{R}}'| \leq m$ and ${\mathcal{R}}^* := {\mathcal{R}}\setminus {\mathcal{R}}'$ is compatible, and let $T({\mathcal{R}}^*)$ be a tree displaying ${\mathcal{R}}^*$. Note that $|{\mathcal{R}}^*| \geq t - m$. We obtain a [`AST-LR`]{} solution by first deleting, in each $T_i \in {\mathcal{T}}$, all the leaves labeled by $L \setminus {\mathcal{X}}(R_i)$ (thus $T_i$ becomes the tree obtained by grafting $R_i$ above $Z$). Then for each deleted triplet $R_i \in {\mathcal{R}}'$, we remove any single leaf of $T_i$ labeled by some element in ${\mathcal{X}}(R_i)$. In this manner, no more than $t(|L| - 3) + m$ leaves get deleted. Moreover, grafting $T({\mathcal{R}}^*)$ above $Z$ yields a tree displaying the modified set of trees, showing that they are compatible. ($\Leftarrow$) We first argue that if ${\mathcal{T}}$ admits a leaf-disagreement ${\mathcal{C}}= ({\mathcal{X}}_1, \ldots, {\mathcal{X}}_t)$ of size at most $t(|L| - 3) + m$, then there is a better or equal solution that removes, in each $T_i$, all the leaves labeled by $L \setminus {\mathcal{X}}(R_i)$ (*i.e.* those grafted in the $Z_i$ tree). For each $i \in [t]$, let $T_i' = T_i - {\mathcal{X}}_i$, and denote ${\mathcal{T}}' = \{T_1', \ldots, T_t'\}$. Suppose that there is some $i \in [t]$ and some ${\ell}\in L \setminus {\mathcal{X}}(R_i)$ such that ${\ell}\in {\mathcal{X}}(T_i')$. We claim that ${\ell}\notin {\mathcal{X}}(T_j')$ for every $i \neq j \in [t]$. Suppose otherwise that ${\ell}\in {\mathcal{X}}(T_j')$ for some $j \neq i$. Consider first the case where ${\ell}\notin {\mathcal{X}}(R_j)$. Note that by the construction of $Z_i$ and $Z_j$, for every $x_i \in {\mathcal{X}}(X_i) \cap {\mathcal{X}}(T_i') \cap {\mathcal{X}}(T_j')$ and every $x_j \in {\mathcal{X}}(X_j) \cap {\mathcal{X}}(T_i') \cap {\mathcal{X}}(T_j')$, $T_i'$ contains the ${\ell}x_i | x_j$ triplet whereas $T_j'$ contains the ${\ell}x_j | x_i$ triplet. Since these triplets are conflicting, no supertree can contain both and so no such $x_i, x_j$ pair can exist, as we are assuming that a supertree for $T'_i$ and $T'_j$ exists. This implies that one of ${\mathcal{X}}(X_i) \cap {\mathcal{X}}(T_i') \cap {\mathcal{X}}(T_j')$ or ${\mathcal{X}}(X_j) \cap {\mathcal{X}}(T_i') \cap {\mathcal{X}}(T_j')$ must be empty. Suppose without loss of generality that the former is empty. Then each $x_i \in X_i$ must have been deleted in at least one of $T_i$ or $T_j$. As $|{\mathcal{X}}(X_i)| = (|L|t)^{10} > t(|L| - 3) + m$, this contradicts the size of the solution ${\mathcal{C}}$. In the second case, we have ${\ell}\in {\mathcal{X}}(R_j)$. But this time, if there are $x_i \in {\mathcal{X}}(X_i) \cap {\mathcal{X}}(T_i') \cap {\mathcal{X}}(T_j')$ and $x_j \in {\mathcal{X}}(X_j) \cap {\mathcal{X}}(T_i') \cap {\mathcal{X}}(T_j')$, then $T_j'$ contains the $x_ix_j|{\ell}$ triplet, again conflicting with the ${\ell}x_i | x_j$ triplet found in $T_i$. As before, we run into a contradiction since too many $X_i$ or $X_j$ leaves need to be deleted. This proves our claim. We thus assume that ${\ell}$ only appears in $T_i'$. Let $R_j \in {\mathcal{R}}$ such that ${\ell}\in {\mathcal{X}}(R_j)$, noting that ${\ell}$ does not appear in $T'_j$. Consider the solution ${\mathcal{T}}''$ obtained from ${\mathcal{T}}'$ by removing ${\ell}$ from $T_i'$, and placing it back in $T_j'$ where it originally was in $T_j$. Formally this is achieved by replacing, in the leaf-disagreement ${\mathcal{C}}$, ${\mathcal{X}}_i$ by ${\mathcal{X}}_i \cup \{{\ell}\}$ and ${\mathcal{X}}_j$ by ${\mathcal{X}}_j \setminus \{{\ell}\}$. Since ${\ell}$ still appears only in one tree, no conflict is created and we obtain another solution of equal size. By repeating this process for every such leaf ${\ell}$, we obtain a solution in which every leaf labeled by $L \setminus {\mathcal{X}}(R_i)$ is removed from $T_i'$. We now assume that the solution ${\mathcal{T}}'$ has this form. Consider the subset ${\mathcal{R}}' = \{R_i \in {\mathcal{R}}: |{\mathcal{X}}(T_i') \cap {\mathcal{X}}(R_i)| < 3\}$, that is those triplets $R_i$ for which the corresponding tree $T_i$ had a leaf removed outside of the $Z_i$ tree. By the form of the ${\mathcal{T}}'$ solution, at least $t(|L| - 3)$ removals are done in the $Z_i$ trees, and as only $m$ removals remain, ${\mathcal{R}}'$ has size at most $m$. We show that ${\mathcal{R}}\setminus {\mathcal{R}}'$ is a compatible set of triplets. Since ${\mathcal{T}}'$ is compatible, there is a tree $T$ that displays each $T_i' \in {\mathcal{T}}'$, and since each triplet of ${\mathcal{R}}\setminus {\mathcal{R}}'$ belongs to some $T_i'$, $T$ also displays ${\mathcal{R}}\setminus {\mathcal{R}}'$. This concludes the proof. **Theorem \[lem:2-approx\]** *(restated). The following is a factor $2$ approximation algorithm for [`LR-Consensus`]{}: return the tree $T \in {\mathcal{T}}$ that minimizes $\sum_{T_i \in {\mathcal{T}}}d_{LR}(T, T_i)$.* Let $T^*$ be an optimal solution for [`LR-Consensus`]{}, *i.e.* $T^*$ is a tree minimizing $\sum_{T_i \in {\mathcal{T}}}d_{LR}(T_i, T^*)$, and let $T$ be chosen as described in the theorem statement. Moreover let $T'$ be the tree of ${\mathcal{T}}$ minimizing $d_{LR}(T', T^*)$. By the triangle inequality, $$\sum_{T_i \in {\mathcal{T}}}d_{LR}(T', T_i) \leq \sum_{T_i \in {\mathcal{T}}}\left( d_{LR}(T', T^*) + d_{LR}(T^*, T_i) \right) \leq 2 \sum_{T_i \in {\mathcal{T}}}d_{LR}(T^*, T_i)$$ where the last inequality is due to the fact that $d_{LR}(T', T^*) \leq d_{LR}(T^*, T_i)$ for all $i$, by our choice of $T'$. Our choice of $T$ implies $\sum_{T_i \in {\mathcal{T}}}d_{LR}(T, T_i) \leq \sum_{T_i \in {\mathcal{T}}}d_{LR}(T', T_i) \leq 2 \sum_{T_i \in {\mathcal{T}}}d_{LR}(T_i, T^*)$. **Corollary \[thm:worst-case\]** *(restated). There are instances of [`AST-LR`]{} in which $\Omega(t(n - \sqrt{n}))$ leaves need to be deleted.* Let ${\mathcal{T}}= \{T_1, \ldots, T_t\}$ be a random set of $t$ trees chosen uniformly and independently. For large enough $n$, the expected sum of distances between each pair of trees is $$\mathbb{E}\left[\sum_{1 \leq i < j \leq t}d_{LR}(T_i, T_j)\right] = \sum_{1 \leq i < j \leq t} \mathbb{E}[d_{LR}(T_i, T_j)] \geq {t \choose 2} (n - c\sqrt{n})$$ for some constant $c$, by Theorem \[thm:expectedmast\]. Let $S := \min_{T} \sum_{i = 1}^t d_{LR}(T, T_i)$ be the random variable corresponding to the minimum sum of distances. By Theorem \[lem:2-approx\], there is a tree $T' \in {\mathcal{T}}$ such that $\sum_{i = 1}^t d_{LR}(T', T_i) \leq 2S$. We have $$\begin{aligned} \sum_{1 \leq i < j \leq t}d_{LR}(T_i, T_j) &\leq \sum_{1 \leq i < j \leq t} d_{LR}(T_i, T') + d_{LR}(T', T_j) \\ &= (t - 1) \sum_{i = 1}^t d_{LR}(T_i, T') \\ &\leq (t - 1)2S\end{aligned}$$ Since, in general for two random variables $X$ and $Y$, always having $X \leq Y$ implies $\mathbb{E}[X] \leq \mathbb{E}[Y]$, we get $${t \choose 2}(n - c\sqrt{n}) \leq \mathbb{E}\left[\sum_{1 \leq i < j \leq t}d_{LR}(T_i, T_j)\right] \leq \mathbb{E}[(t - 1)2S] = 2(t - 1)\mathbb{E}[S]$$ yielding $\mathbb{E}[S] \geq t/4 (n - c\sqrt{n}) = \Omega(t(n - \sqrt{n}))$, and so there must exist an instance ${\mathcal{T}}$ satisfying the statement. **Theorem \[thm:fpt-in-q\]** *(restated). [`AST-LR`]{} can be solved in time $O(12^q t n^3)$.* We provide an algorithm, Algorithm \[alg:fpt2\], for [`AST-LR`]{}, and prove its correctness and complexity. ${\mathcal{T}}$ is the set of input trees, $q$ is the maximum number of leaves to delete, $phase$ is the current phase number (initially $1$), $F$ is the set of locked triplets so far Return FALSE \[fpt-q:line-remleaf\] Branching: If one of the following calls returns True $\textsc{mastrl}(({\mathcal{T}}\setminus \{T_i\}) \cup \{T_i - \{a\}\} ,q - 1, phase, F)$ //remove $a$ from $T_i$ $\textsc{mastrl}(({\mathcal{T}}\setminus \{T_i\}) \cup \{T_i - \{b\}\},q - 1, phase, F)$ //remove $b$ from $T_i$ $\textsc{mastrl}(({\mathcal{T}}\setminus \{T_i\}) \cup \{T_i - \{c\}\},q - 1, phase, F)$ //remove $c$ from $T_i$ then Return True, otherwise Return False \[fpt-q:line-direct\] Branching: If one of the following calls returns True $\textsc{mastrl}({\mathcal{T}}, q, phase, F \cup \{ab|c\})$ $\textsc{mastrl}({\mathcal{T}}, q, phase, F \cup \{ac|b\})$ $\textsc{mastrl}({\mathcal{T}}, q, phase, F \cup \{bc|a\})$ then Return True, otherwise Return False Return $\textsc{mastrl}({\mathcal{T}}, q, 2, F)$ //enter phase 2 \[fpt-q:line-indirect\] Branching: If one of the following calls returns True $\textsc{mastrl}({\mathcal{T}}, q, phase, F \cup \{ac|b\})$ $\textsc{mastrl}({\mathcal{T}}, q, phase, F \cup \{bc|a\})$ $\textsc{mastrl}({\mathcal{T}}, q, phase, F \cup \{bd|c\})$ $\textsc{mastrl}({\mathcal{T}}, q, phase, F \cup \{ab|d\})$ then Return True, otherwise Return False Return True //There are no conflicts $\Rightarrow tr({\mathcal{T}}) \cup F$ is compatible We first argue that the algorithm is correct. First observe that the algorithm only returns TRUE when a conflict-free set of triplets is attained without deleting more than $q$ leaves, and so there are no false-positives. Moreover, it is not hard to see that the first phase of the algorithm tries every possible way of obtaining a full set of triplets from $tr({\mathcal{T}})$ using at most $q$ leaf removals. Indeed, for every set of $3$ labels $a,b,c$ that are present in a direct conflict, the algorithm branches into the $3$ ways of locking $ab|c$, $ac|b$ or $bc|a$ and for each tree $T_i$ disagreeing with the chosen triplet, all three ways of removing a leaf to agree with the chosen triplet are tested. In a similar manner, for each dense set $D$ of triplets that is attained, each way of freeing $D$ from conflicting $4$-sets is evaluated (not only in the ways of choosing a triplet to resolve the conflict, but also in the ways of removing leaves from the trees of ${\mathcal{T}}$ so that they agree with the locked triplets). It follows that if $tr({\mathcal{T}})$ can be made compatible by deleting at most $q$ leaves, then some leaf in the branch tree created by the algorithm will return TRUE, and so there are no false-negatives. As for the complexity, when the algorithm enters the ‘else if’ block of line \[fpt-q:line-remleaf\], it branches into $3$ cases that decrement $q$. When it enters the ‘if’ block of line \[fpt-q:line-direct\], it branches into $3$ cases but $q$ is not decremented. However each of these $3$ recursive calls immediately leads to the ‘else if’ block on line \[fpt-q:line-remleaf\], and so this case can be seen as branching into $9$ cases. Similarly, when the algorithm enters the ‘if’ block of line \[fpt-q:line-indirect\], it branches into $4$ cases, each of which leads to the $3$ subcases following line \[fpt-q:line-remleaf\]. Thus $12$ cases are considered. Therefore, the branching tree created by the algorithm has degree at most $12$ and depth at most $q$, and so at most $12^q$ cases are considered. Finally, each call to the algorithm requires time $O(t n^3)$ since this is the time required to identify conflicting sets of triplets within the $t$ trees. Leaf Prune-and-Regraft Moves ============================ Here we introduce the notion of leaf prune-and-regraft (LPR) moves, which will be used in the proof of Theorem \[thm:fpt-in-d\], and which may be of independent interest. In an LPR move, we prune a leaf from a tree and then regraft it another location (formal definitions below). LPR moves provide an alternate way of characterizing the distance function $d_{LR}$ - indeed, we will show that $d_{LR}(T_1, T_2) \leq k$ if and only if there is a sequence of at most $k$ LPR moves transforming $T_1$ into $T_2$. Let $T$ be a tree on label set ${\mathcal{X}}$. A LPR move on $T$ is a pair $({\ell}, e)$ where ${\ell}\in {\mathcal{X}}$ and $e \in \{E(T - \{{\ell}\}), {\perp}\}$. Applying $({\ell}, e)$ consists in grafting ${\ell}$ on the $e$ edge of $T - \{{\ell}\}$ if $e \neq {\perp}$, and above the root of $T - \{{\ell}\}$ if $e = {\perp}$. An *LPR sequence* $L = (({\ell}_1, e_1), \ldots, ({\ell}_k, e_k))$ is an ordered tuple of LPR moves, where for each $i \in [k]$, $({\ell}_i, e_i)$ is an LPR move on the tree obtained after applying the first $i - 1$ LPR moves of $L$. We may write $L = ({\ell}_1, \ldots, {\ell}_k)$ if the location at which the grafting takes place does not need to be specified. We say that $L$ turns $T_1$ into $T_2$ if, by applying each LPR move of $L$ in order on $T_1$, we obtain $T_2$. See Figure \[fig:LPRsequence\] for an example of an LPR sequence. In the following statements, we assume that $T_1$ and $T_2$ are two trees on label set ${\mathcal{X}}$. We exhibit an equivalence between leaf removals and LPR sequences, then show that the order of LPR moves in a sequence do not matter in terms of turning one tree into another - in particular any leaf can be displaced first. \[lem:lpr-seq-equiv\] There is a subset $X \subseteq {\mathcal{X}}$ such that $T_1 - X = T_2 - X$ if and only if there is an LPR sequence $(x_1, x_2, \ldots, x_k)$ turning $T_1$ into $T_2$ such that $X = \{x_1, \ldots, x_k\}$. If $T_1 = T_2$ then the proof is trivial, so we will assume this is not the case. We prove the lemma by induction on $|X|$. For the base case, suppose that $X = \{x\}$. If $T_1 - X = T_2 - X$, then let $T_m = T_1 - X = T_2 - X$. We find an LPR move $(x, e)$ with $e \in E(T_m) \cup \{{\perp}\}$ turning $T_1$ into $T_2$. Observe that $T_2$ can be obtained by grafting $x$ on $T_m$, either on an edge $uv$, in which case we set $e = uv$, or above the root, in which case we set $e = {\perp}$. Since $T_m = T_1 - \{x\}$, it follows that $(x, e)$ is an LPR move turning $T_1$ into $T_2$. In the other direction, assume there is an LPR move $(x,e)$ turning $T_1$ into $T_2$. Observe that for any tree $T'$ derived from $T_1$ by an LPR move using $x$, $T' - \{x\} = T_1 - \{x\}$. In particular, $T_2 - \{x\} = T_1 - \{x\}$ and we are done. For the induction step, assume that $|X|> 1$ and that the claim holds for any $X'$ such that $|X'| < |X|$. If $T_1 - X = T_2 - X$, then define $T_m = T_1 - X$, and let $x$ be an arbitrary element of $X$. We will first construct a tree $T_1'$ such that $T_1 - \{x\} = T_1' - \{x\}$ and $T_1' - (X\setminus \{x\}) = T_2 - (X\setminus\{x\})$. Observe that $T_2 - (X\setminus\{x\})$ can be obtained by grafting $x$ in $T_m$. Let $e = uv$ if this grafting takes place on an edge of $T_m$ with $v$ being the child of $u$, or $e = {\perp}$ if $x$ is grafted above $T_m$, and in this case let $v = r(T_m)$. Let $v' = v_{T_1 - \{x\}}$ be the node in $T_1 - \{x\}$ corresponding to $v$. Let $T_1'$ be derived from $T_1 - \{x\}$ by grafting $x$ onto the edge between $v'$ and its parent if $v'$ is non-root, and grafting above $v'$ otherwise. It is clear that $T_1 - \{x\} = T_1' - \{x\}$. Furthermore, by our choice of $v'$ we have that $T_1' - (X\setminus \{x\}) = T_2 - (X\setminus\{x\})$. Now that we have $T_1 - \{x\} = T_1' - \{x\}$ and $T_1' - (X\setminus \{x\}) = T_2 - (X\setminus\{x\})$, by the inductive hypothesis there is an LPR sequence turning $T_1$ into $T_1'$ consisting of a single move $(x,e)$, and an LPR sequence $(x_1, x_2, \ldots, x_{k-1})$ turning $T_1'$ into $T_2$ such that $\{x_1, \ldots, x_{k'}\} = (X\setminus \{x\})$. Then by concatenating these two sequences, we have an LPR sequence $(x_1, x_2, \ldots, x_k)$ turning $T_1$ into $T_2$ such that $X = \{x_1, \ldots, x_k\}$. For the converse, suppose that there is an LPR sequence $(x_1, x_2, \ldots, x_k)$ turning $T_1$ into $T_2$ such that $X = \{x_1, \ldots, x_k\}$. Let $T_1'$ be the tree derived from $T_1$ by applying the first move in this sequence. That is, there is an LPR move $(x_1,e)$ turning $T_1$ into $T_1'$, and there is an LPR sequence $(x_2, \ldots, x_k)$ turning $T_1'$ into $T_2$. Then by the inductive hypothesis $T_1 - \{x_1\} = T_1' - \{x_1\}$ and $T_1' - \{x_2, \ldots, x_k\} = T_2 - \{x_2, \ldots, x_k\}$. Thus, $T_1- X = T_1' - X = T_2 - X$, as required. \[lem:lpr-order\] If there is an LPR sequence $L = (x_1, \ldots, x_k)$ turning $T_1$ into $T_2$, then for any $i \in [k]$, there is an LPR sequence $L' = (x'_1, \ldots, x'_k)$ turning $T_1$ into $T_2$ such that $x'_1 = x_i$ and $\{x_1, \ldots, x_k\} = \{x'_1, \ldots, x'_k\}$. Consider again the proof that if $T_1 - X = T_2 - X$ then there is an LPR sequence $(x_1, \dots x_k)$ turning $T_1$ into $T_2$ such that $X = \{x_1, \dots, x_k\}$ (given in the proof of Lemma \[lem:lpr-seq-equiv\]). When $|X| > 1$, we construct this sequence by concatenating the LPR move $(x,e)$ with an LPR sequence of length $|X|-1$, where $x$ is an arbitrary element of $X$. As we could have chosen any element of $X$ to be $x$, we have the following: If $T_1 - X = T_2 - X$ then for each $x\in X$, there is an LPR sequence $(x_1, \dots, x_k)$ turning $T_1$ into $T_2$ such that $X = \{x_1, \dots, x_k\}$ and $x_1 = x$. Thus our proof is as follows: Given an LPR sequence $L = (x_1, \ldots, x_k)$ turning $T_1$ into $T_2$ and some $i \in [k]$, Lemma \[lem:lpr-seq-equiv\] implies that $T_1 - \{x_1, \ldots, x_k\} = T_2 - \{x_1, \ldots, x_k\}$. By the observation above, this implies that there is an LPR sequence $(x'_1, \dots, x'_k)$ turning $T_1$ into $T_2$ such that $\{x_1, \ldots, x_k\} = \{x'_1, \ldots, x'_k\}$ and $x'_1 = x$. Proof of Theorem \[thm:fpt-in-d\] ================================= This section makes use of the concept of LPR moves, which are introduced in the previous section. As discussed in the main paper, we employ a branch-and-bound style algorithm, in which at each step we alter a candidate solution by pruning and regrafting a leaf. That is, we apply an LPR move. The technically challenging part is bound the number of possible LPR moves to try. To do this, we will prove Lemma \[lem:disagreementKernel2\], which provides a bound on the number of leaves to consider, and Lemma \[lem:dont-check-too-many-trees\], which bounds the number of places a leaf may be regrafted to. Denote by $tr(T)$ the set of rooted triplets of a tree $T$. Two triplets ${R}_1 \in tr(T_1)$ and ${R}_2 \in tr(T_2)$ are *conflicting* if ${R}_1 = ab|c$ and ${R}_2 \in \{ac|b, bc|a\}$. We denote by $conf(T_1, T_2)$ the set of triplets of $T_1$ for which there is a conflicting triplet in $T_2$. That is, $conf(T_1, T_2) = \{ab|c \in tr(T_1) : ac|b \in tr(T_2)$ or $bc|a \in tr(T_2)\}$. Finally we denote by $confset(T_1, T_2) = \{ \{a,b,c\} : ab|c \in conf(T_1, T_2) \}$, *i.e.* the collection of 3-label sets formed by conflicting triplets. Given a collection $C = \{S_1, \ldots, S_{|C|}\}$ of sets, a *hitting set* of $C$ is a set $S$ such that $S \cap S_i \neq \emptyset$ for each $S_i \in C$. \[lem:hit-triplets\] Let $X \subseteq {\mathcal{X}}$. Then $T_1 - X =T_2 - X$ if and only if $X$ is a hitting set of $confset(T_1, T_2)$. It is known that for two rooted trees $T_1, T_2$ that are leaf-labelled and binary, $T_1 = T_2$ if and only if $tr(T_1) = tr(T_2)$ [@bryant1997building]. Note also that $tr(T - X) = \{ab|c \in tr(T_1): X\cap \{a,b,c\}= \emptyset\}$ for any tree $T$ and $X\subseteq {\mathcal{X}}$. Therefore we have that $T_1 - X = T_2 - X$ if and only if $tr(T_1 - X) = tr(T_2 - X)$, which holds if and only if for every $a,b,c \in {\mathcal{X}}\setminus X$, if $ab|c \in tr(T_1)$ then $ab|c \in tr(T_2)$. This in turn occurs if and only if $X$ is a hitting set for $confset(T_1, T_2)$. In what follows, we call $X \subseteq {\mathcal{X}}$ a *minimal disagreement* between $T_1$ and $T_2$ if $T_1 - X = T_2 - X$ and for any $X' \subset X$, $T_1 - X' \neq T_2 - X'$. \[lem:must-move-x\] Suppose that $d < d_{LR}(T_1, T_2) \leq d' + d$ with $d' \leq d$, and that there is a tree $T^*$ and subsets $X_1, X_2 \subseteq {\mathcal{X}}$ such that $T_1 - X_1 = T^* - X_1$, $T_2 - X_2 = T^* - X_2$ and $|X_1| \leq d',|X_2| \leq d$. Then, there is a minimal disagreement $X$ between $T_1$ and $T_2$ of size at most $d + d'$ and $x \in X$ such that $x \in X_1 \setminus X_2$. Let $X' = X_1 \cup X_2$. Observe that $T_1 - X' = T^* - X'= T_2 - X'$ and $|X'| \leq d+d'$. Letting $X$ be the minimal subset of $X'$ such that $T_1 - X = T_2 - X$, we have that $X$ is a minimal disagreement between $T_1$ and $T_2$ and $|X| \leq d+d'$. Furthermore as $|X| \geq d_{LR}(T_1, T_2) > d$, $|X\setminus X_2| > 0$, and so there is some $x \in X$ with $x \in X\setminus X_2 = X_1 \setminus X_2$. We are now ready to state and prove Lemma \[lem:disagreementKernel2\]. \[lem:disagreementKernel2\] Suppose that $d_{LR}(T_1, T_2) \leq d$ for some integer $d$. Then, there is some $S \subseteq {\mathcal{X}}$ such that $|S| \leq 8d^2$, and for any minimal disagreement $X$ between $T_1$ and $T_2$ with $|X| \leq d$, $X \subseteq S$. Moreover $S$ can be found in time $O(n^2)$. We will call $S$ as described in Lemma \[lem:disagreementKernel2\] a *$d$-disagreement kernel* between $T_1$ and $T_2$. Thus Lemma \[lem:must-move-x\] essentially states that if $T_1$ isn’t a solution and $d_{LR}(T_1, T_2) > d$, then for $T_1$ to get closer to a solution, there is a leaf $x$ in the $d_{LR}(T_1, T_2)$-disagreement kernel that needs to be removed and regrafted in a location that $T_2$ ‘agrees with’. Lemma \[lem:disagreementKernel2\] in turn gives us a set $S$ of size at most $8d^2$ such that the desired $x$ must be contained in $S$. By Lemma \[lem:hit-triplets\], it is enough to find a set $S$ such that $S$ contains every minimal hitting set of $confset(T_1,T_2)$ of size at most $d$. We construct $S$ as follows. Let $X$ be a subset of ${\mathcal{X}}$ of size at most $d$ such that $T_1 - X = T_2 - X$. As previously noted, this can found in time $O(n \log n)$ [@DBLP:journals/siamcomp/ColeFHPT00]. For notational convenience, for each $x \in X$ we let $x_1,x_2$ be two new labels, and set $X_1 = \{x_1: x \in X\}$, $X_2 = \{x_2: x \in X\}$. Thus, $X_1,X_2$ are disjoint “copies” of $X$. Let $T_1'$ be derived from $T_1$ by replacing every label from $X$ with the corresponding label in $X_1$, and similarly let $T_2'$ be derived from $T_2$ by replacing every label from $X$ with the corresponding label in $X_2$. Let $T_J$ be a tree with label set $({\mathcal{X}}\setminus X) \cup X_1 \cup X_2$ such that $T_J - X_2 = T_1'$ and $T_J -X_1 =T_2'$. The tree $T_J$ always exists and can be found in polynomial time. Intuitively, we can start from $T_1'$, and graft the leaves of $X_2$ where $T_2$ “wants” them to be. See Figure \[fig:joinTree\] for an example. Algorithm \[alg:join-trees\] gives a method for constructing $T_J$, and takes $O(n^2)$ time. \[fig:joinTree\] $T_1'$ is a tree on $L' \cup X_1'$, $T_2'$ is a tree on $L' \cup X_2'$, $T_1'|_{L'} = T_2'|_{L'}$. Output: A tree $T_J$ on $L' \cup X_1' \cup X_2'$ such that $T_J|_{L' \cup X_1'} = T_1'$ and $T_J|_{L' \cup X_2'} = T_2'$ Return $T_2'$ Return $T_1'$ Return $T_1'$ Set $r_1 =$ root of $T_1'$, $u,v$ the children of $r_1$ Set $r_2 =$ root of $T_2'$, $w,z$ the children of $r_2$ Set $X_{1u} =$ descendants of $u$ in $X_1'$, $L_{1u} =$ descendants of $u$ in $L'$ Set $X_{1v} =$ descendants of $v$ in $X_1'$, $L_{1v} =$ descendants of $v$ in $L'$ Set $X_{2w} =$ descendants of $w$ in $X_2'$, $L_{2w} =$ descendants of $w$ in $L'$ Set $X_{2z} =$ descendants of $z$ in $X_2'$, $L_{2z} =$ descendants of $z$ in $L'$ Set $T_{left} = \textsc{join-trees}(T_1'|_{L_{1u} \cup X_{1u}}, T_2'|_{L_{1u} \cup X_{2w}}, L_{1u}, X_{1u}, X_{2w})$ Set $T_{right} = \textsc{join-trees}(T_1'|_{L_{1v} \cup X_{1v}}, T_2'|_{L_{1v}\cup X_{2z}}, L_{1v}, X_{1v}, X_{2z})$ Set $T_{left} = \textsc{join-trees}(T_1'|_{L_{1u} \cup X_{1u}}, T_2'|_{L_{1u}\cup X_{2z}}, L_{1u}, X_{1u}, X_{2z})$ Set $T_{right} = \textsc{join-trees}(T_1'|_{L_{1v} \cup X_{1v}}, T_2'|_{L_{1v} \cup X_{2w}}, L_{1v}, X_{1v}, X_{2w})$ Set $T_{left} = \textsc{join-trees}(T_1'|_{X_{1u}}, T_2'|_\emptyset, \emptyset, X_{1u}, \emptyset)$ Set $T_{right} = \textsc{join-trees}(T_1'|_{L' \cup X_{1v}}, T_2', L', X_{1v}, X_2')$ Set $T_{left} = \textsc{join-trees}(T_1'|_{L' \cup X_{1u}}, T_2', L', X_{1u}, X_2')$ Set $T_{right} = \textsc{join-trees}(T_1'|_{X_{1v}}, T_2'|_\emptyset, \emptyset, X_{1v}, \emptyset)$ Set $T_{left} = \textsc{join-trees}(T_1'|_{\emptyset}, T_2'|_{X_{2w}}, \emptyset, \emptyset, X_{2w})$ Set $T_{right} = \textsc{join-trees}(T_1', T_2'|_{L' \cup X_{2z}}, L', X_1', X_{2z})$ Set $T_{left} = \textsc{join-trees}(T_1', T_2'|_{L' \cup X_{2w}}, L', X_1', X_{2w})$ Set $T_{right} = \textsc{join-trees}(T_1'|_{\emptyset}, T_2'|_{X_{2z}}, \emptyset, \emptyset, X_{2z})$ Set $T_J =$ the tree on $L' \cup X_1' \cup X_2'$ whose root has $T_{left}$ and $T_{right}$ as children. Return $T_J$. \[alg:join-trees\] In addition, let $L$ be the set of all labels in ${\mathcal{X}}\setminus X$ that are descended in $T_J$ from ${\textsc{lca}}_{T_J}(X_1 \cup X_2)$, and let $R = {\mathcal{X}}\setminus (L\cup X)$. Thus, $L,X,R$ form a partition of ${\mathcal{X}}$, and $L,X_1,X_2,R$ form a partition of the labels of $T_J$. For the rest of the proof, we call $\{x,y,z\}$ a *conflict triple* if $\{x,y,z\} \in confset(T_1,T_2)$. We first observe that no triple in $confset(T_1,T_2)$ contains a label in $R$. Indeed, consider a triple $\{x,y,z\}$. Any conflict triple must contain a label from $X$, so assume without loss of generality that $x \in X, z \in R$. If $x \in X, y \in L, z \in R$, then we have that $T_J$ contains the triplets $x_1y|z, x_2y|z$, and so $T_1$ and $T_2$ both contain $xy|z$, and $\{x,y,z\}$ is not a conflict triple. Similarly if $x,y \in X, z \in R$, then $T_J$ contains the triplets $x_1y_1|z, x_2y_2|z$, and again $\{x,y,z\}$ is not a conflict triple. If $x \in X$ and $y,z \in R$, then the triplet on $\{x_1,y,z\}$ in $T_J$ depends only on the relative positions in $T_J$ of $y,z$ and ${\textsc{lca}}_{T_J}(X_1 \cup X_2)$. Thus we get the same triplet if we replace $x_1$ with $x_2$, and so $\{x,y,z\}$ is not a conflict triple. This concludes the proof that no triple in $confset(T_1,T_2)$ contains a label in $R$. Having shown this, we may conclude that any minimal disagreement between $T_1$ and $T_2$ is disjoint from $R$, and so our returned set $S$ only needs to contain labels in $L \cup X$. Now consider the tree $T^* = T_J|_{X_1 \cup X_2}$, *i.e.* the subtree of $T_J$ restricted to the labels in $X_1 \cup X_2$. Thus in the example of Figure \[fig:joinTree\], $T^*$ is the subtree of $T_J$ spanned by $\{x_1,x_2,y_1,y_2\}$. We will now use the edges of $T^*$ to form a partition of $L$, as follows. For any edge $uv$ in $T^*$ with $u$ the parent of $v$, let $s(uv)$ denote the set of labels $y \in {\mathcal{X}}$ such that $y$ has an ancestor which is an internal node on the path from $u$ to $v$ in $T_J$, but $y$ is not a descendant of $v$ itself. For example in Figure \[fig:joinTree\], if $u$ is the least common ancestor of $x_1,y_1$ and $v$ is the least common ancestor of $x_1,y_2$, then $uv$ is an edge in $T^*$ and $s(uv) = \{c,d,e\}$. Observe that $\{s(uv):uv \in E(T^*)\}$ forms a partition of $L$. (Indeed, for any $l \in L$, let $u$ be the minimal element in $T^*$ on the path in $T_J$ between $l$ and ${\textsc{lca}}_{T_J}(X_1 \cup X_2)$ (note that $u$ exists as ${\textsc{lca}}_{T_J}(X_1 \cup X_2)$ itself is in $T^*$). As $u$ is in $T^*$, both of its children are on paths in $T_J$ between $u$ and a child of $u$ in $T^*$. In particular, the child of $u$ which is an ancestor of $l$ is an internal node on the path between $u$ and $v$ in $T_J$, for some child $v$ of $u$ in $T^*$, and $l$ is not descended from $v$ by construction. It is clear by construction that all $s(uv)$ are disjoint.) The main idea behind the construction of $S$ is that we will add $X$ to $S$, together with $O(d)$ labels from $s(uv)$ for each edge $uv$ in $T^*$. As the number of edges in $T^*$ is $2(|X_1 \cup X_2| -1) = O(d)$, we have the required bound of $O(d^2)$ on $|S|$. So now consider $s(uv)$ for some edge $uv$ in $T^*$. In order to decide which labels to add to $S$, we need to further partition $s(uv)$. Let $u = u_0u_1\dots u_t = v$ be the path in $T_J$ from $u$ to $v$. For each $i \in [t-1]$ (note that this does not include $i=0$), we call the set of labels descended from $u_i$ but not $u_{i+1}$ a *dangling clade*. Observe that the dangling clades form a partition of $s(uv)$. Thus in the example of Figure \[fig:joinTree\], if $u$ is the least common ancestor of $x_1,y_1$ and $v$ is the least common ancestor of $x_1,y_2$, then for the edge $uv$ the dangling clades are $\{c\}$ and $\{d,e\}$. We now make the following observations about the relation between $s(uv)$ and triples in $confset(T_1,T_2)$. [**Observation 1:**]{} if $\{x,y,z\}$ is a conflict triple and $x \in s(uv), y,z \notin s(uv)$, then $\{x',y,z\}$ is also a conflict triple for any $x' \in s(uv)$. (The intuition behind this is that there are no labels appearing ’between’ $x$ and $x'$ that are not in $s(uv)$.) [**Observation 2:**]{} for any triple $\{x,y,z\}$ with $x,y \in s(uv)$, $\{x,y,z\}$ is a conflict triple if and only if $x,y$ are in different dangling clades and $z \in X$ with $z_i$ descended from $v$, $z_{3-i}$ not descended from $u_1$ for some $i \in [2]$ (recall that $z_1 \in X_1$ and $z_2 \in X_2$). To prove one direction, it is easy to see that if the conditions hold, then $T_i$ displays either $xz|y$ or $yz|x$ (depending on which dangling clade appears ’higher’), and $T_{3-i}$ displays $xy|z$. For the converse, observe first that $z\in X$ as $X$ is a hitting set for $confset(T_1,T_2)$ and $x,y \notin X$. Then if $xy$ are in the same dangling clade, we have that both $T_1$ and $T_2$ display $xy|z$. So $x,y$ must be in different dangling clades. Next observe that each of $z_1,z_2$ must either be descended from $v$ or not descended from $u_1$, as otherwise $v$ would not be the child of $u$ in $T^*$. If $z_1,z_2$ are both descended from $v$ or neither are descended from $u_1$, then $T_1$ and $T_2$ display the same triplet on $\{x,y,z\}$. So instead one must be descended from $v$ and one not descended from $u_1$, as required. Using Observations 1 and 2, we now prove the following: [**Observation 3:**]{} for any minimal disagreement $X'$ between $T_1$ and $T_2$, one of the following holds: - $X' \cap s(uv) = \emptyset$; - $s(uv) \subseteq X'$; - $s(uv) \setminus X'$ forms a single dangling clade. To see this, let $X'$ be any minimal hitting set of $confset(T_1,T_2)$ with $s(uv) \cap X' \neq \emptyset$ and $s(uv) \setminus X' \neq \emptyset$. As $X'$ is minimal, any $x \in s(uv) \cap X'$ must be in a conflict triple $\{x,y,z\}$ with $y,z \notin X'$. As $X$ is a hitting set for $confset(T_1,T_2)$, at least one of $y,z$ must be in $X$. If $y,z \notin s(uv)$, then by Observation 1 $\{x',y,z\}$ is also a conflict triple for any $x' \in s(uv) \setminus X'$. But this is a contradiction as $\{x',y,z\}$ has no elements in $X'$. Then one of $y,z$ must also be in $s(uv)$. Suppose without loss of generality that $y \in s(uv)$. We must also have that $z \in X$, as $X$ is a hitting set for $confset(T_1,T_2)$ and $x,y \notin X$. By Observation 2, we must have that one of $z_1,z_2$ is descended from $v$, and the other is not descended from $u_1$. This in turn implies (again by Observation 2) that for any $x' \in s(uv) \setminus X'$, if $x'$ and $y$ are in different dangling clades then $\{x',y,z\}$ is a conflict triple. Again this is a contradiction as $\{x',y,z\}$ has no elements of $X'$, and so we may assume that all elements of $s(uv) \setminus X'$ are in the same dangling clade. It remains to show that every element of this dangling clade is in $s(uv) \setminus X'$. To see this, suppose there exists some $x \in X'$ in the same dangling clade as the elements of $s(uv) \setminus X'$. Once again we have that $x$ is in some conflict triple $\{x,y,z\}$ with $y,z \notin X'$, and if $y,z \notin s(uv)$ then $\{x',y,z\}$ is also a conflict triple for any $x' \in s(uv) \setminus X'$, a contradiction. So we may assume that one of $y,z$ is in $s(uv) \setminus X'$. But all elements of $s(uv) \setminus X'$ are in the same dangling clade as $x$, and so by Observation 2 $\{x,y,z\}$ cannot be a conflict triple, a contradiction. So finally we have that all elements of $s(uv) \setminus X'$ are in the same dangling clade and all elements of this clade are in $s(uv) \setminus X'$, as required. With the proof of Observation 3 complete, we are now in a position to construct $S$. For any minimal hitting set $X'$ of $confset(T_1,T_2)$ with size at most $d$, by Observation 3 either $X' \cap s(uv) = \emptyset$, or $s(uv) \subseteq X'$ (in which case $|s(uv)|\leq d$), or $s(uv) \setminus X'$ forms a single dangling clade $C$ (in which case $|s(uv) \setminus C| \leq d$). So add all elements of $X$ to $S$. For all $uv \in E(T_J)$ and any dangling clade $C$ of labels in $s(uv)$, add $s(uv) \setminus C$ to $S$ if $|s(uv) \setminus C| \leq d$. Observe that this construction adds at most $2d$ labels from $s(uv)$ to $S$. Thus, in total, we have that the size of $S$ is at most $|X| + 2d|E(T_J)| \leq d+2d(2(|X_1 \cup X_2|-1)) \leq d + 2d(4d-2) = 8d^2-3d \leq 8d^2$. Algorithm \[alg:disagreementKernel2\] describes the full procedure formally. The construction of $T_J$ occurs once and as noted above takes $O(n^2)$ time. As each other line in the algorithm is called at most $n$ times and takes $O(n)$ time, the overall running time of the algorithm $O(n^2)$. $T_1$ and $T_2$ are trees on ${\mathcal{X}}$,$d$ an integer. Output: A set $S \subseteq {\mathcal{X}}$ such that for every minimal disagreement $X$ between $T_1$ and $T_2$ with $|X| \leq d$, $X \subseteq S$. Find $X$ such that $|X| \leq d$ and $T_1 - X = T_2 - X$ Set $S = X$ Let $X_1, X_2$ be copies of $X$ and replace $T_1,T_2$ with corresponding trees $T_1',T_2'$ on $({\mathcal{X}}\setminus X)\cup X_1, ({\mathcal{X}}\setminus X)\cup X_2$. Let $T_J = \textsc{join-trees}(T_1', T_2', ({\mathcal{X}}\setminus X), X_1, X_2)$ Let $T^*= T_J|_{X_1 \cup X_2}$ Let $u = u_0u_1 \dots u_t = v$ be the path in $T_J$ from $u$ to $v$ Let $s(uv) = \{l \in {\mathcal{X}}\setminus X: l$ is descended from $u_1$ but not from $v\}$ Set $p = |s(uv)| - d$ Any clade $C$ has $|C|\geq p$ iff $|s(uv) \setminus C| \leq d$ Set $C = \{l \in s(uv): l$ is descended from $u_i$ but not from $u_{i+1} \}$ Set $S = S \cup (s(uv) \setminus C)$ Return $S$. \[alg:disagreementKernel2\] The last ingredient needed for Theorem \[thm:fpt-in-d\] is Lemma \[lem:dont-check-too-many-trees\], which shows that if a leaf $x$ of $T_1$ as described in Lemma \[lem:must-move-x\] has to be moved, then there are not too many ways to regraft it in order to get closer to $T^*$. In the course of the following proofs, we will want to take observations about one tree and use them to make statements about another. For this reason it’s useful to have a concept of one node “corresponding” to another node in a different tree. In the case of leaf nodes this concept is clear - two leaf nodes are equivalent if they are assigned the same label- but for internal nodes there is not necessarily any such correspondence. However, in the case that one tree is the restriction of another to some label set, we can introduce a well-defined notion of correspondence: Given two trees $T, T'$ such that $T' = T|_X$ for some $X \subseteq {\mathcal{X}}(T)$, and a node $u \in V(T')$, define the node $u_T$ of $T$ by $u_T = {\textsc{lca}}_T({\mathcal{L}}_{T'}(u))$. That is, $u_T$ is the least common ancestor, in $T$, of the set of labels belonging to descendants of $u$ in $T'$. We call $u_T$ the *node corresponding to $u$ in $T$*. We note two useful properties of $u_T$ here: \[lem:corrAncestor\] For any $T, T', X \subseteq {\mathcal{X}}(T)$ such that $T' = T|_X$ and any $u,v \in V(T')$, $u_T$ is an ancestor of $v_T$ if and only if $u$ is an ancestor of $v$. If $u$ is an ancestor of $v$ then ${\mathcal{L}}_{T'}(v) \subseteq {\mathcal{L}}_{T'}(u)$, which implies that $u_T$ is an ancestor of $v_T$. For the converse, observe that for any $Z \subseteq X$, any label in $X$ descending from ${\textsc{lca}}_T(Z)$ in $T$ is also descending from ${\textsc{lca}}_{T'}(Z)$ in $T'$. In particular letting $Z = {\mathcal{L}}_{T'}(u)$, we have ${\mathcal{L}}_{T}(u_T)\cap X = {\mathcal{L}}_{T}({\textsc{lca}}_T(Z)) \cap X \subseteq {\mathcal{L}}_{T'}({\textsc{lca}}_{T'}(Z)) = {\mathcal{L}}_{T'}({\textsc{lca}}_{T'}({\mathcal{L}}_{T'}(u))) = {\mathcal{L}}_{T'}(u) \subseteq {\mathcal{L}}_{T}(u_T)\cap X$. Thus ${\mathcal{L}}_{T'}(u) = {\mathcal{L}}_T(u_T) \cap X$ and similarly ${\mathcal{L}}_{T'}(v) = {\mathcal{L}}_T(v_T) \cap X$. Then we have that $u_T$ being an ancestor of $v_T$ implies ${\mathcal{L}}_T(v_T) \subseteq L_T(u_T)$, which implies that ${\mathcal{L}}_{T'}(v) = {\mathcal{L}}_T(v_T) \cap X \subseteq L_T(u_T) \cap X = {\mathcal{L}}_{T'}(u)$, which implies that $u$ is an ancestor of $v$. \[lem:corrTransitive\] For any $T'', T', T$ and $Y \subseteq X \subseteq {\mathcal{X}}(T)$ such that $T' = T|_{X}$ and $T'' = T'|_{Y}$, $(u_{T'})_{T} = u_{T}$. It is sufficient to show that any node in $V(T)$ is a common ancestor of ${\mathcal{L}}_{T'}({\textsc{lca}}_{T'}(Z))$ if and only if it is a common ancestor of $Z$, where $Z = {\mathcal{L}}_{T''}(u)$ (as this implies that the least common ancestors of these two sets are the same). It is clear that if $v \in V(T)$ is a common ancestor of ${\mathcal{L}}_{T'}({\textsc{lca}}_{T'}(Z))$ then it is also a common ancestor of $Z$, as $Z \subseteq {\mathcal{L}}_{T'}({\textsc{lca}}_{T'}(Z))$. For the converse, observe that as $T' = T|_X$ and $Z\subseteq X$, any label in $X$ descended from ${\textsc{lca}}_{T'}(Z)$ in $T'$ is also descended from ${\textsc{lca}}_T(Z)$ in $T$. This implies ${\mathcal{L}}_{T'}({\textsc{lca}}_{T'}(Z)) \subseteq {\mathcal{L}}_T({\textsc{lca}}_T(Z))$, and so any common ancestor of $Z$ in $T$ is also a common ancestor of ${\mathcal{L}}_{T'}({\textsc{lca}}_{T'}(Z))$. We are now ready to state and prove Lemma \[lem:dont-check-too-many-trees\] \[lem:dont-check-too-many-trees\] Suppose that $d < d_{LR}(T_1, T_2) \leq d' + d$ with $d' \leq d$, and that there are $X_1, X_2 \subseteq {\mathcal{X}}$, and a tree $T^*$ such that $T_1 - X_1 = T^* - X_1, T_2 - X_2 = T^* - X_2, |X_1| \leq d', |X_2| \leq d$, and let $x \in X_1 \setminus X_2$. Then, there is a set $P$ of trees on label set ${\mathcal{X}}$ that satisfies the following conditions: - for any tree $T'$ such that $d_{LR}(T', T^*) < d_{LR}(T_1,T^*)$ and $T'$ can be obtained from $T_1$ by pruning a leaf $x$ and regrafting it, $T' \in P$; - $|P| \leq 18(d+d')+8$; - $P$ can be found in time $O(n(\log n + 18(d+d')+8))$. The idea behind the proof is as follows: by looking at a subtree common to $T_1$ and $T_2$, we can identify the location that $T_2$ “wants" $x$ to be positioned. This may not be the correct position for $x$, but we can show that if $x$ is moved too far from this position, we will create a large number of conflicting triplets between $T_2$ and the solution $T^*$. As a result, we can create all trees in $P$ by removing $x$ from $T_1$ and grafting it on one of a limited number of edges. For the purposes of this proof, we will treat each tree $T$ as “planted”, *i.e.* as having an additional root of degree $1$, denoted $r(T)$, as the parent of what would normally be considered the “root" of the tree. (That is, $r(T)$ is the parent of ${\textsc{lca}}_T({\mathcal{X}}(T))$. Note that trees are otherwise binary. We introduce $r(T)$ as a notational convenience to avoid tedious repetition of proofs - grafting a label above a tree $T$ can instead be represented as grafting it on the edge between $r(T)$ and its child. For the purposes of corresponding nodes, if $T' = T - X$ then $(r(T'))_T = r(T)$. This allows us to assume that every node in $T$ is a descendant of $u_T$ for some node $u$ in $T'$. A naive method for constructing a tree in $P$ is the following: Apply an LPR move $(x,e)$ on $T_1$, such that $x$ is moved to a position that $T_2$ “wants” $x$ to be in. There are at least two problems with this method. The first is that, since $T_1$ and $T_2$ have different structures, it is not clear where in $T_1$ it is that $T_2$ “wants” $x$ to be. We can partially overcome this obstacle by initially considering a subtree common to both $T_1$ and $T_2$. However, because $T_2$ will want to move leafs that will not be moved in $T_1$, it can still be the case that even though $T_2$ “agrees” with $T^*$ on $x$, $T_2$ may want to put $x$ in the “wrong” place, when viewed from the perspective of $T_1$. For this reason we have to give a counting argument to show that if $x$ is moved “too far” from the position suggested by $T_2$, it will create too many conflicting triplets, which cannot be covered except by moving $x$. We make these ideas precise below. Let $P^*$ be the set of all trees $T'$ such that $d_{LR}(T', T^*) < d_{LR}(T_1, T^*) $ and $T'$ can be obtained from $T_1$ by an LPR move on $x$. Thus, it is sufficient to construct a set $P$ such that $|P| \leq 18(d + d') + 8$ and $P^* \subseteq P$. We first construct a set $X_m \subseteq {\mathcal{X}}$ such that $|X_m| \leq d+ d', x \in X_m$, and $T_1 - X_m= T_2 - X_m$. Note that the unknown set $(X_1 \cup X_2)$ satisfies these properties, as $T_1 - (X_1 \cup X_2) = T^* - (X_1 \cup X_2) = T_2 - (X_1 \cup X_2)$, and so such a set $X_m$ must exist. We can find $X_m$ in time $O(n \log n)$ by applying MAST on $(T_1 - \{x\}, T_2 - \{x\})$ [@DBLP:journals/siamcomp/ColeFHPT00]. Now let $T_m$ be the tree with labelset ${\mathcal{X}}\setminus {X_m}$ such that $T_m = T_1 - X_m = T_2 - X_m$. Note that for any $T'$ in $P^*$, we have that $T' - \{x\} = T_1 - \{x\}$ and therefore $T' - X_m = T_1 - X_m = T_m$. Informally, we now have a clear notion of where $T_2$ “wants” $x$ to go, relative to $T_m$. There is a unique edge $e$ in $T_m$ such that grafting $x$ on $e$ will give the tree $T_2 - (X_m\setminus\{x\})$. If we assume that this is the “correct” position to add $x$, then it only remains to add the remaining labels of $X_m$ back in a way that agrees with $T_1$ (we will describe how this can be done at the end of the proof). Unfortunately, grafting $x$ onto the obvious choice $e$ does not necessarily lead to a graph in $P^*$. This is due to the fact that $T_2$ can be “mistaken” about labels outside of $X_m$. To address this, we have try grafting $x$ on other edges of $T_m$. There are too many edges to try them all. We therefore need the following claim, which allows us to limit the number of edges to try. [**Claim:**]{} Informally, the claim identifies a node $y$ and set of nodes $Z$ in $T_m$, such that $x$ should be added as a descendant of $y$ but not of any node in $Z$, and the number of such positions is bounded. Algorithm \[alg:locationRestrictions\] describes the formal procedure to produce $y$ and $Z$. The proof of the claim takes up most of the remainder of our proof; the reader may wish to skip it on their first readthrough. $T_1, T_2$ are two trees, $T_m$ is a common subtree of $T_1$ and $T_2$ such that $T_m = T_2 - X_m$, $x$ is a label that cannot be moved in $T_2$ (but must be moved in $T_1$), $d$ is the maximum number of leaves we can remove in a tree, $d'$ is the maximum number of leaves we can move in $T_1$. Output is a pair $(y,Z)$ with $y \in V(T_m)$, $Z\subseteq V(T_m)$, such that we may assume $x$ is a descendant of $y$ but not a descendant of any $z' \in Z$, and the number of labels like this in $T_m$ is $O(d)$. For this pseudocode, every tree $T$ has a degree-$1$ root $r(T)$. Set $T_m' = T_2 - (X_m \setminus \{x\})$ Set $z = $ lowest ancestor of $x$ in $T_m'$ such that $|{\mathcal{L}}_{T_m'}(z)\setminus{x}| \geq d+d'$, or return $(r(T_m'), \emptyset)$ if no such $z$ exists. Set $y = $ lowest ancestor of $z$ in $T_m'$ such that $|{\mathcal{L}}_{T_m'}(y) \setminus {\mathcal{L}}_{T_m'}(z)| \geq d+d'$, or $r(T_m')$ if no such ancestor exists. Let $z_1,z_2$ be the children of $z$ such that $x$ is descended from $z_1$ in $T_m'$ Set $Z_1 = \{ z'$ descended from $z_2: |{\mathcal{L}}_{T_m'}(z')| \ge d+d'$ and $|{\mathcal{L}}_{T_m'}(z_2) \setminus {\mathcal{L}}_{T_m'}(z')| \ge d+d'$, and this does not hold for any ancestor of $z'\}$ Let $y_1,y_2$ be the children of $y$ such that $x$ is descended from $y_1$ in $T_m'$ Set $Z_2 = \{ y'$ descended from $y_2: |{\mathcal{L}}_{T_m'}(y')| \ge d+d'$ and $|{\mathcal{L}}_{T_m'}(y_2) \setminus {\mathcal{L}}_{T_m'}(y')| \ge d+d'$, and this does not hold for any ancestor of $y'\}$ Set $y^* = $ node of $T_m$ for which $y$ is the corresponding node in $T_m'$ Set $Z = \{z^*$ in $T_m : z'\in Z_1\cup Z_2$ is the node corresponding to $z^*$ in $T_m\}$ Return $(y^*,Z)$ \[alg:locationRestrictions\] Let $T_m' = T_2 - (X_m\setminus \{x\})$. Note that $T_m' - \{x\} = T_m$. We will use the presence of $x$ in $T_m'$ to identify the node $y$ and set $Z$. (Technically, this means the nodes we find are nodes in $T_m'$ rather than $T_m$. However, we note that apart the parent of $x$ and $x$ itself, neither of which will be added to $\{y\}\cup Z$, every node in $T_m'$ is the node $v_{T_m'}$ corresponding to some node $v$ in $T_m$. For the sake of clarity, we ignore the distinction and write $v$ to mean $v_{T_m'}$ throughout this proof. The nodes in $\{y\}\cup Z$ should ultimately be replaced with the nodes in $T_m$ to which they correspond.) We first identify two nodes $z,y$ of $T_m$ as follows: - Let $z$ be the least ancestor of $x$ in $T_m'$ such that $|{\mathcal{L}}_{T_m'}(z) \setminus \{x\}| \ge d+d'$. If no such $x$ exists, then ${\mathcal{X}}(T_m') \leq d+d'$ and we may return $y = r(T_m'), Z = \emptyset$. - Let $y$ be the least ancestor of $z$ in $T_m'$ such that $|{\mathcal{L}}_{T_m'}(y) \setminus {\mathcal{L}}_{T_m'}(z)| \ge d+d'$. If no such $y$ exists, set $y = r(T_m')$. Using this definition, we will show that $x$ must be a descendant of $y_{T'}$ for any $T' \in P^*$. We first describe a general tactic for restricting the position of $x$ in $T'$, as this tactic will be used a number of times. Suppose that for some $T' \in P^*$ there is a set of $d+d'$ triplets in $confset(T',T_2)$ whose only common element is $x$. Then let $X' \subseteq {\mathcal{X}}$ be a set of labels such that $T' - X' = T^* - X'$ and $|X'| = d_{LR}(T', T^*) \leq d_{LR}(T_1, T^*) - 1 \leq d' - 1$. Note that $T_2 - (X' \cup X_2) = T^* - (X' \cup X_2) = T' - (X' \cup X_2)$, and therefore $(X' \cup X_2)$ is a hitting set for $confset(T',T_2)$. As $|X' \cup X_2| \leq d + d-1$ and there are $d+d'$ triplets in $confset(T',T_2)$ whose only common element is $x$, it must be the case that $x \in X'\cup X_2$. As $x\notin X_2$, we must have $x\in X'$. But this implies that $T_1 - X' = T' - X' = T^* - X'$ and therefore $d_{LR}(T_1, T^*) \leq |X'| = d_{LR}(T', T^*) \leq d_{LR}(T_1, T^*) - 1$, a contradiction. Thus we may assume that such a set of triplets does not exist. We now use this idea to show that $x \in {\mathcal{L}}_{T'}(y_{T'})$, for any $T'\in P^*$. Indeed, suppose $x \notin {\mathcal{L}}_{T'}(y_{T'})$. We may assume $y\neq r(T_m')$ as otherwise $y_{T'}= r(T')$ by definition and so ${\mathcal{L}}_{T'}(y_{T'}) = {\mathcal{X}}(T')$. Then let $z_1, \ldots, z_{d+d'}$ be $d+d'$ labels in ${\mathcal{L}}_{T_m'}(z)\setminus \{x\}$. Let $y_1, \ldots, y_{d+d'}$ be $d+d'$ labels in ${\mathcal{L}}_{T_m'}(y)\setminus {\mathcal{L}}_{T_m'}(z)$. Observe that for each $i \in [d+d']$, $T_m'$ (and therefore $T_2$) contains the triplet $(z_ix|y_i)$, but $T'$ contains the triplet $(z_iy_i|x)$. Therefore $confset(T',T_2)$ contains $d+d'$ sets whose only common element is $x$. As this implies a contradiction, we must have $x \in {\mathcal{L}}_{T'}(y_{T'})$. Note however that $|{\mathcal{L}}_{T_m'}(y)|$ maybe be very large. In order to provide a bounded range of possible positions for $x$, we still need to find a set $Z$ of nodes such that $|{\mathcal{L}}_{T_m'}(y) \setminus \bigcup_{z' \in Z} {\mathcal{L}}_{T_m'}(z'))|$ is bounded, and such that we can show $x \notin {\mathcal{L}}_{T'}(z'_{T'})$ for any $z' \in Z$. We now construct a set $Z_1$ of descendants of $z$ as follows: - Let $z_1, z_2$ be the children of $z$ in $T_m'$ such that $x$ is descended from $z_1$. - If $|{\mathcal{L}}_{T_m'}(z_2)| \leq 3(d+d')$ then set $Z_1 = \emptyset$. - Otherwise, let $Z_1$ be the set of highest descendants $z'$ of $z_2$, such that $|{\mathcal{L}}_{T_m'}(z')|\ge d+d'$ and $|{\mathcal{L}}_{T_m'}(z_2) \setminus {\mathcal{L}}_{T_m'}(z')|\ge d+d'$ (*i.e.* by highest descendant we mean such that $z'$ has no ancestor $z''$ with the same properties). Note that $|{\mathcal{L}}_{T_m'}(z_1)| \leq d+d'$ by our choice of $z$. It follows that if $|{\mathcal{L}}_{T_m'}(z_2)| \leq 3(d+d')$ then $|{\mathcal{L}}_{T_m'}(z)| \le 4(d+d')$. If on the other hand $|{\mathcal{L}}_{T_m'}(z_2)| > 3(d+d')$ then $Z_1$ is non-empty. Indeed, let $z'$ be a lowest descendant of $z_2$ with $|{\mathcal{L}}_{T_m'}(z')|\ge d+d'$, and observe that $|{\mathcal{L}}_{T_m'}(z')|\leq 2(d+d')$. Then either $z' \in Z_1$, or $|{\mathcal{L}}_{T_m'}(z_2) \setminus {\mathcal{L}}_{T_m'}(z')|\le d+d'$, in which case $|{\mathcal{L}}_{T_m'}(z_2)| \leq |{\mathcal{L}}_{T_m'}(z_2) \setminus {\mathcal{L}}_{T_m'}(z')| + |{\mathcal{L}}_{T_m'}(z')| \leq d+d' + 2(d+d') = 3(d+d')$. We also have that $|Z_1| \leq 2$. Indeed, let $z_1',z_2',z_3'$ be three distinct nodes in $Z_1$, and suppose without loss of generality that $(z_1' z_2' | z_3') \in tr(T_m')$. Then setting $z' = {\textsc{lca}}_{T_m'}(z_1',z_2')$, we have that $z'$ is an ancestor of $z_1'$ such that $|{\mathcal{L}}_{T_m'}(z')|\ge d+d'$ and $|{\mathcal{L}}_{T_m'}(z_2) \setminus {\mathcal{L}}_{T_m'}(z')|\ge |{\mathcal{L}}_{T_m'}(z_3')| \ge d+d'$, a contradiction by minimality of $z_1$. We have that $|{\mathcal{L}}_{T_m'}(z) \setminus \bigcup_{z' \in Z_1} {\mathcal{L}}_{T_m'}(z'))| \leq 4(d+d')$. Indeed, if $Z_1 = \emptyset$ then $|{\mathcal{L}}_{T_m'}(z)| \leq 4(d+d')$ as described above. Otherwise, let $z'$ be an element of $Z_1$ and $z_p$ its parent, $z_s$ its sibling in $T_m'$. Clearly $|{\mathcal{L}}_{T_m'}(z_p)| \geq |{\mathcal{L}}_{T_m'}(z')| \geq d+d'$, and so as $z_p \notin Z_1$ we have $|{\mathcal{L}}_{T_m'}(z_2) \setminus {\mathcal{L}}_{T_m'}(z_p)| < d+d'$. If $|{\mathcal{L}}_{T_m'}(z_s)| \geq d+d'$ then $z_s \in Z_1$ (since $|{\mathcal{L}}_{T_m'}(z_2) \setminus {\mathcal{L}}_{T_m'}(z_s)| \geq |{\mathcal{L}}_{T_m'}(z')| \geq d+d'$), and so $|{\mathcal{L}}_{T_m'}(z) \setminus \bigcup_{z' \in Z_1} {\mathcal{L}}_{T_m'}(z'))| \leq |{\mathcal{L}}_{T_m'}(z_1) | + |{\mathcal{L}}_{T_m'}(z_2) \setminus {\mathcal{L}}_{T_m'}(z_p)| \leq 2(d+d')$. Otherwise, $|{\mathcal{L}}_{T_m'}(z) \setminus \bigcup_{z' \in Z_1} {\mathcal{L}}_{T_m'}(z'))| \leq |{\mathcal{L}}_{T_m'}(z_1) | + |{\mathcal{L}}_{T_m'}(z_2) \setminus {\mathcal{L}}_{T_m'}(z_p)| + |{\mathcal{L}}_{T_m'}(z_s)| \leq 3(d+d')$. We have now shown that $|Z_1|\le 2$ and that $|{\mathcal{L}}_{T_m'}(z) \setminus \bigcup_{z' \in Z_1} {\mathcal{L}}_{T_m'}(z'))| \leq 4(d+d')$. The final property of $Z_1$ we wish to show is that for any $z'\in Z_1$ and any $T' \in P$, $x \notin {\mathcal{L}}_{T'}(z'_{T'})$. So suppose $x \in {\mathcal{L}}_{T'}(z'_{T'})$. Let $\hat{z}_1, \dots, \hat{z}_{d+d'}$ be $d+d'$ labels in ${\mathcal{L}}_{T_m'}(z_2)\setminus L_{T_m'}(z')$. Also, $z_1$ and $z_2$ were already taken. Let $w_1, \dots, w_{d+d'}$ be $d+d'$ labels in ${\mathcal{L}}_{T_m'}(z')$. Then for each $i \in [d+d']$, $T_m'$ (and therefore $T_2$) contains the triplet $(\hat{z}_iw_i|x)$, but $T'$ contains the triplet $(xw_i|\hat{z}_i)$. Therefore $confset(T',T_2)$ contains $d+d'$ sets whose only common element is $x$. As this implies a contradiction, we must have $x \notin {\mathcal{L}}_{T'}(z'_{T'})$. We now define a set $Z_2$ of descendants of $y$: - If $y = r(T_m')$, set $Z_2 = \emptyset$. - Otherwise, let $y_1, y_2$ be the children of $y$ in $T_m'$ such that $z$ is descended from $y_1$. - If $|{\mathcal{L}}_{T_m'}(y_2)| \leq 3(d+d')$ then set $Z_2 = \emptyset$. - Otherwise, let $Z_2$ be the set of highest descendants $y'$ of $y_2$, such that $|{\mathcal{L}}_{T_m'}(y')|\ge d+d'$ and $|{\mathcal{L}}_{T_m'}(y_2) \setminus {\mathcal{L}}_{T_m'}(y')|\ge d+d'$ (*i.e.* such that $y'$ has no ancestor $y''$ with the same properties). In a similar way to the proofs for $Z_1$, we can show that $|Z_2|\leq 2$, that $|({\mathcal{L}}_{T_m'}(y) \setminus {\mathcal{L}}_{T_m'}(z)) \setminus \bigcup_{y' \in Z_2} {\mathcal{L}}_{T_m'}(y'))| \leq 4(d+d')$, and that $x \notin {\mathcal{L}}_{T'}(y'_{T'})$ for any $y' \in Z_2$ and any $T' \in P^*$. Note that $|{\mathcal{L}}_{T_m'}(y_1)\setminus {\mathcal{L}}_{T_m'}(z) | \leq d+d'$ by our choice of $y$. It follows that if $|{\mathcal{L}}_{T_m'}(y_2)| \leq 3(d+d')$ then $|{\mathcal{L}}_{T_m'}(y) \setminus {\mathcal{L}}_{T_m'}(z)| \le 4(d+d')$. If on the other hand $|{\mathcal{L}}_{T_m'}(y_2)| > 3(d+d')$, then $Z_2$ is non-empty. Indeed, let $y'$ be a lowest descendant of $y_2$ with $|{\mathcal{L}}_{T_m'}(y')|\ge d+d'$, and observe that $|{\mathcal{L}}_{T_m'}(y')|\leq 2(d+d')$. Then either $y' \in Z_2$, or $|{\mathcal{L}}_{T_m'}(y_2) \setminus {\mathcal{L}}_{T_m'}(y')|\le d+d'$, in which case $|{\mathcal{L}}_{T_m'}(y_2)| \leq |{\mathcal{L}}_{T_m'}(y_2) \setminus {\mathcal{L}}_{T_m'}(y')| + |{\mathcal{L}}_{T_m'}(y')| \leq d+d' + 2(d+d') = 3(d+d')$. We also have that $|Z_2| \leq 2$. Indeed, let $y_1',y_2',y_3'$ be three distinct nodes in $Z_2$, and suppose without loss of generality that $(y_1' y_2' | y_3') \in tr(T_m')$. Then setting $y' = {\textsc{lca}}_{T_m'}(y_1',y_2')$, we have that $y'$ is an ancestor of $y_1'$ such that $|{\mathcal{L}}_{T_m'}(y')|\ge d+d'$ and $|{\mathcal{L}}_{T_m'}(y_2) \setminus {\mathcal{L}}_{T_m'}(y')|\ge |{\mathcal{L}}_{T_m'}(y_3')| \ge d+d'$, a contradiction by minimality of $y_1$. We have that $|({\mathcal{L}}_{T_m'}(y) \setminus {\mathcal{L}}_{T_m'}(z)) \setminus \bigcup_{y' \in Z_2} {\mathcal{L}}_{T_m'}(y'))| \leq 4(d+d')$. Indeed, if $y = r(T_m')$ then by construction $|{\mathcal{L}}_{T_m'}(\hat{y}) \setminus {\mathcal{L}}_{T_m'}(z)| < d+d'$ for any ancestor $\hat{y}$ of $z$ (noting that otherwise there would be no reason to set $y$ as $r(T_m')$ rather than the child of $r(T_m')$), and so in particular $|{\mathcal{L}}_{T_m'}(y) \setminus {\mathcal{L}}_{T_m'}(z)| < d+d'$. If $y\neq r(T_m')$ and $Z_2 = \emptyset$ then $|{\mathcal{L}}_{T_m'}(y) \setminus {\mathcal{L}}_{T_m'}(z)| \leq 4(d+d')$ as described above. Otherwise, let $y'$ be an element of $Z_2$ and $y_p$ its parent, $y_s$ its sibling in $T_m'$. Clearly $|{\mathcal{L}}_{T_m'}(y_p)| \geq |{\mathcal{L}}_{T_m'}(y')| \geq d+d'$, and so as $y_p \notin Z_2$ we have $|{\mathcal{L}}_{T_m'}(y_2) \setminus {\mathcal{L}}_{T_m'}(y_p)| < d+d'$. If $|{\mathcal{L}}_{T_m'}(y_s)| \geq d+d'$ then $y_s \in Z_2$ (since $|{\mathcal{L}}_{T_m'}(y_2) \setminus {\mathcal{L}}_{T_m'}(y_s)| \geq |{\mathcal{L}}_{T_m'}(y')| \geq d+d'$), and so $|({\mathcal{L}}_{T_m'}(y) \setminus {\mathcal{L}}_{T_m'}(z)) \setminus \bigcup_{y' \in Z_2} {\mathcal{L}}_{T_m'}(y'))| \leq |{\mathcal{L}}_{T_m'}(y_1) \setminus {\mathcal{L}}_{T_m'}(z)| + |{\mathcal{L}}_{T_m'}(y_2) \setminus {\mathcal{L}}_{T_m'}(y_p)| \leq 2(d+d')$. Otherwise, $|({\mathcal{L}}_{T_m'}(y) \setminus {\mathcal{L}}_{T_m'}(z)) \setminus \bigcup_{y' \in Z_2} {\mathcal{L}}_{T_m'}(y'))| \leq |{\mathcal{L}}_{T_m'}(y_1) \setminus {\mathcal{L}}_{T_m'}(z)| + |{\mathcal{L}}_{T_m'}(y_2) \setminus {\mathcal{L}}_{T_m'}(y_p)| + |{\mathcal{L}}_{T_m'}(y_s)| \leq 3(d+d')$. We have now shown that $|Z_2| \leq 2$ and $|({\mathcal{L}}_{T_m'}(y) \setminus {\mathcal{L}}_{T_m'}(z)) \setminus \bigcup_{y' \in Z_2} {\mathcal{L}}_{T_m'}(y'))| \leq 4(d+d')$. The final property of $Z_2$ we wish to show is that for any $y' \in Z_2$ and any $T' \in P^*$, we have that $x \notin {\mathcal{L}}_{T'}(y'_{T'})$. So suppose $x \in {\mathcal{L}}_{T'}(y'_{T'})$. Let $\hat{y}_1, \dots, \hat{y}_{d+d'}$ be $d+d'$ labels in ${\mathcal{L}}_{T_m'}(y_2)\setminus L_{T_m'}(y')$. Let $w_1, \dots, w_{d+d'}$ be $d+d'$ labels in ${\mathcal{L}}_{T_m'}(y')$. Then for each $i \in [d+d']$, $T_m'$ (and therefore $T_2$) contains the triplet $(\hat{y}_iw_i|x)$, but $T'$ contains the triplet $(xw_i|\hat{y}_i)$. Therefore $confset(T',T_2)$ contains $d+d'$ sets whose only common element is $x$. As this implies a contradiction, we must have $x \notin {\mathcal{L}}_{T'}(y'_{T'})$. Now that $Z_1$ and $Z_2$ have been constructed, let $Z = Z_1\cup Z_2$. Note that $|Z|\leq 4$. Algorithm \[alg:locationRestrictions\] describes the construction of $y$ and $Z$ formally (see Figure \[fig:anchors\]). {width="50.00000%"} We have shown above that for any $T' \in P^*$, $x$ is descended from $y_{T'}$ in $T'$ and not from $z'_{T'}$ for any $z' \in Z$, and so $x \in {\mathcal{L}}_{T'}(y_{T'}) \setminus \bigcup_{z'\in Z}{\mathcal{L}}_{T'}(z'_{T'})$. As $|{\mathcal{L}}_{T_m'}(z) \setminus \bigcup_{z' \in Z_1} {\mathcal{L}}_{T_m'}(z'))| \leq 4(d+d')$ and $|({\mathcal{L}}_{T_m'}(y) \setminus {\mathcal{L}}_{T_m'}(z)) \setminus \bigcup_{y' \in Z_2} {\mathcal{L}}_{T_m'}(y'))| \leq 4(d+d')$, we have $|{\mathcal{L}}_{T_m'}(y) \setminus \bigcup_{z' \in Z} {\mathcal{L}}_{T_m'}(z'))| \leq 8(d+d')$. To analyze the complexity, note that we can calculate the value of $|{\mathcal{L}}_{T_m'}(u)|$ for all $u$ in $O(n)$ time using a depth-first search approach, together with the fact that $|{\mathcal{L}}_{T_m'}(u)| = |{\mathcal{L}}_{T_m'}(u_1)| + |{\mathcal{L}}_{T_m'}(u_2)|$ for any node $u$ with children $u_1, u_2$. Then we can find $z$ in $O(n)$ time, and once we have found $z$ we can find $y$, and thence $z_1,z_2,y_1,y_2$, in $O(n)$ time. Similarly, once these nodes are found we can find the members of $Z$ in $O(n)$ time. Using the claim, we may now construct a set $P'$ of $O\leq 16(d+d')+8$ trees on ${\mathcal{X}}\setminus (X_m\setminus \{x\})$, such for any $T' \in P^*$, $P'$ contains the tree $T'' = T' - (X_m\setminus \{x\})$. Indeed, let $F$ be the set of arcs $uv$ in $T_m$ that exist on a path from $y$ to a node in $({\mathcal{L}}_{T_m}(y) \setminus \bigcup_{z \in Z}{\mathcal{L}}_{T_m}(z')) \cup Z$. As $|{\mathcal{L}}_{T_m}(y) \setminus \bigcup_{z'\in Z}{\mathcal{L}}_{T_m}(z')| \leq 8(d+d')$, $|Z| \leq 4$ and $T_m$ is a binary tree, we have $|F| \leq 16(d+d')+8$. For each $e \in F$, let $T_e$ be the tree obtained from $T_m$ by grafting $x$ onto the arc $e$ . Let $P' = \{T_e: e \in F\}$. Let $T'$ be a tree in $P^*$ and consider $T'' = T' - (X_m\setminus \{x\})$. Note that $T'' - \{x\} = T' - X_m = T_m$. Therefore $u_{T''}$ is well-defined for every node $u \in V(T_m)$, and every node in $T''$ is equal to $u_{T''}$ for some $u\in V(T_m)$, except for $x$ and its parent in $T''$. So let $w$ be the parent of $x$ in $T''$, $u_{T''}$ the parent of $w$ in $T''$, and $v_{T''}$ the child of $w$ in $T''$ that is not $x$. Observe that $T''$ can be obtained from $T_m$ by grafting $x$ onto the arc $uv$. Then it is enough to show that $uv \in F$. To see that $uv \in F$, first note that for each $z' \in \{y\} \cup Z$, $z'_{T''}$ is well-defined and $ (z'_{T''})_{T'} = z'_{T'}$ (see Lemma \[lem:corrTransitive\]). Then as $x$ is descended from $(y_{T''})_{T'} = y_{T'}$ in $T'$, $x$ is descended from $y_{T''}$ in $T''$ (Lemma \[lem:corrAncestor\]). Similarly, as $x$ is not descended from $(z'_{T''})_{T'} = z'_{T'}$ in $T'$ for any $z' \in Z$, $x$ is not descended from $z_{T''}$ in $T''$. Thus $x \in {\mathcal{L}}_{T''}(y_{T''}) \setminus \bigcup_{z' \in Z} {\mathcal{L}}_{T''}(z'_{T''})$. It follows that $u_{T''}$ is a descendant of $y_{T''}$ in $T''$ (note that $y_{T''} \neq w$, as $w$ is not the least common ancestor of any set of labels in ${\mathcal{X}}(T_m)$). Also, $v_{T''}$ is not a descendant of $z_{T''}$ for any $z' \in Z$, unless $v_{T''} \in \bigcup_{z\in Z} z_{T''}$, as otherwise $x$ would be a descendant of such a $z_{T''}$. Thus, $v_{T''}$ is either a member or an ancestor of $({\mathcal{L}}_{T''}(y_{T''}) \setminus \bigcup_{z' \in Z}{\mathcal{L}}_{T''}(z'_{T''}))) \cup \bigcup_{z' \in Z}z'_{T''}$. It follows using Lemma \[lem:corrAncestor\] that $u$ is a descendant in $T_m$ of $y$, and $v$ is an ancestor of $({\mathcal{L}}_{T_m}(y) \setminus \bigcup_{z' \in Z}{\mathcal{L}}_{T_m}(z'))) \cup \bigcup_{z' \in Z}z'$. Then $uv \in F$, as required. Now that we have constructed our set $P'$, it remains to find, for each $T_e \in P'$, every tree $T'$ on ${\mathcal{X}}$ such that $T' - (X_m\setminus \{x\}) = T_e$ and $T' - \{x\} = T_1 - \{x\}$. This will give us our set $P$, as for every $T' \in P^*$, $T' - (X_m\setminus \{x\})$ is a tree $T_e$ in $P'$, and $T' - \{x\} = T_1 - \{x\}$. Let $e = uv$, where $u,v \in V(T_m)$, and let $T_{1e}$ be the subtree of $T_1 - \{x\}$ whose root is $v$, and has as its label set $v$ together with all labels in $X_m\setminus \{x\}$ descended from $u$ but not $v$. Then we have to try every way of adding $x$ into this tree. If $T_{1e}$ contains $t$ labels from $X_m$, then there are $2t-1$ places to try adding $x$. Therefore $P$ will have at most $2|X_m| \leq 2(d+d')$ additional trees compared to $P'$, and so $|P| \leq 18(d+d')+8$. Algorithm \[alg:candidateTrees\] gives the full procedure to construct $P$. $T_1, T_2$ are two trees, $x$ is a label that cannot be moved in $T_2$ (but must be moved in $T_1$), $d$ is the maximum number of leaves we can remove in a tree, $d'$ is the maximum number of leaves we can move in $T_1$. For this pseudocode, every tree $T$ has a degree-$1$ root $r(T)$. Find $X_m'$ such that $|X_m'| \leq d' + d - 1$ and $(T_1 - \{x\}) - X_m' = (T_2 - \{x\}) - X_m'$ Set $X_m = X_m' \cup \{x\}$ Set $T_m = T_1 - X_m$ Set $(y,Z) = \textsc{location-restriction}(T_m, T_2, X_m, x, d, d')$ Set $U = \{u\in V(T_m): u \in Z$ or $u$ is a leaf descended from $y$ but not from any $z'\in Z\}$ Set $F = \{uv \in E(T_m): uv$ is on a path from $y$ to $U$} Set $P = \emptyset$ Set $T_1' = T_1 - \{x\}$ Set $u_{T_1'} = $ the node in $T_1'$ corresponding to $u$ Set $v_{T_1'} = $ the node in $T_1'$ corresponding to $v$ Set $X_e= $ set of labels $l$ in $X_m\setminus \{x\}$ for which $l$ has an ancestor $v'$ in $T_1'$ with $v'$ descended from $u_{T_1'}$, $v_{T_1'}$ descended from $v'$ Set $U = v_{T_1'} \cup X_e$ Set $E_e = \{u'v' \in E(T_1'): u'v'$ is on a path from $u_{T_1'}$ to $U$} Constuct $T'$ from $T_1'$ by grafting $x$ on $u'v'$ Set $P = P \cup \{T'\}$ Return $P$ \[alg:candidateTrees\] To analyze the complexity, recall that we find $X_m$, and therefore construct $T_m$ and $T_m'$, in $O(n \log n)$ time. As shown above, we can find the node $y$ and set $Z$ in $O(n)$ time. Given $y$ and $Z$, the set of arcs $F$ can be found in $O(n)$ time using a depth-first search approach. For each $e \in F$ it takes $O(n)$ time to construct $T_e$, and so the construction of $P'$ takes $O(|F|n) = O((16(d+d')+8)n)$ time. Finally, the construction of of $P$ from $P'$ takes $O(|P|n) = O((18(d+d')+8)n)$ time. Putting it all together, we have that the construction of $P$ takes $O(n(\log n + 18(d+d')+8))$ time. We will call the set of trees $P$ described in Lemma \[lem:dont-check-too-many-trees\] the set of *candidate trees* for $(T_1, T_2, x)$. We are finally ready to give the proof of Theorem \[thm:fpt-in-d\] **Theorem \[thm:fpt-in-d\]** *(restated). [`AST-LR-d`]{} can be solved in time $O(c^d d^{3d}(n^3 + tn \log n))$, where $c$ is a constant not depending on $d$ or $n$.* The outline for our algorithm is as follows. We employ a branch-and-bound algorithm, in which at each step we attempt to modify the input tree $T_1$ to become close to a solution. We keep track of an integer $d'$, representing the maximum length of an LPR sequence between $T_1$ and a solution. Initially set $d' = d$. At each step, if $d_{LR}(T_1,T_i) \leq d$ for each $T_i \in {\mathcal{T}}$ then $T_1$ is a solutioon, and we are done. Otherwise, there must exist sime $T_i$ for which $d_{LR}(T_1,T_i)\geq d+d'$. In this case, we calculate the $(d+d')$ disagreement kernel $S$ between $T_1$ and $T_i$ (using the procedure of Lemma \[lem:disagreementKernel2\]), and for each $x \in S$, attempt to construct a set $P$ of trees as in Lemma \[lem:dont-check-too-many-trees\]. For each $T' \in P$, we try replacing $T_1$ with $T'$, reducing $d'$ by $1$, and repeating the procedure. Algorithm \[alg:fpt-d\] describes the full procedure formally. We claim that Algorithm \[alg:fpt-d\] is a correct algorithm for [`AST-LR-d`]{}, and runs in time $O(c^d d^{3d}(n^2 + tn \log n))$, for some constant $c$ not depending on $n$ or $d$. First notice that if, in a leaf node of the branch tree created by Algorithm \[alg:fpt-d\], a tree $T^*$ is returned, this occurs at line \[line:returnsol\] in which case it has been verified that $T^*$ is indeed a solution. As an internal node of the branch tree returns a tree if and only if a child recursive call also returns a tree (the for loop on line \[line:for-loop-P\]), this shows that when the algorithm outputs a tree $T^*$, it is indeed a solution. We next show that if a solution exists, then Algorithm \[alg:fpt-d\] will return one. Suppose that ${\mathcal{T}}$ admits a solution, and let $T^*$ be a solution that minimizes $d_1 := d_{LR}(T_1, T^*)$, with $d_1 \leq d'$. We show that one leaf of the branch tree created by the algorithm returns $T^*$ (and thus the root of the branch tree also returns a solution, albeit not necessarily $T^*$). This is done by proving that in one of the recursive calls made to [mastrl-distance]{} on line \[line:reccall\], the tree $T'$ obtained from $T_1$ satisfies $d_{LR}(T', T^*) = d_1 - 1$. By applying this argument inductively, this shows that the algorithm will find $T^*$ at some node of depth $d_1$ in the branch tree of the algorithm. First notice that since $d_{LR}$ is a metric, for each $T_i \in {\mathcal{T}}$, $d_{LR}(T_1, T_i) \leq d_{LR}(T_1, T^*) + d_{LR}(T^*, T_i) \leq d' + d$, and so the algorithm will not return $FALSE$ on line \[line:too-high-dist\]. If $T_1$ isn’t a solution, then there is a tree of ${\mathcal{T}}$, say $T_2$ w.l.o.g., such that $d_{LR}(T_1, T_2) > d$. Notice that in this case, all the conditions of Lemma \[lem:must-move-x\] are satisfied, *i.e.* $d_{LR}(T_1, T_2) > d$, and there are sets $X_1, X_2 \subseteq {\mathcal{X}}$ both of size at most $d$ such that $T_1 - X_1 = T^* - X_1$, $T_2 - X_2 = T^* - X_2$. Thus there is a minimal disagreement $X$ between $T_1$ and $T_2$, $|X| \leq d' + d$, and $x \in X$ such that $x \in X_1 \setminus X_2$. By Lemma \[lem:lpr-seq-equiv\], there is an LPR sequence $L = (x_1, \ldots, x_k)$ turning $T_1$ into $T^*$, where $\{x_1, \ldots, x_k\} = X_1$. As $x \in X_1$, by Lemma \[lem:lpr-order\], the leaves appearing in $L$ can be reordered, and we may assume that $x = x_1$. Finally by Lemma \[lem:dont-check-too-many-trees\], if $T'$ satisfies $d_{LR}(T', T^*) \leq d_1 - 1$ and $T'$ can be obtained from $T_1$ by an LPR move on $x$, then $T' \in P$. As we are making one recursive call to [mastrl-distance]{}for each tree in $P$, this proves that one such call replaces $T_1$ by $T'$ such that $d_{LR}(T', T^*) = d_1 - 1$. As for the complexity, recall from Lemma \[lem:disagreementKernel2\] that the $(d + d')$-disagreement kernel $S$ computed in line 8 contains at most $8d^2$ labels.Therefore when Algorithm \[alg:fpt-d\] enters the ’for’ loop of line 9, it branches into at most $8d^2$ cases, one for each $x \in S$. Within each of these cases, the algorithm enters at most $|P|$ recursive calls, each of which decrements $d'$. As $|P| \leq 18(d+d')+8 \leq 36d + 8$ by Lemma \[lem:dont-check-too-many-trees\], a single call of the algorithm splits into at most $8d^2(36d + 8) = O(d^3)$, each of which decrements $d'$. Therefore, the branching tree created by the algorithm has degree at most $c d^3$ (for some constant $c$) and depth at most $d$, and so $O(c^d d^{3d})$ cases are considered. As $d_{LR}(T_1, T_i)$ can be calculated in $O(n \log n)$ time for each $T_i$, a single call of lines 2-5 of the algorithm takes $O(tn\log n)$ time. A single call of lines 6-8 takes $O(n^2)$ time by Lemma \[lem:dont-check-too-many-trees\]. Thus the total time for all calls of lines 2-8 is $O(c^d d^{3d}n (n^2 + t \log n)$. Each call of line 10 occurs just before a recursive call to the algorithm, as so line 10 is called at most $O(c^d d^{3d})$ times. A single call of line 10 takes $O(n(\log n + 18(d + d') + 8)) = O(n(\log n + 36d)) $ time by Lemma \[lem:dont-check-too-many-trees\], and so the total time for all calls of line 10 is $O(c^d d^{3d}n(\log n + 36d))$. Thus in total, we have that the running time of the algorithm is $O(c^d d^{3d} (n^2 + n (t \log n + 36d))$. As we may assume $d\leq n$, this simplifies to $O(c^d d^{3d}(n^2 + tn \log n))$. ${\mathcal{T}}$ is the set of input trees (represented as a sequence to distinguish $T_1$ from the other trees), $d$ is the maximum number of leaves we can remove in a tree, $d'$ is the maximum number of leaves we can move in $T_1$, which should be initially set to $d$. Return $T_1$ \[line:returnsol\] Return FALSE \#handles the $d' \leq 0$ case \[line:too-high-dist\] Choose $T_i \in {\mathcal{T}}$ such that $d_{LR}(T_1, T_i) > d$ Set $S = \textsc{disagreement-kernel}(d+d',T_1, T_i)$ \[line:dis-kernel\] \[line:for-loop-P\] Set $P = \textsc{candidate-trees}(T_1,T_i,x,d,d')$ \[line:compute-P\] $T^* = FALSE$ $T' = \textsc{mastrl$-$distance}((T, T_2, \ldots, T_t), d, d' - 1)$ \[line:reccall\]  $T'$ is not $FALSE$, let $T^* := T'$ Return $T^*$ [^1]: All trees we consider here are uniquely leaf-labeled, rooted (*i.e.* are out-trees) and binary; see next section for formal definitions. [^2]: All missing proofs are provided in Appendix.

--- abstract: 'We address the effect of disorder geometry on the critical force in disordered elastic systems. We focus on the model system of a long-range elastic line driven in a random landscape. In the collective pinning regime, we compute the critical force perturbatively. Not only our expression for the critical force confirms previous results on its scaling with respect to the microscopic disorder parameters, it also provides its precise dependence on the disorder geometry (represented by the disorder two-point correlation function). Our results are successfully compared to the results of numerical simulations for random field and random bond disorders.' address: - '$^1$ Institut Jean Le Rond d’Alembert (UMR CNRS 7190), Université Pierre et Marie Curie, F-75005 Paris, France' - '$^2$ Department of Physics, University of Massachusetts, Amherst, MA 01003, USA.' - 'Laboratoire Probabilités et Modèles Aléatoires (UMR CNRS 7599), Université Pierre et Marie Curie & Université Paris Diderot, 75013 Paris, France' - 'Laboratoire Physique Th[é]{}orique et Mod[è]{}les Statistiques (UMR CNRS 8626), Université de Paris-Sud, Orsay Cedex, France' author: - 'Vincent D'' emery$^{1,2}$' - Vivien Lecomte - Alberto Rosso bibliography: - 'biblio.bib' title: Effect of disorder geometry on the critical force in disordered elastic systems --- Introduction ============ Disordered elastic systems [@halpin-healy_kinetic_1995; @Kardar1998; @brazovskii_pinning_2004; @Agoritsas2012] are ubiquitous in Nature and condensed matter physics; they encompass a wide range of systems going from vortex lattices in superconductors [@Larkin1979] to ferromagnetic domain walls [@Lemerle1998], wetting fronts [@Joanny1984], imbibition fronts [@Soriano2002; @Santucci2011] or crack fronts in brittle solids [@Bouchaud2002; @Bonamy2008]. In simple models for those phenomena, an elastic object struggles to stay flat while its random environment tries to deform it, in or out of equilibrium. An example is given by an elastic line in a random landscape, that is pictured on Fig. \[fig\_schema\]. As a result of the competition between disorder and elasticity, the elastic object becomes rough and is characterized by a universal roughness exponent [@barabasi_fractal_1995; @krug_origins_1997] that depends on the dimension of the problem, the range of the elastic interaction and the type of disorder, but not on the microscopic details of the system [@Kardar1998; @Agoritsas2012]. ![(Colour online) An elastic line pulled by a spring of stiffness $\kappa$ and position $w$ in a random landscape. The bottom grey surface is the potential $\sigma V(x,u)$. The top blue surface is the effective potential seen by the line, *i.e.* the bare potential plus the parabolic potential $\frac{\kappa}{2}(w-u)^2$ exerted to the spring.[]{data-label="fig_schema"}](schema.pdf){width=".650\columnwidth"} This coupling between disorder and elasticity also has an important consequence on the response of the elastic object to an external force [@Larkin1979]. At zero temperature, there exists a critical force below which it does not move and remains *pinned* by the disorder. If the applied force is larger than this threshold, the elastic object *unpins* and acquires a non-zero average velocity. This describes the *depinning transition* of the elastic line. A finite temperature rounds this behaviour for forces close to the threshold [@Bustingorry2008] and allows the object to move at a finite velocity for forces well below the threshold, by a thermally activated motion called *creep* [@Ioffe1987; @nattermann_scaling_1990; @balents_large-n_1993; @Chauve2000; @Kolton2009]. The critical force plays a crucial role in applications. In [type-II]{} superconductors, it corresponds to the critical current above which the vortex lattice starts moving, leading to a superconductivity breakdown [@Anderson1964]. In brittle solids, it determines the critical loading needed for a crack to propagate through the sample and break it apart [@Bouchaud2002]. Contrarily to roughness and depinning exponents, the critical force is not a universal quantity and its value depends in general on the details of the model. However, scaling arguments allow to find its dependence on the disorder amplitude and the different lengthscales present in the system, such as the size of the defects and the typical distance between them [@Larkin1979; @Nattermann1990]. Unfortunately this approach gives the critical force only up to a numerical prefactor, whose value depends on microscopic quantities such as the geometrical shape of the impurities, and which is essential to determine in view of applications. Recently, a numerical self-consistent scheme [@Roux2003; @Roux2008] and numerical simulations have focused on a precise determination of the critical force [@Demery2012c; @Patinet2013] in the context of brittle failure. Notably, is has been shown that in the *collective* regime, occurring at weak disorder amplitude, the critical force does not depend on the disorder distribution but only on the disorder amplitude and correlation length [@Demery2012c]. Still, the effect of the disorder geometry, that is partly encoded in its two-point correlation function, remains to be determined. In this article, we address the question of the dependence of the critical force on the disorder geometry. We focus on the case of a long-range elastic line in a random potential, that is the relevant model for wetting fronts and crack fronts in brittle failure. We restrict ourselves to the collective pinning regime, that appears when the disorder amplitude is small. The line is driven by a spring pulled at constant velocity, and the *drag force* needed to move the spring is computed perturbatively in the disorder amplitude. In the limit of zero spring stiffness and zero velocity, this force is the critical force and we derive its analytic expression. Our expression depends explicitly on the two-point correlation function of the disorder, and thus on the disorder geometry. Moreover, our computation is valid for a random bond disorder as well as a random field disorder [@Chauve2000]. Numerical simulations are performed for both types of disorder and various disorder geometries. They provide a successful check of our analytical result and show that two systems with the same disorder amplitude and correlation length can have a different critical force if their two-point correlation functions are different. The paper is organized as follows. In section \[sec\_model\], we introduce the model of a long-range elastic line driven in a random landscape. In section \[sec\_main\_result\], we summarize our results. Section \[sec\_analytical\] is devoted to the analytical computation of the drag force, from which we deduce the critical force. Numerical simulations details and results are presented in section \[sec\_numerical\]. We conclude in section \[sec\_conclu\]. Model {#sec_model} ===== We consider a 1+1 dimensional elastic line of internal coordinate $x$ and position $u(x,t)$, pulled by a spring of stiffness $\kappa$ located at position $w(t)$ in a random energy landscape $\sigma V(x,u)$; this system is represented on Fig. \[fig\_schema\]. The parameter $\sigma$ represents the disorder amplitude. The equation of evolution of the line position at zero temperature is [@Kardar1998; @Ferrero2013] $$\label{eq_evol} \partial_t u(x,t)=\kappa[w(t)-u(x,t)]+f{_{\mathrm{el}}}[u(\cdot,t)](x)-\sigma \partial_u V(x, u(x,t)).$$ The elastic force $f{_{\mathrm{el}}}[u(\cdot,t)](x)$ is linear in $u(x,t)$ and we consider the case of a long-range elasticity $$\label{eq_force_el_cont} f{_{\mathrm{el}}}[u(\cdot,t)](x)=\frac{c}{\pi}\int \frac{u(x',t)-u(x,t)}{(x-x')^2}{\mathrm{d}}x'.$$ The disorder has zero mean ($\overline{V(x,u)}=0$) and two-point correlation function $$\overline{V(x,u)V(x',u')} =R_x(x-x')R_u(u-u').$$ The overline represents the average over the disorder. Alternatively, one can also use the force correlation function, $$\begin{aligned} \overline{\partial_u V(x,u)\partial_{u'}V(x',u')} & =- R_x(x-x')\partial_u^2 R_u(u-u') \label{eq:2pointforcecorrelator}\\ & =\Delta_x(x-x')\Delta_u(u-u'). \label{eq:2pointforcecorrelatorDelta}\end{aligned}$$ with $\Delta_u=-\partial_u^2 R_u$. We do not assume that the disorder is Gaussian distributed. For the so-called random bond (RB) case, the potential $V(x,u)$ is short-range correlated both in the variables $x$ and $u$; this implies a global constraint on the force correlation function, namely $\int \Delta_u(u) {\mathrm{d}}u=0$. For the random field (RF) case, $V(x,u)$ is for instance a Brownian motion as a function of $u$, with diffusion constant $\int \Delta_u(u) {\mathrm{d}}u>0$ [@Chauve2000]. Finally, we impose a constant velocity $v$ to the spring, $$w(t)=vt.$$ The drag $f{_{\mathrm{dr}}}$ is defined to be the average force exerted on the line by the spring $$f{_{\mathrm{dr}}}(\kappa,v)=\kappa\left[w(t)- \overline{\left\langle u(x,t) \right\rangle}\right], \label{eq:def_fdrag}$$ where $\langle \cdot \rangle$ denotes the average along the internal coordinate $x$. Since the landscape is statistically translation invariant, this quantity is expected not to depend on time. Besides, it is expected that both the drag force and the critical force, which depend on the realization of the disorder, tend to a limit at large system size which is independent of the particular realization – hence equal to its average over disorder, as we make use of in (\[eq:def\_fdrag\]). Those averages are the so-called *thermodynamic* drag and critical forces. We focus on the computation of the drag as a function of the spring stiffness $\kappa$ and velocity $v$. We then show how to extract the critical force from this force-velocity characteristics, fixing the velocity instead of the force. Main result {#sec_main_result} =========== Our main result is the following expression of the critical force, valid in the collective pinning regime (when the disorder amplitude $\sigma$ is small): $$\label{eq_anal_pred} f{_{\mathrm{c}}} \simeq \frac{\sigma^2\tilde \Delta_x(0)}{4\pi c}\int|k_u|\tilde \Delta_u(k_u){\mathrm{d}}k_u.$$ where $\tilde \Delta_{x,u}$ are Fourier transforms of $\Delta_{x,u}$. The expression of $f{_{\mathrm{c}}}$ holds for a long-range elastic line for both random bond and random field disorders. It is compared to simulations results on Fig. \[fig\_sigma\_fc\], that shows a very good agreement in the collective pinning regime $\Sigma\ll 1$. The dimensionless disorder amplitude $\Sigma$ is defined later (\[eq:disorder\_parameter\]). This analytic result is derived in Section \[sec\_analytical\] and the numerical simulations are detailed in section \[sec\_numerical\]. ![(Colour online) Critical force as a function of the dimensionless disorder amplitude $\Sigma$, defined in (\[eq:disorder\_parameter\]): comparison between numerical simulations and analytical prediction (\[eq\_anal\_pred\]) for random field (models A and B) and random bond (model C) disorders (the disorders are defined precisely in section \[sub\_numerical\_model\]). Points are results of the simulations and the line is the analytical prediction, valid in the small $\Sigma$ limit.[]{data-label="fig_sigma_fc"}](Sigma_Fc3.pdf){width=".650\columnwidth"} Analytical computation {#sec_analytical} ====================== In this section, we start by computing the average force required to drive the line at an average speed $v$ with a spring of stiffness $\kappa$. This provides us a force-velocity characteristic curve that depends on the spring stiffness. From these characteristic curves, we deduce that sending the velocity and stiffness to zero in the appropriate order allows to extract the critical force. Drag force ---------- The line evolution equation (\[eq\_evol\]) is highly non linear due to the presence of the random potential; it is thus very difficult to handle. To evaluate the average drag, we resort to a perturbative analysis in the disorder amplitude $\sigma$. We expand the line position in powers of $\sigma$ as $$u(t)=\sum_n \sigma^n u_n(t).$$ At order $0$, the solution of (\[eq\_evol\]) is independent of disorder $$\label{eq_zeroth_order} u_0(x,t)=vt-\frac{v}{\kappa},$$ leading from (\[eq:def\_fdrag\]) to the average drag $$\label{eq_drag_0} f{_{\mathrm{dr}}}^{(0)}(\kappa,v)=v.$$ The computation at higher orders is done in Fourier space, with the convention $\tilde g(k_x)=\int g(x) {\mathrm{e}}^{-ik_xx} {\mathrm{d}}x$ (and similarly along direction $u$). We start by Fourier transforming the elastic force, $$f{_{\mathrm{el}}}[u(\cdot,t)](x)=-c\int |k_x|\tilde u(k_x,t){\mathrm{e}}^{ik_xx}\frac{{\mathrm{d}}k_x}{2\pi},$$ and the disorder correlator, $$\label{eq_dis_correl_fourier} \overline{\tilde V(k_x,k_u)\tilde V(k_x',k_u')} =\\(2\pi)^2\delta(k_x+k_x')\delta(k_u+k_u')\tilde R_x(k_x)\tilde R_u(k_u).$$ We also define, corresponding to (\[eq:2pointforcecorrelator\]-\[eq:2pointforcecorrelatorDelta\]) $$\begin{aligned} \tilde \Delta_x(k_x) & = \tilde R_x(k_x),\\ \tilde \Delta_u(k_u) & = k_u^2\tilde R_u(k_u).\end{aligned}$$ The first order contribution to the line position satisfies $$\partial_t u_1(x,t)+\kappa u_1(x,t)-f{_{\mathrm{el}}}[u_1(\cdot,t)](x)=-\partial_u V(x,u_0(t)).$$ Fourier transforming this equation in directions $x$ and $u$ gives the solution in Fourier space $$\label{eq_line_first_order} \tilde u_1(k_x,t)=-\int \frac{ik_u}{ik_u v+\omega(k_x)}{\mathrm{e}}^{ik_u u_0(t)}\tilde V(k_x,k_u)\frac{{\mathrm{d}}k_u}{2\pi},$$ where we have introduced the damping rate $$\label{eq_damping_cont} \omega(k_x)=\kappa+c|k_x|,$$ which fully encompasses the effect of the elasticity. Since the first order correction is linear in the potential $V$, its average over disorder is 0 and it does not contribute to the drag: $f{_{\mathrm{dr}}}^{(1)}=0$. At second order, the evolution equation reads $$\begin{aligned} \partial_t u_2(x,t)+&\kappa u_2(x,t)-f{_{\mathrm{el}}}[u_2(\cdot,t)](x) =\nonumber\\ &-\sigma^{-1} [\partial_u V(x,u_0(t)+\sigma u_1(x,t))-\partial_u V(x,u_0(t))].\end{aligned}$$ Following an idea introduced by Larkin [@Larkin1970], we expand the potential around $u_0(t)$, getting $$\begin{aligned} \partial_t u_2(x,t)+\kappa u_2(x,t)-f{_{\mathrm{el}}}[u_2(\cdot,t)](x)=-\partial_u^2 V(x,u_0(t))u_1(x,t). \label{eq_expansion}\end{aligned}$$ It reads in Fourier space $$\begin{aligned} \partial_t\tilde u_2(k_x,t) + &\omega(k_x)\tilde u_2(k_x,t)=\nonumber\\ & \int k_u^2 {\mathrm{e}}^{ik_u u_0(t)}\tilde V(k_x',k_u)\tilde u_1(k_x-k_x',t)\frac{{\mathrm{d}}k_x' {\mathrm{d}}k_u}{(2\pi)^2}.\end{aligned}$$ Inserting the first order result (\[eq\_line\_first\_order\]) and solving gives $$\begin{aligned} \tilde u_2(k_x,t) =& \int \frac{ik_u'^2(k_u'-k_u) {\mathrm{e}}^{ik_u u_0(t)}}{[ik_uv+\omega(k_x)][i(k_u-k_u')v+\omega(k_x-k_x')]} \nonumber\\ & \ \qquad \times \tilde V(k_x-k_x',k_u-k_u')\tilde V(k_x',k_u')\frac{{\mathrm{d}}k_x'{\mathrm{d}}k_u{\mathrm{d}}k_u'}{(2\pi)^3}.\end{aligned}$$ Averaging over disorder with (\[eq\_dis\_correl\_fourier\]) leads to $$\overline{\tilde u_2(k_x)}=2\pi\delta(k_x)\int \frac{ik_u\tilde \Delta_x(k_x')\tilde \Delta_u(k_u)}{\kappa[-ik_uv+\omega(k_x')]} \frac{{\mathrm{d}}k_x'{\mathrm{d}}k_u}{(2\pi)^2}.$$ The proportionality to $\delta(k_x)$ signifies that, in direct space, the average second order correction does not depend on the internal coordinate $x$. It reads $$\overline{u_2}=-\int \frac{vk_u^2\tilde \Delta_x(k_x)\tilde \Delta_u(k_u)}{\kappa[k_u^2v^2+\omega(k_x)^2]} \frac{{\mathrm{d}}k_x {\mathrm{d}}k_u}{(2\pi)^2}.$$ The second order drag is thus: $$f{_{\mathrm{dr}}}^{(2)}(v,\kappa)=\sigma^2\int \frac{vk_u^2\tilde \Delta_x(k_x)\tilde \Delta_u(k_u)}{k_u^2v^2+(\kappa+c|k_x|)^2} \frac{{\mathrm{d}}k_x {\mathrm{d}}k_u}{(2\pi)^2}.$$ Adding this result to the zeroth order drag (\[eq\_drag\_0\]) provides the drag up to the order $\sigma^2$, $$\begin{aligned} f{_{\mathrm{dr,2}}}(v,\kappa)& = f{_{\mathrm{dr}}}^{(0)}(v,\kappa) + f{_{\mathrm{dr}}}^{(1)}(v,\kappa) + f{_{\mathrm{dr}}}^{(2)}(v,\kappa) \\ & = v+\sigma^2\int \frac{vk_u^2\tilde \Delta_x(k_x)\tilde \Delta_u(k_u)}{k_u^2v^2+(\kappa+c|k_x|)^2} \frac{{\mathrm{d}}k_x {\mathrm{d}}k_u}{(2\pi)^2}. \label{eq_drag_line}\end{aligned}$$ This drag gives us access, at the perturbative level, to the crucial force-velocity characteristic. It is plotted on Fig. \[fig\_f\_v\] for a random field disorder with Gaussian two-point functions, at different values of the spring stiffness. When the spring stiffness $\kappa$ goes to zero, the depinning transition appears clearly and becomes sharp when $\kappa=0$. The picture is qualitatively similar for a random bond disorder. Any positive stiffness rounds the transition, analogously to the temperature [@Agoritsas2012; @Chen1995; @Bustingorry2008]. ![(Colour online) Force-velocity curve using the drag (\[eq\_drag\_line\]) computed to the second order in $\sigma$, for different values of the parabola curvature $\kappa$, indicated by the colour scale. The two-point functions $\Delta_x$ and $\Delta_u$ of the disorder are centered Gaussian functions of unit variance, encoding a random field disorder.[]{data-label="fig_f_v"}](f_v.pdf){width=".650\columnwidth"} Critical force {#label} -------------- As noted above, the usual force-velocity characteristic at zero temperature is recovered in the limit $\kappa\rightarrow 0$; its equation is given by $$\label{eq_v_f_k0} f_{\kappa=0,2}(v)=v+\frac{\sigma^2}{c}\int \frac{k_u^2\tilde\Delta_x(vq/c)\tilde\Delta_u(k_u)}{k_u^2+q^2}\frac{{\mathrm{d}}q{\mathrm{d}}k_u}{(2\pi)^2}.$$ We have performed the variable substitution $ck_x=vq$ in order to eliminate the velocity in the denominator. Taking the small velocity limit in this expression gives the critical force (\[eq\_anal\_pred\]) $$f{_{\mathrm{c,2}}}=\frac{\sigma^2\tilde \Delta_x(0)}{4\pi c}\int|k_u|\tilde \Delta_u(k_u){\mathrm{d}}k_u.$$ This expression is our main result, announced in Eq. (\[eq\_anal\_pred\]). The index $2$ indicates that this critical force comes from a second order perturbative computation in the disorder amplitude $\sigma$. The two limits do not commute: since any non-zero stiffness rounds the transition, taking the limit of zero velocity first would give a zero critical force (see Fig. \[fig\_f\_v\]). To get the depinning exponent $\beta$ defined by $v \sim_{f\rightarrow f{_{\mathrm{c}}}^+} (f-f{_{\mathrm{c}}})^\beta$, we have to go one step further in the Taylor expansion of (\[eq\_v\_f\_k0\]) around $v=0$. This can be done analytically for simple correlators $\tilde\Delta_x(k_x)$ and $\tilde\Delta_u(k_u)$, or numerically in the general case (see Fig. \[fig\_f\_v\] for an example). We get $$v\sim f-f{_{\mathrm{c}}},$$ which corresponds to the mean field behaviour ${\beta=\beta{_{\mathrm{MF}}}=1}$ [@Fisher1998] valid above the upper critical dimension $d{_{\mathrm{uc}}}$ (for the long-range elasticity $d{_{\mathrm{uc}}}=2$). Below $d{_{\mathrm{uc}}}$, the mean field value $\beta{_{\mathrm{MF}}}=1$ is an upper bound of the exact value of $\beta$ which can be estimated by a functional renormalization group $\epsilon-$expansion [@Chauve2001; @le_doussal_functional_2004; @wiese_functional_2006] or evaluated numerically to $\beta=0.625 \pm 0.0005$ for the long-range elastic line [@Duemmer2007]. Numerical simulations {#sec_numerical} ===================== We now turn to the comparison of our analytical prediction to numerical simulations of the line. Since our computation remains valid for both random bond and random field disorder, we perform numerical simulations on both cases. Numerical model {#sub_numerical_model} --------------- In our model a line of length $L$ is discretized with a step $a$ and its elasticity is given by $$\label{eq_force_el_disc} f{_{\mathrm{el}}}[u(\cdot,t)]_n=\frac{c}{\pi a}\sum_{n'\neq 0} \frac{u_{n'}(t)-u_n(t)}{(n-n')^2}.$$ Each point of the line moves on a rail with a disordered potential which is uncorrelated with the others rails, so that $\Delta_x(x-x') = \delta_{x,x'}$. Three different models of disorder are considered: - model A: random field disorder obtained by the linear interpolation of the random force drawn at the extremities of segments of length $1$; - model B: random field disorder obtained in the same way as model A, but the segments have length $0.1$ with probability $1/2$ and $1.9$ with probability $1/2$; - model C: random bond disorder obtained by the spline interpolation of the random energies drawn at the extremities of segments of length $1$. Analytical prediction --------------------- The prediction of the critical force for the three models can be obtained observing that for a discrete line the damping rate (\[eq\_damping\_cont\]) changes to $$\omega(k_x)=\kappa+c \left(|k_x|- \frac{ak_x^2}{2\pi}\right).$$ where the wave-vector $k_x$ is restricted to $\left[-\pi/a,\pi/a \right]$. The limits $\kappa\rightarrow 0$ and $v\rightarrow 0$ give exactly the same result as (\[eq\_anal\_pred\]): $$f{_{\mathrm{c,2}}}=\lim_{v\rightarrow 0}\lim_{\kappa\rightarrow 0} f{_{\mathrm{tot,2}}}(v,\kappa)=\frac{\sigma^2\tilde \Delta_x(0)}{4\pi c}\int|k_u|\tilde \Delta_u(k_u){\mathrm{d}}k_u.$$ It is remarkable that discretizing the line does not change the critical force. In all models we set $a=1$ so that $\tilde\Delta_x(0)=1$ and the functional of $\Delta_u(u)$ appearing in our expression (\[eq\_anal\_pred\]) for the critical force is computed in \[ap\_disorder\_correlations\] for the three models. The final prediction for model A is: $$\label{eq_anal_pred_044} f{_{\mathrm{c,2}}}=\frac{2\log(2)}{\pi}\frac{\sigma^2}{c}\simeq 0.44\frac{\sigma^2}{c}.$$ while for model B a numerical computation gives $$f{_{\mathrm{c,2}}}\simeq 0.55\frac{\sigma^2}{c},$$ and for model C we have $$f{_{\mathrm{c,2}}}\simeq 2.83 \frac{\sigma^2}{c}.$$ Measurement of the critical force --------------------------------- We start our numerical procedure with a flat configuration $u(x)=0$ and $w=0$. Then the interface moves to a state, $u_{w=0}(x)$ which is stable with respect to small deformations. Increasing $w$, the interface position increases and a sequence of stable states can be recorded. For each $w$, the stable state $u_w(x)$ can be found using the algorithm proposed in [@Rosso2002] and we measure the pinning force $$\label{eq_def_pin_force} f_w(\kappa)=\kappa [w-\langle u_w(x)\rangle].$$ This pinning force depends on the realization of the disordered potential (its fluctuations have been studied in [@bolech_universal_2004]). An example of the evolution of the pinning force with $w$ is shown on Fig. \[fig\_w\_f\]. The pinning force is in general dependent on the initial condition, however, due to the Middleton no-passing rule [@Middleton1992], we can prove that there exists a $w^*>0$ such that the sequence of stable states $u_{w>w^*}(x)$ becomes independent of the initial condition. A stationary state is thus reached, where the pinning force oscillates around its average value $\overline{f(\kappa)}$ and displays correlation in $w$. Thus in order to estimate correctly $\overline{f(\kappa)}$ we sample $f_w(\kappa)$ far enough from the origin $w=0$ and for values of $w$ far enough from each other. ![(Colour online) Pinning force (\[eq\_def\_pin\_force\]) as a function of the position $w$ of the parabola, for different values of the spring stiffness $\kappa$. The dashed vertical lines represent approximate values $w^*$ after which the pinning force becomes independent of the initial condition. Model A disorder with $\sigma=1$ is used here.[]{data-label="fig_w_f"}](w_f.pdf){width=".650\columnwidth"} An exact relation (the statistical tilt symmetry [@Schulz1988]) assures that the quadratic part of the Hamiltonian (and thus the constant $\kappa$) is not renormalized. This means that the length associated by a simple dimensional analysis to the bare constant $\kappa$, namely $L_\kappa=c/\kappa$, corresponds to the correlation length of the system: above $L_\kappa$ the interface is flat and feels the harmonic parabola only, while below $L_\kappa$ the interface is rough with the characteristic roughness exponent at depinning $\zeta \simeq 0.39$ [@Rosso2002]. The separation from the critical depinning point (located exactly at the critical driving force $f{_{\mathrm{c}}}$) is described by the power law scaling [@Nattermann1992] $f-f{_{\mathrm{c}}}\sim\xi^{-1/\nu}$ where $\xi$ is the correlation length, given in our case by $L_\kappa$. When $\kappa \to 0$ (while keeping $ L\gg L_\kappa$) the pinning force tends to the thermodynamical critical force $f{_{\mathrm{c}}}$. Gathering the previous scalings, we thus have that the finite size effects on the force take the form: $$\begin{aligned} \label{finitesize} && \overline{f(\kappa)}=f{_{\mathrm{c}}} + c_1 \kappa^{1/\nu}+\cdots .\end{aligned}$$ The fluctuations around this value, $\overline{\delta f(\kappa)^2}$, depend on $L$ and $\kappa$. In the limit $L\gg L_\kappa=1/\kappa$, the interface can be modeled as a collection of independent interfaces of size $L_\kappa$ and the central limit theorem assures that the variance $\overline{\delta f(\kappa)^2}$ should scale as $\sim \kappa^{2/\nu}$, but with an extra factor $L_\kappa/L$. This allows us to write an extrapolation formula for $f{_{\mathrm{c}}}$ which is independent of the critical exponent $\nu$: $$\overline{f(\kappa)}=f{_{\mathrm{c}}}+c_1\sqrt{\kappa L \overline{\delta f(\kappa)^2}}+\cdots \label{eq:fkappafc}$$ Our determination of $f{_{\mathrm{c}}}$ is performed using this relation, by extrapolating the numerical measurements of $\overline{f(\kappa)}$ for different values of $\kappa$ to the limit $\kappa\to 0$, as shown on Fig. \[fig\_kappa\_f\]. It is worth noticing that most of the details of the finite size system such as the boundary conditions or the presence of the parabolic well does not affect the thermodynamic value of $f{_{\mathrm{c}}}$ which depends only on the elastic constant $c$ and on the disorder statistics [@Kolton2013; @Budrikis2013]. Our extrapolation of $f{_{\mathrm{c}}}$, shown on Fig. \[fig\_kappa\_f\], has been performed on samples of size $L=1000, 4000$ (depending on the value of sigma) and for parabola curvatures down to $\kappa=10^{-4}$. ![(Colour online) Pinning force averaged over the parabola position $w$, versus a function of the spring stiffness. A fit of the linear part gives the critical force, see Eq. . This plot is for model A disorder with $\sigma=0.8$ and we found $f{_{\mathrm{c}}}=0.285$.[]{data-label="fig_kappa_f"}](kappa_f.pdf){width=".650\columnwidth"} Results ------- The dimensionless critical force $F{_{\mathrm{c}}}=f{_{\mathrm{c}}}/\sigma$ is plotted versus the dimensionless disorder parameter $$\label{eq:disorder_parameter} \Sigma=\frac{\sigma\tilde\Delta_x(0)}{4\pi c}\int|k_u|\tilde\Delta_u(k_u){\mathrm{d}}k_u$$ on Fig. \[fig\_sigma\_fc\]. In all three cases, the results are very close to the theoretical prediction $F{_{\mathrm{c}}}=\Sigma$ (equivalent to ) when the disorder parameter is small. Conclusion {#sec_conclu} ========== We have shown that the critical force for a long-range elastic line in a random landscape can be computed perturbatively in the collective pinning regime, yielding the expression (\[eq\_anal\_pred\]). Our result for the critical force gives, together with its scaling with respect to the microscopic parameters, its dependence on the disorder geometry. Indeed, we have shown that two disorders that can be attributed the same correlation lengths (as in model A and B) may present a different critical force that is precisely predicted by our theory. Some previous studies have studied the scaling of the critical force with respect to microscopic parameters such as the disorder amplitude $\sigma$, the elastic constant $c$ and disorder correlation lengths $\xi_x$ and $\xi_u$ in the directions $x$ and $u$ [@Larkin1979; @Nattermann1990; @Demery2012c]. In particular for the long-range elastic line, the following scaling has been found for the critical force in the collective pinning regime [@Demery2012c]: $$\label{eq_fc_pheno} f{_{\mathrm{c}}}\sim \frac{\sigma^2\xi_x}{c\xi_u}.$$ The lengths $\xi_x$ and $\xi_u$ characterize the typical scale of the disorder correlation along $x$ and $u$, but these scales cannot be uniquely defined. Different definitions lead to correlation lengths that differ only by a numerical factor, so the scaling law (\[eq\_fc\_pheno\]) holds independently of the chosen definitions. However, this prevents the use this scaling law to make a quantitative prediction. Our formula allows to overcome this problem. In particular, starting from Eq. (\[eq\_anal\_pred\]) and writing $$\label{eq_correl_adim} \Delta_x(x) = \Delta_{x1}(x/\xi_x)$$ where $\Delta_{x1}$ is a function of the dimensionless variable $x/\xi_x$ and a similar relation defines $\Delta_{u1}$, one gets $$f{_{\mathrm{c}}}= \left(\frac{\tilde\Delta_{x1}(0)}{4\pi}\int |q_u| \tilde\Delta_{u1}(q_u){\mathrm{d}}q_u\right) \times\frac{\sigma^2\xi_x}{c\xi_u}.$$ This shows that our analytical prediction (\[eq\_anal\_pred\]) allows to recover the scaling law (\[eq\_fc\_pheno\]) and gives additionally the prefactor as a function of the correlation functions, *i.e.* it yields the explicit dependence of the critical force on the disorder geometry. The present work is not the first attempt to compute the critical force perturbatively: expansions have been performed at weak disorder [@Efetov1977; @Chauve2000], small temperature [@Chen1995] or large velocity [@Schmid1973; @Larkin1974]. Weak disorder expansions are valid up to the Larkin length [@Larkin1979], $L_c$ defined as the distance at which the line wanders enough to see the finite disorder correlation length $\xi_u$, (namely $|u(L_c)-u(0)|\simeq \xi_u$). Above the Larkin length, they however predict an incorrect roughness exponent [@Kardar1987]. Last, large velocity expansions give an estimation of the critical force that is obtained by continuing a large-velocity asymptotic result, which lies very far from the depinning regime, to zero velocity. Our computation does not need such continuation, and is compatible with the fact that perturbative expansions in the disorder amplitude are incorrect above the Larkin length, since the critical force can be evaluated from the line behaviour at the scale of the Larkin length [@Larkin1979]. Our analysis is a first step towards a more general understanding of the critical force dependence, and it can be extended in several directions. First, the opposite *individual* pinning regime occurring at a high disorder amplitude is worth investigating. The perturbative analysis used here is not suited for its study, but a few comments can be made on the grounds of former numerical studies [@Demery2012c; @Patinet2013]. First, its scaling with respect to the disorder amplitude and correlation length has been elucidated for a long-range elastic line, giving [@Demery2012c] $$f{_{\mathrm{c}}}\sim\sigma;$$ thus the critical force is now proportional to the disorder amplitude and does not depend on the disorder correlation lengths. Moreover, we have shown in a previous study [@Demery2012c] that the critical force is given by the strongest pinning sites if the pinning force is bounded. In this case, it is likely that the dependence on the disorder geometry is very weak. The case of unbounded pinning force requires further investigation. Another issue arising from our study is the question of the landscape smoothness. Our analysis requires an expansion of the potential to the second order around the position of the unperturbed line (see Eq. (\[eq\_expansion\])): the force generated by the potential must be continuous. When one tries to apply the analytical prediction (\[eq\_anal\_pred\]) to a discontinuous force landscape, it diverges because of a cusp present in the correlation function $\Delta_u(u)$. On the other hand, our previous numerical study [@Demery2012c] used a discontinuous force landscape and did not reveal any divergence, while the dependence on the disorder amplitude, $f{_{\mathrm{c}}}\sim\sigma^2$, was the same as the one observed here. This suggests that the divergence obtained when we try to apply our result to a discontinuous force landscape is regularized by a mechanism that is out of reach of the present perturbative computation. Understanding the behaviour of the elastic line and the critical force in a rougher force landscape would be an important advance on a theoretical point of view, but also for experiments where discontinuous force landscapes are ubiquitous [@Soriano2002; @Bonamy2008]. Last, the case of a short-range instead of long-range elasticity remains to be understood within our approach; interesting comparison could be established with the characteristic force of the creep-regime, whose dependency in the details of the disorder correlator (for RB disorder) has been examined recently [@agoritsas_temperature-induced_2010; @agoritsas_static_2013; @agoritsas_static_numeric_2013]. We would like to thank E. Agoritsas for fruitful discussions and for a critical reading of the manuscript. V. D. acknowledges support from the Institut des Systèmes Complexes de Paris Île de France. Disorder generation and correlation {#ap_disorder_correlations} =================================== We detail here the procedures used to generate the different models of disorder, and how to compute the disorder correlation function, that is needed to evaluate the critical force (\[eq\_anal\_pred\]). Random field disorder: models A and B ------------------------------------- ![(Colour online) Random force disorder on a rail: the force is continuous and piecewise linear.[]{data-label="fig_disorder_scheme"}](des_cont.pdf){width=".750\columnwidth"} For a random force disorder, the disorder is generated on each rail using the following procedure (see Fig. \[fig\_disorder\_scheme\]): - The rail is divided into segments of random length $l$ drawn in the distribution $P(l)$. - At the point linking the segment $j-1$ and the segment $j$, a random force $f_j$ is drawn from a Gaussian distribution with zero mean and unit variance. - Inside the segment $j$, at a generic point $u$, the force $f(u)$, is obtained by the linear interpolation such that $f(u_j)=f_j$ and $f(u_{j+1})=f_{j+1}$. ![(Colour online) Two-point correlation function of the force $\Delta_u(u)=\overline{\partial_u V(0)\partial_u V(u)}$ for the disorder models A, B and C. The model C correlation function has been rescaled by a factor $0.25$.[]{data-label="fig_correls"}](correls.pdf){width=".650\columnwidth"} We want to know the correlation of the forces at two points separated by a distance $u\geq 0$, say $f(0)$ and $f(u)$. This correlation is non-zero if the two points lie on the same segment or on neighbour segments. We introduce the length $l$ of the segment where the point $0$ lies, the length $l'$ of its right neighbour, and the left end $u_0$ of the first segment. The probability distribution for $l$ is $Q(l)=lP(l)/\bar l$, where $\bar l=\int_0^\infty P(l)dl$; the probability distribution for $l'$ is simply $P(l')$ and the one of $u_0$ is $l^{-1}\chi_{[-l,0]}(u_0)$ (meaning that the point $0$ is uniformly distributed in its segment). Putting these probabilities together, we get the probability distribution for $(l,l',u_0)$: $$\label{eq_distrib_seglengths} \mathbb{P}(l,l',u_0)=\bar l^{-1}P(l)P(l')\chi_{[-l,0]}(u_0).$$ The points $0$ and $u$ are on the same segment if ${u\leq u_0+l}$. The force at $u_0$ is $f_0$ and the force at $u_0+l$ is $f_1$; $f_0$ and $f_1$ are uncorrelated random variables with zero mean and unit variance. The forces at $0$ and $u$ are $$\begin{aligned} f(0)&=f_0 \frac{u_0+l}{l}+f_1 \frac{-u_0}{l},\\ f(u)&=f_0 \frac{u_0+l-u}{l}+f_1 \frac{u-u_0}{l}.\\\end{aligned}$$ The correlation between these two forces is $$\label{eq_correl_sameseg} \overline{f(0)f(u)}=\frac{2u_0^2+2(l-u)u_0+l^2}{l^2}.$$ Here, the average is restricted to the forces $f_0$ and $f_1$, the other variables $l$, $l'$ and $u_0$ are fixed. On the other hand, when $u_0+l\leq u\leq u_0+l+l'$, the two points lie on neighbour segments. The same argument gives for the force correlation $$\label{eq_correl_neighbourseg} \overline{f(0)f(u)}=\frac{-u_0^2-(l+l'-u)u_0}{ll'}.$$ Gathering the results (\[eq\_distrib\_seglengths\],\[eq\_correl\_sameseg\],\[eq\_correl\_neighbourseg\]) and integrating over $u_0$ gives for the correlation function: $$\begin{aligned} \label{eq_gen_correl_continuous force} \hspace*{-25mm} \Delta_u(u)=\frac{1}{\bar l}\int_0^\infty {\mathrm{d}}l P(l)\left(\chi_{[0,l]}(u) \frac{u^3-3l^2u+2l^3}{3l^2}+ \vphantom{\left[\frac{-2u_0^3-3(l+l'-u)u_0^2}{6ll'} \right]^{\min(u-l,0)}_{\max(u-l-l',-l)}} \right. \nonumber\\ \hspace*{-2mm} \left. \int_0^\infty {\mathrm{d}}l'P(l')\chi_{[0,l+l']}(u) \left[\frac{-2u_0^3-3(l+l'-u)u_0^2}{6ll'} \right]^{\min(u-l,0)}_{\max(u-l-l',-l)} \right),\end{aligned}$$ where we have used the notation $[g(u_0)]_a^b=g(b)-g(a)$. For the model A, all the segments have the same length $l=1$, corresponding to the probability density $$\label{eq_distrib_A} P(l)=\delta(l-1),$$ For the model B, the segment lengths can take two values, $0.1$ and $1.9$, with probability $1/2$ each: $$\label{eq_distrib_B} P(l)=\frac{1}{2}\delta(l-0.1)+\frac{1}{2}\delta(l-1.9).$$ For the model A, inserting the probability density (\[eq\_distrib\_A\]) in the general formula (\[eq\_gen\_correl\_continuous force\]) gives the correlation function for $u\geq 0$, $$\Delta_u(u)=\chi_{[0,1]}(u)\frac{3u^3-6u^2+4}{6}+\chi_{(1,2]}(u)\frac{(2-u)^3}{6},$$ it is plotted on Fig. \[fig\_correls\]. To compute the critical force (\[eq\_anal\_pred\]), we need the following quantity, $$\begin{aligned} \int |k_u|\tilde\Delta_u(k_u){\mathrm{d}}k_u & =4\int_0^\infty \frac{\Delta_u(0)-\Delta_u(u)}{u^2}{\mathrm{d}}u\nonumber\\ & =8\log(2).\end{aligned}$$ For the model B, the correlation function is more complex and is plotted on Fig. \[fig\_correls\]. The integral entering the expression (\[eq\_anal\_pred\]) of the critical force has to be computed numerically; we get $$\int |k_u|\tilde\Delta_u(k_u){\mathrm{d}}k_u \simeq 6.91.$$ Random bond disorder: model C ----------------------------- A random bond disorder can be generated on a rail by drawing random energies for points on a grid of step $l$. A spline interpolation of these energies then allows to get a smooth landscape of potential. We determine here the two-point correlation function of such a disorder (see [@agoritsas_static_numeric_2013] for a similar study for a two-dimensional spline). Specifically, let us consider a grid of spacing $l$ with $2n+1$ points indexed from $-n$ to $n$. A random value $V_i$ is attached to each site $u_i=il$ of the grid. The function $V(u)$ is a cubic spline of the $(V_i)_{-n\leq i\leq n}$, that is: - $V(u)$ is a cubic polynomial on each lattice segment $[u_i,u_{i+1}]$ for $-n\leq i<n$, - $V(u)$ is continuous on each lattice site $u_i$, and equal to $V_i$: $V(u_i^+)=V(u_i^-)=V_i$, - the first and second derivatives of $V(u)$ are continuous: $V'(u_i^+)=V'(u_i^-)$ and $V''(u_i^+)=V''(u_i^-)$. One defines the coefficients $A_i^0,\ldots,A_i^3$ ($-n\leq i<n$) of the polynomials as $$\label{eq:defAiinterp} V(u)=A_i^0+A_i^1(u-u_i)+\frac{A_i^2}2(u-u_i)^2+\frac{A_i^3}{3!}(u-u_i)^3$$ for $u_i\leq u<u_{i+1}$. One has $A_i^0=V_i$. Denoting $l_i=u_{i+1}-u_i$ (not needed to be constant, we will keep it generic for a while), the continuity conditions write: $$\begin{aligned} A_{i+1}^0&=A_i^0+l_iA_i^1+\frac 12l_i^2A_i^2+\frac 1{3!}l_i^3A_i^3 \:, \label{eq:splineA0} \\ A_{i+1}^1&=A_i^1+l_iA_i^2+\frac 12l_i^2A_i^3 \:, \label{eq:splineA1} \\ A_{i+1}^2&=A_i^2+l_iA_i^3 \:. \label{eq:splineA2}\end{aligned}$$ There are $6n$ unknown variables and $6n-2$ of those bulk equations. They have to be complemented by boundary conditions (*e.g.* fixing the values of the derivatives at extremities, or imposing periodic boundary conditions). The simplest way to solve the set of equations is to eliminate the $A_i^1$’s and the $A_i^3$’s to obtain equations on the $A_i^2$’s only, as a function of the parameters $l_i$ and $A_i^0=V_i$. From  one has $ A_i^3=({A_{i+1}^2-A_i^2})/{l_i} $ and substituting into  one obtains $A_i^1$: $$A_i^1=\frac{A_{i+1}^0-A_i^0}{l_i}-l_i\frac{2A_i^2+A_{i+1}^2}{6}. \label{eq:A2toA1}$$ Using these expressions in  one gets the equations on the $A_i^2$’s: $$\begin{aligned} l_i A_i^2+2(l_{i}+l_{i+1})A_{i+1}^2+&l_{i+1}A_{i+2}^2= \nonumber\\ &6\frac{A_{i+2}^0-A_{i+1}^0}{l_{i+1}}-6\frac{A_{i+1}^0-A_{i}^0}{l_{i}}.\end{aligned}$$ Those are quite complex to solve in general but simplifications occur for an uniform spacing $l_i=l$ and in the infinite grid size limit $n\to\infty$. *Solution for constant $l_i=l$*: The equations write $$A_i^2+4A_{i+1}^2+A_{i+2}^2=\frac 6{h^2}\big(A_{i}^0-2A_{i+1}^0+A_{i+2}^0\big).$$ They take the form $M\vec {A^2}=\frac{6}{h^2}\Delta\vec {A^0} $ where $\Delta$ is the discrete Laplacian and $M$ is a tridiagonal $(2n+1)\times (2n+1)$ matrix. It is best represented as $M=6(\mathbf{1}+\frac{1}{6}\Delta) $ with $$\Delta= \left( \begin{array}{ccccc} -2&1&0&0&\ldots \\ 1&-2&1&0&\ldots \\ 0&1&-2&1& \\ \vdots&&\ddots&\ddots&\ddots \end{array} \right)$$ which allows to invert $M$ by writing: $$M^{-1}=\frac 16 \sum_{p\geq 0} \frac{(-1)^p}{6^p}\Delta^p .$$ Hence, the vector $\vec {A^2}$ of the $A_i^2$’s is obtained as $$\vec {A^2}= \frac 1{l^2}\sum_{p\geq 0}\frac{(-1)^p}{6^p}\Delta^{p+1} \vec {A^0}. \label{eq:A0A2}$$ Each of the $A_i^2$’s is a linear combination of all the fixed potentials $A_i^0=V_i$’s. It is known that the coefficients of $\Delta^p$ are given in the infinite size limit $n\to\infty$ by the binomial coefficients, up to a sign. For instance the diagonal and subdiagonal elements are $$\big(\Delta^p)_{ii}= (-1)^p {\left( {\begin{array}{c} \!\!\!{2p}\!\!\! \\ \!\!\!{p}\!\!\! \end{array}} \right) } \qquad \big(\Delta^p)_{i,i+1}= (-1)^{p+1} {\left( {\begin{array}{c} \!\!\!{2p}\!\!\! \\ \!\!\!{p-1}\!\!\! \end{array}} \right) }\nonumber.$$ One is now ready to determine the correlator of the potential. On a generic interval $il\leq y\leq (i+1)l$ $(i>0)$ one has $$\begin{aligned} V(u+\eta)=&\Big(i+1-\frac ul\Big)A_i^0-\Big(i-\frac ul\Big)A_{i+1}^0+\nonumber\\ &\frac{(u-il)(u-(i+1)l)}{6l}\times\nonumber\\ &\big[((2+i)l-u)A_i^2+(u-(i-1)l)A_{i+1}^2 \big]\, \label{eq:VuAi02}\end{aligned}$$ where $\eta$ is uniformly distributed on $[0,l]$ and allows to implement the statistical invariance by translation of the disorder (and generalizes the result of [@agoritsas_static_numeric_2013]). To determine the correlation function $\overline{V(u)V(u')}$, one thus has to identify the segments to which $u$ and $u'$ belong, and expanding , to determine averages of the form $\overline{A_i^0A_j^2}$. Those are obtained from the large-$n$ limit explicit form of , which reads $$\begin{aligned} A^2_j &= \ldots+(-1)^i\frac 1{l^2}\sum_{p\geq 0}\frac{1}{6^p}{\left( {\begin{array}{c} \!\!\!{2p+2}\!\!\! \\ \!\!\!{p-i+1}\!\!\! \end{array}} \right) } A^0_{j+i} + \ldots\nonumber\\ &= \ldots+(-1)^{i+1}\frac {6\sqrt3}{l^2}\left(2-\sqrt3\right)^{i} A^0_{j+i} + \ldots \label{eq:resA2iA0iplusj}\end{aligned}$$ which yields for instance $\overline{A_0^0A_i^2}= (-1)^{i+1}\frac{6\sqrt3}{l^2}\left(2-\sqrt3\right)^{i}$. One obtains a cumbersome expression in real space, defined piecewise, that we do not reproduce here for clarity. After Fourier transformation, the correlator $\tilde R_u$ is found to take a simple form $$\tilde R_u(k_u)=\frac{9{{\rm \,sinc\!}}\left(\frac{k_u}{2} \right)^8}{(2+\cos(k_u))^2}\:,$$ which we have checked numerically. The force correlation function is shown on Fig. \[fig\_correls\]; unlike the random field correlation functions, it presents negative parts indicating anticorrelations of the disorder (due to the spline continuity constraints). References {#references .unnumbered} ==========

--- author: - | A. Behring, J. Blümlein, , T. Pfoh, C. Raab, M. Round\ Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, D-15738 Zeuthen, Germany\ E-mail: - | J. Ablinger, , C. Schneider, F. Wißbrock\ Research Institute for Symbolic Computation (RISC),\ Johannes Kepler University, Altenbergerstraße 69, A–4040, Linz, Austria\ E-mail: - | A. von Manteuffel\ PRISMA Cluster of Excellence and Institute of Physics, J. Gutenberg University, D-55099 Mainz. Germany. title: | [DESY 13-223,   DO-TH 13/31,   MITP/13-073,   SFB/CPP-13-107,    LPN 13-097]{}\ New Results on the 3-Loop Heavy Flavor Corrections in Deep-Inelastic Scattering[^1] --- Introduction ============ The Wilson coefficients for the heavy quark contributions are known to 2-loop order in semi-analytic form [@Laenen:1992zk; @Bierenbaum:2009zt].[^2] In the asymptotic region of large virtualities $Q^2 \gg m^2$, the Wilson coefficients were calculated analytically in [@Buza:1995ie; @Buza:1996wv; @Bierenbaum:2007dm; @Bierenbaum:2007qe; @Bierenbaum:2008yu]. This approximation holds in case of $F_2(x,Q^2)$ for scales of $Q^2/m^2 {\raisebox{-0.07cm } {$\, \stackrel{>}{{\scriptstyle\sim}}\, $}}10$ at the 1% level [@Buza:1995ie]. In 2009 a series of Mellin moments $N = 2 ... 10 (12,14)$ was calculated for all massive operator matrix elements (OMEs) in [@Bierenbaum:2009mv] mapping these moments to massive tadpoles, which could be calculated using [MATAD]{}, [@Steinhauser:2000ry]. In the asymptotic region also the 3-loop corrections for $F_L(x,Q^2)$ were computed [@Blumlein:2006mh], which are, however, only applicable at much higher scales. Including the case of transversity [@Blumlein:2009rg] there are eight unpolarized massive Wilson coefficients at three loop order to be calculated. In 2010 the Wilson coefficients $L_{qg,Q}^{(3)}$ and $L_{qq,Q}^{(3),\rm PS}$ were computed [@Ablinger:2010ty]. Here the exchanged gauge boson couples to a massless fermion line. Furthermore, all contributions due to the color factors $C_{F,A} T_F^2 N_F$ have been computed in [@Ablinger:2010ty; @Blumlein:2012vq]. 3-loop ladder and $V$-topologies have been studied in detail in [@Ablinger:2012qm; @Ablinger:2012sm]. In all these calculations after performing the Feynman parameter integrals, nested finite and infinite sums over hypergeometric terms, cf. [@Blumlein:2010zv], occur, which have to be solved by applying modern summation technologies.[^3] These are encoded in the packages [[Sigma]{}]{}[ [@Schneider08JSC; @Schneider05AC; @Schneider06JDEA; @Schneider10AC; @Schneider10PW; @Schneider06SL; @Schneider07Hab; @Ablinger:2010pb; @Blumlein:2012hg]]{}, [[EvaluateMultiSums]{}]{}[[@Blumlein:2012hg; @Ablinger:2010ha; @Schneider2013]]{}, [[SumProduction]{}]{}[[@Blumlein:2012hg]]{}, and $\rho$[-Sum]{} [@ROUND]. Algebraic and structural relations between sums of specific types, such as harmonic sums [@HSUM; @Blumlein:2003gb; @Blumlein:2009ta], multiple zeta values [@Blumlein:2009cf], harmonic polylogarithms [@Remiddi:1999ew], generalized harmonic sums and associated polylogarithms [@Moch:2001zr; @Ablinger:2013cf], cyclotomic harmonic sums and polylogarithms [@Ablinger:2011te], as well as binomially weighted finite sums and the associated polylogarithms [@RAAB], are mutually applied in these calculations.[^4] The corresponding relations and algorithms are encoded in the package [HarmonicSums]{} [@Ablinger:2013cf; @Ablinger:2013hcp]. During the last year we have calculated four more OMEs at three loop order, $A_{qq,Q}^{(3), \rm NS},~A_{qq,Q}^{(3), \rm NS,TR},~A_{gq,Q}^{(3)}$ and $A_{Qq}^{(3), \rm PS}$ and the associated massive Wilson coefficients for $A_{qq,Q}^{(3), \rm NS}$ and $A_{Qq}^{(3), \rm PS}$ at large $Q^2$. The corresponding topologies were first reduced to master integrals using integration-by-parts relations [@IBP] using the package [Reduze2]{} [@Studerus:2009ye; @vonManteuffel:2012np]. The master integrals were finally calculated using different summation technologies being described above. Furthermore, we computed the bubble topologies for all OMEs containing one massless bubble. Progress has also been made in the calculation of the topologies with two massive lines of the same mass. In this note we report on these series of results obtained recently. In Section 2 the yet missing results for the 2-bubble topologies, with one massless line beyond the results given in [@Ablinger:2010ty; @Ablinger:2012qm], are presented. In Section 3 we discuss complete results obtained for four new massive OMEs and Wilson coefficients. Results on graphs with two massive quark lines of equal masses are discussed in Section 4. The asymptotic massive two-loop Wilson coefficients for charged current reactions are given in Section 5, and Section 6 contains the conclusions. Results for Bubble Graphs ========================= All 2-bubble topologies have been calculated for all the massive operator matrix elements at general $N$. The contributions $\propto N_F T_F^2$ for $A_{Qq}^{(3) \rm PS}$ and $A_{Qg}^{(3)}$ have been obtained before in [@Ablinger:2010ty] and for $A_{gg}^{(3)}$ in [@Blumlein:2012vq]. Likewise, the terms $\propto T_F^2$ were given in [@Ablinger:2011pb] for $A_{Qq}^{(3) \rm PS}$.[^5] In the following we list the corresponding results for $a_{ij}^{k, {\sf b}}$ in the pure-singlet-, $gg$- and $Qg$-cases. Here harmonic sums $S_{\vec{a}}(N) \equiv S_{\vec{a}}$ up to weight [w = 5]{}, including negative indices contribute. The calculation of the corresponding graphs has been carried out directly using (generalized) hypergeometric function techniques for the whole diagrams to convert them into sum-representations. The latter were solved using modern summation techniques as encoded in the packages [[Sigma]{}]{}[ [@Schneider08JSC; @Schneider05AC; @Schneider06JDEA; @Schneider10AC; @Schneider10PW; @Schneider06SL; @Schneider07Hab; @Ablinger:2010pb; @Blumlein:2012hg]]{}, [[EvaluateMultiSums]{}]{}[[@Blumlein:2012hg; @Ablinger:2010ha; @Schneider2013]]{}, [[SumProduction]{}]{}[[@Blumlein:2012hg]]{}, and $\rho$[-Sum]{} [@ROUND]. For the pure-singlet OME all contributions are given. The constant part of the unrenormalized OME reads : $$\begin{aligned} \label{eq:BUB1} a_{Qq}^{\rm PS,{\sf b}}(N) &=& \textcolor{blue}{C_F^2 T_F} \frac{1}{(N-1) N^2} \Biggl\{ \frac{4 P_{22}}{3 (N-1)^3 N^3 (N+1)^5 (N+2)^4} -\frac{24 \big(N^2+N+2\big)^2}{(N+1)^2 (N+2)} S_3 \nonumber \\ && +\frac{2 P_9}{(N-1) N (N+1)^3 (N+2)^2} \big[\zeta_2 + 4 S_2\big] +\frac{28 \big(N^2+N+2\big)^2 \zeta_3}{3 (N+1)^2 (N+2)} \Biggr\} \nonumber \\ && +\textcolor{blue}{C_F T_F^2} \Biggl\{ \frac{1}{(N-1) N^2} \Biggl[ \frac{32 \big(N^2+N+2\big)^2 }{27 (N+1)^2 (N+2)}S_1^2 -\frac{160 \big(N^2+N+2\big)^2 }{9 (N+1)^2 (N+2)}S_2 \nonumber \\ && +\frac{64 P_{18}}{81 N^2 (N+1)^4 (N+2)^3 (N+3) (N+4) (N+5)} \Biggr] S_1 \nonumber \\ && -\frac{64 P_{21}}{243 (N-1) N^5 (N+1)^4 (N+2)^4 (N+3) (N+4) (N+5)} \nonumber \\ && +\frac{32}{27 (N-1) N^3 (N+1)^2 (N+2)^2 (N+3) (N+4) (N+5)} \left[P_{12} S_2 - \frac{P_{14} S_1^2}{N+1}\right] \nonumber \\ && -\frac{512 \big(N^2+N+2\big)^2 }{27 (N-1) N^2 (N+1)^2 (N+2)}S_3 +\frac{128 \big(N^2+N+2\big)^2}{3 (N-1) N^2 (N+1)^2 (N+2)} S_{2,1} \nonumber \\ && +\Biggl[ \frac{32 \big(N^2+N+2\big)^2 }{3 (N-1) N^2 (N+1)^2 (N+2)}S_1 -\frac{32 P_2}{9 (N-1) N^3 (N+1)^2 (N+2)^2} \Biggr] \zeta_2 \nonumber \\ && -\frac{1024 \big(N^2+N+2\big)^2 \zeta_3}{9 (N-1) N^2 (N+1)^2 (N+2)} \Biggr\} +\textcolor{blue}{C_F T_F^2 n_f} \Biggl\{ -\frac{16 \big(N^2+N+2\big)^2 }{27 (N-1) N^2 (N+1)^2 (N+2)}S_1^3 \nonumber \\ && +\frac{16 P_5 }{27 (N-1) N^3 (N+1)^3 (N+2)^2}S_1^2 +\Biggl[ -\frac{208 \big(N^2+N+2\big)^2}{9 (N-1) N^2 (N+1)^2 (N+2)} S_2 \nonumber \\ && -\frac{32 P_{15}}{81 (N-1) N^4 (N+1)^4 (N+2)^3} \Biggr] S_1 +\frac{32 P_{19}}{243 (N-1) N^5 (N+1)^5 (N+2)^4} \nonumber \\ && +\frac{208 P_5 }{27 (N-1) N^3 (N+1)^3 (N+2)^2}S_2 -\frac{1760 \big(N^2+N+2\big)^2 }{27 (N-1) N^2 (N+1)^2 (N+2)}S_3 \nonumber \\ && +\Biggl[ \frac{16 P_5}{9 (N-1) N^3 (N+1)^3 (N+2)^2} -\frac{16 \big(N^2+N+2\big)^2 }{3 (N-1) N^2 (N+1)^2 (N+2)}S_1 \Biggr] \zeta_2 \nonumber \\ && +\frac{224 \big(N^2+N+2\big)^2 \zeta_3}{9 (N-1) N^2 (N+1)^2 (N+2)} \Biggr\} +\textcolor{blue}{C_A C_F T_F} \Biggl\{ \frac{\big(2 N^3+5 N^2-14 N-24\big) }{72 N^2 (N+1)^2 (N+2)}S_1^4 \nonumber \\ && +\frac{P_6 }{54 (N-1) N^3 (N+1)^3 (N+2)^2}S_1^3 +\Biggl[ \frac{P_{13}}{54 (N-1) N^4 (N+1)^4 (N+2)^3} \nonumber \\ && +\frac{\big(74 N^4+523 N^3+733 N^2+374 N+440\big) }{12 (N-1) N^2 (N+1)^2 (N+2)}S_2 \Biggr] S_1^2 +\Biggl[ \frac{P_{20}}{81 (N-1) N^5 (N+1)^5 (N+2)^4} \nonumber \\ && +\frac{P_7 }{18 (N-1) N^3 (N+1)^3 (N+2)^2}S_2 +\frac{\big(134 N^4+1971 N^3+3857 N^2+2270 N+936\big) }{9 (N-1) N^2 (N+1)^2 (N+2)}S_3 \nonumber \\ && -\frac{2 \big(4 N^4+107 N^3+255 N^2+122 N-32\big) }{3 (N-1) N^2 (N+1)^2 (N+2)}S_{2,1} -\frac{64 \big(N^3+6 N^2+3 N+2\big) }{3 (N-1) N^2 (N+1)^2}S_{-2,1} \Biggr] S_1 \nonumber \\ && +\frac{\big(82 N^4+1211 N^3+3061 N^2+2982 N+2008\big) }{24 (N-1) N^2 (N+1)^2 (N+2)}S_2^2 \nonumber \\ && +\frac{16 \big(N^4+2 N^3+7 N^2+6 N+4\big) }{3 (N-1) N^2 (N+1)^2 (N+2)}S_{-2}^2 +\frac{P_{23}}{243 (N-1)^5 N^5 (N+1)^6 (N+2)^5} \nonumber \\ && +\Biggl[ \frac{8 P_4}{3 (N-1) N^3 (N+1)^3 (N+2)^2} +\frac{32 \big(N^3+6 N^2+3 N+2\big) }{(N-1) N^2 (N+1)^2}S_1 \Biggr] S_{-3} \nonumber \\ && +\frac{P_{17}}{54 (N-1)^3 N^4 (N+1)^4 (N+2)^3} S_2 +\Biggl[ \frac{32 \big(N^3+6 N^2+3 N+2\big) }{3 (N-1) N^2 (N+1)^2}S_1^2 \nonumber \\ && +\frac{32 P_1 }{3 (N-1) N^3 (N+1)^3 (N+2)}S_1 -\frac{4 P_{11}}{3 (N-1) N^4 (N+1)^4 (N+2)^3} \nonumber \\ && +\frac{16 \big(5 N^4+58 N^3+99 N^2+46 N+20\big) }{3 (N-1) N^2 (N+1)^2 (N+2)}S_2 \Biggr] S_{-2} +\frac{P_{10}}{27 (N-1)^2 N^3 (N+1)^3 (N+2)^2} S_3 \nonumber \\ && +\frac{\big(194 N^4+4719 N^3+10489 N^2+6814 N+2136\big) }{12 (N-1) N^2 (N+1)^2 (N+2)}S_4 \nonumber \\ && +\frac{32 \big(3 N^4+36 N^3+61 N^2+28 N+12\big) }{3 (N-1) N^2 (N+1)^2 (N+2)}S_{-4} -\frac{2 P_3 }{3 (N-1) N^3 (N+1)^3 (N+2)^2}S_{2,1} \nonumber \\ && -\frac{2 \big(293 N^3+813 N^2+470 N-80\big) }{3 (N-1) N^2 (N+1)^2 (N+2)}S_{3,1} -\frac{32 P_1 }{3 (N-1) N^3 (N+1)^3 (N+2)}S_{-2,1} \nonumber \\ && -\frac{32\big(N^3+6 N^2+3 N+2\big) }{3 (N-1) N^2 (N+1)^2} \big[2 S_{-2,2} + 3 S_{-3,1}\big] +\frac{64 \big(N^3+6 N^2+3 N+2\big) }{3 (N-1) N^2 (N+1)^2}S_{-2,1,1} \nonumber \\ && -\frac{2 \big(2 N^4-97 N^3-267 N^2-118 N+88\big) }{3 (N-1) N^2 (N+1)^2 (N+2)}S_{2,1,1} +\Biggl[ \frac{\big(2 N^3+5 N^2-14 N-24\big) }{4 N^2 (N+1)^2 (N+2)}S_1^2 \nonumber \\ && +\frac{P_6 }{6 (N-1) N^3 (N+1)^3 (N+2)^2}S_1 +\frac{P_{16}}{18 (N-1)^3 N^3 (N+1)^4 (N+2)^3} \nonumber \\ && +\frac{\big(2 N^4+95 N^3+265 N^2+222 N+88\big) }{4 (N-1) N^2 (N+1)^2 (N+2)}S_2 +\frac{4 \big(N^4+14 N^3+23 N^2+10 N+4\big) }{(N-1) N^2 (N+1)^2 (N+2)}S_{-2} \Biggr] \zeta_2 \nonumber \\ && +\Biggl[ \frac{P_8}{9 (N-1)^2 N^3 (N+1)^3 (N+2)^2} +\frac{\big(34 N^3+109 N^2+98 N-24\big) }{3 N^2 (N+1)^2 (N+2)}S_1 \Biggr] \zeta_3 \Biggr\}~. $$ The polynomials $P_i$ read : $$\begin{aligned} P_1&=&N^6+11 N^5+64 N^4+87 N^3+33 N^2+16 N+4 \\ P_2&=&8 N^6+29 N^5+84 N^4+193 N^3+162 N^2+124 N+24 \\ P_3&=&2 N^7+65 N^6+591 N^5+1904 N^4+2554 N^3+1132 N^2-120 N-48 \\ P_4&=&4 N^7+67 N^6+505 N^5+1277 N^4+1227 N^3+476 N^2+156 N+16 \\ P_5&=&8 N^7+37 N^6+68 N^5-11 N^4-86 N^3-56 N^2-104 N-48 \\ P_6&=&43 N^7+221 N^6+694 N^5+722 N^4-496 N^3-424 N^2+464 N+96 \\ P_7&=&631 N^7+3965 N^6+17170 N^5+33194 N^4+22160 N^3+5912 N^2+5456 N+864 \\ P_8&=&-331 N^8-1540 N^7-3434 N^6-1648 N^5+4089 N^4+1452 N^3-2316 N^2 \nonumber \\ &&+560 N+1152 \\ P_9&=&2 N^8+6 N^7-N^6-51 N^5-59 N^4-19 N^3-26 N^2+28 N+24 \\ P_{10}&=&2365 N^8+13228 N^7+55085 N^6+90910 N^5+4596 N^4-91944 N^3-55632 N^2 \nonumber \\ &&-2960 N-96 \\ P_{11}&=&N^9-215 N^8-2293 N^7-7913 N^6-12020 N^5-8528 N^4-3048 N^3-848 N^2 \nonumber \\ &&+96 N+128 \\ P_{12}&=&40 N^9+625 N^8+3284 N^7+5392 N^6-7014 N^5-33693 N^4-47454 N^3 \nonumber \\ &&-46100 N^2-26280 N+7200 \\ P_{13}&=&-359 N^{10}-2734 N^9-9528 N^8-14379 N^7-11852 N^6-28608 N^5-46716 N^4 \nonumber \\ &&-8528 N^3+22240 N^2+7296 N+576 \\ P_{14}&=&8 N^{10}+133 N^9+1095 N^8+5724 N^7+18410 N^6+34749 N^5+40683 N^4 \nonumber \\ &&+37370 N^3+22748 N^2-3960 N-7200 \\ P_{15}&=&25 N^{10}+176 N^9+417 N^8+30 N^7-20 N^6+1848 N^5+2244 N^4+1648 N^3 \nonumber \\ &&+3040 N^2+2112 N+576 \\ P_{16}&=&-359 N^{11}-2025 N^{10}-4518 N^9+2510 N^8+21229 N^7+14611 N^6-14384 N^5 \nonumber \\ &&-16352 N^4-6592 N^3+152 N^2+6784 N+4128 \\ P_{17}&=&-4739 N^{12}-27252 N^{11}-62919 N^{10}+29003 N^9+277786 N^8+167821 N^7 \nonumber \\ &&-215504 N^6-163112 N^5-31660 N^4-19696 N^3+67744 N^2+46464 N -1728 \\ P_{18}&=&52 N^{13}+746 N^{12}+4658 N^{11}+20431 N^{10}+79990 N^9+251778 N^8+553796 N^7 \nonumber \\ &&+837697 N^6+886552 N^5+599060 N^4+155864 N^3-82368 N^2-76896 N\nonumber\\ && -17280 \\ P_{19}&=&158 N^{13}+1663 N^{12}+7714 N^{11}+23003 N^{10}+56186 N^9+89880 N^8+59452 N^7 \nonumber \\ &&-8896 N^6-12856 N^5-24944 N^4-84608 N^3-77952 N^2-35712 N-6912 \\ P_{20}&=&1474 N^{13}+15137 N^{12}+67586 N^{11}+156550 N^{10}+284233 N^9+530832 N^8 \nonumber \\ &&+412460 N^7-695900 N^6-1291340 N^5-157480 N^4+639968 N^3+318720 N^2 \nonumber \\ &&+72000 N+3456 \\ P_{21}&=&293 N^{15}+4670 N^{14}+32280 N^{13}+145948 N^{12}+559575 N^{11}+1871440 N^{10} \nonumber \\ &&+4877344 N^9+9333994 N^8+12958212 N^7+12693884 N^6+8472792 N^5 \nonumber \\ &&+4514336 N^4+3109248 N^3+2192832 N^2+1026432 N+207360 \\ P_{22}&=&4 N^{16}+30 N^{15}+101 N^{14}+301 N^{13}+561 N^{12}-1385 N^{11}-4474 N^{10}+324 N^9 \nonumber \\ &&+4667 N^8-4115 N^7-2529 N^6+6629 N^5-330 N^4-3672 N^3-1024 N^2 \nonumber \\ &&+976 N+480 \\ P_{23}&=&-8780 N^{19}-83054 N^{18}-302761 N^{17}-396603 N^{16}+104969 N^{15}+2043037 N^{14} \nonumber \\ &&+5908471 N^{13}+1725207 N^{12}-16095317 N^{11}-11836443 N^{10}+21978990 N^9 \nonumber \\ &&+16243568 N^8-18166796 N^7-7483912 N^6+11581992 N^5+1162152 N^4 \nonumber \\ &&-5841152 N^3-833088 N^2+1415808 N+563328~.\end{aligned}$$ The new contributions to the finite part of the OME $A_{gg}^{(3)}$ read : $$\begin{aligned} a_{gg}^{\sf b}(N) &=& \textcolor{blue}{T_F C_A^2} \Biggl\{ -\frac{13}{36} S_1^5 +\frac{R_{14}}{864 (N-1) N (N+1) (N+2)} S_1^4 +\Biggl[ \frac{R_{20}}{1944 (N-1)^2 N^2 (N+1)^2 (N+2)^2} \nonumber \\ && -\frac{235 }{54}S_2 \Biggr] S_1^3 +\Biggl[ \frac{R_{29}}{648 (N-1)^3 N^3 (N+1)^3 (N+2)^3} +\frac{R_{15}}{432 (N-1) N (N+1) (N+2)} S_2 \nonumber \\ && -5 S_3+\frac{4}{9} S_{2,1} +\frac{32}{9} S_{-2,1} \Biggr] S_1^2 +\Biggl[ -\frac{49}{12} S_2^2 +\frac{R_{21}}{216 (N-1)^2 N^2 (N+1)^2 (N+2)^2} S_2 \nonumber \\ && +\frac{R_{30}}{972 (N-1)^4 N^4 (N+1)^4 (N+2)^4} +\frac{R_{13}}{108 (N-1) N (N+1) (N+2)} S_3 -\frac{179 }{18}S_4 \nonumber \\ && +\frac{R_6 }{27 (N-1) N (N+1) (N+2)}S_{2,1} -\frac{32}{9} S_{3,1} +\frac{64 (5 N+22)}{27 (N+2)} S_{-2,1} +\frac{32}{9} S_{-2,2} -\frac{8}{9} S_{2,1,1} \nonumber \\ && -\frac{128}{9} S_{-2,1,1} \Biggr] S_1 +\frac{R_{16}}{864 (N-1) N (N+1) (N+2)} S_2^2 \nonumber \\ && +\frac{R_{32}}{46656 (N-1)^5 N^5 (N+1)^5 (N+2)^5} +\frac{R_{22}}{972 (N-1)^2 N^2 (N+1)^2 (N+2)^2} S_3 \nonumber \\ && +\frac{R_{12}}{432 (N-1) N (N+1) (N+2)} S_4 +6 S_5 +\frac{448}{81} S_{-3} -\frac{80}{27} S_{-4} +\frac{16}{9} S_{-5} \nonumber \\ && +\frac{R_{17}}{81 (N-1) N (N+1)^2 (N+2)^2} S_{2,1} + \Biggl[ -\frac{16}{27} S_1^3 -\frac{64 (N+3) (41 N+56)}{81 (N+1) (N+2)} \nonumber \\ && -\frac{16 (5 N+22) }{27 (N+2)}S_1^2 +\Biggl(\frac{16 }{9}S_2-\frac{128 \big(7 N^2+36 N+38\big)}{81 (N+1) (N+2)}\Biggr) S_1 +\frac{16 (5 N+22) }{27 (N+2)}S_2 -\frac{32 }{27}S_3 \nonumber \\ && +\frac{64}{9} S_{2,1} \Biggr] S_{-2} +\frac{20}{9} S_{2,3} +\frac{64}{9} S_{2,-3} +\frac{R_3 }{27 (N-1) N (N+1) (N+2)}S_{3,1} -\frac{34}{9} S_{4,1} \nonumber \\ && + \Biggl[ \frac{R_{28}}{648 (N-1)^3 N^3 (N+1)^3 (N+2)^3} -\frac{349 }{27}S_3 -\frac{8}{9} S_{2,1} -\frac{32}{3} S_{-2,1} \Biggl] S_2 \nonumber \\ && +\frac{256 \big(7 N^2+36 N+38\big)}{81 (N+1) (N+2)} S_{-2,1} +\frac{32 (5 N+22)}{27 (N+2)} S_{-2,2} +\frac{R_1 }{27 (N-1) N (N+1) (N+2)}S_{2,1,1} \nonumber \\ && -\frac{64}{9} S_{-2,3} -\frac{64}{9} S_{2,1,-2} -\frac{4}{9} S_{2,2,1} +\frac{52}{9} S_{3,1,1} -\frac{128 (5 N+22)}{27 (N+2)} S_{-2,1,1} -\frac{64}{9} S_{-2,2,1} \nonumber \\ && +\frac{64}{9} S_{2,1,1,1} +\frac{256}{9} S_{-2,1,1,1} +\Biggl[ -\frac{13}{6} S_1^3 +\frac{R_{11} S_1^2}{48 (N-1) N (N+1) (N+2)} \nonumber \\ && +\Biggl(\frac{R_{18}}{72 (N-1) N^2 (N+1)^2 (N+2)^2}-\frac{55 }{6}S_2\Biggr) S_1 +\frac{R_{26}}{864 (N-1)^3 N^3 (N+1)^3 (N+2)^3} \nonumber \\ && + \Biggl(-\frac{2 (5 N+18)}{3 (N+2)} -\frac{4 }{3}S_1\Biggr) S_{-2} +\frac{R_9 }{48 (N-1) N (N+1) (N+2)}S_2 +\frac{8 }{3}S_3+\frac{2}{3} S_{-3} \nonumber \\ && +\frac{5}{3} S_{2,1} +\frac{8}{3} S_{-2,1} \Biggr] \zeta_2 +\Biggl[ -\frac{49}{3} S_1^2 +\frac{7 R_{10} S_1}{36 (N-1) N (N+1) (N+2)} +7 S_2 +\frac{14}{3} S_{-2} \nonumber \\ && -\frac{7 R_{25}}{216 (N-1)^2 N^2 (N+1)^2 (N+2)^2} \Biggr] \zeta_3 \Biggr\}~.\end{aligned}$$ The polynomials $R_i$ are $$\begin{aligned} R_1&=&-353 N^4-832 N^3+443 N^2+934 N+96 \\ R_3&=&7 N^4-280 N^3-277 N^2-26 N-288 \\ R_6&=&121 N^4+368 N^3-211 N^2-470 N-96 \\ R_9&=&343 N^4+1036 N^3+2033 N^2+2908 N+3316 \\ R_{10}&=&351 N^4+824 N^3+385 N^2+456 N+1068 \\ R_{11}&=&815 N^4+1932 N^3+537 N^2+60 N+1684 \\ R_{12}&=&1957 N^4+14588 N^3+38867 N^2+50236 N+55020 \\ R_{13}&=&3259 N^4+9004 N^3+4069 N^2+980 N+8596 \\ R_{14}&=&3311 N^4+8148 N^3+4041 N^2+3204 N+9508 \\ R_{15}&=&6869 N^4+16876 N^3+18403 N^2+22892 N+33420 \\ R_{16}&=&12653 N^4+32092 N^3+4987 N^2-8404 N+20268 \\ R_{17}&=&1270 N^6+8222 N^5+16333 N^4+7130 N^3-7481 N^2-2050 N+2928 \\ R_{18}&=&-1571 N^7-9661 N^6-26791 N^5-49153 N^4-67528 N^3-55096 N^2 -11384 N \nonumber\\ && +8064 \\ R_{20}&=&-40553 N^8-185774 N^7-259150 N^6-122366 N^5-173461 N^4-129392 N^3 \nonumber\\ &&+366064 N^2+293208 N-59616 \\ R_{21}&=&-17939 N^8-83762 N^7-104386 N^6+5070 N^5-24959 N^4-144464 N^3 \nonumber\\ &&+25472 N^2+106408 N-3360 \\ R_{22}&=&-3185 N^8-62618 N^7-314842 N^6-626834 N^5-451705 N^4+424288 N^3 \nonumber\\ &&+1063528 N^2+416328 N-133920 \\ R_{25}&=&763 N^8+4224 N^7+10030 N^6+18476 N^5+22927 N^4+8348 N^3-9888 N^2 \nonumber\\ &&-8944 N+5904 \\ R_{26}&=&2977 N^{12}+27302 N^{11}+121749 N^{10}+400754 N^9+654511 N^8-19518 N^7-1184809 N^6 \nonumber\\ &&-1028282 N^5-42444 N^4-316832 N^3-1364944 N^2-481248 N+120384 \\ R_{28}&=&23572 N^{12}+191008 N^{11}+720025 N^{10}+1584161 N^9+1802946 N^8-206032 N^7 \nonumber\\ &&-3384669 N^6-3005177 N^5+1544742 N^4+2995680 N^3+265168 N^2 \nonumber\\ &&-238944 N+195840 \\ R_{29}&=&96724 N^{12}+618064 N^{11}+1208881 N^{10}+170137 N^9-1519638 N^8-213008 N^7 \nonumber\\ &&+897211 N^6-1994577 N^5-2283122 N^4+584016 N^3+268880 N^2-319584 N \nonumber\\ &&-2304 \\ R_{30}&=&-290424 N^{16}-2538872 N^{15}-8370109 N^{14}-11564871 N^{13}-1339582 N^{12} \nonumber\\ &&+15569491 N^{11}+25755378 N^{10}+32901569 N^9+17718754 N^8-29577639 N^7 \nonumber\\ &&-33515273 N^6+12365378 N^5+17499480 N^4+73824 N^3-625344 N^2+266688 N \nonumber\\ &&-736128 \\ R_{32}&=&3752873 N^{20}+44889498 N^{19}+231324635 N^{18}+699986798 N^{17}+1323895202 N^{16} \nonumber\\ &&+1047978356 N^{15}-2040696426 N^{14}-6981260596 N^{13}-6762635091 N^{12} \nonumber\\ &&+2129282098 N^{11}+9092892879 N^{10}+5284986214 N^9-791167784 N^8-3464256800 N^7 \nonumber\\ &&-6299611472 N^6-5601882048 N^5-1407467456 N^4+248136192 N^3-362165760 N^2 \nonumber\\ &&-161782272 N+60134400~.\end{aligned}$$ Likewise, the new contributions to $a_{Qg}$ beyond the results given in [@Ablinger:2010ty], are given by : $$\begin{aligned} a_{Qg}^{\sf b}(N) &=& \textcolor{blue}{T_F C_A^2} \Biggl\{ \frac{(8-N) }{12 N (N+1) (N+2)}S_1^5 +\frac{Q_9 }{108 (N-1) N (N+1)^2 (N+2)^2}S_1^4 \nonumber \\ && +\Biggl[ \frac{Q_{20}}{54 (N-1) N^2 (N+1)^3 (N+2)^3} +\frac{(88-39 N) }{18 N (N+1) (N+2)}S_2 \Biggr] S_1^3 \nonumber \\ && +\Biggl[ \frac{Q_{28}}{162 (N-1) N^3 (N+1)^4 (N+2)^4} +\frac{Q_{11}}{18 (N-1) N (N+1)^2 (N+2)^2} S_2 \nonumber \\ && +\frac{(8-5 N) }{3 N (N+1) (N+2)}S_3 -\frac{2 (N-72) }{3 N (N+1) (N+2)}S_{2,1} +\frac{32 }{(N+1) (N+2)}S_{-2,1} \Biggr] S_1^2 \nonumber \\ && +\Biggl[ \frac{(184-119 N) }{12 N (N+1) (N+2)}S_2^2 +\frac{Q_{19}}{54 (N-1) N^2 (N+1)^3 (N+2)^3} S_2 \nonumber \\ && +\frac{Q_{32}}{81 (N-1) N^4 (N+1)^5 (N+2)^5} +\frac{2 Q_5 }{27 (N-1) N (N+1)^2 (N+2)^2}S_3 \nonumber \\ && +\frac{(53 N-8) }{6 N (N+1) (N+2)}S_4 +\frac{4 \big(95 N^3-787 N^2-2504 N-1310\big) }{9 N (N+1)^2 (N+2)^2}S_{2,1} \nonumber \\ && +\frac{4 (N+48) }{3 N (N+1) (N+2)}S_{3,1} -\frac{32 \big(3 N^2+N-6\big) }{(N+1)^2 (N+2)^2}S_{-2,1} +\frac{160 }{3 (N+1) (N+2)}S_{-2,2} \nonumber \\ && +\frac{64 }{(N+1) (N+2)}S_{-3,1} -\frac{4 (5 N+48) }{3 N (N+1) (N+2)}S_{2,1,1} -\frac{256 }{3 (N+1) (N+2)}S_{-2,1,1} \Biggr] S_1 \nonumber \\ && +\frac{Q_{10}}{36 (N-1) N (N+1)^2 (N+2)^2} S_2^2 -\frac{16 \big(3 N^2-23 N-20\big) }{3 (N-1) N (N+1)^2 (N+2)}S_{-2}^2 \nonumber \\ && +\frac{Q_{34}}{243 (N-1)^4 N^5 (N+1)^5 (N+2)^6} +\Biggl[ \frac{8 Q_7}{3 (N-1) N (N+1)^2 (N+2)^2} \nonumber \\ && -\frac{176 }{3 (N+1) (N+2)}S_1 \Biggr] S_{-4} +\Biggl[ -\frac{16 }{(N+1) (N+2)}S_1^2 +\frac{16 \big(3 N^2+N-6\big) }{(N+1)^2 (N+2)^2}S_1 \nonumber \\ && -\frac{8 Q_{17}}{3 (N-1) N (N+1)^3 (N+2)^3} -\frac{32 }{(N+1) (N+2)}S_2 \Biggr] S_{-3} \nonumber \\ && +\frac{Q_{23}}{27 (N-1) N^2 (N+1)^3 (N+2)^3} S_3 +\frac{Q_1 }{18 (N-1) N (N+1)^2 (N+2)^2}S_4 \nonumber \\ && +\frac{16 (11 N-16) }{3 N (N+1) (N+2)}S_5 -\frac{296 }{3 (N+1) (N+2)}S_{-5} -\frac{2 Q_{15}}{27 N (N+1)^3 (N+2)^3} S_{2,1} \nonumber \\ && +\Biggl[ -\frac{32 }{9 (N+1) (N+2)}S_1^3 +\frac{16 \big(3 N^2+N-6\big) }{3 (N+1)^2 (N+2)^2}S_1^2 +\Biggl[ -\frac{16 Q_3}{3 (N+1)^3 (N+2)^3} \nonumber \\ && -\frac{32 }{(N+1) (N+2)}S_2 \Biggr] S_1 +\frac{8 Q_{24}}{3 (N-1) N (N+1)^4 (N+2)^4} +\frac{16 Q_4 }{(N-1) N (N+1)^2 (N+2)^2}S_2 \nonumber \\ && -\frac{448 }{9 (N+1) (N+2)}S_3 +\frac{224 }{3 (N+1) (N+2)}S_{2,1} \Biggr] S_{-2} +\frac{4 (N-8) }{3 N (N+1) (N+2)}S_{2,3} \nonumber \\ && +\frac{176 }{3 (N+1) (N+2)}S_{2,-3} +\frac{4 \big(87 N^3-157 N^2-844 N-498\big) }{3 N (N+1)^2 (N+2)^2}S_{3,1} -\frac{56 }{3 (N+1) (N+2)}S_{4,1} \nonumber \\ && +\frac{16 Q_3 }{(N+1)^3 (N+2)^3}S_{-2,1} +\Biggl[ \frac{Q_{31}}{162 (N-1)^2 N^3 (N+1)^4 (N+2)^4} +\frac{(-39 N-584) }{9 N (N+1) (N+2)}S_3 \nonumber \\ && +\frac{2 (9 N-8) }{3 N (N+1) (N+2)}S_{2,1} +\frac{64 }{3 (N+1) (N+2)}S_{-2,1} \Biggr] S_2 -\frac{80 \big(3 N^2+N-6\big) }{3 (N+1)^2 (N+2)^2}S_{-2,2} \nonumber \\ && +\frac{48 }{(N+1) (N+2)}S_{-2,3} -\frac{32 \big(3 N^2+N-6\big) }{(N+1)^2 (N+2)^2}S_{-3,1} +\frac{400 }{3 (N+1) (N+2)}S_{-4,1} \nonumber \\ && -\frac{8 \big(10 N^3-344 N^2-991 N-523\big) }{9 N (N+1)^2 (N+2)^2}S_{2,1,1} -\frac{224 }{3 (N+1) (N+2)}S_{2,1,-2} \nonumber \\ && -\frac{4 (17 N-72) }{3 N (N+1) (N+2)}S_{2,2,1} -\frac{4 (23 N+8) }{3 N (N+1) (N+2)}S_{3,1,1} +\frac{128 \big(3 N^2+N-6\big) }{3 (N+1)^2 (N+2)^2}S_{-2,1,1} \nonumber \\ && -\frac{64 }{(N+1) (N+2)}S_{-2,2,1} -\frac{80 }{(N+1) (N+2)}S_{-3,1,1} +\frac{8 (3 N+4) }{3 N (N+1) (N+2)}S_{2,1,1,1} \nonumber \\ && +\frac{96 }{(N+1) (N+2)}S_{-2,1,1,1} +\Biggl[ \frac{(8-N) }{2 N (N+1) (N+2)}S_1^3 +\frac{Q_8 }{6 (N-1) N (N+1)^2 (N+2)^2}S_1^2 \nonumber \\ && +\Biggl[ \frac{Q_{21}}{6 (N-1) N^2 (N+1)^3 (N+2)^3} +\frac{(-3 N-8) }{2 N (N+1) (N+2)}S_2 \Biggr] S_1 \nonumber \\ && +\frac{Q_{29}}{18 (N-1)^2 N^3 (N+1)^3 (N+2)^4} +\Biggl[ \frac{4 Q_4}{(N-1) N (N+1)^2 (N+2)^2} \nonumber \\ && -\frac{8 }{(N+1) (N+2)}S_1 \Biggr] S_{-2} +\frac{Q_2 }{6 (N-1) N (N+1)^2 (N+2)^2}S_2 +\frac{2 (3 N-8) }{N (N+1) (N+2)}S_3 \nonumber \\ && -\frac{12 }{(N+1) (N+2)}S_{-3} +\frac{16 }{N (N+1) (N+2)}S_{2,1} +\frac{24 }{(N+1) (N+2)}S_{-2,1} \Biggr] \zeta_2 \nonumber \\ && +\Biggl[ \frac{7 (N-8) }{3 N (N+1) (N+2)}S_1^2 -\frac{14 Q_6 }{3 (N-1) N (N+1)^2 (N+2)^2}S_1 \nonumber \\ && +\frac{7 Q_{22}}{9 (N-1) N^2 (N+1)^2 (N+2)^3} +\frac{7 (7 N+8) }{3 N (N+1) (N+2)}S_2 +\frac{56 }{3 (N+1) (N+2)}S_{-2} \Biggr] \zeta_3 \Biggr\}~, \nonumber\\ \end{aligned}$$ with the polynomials $Q_i$ given by $$\begin{aligned} Q_1&=&-3327 N^4-5641 N^3-5102 N^2-13268 N-7582 \\ Q_2&=&-51 N^4-361 N^3-434 N^2+196 N+290 \\ Q_3&=&3 N^4-6 N^3-21 N^2+24 N+52 \\ Q_4&=&3 N^4+N^3-24 N^2-60 N-40 \\ Q_5&=&5 N^4-427 N^3-4524 N^2-9344 N-5510 \\ Q_6&=&6 N^4-19 N^3-73 N^2-28 N-6 \\ Q_7&=&33 N^4+2 N^3-213 N^2-462 N-320 \\ Q_8&=&47 N^4-313 N^3-708 N^2+244 N+370 \\ Q_9&=&221 N^4-1849 N^3-3642 N^2+2212 N+2698 \\ Q_{10}&=&269 N^4-7345 N^3-13506 N^2+5188 N+6394 \\ Q_{11}&=&357 N^4-1381 N^3-4142 N^2-356 N+842 \\ Q_{15}&=&1504 N^5-8063 N^4-60746 N^3-111983 N^2-79376 N-21632 \\ Q_{17}&=&9 N^6-N^5-19 N^4+87 N^3+488 N^2+940 N+608 \\ Q_{19}&=&-1430 N^7+29061 N^6+168141 N^5+311889 N^4+262827 N^3+154488 N^2 \nonumber\\ &&+113360 N+62400 \\ Q_{20}&=&-778 N^7+8151 N^6+39567 N^5+40819 N^4-14631 N^3-34136 N^2-12368 N \nonumber\\ &&+1600 \\ Q_{21}&=&-122 N^7+1331 N^6+7459 N^5+10911 N^4+4621 N^3+648 N^2+1776 N+1600 \\ Q_{22}&=&20 N^7+18 N^6-333 N^5-1450 N^4-3273 N^3-4570 N^2-3692 N-1480 \\ Q_{23}&=&-1080 N^8+1898 N^7+22443 N^6+92307 N^5+267403 N^4+421473 N^3 \nonumber\\ &&+361900 N^2+222376 N+81520 \\ Q_{24}&=&3 N^8+21 N^7+174 N^6+700 N^5+1506 N^4+1216 N^3-1676 N^2-4024 N-2144 \\ Q_{28}&=&2080 N^{10}-155867 N^9-992144 N^8-2266725 N^7-2100345 N^6+59166 N^5 \nonumber\\ &&+1939307 N^4+1900784 N^3+806384 N^2+252096 N+57600 \\ Q_{29}&=&184 N^{11}+176 N^{10}-4108 N^9-16762 N^8-25657 N^7-15063 N^6-883 N^5 \nonumber\\ &&+14485 N^4+37996 N^3+13360 N^2-32048 N-17040 \\ Q_{31}&=&6624 N^{12}+5292 N^{11}-210971 N^{10}-1104257 N^9-2480453 N^8-2265264 N^7 \nonumber\\ &&+636087 N^6+2871225 N^5+2408885 N^4+828944 N^3-1329328 N^2 \nonumber\\ &&-1961664 N-671040 \\ Q_{32}&=&15604 N^{13}+361847 N^{12}+2453891 N^{11}+8204366 N^{10}+15666936 N^9 \nonumber\\ &&+16766294 N^8+5755934 N^7-9519761 N^6-13953239 N^5-6072896 N^4 \nonumber\\ &&+1787904 N^3+3241984 N^2+1353984 N+230400 \\ Q_{34}&=&6208 N^{19}-86928 N^{18}-1344972 N^{17}-6002889 N^{16}-9808011 N^{15}+4340125 N^{14} \nonumber\\ &&+32811393 N^{13}+24313093 N^{12}-30513058 N^{11}-46961276 N^{10}+3785621 N^9 \nonumber\\ &&+36663986 N^8+1686347 N^7-40115539 N^6-14945624 N^5+25303412 N^4 \nonumber\\ &&+16493728 N^3-1302672 N^2-6643584 N-2376000~.\end{aligned}$$ These quantities will be used in the later calculation of the full massive OMEs. Complete Wilson Coefficients ============================ After the first two massive OMEs, $L_{qg,Q}^{(3)}$ and $L_{qq,Q}^{(3),\rm PS}$, at 3-loop order were calculated in [@Ablinger:2010ty], during the last months we computed four other OMEs and associated massive Wilson coefficients in the asymptotic region $Q^2 \gg m^2$. These are the non-singlet OME $A_{qq,Q}^{(3), \rm NS}$, that of transversity $A_{qq,Q}^{(3), \rm NS, TR}$, $A_{gq,Q}^{(3)}$, and very recently also the pure singlet $A_{gq,Q}^{(3),\sf PS}$. These matrix elements contain topologies up to Benz-graphs with respective local operator insertions. We used [Reduze2]{} [@Studerus:2009ye; @vonManteuffel:2012np] to reduce the diagrams to master integrals applying the integration-by-parts relations for Feynman diagrams containing local operator insertions. In the first three cases the master integrals could be calculated using hypergeometric function techniques and Mellin-Barnes [@Czakon:2005rk; @Smirnov:2009up] representations to map the integrals into nested finite and infinite sums, which were then solved using the summation technologies of . For $A_{qq,Q}^{(3), \rm NS}, A_{qq,Q}^{(3), \rm NS, TR}$ and $A_{gq,Q}^{(3)}$ the results can be represented using harmonic sums only. As an example we show the constant part of the unrenormalized OME for transversity $a_{qq}^{\rm NS,TR(3)}$ for even and odd values of $N$ : $$\begin{aligned} a_{qq}^{\rm NS,TR(3)}(N) &=& \textcolor{blue}{C_F^2 T_F} \Biggl\{ \frac{128}{27} S_2 S_1^3 +\Biggl[\frac{64}{3} S_3 -\frac{128}{9} S_{2,1} -\frac{256}{9} S_{-2,1} -\frac{16}{9 N} -\frac{32 (-1)^N}{9 N (N+1)} \Biggr] S_1^2 \nonumber\\ && +\Biggl[ -\frac{64}{9} S_2^2 +\frac{7168 S_2}{81} +\frac{32 (-1)^N (13 N+7)}{27 N (N+1)^2} -\frac{2560 S_3}{27} +\frac{704 S_4}{9} -\frac{320}{9} S_{3,1} \nonumber\\ && -\frac{2560}{27} S_{-2,1} -\frac{256}{9} S_{-2,2} +\frac{64}{3} S_{2,1,1} +\frac{1024}{9} S_{-2,1,1} \nonumber\\ && +\frac{8 \big(769 N^4+1547 N^3+787 N^2-15 N-12\big)}{27 N^2 (N+1)^2} \Biggr] S_1 -\frac{496}{27} S_2^2 \nonumber\\ && -\frac{16 (-1)^N \big(133 N^4+188 N^3+46 N^2-45 N-18\big)}{81 N^3 (N+1)^3} \nonumber\\ && -\frac{2 \big(6327 N^6+18981 N^5+18457 N^4+5687 N^3-260 N^2+144 N+144\big)}{81 N^3 (N+1)^3} \nonumber\\ && + \Biggl[ 16-\frac{64}{3} S_1 \Biggr] B_4 + \Biggl[ \frac{256}{9} S_1 -\frac{1280}{27}\Biggr] S_{-4} +\Biggl[96 S_1-72\Biggr] \zeta_4 +\Biggl[ \frac{128}{9} S_1^2-\frac{1280}{27} S_1 \nonumber\\ && +\frac{128}{9} S_2 +\frac{7168}{81} \Biggr] S_{-3} +\frac{10408}{81} S_3 -\frac{2992}{27} S_4 +\frac{512}{9} S_5 +\frac{256}{9} S_{-5} + \Biggl[ \frac{256}{27} S_1^3 \nonumber\\ && +\frac{14336}{81} S_1 -\frac{1280}{27} S_2 +\frac{512}{27} S_3 -\frac{512}{9} S_{2,1}-\frac{64}{9 N (N+1)} \Biggr] S_{-2} +\frac{112}{9} S_{2,1} +\frac{256}{9} S_{2,3} \nonumber\\ && -\frac{512}{9} S_{2,-3} +\frac{1424}{27} S_{3,1} -\frac{512}{9} S_{4,1} -\frac{14336}{81} S_{-2,1} +\Biggl[ -\frac{16 \big(169 N^2+169 N+6\big)}{27 N (N+1)} \nonumber\\ && +\frac{256 S_3}{27} +\frac{256}{3} S_{-2,1} -\frac{32 (-1)^N}{9 N (N+1)}\Biggr] S_2 -\frac{1280}{27} S_{-2,2} +\frac{512}{9} S_{-2,3} -16 S_{2,1,1} +\frac{512}{9} S_{2,1,-2} \nonumber\\ && +\frac{256}{9} S_{3,1,1} +\frac{5120}{27} S_{-2,1,1} +\frac{512}{9} S_{-2,2,1} -\frac{2048}{9} S_{-2,1,1,1} \nonumber\\ && +\Biggl[-\frac{2 \big(45 N^2+45 N-4\big)}{3 N (N+1)} +\frac{64}{3} S_{-2} S_1 -8 S_2 + \Biggl[\frac{32}{3} S_2+40\Biggr] S_1 +\frac{32}{3} S_3 +\frac{32}{3} S_{-3} \nonumber\\ && -\frac{64}{3} S_{-2,1} -\frac{8 (-1)^N}{3 N (N+1)}\Biggr]\zeta_2 +\Biggl[-\frac{1208}{9} S_1 -\frac{64}{3} S_2 +\frac{350}{3}\Biggr] \zeta_3 \Biggr\} \nonumber\\ && + \textcolor{blue}{C_F T_F^2} \Biggl\{ \frac{8 \big(157 N^4+314 N^3+277 N^2-24 N-72\big)}{243 N^2 (N+1)^2} -\frac{19424}{729} S_1 +\frac{1856}{81} S_2 -\frac{640}{81} S_3 \nonumber\\ && +\frac{128}{27} S_4 + \textcolor{blue}{N_F} \Biggl[ \frac{32 \big(308 N^4+616 N^3+323 N^2-3 N-9\big)}{243 N^2 (N+1)^2} -\frac{55552}{729} S_1 +\frac{640}{27} S_2 \nonumber\\ && -\frac{320}{81} S_3 +\frac{64}{27} S_4 \Biggr] +\Biggl[-\frac{320}{27} S_1 +\frac{64}{9} S_2 + \textcolor{blue}{N_F} \Biggl[ -\frac{160}{27} S_1 +\frac{32}{9} S_2 +\frac{16}{9}\Biggr] +\frac{32}{9} \Biggr] \zeta_2 \nonumber\\ && +\Biggl[ -\frac{1024}{27} S_1 + \textcolor{blue}{N_F} \Biggl[ \frac{448}{27} S_1 -\frac{112}{9}\Biggr] +\frac{256}{9}\Biggr] \zeta_3 \Biggr\} \nonumber\\ && + \textcolor{blue}{C_A C_F T_F} \Biggl\{ -\frac{64}{27} S_2 S_1^3 +\Biggl[ \frac{4 (3 N+2)}{9 N (N+1)} -\frac{80}{9} S_3 +\frac{128}{9} S_{2,1} +\frac{128}{9} S_{-2,1} +\frac{16 (-1)^N}{9 N (N+1)} \Biggr] S_1^2 \nonumber\\ && +\Biggl[ \frac{112}{9} S_2^2 -\frac{16 (N-2) (2 N+3)}{9 (N+1) (N+2)} S_2 -\frac{16 (-1)^N (13 N+7)}{27 N (N+1)^2} \nonumber\\ && +\frac{4 \big(6197 N^3+18591 N^2+15850 N+4320\big)}{729 N (N+1) (N+2)} +\frac{320}{9} S_3 -\frac{208}{9} S_4 -8 S_{2,1} +\frac{64}{3} S_{3,1} \nonumber\\ && +\frac{1280}{27} S_{-2,1} +\frac{128}{9} S_{-2,2} -32 S_{2,1,1} -\frac{512}{9} S_{-2,1,1} \Biggr] S_1 -\frac{20}{3} S_2^2 \nonumber\\ && +\frac{8 (-1)^N \big(133 N^4+188 N^3+46 N^2-45 N-18\big)}{81 N^3 (N+1)^3} \nonumber\\ && +\frac{-1013 N^6-3039 N^5-5751 N^4-2981 N^3+1752 N^2+1872 N+432}{243 N^3 (N+1)^3} \nonumber\\ && + \Biggl[72-96 S_1\Biggr] \zeta_4 + \Biggl[\frac{640}{27} - \frac{128}{9} S_1\Biggr] S_{-4} + \Biggl[ \frac{32}{3} S_1-8\Biggr] B_4 + \Biggl[ -\frac{64}{9} S_1^2 +\frac{640}{27} S_1 \nonumber\\ && -\frac{64}{9} S_2 -\frac{3584}{81}\Biggr] S_{-3} -\frac{8 \big(27 N^3+560 N^2+1365 N+778\big)}{81 (N+1) (N+2)} S_3 +\frac{1244}{27} S_4 -\frac{224}{9} S_5 \nonumber\\ && -\frac{128}{9} S_{-5} -\frac{32 \big(3 N^3+7 N^2+7 N+6\big)}{9 (N+1) (N+2)} S_{2,1} +\Biggl[-\frac{128}{27} S_1^3-\frac{7168}{81} S_1 \nonumber\\ && +\frac{640}{27} S_2 -\frac{256}{27} S_3 +\frac{256}{9} S_{2,1} +\frac{32}{9 N (N+1)}\Biggr] S_{-2} -\frac{128}{3} S_{2,3} +\frac{256}{9} S_{2,-3} -\frac{1352}{27} S_{3,1} \nonumber\\ && +\frac{256}{9} S_{4,1} + \Biggl[-\frac{4 \big(364 N^3+1227 N^2+872 N+36\big)}{81 N (N+1) (N+2)} +\frac{496}{27} S_3 -\frac{64}{3} S_{2,1} \nonumber\\ && -\frac{128}{3} S_{-2,1} +\frac{16 (-1)^N}{9 N (N+1)}\Biggr] S_2 +\frac{7168}{81} S_{-2,1} +\frac{640}{27} S_{-2,2} -\frac{256}{9} S_{-2,3} +24 S_{2,1,1} \nonumber\\ && -\frac{256}{9} S_{2,1,-2} +\frac{64}{3} S_{2,2,1} -\frac{256}{9} S_{3,1,1} -\frac{2560}{27} S_{-2,1,1} -\frac{256}{9} S_{-2,2,1} +\frac{224}{9} S_{2,1,1,1} \nonumber\\ && +\frac{1024}{9} S_{-2,1,1,1} +\Biggl[ \frac{2 \big(35 N^2+35 N-6\big)}{9 N (N+1)} -\frac{32}{3} S_{-2} S_1 -\frac{16}{27} S_1 -\frac{88}{9} S_2 -\frac{16}{3} S_3 \nonumber\\ && -\frac{16}{3} S_{-3} +\frac{32}{3} S_{-2,1} +\frac{4 (-1)^N}{3 N (N+1)}\Biggr] \zeta_2 +\Biggl[ -16 S_1^2 +\frac{2548 S_1}{27} \nonumber\\ && +\frac{2 \big(108 N^3-239 N^2-1137 N-646\big)}{9 (N+1) (N+2)} +16 S_2\Biggr] \zeta_3 \Biggr\}~, \nonumber\end{aligned}$$ with $B_4$ $$\begin{aligned} B_4 = - 4 \zeta_2 \ln^2(2) + \frac{2}{3} \ln^4(2) - \frac{13}{2} \zeta_4 + 16 {\rm Li}_4\left(\frac{1}{2}\right)~.\end{aligned}$$ It is represented by harmonic sums up to [w = 5]{}. The logarithmic contributions and other pieces of the constant term stemming from lower order quantities are given in [@LOG]. The renormalized expression both in $N$ and $x$-space is presented in [@NS1]. The analytic continuation to complex values of $N$ is obtained using the relations being given in Refs. [@Blumlein:2009ta; @ANC]. Very recently also the massive OME of the pure singlet case has been calculated [@PS]. Here the master integrals are somewhat more demanding than for $A_{gq}$ and $A_{qq}^{\sf NS, TR}$ and they were partly solved using differential and difference equations applying the packages [[Sigma]{}]{}, [[EvaluateMultiSums]{}]{}, [[SumProduction]{}]{}and [HarmonicSums]{}. Here the structure of the Wilson coefficient contains a series of generalized harmonic sums as a fully inclusive quantity in QCD. They are of the type $$\begin{aligned} S_{2,1}(2,1;N),~~~S_{1,1,2}\left(2,\frac{1}{2},1;N\right),~~~S_{3}(2;N),~~~~~\text{etc.}\end{aligned}$$ These sums may individually diverge as $N \rightarrow \infty$. However, the asymptotic expansion of the complete expression is well behaved. In the representation in $x$-space, generalized harmonic polylogarithms emerge. For QCD corrections being related to deep-inelastic scattering quantities of this kind are observed for the first time. Graphs with two massive quark lines of equal masses =================================================== Starting from 3-loop order graphs with two distinct internal massive lines occur in the calculation of the massive operator matrix elements. The corresponding contributions to the operator matrix elements $A_{gg}$ and $A_{gq}$ are characterized by the color factors $T_F^2 C_A (C_F)$ without additional factors $Nf$. The challenge in computing these diagrams derives from the fact that identifying hypergeometric series directly, as used in earlier 3-loop calculations [@Ablinger:2010ty; @Ablinger:2012qm; @Blumlein:2012vq], leads to divergent sums. In fact the degree of divergence even grows linearly with $N$ due to factors $$\begin{aligned} B(N+i+\alpha, -i+\beta) = \frac{\Gamma(N+i+\alpha)\Gamma(-i+\beta)}{\Gamma(N+\alpha+\beta)}~,\end{aligned}$$ where $N$ is the Mellin variable, $i$ the summation index of an infinite sum, and $\alpha, \beta$ are independent of $N$ and $i$. To avoid this source of divergence, a Mellin-Barnes representation is introduced and the Beta-function of the above type is kept in the form of a Feynman parameter integral. As a result, the divergent pattern can be removed by observing that the contour of the Mellin-Barnes integral either must be closed to the right [*or*]{} to the left, depending on the value of the remaining Feynman parameter. Due to this distinction the remaining integral does not represent a Beta-function anymore, but will be performed in the space of certain iterated integrals at a later stage. The sum of residues is simplified using symbolic summation technologies [@Schneider08JSC; @Schneider05AC; @Schneider06JDEA; @Schneider10AC; @Schneider10PW; @Schneider06SL; @Schneider07Hab; @Ablinger:2010pb; @Ablinger:2010ha; @Schneider2013; @Blumlein:2012hg]. In order to perform the last integral in $x$, say, a generating function for the Mellin moments is introduced with $$\begin{aligned} \sum_{N=0}^{\infty} (\kappa R(x))^N = \frac{1}{1-\kappa R(x)} {\,,}\end{aligned}$$ where $R(x)\in \{1/(1+x^2),x^2/(1+x^2)\}$. This introduces cyclotomic letters weighted by the tracing parameter $\kappa$. The resulting expression involves cyclotomic harmonic polylogarithms (HPLs) which depend on the variable $\kappa$. In order to extract the Mellin-space expression the $N$th Taylor coefficient has to be calculated, which is possible using the packages [[HarmonicSums]{}]{} [[@Ablinger:2013hcp; @Ablinger:2011te; @Ablinger:2013cf]]{} and [[Sigma]{}]{} [ [@Schneider08JSC; @Schneider05AC; @Schneider06JDEA; @Schneider10AC; @Schneider10PW; @Schneider06SL; @Schneider07Hab; @Ablinger:2010pb; @Blumlein:2012hg]]{}. The resulting multi-sum expressions are simplified using the package [[EvaluateMultiSums]{}]{} [[@Blumlein:2012hg; @Ablinger:2010ha; @Schneider2013]]{} and expressed in a basis of indefinite (nested) sums. As a proof of principle we calculated all scalar graphs which correspond to the $T_F^2$-contributions to $A_{gg}$. Also the calculation of the full $T_F^2$ contributions will be finished soon. One of the QCD-graphs is shown in Figure \[fig:D560\]. ![A digram with two massive quark cycles contributing to $A_{gg}$.[]{data-label="fig:D560"}](D560.pdf){width="20.00000%"} The $N$-space result has the following form: $$\begin{aligned} \label{eq:D560Res} I_{560} ={}& \frac{2\overline{P}_4}{3 N (N+1)^2 (N+2) (2 N-5) (2 N-3) (2 N-1)}\frac{1}{4^N}\binom{2N}{N} \Bigg[ \sum_{j=1}^N \frac{4^{j} S_1\big(j\big)}{\binom{2 j}{j} j^2} -\sum_{j=1}^N \frac{4^{j}}{\binom{2 j}{j} j^3} -7 \zeta_3 \Bigg] {\nonumber}\\ & +\frac{N^2+N+2}{27 (N-1) N^2 (N+1)} \Big[-144 S_{2,1} -36 \zeta_2 S_1-4 S_1^3+36 S_2 S_1 + 88 S_3 {\nonumber}\\ & +312 \zeta_3\Big] -\frac{4 \overline{P}_2}{6075 (N-2)^2 (N-1)^4 N^5 (N+1)^4 (N+2) (2 N-5) (2 N-3) (2 N-1)} {\nonumber}\\& -\frac{64 \big(N^2+N+2\big)}{9 {\varepsilon}^3 (N-1) N^2 (N+1)} +\frac{1}{{\varepsilon}^2}\Big\{ -\frac{32 \overline{P}_6}{27 (N-1)^2 N^3 (N+1)^2 (N+2)} -\frac{32 \big(N^2+N+2\big)}{9 (N-1) N^2 (N+1)} S_1 \Big\} {\nonumber}\\& +\frac{1}{{\varepsilon}}\Big\{ -\frac{8 \overline{P}_7}{405 (N-2) (N-1)^3 N^4 (N+1)^3 (N+2)} -\frac{16 \overline{P}_6}{27 (N-1)^2 N^3 (N+1)^2 (N+2)} S_1 {\nonumber}\\& -\frac{8 \big(N^2+N+2\big) \big(S_1^2-3 S_2 + 3 \zeta_2\big)}{9 (N-1) N^2 (N+1)} -\frac{8 \big(55 N^3+235 N^2-52 N+20\big)}{15 (N-2) (N-1) N (N+1)^2 (N+2)} \Big\} {\nonumber}\\& -\frac{4 \overline{P}_5 S_1}{81 (N-1)^3 N^4 (N+1)^3 (N+2) (2 N-5) (2 N-3) (2 N-1)} {\nonumber}\\& -\frac{4 \overline{P}_6 \big(S_1^2-3 S_2+3 \zeta_2\big)}{27 (N-1)^2 N^3 (N+1)^2 (N+2)} -\frac{4 \overline{P}_1}{225 (N-2)^2 (N-1)^2 N^2 (N+1)^3 (N+2)}~.\end{aligned}$$ Here the functions $\overline{P}_i$ are polynomials in $N$ up to degree $d = 17$ and we used the shorthand notation $S_{\vec{a}}(N) \equiv S_{\vec{a}}$ for the harmonic sums. Besides the well-known harmonic sums the above diagram depends on the new structure $$\begin{aligned} \label{eq:invbinsumcomb} \frac{1}{4^N} \binom{2N}{N} \Biggl[ \sum_{j=1}^N \frac{4^{j} S_{1}(j)} {\binom{2j}{j} j^2} - \sum _{j=1}^N \frac{4^{j} } {\binom{2j}{j} j^3} - 7 \zeta_3 \Biggr] {\,,}\end{aligned}$$ which involves binomially weighted harmonic sums within finite sums.[^6] These objects cannot be represented in terms of (generalized) harmonic sums or (generalized) cyclotomic sums. In all scalar diagrams, and all considered QCD diagrams contributing to the color factors $T_F^2 C_A (C_F)$ these sums occur in the same combination, which is hence a property of the corresponding Feynman diagrams. In some terms denominators occur, which introduce poles at points $N=\frac{1}{2}, \frac{3}{2}, 2, \frac{5}{2}$. However, the rightmost singularity expected for these diagrams is $N = 1$. Interestingly, these poles can be shown to be removable by expanding (\[eq:D560Res\]) in a Laurent series around these points. Massive quark production in charged current DIS at 2-loop order =============================================================== The $O(\alpha_s)$ corrections to heavy flavor production in charged current deep-inelastic scattering have been calculated in [@Gottschalk:1980rv; @Kretzer99; @Blumlein:2011zu]. Here the $O(\alpha_s^2)$ corrections are presented in the asymptotic region $Q^2 \gg m^2$ [@BHP13], comparing to an earlier calculation in Ref. [@Buza:1997mg]. This process is particularly important because of its sensitivity to the sea quark densities $\bar{s}(x,Q^2), \bar{d}(x,Q^2)$ and $\bar{u}(x,Q^2)$. Furthermore the asymptotic representation is fully justified since the corresponding data are measured mostly at high virtualities $Q^2~{\raisebox{-0.07cm } {$\, \stackrel{>}{{\scriptstyle\sim}}\, $}}~100~{${\rm GeV}$}^2$.[^7] The charged current cross sections for deep inelastic lepton-nucleon scattering is commonly parameterized in three structure functions $F_1$, $F_2$, $F_3$ : $$\begin{aligned} \label{eq:XSa} \frac{d\sigma^{\nu(\bar{\nu})}}{dx dy} ={}& \frac{G_F^2 s}{4 \pi} \left\{ (1+(1-y)^2) F_2^{W^{\pm}} - y^2 F_L^{W^{\pm}} \pm (1-(1-y)^2) x F_3^{W^{\pm}} \right\} {\,,}\\ \label{eq:XS} \frac{d\sigma^{e^-(e^+)}}{dx dy} ={}& \frac{G_F^2 s}{4 \pi} \left\{ (1+(1-y)^2) F_2^{W^{\mp}} - y^2 F_L^{W^{\mp}} \pm (1-(1-y)^2) x F_3^{W^{\mp}} \right\} {\,.}\end{aligned}$$ The expressions of the heavy flavor Wilson coefficients in the asymptotic region are constructed in terms of light flavor Wilson coefficients and massive operator matrix elements (OMEs). This is achieved by exploiting the process independence of the PDFs and OMEs and constructing the 4-flavor expressions in the variable flavor number scheme in [@Buza:1996wv; @Bierenbaum:2009mv]. By matching them back onto the 3-flavor scheme one finds the factorization formulae in the asymptotic region : $$\begin{aligned} L_{i,q}^{W^+\pm W^-,\text{NS},(2)} =\;& \delta_{i,2} A_{qq,Q}^{\text{NS},(2)} + C_{i,q}^{W^+\pm W^-,\text{NS},(2)}(n_f+1) - C_{i,q}^{W^+\pm W^-,\text{NS},(2)} (n_f) {\,,}{\nonumber}\\ H_{i,q}^{W^+\pm W^-,\text{NS},(2)} =\;& \delta_{i,2} A_{qq,Q}^{\text{NS},(2)} +C_{i,q}^{W^+\pm W^-,\text{NS},(2)}(n_f+1) {\,,}{\nonumber}\\ L_{i,q}^{W,\text{PS},(2)} =\;& C_{i,q}^{W,\text{PS},(2)}(n_f+1) - C_{i,q}^{W,\text{PS},(2)}(n_f) =0 {\,,}{\nonumber}\\ H_{i,q}^{W,\text{PS},(2)} =\;& \frac{1}{2} \delta_{i,2} A_{Qq}^{\text{PS},(2)} +C_{i,q}^{W,\text{PS},(2)}(n_f+1) {\,,}{\nonumber}\\ L_{i,g}^{W,(2)} =\;& A_{gg,Q}^{(1)} C_{i,g}^{W,(1)}(n_f+1) +C_{i,g}^{W,(2)}(n_f+1) -C_{i,g}^{W,(2)}(n_f) {\,,}{\nonumber}\\ H_{i,g}^{W,(2)} =\;& A_{gg,Q}^{(1)} C_{i,g}^{W,(1)}(n_f+1) +C_{i,g}^{W,(2)}(n_f+1) {\nonumber}\\& +\frac{1}{2} \left( \delta_{i,2} A_{Qg}^{(2)} +A_{Qg}^{(1)} C_{i,q}^{W^++ W^-,\text{NS},(1)}(n_f+1) \right) {\,,}{\nonumber}\\ L_{3,q}^{W^+\pm W^-,\text{NS},(2)} =\;& A_{qq,Q}^{\text{NS},(2)} +C_{3,q}^{W^+\pm W^-,\text{NS},(2)}(n_f+1) -C_{3,q}^{W^+\pm W^-,\text{NS},(2)}(n_f) {\,,}{\nonumber}\\ H_{3,q}^{W^+\pm W^-,\text{NS},(2)} =\;& A_{qq,Q}^{\text{NS},(2)} +C_{3,q}^{W^+\pm W^-,\text{NS},(2)}(n_f+1) {\,,}{\nonumber}\\ H_{3,q}^{W,\text{PS},(2)} =\;& -\frac{1}{2} A_{Qq}^{\text{PS},(2)} {\,,}{\nonumber}\\ H_{3,g}^{W,(2)} =\;& \frac{1}{2} \left( -A_{Qg}^{(2)} -A_{Qg}^{(1)} C_{3,q}^{W^++ W^-,\text{NS},(1)}(n_f+1) \right) {\,.}\label{eq:WC}\end{aligned}$$ Here $A_{ij}^{(k)},~k =1,2$ denote the massive OMEs [@Buza:1995ie; @Bierenbaum:2007dm; @Bierenbaum:2007qe; @Bierenbaum:2008yu; @Bierenbaum:2009zt] and $C_{l,m}$ the massless Wilson coefficients [@Zijlstra:1991qc; @vanNeerven:1991nn; @Zijlstra:1992kj; @Zijlstra:1992qd; @Moch:1999eb] up to 2-loop order. Note that the $O(\alpha_s^2)$ corrections contain, besides a single heavy quark excitation also contributions due to heavy quark pair-production. The Wilson coefficients (\[eq:WC\]) correct and complete an earlier derivation of the heavy flavor Wilson coefficients [@Buza:1997mg]; for details see Ref. [@BHP13]. The difference lies in factors $(-1)$ in the expressions for $H_{3,q}^{W,\text{PS},(2)}$ and $H_{3,g}^{W,(2)}$. The correctness of the present result was checked by an explicit calculation of the leading logarithmic parts using the same idea as in [@Altarelli:1977zs], referring to the ladder graph contributions in physical gauge. Furthermore the construction of $H_{3,g}^{W,(1)}$ in the same way delivers a minus sign that can be reproduced by the asymptotic expansion of the exact 1-loop result [@Blumlein:2011zu]. We also calculated the terms to $O(\alpha_s^2)$ having been left out in Ref. [@Buza:1997mg]. The representation of the Wilson coefficients (\[eq:WC\]) both in Mellin-$N$ and $x$-space have been derived [@BHP13] using the package [[HarmonicSums]{}]{} [[@Ablinger:2013hcp; @Ablinger:2011te; @Ablinger:2013cf]]{}. For the use in phenomenological applications we implemented both these expressions into [FORTRAN]{}-programs, which are available on request. In Figures \[fig:F2cWp\] numerical illustrations of the charm quark corrections to $F_2(x,Q^2)$ and $xF_3(x,Q^2)$ are given in leading (LO), next-to-leading (NLO) and next-to-next-to-leading order (NNLO) at different scales of $Q^2$. The difference between LO and NLO turns out to be large since the NLO corrections are dominated by the gluon-$W$ fusion process, which contributes for the first time, and which reflects the size of the gluon distribution. They get much smaller comparing NLO and NNLO, where also the factorization scale uncertainty is expected to stabilize. ![Charm contributions to the structure functions $F_2$ and $xF_3$ of deep-inelastic scattering via $W^+$-exchange at LO, NLO, NNLO.[]{data-label="fig:F2cWp"}](F2WpABM11.pdf "fig:"){width=".49\textwidth"} ![Charm contributions to the structure functions $F_2$ and $xF_3$ of deep-inelastic scattering via $W^+$-exchange at LO, NLO, NNLO.[]{data-label="fig:F2cWp"}](F3WpABM11.pdf "fig:"){width=".49\textwidth"} Here the ABM11 PDF set [@Alekhin:2012ig] at NNLO in the 3-flavor scheme was used. Conclusions =========== We reported on recent progress in the calculation of massive 3-loop operator matrix elements and Wilson Coefficients for deep-inelastic scattering for general values of the Mellin variable $N$. Four years after a larger amount of Mellin moments for these quantities had been computed, six out of eight OMEs and corresponding Wilson coefficients in the region $Q^2 \gg m^2$ have been calculated analytically. In parallel, quite a series of theoretical, mathematical and computer-algebraic technologies had to be newly developed and put significantly forward to make the present results possible. Here we would like to note in particular the automated use of IBP-identities for massive 3-loop diagrams also containing local operator insertions in [Reduze2]{} and modern summation technologies built in several advanced summation packages by the Linz-Group along with the development of the present project. These and related technologies are assumed to have a significant potential to be used in many other calculations in quantum field theory in the future. We also obtained a better insight into the calculation of the more difficult topologies, like those of $V$-graphs and the treatment of graphs with two massive fermion lines, being necessary to perform the forthcoming calculations. Furthermore, we obtained the asymptotic $O(\alpha_s^2)$ heavy-flavor corrections for deep-inelastic charged current scattering. The remaining part of the present project is still putting a series of very interesting challenges to be mastered. [99]{} E. Laenen, S. Riemersma, J. Smith and W. L. van Neerven, Nucl. Phys. B [**392**]{} (1993) 162; 229. I. Bierenbaum, J. Blümlein and S. Klein, Phys. Lett. B [**672**]{} (2009) 401 \[arXiv:0901.0669 \[hep-ph\]\]. S. I. Alekhin and J. Blümlein, Phys. Lett. B [**594**]{} (2004) 299 \[hep-ph/0404034\]. M. Buza, Y. Matiounine, J. Smith, R. Migneron and W. L. van Neerven, Nucl. Phys. B [**472**]{} (1996) 611 \[hep-ph/9601302\]. M. Buza, Y. Matiounine, J. Smith and W. L. van Neerven, Eur. Phys. J. C [**1**]{} (1998) 301 \[hep-ph/9612398\]. I. Bierenbaum, J. Blümlein and S. Klein, Phys. Lett. B [**648**]{} (2007) 195 \[hep-ph/0702265\]. I. Bierenbaum, J. Blümlein and S. Klein, Nucl. Phys. B [**780**]{} (2007) 40 \[hep-ph/0703285\]. I. Bierenbaum, J. Blümlein, S. Klein and C. Schneider, Nucl. Phys. B [**803**]{} (2008) 1 \[arXiv:0803.0273 \[hep-ph\]\]. I. Bierenbaum, J. Blümlein and S. Klein, Nucl. Phys. B [**820**]{} (2009) 417 \[arXiv:0904.3563 \[hep-ph\]\]. M. Steinhauser, Comput. Phys. Commun.  [**134**]{} (2001) 335. J. Blümlein, A. De Freitas, W. L. van Neerven and S. Klein, Nucl. Phys. B [**755**]{} (2006) 272. J. Blümlein, S. Klein and B. Tödtli, Phys. Rev. D [**80**]{} (2009) 094010. J. Ablinger, J. Blümlein, S. Klein, C. Schneider, and F. Wi[ß]{}brock, Nucl. Phys. B, [**844**]{} (2011) 26 \[arXiv:1008.3347 \[hep-ph\]\]. J. Blümlein, A. Hasselhuhn, S. Klein, and C. Schneider, Nucl. Phys. B, [**866**]{} (2013) 196 \[arXiv:1205.4184 \[hep-ph\]\]. J. Ablinger, J. Blümlein, A. Hasselhuhn, S. Klein, C. Schneider, and F. Wi[ß]{}brock, Nucl. Phys. B, [**864**]{} (2012) 52 \[arXiv:1206.2252 \[hep-ph\]\]. J. Ablinger et al. arXiv:1212.5950 \[hep-ph\];\ J. Ablinger, J. Blümlein, C.Raab, C. Schneider and F. Wißbrock, DESY 13-063. J. Blümlein, S. Klein, C. Schneider and F. Stan, J. Symbolic Comput. [**47**]{} (2012) 1267-1289 \[arXiv:1011.2656 \[cs.SC\]\]. C. Schneider, arXiv:1310.0160 \[cs.SC\]. C. Schneider, J. Symbolic Comput., [**43**]{}(9) (2008) 611 \[arXiv:0808.2543 \[cs.SC\]\]. C. Schneider, Ann. Comb., [**9**]{}(1) (2005) 75. C. Schneider, J. Differ. Equations Appl., [**11**]{}(9) (2005) 799. C. Schneider, Ann. Comb., [**14**]{}(4) (2010) 533 \[arXiv:0808.2596 \[cs.SC\]\]. C. Schneider, In: A. Carey, D. Ellwood, S. Paycha, and S. Rosenberg, Eds., [Proceedings of the Workshop Motives, Quantum Field Theory, and Pseudodifferential Operators, Boston, 2008]{}. Clay Mathematics Proceedings, (2010). C. Schneider, Sém. Lothar. Combin., [**56**]{} (2006) Article B56b. C. Schneider, [*Multi-Summation in Difference Fields*]{}, Habilitationsschrift, Johannes Kepler Universität Linz, Austria, 2007, and references therein. J. Ablinger, J. Blümlein, S. Klein, and C. Schneider, Nucl. Phys. Proc. Suppl., [**205–206**]{} (2010) 110 \[arXiv:1006.4797 \[math-ph\]\]. J. Blümlein, A. Hasselhuhn, and C. Schneider, PoS RADCOR (2011) 032 \[arXiv:1202.4303 \[math-ph\]\]. J. Ablinger, I. Bierenbaum, J. Blümlein, A. Hasselhuhn, S. Klein, C. Schneider, and F. Wi[ß]{}brock, Nucl. Phys. Proc. Suppl., [**205–206**]{} (2010) 242 \[arXiv:1007.0375 \[hep-ph\]\]. C. Schneider, In: J. Blümlein and C. Schneider, Eds., [Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions]{}. Springer, Berlin, (2013) 325. C. Schneider, Advances in Applied Math [**34**]{}(4) (2005) 740;\ J. Ablinger, J. Blümlein, M. Round and C. Schneider, PoS LL [**2012**]{} (2012) 050 \[arXiv:1210.1685 \[cs.SC\]\]; M. Round et al., in preparation. J. A. M. Vermaseren, Int. J. Mod. Phys.  A [**14**]{} (1999) 2037;\ J. Blümlein and S. Kurth, Phys. Rev.  D [**60**]{} (1999) 014018. J. Blümlein, Comput. Phys. Commun.  [**159**]{} (2004) 19 \[hep-ph/0311046\]. J. Blümlein, Comput. Phys. Commun.  [**180**]{} (2009) 2218 \[arXiv:0901.3106 \[hep-ph\]\]. J. Blümlein, D. J. Broadhurst and J. A. M. Vermaseren, Comput. Phys. Commun.  [**181**]{} (2010) 582 \[arXiv:0907.2557 \[math-ph\]\]. E. Remiddi and J. A. M. Vermaseren, Int. J. Mod. Phys. A [**15**]{} (2000) 725. S. Moch, P. Uwer and S. Weinzierl, J. Math. Phys.  [**43**]{} (2002) 3363 \[hep-ph/0110083\]. J. Ablinger, J. Blümlein and C. Schneider, arXiv:1302.0378 \[math-ph\], J. Math. Phys, in print. J. Ablinger, J. Blümlein and C. Schneider, J. Math. Phys.  [**52**]{} (2011) 102301. J. Ablinger, J. Blümlein, C. Raab and C. Schneider, in preparation and Refs. [@Ablinger:2012sm]. J. Ablinger and J. Blümlein, In: J. Blümlein and C. Schneider, Eds., [Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions]{}. Springer, Berlin, (2013) 1 \[arXiv:1304.7071 \[math-ph\]\]. J. Ablinger, [*Computer Algebra Algorithms for Special Functions in Particle Physics*]{}, PhD Thesis Univ. Linz, Austria, 2013 arXiv:1305.0687 \[math-ph\]; [*A Computer Algebra Toolbox for Harmonic Sums Related to Particle Physics*]{}, Diploma Thesis Univ. Linz, Austria, 2010 arXiv:1011.1176 \[math-ph\]. J. Lagrange, [Nouvelles recherches sur la nature et la propagation du son]{}, Miscellanea Taurinensis, t. II, 1760-61; Oeuvres t. I, p. 263;\ C.F. Gauss, [Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo novo tractate]{}, Commentationes societas scientiarum Gottingensis recentiores, Vol III, 1813, Werke Bd. [**V**]{} pp. 5-7;\ G. Green, [Essay on the Mathematical Theory of Electricity and Magnetism]{}, Nottingham, 1828 \[Green Papers, pp. 1-115\];\ M. Ostrogradski, Mem. Ac. Sci. St. Peters., [**6**]{}, (1831) 39;\ K. G. Chetyrkin, A. L. Kataev and F. V. Tkachov, Nucl. Phys.  B [**174**]{} (1980) 345. C. Studerus, Comput. Phys. Commun.  [**181**]{} (2010) 1293. A. von Manteuffel and C. Studerus, arXiv:1201.4330 \[hep-ph\]. J. Ablinger, J. Blümlein, S. Klein, C. Schneider and F. Wissbrock, arXiv:1106.5937 \[hep-ph\]. M. Czakon, Comput. Phys. Commun.  [**175**]{} (2006) 559 \[hep-ph/0511200\]. A. V. Smirnov and V. A. Smirnov, Eur. Phys. J. C [**62**]{} (2009) 445 \[arXiv:0901.0386 \[hep-ph\]\]. I. Bierenbaum, J. Blümlein, S. Klein, and F. Wißbock, DESY 11–144. J. Ablinger et al., in preparation. J. Blümlein, Comput. Phys. Commun.  [**133**]{} (2000) 76 \[hep-ph/0003100\];\ J. Blümlein and S.-O. Moch, Phys. Lett. B [**614**]{} (2005) 53 \[hep-ph/0503188\]. J. Ablinger et al., in preparation. J. Fleischer, A. V. Kotikov, and O. L. Veretin, Nucl. Phys. B, [**547**]{} (1999) 343 \[hep-ph/9808242\]. M. Yu. Kalmykov and O. Veretin, Phys. Lett. B, [**483**]{} (2000) 315 \[hep-th/0004010\]. A. I. Davydychev and M. Yu. Kalmykov, Nucl. Phys. B, [**699**]{} (2004) 3 \[hep-th/0303162\]. S. Weinzierl, J. Math. Phys.  [**45**]{} (2004) 2656 \[hep-ph/0402131\]. T. Gottschalk, Phys. Rev. D [**23**]{} (1981) 56. S. Kretzer. . PhD thesis, Universität Dortmund, Germany, 1999. J. Blümlein, A. Hasselhuhn, P. Kovacikova and S. Moch, Phys. Lett. B [**700**]{} (2011) 294 \[arXiv:1104.3449 \[hep-ph\]\]. J. Blümlein, A. Hasselhuhn, and T. Pfoh, in preparation. M. Buza and W. L. van Neerven, Nucl. Phys. B [**500**]{} (1997) 301 \[hep-ph/9702242\]. E. B. Zijlstra and W. L. van Neerven, Phys. Lett. B [**273**]{} (1991) 476. W. L. van Neerven and E. B. Zijlstra, Phys. Lett. B [**272**]{} (1991) 127. E. B. Zijlstra and W. L. van Neerven, Phys. Lett. B [**297**]{} (1992) 377. E. B. Zijlstra and W. L. van Neerven, Nucl. Phys. B [**383**]{} (1992) 525. S. Moch and J. A. M. Vermaseren, Nucl. Phys. B [**573**]{} (2000) 853 \[hep-ph/9912355\]. G. Altarelli and G. Parisi, Nucl. Phys. B [**126**]{} (1977) 298. S. Alekhin, J. Blümlein and S. Moch, Phys. Rev. D [**86**]{} (2012) 054009 \[arXiv:1202.2281 \[hep-ph\]\]. [^1]: This work has been supported in part by DFG Sonderforschungsbereich Transregio 9, Computergestützte Theoretische Teilchenphysik, by Studienstiftung des Deutschen Volkes, by the Austrian Science Fund (FWF) grants P20347-N18, SFB F50 (F5009-N15), by the EU Network [LHCPHENOnet]{} PITN-GA-2010-264564, by the Reserach Center ‘Elementary Forces and Mathematical Foundations (EMG) of J. Gutenberg University Mainz and DFG, and by FP7 ERC Starting Grant 257638 PAGAP. [^2]: For a precise numerical implementation in Mellin space see [@Alekhin:2003ev]. [^3]: For a recent survey article of the summation packages see see Ref. [@Schneider:2013zna]. [^4]: For a recent review see [@Ablinger:2013jta]. [^5]: In (\[eq:BUB1\]) We corrected a typograpgical error in [@Ablinger:2011pb]. [^6]: Infinite binomial and inverse binomial sums have been considered in Refs. [@Fleischer:1998nb; @Kalmykov:2000qe; @Davydychev:2003mv; @Weinzierl:2004bn]. [^7]: For a discussion of the accuracy of the asymptotic representation see [@Buza:1995ie].

--- abstract: 'Recently, reinforcement learning models have achieved great success, completing complex tasks such as mastering Go and other games with higher scores than human players. Many of these models collect considerable data on the tasks and improve accuracy by extracting visual and time-series features using convolutional neural networks (CNNs) and recurrent neural networks, respectively. However, these networks have very high computational costs because they need to be trained by repeatedly using a large volume of past playing data. In this study, we propose a novel practical approach called reinforcement learning with convolutional reservoir computing (RCRC) model. The RCRC model has several desirable features: 1. it can extract visual and time-series features very fast because it uses random fixed-weight CNN and the reservoir computing model; 2. it does not require the training data to be stored because it extracts features without training and decides action with evolution strategy. Furthermore, the model achieves state of the art score in the popular reinforcement learning task. Incredibly, we find the random weight-fixed simple networks like only one dense layer network can also reach high score in the RL task.' author: - | Hanten Chang\ Graduate school of Systems and\ Information Engineering,\ University of Tsukuba, Japan\ `s1820554@s.tsukuba.ac.jp`\ Katsuya Futagami\ Graduate school of Systems and\ Information Engineering,\ University of Tsukuba, Japan\ `s1820559@s.tsukuba.ac.jp`\ title: Convolutional Reservoir Computing for World Models --- Introduction ============ Recently, reinforcement learning (RL) models have achieved great success, mastering complex tasks such as Go[@silver2016mastering; @silver2017mastering] and other games[@DBLP:journals/corr/MnihKSGAWR13; @DBLP:journals/corr/abs-1803-00933; @kapturowski2018recurrent] with higher scores than human players. Many of these models use convolutional neural networks (CNNs) to extract visual features directly from the environment state pixels[@DBLP:journals/corr/abs-1708-05866]. Some models use recurrent neural networks (RNNs) to extract time-series features and achieved higher scores[@kapturowski2018recurrent; @hausknecht2015deep]. However, these deep neural network (DNN) based models are very computationally expensive in that they train networks weights by repeatedly using a large volume of past playing data. Certain techniques can alleviate these costs, such as the distributed approach[@DBLP:journals/corr/abs-1803-00933; @mnih2016asynchronous] which efficiently uses multiple agents, and the prioritized experienced replay[@schaul2015prioritized] which selects samples that facilitate training. However, the cost of a series of computations, from data collection to action determination, remains high. World model[@DBLP:journals/corr/abs-1803-10122; @NIPS2018_7512] can also reduce computational costs by completely separating the training process between the feature extraction model and the action decision model. World model replaces the feature extraction model training process with the supervised learning, by using variational auto-encoder (VAE)[@2013arXiv1312.6114K; @2014arXiv1401.4082J] and mixture density network combined with an RNN (MDN-RNN) [@DBLP:journals/corr/Graves13; @ha2017recurrent]. After extracting the environment state features, it uses an evolution strategy called the covariance matrix adaptation evolution strategy (CMA-ES)[@doi:10.1162/106365601750190398; @DBLP:journals/corr/Hansen16a] to train an action decision model, which achieved outstanding scores in popular RL tasks. The separation of these two models results in the stabilization of feature extraction and omission of parameters to be trained based on task-dependent rewards. From the success of world model, it is implied that in the RL feature extraction process, it is necessary to extract the features that express the environment state rather than features trained to solve the tasks. In this study, adopting this idea, we propose a new method called “reinforcement learning with convolutional reservoir computing (RCRC)”. The RCRC model is inspired by the reservoir computing. Reservoir computing[@verstraeten2007experimental; @lukovsevivcius2009reservoir] is a kind of RNNs, but the model weights are set to random. One of the models of the reservoir computing model, the echo state network (ESN)[@jaeger2001echo; @jaeger2004harnessing; @lukovsevivcius2012practical] is used to solve time-series tasks such as future value prediction. For this, the ESN extracts features for the input based on the dot product between the input and a random matrix generated without training. Surprisingly, features obtained in this manner are expressive enough to understand the input signal, and complex tasks such as chaotic time-series prediction can be solved by using them as the input for a linear model. In addition, the ESN has solved the tasks in multiple fields such as time-series classification [@tanisaro2016time; @ma2016functional] and Q-learning-based RL[@szita2006reinforcement]. Thus, even if the ESN uses random weights, it can extract sufficient expressive features of the input and can solve the task using the linear model. Similarly, in image classification, the model that uses features extracted by the CNN with random fixed-weights as the ESN input achieves high accuracy classification with a smaller number of parameters[@Tong2018]. Based on the success of the above random fixed-weight models, RCRC extracts the visual features of the environment state using random fixed-weight CNN and, using these features as the ESN input, extracts time-series features of the environment state transitions. In the feature extraction process, all features are extracted based on matrices with random elements. Therefore, no training process is required, and feature extraction can be performed very fast. After extracting the environment state features, we use CMA-ES[@DBLP:journals/corr/Hansen16a; @doi:10.1162/106365601750190398] to train a linear combination of extracted features to perform the actions, as in world model. This model architecture results in the omission of the training process of feature extraction and recuded computational costs; there is also no need to store a large volume of past playing data. Furthermore, we show that RCRC can achieve state of the art score in popular RL task. Our contribution in this paper is as follow: - We developed a novel and highly efficient approach to extract visual and time-series features of an RL environment state using a fixed random-weight model with no training. - By combining random weight networks with an evolution strategy method, we eliminated the need to store any past playing data. - We showed that a model with these desirable characteristics can achieve state of the art score in popular continuous RL task. - We showed that simple random weight-fixed networks, for example one dense layer network, can also extract visual features and achieve high score in continuous RL task. Related Work ============ Reservoir Computing ------------------- ![Reservoir Computing overview for the time-series prediction task.[]{data-label="fig:fig1"}](figure/fig1){width="12cm"} Reservoir computing is a promising model that can solve complex tasks, such as chaotic time-series prediction, without training for the feature extraction process. In this study, we focus on the reservoir computing model, ESN[@jaeger2001echo; @jaeger2004harnessing; @lukovsevivcius2012practical]. ESN was initially proposed to solve time-series tasks[@jaeger2001echo] and is regarded as an RNN model[@lukovsevivcius2009reservoir], it can be applied to multiple fields. Let the $N$-length, $D_u$-dimensional input signal be $u = \{u(1), u(2), ..., u(t),...,u(N)\} \in \mathbb{R}^{N \times D_u}$ and the signal that adds input signal to one bias term be $U = [u;1] = \{U(1),U(2),...,U(T),...,U(N)\} \in \mathbb{R}^{N \times (D_u+1)}$. \[;\] is a vector concatenation. ESN gets features called the reservoir state $X = \{X(1),, ..., X(t),...,X(N)\} \in \mathbb{R}^{N \times D_x}$ as follows: $$\begin{aligned} \tilde{X}(t+1) & = & f(W^{\text{in}}U(t) + WX(t)))\\ X(t+1) & = & (1-\alpha)X(t) + \alpha\tilde{X}(t+1) \end{aligned}$$ where the matrices $W^{\text{in}} \in \mathbb{R}^{(D_u+1) \times D_x}$ and $W \in \mathbb{R}^{D_x \times D_x}$ are random sampled from a probability distribution such as a Gaussian distribution, and $f$ is the activation function which is applied element-wise. As the activation function, $linear$ and $tanh$ functions are generally used; it is also known that changing the activation function according to the task improves accuracy[@inubushi2017reservoir; @DBLP:journals/corr/abs-1905-09419]. The leakage rate $\alpha \in [0,1]$ is a hyper parameter that tunes the weight between the current and the previous values, and $W$ has two major hyper parameters called sparsity which is ratio of 0 elements in matrix $W$ and the spectral radius that is memory capacity hyper parameter which is calculated by the maximal absolute eigenvalue of $W$. Finally, ESN estimates the target signal $y = \{y(1), y(2), ..., y(t),...,y(N)\} \in \mathbb{R}^{N \times D_y}$ as $$\begin{aligned} y(t) = W^{\text{out}} [X(t); U(t); 1].\end{aligned}$$ The weight matrix $W^{\text{out}} \in \mathbb{R}^{D_y \times (D_x + D_u + 1)}$ is estimated by a linear model such as ridge regression. An overview of reservoir computing is shown in Figure\[fig:fig1\]. The unique feature of the ESN is that the two matrices $W ^{\text{in}}$ and $W$ used to update the reservoir state are randomly generated from a probability distribution and fixed without training. Therefore, the training process in the ESN consists only of a linear model to estimate $W ^{\text{out}}$; therefore, the ESN model has a very low computational cost. In addition, the reservoir state reflects complex dynamics despite being obtained by random matrix transformation, and it is possible to use it to predict complex time-series by simple linear transformation[@verstraeten2007experimental; @jaeger2001echo; @DBLP:journals/corr/GoudarziBLTS14]. Because of the low computational cost and high expressiveness of the extracted features, the ESN is also used to solve other tasks such as time-series classification[@tanisaro2016time; @ma2016functional], Q-learning-based RL[@szita2006reinforcement] and image classification[@Tong2018]. World models ------------ Recently, most RL models use DNNs to extract features and solved several complex tasks. However, these models have high computational costs because a large volume of past playing data need to be stored, and network parameters need to be updated using the back propagation method. There are certain techniques[@DBLP:journals/corr/abs-1803-00933; @mnih2016asynchronous; @schaul2015prioritized] and models that can reduce this cost; some models[@DBLP:journals/corr/abs-1803-10122; @NIPS2018_7512; @DBLP:journals/corr/abs-1811-04551] separate the training process of the feature extraction and action decision models to more efficiently train the action decision model. The world model[@DBLP:journals/corr/abs-1803-10122; @NIPS2018_7512] is one such model, and uses VAE[@2013arXiv1312.6114K; @2014arXiv1401.4082J] and MDN-RNN[@DBLP:journals/corr/Graves13; @ha2017recurrent] as feature extractors. They are trained using supervised learning with randomly played 10000 episodes data. As a result, in the feature extraction process, the task-dependent parameters are omitted, and there remains only one weight parameter to be trained that decides the action in the model. Therefore, it becomes possible to use the evolution strategy algorithm CMA-ES[@doi:10.1162/106365601750190398; @DBLP:journals/corr/Hansen16a] efficiently to train that weight parameter. The process of optimizing weights of action decision model using CMA-ES can be parallelized. Although the feature extraction model is trained in a task-independent manner, world model achieved outstanding scores and masterd popular RL task [CarRacing-v0]{}[@carracingv0]. CMA-ES is one of the evolution strategy methods used to optimize parameters using a multi-candidate search generated from a multivariate normal distribution $\mathcal{N}(m, \sigma^2C)$. The parameters $m$, $\sigma$, and $C$ are updated with a formula called the evolution path. Evolution paths are updated according to the previous evolution paths and evaluation scores. Because CMA-ES updates parameters using only the evaluation scores calculated by actual playing, it can be used regardless of whether the actions of the environment are continuous or discrete values[@doi:10.1162/106365601750190398; @DBLP:journals/corr/Hansen16a]. Furthermore, training can be faster because the calculations can be parallelized by the number of solution candidates. In world model, the action decision model is simplified to reduce the number of task-dependent parameters, making it possible to use CMA-ES efficiently[@DBLP:journals/corr/abs-1803-10122; @NIPS2018_7512]. World model improves the computational cost of the action decision model and accelerates the training process by separating models and applying CMA-ES. However, in world model, it is necessary to independently optimize VAE, MDN-RNN, and CMA-ES. Further, because the feature extraction model is dependent on the environment, a large amount of past data must be stored to train the feature extraction model each time the environment changes. ![RCRC overview to choose the action for [CarRacing-v0]{}: the first and second layers are collectively called the convolutional reservoir computing layer, and both layers’ model weights are sampled from Gaussian distribution and then fixed. []{data-label="fig:fig2"}](figure/fig2){width="16.5cm"} Proposal Model ============== Basic Concept ------------- World model[@DBLP:journals/corr/abs-1803-10122; @NIPS2018_7512] extracts visual features and time-series features of environment states by using VAE[@2013arXiv1312.6114K; @2014arXiv1401.4082J] and MDN-RNN[@DBLP:journals/corr/Graves13; @ha2017recurrent] without using environment scores. The models achieve outstanding scores through the linear transformation of these features. This implies that it only requires features that sufficiently express the environment state, rather than features trained to solve the task. We thus focus on extracting features that sufficiently express environment state by networks with random fixed-weights. Using networks whose weights are random and fixed has some advantages, such as having very low computational costs and no data storage requirements, while being able to sufficiently extract features. For example, a simple CNN with random fixed-weights can extract visual features and achieve high accuracy[@Tong2018]. Although the MDN-RNN is fixed in the world model, it can achieve outstanding scores[@Corentin2018rnnfix]. In the case of ESN, the model can predict complex time-series using features extracted by using random matrices transformations[@verstraeten2007experimental; @jaeger2001echo; @DBLP:journals/corr/GoudarziBLTS14]. Therefore, it can be considered that CNN can extract visual features and ESN can extract time-series features, even if their weights are random and fixed. From this hypothesis, we propose reinforcement learning with the RCRC model, which includes both random fixed-weight CNN and ESN. Proposal model overview ----------------------- The RCRC model is divided into three model layers. In the first layer, it extracts visual features by using a random fixed-weight CNN. In the second layer, it uses a series of visual features extracted in the first layer as input to the ESN to extract the time-series features. In the two layers above, collectively called the convolutional reservoir layer, visual and time-series features are extracted with no training. In the final layer, the linear combination matrix is trained from the outputs of the convolutional reservoir layer to the actions. An overview is shown in Figure\[fig:fig2\]. In the previous study, there is a similar world model–based approach[@DBLP:journals/corr/abs-1906-08857] that uses fixed weights in VAE and a memory component based on recurrent long short-term memory (LSTM)[@hochreiter1997long]. However, this approach is ineffective in solving [CarRacing-v0]{}. In the training process, the best average score over 20 randomly created tracks of each generation were less than 200. However, as mentioned further on, we achieve an average score above 900 over 100 randomly created tracks by taking reservoir computing knowledge in the RCRC model. The characteristics of the RCRC model are as follows: - The computational cost of this model is very low because visual and time-series features of game states are extracted using a convolutional reservoir computing layer whose weights are fixed and random. - In RCRC, only a linear combination in the controller layer needs to be trained because the feature extraction model (convolutional reservoir computing layer) and the action training model (controller layer) are separated. - RCRC can take a wide range of actions regardless of continuous or discrete, because of maximizing the scores that is measured by actually playing. - Past data storage is not required, as neither the convolutional reservoir computing layer nor the controller layer need to repeatedly train the past data as in backpropagation. - The convolutional reservoir computing layer can be applied to other tasks without further training, because the layer is fixed with task-independent random weights. Convolutional Reservoir Computing layer --------------------------------------- In the convolutional reservoir computing layer, the visual and time-series features of the environment state image are extracted by a random fixed-weight CNN and an ESN which has random fixed-weight, respectively. A study using CNN with fixed random weights for each single-image as input to ESN has been previously conducted, and has shown its ability to classify MNIST dataset[@mnistdata] with high accuracy[@Tong2018]. Based on this study, we developed a novel approach to perform RL tasks. By taking advantage of the RL characteristic by which the current environment state and action determine the next state, RCRC updates the reservoir state with current and previous features. This updating process enables the reservoir state to have time-series features. More precisely, consider the $D_\text{conv}$-dimensional visual features extracted by fixed random weight CNN for $t$-th environment state pixels $X_{\text{conv}}(t) \in \mathbb{R}^{D_\text{conv}}$ and the $D_\text{esn}$-dimensional reservoir state $X_{\text{esn}}(t) \in \mathbb{R}^{D_\text{esn}}$. The reservoir state $X_{\text{esn}}$ is time-series features and is updated as follows: $$\begin{aligned} \tilde{X}_{\text{esn}}(t+1) & = & f(W^{\text{in}}X_{\text{conv}}(t) + WX_{\text{esn}}(t)))\\ X_{\text{esn}}(t+1) & =& (1-\alpha)X_{\text{esn}}(t) + \alpha\tilde{X}_{\text{esn}}(t+1).\end{aligned}$$ This updating process has no training requirement, and is very fast, because $W^{\text{in}}$ and $W$ are random matrices sampled from the probability distribution and fixed. Controller layer ---------------- The controller layer decides the action by using the output of the convolutional reservoir computing layer, $X_{\text{conv}}$ and $X_{\text{esn}}$. Let $t$-th environment state input vector which added one bias term be $S(t) = [X_{\text{conv}}(t); X_{\text{esn}}(t); 1]\in \mathbb{R}^{D_{\text{conv}}+D_{\text{esn}}+1})$. In the action decision, we suppose that the feature $S(t)$ has sufficient expressive information and it can take action by a linear combination of $S(t)$. Therefore, we obtain action $A(t) \in \mathbb{R}^{N_{\text{act}}}$ as follows: $$\begin{aligned} \tilde{A}(t) & = & W^\text{out}S(t)\\ A(t) & = & g(\tilde{A}(t)) \end{aligned}$$ where, $W^{\text{out}} \in \mathbb{R}^{(D_{\text{conv}}+D_{\text{esn}}+1) \times N_{\text{act}}}$ is the weight matrix and $N_{\text{act}}$ is the number of actions in the task environment; $g$ is applied to each action to put each $\tilde{A}(t)$ in the range of possible values in the task environment. Because the weights of the convolutional reservoir computing layer are fixed, only the weight parameter $W^{\text{out}}$ requires training. We optimize $W^{\text{out}}$ by using CMA-ES, as in world model. Therefore, it is possible to parallelize the training process and handle both discrete and continuous values as actions[@doi:10.1162/106365601750190398; @DBLP:journals/corr/Hansen16a]. The process of optimizing $W^\text{out}$ by CMA-ES are shown as follows: 1. Generate $n$ solution candidates $W_{T,i}^{\text{out}} (i=1,2,...,n)$ from a multivariate normal distribution $\mathcal{N}(m(T), \sigma(T)^2C(T))$ 2. Create $n$ environments and agents $\text{worker}_i$ that implement RCRC 3. Set $W_i^{\text{out}}$ to the controller layer of $\text{worker}_i$ 4. In each execution environment, each $\text{worker}_i$ plays $m$ episodes and receives $m$ scores $G_{i, j}(j =1, 2, ..., m)$ 5. Update evolution paths with the score of each $W_i^{\text{out}}$ which is $G_i = 1/m\sum_{j=1}^m{G_{i, j}}$ 6. Update $m$, $\sigma$, $C$ using evolution paths 7. Generate a new $n$ solution candidate $W_{(T+1),i}^{\text{out}}(i=1, 2, ..., n)$ from the updated multivariate normal distribution $\mathcal{N}(m(T+1), \sigma(T+1)^2C(T+1))$ 8. Repeat 2 to 7 until the convergence condition is satisfied or the specified number of repetitions are completed In this process, $T$ represents an update step of the weight matrix $W^{\text{out}}$, and $n$ is the number of solution candidates $W^{\text{out}}$ generated at each step. The worker is an agent that implements RCRC, and each worker extracts features, takes the action and plays in each independent environment to obtain scores. Therefore, it is possible to parallelize $n$ processes to calculate each score. Experiments =========== CarRacing-v0 ------------ We evaluate the RCRC model in the popular RL task [CarRacing-v0]{}[@carracingv0] in OpenAI Gym[@1606.01540]. This is a car racing game environment that was known as a difficult continuous actions task[@DBLP:journals/corr/abs-1803-10122; @NIPS2018_7512]. The goal of this game is to go around the course without getting out by operating a car with three continuous parameters: steering wheel $[-1, 1]$, accelerator $[0, 1]$, and brake $[0, 1]$. Each frame of the game screen is given by RGB 3 channels and 96 $\times$ 96 pixels. The course is filled with tiles as shown in Figure\[fig:fig3\]. Each time the car passes a tile on the course, $1000 / N$ is added to the score. $N$ is the total number of tiles on the course. The course is randomly generated each time, and the total number of tiles in the course varies around 300. If all the tiles are passed, the total reward will be 1000, but it is subtracted by 0.1 for each frame. The episode ends when all the tiles are passed or when 1000 frames are played. If the player can pass all the tiles without getting out of the course, the reward will be over 900. The definition of “solve” in this game is to get an average of 900 per 100 consecutive trials. ![Example environment state image of [CarRacing-v0]{} and three parameters in the enviroments. The score is added when the car passes through a tile laid on the course.[]{data-label="fig:fig3"}](figure/fig3){width="8cm"} Precedure --------- In the convolutional reservoir computing layer, we set 3 convolution layers and 1 dense layer. The filter sizes in the convolution layers are 31, 14, and 6, and the strides are all 2. We set $D_\text{conv}$ and $D_\text{esn}$ to 512 to expand the features. In the reservoir computing layer, we also set the sparsity of $W$ to 0.8; the spectral radius of $W$ to 0.95. All activation functions are set to $tanh$, which is often used in reservoir computing and achieves higher scores. As in world model, we set three units ($\tilde{A}_{\text{1}}(t)$, $\tilde{A}_{\text{2}}(t)$, and $\tilde{A}_{\text{3}}(t)$) as output of the controller layer, and each of them corresponds to an action: steering wheel $A_{\text{1}}(t)$, accelerator $A_{\text{2}}(t)$, and brake $A_{\text{3}}(t)$ [@DBLP:journals/corr/abs-1803-10122; @NIPS2018_7512]. Also as in world model, each action $A(t)$ is determined by converting each $\tilde{A}(t)$ by $g$ shown as follows[@DBLP:journals/corr/abs-1803-10122; @NIPS2018_7512]: $$\begin{aligned} g(\tilde{A}(t)) = \begin{cases} tanh\left(\tilde{A}_{\text{1}}(t)\right) \\ [tanh\left(\tilde{A}_{\text{2}}(t)\right) + 1.0] / 2.0 \\ clip[tanh\left(\tilde{A}_{\text{3}}(t)\right), 0, 1] \end{cases} .\end{aligned}$$ The function $clip[x, \lambda_{\text{min}}, \lambda_{\text{max}}]$ is a function that limits the value of $x$ in range from $\lambda_{\text{min}}$ to $\lambda_{\text{max}}$ by clipping. In the experiment, 16 workers $(n=16)$ with different $W^{\text{out}}$ parameters are prepared for each update step, and each worker is set to simulate over 8 randomly generated tracks $(m=8)$, and update $W^{\text{out}}$ with an average of these scores. As the input value, each frame is resized to 3 channels of 64 $\times$ 64 pixels. As in world model[@DBLP:journals/corr/abs-1803-10122; @NIPS2018_7512], we evaluate an average score over 100 randomly created tracks score as the generalization ability of the models. To investigate the ability of each network structures, we evaluate three models: full RCRC model, the RCRC model that removes the reservoir computing layer from convolutional reservoir computing layer (visual model), the RCRC model that has only one dense layer as feature extractor (dense model). The dense model uses flatten vector of 64$\times$64 with 3 channels image, and weights of all models are random and fixed. The visual model extracts visual features and the dense model extracts only visual features with no convolutional process. In both the visual model and the dense model, the inputs to the controller layer are the $D_{\text{conv}}$-dimensional outputs from the dense layer shown in Figure\[fig:fig2\], and one bias term. ![The best average score over 8 randomly created tracks among 16 workers at [CarRacing-v0]{}.[]{data-label="fig:fig4"}](figure/fig4){width="14cm"} Method Average Score ----------------------------------------------------------- ---------------------- -- DQN[@DQNscore] 343 $\pm$ 18 DQN + Dropout[@dqndropscore] 892 $\pm$ 41 A3C (Continuous)[@A3Ccont] 591 $\pm$ 45 World model with random MDN-RNN[@Corentin2018rnnfix] 870 $\pm$ 120 World model (V model)[@DBLP:journals/corr/abs-1803-10122] 632 $\pm$ 251 World model[@DBLP:journals/corr/abs-1803-10122] [**906**]{} $\pm$ 21 GA[@DBLP:journals/corr/abs-1906-08857] [**903**]{} $\pm$ 73 RCRC model (Visual model) 864 $\pm$ 79 RCRC model [**901**]{} $\pm$ 20 : [CarRacing-v0]{} scores of various methods. \[tab:table\] Result ------ The best scores among 16 workers are shown in Figure\[fig:fig4\]. Each workers score are evaluated as average score over 8 randomly generated tracks. Although the world model improved score faster than the full RCRC model, the full RCRC model also reached high score. The full RCRC model reached an average score around 900 at 200 generations and stable high score after 400 generations, while the world model reached stable high score after 250 generations. This result shows that the full RCRC model is comparable to world model at the same condition, regardless of no training process in feature extractions. However, the full RCRC model was slower than the world model to achieve stable high scores. Incredibly, the dense model reached an average score above 880 over 8 randomly generated tracks, and the visual model reached above 890. The dense model’s score transition has higher volatility than the visual model’s score transition. Furthermore, the visual model’s score is less stable than the full RCRC model’s score. These results shows that only one dense process can extract visual features even though the weight are random and fixed, and the features extracted by convolutional process and ESN improved scores. We also test the ability of single dense network in the MNIST datasets[@mnistdata] which is benchmark dataset of image recognition task, including 28$\times$28 gray-scaled handwritten images. The MNIST dataset contains 60000 training data and 10000 testing data. In experiments, we merged these data, and randomly sampled 60000 data as training data and 10000 data as testing data with no duplication to evaluate model ability by multiple datasets. As input to the dense layer, we used 784 vector that is flatten representation of 28$\times$28 gray-scaled handwritten images, and set the dense layer has 512 units. Each input vector is divided by 255 to normalize value. The weight of the dense layer is randomly generated from a Gaussian distribution $\mathcal{N}(0, 0.06^2)$ and then fixed. After extracting features by the dense layer, we uses these features as input to logistic regression with L2 penalty to classify images into 10 classes. As a result of this experiments, we confirmed the feature extracted by random fixed-weight single dense layer has ability to achieve average accuracy score $91.58 \pm 0.27 (\%)$ over 20 trials by linear model. Surprisingly, this result shows that only single dense layer with random fixed-weight has ability to extract visual features. The generalization ability of the visual model and the full RCRC model that evaluated as average score over 100 randomly generated tracks are shown in Table\[tab:table\]. The full RCRC model achieved above 901$\pm$20 that is comparable to state of the art approaches such as the world model approach[@DBLP:journals/corr/abs-1803-10122] and GA approach[@DBLP:journals/corr/abs-1906-08857]. To achieve over 900 score of 100 randomly generated tracks, the models is only allowed to mistake a few driving. Therefore the full RCRC model can be regarded as having ability to solve [CarRacing-v0]{}. Although the visual model extracts 512-dimensional visual features with random fixed-weight CNN, it achieves 864$\pm$79 which is better than the V model that uses only 32-dimensional features extracted by VAE as input to controller layer in world model. Furthermore, the time-series features extracted by ESN improves driving. Discussion and Future work ========================== In this study, we focused on extracting features that sufficiently express the environment state, rather than those that are trained to solve the RL task. To this end, we developed a novel model called RCRC, which, using random fixed-weight CNN and a novel ESN method, respectively, extracts visual features from environment state pixels and time-series features from a series of the state pixels. Therefore, no training process is required, and features can be efficiently extracted. In the controller layer, a linear combination of both features and actions is trained using CMA-ES. This model architecture results in highly practical features that omit the training process and reduce computational costs, and there is no need to store large volumes of data. We also show that RCRC achieves state of the art scores in a popular continuous RL task, [CarRacing-v0]{}. This result brings us to the conclusion that network structures themselves, such as CNN and ESN, have the capacity to extract features. We also found that the single dense network and simple CNN model with random fixed-weight can extract visual features, and these models achieved high scores. Although VAE has desirable features such as ability to reconstruct the input and high interpretability of latent space by using reparameterization trick which uses Gaussian noise to use backpropagation, we consider that large definitive features can also extract expressive enough visual features. Because of our limited computing resources, we were unable to assign more workers to CMA-ES. There is a possibility that more efficient and stable training could be performed by assigning more workers. Although RCRC can take wide range of actions and parallelization by using CMA-ES, it is not suitable for the task that is hard to real simulation because it has to evaluate parameters by real simulation. As a further improvement, there is a possibility that the score can be improved and made it more stable by using a multi-convolutional reservoir computing layer to extract multiple features[@massar2013mean]. The current convolutional reservoir computing layer uses random weight samples generated from Gaussian distributions. Therefore, it can easily obtain multiple independent features by using different random seeds. Our results have the potential to make RL widely available. Recently, many RL models have achieved high accuracy in various tasks, but most of them have high computational costs and often require significant time for training. This makes the introduction of RL inaccessible for many. However, by using our RCRC model, anyone can train the model at a high speed with much lower computational costs, and importantly, anyone can build a highly accurate model. In addition, RCRC can handle both continuous- and discrete-valued tasks, and even when the environment changes, training can be performed without any prior learning such as the VAE and MDN-RNN in world model. Therefore, it can be used easily by anyone in many environments. In future work, we consider making predictions from previous extracted features and actions to the next ones to be an important and promising task. Because the ESN was initially proposed to predict complex time-series, it can be assumed to have capacity to predict future features. If this prediction is achieved with high accuracy, it can self-simulate RL tasks by making iterative predictions from initial state pixels. This will help to broaden the scope of RL applications. Acknowledgements ================ The authors are grateful to Takuya Yaguchi for the discussions on reinforcement learning. We also thank Hiroyasu Ando for helping us to improve the manuscript. [10]{} \[1\][`#1`]{} \[1\][https://doi.org/\#1]{} David Silver, Aja Huang, Chris J Maddison, Arthur Guez, Laurent Sifre, George Van Den Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershelvam, Marc Lanctot, et al. Mastering the game of go with deep neural networks and tree search. , 529(7587):484, 2016. David Silver, Julian Schrittwieser, Karen Simonyan, Ioannis Antonoglou, Aja Huang, Arthur Guez, Thomas Hubert, Lucas Baker, Matthew Lai, Adrian Bolton, et al. Mastering the game of go without human knowledge. , 550(7676):354, 2017. Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Alex Graves, Ioannis Antonoglou, Daan Wierstra, and Martin A. Riedmiller. Playing atari with deep reinforcement learning. , 2013. Dan Horgan, John Quan, David Budden, Gabriel Barth[-]{}Maron, Matteo Hessel, Hado van Hasselt, and David Silver. Distributed prioritized experience replay. , 2018. Steven Kapturowski, Georg Ostrovski, Will Dabney, John Quan, and Remi Munos. Recurrent experience replay in distributed reinforcement learning. In [*International Conference on Learning Representations*]{}, 2019. Kai Arulkumaran, Marc Peter Deisenroth, Miles Brundage, and Anil Anthony Bharath. Deep reinforcement learning: A brief survey. , 34(6):26–38, 2017. Matthew Hausknecht and Peter Stone. Deep recurrent q-learning for partially observable mdps. In [*2015 AAAI Fall Symposium Series*]{}, 2015. Volodymyr Mnih, Adria Puigdomenech Badia, Mehdi Mirza, Alex Graves, Timothy Lillicrap, Tim Harley, David Silver, and Koray Kavukcuoglu. Asynchronous methods for deep reinforcement learning. In [*International conference on machine learning*]{}, pages 1928–1937, 2016. Tom Schaul, John Quan, Ioannis Antonoglou, and David Silver. Prioritized experience replay. , 2015. David Ha and J[ü]{}rgen Schmidhuber. World models. , 2018. David Ha and Jürgen Schmidhuber. Recurrent world models facilitate policy evolution. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, editors, [*Advances in Neural Information Processing Systems 31*]{}, pages 2450–2462. Curran Associates, Inc., 2018. Diederik P [Kingma]{} and Max [Welling]{}. Auto-encoding variational bayes. , 2013. Danilo [Jimenez Rezende]{}, Shakir [Mohamed]{}, and Daan [Wierstra]{}. Stochastic backpropagation and approximate inference in deep generative models. , 2014. Alex Graves. Generating sequences with recurrent neural networks. , 2013. David Ha. Recurrent neural network tutorial for artists. , 2017. Nikolaus Hansen and Andreas Ostermeier. Completely derandomized self-adaptation in evolution strategies. , 9(2):159–195, 2001. Nikolaus Hansen. The [CMA]{} evolution strategy: [A]{} tutorial. , 2016. David Verstraeten, Benjamin Schrauwen, Michiel d’Haene, and Dirk Stroobandt. An experimental unification of reservoir computing methods. , 20(3):391–403, 2007. Mantas Luko[š]{}evi[č]{}ius and Herbert Jaeger. Reservoir computing approaches to recurrent neural network training. , 3(3):127–149, 2009. Herbert Jaeger. The “echo state” approach to analysing and training recurrent neural networks-with an erratum note. , 148(34):13, 2001. Herbert Jaeger and Harald Haas. Harnessing nonlinearity: Predicting chaotic systems and saving energy in wireless communication. , 304(5667):78–80, 2004. Mantas Luko[š]{}evi[č]{}ius. A practical guide to applying echo state networks. In [*Neural networks: Tricks of the trade*]{}, pages 659–686. Springer, 2012. Pattreeya Tanisaro and Gunther Heidemann. Time series classification using time warping invariant echo state networks. In [*2016 15th IEEE International Conference on Machine Learning and Applications (ICMLA)*]{}, pages 831–836. IEEE, 2016. Qianli Ma, Lifeng Shen, Weibiao Chen, Jiabin Wang, Jia Wei, and Zhiwen Yu. Functional echo state network for time series classification. , 373:1–20, 2016. Istv[á]{}n Szita, Viktor Gyenes, and Andr[á]{}s L[ő]{}rincz. Reinforcement learning with echo state networks. In [*International Conference on Artificial Neural Networks*]{}, pages 830–839. Springer, 2006. Zhiqiang Tong and Gouhei Tanaka. Reservoir computing with untrained convolutional neural networks for image recognition. In [*2018 24th International Conference on Pattern Recognition (ICPR)*]{}, pages 1289–1294. IEEE, 2018. Masanobu Inubushi and Kazuyuki Yoshimura. Reservoir computing beyond memory-nonlinearity trade-off. , 7(1):10199, 2017. Hanten Chang, Shinji Nakaoka, and Hiroyasu Ando. Effect of shapes of activation functions on predictability in the echo state network. , 2019. Alireza Goudarzi, Peter Banda, Matthew R. Lakin, Christof Teuscher, and Darko Stefanovic. A comparative study of reservoir computing for temporal signal processing. , 2014. Danijar Hafner, Timothy P. Lillicrap, Ian Fischer, Ruben Villegas, David Ha, Honglak Lee, and James Davidson. Learning latent dynamics for planning from pixels. , 2018. Oleg Klimov. Carracing-v0. <https://gym.openai.com/envs/CarRacing-v0/>, 2016. Corentin Tallec, L[é]{}onard Blier, and Diviyan Kalainathan. Reproducing “world models”. is training the recurrent network really needed ? <https://ctallec.github.io/world-models/>, 2018. Sebastian Risi and Kenneth O. Stanley. Deep neuroevolution of recurrent and discrete world models. , 2019. Sepp Hochreiter and J[ü]{}rgen Schmidhuber. Long short-term memory. , 9(8):1735–1780, 1997. Yann LeCun. . <http://yann.lecun.com/exdb/mnist/>, 1998. Greg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and Wojciech Zaremba. Openai gym, 2016. Luc. Prieur. . <https://goo.gl/VpDqSw>, 2017. Patrik Gerber, Jiajing Guan, Elvis Nunez, Kaman Phamdo, Tonmoy Monsoor, and Nicholas Malaya. Solving openai’s car racing environment with deep reinforcement learning and dropout. <https://github.com/AMD-RIPS/RL-2018/blob/master/documents/nips/nips_2018.pdf>, 2018. Se Won Jang, Jesik Min, and Chan Lee. . <https://www.scribd.com/document/358019044/>, 2017. Marc Massar and Serge Massar. Mean-field theory of echo state networks. , 87(4):042809, 2013.

--- abstract: 'Let $G$ be a connected and non-necessarily compact Lie group acting on a connected manifold $M$. In this short note we announce the following result: for a $G$-invariant closed differential form on $M$, the existence of a closed equivariant extension in the Cartan model for equivariant cohomology is equivalent to the existence of an extension in the homotopy quotient.' address: - 'Escuela de Matemáticas, Universidad Nacional de Colombia, Calle 59A No 63 - 20, Medellín, Colombia ' - 'Departamento de Matemáticas y Estadística, Universidad del Norte, Km. 5 vía Puerto Colombia, Barranquilla, Colombia' author: - Camilo Arias Abad - Bernardo Uribe bibliography: - 'Cartan-non-compact-Lie.bib' title: 'On the equivarant de Rham cohomology for non-compact Lie groups' --- Introduction ============ The Cartan model for the equivariant cohomology of the manifold $M$ $$\Omega_G^*M:= (S({{\mathfrak g}}^*) \otimes \Omega^*M)^G, \ \ d_G= d + \Omega^a\iota_{X_a}$$ can be seen as the de Rham version for the equivariant cohomology. Whenever the Lie group $G$ is compact, Cartan proved an equivariant version of the De-Rahm Theorem, stating that the cohomology of the Cartan complex is canonically isomorphic to the cohomology with real coefficients of the homotopy quotient $H*(\Omega_G^*M) \cong H^*(M \times_G EG; {\mathbb{R}})$ [@Cartan] cf. [@Guillemin-Sternberg Thm. 2.5.1]. When the Lie group $G$ is not compact, the cohomology of the complex $\Omega_G^*M$ (which we also call the Cartan complex) fails in many situations to be isomorphic to the cohomology of the homotopy quotient, and the explicit relation between the two has been very scarcely addressed. Nevertheless, the Cartan complex is very well suited for studying equivariant conditions at the infinitesimal level. Of particular interest is the study of the conditions under which there is absence of anomalies in gauged WZW actions on Lie groups. In [@Witten] Witten showed that the absence of anomalies in gauged WZW actions on compact Lie groups was equivalent to the existence of closed equivariant extension of the WZW term on the Cartan complex, further showing that the existence or absence of anomalies is purely topological. The arguments of Witten could be extended without trouble to the non-compact case (see [@Uribe Chapter 4]), and together with the main result of this paper, we conclude that the absence or existence of anomalies is purely topological fact, independent of the compacity of the Lie group. In this short note we investigate the relation between the cohomology of the $G$-equivariant Cartan complex of $M$ and the cohomology of the homotopy quotient $M \times_G EG$, and we show that indeed there is a surjective map from the former to the latter. In particular this result implies that for a $G$-invariant closed differential form on $M$, the existence of a closed equivariant extension in the Cartan model for equivariant cohomology is equivalent to the existence of an extension in the homotopy quotient. Equivariant Cartan complex for connected Lie groups =================================================== Let $G$ be a connected Lie group with lie algebra ${{\mathfrak g}}$. Let $K \subset G$ be a maximal compact subgroup of $G$ and denote by ${\mathfrak}k$ its Lie algebra. The inclusion of Lie algebras ${{\mathfrak}k} \hookrightarrow {{\mathfrak g}}$ induces a dual map ${{\mathfrak g}}^* \to {{\mathfrak}k}^*$ which is ${{\mathfrak}k}$-equivariant. Therefore we have the $K$-equivariant map $$S({{\mathfrak g}}^*) \to S({\mathfrak}k^*)$$ from the symmetric algebra on ${{\mathfrak g}}^*$ to the symmetric algebra on ${\mathfrak}k^*$. Consider a manifold $M$ endowed with an action of $G$. The Cartan complex associated to the $G$-manifold $M$ is $$\Omega_G^*M:= (S({{\mathfrak g}}^*) \otimes \Omega^*M)^G, \ \ d_G= d + \Omega^a\iota_{X_a}$$ where $a$ runs over a base of ${{\mathfrak g}}$, $\Omega^a$ denotes the element in ${{\mathfrak g}}^*$ dual to $a$ and $X_a$ is the vector field on $M$ that defines the element $a \in {{\mathfrak g}}$. In the literature, whenever the Cartan complex is used, it is assumed that the Lie group is compact. In this note we extend the notation of Cartan to the non-compact case. The composition of the natural maps $$(S({{\mathfrak g}}^*) \otimes \Omega^*M)^G \hookrightarrow (S({{\mathfrak g}}^*) \otimes \Omega^*M)^K \to (S({\mathfrak}k^*) \otimes \Omega^*M)^K$$ induces a homomorphism of Cartan complexes $$\Omega_G^* M \to \Omega_K^*M.$$ Let $G$ be a connected Lie group with Lie algebra ${{\mathfrak g}}$, let ${\mathfrak}k$ be the Lie algebra of the maximal compact subgroup $K$ of $G$ and consider a $G$-manifold $M$. Then the map $$\Omega_G^* M \to \Omega_K^*M$$ induces a surjective map in cohomology $$H^*(\Omega_G^* M, d_G) \twoheadrightarrow H^*(\Omega_K^* M, d_K).$$ Since there are canonical isomorphisms $ H^*(\Omega_K^* M, d_K)\cong H^*(M \times_K EK,{\mathbb{R}}) \cong H^*(M \times_G EG,{\mathbb{R}})$, we conclude that the canonical map $$H^*(\Omega_G^* M, d_G) \twoheadrightarrow H^*(M \times_G EG,{\mathbb{R}})$$ is surjective. Consider the complex $C^k(G, S({{\mathfrak}g}^*) \otimes \Omega^\bullet M )$ defined in [@Getzler Section 2.1] whose elements are smooth maps $$f(g_1, \dots , g_k | X) : G^k \times {\mathfrak}{g} \to \Omega^\bullet M,$$ which vanish if any of the arguments $g_i$ equals the identity of $G$. The differentials $d$ and $\iota$ are defined by the formulas $$\begin{aligned} (df)(g_1, \dots , g_k | X) &=& (-1)^k df(g_1, \dots , g_k | X) \ \ \ \ \ \ {\rm{and}}\\ (\iota f) (g_1, \dots , g_k | X) &=& (-1)^k \iota(X) f(g_1, \dots , g_k | X),\end{aligned}$$ as in the case of the differentials in Cartan’s model for equivariant cohomology [@Cartan; @Guillemin-Sternberg]. The differential $\bar{d}: C^k \to C^{k+1}$ is defined by the formula $$\begin{aligned} (\bar{d}f)(g_0, \dots , g_k|X) & = & f( g_1, \dots , g_k | X ) + \sum_{i=1}^k (-1)^i f(g_0, \dots, g_{i-1}g_i, \dots , g_k | X)\\ & & +(-1)^{k+1} g_k f(g_0, \dots , g_{k-1} | {\rm{Ad}}(g_k^{-1})X),\end{aligned}$$ and the fourth differential $\bar{\iota} : C^k \to C^{k-1}$ is defined by the formula $$\begin{aligned} (\bar{\iota}f)(g_1, \dots , g_{k-1}|X) & = & \sum_{i=0}^{k-1} (-1)^i \frac{\partial}{\partial t} f(g_1, \dots, g_i, e^{tX_i}, g_{i+1} \dots , g_{k-1} | X),\end{aligned}$$ where $X_i= {\rm{Ad}}(g_{i+1} \dots g_{k-1})X$. If the map $$f: G^k \to S( {\mathfrak}g^*) \otimes \Omega^\bullet M$$ has for image a homogeneous polynomial of degree $l$, then the total degree of the map $f$ is $deg(f)=k+l$. The structural maps $d, \iota, \bar{d}$ and $\bar{\iota}$ are all of degree 1, and the operator $$d_G = d + \iota +\bar{d} + \bar{\iota}$$ becomes a degree 1 map that squares to zero. The cohomology of the complex $$\left( C^*(G, S({{\mathfrak}g}^*) \otimes \Omega^\bullet M ) , d_G \right)$$ will be denoted by $$H^*(G, S({{\mathfrak}g}^*) \otimes \Omega^\bullet M )$$ and in [@Getzler Thm. 2.2.3] it was shown that there is a canonical isomorphism of rings $$H^*(G, S({{\mathfrak}g}^*) \otimes \Omega^\bullet M ) \cong H^*(M \times_G EG ; {\mathbb{R}})$$ Note that there are natural maps of complexes $$C^*(G, S({{\mathfrak g}}^*) \otimes \Omega^*M) \to C^*(K,S({\mathfrak}k^*)\otimes \Omega^*M)$$ inducing an isomorphism on cohomology groups $$H^*(G, S({{\mathfrak g}}^*) \otimes \Omega^*M) \stackrel{\cong}{\to} H^*(K,S({\mathfrak}k^*) \otimes \Omega^*M).$$ This isomorphism follows from the fact that the inclusion $K \subset G$ is a homotopy equivalence inducing a homotopy equivalence $$M\times_KEK \simeq M \times_G EG$$ and the fact that $$H^*(M\times_G EG,{\mathbb{R}}) \cong H^*(G, S({{\mathfrak g}}^*)\otimes \Omega^*M)$$ for any connected Lie group $G$. Filtering the double complex $C^*(G, S({{\mathfrak g}}^*)\otimes \Omega^*M)$ by the degree of the elements in $S({{\mathfrak g}}^*)\otimes \Omega^*M$ we obtain a spectral sequence whose first page is $$E_1=H^*_d(G,S({{\mathfrak g}}^*)\otimes \Omega^*M),$$ the differentiable cohomology of $G$ with values in the graded representation $S({{\mathfrak g}}^*)\otimes \Omega^*M$. Note that in the 0-th row we obtain $$E_1^{*,0}= (S({{\mathfrak g}}^*)\otimes \Omega^*M)^G=\Omega_G^*M.$$ The same degree filtration applied to the complex $C^*(K,S({\mathfrak}k^*)\otimes \Omega^*M)$ produces a spectral sequence which at the first page is $\overline{E}_1=H^*_d(K,S({\mathfrak}k^*)\otimes \Omega^*M)$, and since $K$ is compact this simply becomes $$\overline{E}_1^{*,0}=(S({\mathfrak}k^*)\otimes \Omega^*M)^K=\Omega_K^*M$$ with $\overline{E}_1^{p,q}=0$ for $q\neq 0$. The first differential of the spectral sequence once restricted to the 0-th row $E_1^{*,0}=\Omega_G^*M$ is precisely the differential of the Cartan complex; therefore we obtain $$E_2^{*,0}= H^*(\Omega_G^*M).$$ Equivalently we obtain $$\overline{E}_2^{*,0}= H^*(\Omega_K^*M)\cong H^*(M\times_K EK, {\mathbb{R}}),$$ but in this case the spectral sequence collapses at the second page and the only non zero elements in $\overline{E}_\infty$ appear on the 0-th row $\overline{E}_\infty^{*,0}\cong H^*(M\times_K EK, {\mathbb{R}})$. The canonical map between the complexes $$C^*(G, S({{\mathfrak g}}^*) \otimes \Omega^*M) \to C^*(K,S({\mathfrak}k^*)\otimes \Omega^*M)$$ induces a map of spectral sequences $E_\bullet \to \overline{E}_\bullet$, and we know that at the pages at infinity it should induce an isomorphism $E_\infty^{*,\star} \stackrel{\cong}{\to} \overline{E}_\infty^{*,\star} $. Therefore the map $$E_2^{*,0} \to \overline{E}_2^{*,0}$$ must be a surjective map, and hence we have the canonical map $$\Omega_G^*M = E_1^{*,0} \to \overline{E}_1^{*,0}= \Omega_K^*M$$ inducing the desired surjective map in cohomology $$H^*(\Omega_G^* M, d_G) \twoheadrightarrow H^*(\Omega_K^* M, d_K).$$ Finally, from the previous theorem we may conclude: For a $G$-invariant closed differential form on $M$, the existence of a closed equivariant extension in the Cartan model for equivariant cohomology is equivalent to the existence of an extension in the homotopy quotient. A $G$-invariant closed differential form on $M$ may be extended to a closed form in the Cartan complex if and only if its cohomology class lies in the image of the projection map $$H^*(\Omega_G^* M, d_G) \to H^*(M).$$ This projection map can be seen as the composition of the maps $$H^*(\Omega_G^* M, d_G) \twoheadrightarrow H^*(\Omega_K^* M, d_K) \to H^*(M).$$ Since the left hand side map is surjective, a $G$-invariant closed differential form on $M$ may be extended to a closed form in the Cartan complex if and only if its cohomology class lies in the image of the right hand side map. The canonical isomorphisms $$H^*(M\times_GEG; {\mathbb{R}}) \cong H^*(M\times_KEK; {\mathbb{R}}) \cong H^*(\Omega_K^* M, d_K)$$ imply the result.

--- abstract: | In this article, we study dynamic chiral symmetry breaking at zero temperature, finite chemical potential and external magnetic field with massless NJL model. We have proposed a mathematical method to classify phases in phase diagram of cold dense quark matter, and use mathematical analysis to identify the multi phase phenomenon among solutions for gap equation which means with fixed chemical potential and magnetic field, there could be two phases coexisting. Key-words: NJL model, magnetic field, dynamical mass, gap equation, quark condensate PACS Numbers: 11.10.Wx, 26.60.Kp, 21.65.Qr, 25.75.Nq, 12.39.Ki author: - Song Shi$^1$ - Juan Liu$^1$ - 'Zhu-fang Cui$^2$' - 'Hong-Shi Zong$^{2,3}$' title: Multi Phase in Cold Dense Quark Matter --- Introduction ============ The phase structure of QCD matter has always been an important and attractive topic in theoretical physics [@fukushima; @costa; @chao; @zhao; @jiang; @xu; @fu]. In relativistic heavy-ion collisions, the produced QCD matter will go though a phase transition or a crossover as time goes by. Either way, the state of QCD matter is believed to change from quark-gluon plasma to hadronic matter in this process. Its physical properties and dynamical behaviors such as chiral symmetry and confinement are altered along with the change of the state. At the early stage of noncentral collision, the QCD matter produces extremely strong magnetic field [@kharzeev; @kha; @sko; @vor], which brings about obvious magnetic effects. Moreover, the compact stellar objects such as magnetars are believed had strong magnetic field around $10^{15}$G at their surface [@dun; @kou], Therefore studying QCD matter’s properties under the influence of magnetic field becomes a meaningful and important subject. So far, many relevant theories and models have been proposed and it is shown that the quark condensate are strengthened by magnetic field, which is known as ‘Magnetic Catalysis’ [@gusynin; @gusynin2; @ebert; @shov; @alan]. Consequently, the QCD phase diagram is related to magnetic field [@andersen; @endrodi]. In this article, we will study the phase property of so called “cold dense quark matter”, unlike the quark matter in high energy experiments, this matter has low temperature, hence we can establish models at zero temperature limitation. This kind of research could facilitate the study of compact stellar objects. Although similar projects have been thoroughly studied in early articles [@ebert2; @ebert3; @ina; @men; @boom; @faya; @ferr; @ferrer; @mand; @mand2; @pre; @pre2; @allen], but in this article we try to study phase diagram of cold dense matter in a different point of view, we have developed a mathematical method to classify phases in phase diagram, and find out new property from gap equation. The model we employ is the two-flavor NJL model at chiral limitation with mean-field approximation [@vogl; @klev; @hats; @bub], it is a good tool to investigate dynamical chiral symmetry breaking of nonperturbative QCD matter. One thing need mentioning here, in Asakawa’s work [@asa], it was pointed that with the presence of chemical potential, the self-energy does not simply equal dynamical mass, which reveals with the help of the Fierz transformation. The actual self-energy should be written as $\Sigma=\sigma+a\gamma^0$ to guarantee the self-consistency of gap equation. In our case, this problem becomes much more complicate, the external magnetic field and chemical potential render self energy has four kinds of mean fields, $\Sigma=\sigma+a\gamma^0+b\gamma^5\gamma^3+c\sigma^{12}$. At zero temperature limitation, this will cause very chaotic situation in gap equation. But fortunately, except $\sigma$, the other three quantities are very small comparing to nonzero solutions of $\sigma$, we can ignore them for a schematic view of phase diagram. The article is arranged as below, Section \[ge\] is the deduction of gap equation with our new developed method different from Schwinger’s “proper time”, this method could handle more complicate models or ansatz such as $\Sigma=\sigma+a\gamma^0+b\gamma^5\gamma^3+c\sigma^{12}$, one can refer to Appendix \[prop\] and \[eigen\] for more details of this method. Section \[analysis\] is the classification of phase diagram by using mathematical analysis and numerical analysis, in this section we mathematically define several areas in phase diagram which have different phase properties. Section \[concl\] is the conclusion. Gap Equation\[ge\] ================== We start from the lagrangian below (which is the NJL model with mean field approximation) $$\mathcal{L}=\bar\psi(\slashed{D}+\mu\gamma^0-\sigma)\psi-\frac{N_\text{c}}{2G}\sigma^2,\label{la0}$$ $$D_\mu=i\partial_\mu+eA_\mu\otimes Q,\qquad A_\mu=(0,\frac{B}{2}x^2,-\frac{B}{2}x^1,0),\qquad Q=\operatorname{diag}(q_\text{u}, q_\text{d}),$$ here $A_\mu$ gives us the external magnetic field with the strength $B$ that parallels to $x^3$ axis. $q_\text{u}$, $q_\text{d}$ separately represent electric charge numbers of up quark and down quark, hence $q_\text{u}=\frac{2}{3}$, $q_\text{d}=-\frac{1}{3}$. We employ $q_\text{f}$ to generally represent $q_\text{u}$, $q_\text{d}$ in following discussions, and the index ‘f’ can be assigned to ‘u’ or ‘d’. In the Lagrangian Eq. (\[la0\]) we have employed the ansatz that $\Sigma=\sigma I_4$, while actually, with nonzero external magnetic field and chemical potential, $\Sigma$ should have not only quark condensate but also vector, axial vector and tensor condensates. The last there kinds of condensates are thought to be too small to affect the general properties of QCD matter but some subtleties, therefore in this paper we assume $\Sigma=\sigma I_4$. Now we manage to get the fermion propagator with magnetic field and chemical potential, in the mean time, to identify the appropriate form of $\varepsilon$ term (because it is zero temperature). According to Appendix A, in order to acquire the correct fermion propagator simply, one has to multiply Hamiltonian density with a factor $(1-i\eta)$, $$\mathcal{H}'=-(1-i\eta)\bar\psi(\gamma^iD_i+\mu\gamma^0-\sigma)\psi+\frac{N_\text{c}}{2G}\sigma^2(1-i\eta).$$ Consequently, the Lagrangian has changed to, $$\mathcal{L}'=\sum_\text{f}\bar\psi\hat S_\text{f}'^{-1}\psi-\frac{N_\text{c}}{2G}\sigma^2(1-i\eta),\qquad\hat S'_\text{f}=\frac{1}{/\kern-0.55em\hat\Pi^\text{f}-\sigma'},\label{la1}$$ $$\sigma'=(1-i\eta)\sigma,\quad\hat\Pi^\text{f}_\mu=\hat p'_\mu+q_\text{f}eA_\mu(1-i\eta),\quad\hat p'_0=\hat p_0+(1-i\eta)\mu,\quad\hat p'_i=(1-i\eta)\hat p_i,$$ the $\sum_\text{f}$ here means we have separated flavor space, and the fermion field operator $\psi$ in Eq. (\[la1\]) is $4$ components single flavor spinor rather than $8$ components two flavor spinor in Eq. (\[la0\]). Through the definition of partition function $$\mathcal{Z}=\lim_{\eta\to0^+}\int\mathrm{D}\bar\psi\,\mathrm{D}\psi\,e^{i\int\mathcal{L}'\,\mathrm{d}^4x}=\lim_{\eta\to0^+}e^{-iW'(\sigma,\mu,eB,\eta)},$$ we have the effective action $W'$ $$W'=\frac{N_\text{c}}{2G}\sigma^2(1-i\eta)\int\mathrm{d}^4x+iN_\text{c}\sum_\text{f}\operatorname{Tr}(\ln\hat S_\text{f}'^{-1}).\label{free}$$ The principle of gap equation is to identify the least or local minimum value for effective action, which is $$\frac{\delta W'}{\delta\sigma}=0,$$ hence we have $$(1-i\eta)\frac{\sigma}{G}\int\mathrm{d}^4x=i\sum_\text{f}\operatorname{Tr}\hat S'_\text{f}.\label{gap1}$$ in following discussion we can safely set factor $(1-i\eta)$ to be $1$ at the left hand side of Eq. (\[gap1\]) with the limitation $\eta\to0^+$. Now we need to take care of $\operatorname{Tr}\hat S'_\text{f}$ in Eq. (\[gap1\]), because there is chemical potential in the propagator, it is not convenient to use Schwinger’s ‘proper time’ method. We have developed a new method to deal such situation. One can refer to our previous work [@shi] for a detailed introduction or to Appendix B for an overview. According to Eq. (\[gapb\]), we have $$\operatorname{Tr}\hat S'_\text{f}=\frac{|q_\text{f}|eB\sigma}{\pi}\int\frac{\mathrm{d}p_0\,\mathrm{d}p_3}{(2\pi)^2}\, \sum_n\frac{2-\delta_{0n}}{p'^2_0-2n|q_\text{f}|eB'-p'^2_3-\sigma'^2}\int\mathrm{d}^4x,\label{trace1}$$ $$p'_0=p_0+(1-i\eta)\mu,\qquad p'_3=(1-i\eta)p_3,\qquad B'=(1-i\eta)^2B.$$ Noticing, the complex factor in $B'$ is $(1-i\eta)^2$ rather than $(1-i\eta)$, because $2n|q_\text{f}|eB'$ comes from the combination of quantization of $\hat\Pi^2_\perp$ and $q_\text{f}eB\sigma^{12}$ which is the outcome of $[\hat\Pi^\text{f}_1,\hat\Pi^\text{f}_2]$, both of them provide the factor $(1-i\eta)^2$. Secondly one should be aware due to nonzero chemical potential, $\operatorname{Tr}(\hat S'_\text{f}\gamma^0)$ is not $0$, the consequence is that we should introduce a shift for chemical potential to prevent the inconsistency. But as we have mentioned at the beginning, such shift relates to a vector condensate and it is very small, we could exclude its effect in our qualitative results. Continuing to adjust the expression of Eq. (\[trace1\]) $$\begin{split} \operatorname{Tr}\hat S'_\text{f}&=\frac{|q_\text{f}|eB\sigma}{\pi}\int\frac{\mathrm{d}p_0\,\mathrm{d}p_3}{(2\pi)^2}\, \sum_n\frac{2-\delta_{0n}}{(p_0+\mu)^2-\omega_{n\text{f}}^2+i\varepsilon[\omega_{n\text{f}}^2-\mu(p_0+\mu)]}\int\mathrm{d}^4x\\ &=\frac{|q_\text{f}|eB\sigma}{\pi}\int\frac{\mathrm{d}p_0\,\mathrm{d}p_3}{(2\pi)^2}\, \sum_n\frac{2-\delta_{0n}}{p_0-\omega_{n\text{f}}^2+i\varepsilon(\omega_{n\text{f}}^2-\mu p_0)}\int\mathrm{d}^4x, \end{split}\label{trace2}$$ $$\omega_{n\text{f}}=\sqrt{p_3^2+2n|q_\text{f}|eB+\sigma^2},\qquad \varepsilon=2\eta\to0^+.$$ Making a Wick rotation to Eq. (\[trace2\]), and applying proper time method $$\frac{\operatorname{Tr}\hat S'_\text{f}}{\int\mathrm{d}^4x}=-i\frac{|q_\text{f}|eB\sigma}{4\pi^2}\int_0^{+\infty}\frac{e^{-\sigma^2s}}{s}\coth(|q_\text{f}|eBs)\,\mathrm{d}s +i2\sum_\text{f}|q_\text{f}|eB\sigma\sum_n(2-\delta_{0n})\operatorname{\uptheta}(\mu-\lambda_{n\text{f}})\ln\frac{\mu+\sqrt{\mu^2-\lambda_{n\text{f}}^2}}{\lambda_{n\text{f}}},\label{trace3}$$ $$\lambda_{n\text{f}}=\sqrt{\sigma^2+2n|q_\text{f}|eB}.$$ Combining Eqs. (\[gap1\]) and (\[trace3\]), making a truncation to proper time ‘$s$’ ($\int_0^{+\infty}\mathrm{d}s\to\int_{1/\Lambda^2}^{+\infty}\mathrm{d}s$), the gap equation could be simplified to (despite the trivial solution $\sigma=0$) $$\begin{split} \frac{4\pi^2}{G}=&\sum_\text{f}|q_\text{f}|eB\int_{1/\Lambda^2}^{+\infty}\frac{e^{-\sigma^2s}}{s}\coth(|q_\text{f}|eBs)\,\mathrm{d}s\\ &-2eB\operatorname{\uptheta}(\mu-\sigma)\ln\frac{\mu+\sqrt{\mu^2-\sigma^2}}{\sigma}-4\sum_\text{f}|q_\text{f}|eB\sum_{n=1}^{+\infty}\operatorname{\uptheta}(\mu-\lambda_{n\text{f}}) \ln\frac{\mu+\sqrt{\mu^2-\lambda_{n\text{f}}^2}}{\lambda_{n\text{f}}}, \end{split}\label{gap2}$$ At the $B\to0^+$ and $\mu\to0^+$ limits, Eq. (\[gap2\]) degenerates to the classic gap equation $$\frac{4\pi^2}{G}=2\int_{1/\Lambda^2}^{+\infty}\frac{e^{-\sigma^2s}}{s^2}\mathrm{d}s,$$ which we can use to determine the value of $\Lambda$ and $G$ [@inagaki], $$\Lambda=0.99\text{GeV},\qquad G=25.4\text{GeV}^{-2}.$$ Analysis and Numerical Results\[analysis\] ========================================== The Boundary of Chemical Potential ---------------------------------- There is a boundary to chemical potential. Before explain that, we have to vary gap equation Eq. (\[gap2\]) a little firstly $$\begin{aligned} f(\sigma,eB)&=&h(\sigma,\mu,eB),\nonumber\\ f(\sigma,eB)&=&\sum_\text{f}|q_\text{f}|eB\int_{1/\Lambda^2}^{+\infty}\frac{e^{-\sigma^2s}}{s}\coth(|q_\text{f}|eBs)\,\mathrm{d}s,\nonumber\\ h(\sigma,\mu,eB)&=&2eB\operatorname{\uptheta}(\mu-\sigma)\ln\frac{\mu+\sqrt{\mu^2-\sigma^2}}{\sigma}+4\sum_\text{f}|q_\text{f}|eB\sum_{n=1}^{+\infty}\operatorname{\uptheta}(\mu-\lambda_{n\text{f}}) \ln\frac{\mu+\sqrt{\mu^2-\lambda_{n\text{f}}^2}}{\lambda_{n\text{f}}}+\frac{4\pi^2}{G}\label{gap3}\end{aligned}$$ The first row is the variation. In this way, we can study two functions $f(\sigma,eB)$ and $h(\sigma,\mu,eB)$ separately beside gap equation. From the expression of $h(\sigma,\mu,eB)$ in Eq. (\[gap3\]) we know if chemical potential $\mu$ is smaller than dynamic mass $\sigma$, then the gap equation is simplified to $$f(\sigma,eB)=\frac{4\pi^2}{G},\label{gap0}$$ there will be no $\mu$-dependent dynamic mass in existence, we define these ordinary ‘only-magnet-dependent’ dynamic mass (ODM) as $\sigma_0$. $\sigma_0$ roughly gives us lower boundary of $\mu$ that the solutions of gap equation Eq. (\[gap3\]) are beyond ‘ordinary’. In the matter of fact, the actual lower boundary is generally smaller than $\sigma_0$ with all $eB$s, take Fig. (\[elu1\]) for example, we have three solutions for gap equation in the case of $\mu_1$, $\sigma_0$ is the ODM, $\sigma'$ is also a valid solution for Eq. (\[gap3\]) while $\sigma''$ is not, because from mathematical analysis we know at $\sigma''$ the effective action Eq. (\[free\]) has a local maximum rather than minimum, it is not what we need. In following discussion we will ignore these maximum points. The contacting points of $f(\sigma,eB)$ and $h(\sigma,\mu,eB)$ in Fig. (\[elu1\]) is not a valid solution neither, but it is a boundary point for chemical potential. Comparing $\mu_1$ line with $\mu_2$, we can see when $\mu>\mu_2$, Eq. (\[gap3\]) has valid solutions beyond ODM, and analytically it is true, because $h(\sigma,\mu,eB)$ is a monotonically increasing function by $\mu$, as long as $\mu$ is bigger than a specific value at different $eB$s, the solution will be not just ODM. Therefore the real lower boundary of $\mu$, defined as $\mu_\text{low}$, is always lower than $\sigma_0$. ![In this figure, we treat $\sigma$ as variable, $eB$ and $\mu$ as preset parameters, and the vertical axis as function of $\sigma$. $eB=0.01$GeV$^2$, $\mu_1\approx0.25394$GeV, $\mu_2\approx0.253886$GeV. The horizontal dotted line represents $\frac{4\pi^2}{G}$. $f(\sigma,eB)$ intersects with $h(\sigma,\mu_1,eB)$ at there points, $\sigma'$, $\sigma''$ and $\sigma_0$. $f(\sigma,eB)$ contacts with $h(\sigma,\mu_2,eB)$ at $\sigma'''$.\[elu1\]](elucidate_mu_infmum_1.eps){width="3in"} To identify $\mu_\text{low}$, we have to solve the simultaneous equations of $\sigma$, $$f(\sigma,eB)|_{\sigma=\mu_\text{low}}=h(\sigma,\mu,eB)|_{\sigma=\mu_\text{low}}, \qquad\left.\frac{\partial f(\sigma,eB)}{\partial\sigma}\right|_{\sigma=\mu_\text{low}}=\left.\frac{\partial h(\sigma,\mu,eB)}{\partial\sigma}\right|_{\sigma=\mu_\text{low}}.\label{mulow}$$ Beside lower boundary, we also have upper boundary for chemical potential, defined as $\mu_\text{up}$, when $\mu$ exceeds such boundary, Eq. (\[gap3\]) has no valid solution ($f(\mu,eB)$ has no intersection with $h(\sigma,\mu,eB)$, or the intersection point represents maximum rather minimum of effective action), which means chiral symmetry restores, Wigner phase arises. How to identify the upper boundary depends on the asymptotic behaviors of $f(\mu,eB)$ and $h(\sigma,\mu,eB)$ at $\sigma\to0^+$. First of all, when $\sigma$ is big enough, we will eventually have $h(\sigma,\mu,eB)>f(\sigma,eB)$, because $h(+\infty,\mu,eB)=4\pi^2/G$ and $f(+\infty,eB)=0$, so if $h(0^+,\mu,eB)<f(0^+,eB)$, by Bolzano’s Theorem, there must be a valid solution of $\sigma$ in somewhere between $0$ and $+\infty$. But what if $h(0^+,\mu,eB)>f(0^+,eB)$? At $\sigma\to0^+$ and $\sigma\to+\infty$, we both have $h(\sigma,\mu,eB)>f(\sigma,eB)$, it seems using mathematica analysis to identify the intersections is impossible. Nevertheless, we should find out the asymptotic properties at $\sigma\to0^+$ first. For the function $f(\sigma,eB)$, $$f(\sigma,eB)=\sum_\text{f}|q_\text{f}|eB\int_{1/\Lambda^2}^{+\infty}\frac{e^{-\sigma^2s}}{s}[\coth(|q_\text{f}|eBs)-1]\,\mathrm{d}s +eB\int_{1/\Lambda^2}^{+\infty}\frac{e^{-\sigma^2s}}{s}\,\mathrm{d}s,$$ by L’Hôpital’s rule, $$\lim_{\sigma\to0^+}\frac{\int_{1/\Lambda^2}^{+\infty}e^{-\sigma^2s}/s\,\mathrm{d}s}{\ln\sigma} =\lim_{\sigma\to0^+}\frac{\int_{\sigma^2/\Lambda^2}^{+\infty}e^{-s}/s\,\mathrm{d}s}{\ln\sigma}=-2,$$ therefore we have $$\begin{split} f(\sigma,eB)&\sim f_0(\sigma,eB),\qquad\sigma\to0^+,\\ f_0(\sigma,eB)&=-2eB\ln\sigma+eBC +\sum_\text{f}|q_\text{f}|eB\int_{1/\Lambda^2}^{+\infty}\frac{\coth(|q_\text{f}|eBs)-1}{s}\,\mathrm{d}s,\\ C&=\lim_{\sigma\to0^+}\left(\int_{1/\Lambda^2}^{+\infty}\frac{e^{-\sigma^2s}}{s}\,\mathrm{d}s+2\ln\sigma\right)\approx-0.595297,\label{asym1} \end{split}$$ here ‘$\sim$’ reads “$f(\sigma,eB)$ is asymptotic to $f_0(\sigma,eB)$ as $\sigma$ tends to $0^+$" [@bender]. If someone get confused on the units problem in the logarithm function of $\sigma$, one can use $$-2\ln\sigma+C=-2\ln\frac{\sigma}{C'},\qquad C'\approx0.742562\text{GeV},$$ to replace the one in Eq. (\[asym1\]). For the function $h(\sigma,\mu,eB)$, $$\begin{split} h(\sigma,\mu,eB)&\sim h_0(\sigma,\mu,eB),\qquad\sigma\to0^+,\\ h_0(\sigma,\mu,eB)&=-2eB\ln\frac{\sigma}{2\mu} +4\sum_\text{f}|q_\text{f}|eB\sum_{n=1}^{+\infty}\operatorname{\uptheta}(\mu-\lambda^0_{n\text{f}})\ln\frac{\mu+\sqrt{\mu^2-(\lambda^0_{n\text{f}})^2}}{\lambda^0_{n\text{f}}} +\frac{4\pi^2}{G},\nonumber\\ \lambda^0_{n\text{f}}&=\sqrt{2n|q_\text{f}|eB}. \end{split}$$ Noticing, as $\sigma\to0^+$, both $f(\sigma,eB)$ and $h(\sigma,\mu,eB)$ have the same dominant asymptotic behaviors, closing to $(-2eB\ln\sigma)$, this renders quite complicate relations between them, and the solutions too, when they are close enough. But still, we can make a use of their asymptotic behaviors to ‘roughly’ identify the upper boundary of chemical potential by equation of $\mu$, $$f_0(\sigma,eB)=h_0(\sigma,\mu,eB)|_{\mu=\mu_\text{up}},\label{muup}$$ the $\ln\sigma$ terms in Eq. (\[muup\]) are perfectly canceled on both side, which leaves us an implicit function with respect to $\mu_\text{up}$ and $eB$. In the above discussion we have mentioned “roughly identify the upper boundary", that’s because the actual upper boundary is beyond Eq. (\[muup\]). There are two kinds of situations. First, $f(\sigma,eB)$ is a smooth function of $\sigma$ but $h(\sigma,\mu,eB)$ is not, beside that, $f(\sigma,eB)$ and $h(\sigma,\mu_\text{up},eB)$ are not only very close as $\sigma\to0^+$, but also close enough when $\sigma$ stretches to a finite value, say $0.1$GeV, these two reasons cause multiple intersections before they distinctly separate, and of cause some of the intersections account for nonzero dynamic mass, e.g. Fig. (\[inter1\]). But that doesn’t say Eq. (\[muup\]) is of no use, we can add a tiny modification to $\mu_\text{up}$ from Eq. (\[muup\]), which brings us the actual upper boundary $\mu_\text{ub}=\mu_\text{up}+\!{\vartriangle\!\!}\mu$, in the case of Fig. (\[inter1\]), ${\vartriangle\!\!}\mu\approx0.0018$GeV, it is a small quantity, therefore we can say Eq. (\[muup\]) roughly identifies upper boundary. ![$eB=0.05$GeV$^2$, $\mu_\text{up}\approx0.317547$GeV. $\sigma'$ is a valid intersection.\[inter1\]](intersections_1.eps){width="3in"} The second situation, through numerical result, we find that $\mu_\text{up}$ decreases along with increasing magnetic field, while the ODM $\sigma_0$ is increasing, therefore when $\mu_\text{up}$, as the function of $eB$, exceeds a threshold, $f(\sigma,eB)$ will always intersect with $h(\sigma,\mu_\text{up},eB)$ at a ODM point, e.g. Fig. (\[inter2\]), this implies that the actual upper boundary, defined as $\mu_\text{ub}$, depends on $\sigma_0$ rather than $\mu_\text{up}$. ![$eB=0.25$GeV$^2$, $\mu_\text{up}\approx0.227481$GeV, $\sigma_0\approx0.2978$GeV. $\sigma_0$ is a valid intersection between $f(\sigma,eB)$ and $h(\sigma,\mu_\text{up},eB)$, but is not valid between $f(\sigma,eB)$ and $h(\sigma,\sigma_0,eB)$.\[inter2\]](intersections_2.eps){width="3in"} There is an equation to summarize above discussions, $$\mu_\text{ub}=\max(\sigma_0,\mu_\text{up}+\!{\vartriangle\!\!}\mu).$$ So far, we have discussed the properties of $\sigma_0$, $\mu_\text{low}$ and $\mu_\text{up}$, we put them in Fig. (\[bound\]). The modifications of $\mu_\text{up}$ is also sketchily plotted in this figure as error bars, a precise demonstration of these modifications is shown in Fig. (\[error\]), which, as we can see, is relatively small and ruleless, therefore for a qualitative discussion, we can just talk about $\mu_\text{up}$ instead of $\mu_\text{ub}$ when $\mu_\text{ub}=\mu_\text{up}+\!{\vartriangle\!\!}\mu$. Noticeably, when magnetic field is not strong enough, relatively, the upper boundary of chemical potential, which separates chiral restored phase and chiral broken phase, is highly ruleless, it has irregular oscillation, but when $eB$ keeps increasing, $\mu_\text{up}$ begins regularly decreasing, we will discuss this in the conclusion. From Fig. (\[bound\]), the three functions of $eB$ have roughly divided the diagram into several areas, this gives us phase diagram with nonzero chemical potential and magnetic field at zero temperature, as shown in Fig. (\[phase\]). The diagram is mainly divided to three kinds of areas, the ordinary Nambu phase, the multi phase area and the Wigner phase, and in multi phase area (MPA), it is divided into three subareas by function $\min(\mu_\text{up},\sigma_0)$. The ordinary Nambu phase has only one solution, the ODM, determined by Eq. (\[gap0\]), it depends only on magnetic field, has no relationship with chemical potential. The Wigner phase is the chiral restored phase, it always has zero dynamic mass $\sigma=0$ in this model (chiral limit NJL model) we study. The most interesting part is the MPA, its three subareas have different properties of the solutions of gap equation Eq. (\[gap3\]). We will discuss this in detail next subsection. ![$\sigma_0$, $\mu_\text{up}$, $\mu_\text{low}$ are implicit functions of $eB$. The error bars of $\mu_\text{up}$ represent modification ${\vartriangle\!\!}\mu$ to $\mu_\text{up}$, they are only sketch, the actual modifications are relatively small quantities. When $eB>0.16$GeV$^2$, $\mu_\text{up}<\sigma_0$, the actual upper boundary of chemical potential depends on $\sigma_0$ rather than $\mu_\text{up}$. Along with $eB$’s increasing, $\mu_\text{low}$ and $\mu_\text{up}$ are decreasing, and getting closer and closer, this property could be easily understood through mathematics analysis. Noticing, when $eB$ ranges from $0$ to $0.15$GeV$^2$, $\mu_\text{up}$ has irregular oscillation.\[bound\]](boundary.eps){width="3in"} ![The modifications of $\mu_\text{up}$, lead to actual upper boundaries of chemical potential. The range of $eB$ is from $0$ to $0.16$GeV$^2$, because when $eB>0.16$GeV$^2$, the actual upper boundaries is determined by $\sigma_0$, there is no need to consider modification for $\mu_\text{up}$. The biggest modification happens at $eB=0.07$GeV$^2$, ${\vartriangle\!\!}\mu\approx0.017$GeV, while the smallest modification, except $0$, is ${\vartriangle\!\!}\mu\approx0.0003$GeV at $eB=0.03$GeV$^2$. At some points, ${\vartriangle\!\!}\mu=0$, no modification is needed, $\mu_\text{ub}=\mu_\text{up}$.\[error\]](error.eps){width="3in"} ![The phase diagram of $eB$ and $\mu$. Below the dashed line is ordinary Nambu phase, the dynamic mass is ODM, determined by Eq. (\[gap0\]). Above the solid line is Wigner phase, $\sigma=0$. In between is the multi phase area (MPA), and it is separated to three areas by dotted line, the function for dotted line is $\min(\mu_\text{up},\sigma_0)$. These areas have different properties of dynamic mass.\[phase\]](phase.eps){width="3in"} The Solutions in Multi Phase Area --------------------------------- In order to clarify the phase properties in MPA of Fig. (\[phase\]), we have to treat $h(\sigma,\mu,eB)$ in a proper way. Due to the Heaviside step function, $h(\sigma,\mu,eB)$, as a function of $\sigma$, is continue but not smooth, it has stages, and at the lowest stage, $h$ becomes a constant $4\pi^2/G$, therefore we consider $h(\sigma,\mu,eB)$ as a two sections function, section I, the stairs ($h>4\pi^2/G$), in this section, $h$ could have many stairs, inside the stair, $h$ is smooth, at the point of two stairs contacting, $h$ is continue but not smooth, the $h$ function in Fig. (\[inter1\]) is a good example, section II, the ground ($h=4\pi^2/G$), of cause at the point of stairs and ground contacting, $h$ is continue but not smooth, too. When $f$ and $h$ intersect at the ground, we have ODM, when at the stairs, we have UDM (unordinary dynamic mass). MPA I is a area that only has UDMs. It quite clear that in this area no ODM solution is allowed, because $\mu>\sigma_0$. In MPA I, we also have $\mu>\mu_\text{low}$, $\mu_\text{low}$ means the point of first contact of $f$ and $h$, when $\mu$ exceeds the first contact point, $f$ and $h$ will have at least one intersection, of cause an UDM, or multi UDMs simultaneously, e.g. Fig. (\[inter3\]). But multi UDMs does not happens all the time, generally, when the solution of gap equation is closing to a contacting point of two stairs of $h$, we can find another valid solution in the other stair. The $\sigma$-$\mu$ relation could be like Fig. (\[sigmu1\]), which is just a small part of a big picture. When $\mu$ ranges from $\mu_\text{low}$ to $\mu_\text{up}$, the cascade could happen many times, and the multi phase areas only exist for a short range. ![$eB=0.03$GeV$^2$, $\mu=0.298$GeV. Both $\sigma_1$ and $\sigma_2$ are valid solutions for gap equation.\[inter3\]](intersections_3.eps){width="3in"} ![Defining ${\vartriangle\!\!}\sigma=\sigma_1-\sigma_2$. The maximum of ${\vartriangle\!\!}\sigma$ is approximately $0.03$GeV at $\mu\approx0.3005$GeV. The minimum of ${\vartriangle\!\!}\sigma$ is approximately $0.019$GeV at $\mu\approx0.2965$GeV. $\mu\in(0.2965, 0.3005)$ is one multi phase interval for $\mu$ at $eB=0.03$GeV$^2$.\[sigmu1\]](sigmamu_1.eps){width="3in"} MPA II is the area that definitely has multi phase simultaneously, in this area, we always have at least two valid solutions from the gap equation, and one of which is an ODM. Because $\mu<\sigma_0$, $f$ has an intersection with $h$ at the ground, while $\mu_\text{low}<\mu<\mu_\text{up}$, $f$ always cuts through the stairs of $h$, that brings us another valid solution. The $f$ and $h$’s relation is quite like Fig. (\[inter3\]), except the lowest stair of $h$ is ground $4\pi^2/G$. The $\sigma$-$\mu$ relation about this area is shown in Fig. (\[sigmu2\]). ![$\sigma_0$ is the ODM at $eB=0.15$GeV$^2$, $\sigma_1$ is the UDM. The dotted lines divide MPA II from other areas.\[sigmu2\]](sigmamu_2.eps){width="3in"} MPA III is just another ordinary Nambu phase area, in this area, all dynamic mass are determined by Eq. (\[gap0\]). From Fig. (\[phase\]), it seems MPA III and ordinary Nambu phase from the bottom are separated by MPA II, but considering $\mu_\text{up}$ and $\mu_\text{low}$ are moving really closer with increasing $eB$, maybe they will connect when $eB$ is strong enough. The Conclusion\[concl\] ======================= In this article, we have developed a mathematical analysis method to help drawing phase diagram of cold dense quark matter, and roughly divide phase diagram into several areas, each area has a unique phase property. The phase diagram and division depend on three kinds of quantities, $\sigma_0$, $\mu_\text{low}$, $\mu_\text{up}$, they are all implicit functions of $eB$, the equations that depict these quantities are separately Eqs. (\[gap0\]), (\[mulow\]), (\[muup\]). $\sigma_0$ is the ordinary dynamic mass in NJL model with external magnetic field, when chemical potential is involved, there are some dynamic mass deviating ODM, therefore $\sigma_0$ is the base line of all other phases. $\mu_\text{low}$ is the first contacting point of $f(\sigma,eB)$ and $h(\sigma,\mu,eB)$, when chemical potential is smaller than $\mu_\text{low}$, $f$ and $h$ can only have the ODM solutions for gap equation, therefore $\mu_\text{low}$ is the dividing line of ordinary dynamic mass and unordinary dynamic mass. The meaning of $\mu_\text{up}$ is a bit complicate, generally speaking, it is the dividing line of multi phase area and simple phase area, when magnetic field is relatively weak, say $0.15$GeV$^2$ comparing to $0.3$GeV$^2$, $\mu_\text{up}$ separates the multi phase area from Wigner phase area, when magnetic field is strong, $\mu_\text{up}$ separates multi phase area from another ordinary Nambu phase area. Of cause this classification is not as subtle as the phase diagram in [@allen], but all these dividing lines are mathematically definable. When magnetic field is below about $0.14$GeV$^2$, the upper boundary $\mu_\text{up}$ of chemical potential has an interesting irregular oscillation, this could be caused by quantum fluctuation around the critical point. However, a strong enough magnetic field would smear the fluctuation, that’s why when $eB>0.14$GeV$^2$, $\mu_\text{up}$ has a regular smooth descending. Mathematically speaking, when $eB$ is small, there are several Landau levels play roles in the gap equation due to Heaviside step functions in $h(\sigma,\mu,eB)$, these levels cause the irregular solutions in Eq. (\[muup\]), but when $eB$ is strong enough, that leaves us only the lowest Landau level ($\operatorname{\uptheta}(\mu-\sigma)$ term in $h$), and it is much more regular and predictable. We can roughly estimate from which point on $\mu_\text{up}$ becomes regular, firstly, the phase transition line between chiral restored phase and chiral broken phase is around $0.3$GeV, so we assume $\mu=0.3$GeV, and now if we want only lowest Landau level involving in $h$, it requires $(\mu-\sqrt{2|q_\text{d}|eB})\le0$, which leads to $eB\ge0.135$GeV$^2$, it is pretty close to $0.14$GeV$^2$. In this article we have proved the existence of multi phase, shown in Figs. (\[sigmu1\]) and (\[sigmu2\]). The case in Fig. (\[sigmu1\]) belongs to MPA I of Fig. (\[phase\]), in this area, not any chemical potential guarantees two valid solutions to the gap equation, only when chemical potential belongs to some specific intervals, the two valid solutions can be found. When we use numerical method to solve Eq. (\[gap3\]), if one solution is close to the joint point of two different stairs in $h(\sigma,\mu,eB)$ such as $\sqrt{\mu^2-2n|q_\text{f}|eB}$, then maybe we could find another solution on the other side of the joint point. It probably has no mathematical equations to identify whether or not there is multi solution around $\sigma=\sqrt{\mu^2-2n|q_\text{f}|eB}$, we suspect this phenomenon happens whenever dynamic mass jumps from one stair to another with chemical potential changing, and the smaller dynamic mass is, the shorter intervals chemical potential need going through. The case in Fig. (\[sigmu2\]) belongs to MPA II, as we can see in Fig. (\[phase\]), this area spread from weak magnetic field to strong magnetic field, which means multi phase formed by ODM and UDM always exists as long as magnetic field is nonzero, this is understandable, because no matter how strong or weak the magnetic field is, from Nambu phase to Wigner phase, all particles have to pass through lowest Landau level, and this level causes multi phase. The physical effect of multi phase is energy level transition, we take the cases from Figs. (\[sigmu1\]) and (\[sigmu2\]) as examples, fix magnetic field and chemical potential at specific values which guarantee multi phase, treat $\sigma$ as free variable of free energy density $\mathcal{F}$ ($W=\int\mathcal{F}\,\mathrm{d}^4x$) from Eq. (\[free\]), unsurprisingly there are two minimums of $\mathcal{F}$ in the intervals we choose, seen in Fig. (\[tran\]). Because $\sigma_{1,2}$ (or $\sigma_{0,3}$ exist in the same external conditions (chemical potential and magnetic field), they can transfer to each other accompanied by energy absorption or radiation. If the absorption happens, absorbing photon for instance, some particles jump to higher Landau level (only one level higher for sure, because multi phase happens between adjacent levels), or vice versa. Interestingly when energy level transition happens, not only Landau level, but also dynamic mass changes, phenomenally speaking, the “structure” of quark matter has changed. We know if a thermal system is in equilibrium state with multiple phases, all phases must fulfill three kinds of equilibrium, thermal equilibrium (equal temperature), mechanical equilibrium (equal pressure) and diffusive equilibrium (equal chemical potential). In the multi phase case here, the quark matter remains at zero temperature (thermal equilibrium), with fixing chemical potential (diffusive equilibrium) and magnetic field (it is an external condition and space-time-independent), the free energy density (or pressure, $P=-\mathcal{F}$) is not equal, mechanical equilibrium is not fulfilled, therefore the particles in high energy state such as $\sigma_1$ in Fig. (\[tran\]) will be pushed away by or transfer to particles in lower energy state $\sigma_2$, the multi phase state could not stably exist. ![Examples of energy level transition. These two figures are corresponding to cases in Figs. (\[sigmu1\]) and (\[sigmu2\]) separately. The vertical axis represents free energy density $\mathcal{F}$, hence its nature units is GeV$^4$. $\sigma_1-\sigma_2\approx(0.022\text{GeV})^4$, $\sigma_0-\sigma_3\approx(0.059\text{GeV})^4$. Noticing only $\sigma_{0,1,2,3}$ have physical meaning, the other values of $\sigma$ in the intervals are virtual.\[tran\]](transition_1.eps "fig:"){width="2.5in"} ![Examples of energy level transition. These two figures are corresponding to cases in Figs. (\[sigmu1\]) and (\[sigmu2\]) separately. The vertical axis represents free energy density $\mathcal{F}$, hence its nature units is GeV$^4$. $\sigma_1-\sigma_2\approx(0.022\text{GeV})^4$, $\sigma_0-\sigma_3\approx(0.059\text{GeV})^4$. Noticing only $\sigma_{0,1,2,3}$ have physical meaning, the other values of $\sigma$ in the intervals are virtual.\[tran\]](transition_2.eps "fig:"){width="2.5in"} In this article, we employ zero temperature limitation and the ansatz $\Sigma=\sigma$, but as we mentioned at the beginning, a complete self energy should involve four kinds of condensate, if considering all these condensates, undoubtedly properties of phases for gap equations should be more complicate and abundant, in following work, we would like to do more detailed study, but as to a schematic view of quark matter at zero temperature, the ansatz in this article is adequate. The advantage of zero temperature limitation is that gap equation can be more “clear”, and we are able to rely on mathematical tools to analyse gap equation, but when temperature goes to nonzero, we have to depend on numerical methods, it could be hard to find some properties of the solutions such as multi phase. We believe when temperature is low, multi phase phenomenon could still exist, because from zero temperature to low temperature, it is a continuous process, the properties of phases should be continuously varying. Also in following studies, we would like explore the cases at high temperature, although the particles’ thermal motions at high temperature will smear many effects that can be found at low temperature, but if the multi phase is still there, that could be significant. $\varepsilon$ Term in Propagator\[prop\] ======================================== At zero temperature, there is always a $\varepsilon$ term in the denominator of a particle propagator of momentum space, such propagator is generally have the form as $\frac{1}{p^2-m^2+i\varepsilon}$. Caused by relativistic causality, in momentum space, the $4$ dimensional integral of propagator should choose an appropriate contour in complex space, hence there comes the $\varepsilon$ term. But when chemical potential comes in, such term becomes chemical-potential-relevant, say, $i\varepsilon p_0(p_0-\mu)$, and it will essentially render a different contour in complex momentum space. In the above two examples, identifying $\varepsilon$ term is quite easy, just the regular canonical quantization routine to free particles. But what if the interaction terms come in or other external fields come in, how to identify the $\varepsilon$ term is a problem. In this appendix, we have developed a convenient method to identify $\varepsilon$ term, which is inspired by [@dai]. The original idea is changing the ‘Hamiltonian’ a bit in partition function $$\mathcal{Z}=\langle0|Te^{-i\int\hat H\,\mathrm{d}t}|0\rangle,$$ In order to ensure the partition function is finite at infinite time, one can replace the Hamiltonian operator $\hat H$ with another complex version $\hat H'$, $$\hat H'=(1-i\eta)\hat H,\qquad\eta\to0^+,$$ or replace Hamiltonian density instead $$\mathcal{H}'=(1-i\eta)\mathcal{H},\qquad H=\int\mathcal{H}\,\mathrm{d}^3x.$$ Coincidently, when transferring this new Hamiltonian into Lagrangian, we can instantly get the correct particle propagator in momentum space, take free fermion for example $$\mathcal{H}'[\psi]=-(1-i\eta)\bar\psi(\gamma^ii\partial_i-m)\psi,$$ $$\mathcal{L}'=\frac{\partial\mathcal{L}}{\partial\dot{\psi}}-\mathcal{H}'=\bar\psi\hat D'\psi,\qquad\hat D=\gamma^0\partial_0+(1-i\eta)(\gamma^ii\partial_i-m).$$ In momentum space, the inverse of operator $\hat D'$ is exactly the propagator we need $$\hat D'^{-1}=\frac{1}{\gamma^0p_0+(1-i\eta)(\gamma^ip_i-m)}=\frac{\gamma^0p_0+(1-i\eta)(\gamma^ip_i+m)}{p_0^2-(|\vec p|^2+m^2)(1-i\eta)^2} =\frac{\slashed{p}+m}{p^2-m^2+i\varepsilon},\quad\varepsilon=2(|\vec p|^2+m^2)\eta\to0^+.$$ One can also prove this is effective to free boson propagator. To demonstrate this method we have developed is valid, here we consider another example, the zero temperature and finite chemical potential case of free fermion propagator. $$\mathcal{H}'=-(1-i\eta)\bar\psi(\gamma^ii\partial_i+\mu\gamma^0-m)\psi,\qquad\mathcal{L}'=\bar\psi\hat D'\psi,$$ $$\hat D'=\slashed{\hat p}'-m',\quad \hat p_0'=\hat p_0+\mu',\quad\mu'=(1-i\eta)\mu,\quad \hat p_i'=(1-i\eta)\hat p_i,\quad m'=(1-i\eta)m.$$ In momentum space, we have $$D'^{-1}=\frac{\gamma^0(p_0+\mu)+\gamma^ip_i+m}{(p_0+\mu)^2-\omega^2+i\varepsilon[\omega^2-\mu(p_0+\mu)]},\qquad\omega=\sqrt{|\vec p|^2+m^2},\label{pro0}$$ this looks a bit different from what we need. First of all, in the momentum integral, we can always make a shift to $p_0$, $p_0+\mu\to p_0$. Secondly, in the denominator of Eq. (\[pro0\]), $p_0^2-\omega^2$ implies the poles which require $p_0=\pm\omega$, we could use this relation to replace $\omega$ in $\varepsilon$ term, which gives us $$D'^{-1}\to\frac{\slashed{p}+m}{p_0^2-\omega^2+i\varepsilon p_0(p_0-\mu)}.$$ That is what we need. Of cause in $\varepsilon$ term, the replacement of $\omega$ with $p_0$ seems a little undemanding. One can also prove rigorously from complex analysis that such replacement is legitimate. Eigenstate Method for Nonzero External Magnetic Field in Fermion Propagator\[eigen\] ==================================================================================== Assuming the fermion propagator with nonzero external magnetic field is $$\hat S_\text{f}=\frac{1}{/\kern-0.55em\hat\Pi^\text{f}-m},\qquad\hat\Pi^\text{f}_\mu=\hat p_\mu+q_\text{f}eA_\mu,\qquad A_\mu=(0,\frac{B}{2}x^2,-\frac{B}{2}x^1,0).$$ Generally we need to deal with the cases such as $\operatorname{Tr}(\hat S_\text{f}\Gamma^a)$, where $\Gamma^a\in\{\gamma^\mu,\gamma^5,\gamma^5\gamma^\mu,\sigma^{\mu\nu}\}$. With the presence of external magnetic field, neither $|p\rangle$ nor $|x\rangle$ is $\hat S_\text{f}$’s eigenstate, therefore we need to find an appropriate representation through which we could avoid to confront the annoying noncommutative relation in $\hat S_\text{f}$ (eg. $[\hat\Pi_1,\hat\Pi_2]=-iq_\text{f}eB$). First, we try to scalarize the denominator of $\hat S_\text{f}$, $$\hat S_\text{f}=\frac{/\kern-0.55em\hat\Pi^\text{f}+m}{\hat p_0^2-\hat\Pi_\perp^2-\hat p_3^2-q_\text{f}eB\sigma^{12}-m^2},\qquad\hat\Pi^2_\perp=(\hat\Pi^\text{f}_1)^2+(\hat\Pi^\text{f}_2)^2.$$ From the new propagator, we extract a series operators $(\hat p_0,\hat p_3,\hat\Pi_\perp^2)$ that commute with each others. Now we are able to define a eigenstate $|p_0,p_3\rangle\otimes|n,\lambda\rangle$ for these operators. $|p_0,p_3\rangle$ is obviously the eigenstate of $\hat p_{0,3}$, $$\hat p_{0,3}|p_0,p_3\rangle=p_{0,3}|p_0,p_3\rangle,$$ $|n,\lambda\rangle$ is eigenstate of $\hat\Pi^2_\perp$, $$\hat\Pi^2_\perp|n,\lambda\rangle=(2n+1)|q_\text{f}|eB|n,\lambda\rangle,\qquad n\in\mathbb{N}^0,\qquad\lambda\in\mathbb{R},$$ $\lambda$ is a free variable in the eigenstate, like $\theta$ in $e^{i\theta}$ as the free phase of wave function, it will not participate in the gap equation. Then for example the trace of $\hat S_\text{f}$ is $$\begin{split} \operatorname{Tr}\hat S_\text{f}&=\int\mathrm{d}p_0\,\mathrm{d}p_3\,\sum_{n=0}^{+\infty}\int_{-\infty}^{+\infty}\mathrm{d}\lambda\,\frac{\langle p_0,p_3;n,\lambda|/\kern-0.55em\hat\Pi^\text{f}+m|p_0,p_3;n,\lambda\rangle}{p_0^2-(2n+1)|q_\text{f}|eB-p_3^2-q_\text{f}eB\sigma^{12}-m^2}\\ &=\frac{|q_\text{f}|eBm}{\pi}\int\frac{\mathrm{d}p_0\,\mathrm{d}p_3}{(2\pi)^2}\,\sum_n\frac{2-\delta_{0n}}{p_0^2-2n|q_\text{f}|eB-p_3^2-m^2}\int\mathrm{d}^4x \end{split}\label{gapb}$$ for detailed deduction of above equations one can refer to the appendix in our previous work [@shi]. [99]{} Kenji Fukushima, Phys. Rev. D **77**, 114028 (2008). P. Costa, C. A. de Sousa, M. C. Ruivo et al, Phys. Lett. B **647**, 431 (2007). C. Shi, Y.-L. Wang, Y. Jiang et al, JHEP **07**(2014)014. A-M. Zhao, Z.-F. Cui, Y. Jiang et al, Phys. Rev. D **90**, 114031 (2014). Y. Jiang, L.-J. Luo and H.-S. Zong, JHEP **66**(2011)1. S.-S. Xu, Z.-F. Cui, B. Wang, Phys. Rev. D **91**(5), 056003 (2015). W.-J. Fu, Z. Zhang and Y.-X. Liu, Phys. Rev. D **77**, 014006 (2008). D. E. Kharzeev, L. D. McLerran, and H. J. Warringa, Nucl. Phys. **A803**, 227 (2008). D. Kharzeev, K. Landsteiner, A. Schmitt, and H.-U. Yee, Lect. Notes Phys. **871**, 1 (2013). V. Skokov, A. Y. Illarionov and V. Toneev, Int. J. Mod. Phys. A **24**, 5925 (2009). V. Voronyuk, V. D. Toneev, W. Cassing et al., Phys. Rev. C **83**, 054911 (2011). R. C. Duncan and C. Thompson, Astrophys. J. **392**, L9 (1992) C. Kouveliotou et al., Nature (London) **393**, 235 (1998). V. P. Gusynin, V. A. Miransky and I. A. Shovkovy, Phys. Lett. B **349**, 477 (1995). V. P. Gusynin, V. A. Miransky and I. A. Shovkovy, Phys. Rev. D **52**(8), 4747 (1995). D. Ebert, Phys. Rev. D **61**, 025005 (1999). I. A. Shovkovy, Lect. Notes Phys. **871**, 13 (2013), also arXiv:1207.5081 \[hep-ph\]. A. Chodos and K. Everding, Phys. Rev. D **42**, 2881 (1990). J. O. Andersen, W. R. Naylor, A. Tranberg, arXiv:1411.7176 \[hep-ph\]. G. Endrödi, JHEP07(2015)173. D. Ebert, K. G. Klimenko, M. A. Vdovichenko et al., Phys. Rev. D **61**, 025005 (1999). D. Ebert and K. G. Klimenko, Nucl. Phys. **A728**, 203 (2003). T. Inagaki, D. Kimura and T. Murata, Prog. Theor. Phys. **111**, 371 (2004). D. P. Menezes, M. Benghi Pinto, S. S. Avancini et al., Phys. Rev. C **79**, 035807 (2009). J. K. Boomsma and D. Boer, Phys. Rev. D **81**, 074005 (2010). S. Fayazbakhsh and N. Sadooghi, Phys. Rev. D **82**, 045010 (2010). G. N. Ferrari, A. F. Garcia and M. B. Pinto, Phys. Rev. D **86**, 096005 (2012). E. J. Ferrer and V. de la Incera, Lect. Notes Phys. **871**, 399 (2013). T. Mandal, P. Jaikumar and S. Digal, arXiv:0912.1413 T. Mandal and P. Jaikumar, Phys. Rev. C **87**, 045208 (2013). F. Preis, A. Rebhan, and A. Schmitt, J. High Energy Phys. **03**, 033 (2011). F. Preis, A. Rebhan, and A. Schmitt,Lect. Notes Phys. **871**, 51 (2013). Pablo G. Allen and N.N. Scoccola, Phys. Rev. D **88**, 094005 (2013). U. Vogl and W. Weise, Prog. Part. Nucl. Phys. **27**, 195 (1991). S. Klevansky, Rev. Mod. Phys. **64**, 649 (1992) T. Hatsuda and T. Kunihiro, Phys. Rep. **247**, 221 (1994) M. Buballa, Phys. Rep. **407**, 205 (2005). M. Asakawa and K. Yazaki, Nucl. Phys. **A504**, 668 (1989). Yuan-ben Dai, ‘*Gauge Theory of Interactions*’, 2nd ed., Beijing: Science Press, 2005, 68-70, ISBN 7-03-014751-0. Song Shi, You-chang Yang, Yong-hui Xia et al., Phys. Rev. D **91**, 036006 (2015). T. Inagaki, D. Kimura, and T. Murata, Prog. Theor. Phys. 111, 371 (2004). Carl M. Bender and Steve A. Orszag, ‘*Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory*’, 1st ed., New York: Springer-Verlag, 1999, 78, DOI 10.1007/978-1-4757-3069-2.

--- abstract: 'We consider a nonlinear variational wave equation that models the dynamics of nematic liquid crystals. Finite difference schemes, that either conserve or dissipate a discrete version of the energy, associated with these equations, are designed. Numerical experiments, in both one and two-space dimensions, illustrating the stability and efficiency of the schemes are presented. An interesting feature of these schemes is their ability to approximate both the conservative as well as the dissipative weak solution of the underlying system.' address: - ' Institut für Mathematik, Julius-Maximilians-Universität Würzburg, Campus Hubland Nord, Emil-Fischer-Strasse 30, 97074, Würzburg, Germany.' - ' Center of Mathematics for Applications (CMA), University of Oslo, P.O.Box -1053, Blindern, Oslo-0316, Norway (and) Seminar for Applied Mathematics (SAM) Department of Mathematics, ETH Zürich, HG G 57.2, Zürich -8092, Switzerland' - 'Centre of Mathematics for Applications (CMA) University of OsloP.O. Box 1053, Blindern N–0316 Oslo, Norway' - 'Centre of Mathematics for Applications (CMA) University of OsloP.O. Box 1053, Blindern N–0316 Oslo, Norway' author: - 'U. Koley' - 'S. Mishra' - 'N. H. Risebro' - 'F. Weber' title: | Robust finite difference schemes for a\ Nonlinear variational wave equation\ modeling liquid crystals --- Introduction {#sec:intro} ============ The model --------- The dynamics of nematic liquid crystals is an important object of study in both physics and engineering. Many popular models of nematic liquid crystals consider a medium consisting of thin rods that are allowed to rotate about their center of mass but are not allowed to translate. Under the assumption that the medium is not flowing and deformations only occur when the mean orientation of long molecules is changed, one can describe the the orientation of the molecules at each location ${\bf x} \in {\mathbb{R}}^3$ and time $t \in {\mathbb{R}}$ using a field of unit vectors $$\begin{aligned} {{\mathbf{n}}}= {{\mathbf{n}}}({\bf x},t) \in \mathcal{S}^2.\end{aligned}$$ This field ${{\mathbf{n}}}$ is termed as the *director field*. Given a director field ${{\mathbf{n}}}$, the well-known Oseen-Frank potential energy density ${{\mathbf{W}}}$, associated with this field, is given by $$\begin{aligned} {{\mathbf{W}}}({{\mathbf{n}}}, \nabla {{\mathbf{n}}}) = \alpha {\left|{{\mathbf{n}}}\times (\nabla \times {{\mathbf{n}}})\right|}^2 + \beta \left(\nabla \cdot {{\mathbf{n}}}\right)^2 + \gamma \left( {{\mathbf{n}}}\cdot (\nabla \times {{\mathbf{n}}}) \right)^2. \end{aligned} \label{eq:osf}$$ The positive constants $\alpha, \beta$ and $\gamma$ are elastic constants of the liquid crystal. Note that each term on the right hand side of arises from different types of distortions. For instance, the term $\alpha {\left|{{\mathbf{n}}}\times (\nabla \times {{\mathbf{n}}})\right|}^2$ corresponds to the bending of the medium, the term $\beta \left(\nabla \cdot {{\mathbf{n}}}\right)^2$ corresponds to a type of deformation called splay, and the term $\gamma \left( {{\mathbf{n}}}\cdot (\nabla \times {{\mathbf{n}}}) \right)^2$ corresponds to the twisting of the medium. For the special case of $\alpha = \beta = \gamma$, the potential energy density reduces to $$\begin{aligned} {{\mathbf{W}}}({{\mathbf{n}}}, \nabla {{\mathbf{n}}}) = \alpha {\left|\nabla {{\mathbf{n}}}\right|}^2,\end{aligned}$$ which corresponds to the potential energy density used in harmonic maps into the sphere $\mathcal{S}^2$. The constrained elliptic system of equations for ${{\mathbf{n}}}$, derived from the potential using a variational principle, and the parabolic flow associated with it, are widely studied, see [@Beres; @coron; @ericksen] and references therein. However in the regime where inertial effects are dominant (over the viscosity), it is more natural to model the propagation of orientation waves in the director field by employing the principle of least action [@saxton] i.e, $$\begin{aligned} \frac{\delta}{\delta {{\mathbf{n}}}} \iint \left({{\mathbf{n}}}_t^2 - {{\mathbf{W}}}({{\mathbf{n}}}, \nabla {{\mathbf{n}}}) \right) \,dx\,dt = 0, \qquad {{\mathbf{n}}}\cdot {{\mathbf{n}}}=1. \end{aligned} \label{eq:lap}$$ Again in the special case of $\alpha = \beta = \gamma$, this variational principle yields the equation for harmonic wave maps from $(1+3)$-dimensional Minkowski space into the two sphere, see [@chris; @shatah; @shatah1] and references therein. ### One-dimensional planar waves Planar deformations are of great interest in the study of nematic liquid crystals. In particular, if we assume that the deformation depends on a single space variable $x$ and that the director field ${{\mathbf{n}}}$ has a special form (which means that the director field is in a plane containing the $x$ axis): $$\begin{aligned} {{\mathbf{n}}}= \cos u(x,t) \mathbf{e}_x + \sin u(x,t) \mathbf{e}_y.\end{aligned}$$ Here, the unknown $u \in {\mathbb{R}}$ measures the angle of the director field to the $x$-direction, and $\mathbf{e}_x$ and $\mathbf{e}_y$ are the coordinate vectors in the $x$ and $y$ directions, respectively. In this case, the variational principle reduces to $$\label{eq:main} \begin{cases} u_{tt} - c(u)\left( c(u) u_x\right)_x =0, \quad (x,t) \in {\Pi_T}, & \\ u(x,0) = u_0(x),\quad x \in {\mathbb{R}}, & \\ u_t(x,0) = u_1(x), \quad x \in {\mathbb{R}}. & \end{cases}$$ where $ {\Pi_T}= {\mathbb{R}}\times [0,T]$ with fixed $T>0$ , and the wave speed $c(u)$ given by $$\label{eq:wavespeed} c^2(u) = \alpha \cos^2 u + \beta \sin^2u,$$ for some positive constants $\alpha,\beta$. The form is the standard form of the nonlinear variational wave equation considered in the literature. If we consider the following *energy*: $$\label{eq:energy} \mathcal{E}(t) = \int_{{\mathbb{R}}} \left( u_t^2 + c^2(u) u_x^2 \right) \,dx,$$ a simple calculation shows that smooth solutions of the variational wave equation *conserve* this energy i.e, they satisfy $$\label{eq:econ} \frac{d \mathcal{E}(t)}{dt} \equiv 0.$$ ### Two-dimensional planar waves Similarly, if the deformation depends on two space variables $x,y$, the director field has the form: $$\begin{aligned} {{\mathbf{n}}}= \cos u(x,y,t) \mathbf{e}_x + \sin u(x,y,t) \mathbf{e}_y,\end{aligned}$$ with $u$ being the angle to the $x-y$ plane. The corresponding wave equation is given by, $$\label{eq:main_2d} \begin{cases} u_{tt} - c(u)\left( c(u) u_x\right)_x -b(u)\left( b(u) u_y\right)_y - a'(u) u_x u_y - 2 a(u) u_{xy} =0, \quad (x,y,t) \in \mathbb{Q}_T, & \\ u(x,y,0) = u_0(x,y),\quad (x,y) \in {\mathbb{R}}^2, & \\ u_t(x,y,0) = u_1(x,y), \quad (x,y) \in {\mathbb{R}}^2. & \end{cases}$$ where $ \mathbb{Q}_T = {\mathbb{R}}^2 \times [0,T]$ with $T>0$ fixed, $u: \mathbb{Q}_T \rightarrow {\mathbb{R}}$ is the unknown function and $a,b,c$ are given by $$\begin{aligned} c^2(u) & = \alpha \cos^2 u + \beta \sin^2 u, \\ b^2(u) &= \alpha \sin^2 u + \beta \cos^2 u, \\ a(u) &= \frac{\alpha - \beta}{2} \sin(2u).\end{aligned}$$ for some constants $\alpha$ and $\beta$. Furthermore, smooth solutions of also *conserve* the following energy: $$\begin{aligned} \label{eq:energy_2d} \mathcal{E}(t) &= \iint_{{\mathbb{R}}^2} \left( u_t^2 + c^2(u) u_x^2 + b^2(u) u_y^2 + 2 a(u) u_x u_y \right) \,dx \,dy \\ &= \iint_{{\mathbb{R}}^2} \left( u_t^2 + \alpha( \cos(u) u_x + \sin(u) u_y)^2 + \beta( \sin(u) u_x - \cos(u) u_y)^2 \right)\,dx\,dy, \end{aligned}$$ i.e, smooth solutions satisfy with respect to the above energy. Mathematical issues ------------------- It is well known that the solution of the initial value problem, even for the one-dimensional planar wave equation develops singularities in finite time, even if the initial data are smooth [@glassey]. Hence, solutions of are defined in the sense of distributions, i.e., Set ${\Pi_T}={\mathbb{R}}\times (0,T)$. A function $$u(t,x) \in L^{\infty}\left([0,T];W^{1,p}({\mathbb{R}})\right) \cap C({\Pi_T}), u_t \in L^{\infty}\left([0,T];L^{p}({\mathbb{R}})\right),$$ for all $p \in [1, 3+q]$, where $q$ is some positive constant, is a weak solution of the initial value problem if it satisfies: 1. For all test functions ${\varphi}\in \mathcal{D}({\mathbb{R}}\times [0,T))$ \[def:w3\] $$\label{eq:weaksol} \iint_{{\Pi_T}} \left( u_t {\varphi}_t -c^2(u) u_x {\varphi}_x - c(u) c'(u) (u_x)^2 {\varphi}\right)\,dx \,dt = 0.$$ 2. $u(\cdot, t) \rightarrow u_0$ in $C\left( [0,T];L^2({\mathbb{R}}) \right)$ as $t \rightarrow 0^{+}$. 3. $u_t(\cdot, t) \rightarrow v_0$ as a distribution in ${\Pi_T}$ when $t \rightarrow 0^{+}$. It is highly non-trivial to extend the solution after the appearance of singularities. In particular, the choice of this extension is not unique. Two distinct types of solutions, the so-called *conservative* and *dissipative* solutions are known. To illustrate this difference, one considers initial data for which the solution vanishes identically at some specific (finite) time. At this point, at least two possibilities exist: to continue with the trivial zero solution, termed as the dissipative solution. As an alternative, one can show that there exists a nontrivial solution that appears as a natural continuation of the solution prior to the critical time. This solution is denoted the conservative solution as it preserves the total energy of the system. This dichotomy makes the question of well-posedness of the initial value problem very difficult. Additional admissibility conditions are needed to select a physically relevant solution. The specification of such admissibility criteria is still open. Although the problem of global existence and uniqueness of solutions to the Cauchy problem of the nonlinear variational wave equation is still open, several recent papers have explored related questions or particular cases of . It has been demonstrated in [@ghz1996] that is rich in structural phenomena associated with weak solutions. In fact rewriting the highest derivatives of in conservative form $$\begin{aligned} u_{tt} -\left( c^2(u) u_x\right)_x = - c(u) c'(u) u_x^2,\end{aligned}$$ we see that the strong precompactness in $L^2$ of the derivatives $\lbrace u_x \rbrace$ of a sequence of approximate solutions is essential in establishing the existence of a global weak solution. However, the equation shows the phenomenon of persistence of oscillations [@diperna] and annihilation in which a sequence of exact solutions with bounded energy can oscillate forever so that the sequence $\lbrace u_x \rbrace$ is not precompact in $L^2$, but the weak limit of the sequence is still a weak solution. There has been a number of papers concerning the existence of weak solutions of the Cauchy problem , starting with the papers by Zhang and Zheng [@zhang1; @zhang2; @zhang3; @zhang4; @zhang5; @zhang6], Bressan and Zheng [@bressan] and Holden *et al.* [@holden]. In [@zhang5], the authors show existence of a global weak solution, using method of Young measures, for initial data $u_0 \in H^1({\mathbb{R}})$ and $u_1 \in L^2({\mathbb{R}})$. The function $c$ is assumed to be smooth, bounded, positive with derivative that is non-negative and strictly positive on the initial data $u_0$. This means that the analysis in [@holden; @zhang1; @zhang2; @zhang3; @zhang4; @zhang5; @zhang6] does not directly apply to . A different approach to the study of was taken by Bressan and Zheng [@bressan]. Here, they rewrote the equation in new variables such that the singularities disappeared. They show that for $u_0$ absolutely continuous with $(u_0)_x , u_1 \in L^2 ({\mathbb{R}})$, the Cauchy problem allows a global weak solution with the following properties: the solution $u$ is locally Lipschitz continuous and the map $t \rightarrow u(t,\cdot)$ is continuously differentiable with values in $L^p_{\mathrm{loc}}({\mathbb{R}})$ for $1 \le p < 2$. In [@holden], Holden and Raynaud prove the existence of a global semigroup for conservative solutions of , allowing for concentration of energy density on sets of zero measure. Furthermore they also allow for initial data $u_0, u_1$ that contain measures. The proof involves constructing the solution by introducing new variables related to the characteristics, leading to a characterization of singularities in the energy density. They also prove that energy can only focus on a set of times of zero measure or at points where $c'(u)$ vanishes. In contrast to the one-dimensional case, hardly any rigorous wellposedness or even qualitative results are available for the two-dimensional version of the variational wave equation . Numerical schemes ----------------- Given the nonlinear nature of the variational wave equations and , explicit solution formulas are hard to obtain. Consequently, robust numerical schemes for approximating the variational wave equation, are very important in the study of nematic liquid crystals. However, there is a paucity of efficient numerical schemes for these equations. Within the existing literature, we can refer to [@ghz1997], where the authors present some numerical examples to illustrate their theory. In recent years, a semi-discrete finite difference scheme for approximating one-dimensional equation was considered in [@hkr2009]. The authors were even able to prove convergence of the numerical approximation, generated by their scheme, to the *dissipative* solutions of . However, the underlying assumptions on the wave speed $c$ (positivity of the derivative of $c$) precludes consideration of realistic wave speeds given by . Another recent paper dealing with numerical approximation of is [@holden]. Here, the authors use their analytical construction to define a numerical method that can approximate the *conservative* solution. However, the method is computationally very expensive as there is no time marching. Furthermore, there are some works on the Ericksen–Leslie (EL) equations [@Badia1] (essentially the simplest set of equations describing the motion of a nematic liquid crystal). In [@Becker], authors have presented a finite element scheme for the EL equations. Their approximations are based on the ideas given in [@Bartels] which utilizes the Galerkin method with Lagrange finite elements of order $1$. Convergence, even convergence to measure-valued solutions, of such schemes is an open problem. In [@Badia2], a saddle-point formulation was used to construct finite element approximate solutions to the EL equations. A penalty method based on well-known penalty formulation for EL equations has been introduced in [@Lin] which uses the *Ginzburg–Landau* function. Convergence of such approximate solutions, based on an energy method and a compactness result, towards measure valued solutions has been proved in [@Liu]. Aims and scope of the current paper ----------------------------------- The above discussion clearly highlights the lack of robust and efficient numerical schemes to simulate the nonlinear variational wave equation . In particular, one needs a scheme that is both efficient, simple to implement and is able to approximate the solutions of accurately. Furthermore, one can expect both *conservative* as well as *dissipative* solutions of the variational wave equation , after singularity formation. Hence, it is essential to design schemes that approximate these different types of solutions. To this end, we will construct robust finite difference schemes for approximating the variational wave equation in both one and two space dimensions. The key design principle will be energy conservation (dissipation). As pointed out before, smooth solutions of and are energy conservative. After singularity formation, either this energy is conserved or dissipated. We will design numerical schemes that imitate this energy principle. In other words, our schemes will either conserve a discrete form of the energy or dissipative it. Hence, we construct both energy conservative schemes as well as energy dissipative schemes for the variational wave equation, in both one and two space dimensions. Extensive numerical experiments are presented to illustrate that the energy conservative (dissipative) schemes converge to the conservative (dissipative) solution of the variational wave equation as the mesh is refined. To the best of our knowledge, these are the first finite difference schemes that can approximate the conservative solutions of the one-dimensional variational wave equation. Furthermore, we present the first set of numerical schemes to approximate the two-dimensional version of these equations. Our energy conservative (dissipative) schemes are based on either rewriting the wave equation as a first-order system of equations or using a Hamiltonian formulation of our system. The rest of the paper is organized as follows: In section \[sec:num\], we present energy conservative and energy dissipative schemes for the one-dimensional equation . Numerical experiments illustrating these schemes are presented in section \[sec:numex\]. The two-dimensional schemes are presented in section \[sec:2d\]. Numerical schemes for the one-dimensional variational wave equation {#sec:num} ==================================================================== The grid and notation --------------------- We begin by introducing some notation needed to define the finite difference schemes. Throughout this paper, we reserve ${\Delta x}$ and ${\Delta t}$ to denote two small positive numbers that represent the spatial and temporal discretization parameters, respectively, of the numerical schemes. For $j \in {{\mathbb{N}}}_{0} = {{\mathbb{N}}}\cup \lbrace 0 \rbrace$, we set $x_j = j{\Delta x}$, and for $n = 0, 1, . . .,N$, where $N {\Delta t}= T$ for some fixed time horizon $T > 0$, we set $t_n = n{\Delta t}$. For any function $g = g(x)$ admitting point values we write $g_j = g(x_j )$, and similarly for any function $h = h(x, t)$ admitting point values we write $h^n_j= h(x_j, t_n)$. We also introduce the spatial and temporal grid cells $$\begin{aligned} I_j = [x_{j -\frac{1}{2}}, x_{j+\frac{1}{2}}), \qquad I^n_j = I_j \times [t_n, t_{n+1}).\end{aligned}$$ Furthermore we introduce the jump, and respectively, the average of any grid function $\rho$ across the interface $x_{j + \frac{1}{2}}$ $$\begin{aligned} \overline{\rho}_{j+\frac{1}{2}} &:= \frac{\rho_j + \rho_{j+1}}{2}, \\ {\llbracket \rho \rrbracket}_{j + \frac{1}{2}} & := \rho_{j+1} - \rho_j.\end{aligned}$$ The following identities are readily verified: $$\begin{aligned} \label{eq:useful} { \llbracket u v \rrbracket}_{j + \frac{1}{2}} & = \overline{u}_{j+\frac{1}{2}} {\llbracket v \rrbracket}_{j + \frac{1}{2}} + { \llbracket u \rrbracket}_{j + \frac{1}{2}} \overline{v}_{j+\frac{1}{2}},\\ v_j &= \overline{v}_{j \pm \frac{1}{2}} \mp \frac{1}{2} {\llbracket v \rrbracket}_{j \pm \frac{1}{2}}. \end{aligned}$$ A first-order system for ------------------------- It is easy to check that the variational wave equation can be rewritten as a first-order system by introducing the independent variables: $$\begin{aligned} v &:= u_t \\ w &:= c(u) u_x.\end{aligned}$$ Then, for smooth solutions, equation is equivalent to the following system for $(v,w,u)$, $$\begin{cases} v_t - (c(u) w)_x = - c_x(u) w & \\ w_t - (c(u) v)_x = 0, & \\ u_t = v. & \end{cases} \label{eq:main3}$$ Furthermore, the energy associated with the above equation is $$\label{eq:energy1} \mathcal{E}(t) = \int_{{\mathbb{R}}} \left( v^2 + w^2 \right) dx.$$ Again, we can check that smooth solutions of preserve this energy. Weak solutions can be either energy conservative or energy dissipative. Energy Preserving Scheme Based On System $\eqref{eq:main3}$ ----------------------------------------------------------- Our objective is to design a (semi-discrete) finite difference scheme such that the numerical approximations conserve a discrete version of the energy . To this end, we suggest the following finite difference scheme: $$\begin{aligned} (v_j)_t - \frac{1}{{\Delta x}} \left( \overline{c}_{j+\frac{1}{2}} \overline{w}_{j+\frac{1}{2}} - \overline{c}_{j-\frac{1}{2}} \overline{w}_{j-\frac{1}{2}}\right) &= -\frac{1}{2 {\Delta x}} \left( {\llbracket c \rrbracket}_{j + \frac{1}{2}} \overline{w}_{j+\frac{1}{2}} + {{\llbracket c \rrbracket}}_{j - \frac{1}{2}} \overline{w}_{j-\frac{1}{2}} \right), \\ (w_j)_t - \frac{1}{{\Delta x}} \left( \overline{cv}_{j+\frac{1}{2}} - \overline{cv}_{j-\frac{1}{2}} \right) &=0,\\ (u_j)_t &=v_j. \end{aligned} \label{eq:scheme1}$$ The energy conservative property of this semi-discrete scheme is presented in the following theorem: \[theo:cons\] Let $v_j(t)$ and $w_j(t)$ be approximate solutions generated by the scheme . Then $$\begin{aligned} \frac{d}{dt} \left( \frac{{\Delta x}}{2} \sum_{j} (v_j^2(t) + w_j^2(t)) \right) =0. \end{aligned}$$ We start by multiplying the first equation of by $v_j$ and second equation by $w_j$ respectively. Using relations , we have $$\begin{aligned} \frac{1}{2} &\frac{d}{dt} (v_j^2 + w_j^2) \\ & = \frac{1}{{\Delta x}} \overline{v}_{j + \frac{1}{2}} \overline{c}_{j + \frac{1}{2}} \overline{w}_{j + \frac{1}{2}} - \frac{1}{2{\Delta x}} {\llbracket v \rrbracket}_{j+\frac{1}{2}} \overline{c}_{j + \frac{1}{2}} \overline{w}_{j + \frac{1}{2}} \\ & \qquad - \frac{1}{{\Delta x}} \overline{v}_{j - \frac{1}{2}} \overline{c}_{j - \frac{1}{2}} \overline{w}_{j - \frac{1}{2}} - \frac{1}{2{\Delta x}} {\llbracket v \rrbracket}_{j-\frac{1}{2}} \overline{c}_{j - \frac{1}{2}} \overline{w}_{j - \frac{1}{2}}\\ & \qquad \qquad - \frac{1}{2{\Delta x}} \overline{v}_{j+\frac{1}{2}} {\llbracket c \rrbracket}_{j + \frac{1}{2}} \overline{w}_{j + \frac{1}{2}} + \frac{1}{4{\Delta x}} {\llbracket v \rrbracket}_{j+\frac{1}{2}} {\llbracket c \rrbracket}_{j + \frac{1}{2}} \overline{w}_{j + \frac{1}{2}} \\ & \qquad \qquad \qquad - \frac{1}{2{\Delta x}} \overline{v}_{j-\frac{1}{2}} {\llbracket c \rrbracket}_{j - \frac{1}{2}} \overline{w}_{j - \frac{1}{2}} - \frac{1}{4{\Delta x}} {\llbracket v \rrbracket}_{j-\frac{1}{2}} {\llbracket c \rrbracket}_{j - \frac{1}{2}} \overline{w}_{j - \frac{1}{2}} \\ & + \frac{1}{{\Delta x}} \overline{w}_{j + \frac{1}{2}} \overline{cv}_{j+\frac{1}{2}} - \frac{1}{2{\Delta x}} {\llbracket w \rrbracket}_{j + \frac{1}{2}} \overline{cv}_{j+\frac{1}{2}} - \frac{1}{{\Delta x}} \overline{w}_{j - \frac{1}{2}} \overline{cv}_{j-\frac{1}{2}} - \frac{1}{2{\Delta x}} {\llbracket w \rrbracket}_{j - \frac{1}{2}} \overline{cv}_{j-\frac{1}{2}}, \end{aligned}$$ which implies $$\label{eq:dis1} \frac{1}{2} \frac{d}{dt} (v_j^2 + w_j^2) = \frac{1}{{\Delta x}}\left( H_{j +\frac{1}{2}} - H_{j -\frac{1}{2}} \right) - \frac{1}{2{\Delta x}} \left( {\llbracket cvw \rrbracket}_{j +\frac{1}{2}} + {\llbracket cvw \rrbracket}_{j -\frac{1}{2}} \right),$$ where $$\begin{aligned} H_{j +\frac{1}{2}}= \overline{v}_{j + \frac{1}{2}} \overline{c}_{j + \frac{1}{2}} \overline{w}_{j + \frac{1}{2}} + \frac{1}{4} {\llbracket v \rrbracket}_{j+\frac{1}{2}} {\llbracket c \rrbracket}_{j + \frac{1}{2}} \overline{w}_{j + \frac{1}{2}} +\overline{w}_{j + \frac{1}{2}} \overline{cv}_{j+\frac{1}{2}}. \end{aligned}$$ Next, multiplying by ${\Delta x}$ and summing over $j$ gives $$\begin{aligned} \frac{d}{dt} \left(\frac{{\Delta x}}{2} \sum_{j} (v_j^2 + w_j^2)\right) & = \sum_{j} \left( H_{j +\frac{1}{2}} - H_{j -\frac{1}{2}} \right) - \frac{1}{2} \sum_{j} \left( {\llbracket cvw \rrbracket}_{j +\frac{1}{2}} + {\llbracket cvw \rrbracket}_{j -\frac{1}{2}} \right) \\ & =0. \end{aligned}$$ This proves the theorem. Energy dissipating Scheme Based On System $\eqref{eq:main3}$ ------------------------------------------------------------ We expect the above designed energy conservative scheme to approximate a conservative solution of the underlying system . In order to be able to approximate a dissipative solution of , we propose the following modification of the energy conservative scheme : $$\begin{aligned} & (v_j)_t - \frac{1}{{\Delta x}} \left( \overline{c}_{j+\frac{1}{2}} \overline{w}_{j+\frac{1}{2}} - \overline{c}_{j-\frac{1}{2}} \overline{w}_{j-\frac{1}{2}}\right) \\ & \qquad = -\frac{1}{2 {\Delta x}} \left( {\llbracket c \rrbracket}_{j + \frac{1}{2}} \overline{w}_{j+\frac{1}{2}} + {\llbracket c \rrbracket}_{j - \frac{1}{2}} \overline{w}_{j-\frac{1}{2}} \right) + \frac{1}{2 {\Delta x}} \left(s_{j+\frac{1}{2}} {\llbracket v \rrbracket}_{j+\frac{1}{2}} -s_{j-\frac{1}{2}} {\llbracket v \rrbracket}_{j-\frac{1}{2}} \right)\\ & (w_j)_t - \frac{1}{{\Delta x}} \left( \overline{cv}_{j+\frac{1}{2}} - \overline{cv}_{j-\frac{1}{2}} \right) = \frac{1}{2 {\Delta x}} \left(s_{j+\frac{1}{2}} {\llbracket w \rrbracket}_{j+\frac{1}{2}} -s_{j-\frac{1}{2}} {\llbracket w \rrbracket}_{j-\frac{1}{2}} \right),\\ & (u_j)_t = v_j, \end{aligned} \label{eq:scheme2}$$ where we have chosen $s_{j\pm\frac{1}{2}}=\max\{c_j,c_{j\pm 1}\}$ i.e, the maximum local wave speed. We show that the above scheme dissipates energy in the following theorem: \[theo:diss\] Let $v_j(t)$ and $w_j(t)$ be approximate solutions generated by the scheme . Then one can prove that $$\begin{aligned} \frac{d}{dt} \left( \frac{{\Delta x}}{2} \sum_{j} (v_j^2(t) + w_j^2(t)) \right) \le 0, \end{aligned}$$ with strict inequality if $w$ or $v$ is not constant. First note that $s_{j\pm\frac{1}{2}}$ are positive for all $j$, since $c_j>0$ for all $j$. Emulating the calculations of previous theorem, i.e., first multiplying the first equation of by ${\Delta x}v_j$ and second equation by ${\Delta x}w_j$ respectively, and summing over all $j$ we have $$\begin{aligned} \frac{d}{dt} & \left(\frac{{\Delta x}}{2} \sum_{j} (v_j^2 + w_j^2)\right) \\ & = \sum_{j} \left( H_{j +\frac{1}{2}} - H_{j -\frac{1}{2}} \right) - \frac{1}{2} \sum_{j} \left( {\llbracket cvw \rrbracket}_{j +\frac{1}{2}} + {\llbracket cvw \rrbracket}_{j -\frac{1}{2}} \right) \\ & + \frac{1}{2} \sum_{j} s_{j+\frac{1}{2}} \overline{v}_{j + \frac{1}{2}} {\llbracket v \rrbracket}_{j+ \frac{1}{2}} - \frac{1}{4} \sum_{j} s_{j+\frac{1}{2}} {\llbracket v \rrbracket}^2_{j+ \frac{1}{2}} - \frac{1}{2} \sum_{j} s_{j-\frac{1}{2}} \overline{v}_{j - \frac{1}{2}} {\llbracket v \rrbracket}_{j- \frac{1}{2}} \\ & \qquad - \frac{1}{4} \sum_{j} s_{j-\frac{1}{2}} {\llbracket v \rrbracket}^2_{j- \frac{1}{2}} + \frac{1}{2} \sum_{j} s_{j+\frac{1}{2}} \overline{w}_{j + \frac{1}{2}} {\llbracket w \rrbracket}_{j+ \frac{1}{2}} - \frac{1}{4} \sum_{j} s_{j+\frac{1}{2}} {\llbracket w \rrbracket}^2_{j+ \frac{1}{2}} \\ & \qquad \qquad - \frac{1}{2} \sum_{j} s_{j-\frac{1}{2}} \overline{w}_{j - \frac{1}{2}} {\llbracket w \rrbracket}_{j- \frac{1}{2}} - \frac{1}{4} \sum_{j} s_{j-\frac{1}{2}} {\llbracket w \rrbracket}^2_{j- \frac{1}{2}} \end{aligned}$$ Next, define $$\begin{aligned} K_{j+\frac{1}{2}} & = H_{j + \frac{1}{2}} + \frac{1}{2} s_{j+\frac{1}{2}} \overline{v}_{j + \frac{1}{2}} {\llbracket v \rrbracket}_{j+ \frac{1}{2}} + \frac{1}{2} s_{j+\frac{1}{2}} \overline{w}_{j + \frac{1}{2}} {\llbracket w \rrbracket}_{j+ \frac{1}{2}} \\ M_{j+\frac{1}{2}} & = \frac{1}{4} s_{j+\frac{1}{2}} \left( {\llbracket v \rrbracket}^2_{j+ \frac{1}{2}} + {\llbracket w \rrbracket}^2_{j+ \frac{1}{2}} \right). \end{aligned}$$ Then we have $$\begin{aligned} \frac{d}{dt} \left(\frac{{\Delta x}}{2} \sum_{j} (v_j^2 + w_j^2)\right) & = \sum_{j} \left( K_{j +\frac{1}{2}} - K_{j -\frac{1}{2}} \right) - \frac{1}{2} \sum_{j} \left( {\llbracket cvw \rrbracket}_{j +\frac{1}{2}} + {\llbracket cvw \rrbracket}_{j -\frac{1}{2}} \right) \\ & \qquad \qquad - \sum_{j} \left(M_{j+\frac{1}{2}} + M_{j-\frac{1}{2}} \right) \le 0. \end{aligned}$$ This proves the theorem. Hence, the scheme is energy stable (dissipating) and we expect it to converge to a dissipative solution of as the mesh is refined. We remark that energy dissipation results by adding *numerical viscosity* (scaled by the maximum wave speed) to the energy conservative scheme . A first-order system for based on Riemann invariants ---------------------------------------------------- We can also rewrite the one-dimensional variational wave equation as a first-order system of equations by introducing the Riemann invariants: $$\begin{aligned} R :&= u_t + c(u) u_x \\ S :&= u_t - c(u) u_x.\end{aligned}$$ Again, for smooth solutions, equation is equivalent to the following system in non-conservative form for $(R,S,u)$, $$\begin{cases} R_t - c(u) R_x = \frac{c'(u)}{4 c(u)} \left( R^2 -S^2 \right), & \\ S_t + c(u) S_x = -\frac{c'(u)}{4 c(u)} \left( R^2 -S^2 \right), & \\ u_t = \frac{R+S}{2}. & \end{cases} \label{eq:main1}$$ Observe that one can also rewrite the equation in conservative form for $(R,S,u)$, $$\begin{cases} R_t - (c(u) R)_x = -\frac{c_x(u)}{2} \left( R -S \right), & \\ S_t + (c(u) S)_x = -\frac{c_x(u)}{2} \left( R -S \right), & \\ u_t = \frac{R+S}{2}. & \end{cases} \label{eq:main2}$$ The corresponding energy associated with the system is $$\label{eq:energy2} \mathcal{E}(t) = \frac{1}{2}\int_{{\mathbb{R}}} \left( R^2 + S^2 \right) \,dx.$$ A simple calculation shows that smooth solutions of satisfy the energy identity: $$\label{eq:egypreserve2} (R^2 + S^2)_t - \left( c(u) (R^2 -S^2) \right)_x =0.$$ Hence, the fact that the total energy is conserved follows from integrating the above identity in space and assuming that the functions $R,S$ decay at infinity. Energy Preserving Scheme Based On System $\eqref{eq:main2}$ ----------------------------------------------------------- We also propose the following energy conservative scheme based on the Riemann invariant system : to $$\begin{aligned} & (R_j)_t - \frac{1}{{\Delta x}} \left( \overline{c}_{j+\frac{1}{2}} \overline{R}_{j+\frac{1}{2}} - \overline{c}_{j-\frac{1}{2}} \overline{R}_{j-\frac{1}{2}}\right) \\ & \qquad = -\frac{R_j}{4 {\Delta x}} \left( {\llbracket c \rrbracket}_{j + \frac{1}{2}} + {\llbracket c \rrbracket}_{j - \frac{1}{2}} \right) + \frac{S_j}{4 {\Delta x}} \left( {\llbracket c \rrbracket}_{j + \frac{1}{2}} + {\llbracket c \rrbracket}_{j - \frac{1}{2}} \right) \\ & (S_j)_t + \frac{1}{{\Delta x}} \left( \overline{c}_{j+\frac{1}{2}} \overline{S}_{j+\frac{1}{2}} - \overline{c}_{j-\frac{1}{2}} \overline{S}_{j-\frac{1}{2}}\right) \\ & \qquad = -\frac{R_j}{4 {\Delta x}} \left( {\llbracket c \rrbracket}_{j + \frac{1}{2}} + {\llbracket c \rrbracket}_{j - \frac{1}{2}} \right) + \frac{S_j}{4 {\Delta x}} \left( {\llbracket c \rrbracket}_{j + \frac{1}{2}} + {\llbracket c \rrbracket}_{j - \frac{1}{2}} \right), \\ & (u_j)_t = \frac{R_j + S_j}{2}. \end{aligned} \label{eq:scheme3}$$ We have the following theorem for the scheme: Let $R_j(t)$ and $S_j(t)$ be approximate solutions generated by the scheme . Then $$\begin{aligned} \frac{d}{dt} \left( \frac{{\Delta x}}{2} \sum_{j} (R_j^2(t) + S_j^2(t)) \right) =0. \end{aligned}$$ Proof of this theorem is very similar to Theorem  \[theo:cons\]. We multiply $R_j$ to the first equation of , $S_j$ to the second equation of and add resulting equations. This yields, $$\begin{aligned} \frac{1}{2} &\frac{d}{dt} (R_j^2 + S_j^2) \\ & = \frac{1}{{\Delta x}} \overline{R}_{j + \frac{1}{2}} \overline{c}_{j + \frac{1}{2}} \overline{R}_{j + \frac{1}{2}} - \frac{1}{2{\Delta x}} {\llbracket R \rrbracket}_{j+\frac{1}{2}} \overline{c}_{j + \frac{1}{2}} \overline{R}_{j + \frac{1}{2}} \\ & \qquad - \frac{1}{{\Delta x}} \overline{R}_{j - \frac{1}{2}} \overline{c}_{j - \frac{1}{2}} \overline{R}_{j - \frac{1}{2}} - \frac{1}{2{\Delta x}} {\llbracket R \rrbracket}_{j-\frac{1}{2}} \overline{c}_{j - \frac{1}{2}} \overline{R}_{j - \frac{1}{2}} \\ & \qquad \qquad - \frac{R^2_j}{4 {\Delta x}} \left( {\llbracket c \rrbracket}_{j + \frac{1}{2}} + {\llbracket c \rrbracket}_{j - \frac{1}{2}} \right) + \frac{R_jS_j}{4 {\Delta x}} \left( {\llbracket c \rrbracket}_{j + \frac{1}{2}} + {\llbracket c \rrbracket}_{j - \frac{1}{2}} \right) \\ & \qquad \qquad \qquad - \frac{1}{{\Delta x}} \overline{S}_{j + \frac{1}{2}} \overline{c}_{j + \frac{1}{2}} \overline{S}_{j + \frac{1}{2}} + \frac{1}{2{\Delta x}} {\llbracket S \rrbracket}_{j+\frac{1}{2}} \overline{c}_{j + \frac{1}{2}} \overline{S}_{j + \frac{1}{2}} \\ & \qquad \qquad \qquad \qquad + \frac{1}{{\Delta x}} \overline{S}_{j - \frac{1}{2}} \overline{c}_{j - \frac{1}{2}} \overline{S}_{j - \frac{1}{2}} - \frac{1}{2{\Delta x}} {\llbracket S \rrbracket}_{j-\frac{1}{2}} \overline{c}_{j - \frac{1}{2}} \overline{S}_{j - \frac{1}{2}} \\ & \qquad \qquad \qquad \qquad \qquad + \frac{S^2_j}{4 {\Delta x}} \left( {\llbracket c \rrbracket}_{j + \frac{1}{2}} + {\llbracket c \rrbracket}_{j - \frac{1}{2}} \right) - \frac{R_jS_j}{4 {\Delta x}} \left( {\llbracket c \rrbracket}_{j + \frac{1}{2}} + {\llbracket c \rrbracket}_{j - \frac{1}{2}} \right) \end{aligned}$$ Next, multiplying by ${\Delta x}$ and summing over all $j$ gives $$\begin{aligned} \frac{d}{dt} \left(\frac{{\Delta x}}{2} \sum_{j} (R_j^2 + S_j^2)\right) & = \sum_{j} \left( F_{j +\frac{1}{2}} - F_{j -\frac{1}{2}} \right) - \frac{1}{2} \sum_{j} \left( {\llbracket c \frac{R^2}{2} \rrbracket}_{j +\frac{1}{2}} + {\llbracket c \frac{R^2}{2} \rrbracket}_{j -\frac{1}{2}} \right) \\ & \qquad \qquad + \frac{1}{2} \sum_{j} \left( {\llbracket c \frac{S^2}{2} \rrbracket}_{j +\frac{1}{2}} + {\llbracket c \frac{S^2}{2} \rrbracket}_{j -\frac{1}{2}} \right) = 0, \end{aligned}$$ where $$\begin{aligned} F_{j +\frac{1}{2}}= \overline{R}_{j + \frac{1}{2}} \overline{c}_{j + \frac{1}{2}} \overline{R}_{j + \frac{1}{2}} - \overline{R}_{j + \frac{1}{2}} \overline{c}_{j + \frac{1}{2}} \overline{R}_{j + \frac{1}{2}}. \end{aligned}$$ This proves the theorem. Energy Dissipating Scheme Based On System $\eqref{eq:main2}$ ------------------------------------------------------------ In-order to approximate dissipative solutions, we add some numerical viscosity to the energy conservative scheme to obtain, $$\begin{aligned} & (R_j)_t - \frac{1}{{\Delta x}} \left( \overline{c}_{j+\frac{1}{2}} \overline{R}_{j+\frac{1}{2}} - \overline{c}_{j-\frac{1}{2}} \overline{R}_{j-\frac{1}{2}}\right) \\ & \qquad = -\frac{R_j}{4 {\Delta x}} \left( {\llbracket c \rrbracket}_{j + \frac{1}{2}} + {\llbracket c \rrbracket}_{j - \frac{1}{2}} \right)\\ & \qquad + \frac{S_j}{4 {\Delta x}} \left( {\llbracket c \rrbracket}_{j + \frac{1}{2}} + {\llbracket c \rrbracket}_{j - \frac{1}{2}} \right) + \frac{1}{2 {\Delta x}} \left( s_{j+\frac{1}{2}} {\llbracket R \rrbracket}_{j+\frac{1}{2}} - s_{j-\frac{1}{2}} {\llbracket R \rrbracket}_{j-\frac{1}{2}} \right), \\ & (S_j)_t + \frac{1}{{\Delta x}} \left( \overline{c}_{j+\frac{1}{2}} \overline{S}_{j+\frac{1}{2}} - \overline{c}_{j-\frac{1}{2}} \overline{S}_{j-\frac{1}{2}}\right) \\ & \qquad = -\frac{R_j}{4 {\Delta x}} \left( {\llbracket c \rrbracket}_{j + \frac{1}{2}} + {\llbracket c \rrbracket}_{j - \frac{1}{2}} \right) \\ & \qquad + \frac{S_j}{4 {\Delta x}} \left( {\llbracket c \rrbracket}_{j + \frac{1}{2}} + {\llbracket c \rrbracket}_{j - \frac{1}{2}} \right)+ \frac{1}{2 {\Delta x}} \left(s_{j+\frac{1}{2}} {\llbracket S \rrbracket}_{j+\frac{1}{2}} -s_{j-\frac{1}{2}} {\llbracket S \rrbracket}_{j-\frac{1}{2}} \right),\\ & (u_j)_t = \frac{R_j + S_j}{2}. \end{aligned} \label{eq:scheme4}$$ where we have chosen $s_{j\pm\frac{1}{2}}=\max\{c_j,c_{j\pm 1}\}$, i.e, the maximum local wave speed. We have the following theorem illustrating the energy dissipation associated with Let $R_j(t)$ and $S_j(t)$ be approximate solutions generated by the scheme . Then, $$\begin{aligned} \frac{d}{dt} \left( \frac{{\Delta x}}{2} \sum_{j} (R_j^2(t) + S_j^2(t)) \right) \le 0, \end{aligned}$$ where the inequality is strict if $R$ or $S$ is not constant. The proof of this is similar to the proof of Theorem \[theo:diss\] and is therefore omitted. Energy Preserving Scheme Based On a Variational Formulation ----------------------------------------------------------- All the above schemes were designed by rewriting the variational wave equation as first-order systems and approximating these systems. However, one also design an energy conservative scheme by approximating the nonlinear wave equation directly. To this end. we write the nonlinear wave equation in the general form: $$\label{eq:vari} u_{tt} = - \frac{\delta H}{\delta u},$$ with $H = H(u, u_x):= \frac{1}{2} c^2(u) u_x^2$ being a part of the “Hamiltonian”, and $\frac{\delta H}{\delta u}$ being the variational derivative of function $H(u, u_x)$ with respect to $u$. In general, it is easy to show that for $$\label{eq:energycont} \frac{d}{dt} \int_{{\mathbb{R}}} \left( \frac{1}{2} u_t^2 + H(u, u_x) \right) \,dx =0.$$ In fact, this is a direct consequence of the fact that $$\label{eq:varderivative} \frac{\delta H}{\delta u} = \frac{\partial H}{\partial u} - \frac{d}{dx}\left(\frac{\partial H}{\partial u_x}\right).$$ We also note that for equation , $$\begin{aligned} \frac{\delta H}{\delta u} = c(u) c'(u) u_x^2 - \left( c^2(u) u_x\right)_x =- c^2(u) u_{xx} - c(u) c'(u) u_x^2 = - c(u) \left( c(u) u_x \right)_x.\end{aligned}$$ Based on above observations, we propose the following scheme for $$(u_j)_{tt} + c(u_j) c'(u_j) (D^{0} u_j)^2 - D^{0} \left(c^2(u_j) D^{0} u_j \right) = 0, \label{eq:scheme9}$$ where the central difference $D^0$ is defined by $$D^0 z_j = \frac{z_{j+1}-z_{j-1}}{2{\Delta x}}.$$ This scheme is energy preserving as shown in the following theorem: Let $u_j(t)$ be approximate solution generated by the scheme . Then we have $$\begin{aligned} \frac{d}{dt} \left( \frac{{\Delta x}}{2} \sum_{j} ((u_j)_t^2 + c^2(u_j) \left( D^{0} u_j \right)^2) \right) = 0. \end{aligned}$$ We start by calculating $$\begin{aligned} \frac{d}{dt} &\left( \frac{{\Delta x}}{2} \sum_{j} ((u_j)_t^2 + c^2(u_j) \left( D^{0} u_j \right)^2) \right) \\ &= {\Delta x}\sum_{j} \left( (u_j)_t (u_j)_{tt} + c(u_j) c'(u_j) \left( D^{0} u_j \right)^2 (u_j)_t + c^2(u_j) D^{0} u_j D^{0} (u_j)_t \right) \\ & = {\Delta x}\sum_{j} \left( (u_j)_t (u_j)_{tt} + c(u_j) c'(u_j) \left( D^{0} u_j \right)^2 (u_j)_t - D^{0} \left(c^2(u_j) D^{0} u_j\right) (u_j)_t \right) \\ & = 0. \end{aligned}$$ Numerical experiments {#sec:numex} ===================== In this section, we will test the numerical schemes developed in the previous section on several examples. The semi-discrete schemes , , and are integrated in time using a third order SSP-Runge Kutta method, see [@gottliebetal]. The timestep $\Delta t$ is chosen such that it satisfies the CFL-condition $$\label{eq:cfl} {\Delta t}=\theta \frac{{\Delta x}}{\sup_j c(u_j)}$$ for some $0\leq \theta\leq 0.5$. We denote by $N$ the number of gridpoints in the spatial dimension. Gaussian pulse {#ssec:gauss} -------------- As a first test problem we consider with the initial data $$\begin{aligned} \label{eq:init1} \begin{split} u_0(x)&=\frac{\pi}{4}+\exp(-x^2),\\ u_1(x)&=-c(u_0(x))\, (u_0)_x(x),\\ \end{split}\end{aligned}$$ on the domain $D=[-15,15]$ with periodic boundary conditions and the function $c(u)$ given by $$\label{eq:c} c(u)=\sqrt{\alpha \cos^2(u)+ \beta \sin^2(u)},$$ where $\alpha,\beta$ are positive constants. For this experiment, we choose $\alpha=0.5$ and $\beta=4.5$. This Cauchy problem has already been numerically investigated in [@ghz1997] and [@hkr2009]. We compute approximations by the schemes at times, $T=1,10$. --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ![Approximations of $u$ in , , computed by scheme and scheme on a grid with $N_x=15 \cdot 2^{12}$ points at time $T=1,10$ and with CFL-number $\theta=0.05$. Left: Approximation by energy conservative scheme , Right: Approximation by energy dissipative scheme .[]{data-label="fig:gauss_vw"}](ucons_eg1_gauss "fig:"){width="50.00000%"} ![Approximations of $u$ in , , computed by scheme and scheme on a grid with $N_x=15 \cdot 2^{12}$ points at time $T=1,10$ and with CFL-number $\theta=0.05$. Left: Approximation by energy conservative scheme , Right: Approximation by energy dissipative scheme .[]{data-label="fig:gauss_vw"}](udiff_eg1_gauss "fig:"){width="50.00000%"} --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- In Figure \[fig:gauss\_vw\] the approximations computed by scheme and with CFL-number $\theta=0.05$ at times $T=1,10$ are shown. We observe that at time $T=1$ the approximated solution appears smooth whereas at time $T=10$, we observe kinks in the solution indicating that singularities have appeared by this time. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ![Approximations of the quantity $v$ in , , computed by scheme and scheme on a grid with $N_x=15 \cdot 2^{12}$ points at time $T=1,10$ and with CFL-number $\theta=0.05$. Left: Approximation by scheme , Right: Approximation by scheme .[]{data-label="fig:gauss_vwv"}](vcons_eg1_gauss "fig:"){width="50.00000%"} ![Approximations of the quantity $v$ in , , computed by scheme and scheme on a grid with $N_x=15 \cdot 2^{12}$ points at time $T=1,10$ and with CFL-number $\theta=0.05$. Left: Approximation by scheme , Right: Approximation by scheme .[]{data-label="fig:gauss_vwv"}](vdiff_eg1_gauss "fig:"){width="50.00000%"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ![Approximations of the quantity $w$ in , , computed by scheme and scheme on a grid with $N_x=15 \cdot 2^{12}$ points at time $T=1,10$ and with CFL-number $\theta=0.05$. Left: Approximation by scheme , Right: Approximation by scheme .[]{data-label="fig:gauss_vww"}](wcons_eg1_gauss "fig:"){width="50.00000%"} ![Approximations of the quantity $w$ in , , computed by scheme and scheme on a grid with $N_x=15 \cdot 2^{12}$ points at time $T=1,10$ and with CFL-number $\theta=0.05$. Left: Approximation by scheme , Right: Approximation by scheme .[]{data-label="fig:gauss_vww"}](wdiff_eg1_gauss "fig:"){width="50.00000%"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- Since the schemes and , are based on the first-order system , we plot the quantities $v$ and $w$ in Figures \[fig:gauss\_vwv\] and \[fig:gauss\_vww\], and observe high frequency oscillations in the approximations computed by the energy-conservative scheme . This is not unexpected as there is no numerical viscosity in this approximations and the high frequency oscillations are a manifestation of this effect. Furthermore, at time $T=1$, the two approximations computed by and look alike, whereas we observe visible differences at time $T=10$. Similarly, the approximations computed with schemes and are qualitatively very close to those computed by the energy conservative scheme , whereas the approximations computed with resemble those computed with . Furthermore, as predicted by our analysis and shown in Figure \[fig:energy1\], the conservative schemes almost preserve the discrete $L^2$-energy over time whereas the approximations computed by the dissipative schemes lose energy by a significant amount. Note the difference in scales (in the Y-axis) between the adjacent plots in Figure\[fig:energy1\]. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -- ![Evolution of discrete energies over time, Left: Energies for the conservative schemes , , , Right: Energies for the diffusive schemes , .[]{data-label="fig:energy1"}](energy_conservative_eg1_gauss "fig:"){width="50.00000%"} ![Evolution of discrete energies over time, Left: Energies for the conservative schemes , , , Right: Energies for the diffusive schemes , .[]{data-label="fig:energy1"}](energy_diffusive_eg1_gauss "fig:"){width="50.00000%"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -- To investigate the possibility of different limit solutions approximated by the conservative and dissipative schemes, we compute reference approximations by schemes and at times $T=1,10$, with CFL-number $\theta=0.05$ on a grid with cell size ${\Delta x}=2^{-11}$ (i.e., the number of grid points is $N_x=30\cdot 2^{11}$) and test the convergence of the schemes towards these reference solutions. We measure the distance to the reference solutions in the following discrete relative $L^2$-distance, $$\label{eq:reldist} d^2(a,b):=200\times \frac{\bigl( \sum_j (a_j-b_j)^2\bigr)^{1/2}}{\bigl( \sum_j (a_j)^2\bigr)^{1/2}+\bigl( \sum_j (b_j)^2\bigr)^{1/2}}$$ for vectors $\mathbf{a}^{N}=(\dots,a^{N}_{j-1},a^{N}_{j},a^{N}_{j+1},\dots)$, $\mathbf{b}^{N}=(\dots,b^{N}_{j-1},b^{N}_{j},b^{N}_{j+1},\dots)$. The distances to the conservative and dissipative reference solution respectively, are shown in Tables \[tab:convT1cons\], \[tab:convT1diff\], \[tab:convT10cons\] and \[tab:convT10diff\]. $\Delta x$ ------------ -------- -------- -------- -------- -------- $2^{-2}$ 3.2367 8.4373 3.2710 8.5292 2.8605 $2^{-3}$ 1.1274 4.9475 1.1314 4.9974 1.0724 $2^{-4}$ 0.4723 2.7432 0.4727 2.7693 0.4644 $2^{-5}$ 0.2217 1.4592 0.2217 1.4726 0.2206 $2^{-6}$ 0.1088 0.7557 0.1088 0.7625 0.1087 $2^{-7}$ 0.0541 0.3852 0.0541 0.3886 0.0541 $2^{-8}$ 0.0269 0.1946 0.0269 0.1963 0.0269 $2^{-9}$ 0.0131 0.0978 0.0131 0.0986 0.0131 : $d^2(\mathbf{u}_{{\Delta x}}^N,\mathbf{u}_{\mathrm{ref}}^N)$ for different mesh resolutions, $T=1$, CFL-number $\theta=0.05$, $\mathbf{u}_{{{\Delta x}}}^N$ approximation computed by the various schemes at different mesh resolutions, $\mathbf{u}_{\mathrm{ref}}^N$ the reference solution computed by scheme .[]{data-label="tab:convT1cons"} $\Delta x$ ------------ -------- -------- -------- -------- -------- $2^{-2}$ 3.2353 8.4145 3.2700 8.5065 2.8565 $2^{-3}$ 1.1280 4.9240 1.1322 4.9740 1.0713 $2^{-4}$ 0.4735 2.7194 0.4741 2.7455 0.4647 $2^{-5}$ 0.2234 1.4352 0.2235 1.4486 0.2219 $2^{-6}$ 0.1117 0.7316 0.1118 0.7384 0.1114 $2^{-7}$ 0.0594 0.3611 0.0594 0.3645 0.0593 $2^{-8}$ 0.0363 0.1705 0.0363 0.1722 0.0362 $2^{-9}$ 0.0276 0.0737 0.0276 0.0746 0.0276 : $d^2(\mathbf{u}_{{\Delta x}}^N,\mathbf{u}_{\mathrm{ref}}^N)$ for different mesh resolutions, $T=1$, CFL-number $\theta=0.05$, $\mathbf{u}_{{\Delta x}}^N$ approximation computed by the various schemes at different mesh resolutions, $\mathbf{u}_{\mathrm{ref}}^N$ the dissipative reference solution computed by scheme .[]{data-label="tab:convT1diff"} From Tables \[tab:convT1cons\] and \[tab:convT1diff\], we see that at time $T=1$, all approximations to the variable $u$ seem to converge to both reference solutions, so the two reference solutions are very close to each other. $\Delta x$ ------------ --------- --------- --------- --------- --------- $2^{-2}$ 49.9592 95.6001 52.2856 96.0428 66.2850 $2^{-3}$ 75.9922 75.1585 76.9143 75.2326 78.6955 $2^{-4}$ 71.5391 75.4320 71.9087 75.4286 72.5368 $2^{-5}$ 52.2962 75.9529 52.4645 75.9412 52.7209 $2^{-6}$ 36.9257 74.3883 36.9673 74.3799 36.9809 $2^{-7}$ 25.7497 72.4280 25.7554 72.4241 25.6531 $2^{-8}$ 16.4966 71.5233 16.4962 71.5220 16.2811 $2^{-9}$ 9.4041 71.8735 9.4031 71.8734 9.0068 : $d^2(\mathbf{u}_{{\Delta x}}^N,\mathbf{u}_{\mathrm{ref}}^N)$ for different mesh resolutions, $T=10$, CFL-number $\theta=0.05$, $\mathbf{u}_{{\Delta x}}^N$ approximation computed by the various schemes at different mesh resolutions, $\mathbf{u}_{\mathrm{ref}}^N$ the reference solution computed by the energy-conservative scheme .[]{data-label="tab:convT10cons"} $\Delta x$ ------------ --------- --------- --------- --------- --------- $2^{-2}$ 60.0811 81.5763 59.6149 81.9256 62.0062 $2^{-3}$ 71.5451 50.5324 72.0433 50.5400 72.7812 $2^{-4}$ 67.0084 41.7781 67.1806 41.8145 67.4443 $2^{-5}$ 59.7701 34.6851 59.7986 34.7138 59.8481 $2^{-6}$ 62.0292 27.4402 62.0238 27.4533 62.0309 $2^{-7}$ 64.0325 20.2404 64.0339 20.2445 64.0901 $2^{-8}$ 67.1313 13.6321 67.1327 13.6326 67.2587 $2^{-9}$ 70.3864 7.9905 70.3874 7.9899 70.6403 : $d^2(\mathbf{u}_{{\Delta x}}^N,\mathbf{u}_{\mathrm{ref}}^N)$ for different mesh resolutions, $T=10$, CFL-number $\theta=0.05$, $\mathbf{u}_{{\Delta x}}^N$ approximation computed by the various schemes at different mesh resolutions, $\mathbf{u}_{\mathrm{ref}}^N$ the reference solution computed by the dissipative scheme .[]{data-label="tab:convT10diff"} However, at time $T=10$, the approximations of the conservative schemes still seem to be converging to the reference solution computed by whereas no convergence can be observed for the energy dissipative schemes and (Table \[tab:convT10cons\]). Similarly, the dissipative schemes seem to be converging to the dissipative reference solution whereas the distance of the conservative approximations computed by , and to the dissipative reference solution remains (approximately) constant despite mesh refinement (Table \[tab:convT10diff\]). We conclude that the energy-conservative schemes converge to a different limit solution than the energy-dissipative schemes in this example. Furthermore, as the conservative schemes preserve energy, the limit of these schemes is the conservative solution. Similarly, the dissipative schemes converge to a solution that has lower energy than the initial data. Hence, this solution appears to be a dissipative solution of . This dichotomy of solutions is also illustrated in Figure \[fig:u\_comp\] where the difference of conservative and dissipative solutions (realized as limits of the energy conservative and energy dissipative schemes, respectively) is apparent at time $T=10$ (after singularity formation). ![Approximations of , , computed by schemes , , , and on a grid with $15 \cdot 2^{10}$ points and CFL-number $\theta=0.2$. Left: At time $T=1$, Right: At time $T=10$.[]{data-label="fig:u_comp"}](u_comparisonT1 "fig:"){width="45.00000%"} ![Approximations of , , computed by schemes , , , and on a grid with $15 \cdot 2^{10}$ points and CFL-number $\theta=0.2$. Left: At time $T=1$, Right: At time $T=10$.[]{data-label="fig:u_comp"}](u_comparisonT10 "fig:"){width="45.00000%"} Traveling wave with infinite local energy {#ssec:traveling1} ----------------------------------------- As a second example we consider a traveling wave solution of , that is a solution of the form $$\label{eq:tw1} u(t,x)=\psi(x-s t),$$ where $s\in \mathbb{R}$ is the wave speed. Glassey, Hunter and Zheng have shown in [@ghz1996], that the function $\psi$ is given as the solution of the ODE $$\label{eq:twode} \psi' \sqrt{|s^2-c^2(\psi)|}=k,$$ where $k$ is an integration constant. If $|s|\notin [c_0,c_1]$, where $$c_0=\min_{v\in {\mathbb{R}}} c(v),\quad c_1=\max_{v\in {\mathbb{R}}}c(v),$$ the solution to the ODE is smooth and unbounded on ${\mathbb{R}}$. If $|s|\in [c_0,c_1]$ on the other hand, then exists $u_0\in [0,\pi/2]$ such that $|s|=c(u_0)$ and $\psi'$ has a singularity. One can then construct a bounded traveling wave solution as $$\begin{aligned} \label{eq:btw} \psi(\xi)&=u_0, &\mathrm{for }\,& \xi\leq \xi_0,\nonumber\\ \int_{u_0}^{\psi(\xi)}\sqrt{|c^2(u_0)-c^2(v)|}\, dv &=k_1 (\xi-\xi_0), &\mathrm{for }\,& \xi_0\leq \xi\leq \psi^{-1}(\pi-u_0), \\ \psi(\xi)&=\pi-u_0, &\mathrm{for }\,& \xi\geq \psi^{-1}(\pi-u_0),\nonumber\end{aligned}$$ for any $u_0\in [0,\pi/2]$, any $\xi_0\in {\mathbb{R}}$ and and $k_1> 0$. We choose (for this example) $s=\sqrt{\alpha}$, since for this case, we obtain an explicit expression for $\psi$, namely, for $\alpha=0.5$ and $\beta=1.5$, the function $$\begin{aligned} \label{eq:twexp} \psi(\xi)=\begin{cases} 0&\quad \xi\leq 0,\\ \cos^{-1}(-2 \xi+1),&\quad 0<\xi<1,\\ \pi,&\quad \xi\geq 1, \end{cases}\end{aligned}$$ with $$\begin{aligned} \label{eq:twder} \psi'(\xi)=\begin{cases} 0&\quad \xi\leq 0,\\ \frac{1}{\sqrt{\xi-\xi^2}},&\quad 0<\xi<1,\\ 0,&\quad \xi\geq 1, \end{cases}\end{aligned}$$ is a traveling wave solution. This function has infinite local energy as demonstrated in [@ghz1996]. We compute approximations to the solution at time $T=0.5$ and $T=1$, when it has the form $$\begin{aligned} \label{eq:twT1} u(T,x)=\begin{cases} 0&\quad x\leq \sqrt{\alpha} T,\\ \cos^{-1}(-2 (x-\sqrt{\alpha} T)+1),&\quad \sqrt{\alpha} T<x<1+\sqrt{\alpha} T,\\ \pi,&\quad x\geq 1+\sqrt{\alpha} T. \end{cases}\end{aligned}$$ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ![Approximations of computed by schemes , , , and on a grid with cell size ${\Delta x}=2^{-13}$ points and CFL-number $\theta=0.4$. Left: At time $T=0.5$, Right: At time $T=1$.[]{data-label="fig:tw"}](tw_T0_5_cfl0_4_N2_15_us2 "fig:"){width="50.00000%"} ![Approximations of computed by schemes , , , and on a grid with cell size ${\Delta x}=2^{-13}$ points and CFL-number $\theta=0.4$. Left: At time $T=0.5$, Right: At time $T=1$.[]{data-label="fig:tw"}](tw_T1_cfl0_4_N2_15_us2 "fig:"){width="50.00000%"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- In Figure \[fig:tw\], we plot the approximations of computed by the different schemes on a mesh with $\Delta x=2^{-13}$ gridpoints and CFL-number $\theta=0.4$ at time $T=0.5$ and $T=1$. We observe that in the lower part between $y=0$ and $y=0.4$, the approximations appear to differ from the exact solution, and the discrepancy increases with time $T$ . However, as we can see from Tables \[tab:twerr\_T05\] and \[tab:twerr\_04\] it appears all that all the schemes but the Hamiltonian scheme converge, at however, a very slow rate. This slow rate of convergence is not unexpected given the presence of strong singularities (with infinite local energy). scheme ---------------------- -------- -------- -------- -------- -------- ${\Delta x}=2^{-5}$ 6.2205 6.2292 4.4418 5.0283 5.1028 ${\Delta x}=2^{-6}$ 4.6404 4.6546 3.3480 3.8852 3.9214 ${\Delta x}=2^{-7}$ 3.4806 3.4937 2.5925 2.9698 2.9897 ${\Delta x}=2^{-8}$ 2.7565 2.7672 2.2064 2.3253 2.3367 ${\Delta x}=2^{-9}$ 2.3243 2.3365 2.0482 1.8115 1.8181 ${\Delta x}=2^{-10}$ 2.1222 2.1437 2.0499 1.4261 1.4300 ${\Delta x}=2^{-11}$ 2.0401 2.0828 2.1209 1.1282 1.1306 ${\Delta x}=2^{-12}$ 1.9870 2.0666 2.2118 0.8970 0.8984 ${\Delta x}=2^{-13}$ 1.8997 2.0318 2.2988 0.7167 0.7175 : $d^2(\mathbf{u}_{\Delta x},\mathbf{u}_{\mathrm{exact}})$ between the exact solution and the approximations for different mesh resolutions, $T=0.5$, $\theta=0.2$.[]{data-label="tab:twerr_T05"} scheme ---------------------- --------- --------- --------- --------- --------- ${\Delta x}=2^{-5}$ 11.7380 11.6747 10.2197 11.7075 11.4846 ${\Delta x}=2^{-6}$ 9.2074 9.1694 8.1966 9.5308 9.4172 ${\Delta x}=2^{-7}$ 7.4764 7.4783 6.9984 7.7588 7.6963 ${\Delta x}=2^{-8}$ 6.2859 6.3612 6.4006 6.2545 6.2171 ${\Delta x}=2^{-9}$ 5.4966 5.7117 6.3075 5.0454 5.0213 ${\Delta x}=2^{-10}$ 4.8401 5.2414 6.4866 4.0728 4.0568 ${\Delta x}=2^{-11}$ 4.1818 4.7290 6.7554 3.2958 3.2850 ${\Delta x}=2^{-12}$ 3.5273 4.1234 7.0079 2.6766 2.6695 ${\Delta x}=2^{-13}$ 2.9187 3.4861 7.1997 2.1830 2.1783 : $d^2(\mathbf{u}_{\Delta x},\mathbf{u}_{\mathrm{exact}})$ between the exact solution and the approximations for different mesh resolutions, $T=1$, $\theta=0.4$.[]{data-label="tab:twerr_04"} Multiplicity of dissipative solutions ------------------------------------- The above numerical experiments clearly illustrate that one set of schemes converge to a conservative solution whereas another to a dissipative solution of the variational wave equation . Is there uniqueness within these two classes of solutions? A priori, it seems that in contrast to conservative solutions, one might be able to construct multiple dissipative solutions by varying the amount and rate of energy dissipation. We study this possibility by modifying the dissipative scheme to $$\begin{aligned} & (v_j)_t - \frac{1}{{\Delta x}} \left( \bar{c}_{j+\frac{1}{2}} \bar{w}_{j+\frac{1}{2}} - \bar{c}_{j-\frac{1}{2}} \bar{w}_{j-\frac{1}{2}}\right) \\ & \qquad = -\frac{1}{2 {\Delta x}} \left( {\llbracket c \rrbracket}_{j + \frac{1}{2}} \bar{w}_{j+\frac{1}{2}} + {\llbracket c \rrbracket}_{j - \frac{1}{2}} \bar{w}_{j-\frac{1}{2}} \right) + \frac{\kappa}{2 {\Delta x}} \left(s_{j+\frac{1}{2}} {\llbracket v \rrbracket}_{j+\frac{1}{2}} -s_{j-\frac{1}{2}} {\llbracket v \rrbracket}_{j-\frac{1}{2}} \right)\\ & (w_j)_t - \frac{1}{{\Delta x}} \left( \overline{cv}_{j+\frac{1}{2}} - \overline{cv}_{j-\frac{1}{2}} \right) = \frac{\kappa}{2 {\Delta x}} \left(s_{j+\frac{1}{2}} {\llbracket w \rrbracket}_{j+\frac{1}{2}} -s_{j-\frac{1}{2}} {\llbracket w \rrbracket}_{j-\frac{1}{2}} \right)\\ & (u_j)_t=v_j, \end{aligned} \label{eq:scheme2a}$$ by adding $\kappa$ to scale the numerical viscosity in [^1]. We investigate the above question by setting $\kappa$ to different values, resulting in different amounts of energy-loss and possibly, convergence to different dissipative solutions. We test the convergence of resulting numerical approximations by of , , for $\kappa=0.01,0.05, 0.1,1,2,5,10,20$ towards the reference solutions from Section \[ssec:gauss\], computed by both schemes and . The distances $d^2$ as defined in are displayed in Tables \[tab:diff1\] (dissipative reference solution) for time $T=10$ (after singularity formation) and CFL-number $\theta=0.05$. ${\Delta x}$ \\$\kappa$ $0.01$ $0.05$ $0.1$ $1$ $2$ $5$ $10$ $20$ ------------------------- -------- -------- ------- ------- -------- -------- -------- -------- $2^{-2}$ 54.93 47.42 43.86 81.58 142.49 162.03 166.35 169.59 $2^{-3}$ 71.27 63.95 47.50 50.53 85.11 151.50 162.14 166.37 $2^{-4}$ 66.01 48.50 31.36 41.78 50.76 108.55 151.64 162.18 $2^{-5}$ 58.34 28.33 17.25 34.69 41.71 56.59 108.74 151.68 $2^{-6}$ 51.59 11.03 8.19 27.44 34.54 44.12 56.61 108.79 $2^{-7}$ 37.25 3.40 2.84 20.24 27.31 36.80 44.11 56.61 $2^{-8}$ 24.40 3.62 0.7 13.63 20.15 29.60 36.8 44.11 $2^{-9}$ 18.93 4.88 3.01 7.99 13.59 22.39 29.6 36.8 : $d^2(\mathbf{u}_{\Delta x},\mathbf{u}_{\mathrm{ref}})$ between the dissipative reference solution computed by and the approximations by for different mesh resolutions and diffusion coefficient $\kappa$, $T=10$, $\theta=0.05$.[]{data-label="tab:diff1"} The results, presented Table \[tab:diff1\], clearly show that all the approximations (computed with different values of the diffusion coefficient $\kappa$), clearly converge to the same dissipative reference solution. Furthermore, we have listed the discrete energy ratio $$\label{eq:rele} E_{\mathrm{rel}}=\frac{\sum_j \bigl\{(v_j^M)^2+ (w_j^M)^2\bigr\}}{\sum_j \bigl\{(v_j^0)^2+ (w_j^0)^2\bigr\}},$$ where $v_j^0$, $w_j^0$ are the approximations at the initial time and $v_j^M$, $w_j^M$ are the approximations at the final time $T$. From Table \[tab:rele1\], we observe that these ratios seem to converge to $\approx 0.2$ for all the tested $\kappa$, showing that there is a universal rate of energy dissipation, associated with the dissipative solutions (at least in this example). ${\Delta x}$ \\$\kappa$ $0.01$ $0.05$ $0.1$ $1$ $2$ $5$ $10$ $20$ ------------------------- -------- -------- -------- -------- -------- -------- -------- -------- $2^{-2}$ 0.8438 0.5002 0.3326 0.0740 0.0257 0.0052 0.0014 0.0003 $2^{-3}$ 0.8021 0.4899 0.3124 0.1291 0.0702 0.0171 0.0051 0.0014 $2^{-4}$ 0.7086 0.3760 0.2505 0.1630 0.1269 0.0515 0.0170 0.0051 $2^{-5}$ 0.6174 0.2939 0.214 0.1779 0.1615 0.1095 0.0514 0.0170 $2^{-6}$ 0.5304 0.2368 0.2034 0.1842 0.1769 0.1526 0.1094 0.0514 $2^{-7}$ 0.4216 0.2122 0.2037 0.1891 0.1837 0.1730 0.1525 0.1094 $2^{-8}$ 0.3331 0.2081 0.2051 0.1943 0.1889 0.1817 0.1729 0.1525 $2^{-9}$ 0.2801 0.2076 0.2060 0.1987 0.1942 0.1871 0.1817 0.1729 : $E_\mathrm{rel}$ as in for different mesh resolutions and diffusion coefficient $\kappa$, $T=10$, $\theta=0.05$.[]{data-label="tab:rele1"} Time stepping schemes --------------------- Energy conservation for the schemes and has only been proved in the semi-discrete setting. Some time integration routine needs to be used in order to obtain a fully discrete scheme. The choice of the time integration scheme might lead to some energy dissipation or production. We explore this issue by integrating the energy conservative scheme in time using different Runge-Kutta methods, namely the 2nd-order strong stability preserving (SSPRK2) method, the 3rd-order SSPRK3 ([@gottliebetal]) and the standard RK4 procedure. Furthermore, a standard Leap-Frog (LF) time-stepping procedure has also been used for the sake of comparison. It can be readily shown that the fully discrete scheme, combining leap-frog time stepping with the energy conservative scheme conserves the discrete energy: $${\Delta x}\sum_j \bigl\{v_j^n v_j^{n+1}+ w_j^n w_j^{n+1}\bigr\},$$ for all time levels $n$. To illustrate the energy balance of the above time stepping procedures, we have computed the energy ratio for the variational wave equation , , , $\alpha=0.5$, $\beta=4.5$; with different time-stepping methods, for all the above mentioned time-stepping schemes at CFL-numbers $\theta=0.1,\,0.2,\,0.4$ . The results of the mesh refinement study are displayed in Table \[tab:rele2\]. ${\Delta x}$ $\theta$ SSPRK2 SSPRK3 RK4 lf -------------- ---------- -------- -------- -------- -------- $2^{-2}$ 0.4 1.1006 0.9761 0.9996 1.0018 $2^{-3}$ 0.4 1.1602 0.9660 0.9994 1.0017 $2^{-4}$ 0.4 1.5334 0.9285 0.9984 1.0016 $2^{-5}$ 0.4 2.3257 0.8699 0.9968 1.0011 $2^{-6}$ 0.4 4.8354 0.8022 0.9939 1.0008 $2^{-7}$ 0.4 7.4642 0.7197 0.9886 1.0008 $2^{-8}$ 0.4 6.0891 0.6684 0.9794 1.0008 $2^{-2}$ 0.2 1.0111 0.9969 1.0000 1.0004 $2^{-3}$ 0.2 1.0153 0.9954 1.0000 1.0004 $2^{-4}$ 0.2 1.0361 0.9895 0.9999 1.0004 $2^{-5}$ 0.2 1.0740 0.9790 0.9999 1.0003 $2^{-6}$ 0.2 1.1509 0.9619 0.9998 1.0002 $2^{-7}$ 0.2 1.3280 0.9302 0.9996 1.0002 $2^{-8}$ 0.2 1.8364 0.8822 0.9993 1.0002 $2^{-2}$ 0.1 1.0014 0.9996 1.0000 1.0001 $2^{-3}$ 0.1 1.0018 0.9994 1.0000 1.0001 $2^{-4}$ 0.1 1.0042 0.9987 1.0000 1.0001 $2^{-5}$ 0.1 1.0084 0.9973 1.0000 1.0001 $2^{-6}$ 0.1 1.0158 0.9949 1.0000 1.0000 $2^{-7}$ 0.1 1.0307 0.9904 1.0000 1.0001 $2^{-8}$ 0.1 1.0584 0.9825 1.0000 1.0001 : $E_\mathrm{rel}$ as in for scheme , problem , , , for different mesh resolutions, time stepping methods and CFL-numbers $\theta=0.1,\, 0.2,\, 0.4$, $T=10$.[]{data-label="tab:rele2"} We observe that for ‘higher’ CFL-numbers, such as $0.4$, the leap frog scheme performs best whereas the 2nd and 3rd order SSPRK methods can produce/dissipative energy of significant amplitude. However, for lower CFL numbers, the standard RK4 also performs adequately in terms of energy balance, vis a vis the leap frog time stepping scheme. Numerical schemes in two-space dimensions {#sec:2d} ========================================= As in the one-dimensional case, we will design energy conservative and energy dissipative finite difference discretizations of the two-dimensional version of the nonlinear variational wave equation by rewriting it as a first-order system. To this end, we introduce three new independent variables: $$\begin{aligned} p & := u_t, \\ v & := \cos(u) u_x + \sin(u) u_y,\\ w & := \sin(u) u_x - \cos(u) u_y,\end{aligned}$$ Then, for smooth solutions, equation is equivalent to the following system for $(p,v,w,u)$, $$\begin{cases} p_t - \alpha (\phi(u) v)_x -\alpha (\psi(u) v)_y - \beta (\psi(u) w)_x + \beta (\phi(u) w)_y - \alpha v w + \beta v w = 0, &\\ v_t - (\phi(u) p)_x + p \phi_x - (\psi(u) p)_y + p \psi_y + pw = 0, &\\ w_t - (\psi(u) p)_x + p \psi_x + (\phi(u) p)_y - p \phi_y - pv = 0, & \\ u_t =v &. \end{cases} \label{eq:main1_2d}$$ where $\phi(u):= \cos(u)$, and $\psi(u):= \sin(u)$. The grid -------- We introduce some notation needed to define the finite difference schemes in two dimensions. we reserve ${\Delta x}, {\Delta y}$ and ${\Delta t}$ to denote three small positive numbers that represent the spatial and temporal discretization parameters, respectively, of the numerical schemes. For $i$, $j \in {{\mathbb{Z}} }$, we set $x_i = i{\Delta x}$, $y_j = j {\Delta y}$ and for $n = 0, 1, . . .,N$, where $N {\Delta t}= T$ for some fixed time horizon $T > 0$, we set $t_n = n{\Delta t}$. For any function $g = g(x,y)$ admitting pointvalues we write $g_{i,j} = g(x_i,y_j)$, and similarly for any function $h = h(x,y,t)$ admitting pointvalues we write $h^n_{i,j}= h(x_i, y_j, t_n)$. We also introduce the Cartesian spatial and temporal grid cells $$\begin{aligned} I_{i,j} = [x_{i -\frac{1}{2}}, x_{i+\frac{1}{2}}) \times [y_{j -\frac{1}{2}}, y_{j+\frac{1}{2}}), \qquad I^n_{i,j} = I_{i,j} \times [t_n, t_{n+1}).\end{aligned}$$ Furthermore we introduce the jump, and respectively, the average of a quantity $w$ across the interfaces $x_{i + \frac{1}{2}}$ and $y_{j + \frac{1}{2}}$ $$\begin{aligned} \bar{w}_{i, j + \frac{1}{2}} & := \frac{w_{i,j} + w_{i,j+1}}{2}, \\ \bar{w}_{i+ \frac{1}{2}, j} & := \frac{w_{i,j} + w_{i+1,j}}{2}, \\ {\llbracket w \rrbracket}_{i,j + \frac{1}{2}} & := w_{i,j+1} - w_{i,j}, \\ {\llbracket w \rrbracket}_{i+ \frac{1}{2},j} & := w_{i+1,j} - w_{i,j}.\end{aligned}$$ The following identities are readily verified: $$\begin{aligned} \label{eq:useful_2d} { \llbracket u v \rrbracket}_{i,j + \frac{1}{2}} & = \overline{u}_{i,j+\frac{1}{2}} {\llbracket v \rrbracket}_{i,j + \frac{1}{2}} + { \llbracket u \rrbracket}_{i,j + \frac{1}{2}} \overline{v}_{i,j+\frac{1}{2}},\\ { \llbracket u v \rrbracket}_{i+ \frac{1}{2},j} & = \overline{u}_{i+\frac{1}{2},j} {\llbracket v \rrbracket}_{i+ \frac{1}{2},j} + { \llbracket u \rrbracket}_{i+ \frac{1}{2},j} \overline{v}_{i+\frac{1}{2},j},\\ v_{i,j} &= \overline{v}_{i,j \pm \frac{1}{2}} \mp \frac{1}{2} {\llbracket v \rrbracket}_{i,j \pm \frac{1}{2}}, \\ v_{i,j} &= \overline{v}_{i \pm \frac{1}{2},j} \mp \frac{1}{2} {\llbracket v \rrbracket}_{i \pm \frac{1}{2},j}. \end{aligned}$$ Energy conservative scheme -------------------------- Based on the first-order system , We propose the following semi-discrete difference scheme to approximate the two-dimensional version of the nonlinear variational wave equation : $$\begin{aligned} &(p_{i,j})_t - \frac{\alpha}{{\Delta x}} \left( \overline{\phi v}_{i+\frac{1}{2},j} - \overline{\phi v}_{i-\frac{1}{2}, j} \right) - \frac{\alpha}{{\Delta y}} \left( \overline{\psi v}_{i, j+\frac{1}{2}} - \overline{\psi v}_{i, j-\frac{1}{2}} \right) - \frac{\beta}{{\Delta x}} \left( \overline{\psi w}_{i+\frac{1}{2},j} - \overline{\psi w}_{i-\frac{1}{2}, j} \right) \\ & \qquad + \frac{\beta}{{\Delta y}} \left( \overline{\phi w}_{i, j+\frac{1}{2}} - \overline{\phi w}_{i, j-\frac{1}{2}} \right) -\alpha v_{i,j}\, w_{i,j} + \beta v_{i,j}\, w_{i,j} =0, \\ & (v_{i,j})_t - \frac{1}{{\Delta x}} \left( \bar{\phi}_{i+\frac{1}{2},j} \bar{p}_{i+\frac{1}{2},j} - \bar{\phi}_{i-\frac{1}{2},j} \bar{p}_{i-\frac{1}{2},j}\right) + \frac{1}{2 {\Delta x}} \bar{p}_{i+\frac{1}{2},j} {\llbracket \phi \rrbracket}_{i+\frac{1}{2},j} + \frac{1}{2 {\Delta x}} \bar{p}_{i-\frac{1}{2},j} {\llbracket \phi \rrbracket}_{i-\frac{1}{2},j} \\ & - \frac{1}{{\Delta y}} \left( \bar{\psi}_{i,j+\frac{1}{2}} \bar{p}_{i, j+\frac{1}{2}} - \bar{\psi}_{i, j-\frac{1}{2}} \bar{p}_{i, j-\frac{1}{2}}\right) + \frac{1}{2 {\Delta y}} \bar{p}_{i,j+\frac{1}{2}} {\llbracket \psi \rrbracket}_{i,j+\frac{1}{2}} + \frac{1}{2 {\Delta y}} \bar{p}_{i,j-\frac{1}{2}} {\llbracket \psi \rrbracket}_{i,j-\frac{1}{2}} \\ & \qquad \qquad + p_{i,j}\,w_{i,j} =0, \\ & (w_{i,j})_t - \frac{1}{{\Delta x}} \left( \bar{\psi}_{i+\frac{1}{2},j} \bar{p}_{i+\frac{1}{2},j} - \bar{\psi}_{i-\frac{1}{2},j} \bar{p}_{i-\frac{1}{2},j}\right) + \frac{1}{2 {\Delta x}} \bar{p}_{i+\frac{1}{2},j} {\llbracket \psi \rrbracket}_{i+\frac{1}{2},j} + \frac{1}{2 {\Delta x}} \bar{p}_{i-\frac{1}{2},j} {\llbracket \psi \rrbracket}_{i-\frac{1}{2},j} \\ & + \frac{1}{{\Delta y}} \left( \bar{\phi}_{i,j+\frac{1}{2}} \bar{p}_{i, j+\frac{1}{2}} - \bar{\phi}_{i, j-\frac{1}{2}} \bar{p}_{i, j-\frac{1}{2}}\right) - \frac{1}{2 {\Delta y}} \bar{p}_{i,j+\frac{1}{2}} {\llbracket \phi \rrbracket}_{i,j+\frac{1}{2}} - \frac{1}{2 {\Delta y}} \bar{p}_{i,j-\frac{1}{2}} {\llbracket \phi \rrbracket}_{i,j-\frac{1}{2}} \\ & \qquad \qquad - p_{i,j}\,v_{i,j} =0, \\ & (u_{i,j})_t = v_{i,j}. \end{aligned} \label{eq:scheme1_2d}$$ The above scheme preserves a discrete version of the energy as reported in the following theorem: Let $p_{ij}(t)$, $v_{i,j}(t)$ and $w_{i,j}(t)$ be approximate solutions generated by the scheme . Then $$\begin{aligned} \frac{d}{dt} \left( \frac{{\Delta x}{\Delta y}}{2} \sum_{i}\sum_{j} (p_{i,j}^2(t) + \alpha \, v_{i,j}^2(t) + \beta \, w_{i,j}^2(t)) \right) =0. \end{aligned}$$ The proof of this theorem is analogous to the proof of Theorem \[theo:cons\]. Energy dissipative scheme ------------------------- As in the one-dimensional case, we can add some *numerical viscosity* to the energy conservative scheme to obtain the following energy dissipative scheme for approximating the variational wave equation : $$\begin{aligned} &(p_{i,j})_t - \frac{\alpha}{{\Delta x}} \left( \overline{\phi v}_{i+\frac{1}{2},j} - \overline{\phi v}_{i-\frac{1}{2}, j} \right) - \frac{\alpha}{{\Delta y}} \left( \overline{\psi v}_{i, j+\frac{1}{2}} - \overline{\psi v}_{i, j-\frac{1}{2}} \right) - \frac{\beta}{{\Delta x}} \left( \overline{\psi w}_{i+\frac{1}{2},j} - \overline{\psi w}_{i-\frac{1}{2}, j} \right) \\ & \qquad + \frac{\beta}{{\Delta y}} \left( \overline{\phi w}_{i, j+\frac{1}{2}} - \overline{\phi w}_{i, j-\frac{1}{2}} \right) -\alpha v_{i,j}\, w_{i,j} + \beta v_{i,j}\, w_{i,j} \\ & = \frac{1}{2 {\Delta y}} \left( s_{i,j+\frac{1}{2}} {\llbracket p \rrbracket}_{i,j+\frac{1}{2}} - s_{i,j-\frac{1}{2}} {\llbracket p \rrbracket}_{i,j-\frac{1}{2}} \right) + \frac{1}{2 {\Delta x}} \left( s_{i+\frac{1}{2},j} {\llbracket p \rrbracket}_{i+\frac{1}{2},j} - s_{i-\frac{1}{2},j} {\llbracket p \rrbracket}_{i-\frac{1}{2},j} \right), \\ & (v_{i,j})_t - \frac{1}{{\Delta x}} \left( \bar{\phi}_{i+\frac{1}{2},j} \bar{p}_{i+\frac{1}{2},j} - \bar{\phi}_{i-\frac{1}{2},j} \bar{p}_{i-\frac{1}{2},j}\right) + \frac{1}{2 {\Delta x}} \bar{p}_{i+\frac{1}{2},j} {\llbracket \phi \rrbracket}_{i+\frac{1}{2},j} + \frac{1}{2 {\Delta x}} \bar{p}_{i-\frac{1}{2},j} {\llbracket \phi \rrbracket}_{i-\frac{1}{2},j} \\ & - \frac{1}{{\Delta y}} \left( \bar{\psi}_{i,j+\frac{1}{2}} \bar{p}_{i, j+\frac{1}{2}} - \bar{\psi}_{i, j-\frac{1}{2}} \bar{p}_{i, j-\frac{1}{2}}\right) + \frac{1}{2 {\Delta y}} \bar{p}_{i,j+\frac{1}{2}} {\llbracket \psi \rrbracket}_{i,j+\frac{1}{2}} + \frac{1}{2 {\Delta y}} \bar{p}_{i,j-\frac{1}{2}} {\llbracket \psi \rrbracket}_{i,j-\frac{1}{2}} \\ & \qquad + p_{i,j}\,w_{i,j} =\frac{1}{2 {\Delta y}} \left( {\llbracket v \rrbracket}_{i,j+\frac{1}{2}} - {\llbracket v \rrbracket}_{i,j-\frac{1}{2}} \right) + \frac{1}{2 {\Delta x}} \left( {\llbracket v \rrbracket}_{i+\frac{1}{2},j} - {\llbracket v \rrbracket}_{i-\frac{1}{2},j} \right), \\ & (w_{i,j})_t - \frac{1}{{\Delta x}} \left( \bar{\psi}_{i+\frac{1}{2},j} \bar{p}_{i+\frac{1}{2},j} - \bar{\psi}_{i-\frac{1}{2},j} \bar{p}_{i-\frac{1}{2},j}\right) + \frac{1}{2 {\Delta x}} \bar{p}_{i+\frac{1}{2},j} {\llbracket \psi \rrbracket}_{i+\frac{1}{2},j} + \frac{1}{2 {\Delta x}} \bar{p}_{i-\frac{1}{2},j} {\llbracket \psi \rrbracket}_{i-\frac{1}{2},j} \\ & + \frac{1}{{\Delta y}} \left( \bar{\phi}_{i,j+\frac{1}{2}} \bar{p}_{i, j+\frac{1}{2}} - \bar{\phi}_{i, j-\frac{1}{2}} \bar{p}_{i, j-\frac{1}{2}}\right) - \frac{1}{2 {\Delta y}} \bar{p}_{i,j+\frac{1}{2}} {\llbracket \phi \rrbracket}_{i,j+\frac{1}{2}} - \frac{1}{2 {\Delta y}} \bar{p}_{i,j-\frac{1}{2}} {\llbracket \phi \rrbracket}_{i,j-\frac{1}{2}} \\ & \qquad - p_{i,j}\,v_{i,j} =\frac{\nu}{2 {\Delta y}} \left( {\llbracket w \rrbracket}_{i,j+\frac{1}{2}} - {\llbracket w \rrbracket}_{i,j-\frac{1}{2}} \right) + \frac{\nu}{2 {\Delta x}} \left( {\llbracket w \rrbracket}_{i+\frac{1}{2},j} - {\llbracket w \rrbracket}_{i-\frac{1}{2},j} \right), \\ & (u_{i,j})_t = v_{i,j}. \end{aligned} \label{eq:scheme2_2d}$$ Following similar arguments to those used in the proof of Theorem \[theo:diss\], we can prove: Let $p_{ij}(t)$, $v_{i,j}(t)$ and $w_{i,j}(t)$ be approximate solutions generated by the scheme . Then $$\begin{aligned} \frac{d}{dt} \left( \frac{{\Delta x}{\Delta y}}{2} \sum_{i}\sum_{j} (p_{i,j}^2(t) + \alpha \, v_{i,j}^2(t) + \beta \, w_{i,j}^2(t)) \right) \le 0, \end{aligned}$$ with equality if and only if $p$, $v$, and $w$ are constant. Numerical experiments {#numerical-experiments} --------------------- We illustrate the energy conservative scheme and the energy dissipative scheme by considering the first-order system with the following initial data \[eq:init2d\] $$\begin{aligned} \label{seq1:u0} u_0(x,y)&= 2\cos(2 \pi x)\sin(2 \pi y),\\ \label{seq2:u0t} u_1(x,y)&=\sin(2 \pi (x-y)), \end{aligned}$$ on the domain $D=[0,1]^2$ with periodic boundary conditions and coefficients $\alpha=0.5$, $\beta=1.5$ in $a$, $b$ and $c$ at times $T=2$ and $T=4$. For the time integration we have chosen a 3rd order strong stability preserving Runge Kutta method for the energy-dissipative scheme and leap-frog time stepping for the energy-conservative scheme ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ![Approximations of the solution of computed by scheme on a grid with cell size ${\Delta x}=\Delta y=2^{-11}$, CFL-number $\theta=0.4$ at time $T=2$. Left: the angle $u$, right: the corresponding director ${{\mathbf{n}}}=(\cos(u),\sin(u))$.[]{data-label="fig:2d1"}](eg22d_consT2cfl0_4dx2048 "fig:"){width="50.00000%"} ![Approximations of the solution of computed by scheme on a grid with cell size ${\Delta x}=\Delta y=2^{-11}$, CFL-number $\theta=0.4$ at time $T=2$. Left: the angle $u$, right: the corresponding director ${{\mathbf{n}}}=(\cos(u),\sin(u))$.[]{data-label="fig:2d1"}](eg22d_consT2cfl0_4dx2048_director "fig:"){width="50.00000%"} ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ![Approximations of the solution of computed by scheme on a grid with cell size ${\Delta x}=\Delta y=2^{-11}$, CFL-number $\theta=0.4$ at time $T=2$. Left: the angle $u$, right: the corresponding director ${{\mathbf{n}}}=(\cos(u),\sin(u))$.[]{data-label="fig:2d2"}](eg22d_diffT2cfl0_4dx2048 "fig:"){width="50.00000%"} ![Approximations of the solution of computed by scheme on a grid with cell size ${\Delta x}=\Delta y=2^{-11}$, CFL-number $\theta=0.4$ at time $T=2$. Left: the angle $u$, right: the corresponding director ${{\mathbf{n}}}=(\cos(u),\sin(u))$.[]{data-label="fig:2d2"}](eg22d_diffT2cfl0_4dx2048_director "fig:"){width="50.00000%"} ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- Approximations computed by schemes , respectively, on a mesh with $2048$ points in each coordinate direction can be seen in Figures \[fig:2d1\], \[fig:2d2\], \[fig:2d3\] and \[fig:2d4\]. From the above figures, we observe that both the conservative and dissipative schemes seem to resolve the solution in a stable manner. Although the two schemes seems to converge to the same limit at time $T=2$, the limits, computed by the energy conservative and energy dissipative schemes differ at time $T=4$. These results are also confirmed by a convergence study, performed in a manner analogous to the one-dimensional case. However, we omit the convergence tables for brevity in the exposition. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ![Approximations of the solution of computed by scheme on a grid with cell size ${\Delta x}=\Delta y=2^{-11}$, CFL-number $\theta=0.4$ at time $T=4$. Left: the angle $u$, right: the corresponding director ${{\mathbf{n}}}=(\cos(u),\sin(u))$.[]{data-label="fig:2d3"}](eg22d_consT4cfl0_4dx2048 "fig:"){width="50.00000%"} ![Approximations of the solution of computed by scheme on a grid with cell size ${\Delta x}=\Delta y=2^{-11}$, CFL-number $\theta=0.4$ at time $T=4$. Left: the angle $u$, right: the corresponding director ${{\mathbf{n}}}=(\cos(u),\sin(u))$.[]{data-label="fig:2d3"}](eg22d_consT4cfl0_4dx2048_director "fig:"){width="50.00000%"} ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ![Approximations of the solution of computed by scheme on a grid with cell size ${\Delta x}=\Delta y=2^{-11}$, CFL-number $\theta=0.4$ at time $T=4$. Left: the angle $u$, right: the corresponding director ${{\mathbf{n}}}=(\cos(u),\sin(u))$.[]{data-label="fig:2d4"}](eg22d_diffT4cfl0_4dx2048 "fig:"){width="50.00000%"} ![Approximations of the solution of computed by scheme on a grid with cell size ${\Delta x}=\Delta y=2^{-11}$, CFL-number $\theta=0.4$ at time $T=4$. Left: the angle $u$, right: the corresponding director ${{\mathbf{n}}}=(\cos(u),\sin(u))$.[]{data-label="fig:2d4"}](eg22d_diffT4cfl0_4dx2048_director "fig:"){width="50.00000%"} ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- Conclusion ========== We have considered a nonlinear variational wave equation, that models (both one and two dimensional) planar waves in the dynamics of nematic liquid crystals. Our schemes are based on either the conservation or the dissipation of the energy associated with these equations. In the one-dimensional case, we rewrite the variational wave equation in the form of two equivalent first-order systems. Energy conservative as well as energy dissipative schemes, approximating both these formulations are derived. Furthermore, we also design an energy conservative scheme based on a Hamiltonian formulation of the variational wave equation . Numerical experiments, performed with these schemes, strongly suggest 1. All the designed schemes resolved the solution (including possible singularities in the angle $u$) in a stable manner. 2. The energy conservative schemes converge to a limit solution (as the mesh is refined), whose energy is preserved. This solution is a *conservative* solution of . 3. The energy dissipative schemes also converge to a limit solution with energy being dissipated with time. This solution is a *dissipative* solution of the variational wave equation. Furthermore, this solutions appears to be unique. Varying the amount of numerical viscosity did not affect the limiting rate of entropy dissipation. We also design both energy conservative as well as energy dissipative finite difference schemes for the two-dimensional version of the variational wave equation , based on the first-order system . Again, these schemes approximate the conservative, resp. dissipative, solutions efficiently. Thus, we have designed a stable, simple to implement, set of finite difference schemes that can approximate both the conservative as well as the dissipative solutions of the nonlinear variational wave equations. These schemes will be utilized in a forthcoming paper to study realistic modeling scenarios involving liquid crystals. [10]{} S. Badia, F. Guillén. González and J. V. Gutiérrez–Santacreu. An overview on Numerical Analyses of Nematic Liquid Crystal flows, ,18: 285-313(2011). S. Badia, F. Guillén. González and J. V. Gutiérrez–Santacreu. Finite element approximation of nematic liquid crystal flows using a saddle-point structure, , 230(4): 1686-1706(2011). S. Bartels, A. Prohl. Constraint preserving implicit finite element discretization of harmonic map heat flow into spheres, , 76: 1847-1859(2007). R. Becker, X. Feng and A. Prohl. Finite element approximations of the Ericksen–Leslie model for nematic liquid crystal flow, , 46(4): 1704-1731(2008). H. Berestycki, J. M. Coron and I. Ekeland. Variational Methods, Progress in nonlinear differential equations and their applications, , Birkhäuser, Boston (1990). A.  Bressan and Y.  Zheng. Conservative solutions to a nonlinear variational wave equation, , 266, 471-497. D.  Christodoulou and A.  Tahvildar-Zadeh. On the regularity of Spherically symmetric wave maps, , 46(1993), pp. 1041-1091. J.  Coron, J.  Ghidaglia and F.  Hélein. , Kluwer Academic Publishers, 1991. R.  J. Diperna and A.  Majda. Oscillations and Concentrations in weak solutions of the incompressible fluid equations, , 108(1987), pp. 667-689. J.  L. Ericksen and D.  Kinderlehrer. Theory and application of Liquid Crystals, , Vol 5, Springer Verlag, Newyork (1987). R.  T. Glassey. Finite-time blow-up for solutions of nonlinear wave equations, , 177 (1981), pp. 1761-1794. S. Gottlieb and C.-W. Shu and E. Tadmor. Strong stability preserving high-order time discretization methods, , Vol 43, 1 (2001), pp. 89-112. R.  Glassey, J.  Hunter, and Y.  Zheng. Singularities and Oscillations in a nonlinear variational wave equation. In J. Rauch and M. Taylor, editors, , Volume 91 of the IMA volumes in Mathematics and its Applications, pages 37-60. Springer New York, 1997. R.  T. Glassey, J.  K. Hunter and Yuxi.  Zheng. Singularities of a variational wave equation, , 129 (1996), pp. 49-78. H.  Holden and X.  Raynaud. Global semigroup for the nonlinear variational wave equation, , (DOI) 10.1007/s00205-011-0403-5. H.  Holden, K.  H. Karlsen, and N.  H. Risebro. A convergent finite-difference method for a nonlinear variational wave equation, , 29(3): 539-572, 2009. F.  H. Lin, C. Liu. Non-parabolic dissipative systems modelling the flow of liquid crystals, , , 48(1995), pp. 501-537. F.  H. Lin, C. Liu. Existence of solutions for the Ericksen-Leslie system, , 154, pp. 135-156(2000). R.  A. Saxton. Dynamic instability of the liquid crystal director, , Current Progress in Hyperbolic Systems, pp. 325-330, ed. W. B. Lindquist, AMS, Providence, 1989. J.  Shatah. Weak solutions and development of singularities in the SU(2) $\sigma$-model, , 41(1988), pp. 459-469. J.  Shatah, and A.  Tahvildar-Zadeh. Regularity of harmonic maps from Minkowski space into rotationally symmetric manifolds, , 45(1992), pp. 947-971. P.  Zhang and Y.  Zheng. On oscillations of an asymptotic equation of a nonlinear variational wave equation, , 18, 307-327. P.  Zhang and Y.  Zheng. Singular and rarefactive solutions to a nonlinear variational wave equation, , 22B, 159-170. P.  Zhang and Y.  Zheng. Rarefactive solutions to a nonlinear variational wave equation of liquid crystals, , 26, 381-419. P.  Zhang and Y.  Zheng. Weak solutions to a nonlinear variational wave equation, , 166, 303-319. P.  Zhang and Y.  Zheng. Weak solutions to a nonlinear variational wave equation with general data, , 22, 207-226. P.  Zhang and Y.  Zheng. On the global weak solutions to a nonlinear variational wave equation, , (C. M. Dafermos and E. Feireisl eds), vol. 2. Amsterdam: Elsevier, pp. 561-648. [^1]: In a similar way, we generalize scheme .

--- abstract: 'The excited states of $^{12}$Be, $^{14}$Be and $^{15}$B were studied by an antisymmetrized molecular dynamics method. The theoretical results reproduced the energy levels of recently measured excited states of $^{12}$Be, and also predicted rotational bands with innovative clustering structures in $^{12}$Be, $^{14}$Be and $^{15}$B. Clustering states with new exotic clusters ($^6$He, $^8$He and $^9$Li) were theoretically suggested. One new aspect in very neutron-rich nuclei is a 6-nucleon correlation among 4 neutrons and 2 protons, which plays an important role in the formation of $^6$He clusters during clustering: $^8$He+$^6$He of $^{14}$Be and $^9$Li+$^6$He of $^{15}$B.' address: | Institute of Particle and Nuclear Studies,\ High Energy Accelerator Research Organization,\ 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan author: - 'Y. Kanada-En’yo' title: 'Exotic clusters in the excited states of $^{12}$Be, $^{14}$Be and $^{15}$B' --- Owing to progress of experimental techniques, information concerning the excited states of light unstable nuclei has increased rapidly. Clustering structures should be one of the essential features in unstable nuclei as well as in stable nuclei. However, the characteristics of clustering in unstable nuclei are still mysterious. Do clustering structures often appear in the excited states of general light unstable nuclei ? Searching for exotic clustering structures and other new-type clusters than the well-known $\alpha$ cluster is the focus in studies of the excited states of light unstable nuclei. The clustering structures of unstable nuclei are one of the attractive and important subjects in experimental and theoretical research. Theoretical studies on $^{10}$Be [@ENYOf; @OGAWA; @DOTE; @ITAGAKI; @ENYOg] have suggested molecule-like structures in the $K=1^-$ band and in the excited $K=0^+_2$ band. Other candidates of clustering structures are the excited states of $^{12}$Be observed recently in the inelastic scattering reactions [@TANIHATA] and in break-up reactions into $^6$He+$^6$He and $^8$He+$^4$He channels [@FREER; @FREERb]. However, fully microscopic calculations have not yet been done for these states of $^{12}$Be. Another interesting feature of $^{12}$Be is a vanishing of the neutron magic number $N=8$ [@TSUZUKIa; @IWASAKI]. If the ground state is an intruder state, where are the normal states with closed neutron-shell configurations? It is important to solve the relation between the intruder ground state and the possible clustering states within one microscopic framework in order to systematically investigate the structures of $^{12}$Be. Our first aim is to study the ground and excited states of $^{12}$Be with a microscopic method. Furthermore, it is a challenge to theoretically search light neutron-rich nuclei ($^{14}$Be and $^{15}$B) for new clustering states with exotic clusters We have applied a microscopic method of antisymmetrized molecular dynamics (AMD). The AMD methods have already proved to be a powerful approach for structure studies of general light nuclei [@ENYOg; @ENYObc; @ENYOe; @ENYOsup] because of the flexibility of the wave functions, which can express various kinds of structures from spherical shell-model-like structures to developed clustering structures. It is possible to systematically investigate the ground and excited states of light nuclei with a new version: a variation after spin-parity projections (VAP) of AMD, which was proposed by the author [@ENYOe]. As for theoretical predictions with this method, the predicted $5^-$ and $6^+$ states of $^{10}$Be with developed clustering structures [@ENYOg] have actually been discovered in recent experiments [@FREERb]. In the present work, we have studied the structures of the ground and excited states of $^{12}$Be and compared the theoretical results with the experimental data. We theoretically searched $^{14}$Be and $^{15}$B for unknown excited states with new clustering structures. In the results for $^{14}$Be and $^{15}$B, the prediction of new rotational bands of exotic clustering structures are presented. We discuss the mechanism of clustering development and the origin of the formation of $^6$He clusters. An AMD wave function is a Slater determinant of Gaussian wave packets: $$\begin{aligned} &\Phi_{AMD}={1 \over \sqrt{A!}} {\cal A}\{\varphi_1,\varphi_2,\cdots,\varphi_A\},\\ &\varphi_i=\phi_{{{\bf Z}}_i}\chi_i\tau_i :\left\lbrace \begin{array}{l} \phi_{{{\bf Z}}_i}({\bf r}_j) \propto \exp\left [-\nu\biggl({\bf r}_j-{{\bf Z}_i \over \sqrt \nu}\biggr)^2\right], \label{eqn:single}\\ \chi_{i}= \left(\begin{array}{l} {1\over 2}+\xi_{i}\\ {1\over 2}-\xi_{i} \end{array}\right), \end{array}\right.\end{aligned}$$ where the centers of Gaussians ${\bf Z}_i$’s are complex variational parameters. $\chi_i$ is an intrinsic spin function parameterized by $\xi_{i}$, which is also a variational parameter specifying the direction of the intrinsic spin of the $i$th nucleon. $\tau_i$ is an isospin function which was fixed to be up (proton) or down (neutron) in the present calculations. We varied the parameters ${\bf Z}_i$ and $\xi_{i}$($i=1\sim A$) to minimize the energy expectation value for the spin-parity eigenstate (VAP calculations), $${\langle P^J_{MK'}\Phi^\pm_{AMD}|H|P^J_{MK'}\Phi^\pm_{AMD}\rangle \over \langle P^J_{MK'}\Phi^\pm_{AMD} |P^J_{MK'}\Phi^\pm_{AMD}\rangle },$$ where the operator of total-angular-momentum projection ($P^J_{MK'}$) is $\int d\Omega D^{J*}_{MK'}(\Omega)R(\Omega)$. The integration for the Euler angle $\Omega$ was calculated numerically. We adopted a frictional cooling method [@ENYObc] to obtain the minimum energy states. Here, we represent the intrinsic AMD wave function as $\Phi^{J\pm}_0$, that was obtained by a VAP calculation for the lowest state with a given spin parity of $J^\pm$. Higher excited states were constructed by varying the energy of the orthogonal components to the lower states by superposing wave functions. More details of the AMD method for the excited states with the variation after spin-parity projection are described in Refs [@ENYOg; @ENYOe]. By performing VAP calculations for the lowest and the higher excited states with various sets of total spin and parity $\{J\pm\}$, we obtained many AMD wave functions $\{\Phi_1,\cdots,\Phi_m\}$, each of which approximately represents the intrinsic state of the corresponding $J^\pm_n$ state. The final results were attained by diagonalizing a Hamiltonian matrix, $\langle P^{J\pm}_{MK'}\Phi_i|H| P^{J\pm}_{MK''}\Phi_j\rangle$ ($i,j=1\sim m$). We applied the AMD method for the excited states of $^{12}$Be, $^{14}$Be and $^{15}$B. The adopted interactions in the present work were the central force of the modified Volkov No.1 with case 3 [@TOHSAKI], the spin-orbit force of G3RS [@LS] and the Coulomb force. The Majorana parameter used here was $m=0.65$, and the strength of G3RS force was chosen to be $u_1=-u_2=3700$ MeV. We choose an optimum width parameter ($\nu$) for the Gaussians of the single-particle wave functions of each nucleus. The theoretical results of the binding energies of $^{12}$Be, $^{14}$Be and $^{15}$B are 61.9 MeV, 59.7 MeV and 73.1 MeV, which underestimate the experimental values: 68.65 MeV, 69.77 MeV, 88.19 MeV, respectively. We find that the smaller values of the binding energies in the present results than the experimental data can be improved by changing the Majorana parameter of the interaction small to be $m=0.61$. With this parameter, the binding energy of $^{14}$Be is 67.7 MeV. In spite of the smaller values of binding energies, we adopt the parameter $m=0.65$ in the present work, because this parameter gives a good reproduction of parity inversion of $^{11}$Be, which should be important to well describe the ground-state properties of neighboring nucleus: $^{12}$Be. We have checked that the change of $m$ parameter has no significant effect on the excitation energy $E(0^+_2)$ of $^{14}$Be, at least. Namely, the calculated values of $E(0^+_2)$ in the cases $m=0.61$ and $m=0.65$ are 4.2 MeV and 4.4 MeV, respectively. In the calculated results for $^{12}$Be with AMD, many excited states appear in the low-energy region. The energy levels of $^{12}$Be are presented in Fig. \[fig:be12spe\]. The theoretical levels of the $4^+_2$, $6^+_2$ and $8^+_1$ states correspond well to the recently observed excited states [@FREER]. By analyzing the intrinsic AMD wave functions, we can classify the excited states into rotational bands, such as $K^\pi=0^+_1,0^+_2,0^+_3,1^-_1$. It is surprising that the newly observed levels [@FREER] at an energy region above 10 MeV belong to the third rotational band, $K^\pm=0^+_3$, which is a $2\hbar\omega$ excited state with a well-developed clustering structure. On the other hand, there are many low-lying positive-parity states which belong to the lower rotational bands, $K^\pi=0^+_1$ and $K^\pi=0^+_2$. Even though $^{12}$Be has a neutron magic number of 8, the intrinsic state of the ground $0^+$ state is not an ordinary state with a closed neutron $p$-shell, but a prolately deformed state with a developed clustering structure. The deformed ground state is dominated by another $2p-2h$ configuration than that in the $0^+_3$ state. As a result, the ground $K^\pi=0^+$ band starts from the ground $0^+$ state and reaches the band terminal at the $8^+_1$ state. The rotational bands, $K^\pm=0^+_3$ and $K^\pm=0^+_1$, share the same band terminal state because the terminal $J^\pm=8^+_1$ state is the highest spin state in the $2\hbar\omega$ configurations. On the other hand, the main components of the $0^+_2$ and $2^+$ states are the $0\hbar\omega$ configuration with a closed neutron $p$-shell, and construct the second $K^\pi=0^+_2$ band. For experimental evidence, the strength of the $\beta$ decay from $^{12}$Be(0$^+_1$) to $^{12}$B($1^+$) is helpful to estimate the breaking of the neutron $p$-shell [@TSUZUKIa]. As expected, the theoretical value of $B(GT)= 0.8$ is sufficiently small as the experimental data, $B(GT)=0.59$, because the component of the $2\hbar\omega$ configurations in the parent $^{12}$Be(0$^+_1$) makes the transition matrix of the Gamov-Teller operator to be small. Such a result is consistent with pioneer work [@TSUZUKIa; @ITAGAKIa], where their treatments were not fully microscopic for the freedom of all nucleons. The calculated value of $B(E1)$ from $1^-_1$ to $0^+_1$ is 0.02 $e^2$fm$^2$, which agrees reasonably with the experimental large value, $B(E1)$ = 0.05 $e^2$fm$^2$ [@IWASAKI]. As for the $E2$ transition strength, the predicted values in the present results are $B(E2;2^+_1\rightarrow 0^+_1)$ = 14 $e^2$fm$^4$ and $B(E2;2^+_2\rightarrow 0^+_2)$ = 8 $e^2$fm$^4$, which indicate a larger value of the $E2$ transition strength in the deformed $K=0^+_1$ band than that in the $K=0^+_2$ band. It should be pointed out that this is the first theoretical work which systematically reproduces the energy levels of all spin-assigned states discovered recently, except for the $1^-$ state. Although the excitation energy of the $1^-$ state is overestimated by theory, we regard the theoretical $1^-_1$ state as being the measured $1^-$ state at 2.68 MeV because of the large $B(E1)$ value. A theoretical discovery in the present calculations is the $K^\pi=0^+_3$ rotational band with the developed $^{6}$He+$^6$He clustering structure, which constructs the $0^+_3$, $2^+_3$, $4^+_2$, $6^+_2$ and $8^+_1$ states. The developed clustering in the $0^+_3$ state gradually weakens with increasing total-angular momenta ($J$), and finally it changes to a spin-aligned state at $J^\pm=8^+_1$. Many excited states in the rotational bands $K^\pi=0^+_1,0^+_2$ and $1^-$ are predicted to exist in the low-energy region with an excitation energy below 10 MeV. =0.45 By analyzing the single-particle wave functions in the intrinsic state, the deformation mechanism of the ground state can be understood by the idea of molecular orbits surrounding 2$\alpha$ cores. Although the molecular orbits in Be isotopes have been studied theoretically [@ENYOf; @DOTE; @ITAGAKI; @OERTZEN], one of the important points of the present results is that the development of a $2\alpha+4n$ structure in the ground state of $^{12}$Be can be confirmed without assuming the existence of clusters. On the other hand, the third $0^+$ state seems to be a $^6$He+$^6$He clustering structure instead of the $\alpha$ and surrounding neutrons in molecular orbits, because the distance between two clusters is too large. In this case, an $\alpha$ cluster goes outward far enough to form a $^6$He cluster with 2 valence neutrons. We next present the results for $^{14}$Be. Figure \[fig:be14spe\] shows the energy levels of positive-parity states of $^{14}$Be obtained by VAP calculations. Although few excited states are experimentally known, many excited states are predicted based on the theoretical results. There are a few candidates of levels above the threshold energy (9.1 MeV excitation) for separation into $^8$He+$^6$He in break-up experiments [@SAITO]. =0.45 The excited states in the region $J\le 6$ are classified into 3 rotational bands: $K=0^+_1$, $K=0^+_2$ and $K=2^+$. In most of the states in $^{14}$Be, 2-$\alpha$ cores are formed in the calculated results, although no clusters are assumed in the model. Because of the 2-$\alpha$ cores, the $0^+_1$ and $0^+_2$ states have prolate deformations with deformation parameters of $\beta=0.49$ and $\beta=0.64$, respectively. The latter one with a large deformation constructs the $K^\pi=0^+_2$ rotational band, which reaches the $8^+_2$ state. By analyzing the single-particle wave functions of the intrinsic states, it is found that the states in the $K=0^+_1$ and $K=2^+$ bands are dominated by the normal $0\hbar\omega$ configurations, while the largely deformed states in $K^\pi=0^+_2$ mainly contain $4p$-$2h$ states with $2\hbar\omega$ configurations. The unique result of $^{14}$Be is innovative clustering structures of the excited states in this second $K^\pi=0^+_2$ band. The intrinsic states of $K^\pi=0^+_2$ band have spatially developed $^8$He+$^6$He clustering state, where both clusters are, themselves, very neutron-rich nuclei. In Figure \[fig:be14\](a) for the contour surface for the matter density, $\rho\ge$ 0.16 nucleons/fm$^3$, of the intrinsic state of the $0^+_2$ state, the $^8$He($^6$He) cluster can be clearly recognized in the right(left)-handed side. In order to understand the origin of $^8$He+$^6$He clustering formation, it is valuable to inspect the single-particle behavior of the valence neutrons. As mentioned above, the $^8$He+$^6$He clustering state in the $K^\pi=0^+_2$ band comes from the $4p$-$2h$ configurations. According to analyses of the single-particle energies and wave functions, the highest 4 neutron orbits correspond to two kinds of spatial orbits, which are associated with the $sd$-mixing orbits in the deformed system. Roughly speaking, each of two kinds of spatial orbits is occupied by a spin-up neutron and a spin-down neutron. Figures \[fig:be14\](b) and \[fig:be14\](c) present the higher $sd$-like orbit and the lower $sd$-like orbit, respectively. They contain more than 80% components of the positive parity states. If we call the longitudinal direction of the prolate deformation as the $z$-axis, the highest nucleon orbit seen in Fig. \[fig:be14\](b) originates from the $sd$-orbit with a form of $yz \exp [-\nu r^2]$. On the other hand, the spatial orbit for the lower one is similar to the orbit of $z^2 \exp[-\nu r^2]$, which deforms along the $z$-axis, as shown in Fig.\[fig:be14\](c). The original $sd$-orbits are modified because of the deformation in $^{14}$Be. The deformed neutron orbits are stabilized by 2 stretched pairs of protons, because the orbits gain their kinetic energies with respect to the nodes along the $z$-axis. From the viewpoint of single-particle orbits, the $^6$He cluster in the $^6$He+$^8$He clustering system consists of 4 neutrons in the $sd$-orbits and 2 protons in the $p$-orbits. In other words, the formation of $^6$He cluster originates from the correlation of 4 neutrons in the $sd$-orbits and 2 protons in the $p$-orbits. Once a $^6$He cluster takes shape, the cluster spatially develops out of the core-cluster $^8$He keeping the stability of the modified $sd$-mixing orbits in the deformed system. Unique points of this idea are as follows. Firstly, correlations of more than 4 nucleons have not been found. The $^6$He cluster is related to 6-nucleon correlations (4 neutrons with 2 protons). Second, the nucleons, which compose a cluster out of the core nucleus, rise over 2 shells ($sd$-shell and $p$-shell in this case). This is unusual in light stable nuclei. The third point is that the spatially clustering development is associated with the stability of the single-particle orbits of the system. We suggest one of the novel feature that 4 neutrons with 2 protons correlate to form a developed $^6$He cluster in neutron-rich nuclei. =0.45 In the results concerning the excited states of $^{15}$B, the situation is similar to that of $^{14}$Be. We find an exotic clustering structure with $^9$Li+$^6$He in the second $K^\pi=3/2^-_2$ band, which starts from the $J^\pm=3/2^-_2$ state at about 7 MeV higher than the ground state. The intrinsic structure of the $K^\pi=3/2^-_2$ band consists of two exotic clusters:$^9$Li and $^6$He clusters, which develop as remarkably as those in the $K^\pi=0^+_2$ band of $^{14}$Be. These excited states with $^9$Li+$^6$He clustering are based on $2\hbar\omega$ configurations. The spatial parts of 4 valence neutrons in $sd$-like orbits are very similar to those of $^{14}$Be presented in Figs. \[fig:be14\](b) and \[fig:be14\](c). It is an interesting problem whether or not the $^6$He-cluster may appear in the excited states of other neutron-rich nuclei than $^{14}$Be and $^{15}$B. Although particle decay widths are important for the stability of the excited states, the widths were not considered in the present calculations, which were performed within a bound state approximation. The widths for the particle decays should be carefully discussed using other frameworks, such as a method with a reduced width amplitude, or a complex scaling method beyond the present AMD framework. We just mention the partial widths of the states into He channels in the $K^\pi=0^+_2$ of $^{14}$Be. When the excitation energies are above the threshold energy, 12.2 MeV, the excited states in $K^\pi=0^+_2$ of $^{14}$Be may decay into $^8$He+$^6$He channel. By assuming $^8$He and $^6$He clusters in SU(3) limits, we estimated the partial decay width of $6^+_2$ and $8^+_2$ into $^8$He(0$^+$)+$^6$He(0$^+$) channel to be 6 keV and 50 keV, respectively, by calculating the reduced width amplitude. For the total width, it is required to carefully study the excitation energies and other decay channels, such as neutron decays and excited He cluster decays. In summary, we studied the structure of the excited states of $^{12}$Be, $^{14}$Be and $^{15}$B based on the framework of the AMD method. This is the first microscopic calculation to systematically reproduce the energy levels of all the spin-assigned states in $^{12}$Be, except for the $1^-$ state. One of the discoveries of the present results is the third $K^\pi=0^+$ band with $^6$He+$^6$He clusters, which corresponds well to recently observed states in the break-up reactions. In $^{14}$Be and $^{15}$Be, we predicted low-lying excited states with innovative clustering structures. Namely, new types of clusters, $^8$He+$^6$He and $^{9}$Li+$^6$He, were found in the excited states of $^{14}$Be and $^{15}$Be, respectively. In these well-developed clustering structures with 2$\hbar\omega$ excited configurations, the mechanism of the $^6$He cluster development was discussed in relation to the single-particle picture. It was found that the 6-nucleon correlation among 4 neutrons and 2 protons plays an important role in the formation of $^6$He cluster. The results suggest an important feature, that an exotic clustering structure may very often exist in the excited states of neutron-rich nuclei. The author would like to thank Prof. H. Horiuchi for many discussions. She is also thankful to Dr. N. Itagaki and Prof. W. Von Oertzen for helpful discussions and comments. Valuable comments of Prof. S. Shimoura and A. Saito are also acknowledged. The computational calculations of this work are supported by RCNP in Osaka University, YITP in Kyoto University and the supercomputer in KEK. References {#references .unnumbered} ========== [9]{} Y. Kanada-En’yo, H. Horiuchi and A. Doté, J. Phys. G, Nucl. Part. Phys. [**24**]{} 1499 (1998). Y. Ogawa, K. Arai, Y. Suzuki, and K. Varga, Nucl. Phys. [**A673**]{} 122 (2000). A. Doté, H. Horiuchi, and Y. Kanada-En’yo, Phys. Rev. C [**56**]{}, 1844 (1997). N. Itagaki and S. Okabe, Phys. Rev. C [**61**]{}, 044306 (2000). Y. Kanada-En’yo, H. Horiuchi and A. Doté Phys. Rev. [**C 60**]{}, 064304(1999) A.A. Korsheninnikov, et al., Phys. Lett. [**B**]{} 343, 53(1995). M. Freer, et al., Phys. Rev. Lett. [**82**]{}, 1383 (1999). M. Freer, et al., Phys. Rev. C [**63**]{}, 034301 (2001). T. Suzuki and T. Otsuka, Phys. Rev. [**C**]{} 56, 847(1997). H. Iwasaki, et al. Phys.Lett.[**B**]{}491, 8(2000). Y. Kanada-En’yo, H. Horiuchi and A. Ono, Phys. Rev. C [**52**]{}, 628 (1995); Y. Kanada-En’yo and H. Horiuchi, Phys. Rev. C [**52**]{}, 647 (1995). Y. Kanada-En’yo, Phys. Rev. Lett. [**81**]{}, 5291 (1998). Y. Kanada-En’yo and H. Horiuchi, to be published in Prog. Theor. Phys. Suppl.. T. Ando, K.Ikeda, and A. Tohsaki, Prog. Theor. Phys. [**64**]{}, 1608 (1980). N. Yamaguchi, T. Kasahara, S. Nagata, and Y. Akaishi, Prog. Theor. Phys. [**62**]{}, 1018 (1979); R. Tamagaki, Prog. Theor. Phys. [**39**]{}, 91 (1968). N. Itagaki, S. Okabe and K. Ikeda, Phys. Rev. C [**62**]{}, 034301 (2000). W. von Oertzen, Z. Phys. A [**354**]{}, 37 (1996). A. Saito and S. Shimoura, private cominucations.

--- abstract: | Payment channel networks (PCN) are used in cryptocurrencies to enhance the performance and scalability of off-chain transactions. Except for opening and closing of a payment channel, no other transaction requests accepted by a PCN are recorded in the Blockchain. Only the parties which have opened the channel will know the exact amount of fund left at a given instant. In real scenarios, there might not exist a single path which can enable transfer of high value payments. For such cases, splitting up the transaction value across multiple paths is a better approach. While there exists several approaches which route transactions via several paths, such techniques are quite inefficient, as the decision on the number of splits must be taken at the initial phase of the routing algorithm (e.g., SpeedyMurmur [@speedymurmur]). Algorithms which do not consider the residual capacity of each channel in the network are susceptible to failure. Other approaches leak sensitive information, and are quite computationally expensive [@silentwhispers]. To the best of our knowledge, our proposed scheme *HushRelay* is an efficient privacy preserving routing algorithm, taking into account the funds left in each channel, while splitting the transaction value across several paths. Comparing the performance of our algorithm with existing routing schemes on real instances (e.g., Ripple Network), we observed that *HushRelay* attains a success ratio of 1, with an execution time of 2.4 sec. However, *SpeedyMurmur* [@speedymurmur] attains a success ratio of 0.98 and takes 4.74 sec when the number of landmarks is 6. On testing our proposed routing algorithm on the Lightning Network, a success ratio of 0.99 is observed, having an execution time of 0.15 sec, which is 12 times smaller than the time taken by *SpeedyMurmur*. author: - | Subhra Mazumdar$\ast$, Sushmita Ruj$\ddag$, Ram Govind Singh$\dag$, Arindam Pal$\ddag\bigstar$\ $\ast$ Indian Statistical Institute Kolkata, India, Email: subhram\_r@isical.ac.in\ $\bigstar$Cyber Security CRC, Sydney, New South Wales, Australia\ $\ddag$ CSIRO, Data61, Australia, Email: {sushmita.ruj, arindam.pal}@data61.csiro.au\ $\dag$ ICERT, Ministry of Electronics and Information Technology, India, Email: ramgovind.2010@gmail.com bibliography: - 'PCN.bib' title: 'HushRelay: A Privacy-Preserving, Efficient, and Scalable Routing Algorithm for Off-Chain Payments' --- Payment Channel Network, Off-Chain Payments, Routing, Distributed Push-Relabel Algorithm. Introduction ============ Cryptocurrencies like Bitcoin [@nakamoto2008bitcoin] have gained popularity as an alternative method of payment. Blockchain, a cryptographically secure, tamper proof ledger, forms the backbone of such decentralized network, guaranteeing pseudonymity of participant. The records stored in this distributed ledger can be verified by anyone in the network. Consensus algorithms like Proof-of-Work [@nakamoto2008bitcoin], [@o2014bitcoin], [@bano2017consensus], Proof-of-Stake [@king2012ppcoin], [@li2017securing]) are used for reaching agreement on state change in the ledger across the network participants. However, computation time taken by such consensus algorithm is the major bottleneck in scalability of blockchain based transaction [@croman2016scaling], [@poon2016bitcoin]. To be at par with traditional methods of payment like Visa, PayPal, scaling blockchain transactions is an important concern which needs to be addressed, without compromising on the privacy. Several solutions like sharding [@luu2016secure], [@gencer2016service], alternate consensus architecture [@kiayias2017ouroboros], [@miller2014permacoin], [@buterin2017casper], [@park2018spacemint], [@eyal2016bitcoin], [@pass2017hybrid], side-chains [@back2014enabling] have been proposed in Layer-one. But this requires revamping the trust assumptions of the base layer and changing the codebase. A more modular approach is exploring scalability in Layer-two [@gudgeon2019sok]. It massively cuts down data processing on the blockchain by running computations off-chain. The amount of data storage on Layer-one is minimized. Taking transactions off the base layer, while still anchored to it, would free up processing resources to do other things. Also Layer-two relies on Layer-one for security. Several solutions like [@decker2018eltoo], [@decker2015fast], [@luu2016secure] have been proposed. *Payment Channel* [@decker2015fast] stands out as a practically deployable answer to the scalability issue. Any two users, with mutual consent, can open a payment channel by locking their funds in a deposit. Users can perform several off-chain payments between each other without recording the same on blockchain. This is done by locally agreeing on the new deposit balance, enforced cryptographically by smart contracts [@poon2016bitcoin], key based locking [@malavoltamulti] etc. Whenever one of the party wants to close the payment channel, it broadcasts the transaction on blockchain with the final balance. None of the parties can afford to cheat by claiming payment for an older transaction. Opening of new payment channel between parties which are not connected directly has its overhead where funds get locked for a substantial amount of time. This can be avoided by leveraging on the set of existing payment channels for executing a transaction, proving beneficial in terms of resource utilization. These set of payment channels form the *Payment Channel Network (PCN)*. Several problem such as routing, security and interoperability needs to be addressed in such a network. The major challenge in designing any protocol for PCN is to ensure privacy of payer and payee and hiding the payment value transferred. No party, other than the payer and payee, should get any information about the transaction. Thus any routing algorithm designed for such a network must be decentralized, where individual nodes take decision based on the information received from its neighbourhood. Several distributed routing algorithms exists but they suffer from various disadvantages - Elias et al. [@elias] requires a single node to maintain list of active vertices for executing push relabel algorithm on single source-sink pair, Flare [@prihodko2016flare] requires intermediate users to reveal the current capacity of their payment channels to the sender for computation of the maximum possible value to be routed through a payment path, Canal [@viswanath2012canal] entrusts a single node for computing maximum flow in a graph. Landmark-based routing algorithms [@silentwhispers], [@speedymurmur] decide the number of landmarks by trial and error. If the total number of landmarks is $k$, then the payment value is split into $k$ microtransactions randomly without considering the nature of the graph. Such a myopic approach for routing each microtransaction may result in failure as it does not allow optimal utilization of the available capacities present across multiple paths. It was first mentioned in Elias et al. [@elias] that push relabel fits better as a routing algorithm for PCN because it proceeds locally, taking into account the residual capacity of each payment channel. However, the push relabel algorithm used for single source-sink pair [@elias] is not decentralized in nature. A distributed version of the same was implemented in their paper for multiple source-sink pair but it is not well defined. It is not clear how many payment transfer can be allowed at a time through a channel. Further, it was assumed that each payment value for a source-sink pair is unsplittable. This assumption does not work in real life since the payment value might be higher than the bottleneck capacity of a single path. Deciding feasible routes even for a single payment transfer is an involved process in a distributed network. This motivated us to design a new routing algorithm for PCN which is privacy-preserving, efficient as well as scalable. **Our Contributions** --------------------- The following contributions have been made in this paper : - We have proposed a privacy preserving distributed routing algorithm, *HushRelay*, in payment channel network. - We have implemented the scheme and its performance has been compared with SpeedyMurmur [@speedymurmur] in terms of *success ratio* and *time taken to route (TTR)* a payment. Testing was done on real instances of Ripple Network and Lightning Network [^1] and it is observed that *HushRelay* attains a success ratio of 1 in both the cases. However *SpeedyMurmur* attained a maximum success ratio of 0.9815 and 0.907 respectively, when number of landmarks is 6. The time taken to execute the routing algorithm in Ripple like Network and Lightning Network are 2.4s and 0.15189s for *HushRelay* but it takes 4.736s and 1.937s for *SpeedyMurmur*. These statistics justify our claim of the algorithm being efficient and scalable. The code is given in [@Code]. - The proposed routing algorithm is modular and it can be combined with any other privacy preserving payment protocol. **Organization** ---------------- Section \[2\] discusses the state-of-the-art in PCN. Section \[3\] gives a brief overview of the preliminaries. Section \[4\] defines the problem statement and Section \[5\] provides discusses *HushRelay* with Section \[route\] dealing with Generic Construction and Section \[correct\] providing the proof of correctness. Performance analysis of each subprotocol of *HushRelay* has been stated in Section \[6\] and Section \[7\] concludes the paper. Related Work {#2} ============ A payment channel network is a peer-to-peer, path based transaction (PBT) network, where each party operates independent of other parties. Several P2P path-based transaction networks such as such as the Lightning Network for Bitcoin [@poon2016bitcoin], the Raiden Network for Ethereum [@raiden], SilentWhispers [@silentwhispers], InterLedger [@thomas2015protocol], Atomic-swap [@atomic], TeeChain [@lind2017teechain] etc. have been developed over the years. Perun [@dziembowski2017perun] proposes a more efficient network structure which is built around payment hubs. An extension of payment channel, State Channel Network [@dziembowski2018general], not only supports off chain payment but allows execution of complex smart contract. Spider network [@sivaraman2018routing] adheres to a packet-switched architecture for payment channel network. Payment is split into several transaction units and it is transmitted over time across different paths. However the split does not take into account the bottleneck capacity of each path which might lead to failure of payment. BlAnC [@panwar2019blanc], a fully decentralized blockchain-based network, has been proposed which transfers credit between a sender and receiver on demand. Till date, the routing algorithms proposed for payment channel network are as follows : Canal [@viswanath2012canal] - uses a centralized server for computing the path, Flare [@prihodko2016flare] - requires intermediate nodes to inform source node about their residual capacity, SilentWhispers [@silentwhispers] - a distributed PBT network without using any public ledger, SpeedyMurmur [@speedymurmur] - a privacy preserving embedded based routing, extending Voute [@roos2016voute], depending on presence of landmark nodes. SpeedyMurmur is the most relevant privacy preserving distributed routing algorithm. However, it makes use of repeated trials to figure out a suitable split of the total transaction value across multiple paths. Elias et al. [@elias] proposed an extended push relabel for finding payment flow in the payment network. They are the first to point out the flaw in assumption of considering transaction *unsplittable* for existing routing techniques. In real life, splitting of fund across multiple path is inevitable since the bottleneck capacity of a single path may be lower than the total value of fund transfer. Later, a distributed approach for PCN routing, CoinExpress [@yu2018coinexpress], was proposed for finding routes that fulfill payment with higher success ratio. A routing algorithm based on swarm intelligence, ant colony optimization [@grunspan2018ant] has been explored. Hoenisch et al. [@hoenisch2018aodv] proposed an adaptation of an Ad-hoc On-demand Distance Vector (AODV)-based routing algorithm which supports different cryptocurrencies allowing transactions across multiple blockchains. We observe that none of the past works provide an efficient and secure routing algorithm. It is either susceptible to leaking of sensitive information or there exist a central entity controlling the routing algorithm. Background {#3} ========== In this section, we provide the required background on payment channel network. The terms source/payer means the sender node. Similarly, sink/payee/destination means the receiver node and transaction means payment transfer. **Payment Channel Network** --------------------------- \[basic\] A Payment Channel Network (PCN) [@malavolta] is defined as a bidirected graph $G:=(V,E)$, where $V$ is the set of accounts dealing with cryptocurrency and $E$ is the set of payment channels opened between a pair of accounts. A PCN is defined with respect to a blockchain. Only opening and closing of payment channel gets recorded on blockchain apart from disputed transactions where settlement is done by broadcasting the transaction on blockchain. Basic operations of PCN [@malavolta]- - `openPaymentChannel(v_1,v_2,\alpha,t,m)` : For a given pair of accounts $v_1,v_2 \in V$, channel capacity $\alpha$ (initial balance escrowed), timeout value of $t$ and processing fee charged $m$, `openPaymentChannel` creates a new payment channel $(id_{(v_1,v_2)},\alpha,t,m) \in E$, where $id_{(v_1,v_2)}$ is the channel identifier, provided both $v_1$ and $v_2$ has authorized to do so and the funds contributed by each of them sum up to value $\alpha$. - `closePaymentChannel(id_{(v_1,v_2)},\tilde{\alpha})` : Given a channel identifier $id_{(v_1,v_2)}$ with balance $\tilde{\alpha}$, `closePaymentChannel` removes the channel from $G$ provided it is authorized to do so by both $v_1,v_2 \in V$. The balance $\tilde{\alpha}$ gets written on blockchain and this amount is distributed between $v_1$ and $v_2$ as per the net balance recorded. - `payVal(p(s,r),val)` : $p(s,r)$ denotes a path between sender $s$ and receiver $r$. It is defined by a set of identifiers $id_{(s,v_1)},id_{(v_1,v_2)},\ldots,id_{(v_n,r)}$, $s,v_1,v_2,\ldots,v_n,r \in V$, having enough credit to allow transfer of $val$ from $s$ to $r$, if for each payment channel denoted by $id_{(v_i,v_{i+1})}$ has capacity of at least $\beta \geq val_i', val_i'=val+\Sigma_{j=i+1}^{n} fee(v_j), 0 \leq i \leq n, v_0=s \ and \ v_{n+1}=r$, where $fee(v_j)$ is the processing fee charged by each intermediate node $v_j$ in $p(s,r)$. A successful `payVal` operation leads to a decrease of capacity of each payment channel $id_{(v_i,v_{i+1})}$ by $val_i'$. Else the capacity of the channel remains unaltered. **Payment Flow problem** {#flowpay} ------------------------ Consider a directed graph $G:=(V,E) : \ n=|V|, m=|E|, m \geq n-1$, having two distinguished vertices, source $s \in V$, sink $r \in V, s \neq r$, as a *flow network.* For a pair of vertices $v,w$, distance from $v$ to $w$ in graph $G$ is defined by $d_G(v,w)$, the minimum number of edges on the path from $v$ to $w$; if there is no path from $v$ to $w$, $d_G(v,w)=\infty$. A positive real-valued capacity $c(v,w)$, defined by $c : E \rightarrow \mathbb{R}$, is the amount of funds that can be transferred between two nodes sharing an edge. For every edge $(v,w) \in E$ ; if $(v,w) \not \in E$, then $c(v,w)=0$. A flow $f$ on $G$ is a real-valued function on vertex pairs satisfying the constraints [@tarjan], [@thuy2005distributed] : $$\label{eq1} \begin{matrix} f(v,w) \leq c(v,w), \ \forall (v,w) \in V \times V \ \textrm{(capacity)}, \\ f(v,w)=-f(w,v), \ \forall (v,w) \in V \times V \ \textrm{(antisymmetry)}, \\ \Sigma_{u \in V} f(u,v)=0 \ \forall v \in V-\{s,r\} \ \textrm{(flow-conservation)}, \\ \end{matrix}$$ The net flow into the sink is given by $f$, where: $$f=\Sigma_{v \in V} f(v,r)$$ A payment channel network can be mapped to flow network with channels forming the edges and funds locked on each channel can be considered as the edge capacity. Finding the maximum flow value from source to sink for a flow network is termed as the *Maximum Flow problem*. In the context of PCN, given a payment value $val$, one has to find a feasible flow from payer to payee, which is termed here as *Payment Flow problem*. Any max-flow algorithm with subtle modifications can be applied here, taking into account the preflow $f$ of each vertex (except the source and sink) on the network. A preflow is a real-valued function on a vertex pair which satisfies the first two constraints of Eq. \[eq1\] and a weaker form of the third constraint : $$\Sigma_{u \in V} f(u,v)\geq 0, \ \forall v \in V-\{s,r\} \ \textrm{(non-negativity constraint)},$$ A residual capacity of an edge $(v,w) \in E$ is the amount of capacity remaining after the preflow $f$, i.e. $c(v,w)-f(v,w)$ and it is denoted by $r_f(v,w)$. A residual graph $G_f=(V,E_f)$ for a preflow $f$ is the graph whose vertex set is $V$ and edge set $E_f$ is the set of residual edges $(v,w) \in E \ : \ r_f(v,w)>0$. The *flow excess e(v)* of a vertex $v$ is the net balance of funds in node $v$ denoted by $\Sigma_{u \in V} f(u,v)$. The algorithm ends with all vertices except $s$ and $r$ having zero excess flow. If sink is unreachable or if the network does not have adequate capacity for transferring the amount $val$, then the excess value is pushed back to source $s$. Problem Statement {#4} ================= It is not always possible to route the transaction across a single path as the value may be quite high compared to minimum capacity of the designated path. Hence it is better to find set of paths such that the total amount to be transferred is split across each such path. We define the problem as follows - *Given a payment channel network $G(V,E)$, a transaction request ($s,r,val$) for a source-sink pair $(s,r)$, the objective is to find a set of paths $p_1,p_2,\ldots,p_m$ for transferring the fund from $s$ to $r$ such that $p_1$ transfers $val_1$, $p_2$ transfers $val_2,\ldots,$ $p_m$ transfers $val_m : val=\Sigma_{i=1}^m val_i$ without violating *transaction level privacy* i.e. neither the sender nor the receiver of a particular transaction must be identified as well as hiding the actual transaction value from intermediate parties.* Our Proposed Construction {#5} ========================= In this section we provide a detailed overview of the routing algorithm, *HushRelay*. The payment network comprises set of payment channels denoted by channel identifier $id_{(i,j)}, (i,j)\in E$. We describe state the model and the assumptions made. **Network Model and its Assumptions** {#network-model-and-its-assumptions .unnumbered} ------------------------------------- - The network is static i.e. no opening of new payment channel or closing of existing payment channel is considered. - The topology of the network is known by any node in the network since any opening or closing of channel is recorded on the blockchain. - Atmost one timelock contract is allowed to be established on a payment channel at a time. - Sender of a payment chooses set of paths to the receiver according to her own criteria. - The current value on each payment channel is not published but instead kept locally by the users sharing a payment channel. - Pairs of users sharing a payment channel communicate through secure and authenticated channels (such as TLS). Generic Algorithm {#route} ----------------- Since we consider PCN as a flow network, for solving the *payment flow problem* in the given network for executing a transaction request $(s,r,val)$, we propose a routing algorithm inspired from distributed push relabel algorithm stated in [@tarjan], [@thuy2005distributed]. The algorithm proceeds locally by exchange of messages between neighbouring nodes. No single entity controls the flow in the network. Before discussing the algorithm, we briefly describe the *Push Relabel* algorithm for a single source-sink pair (as stated in [@tarjan]): - The instruction *push* redirects the excess flow of a vertex to the sink via its neighbouring vertices. The amount of excess flow that can be *pushed* from a vertex $v$ to one of its neighbouring vertex $w$ is $\delta=min(e(v),r_{f(v,w)})$, where $r_{f(v,w)}$ is the residual capacity of edge $(v,w)$. The value $\delta$ is added to the preflow value $f(v,w)$ (subtracted from $f(w,v)$) and subtracted from $e(v)$. Any push which results in zero residual capacity of the edge is said to be *saturating*. - A valid labeling function $d : V \rightarrow I^{+}\cup\{0,\infty\}$ is used for estimating the distance of a vertex $v$ from sink $r$. $d(s)=n, d(r)=0$ and $d(v)\leq d(w)+1$ for every residual edge $(v,w)$. The label $d(v) < n$ forms the lower bound on the actual distance from $v$ to $r$ in the residual graph $G_f$ and if $d(v)\geq n$, then $d(v)-n$ is a lower bound on the actual distance from $r$ in the residual graph. - A relabeling operation is initiated when a vertex with excess flow has a label less than or equal to that of the neighbouring vertex. Once relabeling is done, it can initiate a push operation. So one can think labels to denote the potential level, where flow can occur from a region of higher potential to a region of lower potential. - A vertex $v$ is defined as active $v \in V - \{s,r\}, d(v) < \infty$, and $e(v)>0$. The maximum-flow algorithm is initialized with preflow value $f$, which is summation of the edge capacities of all edges incident from the source vertex $s$ and rest all other edges have zero flow. For our distributed algorithm, *HushRelay*, the basic operations is *Push* and *Relabel*, with all the nodes acting as individual processing unit in parallel. The network model considered for payment channel network is asynchronous. Synchronization across all the nodes is achieved via use of *acknowledgements* [@tarjan]. A vertex $v$ tries to push excess flow to one of its neighbouring vertex $w$ if and only if, as per the information maintained by $v$, label $d(v)=d(w)+1$. It first sends a request message with the information $(v,\delta,d(v),e(v))$. Vertex $w$ can either accept the push by sending an acknowledgement or it may reject it by sending a negative acknowledgement $(NAK)$. If $d(v)=d(w)+1$, then $w$ sends to $v$ a message of the form $(accept,w,\delta,d(w))$ and $v$ initiates the push. Otherwise, if $d(w)\geq d(v)$ or $d(v) < d(w)+1$, then it sends a message $(reject,w,\delta,d(w))$ where $d(w)$ is the updated distance label of $w$. A reject message will cause $v$ to update the value of $d(w)$. When a distance label of the vertex increases, it sends the information of new label to all its neighbouring nodes. As seen in *Push Relabel* algorithm [@tarjan], the label initially set for source and sink node reveals the identity of payer and payee. To obfuscate their identity from other intermediate nodes in the network, we use a dummy source vertex $s'$ for node $s$ and a dummy sink vertex $r'$ for node $r$. Note that the existence of dummy node is known only by the source and sink. In the initialization phase of *HushRelay*, a directed virtual edge from $s'$ to $s$ and from $r$ to $r'$ is established. Since $s',r'$ are virtual entities, introduction of edge $(s',s)$ and $(r,r')$ is not recorded in the blockchain. The capacity is initialized to $c(s',s)=val, c(r,r')=val$ and the label is set as $d(s')=n+2, d(s)=0, d(r)=0$ and $d(r')=0$. The flow $f(s',s)$ is set to $val$, $f(r,r')=0$, excess flow $e(s)=val$, $e(r)=e(r')=0$. For all vertices $v \in V-\{s,r\}$, $d(v)=0,e(v)=0$, $f(w,v)=0, (w,v) \in E, w,v \in V-\{s,r\}$. We mention the procedure of *Push*, *Push-request* and *Relabel* for a vertex in Procedure \[algo:v4\], \[algo:vpr\] and \[algo:v5\] respectively. 1. Update $d(v)=min(d(w) ,(v,w) \in E) +1$ 2. Inform all the neighbours of vertex $v$ about the updated label $d(v)$. The algorithm terminates when there are no active vertex left (except the dummy source and dummy sink) in the graph. The number of messages exchanged (*for push request, push accepted/NAK, height updation*) is also bounded. The communication complexity, termination condition of the algorithm and an upper bound on the $d$ value of a node in the given graph is stated in [@thuy2005distributed]. $\mathcal{O}(n^2m)$ messages are exchanged in the asynchronous implementation with $\mathcal{O}(n^2)$ runtime. The overhead lies in the interprocessor communication between a vertex and its neighbours. Consider a network given in Fig. \[fig:example1\]. Sender $S$ intends to make a payment of 15 units to receiver R. Dummy vertices $S'$ and $R'$ is added to the network with edges $(S',S)$ and $(R,R')$. The edge capacities are as follows : $c(S',S)=15, c(S,A)=10, c(S,B)=10, c(A,C)=10, c(B,C)=15, c(C,R)=20$ and $c(R,R')=15$. *HushRelay* is implemented on the network to obtain a feasible flow of value 15 from source node S to sink node R. The initial state is given in Fig. \[fig:example1\] (a). \[img1\] In the initialization phase, each nodes is assigned a label of 0 except dummy vertex $S'$ where d(S’) is the count of the number of nodes (except S’) in the network. As given in Fig. \[fig:example1\] (b), S’ sends a push request of 15 units to S. In Fig. \[fig:example1\] (c), S accepts the push request as $d(S)<d(S')$ and an excess flow of 15 units is assigned to S. S changes its label calling *relabel* function and changes $d=1$. Now S sends a push request of 10 units to A, bounded by the capacity of payment channel SA. \ In Fig. \[fig:example1\] (d), A accepts the push request as $d(A)<d(S)$ and gets an excess flow of 5 units. $d(A)$ is changed to 1. S still has an excess flow of 5. It sends a push request to B. Simultaneously A sends a push request of 10 units to C. In Fig. \[fig:example1\] (e), B accepts the push request as $d(B)<d(S)$ and has excess flow of 5 units. $d(B)$ is changed to 1. Similarly C accepts push request as $d(C)<d(A)$ and has an excess flow of 10 units. $d(C)$ is changed to 1. B sends a push request to C and C sends a push request to R. In Fig. \[fig:example1\] (f), upon receipt of request from B, C finds that $d(C)=d(B)$ and hence it sends a NAK to B. R accepts the push request from C and gets an excess flow of 10. It changes its label to $d(R)=1$ and consequently, it sends a push request of 10 units to R’. In Fig. \[fig:example1\] (g), B changes its label to $d(B)=2$. R’ accepts the push request. In Fig. \[fig:example1\] (h), B sends a push request of 5 units to C. In Fig. \[fig:example1\] (i), C accepts the push request and changes it label to 1. It sends a push request of 5 units to R. In Fig. \[fig:example1\] (j), R finds that $d(R)=d(C)$ and hence it sends a NAK to C. In Fig. \[fig:example1\] (k), C undergoes a relabeling operation and $d(C)$ is changed to 2. In Fig. \[fig:example1\] (l), C again sends a push request of 5 units to R. In Fig. \[fig:example1\] (m), R accepts the push request and it sends a push request of 5 units to R’. In Fig. \[fig:example1\] (n), the algorithm terminates transferring 15 units to R’ Proof of correctness of the HushRelay {#correct} ------------------------------------- We state the following lemmas which justifies the correctness of our routing algorithm. \[lm1\] If $val \leq \textrm{maximum flow}$ in $G$, then $s$ can successfully transfer funds to $r$. **Proof of Lemma \[lm1\]**. Given that $\tilde{d} < \textrm{maximum flow}$, let us assume that transaction from $s$ to $r$ fails due to non existence of sufficient capacity from sender to receiver. Now we execute the distributed push relabel algorithm which will return the maximum flow in the graph. Let us denote it by $f_m$. But our algorithm was able to find augmenting path for flow till $\tilde{d} - \gamma$, where $\gamma >0$ is an integral value, since our transaction failed. If there exists no more augmenting paths, then $f_m = \tilde{d} - \gamma$, which implies $f_m < \tilde{d}$. This contradicts the fact that $f_m > \tilde{d}$. Hence our assumption was wrong. \[lm2\] For $ v \in V\setminus \{s',t'\}, e(v)=0$ on termination. **Proof of Lemma \[lm2\]**. Assume that there exist one vertex $\hat{v} \in V\setminus \{s',t'\} : e(\hat{v})>0$ after termination. But since termination condition has been reached, it means there vertex $s$ is not reachable from this vertex $\hat{v}$. Let set of vertices not reachable from $\hat{v}$ be denoted by $V'$ and those reachable from $\hat{v}$ be $V-V'$. $$\begin{matrix} e(v)=\Sigma_{k \in V, (v,k) \in E} f(k,v) \\ = \Sigma_{k \in V',(v,k) \in E} f(k,v) + \Sigma_{k \in V-V',(v,k) \in E} f(k,v) \\ = \Sigma_{k \in V',(v,k) \in E} f(k,v) \\ (\because \Sigma_{k \in V-V',(v,k) \in E} f(k,v)=0 , \textrm{flow conservation constraint}) \end{matrix}$$ But since $e(v)>0$, then $\Sigma_{k \in V',(v,k) \in E} f(k,v)>0$, which means there still exists some augmenting path from $s$ to $v$ ($s \in V'$). Hence it contradicts the assumption of termination. \[lm3\] For all edges $(v,w) \in E, v,w \in V$, $f(v,w)\leq c(v,w)$. **Proof of Lemma \[lm3\]**. In the algorithm *Push*, the flow value $\delta$ for a given edge $(v,w) \in E$, the flow value $f(v,w)$ from vertex $v$ to vertex $w$ is decided by $\min(e(v),r_f(v,w))$. Since $r_f(v,w)=c(v,w)-f(v,w)$, $f(v,w)\leq c(v,w)$ and $e(v) \leq \tilde{d}$. Flow value will be bounded by $\tilde{d}$, if $\tilde{d} < c(v,w)$ or $c(v,w)$ otherwise. ### **Propagating the flow information to source node** {#hide} Each edge $e \in E$ involved in transfer of payment from source to sink will generate temporary key $k_e$ for encrypting the flow message to be propagated back to the source node. The sink node, on termination, generates a key $k_{sink}$ as well some random message $rm$, equivalent to size of the packet or its multiple. Each such packet contains flow information which is shared with a predecessor node. It constructs a message $m'$ containing the information of identity of preceding vertex $w$, the non negative flow $f_{wv}>0$ along with key $k_{wv}$. It encrypts the packet with $k_{sink}$, $E'=Enc_{k_{sink}}(w,f_{wv},k_{wv})$, and concatenates the randomly generated message $rm$ with the encrypted packet to construct message $E'||rm$. It shares this information with $w$. If $w$ is honest, it will construct a similar message $m'$, for its neighbour say $u$ containing the identity of $u$, flow $f_{uw}$, key $k_{uw}$. It is encrypted with $k_{wv}$ to get $E''$. The encrypted message is concatenated with the message received from its successor i.e. $E''||E'||rm$. This continues till all the packets reach the source vertex $s$. The sink vertex shares $k_{sink}$ and set of randomly generated message $rm$ with source vertex $s$ via secure communication channel. $s$ discards $rm$ from the received message and starts decrypting, beginning with the message encrypted by sink. On decryption, it retrieves the flow information, identity of vertex and key with which it will decrypt the next encrypted packet. All duplicate information on flow is discarded and the remaining one is used for reconstructing the flow across the network. This is the routing information denoted by $\mathcal{P}$. \[0.9\] ------------------- ------ ------ --------------- ------------ ------- ------- ------- ------- ------ ------- Success Ratio Time taken 1 2 4 6 1 2 4 6 Ripple Network 0.38 0.7 0.92 0.98 1.66s 2.2s 3.23s 4.74s 1 2.4s Lightning Network 0.42 0.65 0.83 0.91 0.61s 0.69s 0.83s 1.94s 0.99 0.15s ------------------- ------ ------ --------------- ------------ ------- ------- ------- ------- ------ ------- Performance Analysis of *HushRelay* {#6} =================================== ### Experimental Setup {#experimental-setup .unnumbered} In this section, we define the experimental setup. The code for *HushRelay* is available in [@Code]. System configuration used is : `Intel Core i5-8250U CPU, Kabylake GT2 octa core processor, frequency 1.60 GHz`, OS : *Ubuntu-18.04.1 LTS* (64 bit). The programming language used is C, compiler - gcc version 5.4.0 20160609. The library *igraph* was used for generating random graphs of size ranging from 50 to 25000, based on Barábasi-Albert model [@albert2002statistical], [@barabasi2003scale]. Payment Channel Network follows the scale free network where certain nodes function as hub (like central banks), having higher degree compared to other nodes [@javarone2018bitcoin]. For implementing the cryptographic primitives, we use the library *Libgcrypt*, version-1.8.4 [@libgcrypt], which is based on code from GnuPG. ### Evaluation {#evaluation .unnumbered} Following metrics are used to compare the performance of the routing algorithm, *HushRelay* with *SpeedyMurmur* [@speedymurmur] - Success Ratio : It is the ratio of number of successful payment to the total number of payment transfer request submitted in an epoch. - TTR *(Time Taken to Route)* : Given a payment transfer request, it is the time taken from the start of routing protocol till its completion (returning the set of feasible paths). We allow just one trial (i.e. $a=1$) of *SpeedyMurmur* since *HushRelay* executes just once. The number of landmarks is varied as 1,2,4 and 6. - Real Instances - *HushRelay* and *SpeedyMurmur* has been executed on real instances - Ripple Network [@malavolta], Lightning Network [@seres2019topological]. The results are tabulated in Table \[tab:1\]. - Simulated Instances - The capacity of each payment channel is set between 20 to 100 and each transaction value ranges from 10 to 80. For each synthetic graph, we have executed a set of 2000 transactions, with original state of the graph being restored after a transaction gets successfully executed. The source code for SpeedyMurmur is available in [@crysp]. It is written in Java and makes use of the graph analysis tool GTNA[^2]. From the graphs plotted in Fig. \[fig:example2\] a) and b), it is seen that as the number of landmarks increases, SpeedyMurmur gives better success ratio but at the cost of delayed routing. On the other hand, our routing algorithm, which is independent of any landmark, achieves a better success ratio in less time. From the results, we can infer that random splitting of capacity without any knowledge of residual graph may lead to failure in spite of presence of routes with the required capacity. Conclusion {#7} ========== In this paper, we have proposed a novel privacy preserving routing algorithm for payment channel network, *HushRelay* suitable for simultaneous payment across multiple paths. From the results, it was inferred that our proposed routing algorithm outperforms landmark based routing algorithms in terms of success ratio and the time taken to route. Currently all our implementations assume that the network is static. In future, we would like to extend our work for handling dynamic networks as well. Our algorithms have been defined with respect to a transaction between a single payer and payee but it can extended to handle multiple transaction by enforcing blocking protocol or non blocking protocol to resolve deadlocks in concurrent payments. [@malavolta]. [^1]: In the absence of widespread PCN, we use the statistics of such real instances to create the network [^2]: https://github.com/BenjaminSchiller/GTNA

--- abstract: 'We suggest the procedure of the construction of Baxter $Q$-operators for Toda chain . Apart from the one-paramitric family of $Q$-operators, considered in our recent paper [@Pronko] we also give the construction of two basic $Q$-operators and the derivation of the functional relations for these operators. Also we have found the relation of the basic $Q$-operators with Bloch solutions of the quantum linear problem.' --- [**On Baxter Q-operators for Toda Chain**]{}\ [**G.P.Pronko**]{}\ [*Institute for High Energy Physics , Protvino, Moscow reg., Russia\ International Solvay Institute, Brussels, Belgium*]{} Introduction ============ Long ago, in his famous papers [@Baxter] R.Baxter has introduced the object, which is known now as $Q$-operator. This operator was used initially for the solution of the eigenvalue problem of $XYZ$-spin chain, where usual Bethe ansatz fails. Recently this operator was intensively discussed in the series of papers [@BLZ] in the connection with continuous quantum field theory. In [@Sklyanin1] it was pointed out the relation of $Q$-operator with quantum Bäklund transformations. In [@Pronko] we suggested the construction of the one-parametirc family of $Q$-operators for the most difficult case of isotropic Heisenberg spin chain. (In spite of the obvious simplicity of this model, the original Baxter construction fails here.) The existence of the one-parametric family of $Q$-operators implies the existence of two basic solutions of Baxter equation, whose linear combinations ( with operator coefficients ) form the one-parametric family. In the present paper we extend the investigation started in [@Pronko] to the periodic Toda chain, the other model with rational $R$-matrix. It turns out that apart from the construction of the one-parametric family of $Q$-operators (section 2), in the case of Toda chain it is possible to build also two basic $Q$-operators separately (section 3). These basic operators satisfy to the set of the functional wronskian relations (section 5), first established for certain field theoretical model in [@BLZ]. On the one hand the wronskian relations imply the linear independence of the basic operators, on the other hand they are the origin for numerous fusion relations for the transfer matrix of the model. In our approach we construct the basic $Q$-operators as the trace of the monodromy of certain $M_{n}^{(1,2)}(x)$ operators (section 3). It turns out that these operators also permit us to construct the quantum Bloch functions, the basis of the solutions of the quantum linear problem, which are the eigenvectors of the monodromy matrix (section 6). The defining relation of the $Q$-operator (Baxter equation) for the models with rational $R$-matrix looks as follows: $$t(x)Q(x)=a(x)Q(x+i)+b(x)Q(x-i),$$ where $t(x)$ is the corresponding transfer matrix and $a(x)$ and $b(x)$ are the c-number functions which enter into factorization of quantum determinant of $t(x)$. In case of Toda chain the quantum determinant is unity, therefore we can choose the normalization $a(x)=b(x)=1$, which we shall use below. Toda Chain ========== The periodic Toda Chain is the quantum system described by the Hamiltonian $$H=\sum_{i=1}^{N}\left(p_i^2/2+\exp(q_{i+1}-q_{i})\right),$$ where the canonical variables $p_{i},q_{i}$ satisfy commutation relations $$[p_{i},q_{j}]=i\delta_{ij}$$ and periodic boundary conditions $$\begin{aligned} p_{i+N}&=&p_i\nonumber\\ q_{i+N}&=&q_i\end{aligned}$$ Following Sklyanin [@Sklyanin2] we introduce Lax operator in $2$-dimensional auxiliary space as follows: $$\begin{aligned} L_{n}(x)=\left(\begin{array}{cc} x-p_{n}&e^{q_{n}}\\ -e^{-q_{n}}&0 \end{array}\right),\end{aligned}$$ where x is the spectral parameter. The fundamental commutation relations for Lax operator could be written in $R$-matrix form: $$R_{12}(x-y)L_{n}^{1}(x)L_{n}^{2}(y)=L_{n}^{2}(y)L_{n}^{1}(x)R_{12}(x- y),$$ where indexes $1,2$ indicate different auxiliary spaces and $R$-matrix is given by $$R_{12}(x)=x+iP_{12},$$ where $P$-is the operator of permutation of the auxiliary spaces. The same intertwining relation also holds true and for the monodromy matrix corresponding to the $L$-operator (5): $$T_{ij}(x)=\left(\prod^{N}_1L_{n}(x)\right)_{ij},$$ where the multipliers of the product is ordered from the right to the left. The $Q(x)$-operator we are going to construct will be given as the trace of the monodromy $\hat Q(x)$ appropriate operators $M_{n}(x)$, which acts in n-th quantum space and its auxiliary space, which we will choose to be the representation space $\Gamma$ of the algebra: $$%\label{} [\rho_{i},\rho^{+}_{j}]=\delta_{ij},\qquad i,j=1,2$$ The operator $\hat Q(x)$ will be given by the ordered product: $$%\label{} \hat Q(x)=\prod_{n=1}^{N}M_{n}(x),$$ Further we shall need to consider the product $\left(L_{n}(x)\right)_{ij} M_{n}(x)$, which acts in the auxiliary space $\Gamma\times C^2$ ($\Gamma$ - for $M_{n}(x)$ and $C^2$ - is two-dimensional auxiliary space for $L_{n}(x)$). In this space it is convenient to consider a pair of projectors $\Pi^{\pm}_{ij}$: $$\begin{aligned} %\label{} \Pi^{+}_{ij}&=&(\rho^{+}\rho +1)^{-1}\rho_{i}\rho^{+}_{j}=\rho_{i}\rho^{+}_{j} (\rho^{+}\rho +1)^{-1},\nonumber\\ \Pi^{-}_{ij}&=&(\rho^{+}\rho +1)^{-1}\epsilon_{il}\rho^{+}_{l}\epsilon_{jm} \rho_{m}=\epsilon_{il}\rho^{+}_{l}\epsilon_{jm}\rho_{m}(\rho^{+}\rho +1)^{-1},\end{aligned}$$ where $$\begin{aligned} %\label{} \rho^{+}\rho&=&\rho^{+}_{i}\rho_{i}\nonumber\\ \epsilon_{ij}&=&-\epsilon_{ji}, \quad \epsilon_{12}=1.\end{aligned}$$ These projectors formally satisfy the following relations: $$\begin{aligned} %\label{} \Pi^{\pm}_{ik}\Pi^{\pm}_{kj}&=&\Pi^{\pm}_{ij},\nonumber\\ \Pi^{+}_{ik}\Pi^{-}_{kj}&=&0,\nonumber \\ \Pi^{+}_{ij}+\Pi^{-}_{ij}&=&\delta_{ij}.\end{aligned}$$ Rigorously speaking the r.h.s. of the first equation (13) in the Fock representation has an extra term, proportional to the projector on the vacuum state, but, as we shall see below, this term is irrelevant in the present discussion. In order to define $Q$-operator which satisfies Baxter equation we shall exploit Baxter’s idea [@Baxter], which we reformulate as following: $M_{n}(x)$-[*operator should satisfies the relation*]{}: $$%\label{} \Pi^{-}_{ij}\left(L_{n}(x)\right)_{jl}M_{n}(x)\Pi^{+}_{lk}=0.$$ If this condition is fulfilled, then $$\begin{aligned} %\label{} \left(L_{n}(x)\right)_{ij} M_{n}(x)&=&\Pi^{+}_{ik}\left(M_{n}(x)\right)_{kl} M_{n}(x)\Pi^{+}_{lj} +\nonumber\\ \Pi^{-}_{ik}\left(L(x)_{n}\right)_{kl} M_{n}(x)\Pi^{-}_{lj}&+&\Pi^{+}_{ik}\left(L_{n}(x)\right)_{kl} M_{n}(x)\Pi^{-}_{lj}.\end{aligned}$$ In other words, the condition (14) guaranties that the r.h.s. of (15) in the sense of projectors $\Pi^{\pm}$ has the triangle form and this form will be conserved for products over $n$ due to orthogonality of the projectors. From (14) we obtain $$%\label{} \epsilon_{jm}\rho_{m}\left(L_{n}(x)\right)_{jk} M_{n}(x)\rho_{k}=0.$$ To satisfy this equation it is sufficient if $$%\label{} M_{n}(x)\rho_{k}=\left(L_{n}^{-1}(x)\right)_{kl} \rho_{l}A_{n}(x)$$ or $$%\label{} \epsilon_{jm}\rho_{m}\left(L_{n}(x)\right)_{jk} M_{n}(x)=B_{n}(x)\epsilon_{kl}\rho_{l},$$ where $A_{n}(x)$ and $B_{n}(x)$ are some operators which we shall find now. Note that the operator $L_{n}^{-1}(x)$ is given by $$\begin{aligned} L_{n}^{-1}(x)=\left(\begin{array}{cc} 0&-e^{q_{n}}\\ e^{-q_{n}}&x-i-p_{n} \end{array}\right),\end{aligned}$$ The equation (18) could be rewritten in the following form: $$\left(L_{n}^{-1}(x+i)\right)_{jk}\rho_{k}M_{n}(x)=B_{n}(x)\rho_{j}$$ Comparing the equations (17) and (20) we conclude that they both are satisfied provided $$\begin{aligned} A_{n}(x)&=&M_{n}(x-i),\nonumber\\ B_{n}(x)&=&M_{n}(x+i).\end{aligned}$$ In such a way we obtain the following equation for the $M(x)$-operator: $$\left(L_{n}^{-1}(x+i)\right)_{jk}\rho_{k}M_{n}(x)=M_{n}(x+i)\rho_{j}.$$ If the operator $M_{n}(x)$ satisfies this equation, the product $L_{n}(x)M_{n}(x)$ takes the following form: $$\begin{aligned} %\label{} &\left(L_{n}(x)\right)_{ij}M_{n}(x)=\rho_{i} M_{n}(x-i)\rho^{+}_{j}(\rho^{+}\rho+1)^{-1}&\nonumber\\ &+(\rho^{+}\rho+1)^{-1}\epsilon_{il}\rho^{+}_{l} M_{n}(x+i)\epsilon_{jm}\rho_{m}+ \Pi^{+}_{ik}\left(L_{n}(x)\right)_{kl} M_{n}(x)\Pi^{-}_{lj}&.\end{aligned}$$ We do not detail the last term in (23) because, due to triangle structure of it r.h.s. this term will not enter into the trace of $\hat Q(x)$. Now our task is to solve the equation for $M_{n}(x)$-operator. The detailed investigation of the equation (22) shows that the usual Fock representation for (9) does not fit for our purpose, therefore we shall use less restrictive holomorphic representation. Let the operator $\rho^{+}_{i}$ be the operator of multiplication by the $\alpha_{i}$, while the operator $\rho_{i}$-the operator of differentiation with respect to $\alpha_{i}$: $$\begin{aligned} %\label{} \rho^{+}_{i}\psi(\alpha)&=&\alpha_{i}\psi(\alpha),\nonumber\\ \rho_{i}\psi(\alpha)&=&\frac{\partial}{\partial \alpha}\psi(\alpha).\end{aligned}$$ The operators $\rho^{+}_{i},\rho_{i}$ are canonically conjugated for the scalar product: $$%\label{} (\psi,\phi)=\int \frac{\prod_{i=1,2}d\alpha_{i}d\bar\alpha_{i}}{(2\pi i)^{2}} e^{-\alpha\bar\alpha}\bar\psi(\alpha)\phi(\alpha)$$ The action of an operator in holomorphic representation is defined by its kernel: $$%\label{} \left(K\psi\right)(\alpha)=\int d^2\mu(\beta)K(\alpha,\bar\beta)\psi(\beta),$$ where we have denoted $$%\label{} d^2\mu(\beta)=\frac{\prod_{i=1,2}d\beta_{i}d\bar\beta_{i}} {(2\pi i)^{2}}.$$ Now we are ready to make the following [*Statement*]{} The kernel $M_{n}(x,\alpha,\bar\beta)$ of the operator $M_{n}(x)$ in holomorphic representation has the following form: $$M_{n}(x,\alpha,\bar\beta)=m_{n}(x)\frac{(\alpha\bar\beta)^{2l+ix}}{ \Gamma(2l+ix+1)},$$ where $l$ is arbitrary parameter and the operator $m_{n}(x)$ is given by $$\begin{aligned} &m_{n}(x)=\exp\left[{\pi/2(\rho_{1}^+\rho_{2}e^{q_{n}}- \rho_{2}^+\rho_{1}e^{-q_{n}}}\right] \left(1+i\rho_{2}^+\rho_{1}e^{-q_{n}}\right)^{i(p_{n}-x)+\rho_{1}^+ \rho_{1}}\nonumber\\ &=\left(1-i\rho_{1}^+\rho_{2}e^{q_{n}}\right)^{i(p_{n}-x)+\rho_{1}^+ \rho_{1}}\exp\left[{\pi/2(\rho_{1}^+\rho_{2}e^{q_{n}}- \rho_{2}^+\rho_{1}e^{-q_{n}}}\right]\end{aligned}$$ In (28) the operator $m_{n}(x)$ acts on the argument $\alpha$ of the function $(\alpha\bar\beta)^{2l+ix}$ according to (24). The proof of the [*Statement*]{} is straightforward by direct substitution of (28) into equation (22). This calculation give us also the by-product – the meaning of the operator $m_{n}(x)$. Apparently this operator commutes with the operator $$\hat l=\frac{1}{2}(\rho_{1}^{+}\rho_{1}+\rho_{2}^{+}\rho_{2}).$$ If we shall fix the subspace of $\Gamma$ corresponding to the definite eigenvalue $l$ of the operator $\hat l$ then the operator $m_{n}(x-i(l+1/2))$ becomes Lax operator of Toda chain with auxiliary space, corresponding to the spin $l$. In particular, the operator (5) corresponds to $l=1/2$. Generally speaking, the $m_{n}(x-i(l+1/2))$ represents Lax operator of Toda chain in the auxiliary space $\Gamma$. This statement could be proved by intertwining of operator (5) with $m_{n}(x-i(l+1/2))$. Now, taking the ordered product of the $M_n(x)$ operators we shall obtain the operator $\hat Q(x,l)$ whose kernel is given by $$\begin{aligned} \hat Q(x,l,\alpha,\bar\beta)&=&\int \prod_{i=1}^{N-1}d^2\mu(\gamma_{i}) M_{N}(x,l,\alpha,\bar\gamma_{N-1} ) M_{N-1}(x,l,\gamma_{N-1},\bar\gamma_{N-2})\nonumber\\ \cdots&\times&M_{2}(x,l,\gamma_{2},\bar\gamma_{1}) M_{1}(x,l,\gamma_{1},\bar\beta).\end{aligned}$$ Due to triangle (in the sense of projectors $\Pi^{\pm}$ ) structure of the r.h.s. of (23) we obtain the following rule of multiplication of the monodromy matrix $T(x)$ on operator $\hat Q(x)$: $$\begin{aligned} &\left(T(x)\right)_{ij}\hat Q(x,l,\alpha,\bar\beta)= (x+\frac{i}{2})^{N}\rho_{i}\hat Q(x-i,l,\alpha,\bar\beta)\rho^{+}_{j}(\rho^{+}\rho+1)^{-1}&\nonumber\\ &(x-\frac{i}{2})^{N}(\rho^{+}\rho+1)^{-1}\epsilon_{im}\rho^{+}_{m}\hat Q(x+i,l,\alpha,\bar\beta)\epsilon_{jk}\rho_{k}+\Pi^{+}_{im} \bigl(\cdots\bigr)_{mk}\Pi^{-}_{kj},&\end{aligned}$$ where we omitted the explicit expression of the last term by obvious reason. To proceed further we need to remind the definition of trace of an operator in holomorphic representation. If the operator is given by its kernel $F(\alpha,\bar \beta)$ then, (see e.g. [@Berezin]) $$Tr F= \int d^2\mu(\alpha) F(\alpha,\bar\alpha),$$ where the measure was defined in (27). Now we can perform the trace operation for both sides of (32 )over the holomorphic variables and over $i,j$ indexes, corresponding to the auxiliary $2$-dimensional space of $T(x)$. The result is the desired Baxter equation: $$t(x)Q(x,l)=Q(x-i,l)+Q(x+i,l),$$ where, according to (33) $$Q(x,l)=\int d^2\mu(\alpha)\hat Q(x,l,\alpha,\bar\alpha).$$ Note, that the trace of $\hat Q$ exists due to the exponential factor in holomorphic measure (27) and has cyclic property, therefore $Q(x,l)$ is invariant under cyclic permutation of the quantum variables. Acting as above we can also consider right multiplication $M_{n}(x)L_{n}(x)$ to obtain $$Q(x,l)t(x)=Q(x-i,l)+Q(x+i,l).$$ We shall not consider here the derivations of the intertwining relations for $\hat Q(x,l)$ for different values of $x$ and $l$ and for $\hat Q(x,l)$ and $T_{ij}(y)$. This may be done in the same way as in [@Pronko] and these relations imply the following commutation relations: $$\begin{aligned} \left[Q(x,l),Q(y,m)\right]&=&0\nonumber\\ \left[t(x),Q(y,l)\right]&=&0\end{aligned}$$ In such a way we have constructed the family of solutions of the Baxter equation which are parametrized by the parameter $l$. We can prove that this family may be considered as a linear combinations of two basic solutions with operator coefficients. Here arises the question - is it possible to construct these basic operators separately. The answer is positive and now we shall show how our procedure should be modified in this case. Basic $Q$-operators for Toda Chain. =================================== As above, we shall look for the $Q$-operators in the form of the monodromy of appropriate $M_{n}^{(i)}(x)$-operators, which we now supply with the index $i=1,2$ and which act in $n$-th quantum space. The auxiliary space $\Gamma$ now will be the representation space of one Heisenberg algebra, instead of (9): $$[\rho, \rho^+]=1.$$ The product $(L_{n}(x))_{ij}M_{n}^{(i)}(x)$ is an operator in $n$-th quantum space and in auxiliary space which is tensor product $\Gamma\times C^2$. In this auxiliary space we shall introduce new projectors : $$\begin{aligned} \Pi_{ij}^{+}=\left(\begin{array}{c} 1\\ \rho \end{array}\right) \frac{1}{\rho^+\rho+1}(1,\rho^+),\nonumber\\ \Pi_{ij}^{-}=\left(\begin{array}{c} -\rho^+\\ 1 \end{array}\right) \frac{1}{\rho^+\rho+2}(-\rho,1)\end{aligned}$$ The defining equations for the operators $M_{n}^{(i)}$ ( the analogies of eq. (14) ) are $$\begin{aligned} \Pi_{ik}^-\left(L_{n}(x)\right)_{kl}M_{n}^{(1)}(x)\Pi_{lj}^+&=&0, \nonumber\\ \Pi_{ik}^+\left(L_{n}(x)\right)_{kl}M_{n}^{(2)}(x)\Pi_{lj}^-&=&0.\end{aligned}$$ The solutions of these equations we again will present as the kernels of the corresponding operators in holomorphic representation of the algebra (38): $$\begin{aligned} M_{n}^{(1)}(x, \alpha,\bar\beta)&=&\exp(-i\bar\beta e^{q_{n}})\frac{e^{-\pi x/2}}{\Gamma(-i(x-p_{n})+1)}\exp(i\alpha e^{-q_{n}}),\nonumber\\ M_{n}^{(2)}(x, \alpha,\bar\beta)&=&\exp(-i\alpha e^{-q_{n}})e^{-\pi x/2}e^{(x-p_{n})}\nonumber\\ &&\qquad\times\Gamma(-i(x-p_{n})) \exp(i\bar\beta e^{q_{n}}).\end{aligned}$$ For the right multiplication by $L_{n}(x)$ these operators automatically satisfy the following equations: $$\begin{aligned} \Pi_{ik}^+M_{n}^{(1)}(x)\left(L_{n}(x)\right)_{kl}\Pi_{lj}^-&=&0, \nonumber\\ \Pi_{ik}^-M_{n}^{(2)}(x)\left(L_{n}(x)\right)_{kl}\Pi_{lj}^+&=&0.\end{aligned}$$ The full multiplication rules for the operators $M_{n}^{i}(x)$ and $L_{n}(x)$ have the following form for left multiplication: $$\begin{aligned} &&\left(L_{n}(x)\right)_{ij}M_{n}^{(1)}(x)=\left(\begin{array}{c} 1\\ \rho\end{array}\right)_{i}M_{n}^{(1)}(x-i)\frac{1}{\rho^+\rho+1}(1,\rho^+) _{j}\nonumber\\ &+&\left(\begin{array}{c} -\rho^+\\ 1 \end{array}\right)_{i} \frac{1}{\rho^+\rho+2}M_{n}^{(1)}(x+i)(-\rho,1)_{j}+\Pi_{ik}^+ \left(L_{n}(x)\right)_{kl}M_{n}^{(1)}(x)\Pi_{lj}^-\nonumber\\ &&\left(L_{n}(x)\right)_{ij}M_{n}^{(2)}(x)=\left(\begin{array}{c} 1\\ \rho \end{array}\right)_{i} \frac{1}{\rho^+\rho+1}M_{n}^{(2)}(x+i)(1,\rho^{+})_{j}\\ &+&\left(\begin{array}{c} -\rho^+\\ 1\end{array}\right)_{i}M_{n}^{(2)}(x-i) \frac{1}{\rho^+\rho+2}(-\rho,1)_{j}+\Pi_{ik}^-\left(L_{n}(x)\right)_{kl}M_ {n}^{(2)}(x)\Pi_{lj}^+\nonumber\\ \nonumber\end{aligned}$$ and for right multiplication: $$\begin{aligned} &&M_{n}^{(1)}(x)\left(L_{n}(x)\right)_{ij}=\left(\begin{array}{c} 1\\ \rho\end{array}\right)_{i}\frac{1}{\rho^+\rho+1}M_{n}^{(1)}(x-i)(1,\rho^+) _{j}\\ &+&\left(\begin{array}{c} -\rho^+\\ 1 \end{array}\right)_{i} M_{n}^{(1)}(x+i)\frac{1}{\rho^+\rho+2}(-\rho,1)_{j}+\Pi_{ik}^- \left(L_{n}(x)\right)_{kl}M_{n}^{(1)}(x)\Pi_{lj}^+\nonumber\\ &&M_{n}^{(2)}(x)\left(L_{n}(x)\right)_{ij}=\left(\begin{array}{c} 1\\ \rho \end{array}\right)_{i} M_{n}^{(2)}(x+i)\frac{1}{\rho^+\rho+1}(1,\rho^{+})_{j}\\ &+&\left(\begin{array}{c} -\rho^+\\ 1\end{array}\right)_{i}\frac{1}{\rho^+\rho+2}M_{n}^{(2)}(x-i) (-\rho,1)_{j}+\Pi_{ik}^+\left(L_{n}(x)\right)_{kl}M_ {n}^{(2)}(x)\Pi_{lj}^-\nonumber\\ \nonumber\end{aligned}$$ These relations guaranty that the traces of the monodromies, corresponding to both operators $M_{n}^{(i)}(x)$ satisfy Baxter equations: $$\begin{aligned} t(x)Q^{(i)}(x)&=&Q^{(i)}(x+i)+Q^{(i)}(x-i)\nonumber\\ Q^{(i)}(x)t(x)&=&Q^{(i)}(x+i)+Q^{(i)}(x-i)\end{aligned}$$ We shall conclude this section with the calculation of the operators $Q^{i}(x)$ for the simplest case of one quantum degree of freedom. In this case from (33) we easily obtain $$\begin{aligned} Q^{(1)}(x)&=&\int \frac{d\alpha d \bar \alpha}{2\pi i} e^{-\alpha \bar \alpha} M^{1}(x,\alpha,\bar\alpha) =\sum_{n=0}\frac{e^{-\pi x/2}}{n!} e^{-qn}\frac{1}{\Gamma(-i(x-p)+1)}e^{qn}\nonumber\\ &=&\sum_{n=0}\frac{e^{-\pi x/2}} {n!\Gamma(-i(x-p)+n+1)}=e^{-\pi x/2}I_{-i(x-p)}(2),\end{aligned}$$ where $I_{\nu}(x)$ is the modified Bessel function. The analogues calculations for the second $Q$-operator gives: $$\begin{aligned} Q^{(2)}(x)&=&-e^{-\pi x/2}\frac{\pi e^{\pi (x-p)}}{\sin{\pi i(x-p)}} \sum_{n=0}\frac{1}{n!\Gamma(i(x-p)+n+1)}\nonumber\\ &&=-e^{-\pi x/2}\frac{\pi e^{\pi (x-p)}}{\sin{\pi i(x-p)}}I_{i(x-p)}(2).\end{aligned}$$ These two expressions could be compared with the results of [@Pas.God]. Intertwining Relations. ======================= In this section we shall consider the set of intertwining relations among $L_{n}(x)$-operator and $M_{n}^{(i)}(x)$-operators which will imply the mutual commutativity of transfer matrix and $Q^{(i)}(x)$. Let us start with the simplest relation $$R^{(i)}_{kl}(x-y)\left(L_{n}(x)\right)_{lm}M_{n}^{(i)}(y)=M_{n}^{(i)}(y) \left(L_{n}(x)\right)_{kl}R^{(i)}_{lm}(x-y)$$ From eq. (40) follows that for $x=y$ the $R^{(i)}$-matrixes become the corresponding projectors - $\Pi^{-}$ for $i=1$ and $\Pi^{+}$ for $i=2$. Making use of these properties we easily obtain: $$\begin{aligned} R^{(1)}_{kl}(x-y)=\left(\begin{array}{cc} x-y+i\rho^+\rho&-i\rho^+\\ -i\rho&i \end{array}\right)\\ \nonumber\\ R^{(2)}_{kl}(x-y)=\left(\begin{array}{cc} i&i\rho^+\\ i\rho&x-y+i+i\rho^+\rho \end{array}\right)\end{aligned}$$ Next relation which we shall consider is $$M_{n}^{(1)}(x,\rho)M_{n}^{(2)}(y,\tau)R^{12}(x-y)=R^{12}(x-y)M_{n}^{(2)}(y ,\tau)M_{n}^{(1)}(x,\rho),$$ where both $M$-operators act in different auxiliary spaces $\Gamma^{(i)}$ and mutual quantum space. The $R$-matrix acts in the tensor product of auxiliary spaces $\Gamma^{(1)}\times \Gamma^{(2)}$. In (52) we have denoted the operators which act in the auxiliary space $\Gamma^{(1)}$ as $\rho,\rho^+$ and operators in $\Gamma^{(2)}$ as $\tau,\tau^+$. From explicit expressions for $M$-operators (41) follows that $$\begin{aligned} (\rho+\tau)M_{n}^{(1)}(x,\rho)M_{n}^{(2)}(y,\tau)&=&0,\nonumber\\ M_{n}^{(2)}(y,\tau)M_{n}^{(1)}(x,\rho)(\rho^{+}+\tau^{+})&=&0.\end{aligned}$$ These relations mean that the products of the $M$-operators are triangle operators in the $\Gamma^{(1)}\times \Gamma^{(2)}$ and , as a result the $R$-matrix satisfy the following equations: $$\begin{aligned} &(\rho+\tau)R^{12}(x)=0\nonumber\\ &R^{12}(x)(\rho^{+}+\tau^{+})=0.\end{aligned}$$ The corollary of (54) is that the kernel of $R$-matrix in holomorphic representation depends only on one variable: $$R^{12}(x,\alpha,\bar\beta;\gamma,\bar\delta)=f(x,(\alpha-\gamma)(\bar\beta -\bar\delta)),$$ where the variables $\alpha,\bar\beta$ refer to the operators $\rho,\rho^+$ and variables $\gamma,\bar\delta$ to the operators $\tau,\tau^+$. Taking (55) into account we can write the intertwining relation (52) in holomorphic representation: $$\begin{aligned} &\int d\mu(\beta')d\mu(\delta')M_{n}^{(1)}(x,\alpha,\bar\beta') M_{n}^{(2)}(y,\gamma,\bar\delta')f(x-y,(\beta'-\delta')(\bar\beta- \bar\delta))=\nonumber\\ &\int d\mu(\alpha')d\mu(\gamma')f(x-y,(\alpha-\gamma)(\bar\alpha'- \bar\gamma'))M_{n}^{(2)}(y,\gamma',\bar\delta)M_{n}^{(1)}(x,\alpha', \bar\beta),\end{aligned}$$ where $$d\mu(\alpha)=\frac{d\alpha d\bar\alpha}{2\pi i}e^{-\alpha\bar\alpha}.$$ To simplify this equation let us introduce the new external variables: $$\begin{aligned} \xi_{1}&=\frac{1}{\sqrt{2}}(\alpha+\gamma),\qquad\xi_{1}'&=\frac{1} {\sqrt{2}}(\beta+\delta),\nonumber\\ \xi_{2}&=\frac{1}{\sqrt{2}}(\alpha-\gamma),\qquad\xi_{2}'&=\frac{1} {\sqrt{2}}(\beta-\delta)\end{aligned}$$ and new integration variables for l.h.s. (r.h.s.) integral: $$\begin{aligned} \xi''_{1}&=\frac{1}{\sqrt{2}}(\beta'+\delta')\quad&\left(\xi''_{1}=\frac{1 }{\sqrt{2}}(\alpha'+\gamma')\right)\nonumber\\ \xi''_{2}&=\frac{1}{\sqrt{2}}(\beta'-\delta')\quad&\left(\xi''_{2}=\frac{1 }{\sqrt{2}}(\alpha'-\gamma')\right).\end{aligned}$$ Apparently, due to the structure of $M^{(i)}$-operators and $R$-matrix, both sides of (56) depend only on the variables $\xi_{2},\bar\xi'_{2}$ and integration over $\xi''_{1}$ becomes trivial, resulting in elimination of these variables in the integrands. Further, representing the function $f(x,2\xi''\bar\xi')$ as $$f(x,2\xi''\bar\xi')=\sum_{n=0}C_{n}(x)\frac{(2\xi''\bar\xi')^{n}}{n!},$$ we can perform the integration over $\xi''_{2}$ and, comparing similar terms in both sides of (56), conclude that $$C_{n}(x)=\frac{1}{\Gamma(-ix+n+1)}.$$ Therefore $R$-matrix in (52) has the following form in holomorphic representation $$R^{12}(x,\alpha,\bar\beta;\gamma,\bar\delta)=\sum_{n=0}\frac{\left((\alpha -\gamma)(\bar\beta-\bar\delta)\right)^n}{n!\Gamma(-ix+n+1)}.$$ As the operator in the space $\Gamma^{(1)}\times \Gamma^{(2)}$ the $R$-matrix (62) is pathological because its kernel depends only on part of holomorphic variables. In other words it contains the projector $\pi$ on the subspace of $\Gamma^{(1)}\times \Gamma^{(2)}$ which is formed by the functions depending on the difference of variables. This property may be an obstacle in the derivation of the commutativity of $Q$-operators from the intertwining relation (52). The situation is saved due to the same pathological nature of the product of $M$-operators. Indeed, let us consider the product $$\begin{aligned} &Q^{(1)}(x)Q^{(2)}(y)=Tr_{1}\prod_{k=1}^{N}M_{k}^{(1)}(x) Tr_{2}\prod_{k=1}^{N}M_{k}^{(2)}(y)=\nonumber\\ &=Tr_{1,2}\prod_{k=1}^{N}M_{k}^{(1)}(x)M_{k}^{(2)}(y),\end{aligned}$$ where the indexes $1,2$ mark the corresponding auxiliary space. Due to the property (53) we can supply each term $M_{k}^{(1)}(x)M_{k}^{(2)}(y)$ in the last product with the projector $\pi$. The same holds true also for the product of $Q$-operators taken in the inverse order. In such a way for the commutativity of $Q$-operators we need to consider only the intertwining relations of $M$-operators projected onto the space $\pi\left(\Gamma^{(1)}\times \Gamma^{(2)}\right)$ , where our $R$-matrix is well defined. Next we shall consider the intertwining relation for the $M^{(1)}$-operators with different values of spectral parameter: $$R^{(11)}(x-y)M^{(1)}(x,\rho)M^{(1)}(y,\tau)=M^{(1)}(y,\tau)M^{(1)}(x,\rho) R^{(11)}(x-y).$$ As above, the $R$-matrix in (64) acts in the space $\Gamma^{(1)}\times \Gamma^{(2)}$. From explicit expression for $M^{(1)}$-operator (41) we obtain: $$\rho M^{(1)}(x,\rho)=M^{(1)}(x,\rho)ie^{-q},\qquad -ie^{q}M^{(1)}(x,\rho)=M^{(1)}(x,\rho)\rho^+$$ These properties of $M^{(1)}$-operator imply the following conditions on the $R$-matrix: $$\tau^+ R^{(11)}(x)=R^{(11)}(x)\rho^+,\qquad \rho R^{(11)}(x)=R^{(11)}(x)\tau,$$ which could be satisfied if $R^{(11)}(x)$ has the following form: $$R^{(11)}(x)=P_{\rho\tau}g(x,\rho^+\tau),$$ where $P_{\rho\tau}$ denotes the operator of permutation of $\rho\tau$ variables. Substituting (67) into relation (64) we get the equation for the function $g$: $$g(x-y,\rho^+\tau)M^{(1)}(x,\rho)M^{(1)}(y,\tau)=M^{(1)}(y,\tau) M^{(1)}(x,\rho)g(x-y,\rho^+\tau)$$ Making use of the explicit form of the $M^{(1)}$-operator and the formal power series expansion for function $g$ with respect to it second argument we can solve this equation and find the function $g$: $$g(x,\rho^+\tau)=(1+\rho^+\tau)^{-ix}$$ and therefore $$R^{(11)}(x)=P_{\rho\tau}(1+\rho^+\tau)^{-ix}.$$ As this $R$-matrix intertwines two similar objects, it should satisfies the Yang-Baxer equation (and it really does), but we shall not investigate further this issue. The last relation which we need to discuss is the intertwining of two $M^{(2)}$-operators: $$R^{(22)}(x-y)M^{(2)}(x,\rho)M^{(2)}(y,\tau)=M^{(2)}(y,\tau)M^{(2)}(x,\rho) R^{(22)}(x-y).$$ The $M^{(2)}$-operators also satisfy the relations analogues to (65): $$\rho M^{(2)}(x,\rho)=-ie^{-q}M^{(2)}(x,\rho),\qquad M^{(2)}(x,\rho)ie^{q}=M^{(2)}(x,\rho)\rho^+,$$ from where we obtain the analogue of (66): $$\tau^+ R^{(22)}(x)=R^{(22)}(x)\rho^,\qquad \rho^+ R^{(22)}(x)=R^{(22)}(x)\tau^+,$$ and therefore $R^{(22)}$ has the following form: $$R^{(22)}(x)=P_{\rho\tau}h(x,\tau^+\rho).$$ Further , acting as above we find that the unknown function $h$ does coincide with the function $g$, resulting in the following $R^{(22)}$-matrix: $$R^{(22)}(x)=P_{\rho\tau}(1+\tau^+\rho)^{-ix}.$$ Now we have completed the derivation of all needed intertwining relation The main corollary of these relations is the mutual commutativity of the transfer matrix and both $Q$-operators: $$[t(x),Q^{(i)}(y)]=0, \quad [Q^{(i)}(x),Q^{(j)}(y)]=0, \quad i(j)=1,2.$$ Wronskian-type Functional Relations =================================== It was first pointed out in [@BLZ] that Baxter equation (1) which defines the $Q$ -operator could be viewed as the finite difference analogue of the second order differential equation which admits two independent solution. The linear independence of the solutions could be established through the calculation of the wronskian corresponding to the equation. In the previous section we have constructed two solution of Baxter equation and now our task is to prove its linear independence i.e. to derive the finite difference analogue of the wronskian. To solve this problem let us consider in the details the representation of the product (63) of two different $Q$-operators. In the notations of the previous section the product of two $M$-operators which enters into the r.h.s. of (63) has the following form: $$\begin{aligned} &M_{k}^{(12)}(x,y,\alpha,\bar\beta,\gamma,\bar\delta)= M_{k}^{(1)}(x,\alpha,\bar\beta)M_{k}^{(2)}(y,\gamma,\bar\delta)=\\ &e^{-\pi(x+y)/2}e^{-i\bar\beta e^{q_{k}}} \displaystyle\frac{1}{\Gamma(-i(x-p_{k})+1)}e^{i(\alpha-\gamma)e^{-q_{k}}} e^{\pi(y-p_{k})}\Gamma(-i(y-p_{k}))e^{i\bar\delta e^{q_{k}}}\nonumber\end{aligned}$$ Changing the holomorphic variables according to (58) we obtain: $$\begin{aligned} &M_{k}^{(12)}(x,y,\xi_{1},\xi_{2},\bar\xi'_{1},\bar\xi'_{2})= e^{-\pi(x+y)/2}e^{-i/{\sqrt{2}}(\bar\xi'_{1}+\bar\xi'_{2}) e^{q_{k}}}\nonumber\\ &\displaystyle\frac{1}{\Gamma(-i(x-p_{k})+1)} e^{i\sqrt{2}\xi_{2}e^{-q_{k}}} e^{\pi(y-p_{k})}\Gamma(-i(y-p_{k}))e^{i/{\sqrt{2}} (\bar\xi'_{1}-\bar\xi'_{2})e^{q_{k}}}\end{aligned}$$ This equation demonstrates that the kernel of $M^{(1)}(x)M^{(2)}(y)$ does not depends on the variable $\xi_{1}$ and for calculation of the $Q^{(1)}(x)Q^{(2)}(y)$ the dependence of (78) on the variable $\bar\xi'_{1}$ is irrelevant because the integration over $\xi',\bar\xi'$ in (63) results in deleting $\bar\xi'_{1}$ from (78). In such a way for the calculation of $Q^{(1)}(x)Q^{(2)}(y)$ we can use instead of $M_{k}^{(12)}(x,y,\xi_{1},\xi_{2},\bar\xi'_{1},\bar\xi'_{2})$ the following reduced object: $$\begin{aligned} &\tilde M_{k}^{(12)}(x,y,\xi,\bar\xi')= e^{-\pi(x+y)/2}e^{-i/{\sqrt{2}}\bar\xi'e^{q_{k}}}\nonumber\\ &\displaystyle\frac{1}{\Gamma(-i(x-p_{k})+1)} e^{i\sqrt{2}\xi e^{-q_{k}}} e^{\pi(y-p_{k})}\Gamma(-i(y-p_{k}))e^{-i/{\sqrt{2}} \bar\xi'e^{q_{k}}}.\end{aligned}$$ Note that $\tilde M^{(12)}(x,y)$ is nothing else but the kernel of $M^{(1)}(x)M^{(2)}(y)$ on the space $\pi\left(\Gamma^{(1)}\times \Gamma^{(2)}\right)$. Now let use expand the exponents which contain $\xi,\bar\xi$ in the r.h.s. of (79) and move the all the factors depending on $p_{k}$ to the right: $$\begin{aligned} &\tilde M_{k}^{(12)}(x,y,\xi,\bar\xi')= e^{-\pi(x+y)}\displaystyle\sum_{n,m=0} \frac{(i\sqrt{2}\xi)^n}{n!} (-i\bar\xi'/{\sqrt{2}})^m e^{(m-n)q_{k}} \nonumber\\ &\displaystyle\times\sum_{l=0}^{m}\frac{(-1)^{m-l}}{l!(m-l)!} \frac{\Gamma(-i(y-p_{k})-m+l)}{\Gamma(-i(x-p_{k})+1+n-m+l)} e^{\pi(y-p_{k})}.\end{aligned}$$ The summation over $l$ in (80) gives: $$\begin{aligned} &\displaystyle\sum_{l=0}^{m}\frac{(-1)^{m-l}}{l!(m-l)!} \frac{\Gamma(-i(y-p_{k})-m+l)}{\Gamma(-i(x-p_{k})+1+n-m+l)}=\nonumber\\ &\displaystyle\frac{(-1)^m}{m!}\frac{\Gamma(-i(y-p_{k})-m)} {\Gamma(-i(x-p_{k})+n+1)} \frac{\Gamma(-i(x-y)+m+n+1)}{\Gamma(-i(x-y)+n+1)}\end{aligned}$$ and we arrive at the following expression for the $\tilde M_{k}^{(12)}(x,y,\xi,\bar\xi')$: $$\begin{aligned} &\tilde M_{k}^{(12)}(x,y,\xi,\bar\xi')= e^{-\pi(x+y)}\displaystyle\sum_{n,m=0} \frac{(i\sqrt{2}\xi)^n}{n!}\frac{(i\bar\xi'/{\sqrt{2}})^m}{m!} e^{(m-n)q_{k}}\nonumber\\ &\displaystyle\times\frac{\Gamma(-i(y-p_{k})-m)} {\Gamma(-i(x-p_{k})+n+1)} \frac{\Gamma(-i(x-y)+m+n+1)}{\Gamma(-i(x-y)+n+1)}e^{\pi(y-p_{k})}.\end{aligned}$$ Now let $x$ and $y$ be $$\begin{aligned} x=z_{+}=z+i(l+1/2),\qquad y=z_{-}=z-i(l+1/2),\end{aligned}$$ where $l$ is an integer (half-integer). For these values of spectral parameters (82) takes the following form: $$\begin{aligned} &\tilde M_{k}^{(12)}(z_{+},z_{-},\xi,\bar\xi')= e^{-\pi z}\displaystyle\sum_{n,m=0} \frac{(i\sqrt{2}\xi)^n}{n!}\frac{(i\bar\xi'/{\sqrt{2}})^m}{m!} e^{(m-n)q_{k}}\nonumber\\ &\times \displaystyle\frac{\Gamma(-i(z_{-}-p_{k})-m)} {\Gamma(-i(z_{+}-p_{k})+n+1)} \frac{\Gamma(2l+m+n+2)}{\Gamma(2l+n+2)}e^{\pi(z_{-}-p_{k})}.\end{aligned}$$ Further we need to consider (82) for the opposite shift of spectral parameters $$\begin{aligned} x=z_{-}-i\epsilon,\qquad y=z_{+}+i\epsilon.\end{aligned}$$ We have introduced infinitesimal $\epsilon$ in (85) to remove an ambiguity which arises in (82) for these $x$ and $y$: $$\begin{aligned} &\tilde M_{k}^{(12)}(z_{-},z_{+},\xi,\bar\xi')= e^{-\pi z}\displaystyle\sum_{n,m=0} \frac{(i\sqrt{2}\xi)^n}{n!}\frac{(i\bar\xi'/{\sqrt{2}})^m}{m!} e^{(m-n)q_{k}}\nonumber\\ &\displaystyle\frac{\Gamma(-i(z_{+}-p_{k})-m)} {\Gamma(-i(z_{-}-p_{k})+n+l)} \frac{\Gamma(-2l-2\epsilon+m+n)}{\Gamma(-2l-2\epsilon+n)} e^{\pi(z_{+}-p_{k})}.\end{aligned}$$ For $\epsilon \to 0$ the fraction of $\Gamma$-functions in (86) takes the following values: $$\begin{aligned} \lim_{\epsilon \to 0} \displaystyle\frac{\Gamma(-2l-2\epsilon+m+n)}{\Gamma(-2l-2\epsilon+n)}= \left\{\begin{array}{cc} \displaystyle\frac{\Gamma(-2l+m+n)}{\Gamma(-2l+n)}, &n,m\geq 2l+1,\\ \displaystyle (-1)^m\frac{(2l-n)!}{(2l-n-m)!}, &2l\geq n+m \geq 0,\\ \displaystyle\frac{\Gamma(n+m-2l)}{\Gamma(n-2l)},&n\geq 2l\geq m\\ 0,&otherwise \end{array}\right.\end{aligned}$$ Apparently, the vanishing of the (87) in the fourth region manifests the triangularity of the operator the $\tilde M_{k}^{(12)}(z_{-},z_{+})$, therefore for the calculation of the trace of the product over $k$ of these operators we need to consider only the part of (87), corresponding to the first two regions. Thus, the resulting expression for the twice reduced operator has the following form: $$\begin{aligned} \stackrel{\approx}M_{k}^{(12)}(z_{-},z_{+},\xi, \bar\xi')=A_{k}(z,l,\xi,\bar\xi')+B_{k}(z,l,\xi,\bar\xi'),\end{aligned}$$ where $A$ contains the part of the r.h.s. of (86) with the summation over $n,m$ in the region $n,m\geq 2l+1$, $B$ contains the summation over $n,m$ in the region $2l\geq n+m \geq 0$. In other words, the degrees of $\xi,\bar\xi'$ in $A$ and $B$ have no intersection and therefore while the calculation of the product $Q^{(1)}(z_{-})Q^{(2)}(z_{+})$ these two parts will multiply coherently: $$\begin{aligned} &Q^{(1)}(z_{-})Q^{(2)}(z_{+})=\int \prod_{k=1}^{N}d\mu(\xi_{k}) \stackrel{\approx}M_{N}^{(12)}(z_{-},z_{+},\xi_{1},\bar\xi_{N}) \nonumber\\ &\times\stackrel{\approx}M_{N-1}^{(12)}(z_{-},z_{+},\xi_{N},\bar\xi_{N-1}) \cdots \stackrel{\approx}M_{1}^{(12)}(z_{-},z_{+},\xi_{2},\bar\xi'_{1}) \nonumber\\ \nonumber\\ &=\int \prod_{k=1}^{N}d\mu(\xi_{k})A_{N}(z,l,\xi_{1},\bar\xi_{N}) A_{N-1}(z,l,\xi_{N},\bar\xi_{N-1}) \cdots A_{1}(z,l,\xi_{2},\bar\xi_{1})\nonumber\\ \nonumber\\ &+\int \prod_{k=1}^{N}d\mu(\xi_{k})B_{N}(z,l,\xi_{1},\bar\xi_{N}) B_{N-1}(z,l,\xi_{N},\bar\xi_{N-1}) \cdots B_{1}(z,l,\xi_{2},\bar\xi_{1})\end{aligned}$$ Let us consider first $A$. For the convenience we will shift the values of $n,m$ by $2l+1$, then $$\begin{aligned} &A_{k}(z,l,\xi,\bar\xi)=(\xi\bar\xi')^{2l+1}e^{-\pi z} \displaystyle\sum_{n,m=0} \frac{(i\sqrt{2}\xi)^n}{n!}\frac{(i\bar\xi'/{\sqrt{2}})^m}{(m+2l+1)!} e^{(m-n)q_{k}}\nonumber\\ &\times \displaystyle\frac{\Gamma(-i(z_{-}-p_{k})-m)} {\Gamma(-i(z_{+}-p_{k})+n+l)} \frac{\Gamma(2l+m+n+2)}{\Gamma(2l+n+2)}e^{\pi(z_{-}-p_{k})}.\end{aligned}$$ Comparing (90) with (84), we see that they differ from each other by the factor $(\xi\bar\xi')^{2l+1}$ and shift of the factorial $m!$. This difference may be presented as appropriate transformation of $\tilde M_{k}^{(12)}(z_{+},z_{-},\xi,\bar\xi')$ : $$A_{k}(z,l,\xi,\bar\xi')=\int d\mu(\zeta)d\mu(\zeta')g_{l}(\xi,\bar\zeta) \tilde M_{k}^{(12)}(z_{+},z_{-},\zeta,\bar\zeta')f_{l}(\zeta',\bar\xi'),$$ where $$g_{l}(\xi,\bar\zeta)=(\xi)^{2l+1}e^{\xi\bar\zeta},\qquad f_{l}(\zeta,\bar\xi)= (\bar\xi)^{2l+1}\sum_{n=0}\frac{(\zeta\bar\xi)^{n}}{(n+2l+1)!}.$$ These two functions possess the following property: $$\int d\mu(\xi)f_{l}(\zeta,\bar\xi)g_{l}(\xi,\bar\zeta')= e^{\zeta\bar\zeta'}$$ The r.h.s. of (93) is the $\delta$-function in holomorphic representation. But note that $$\int d\mu(\xi)g_{l}(\zeta,\bar\xi)f_{l}(\xi,\bar\zeta')= \sum_{n=2l+1}\frac{(\zeta\bar\zeta')^n}{n!}.$$ Taking into account (93), we immediately obtain: $$\begin{aligned} &\int \prod_{k=1}^{N}d\mu(\xi_{k})A_{N}(z,l,\xi_{1},\bar\xi_{N}) A_{N-1}(z,l,\xi_{N},\bar\xi_{N-1}) \cdots A_{1}(z,l,\xi_{2},\bar\xi_{1})\nonumber\\ &=Q^{(1)}(z_{+})Q^{(2)}(z_{-})\end{aligned}$$ Our next step is the consideration of $B$ part of the $M^{(12)}(z_{-},z_{+})$. First of all we shall remove the $\sqrt{2}$ from its holomorphic arguments, because in the integral (89) these factors will cancelled out. Therefore we need to consider the following expression for $B$: $$\begin{aligned} B_{k}(z,l,\xi,\bar\xi')=e^{-\pi z}\displaystyle\sum_{t=0}^{2l}\sum_{m=0}^{t} \frac{\xi^{t-m}}{(t-m)!}\frac{\bar\xi'^m}{m!}(-1)^m i^{t+2l+1}e^{(2m-t)q_{k}}\nonumber\\ \times\frac{(2l+m-t)!}{(2l-t)!} \displaystyle\frac{\Gamma(-i(z-p_{k})+l-m+1/2)} {\Gamma(-i(z-p_{k})-l+t-m+l/2)}e^{\pi(z-p_{k})}\end{aligned}$$ We intend to compare this operator with Lax operator $L^{l}_{k}(x)$ of Toda chain with auxiliary space corresponding to the spin $l$. As it follows from the results of the 2-nd section, $L^{l}_{k}(x)$ could be obtain by the reduction of the operator $m_{k}(x)$ defined in (29) to the subspace corresponding to spin $l$. In the holomorphic representation the kernel of $L^{l}_{k}(x)$ could be easily found using the projection: $$L^{l}_{k}(x,\alpha,\bar\beta)=m_{k}(x-i(l+1/2)) \frac{(\alpha\bar\beta)^{2l}}{\Gamma(2l+1)}.$$ (Note that here we again use two-component variables $\alpha_{i},\beta_{i}, i=1,2$). In (97) the operator $m_{k}(x)$ should be understood as the differential operator, acting on the projection kernel $\frac{(\alpha\bar\beta)^{2l}}{\Gamma (2l+1)}$. For the calculation of the r.h.s. of the (97) recall that the operator exponential function in (29) is well defined because $$[i(p-x)+l_{3}, \rho_{1}^+\rho_{2}e^q]=[i(p-x)+l_{3}, \rho_{2}^+\rho_{1}e^{-q}]=0,$$ therefore we can expand the exponential function into formal series and find the action of each term on the projection kernel: $$\begin{aligned} &m_{k}(x-i(l+1/2))\displaystyle\frac{(\alpha\bar\beta)^{2l}} {\Gamma(2l+1)}=\\ \nonumber\\ &\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^n\Gamma(i(p_{k}-x)+\rho_{1}^+ \rho_{1}-l+\frac{1}{2})}{\Gamma(i(p_{k}-x)+\rho_{1}^+\rho_{1}-l-n+\frac{1} {2})} \frac{(i\rho_{1}^+\rho_{2}e^{q_{k}})^n}{n!} \frac{(\alpha_{1}\bar\beta_{2}e^{q_{k}}- \alpha_{2}\bar\beta_{1}e^{-q_{k}})^{2l}}{\Gamma(2l+1)}.\nonumber\end{aligned}$$ Apparently, only $2l$ terms in (99) will survive because the differential operator $(\rho_{2})^n$ acts on the polynomial. The result has the following form: $$\begin{aligned} &L^{l}_{k}(x,\alpha,\bar\beta)=\displaystyle\sum_{t=0}^{2l}\sum_{m=0}^{t} e^{(2m-t)q_{k}}\frac{\Gamma(-i(x-p_{k})-m+l+1/2)}{\Gamma(-i(x-p_{k})-m+t-l +1/2)}\nonumber\\ &(-1)^mi^{2l+t}\displaystyle\frac{\alpha_{1}^{2l-t+m}\alpha_{2}^{t-m} \bar\beta_{1}^{2l-m}\bar\beta_{2}^m}{(2l-t)!(t-m)!m!}\end{aligned}$$ This $L$-operator defines the transfer matrix of Toda chain with auxiliary space, corresponding spin $l$: $$\begin{aligned} t^{l}(x)=\int \prod_{k=1}^{N}d^2\mu(\alpha_{k})L^{l}_{N}(x,\alpha_{1},\bar\alpha_{N}) L^{l}_{N-1}(x,\alpha_{N},\bar\alpha_{N-1})\cdots L^{l}_{1}(x,\alpha_{2},\bar\alpha_{1})\end{aligned}$$ If in this formula we will perform the integration over one pair of the holomorphic variables, corresponding for example $\alpha_{1},\bar\beta_{1}$ in (100), the integrand still will be presented in the factorized form, but with new, reduced kernel of $L$-operator: $$\begin{aligned} &\tilde L^{l}_{k}(x,\alpha,\bar\beta)= \displaystyle\sum_{t=0}^{2l}\sum_{m=0}^{t} e^{(2m-t)q_{k}}\frac{\Gamma(-i(x-p_{k})-m+l+1/2)}{\Gamma(-i(x-p_{k})-m+t-l +1/2)}\nonumber\\ &(-1)^mi^{2l+t}\displaystyle\frac{\alpha_{2}^{t-m}\bar\beta_{2}^m}{(2l-t)! (t-m)!m!}(2l-t+m)!\end{aligned}$$ Comparing (102) with (96) we find that $$B_{k}(z,l,\xi,\bar\xi')=\tilde L^{l}_{k}(z,\xi,\bar\xi)ie^{-\pi p_{k}}.$$ Therefore $$\begin{aligned} &\int \prod_{k=1}^{N}d\mu(\xi_{k})B_{N}(z,l,\xi_{1},\bar\xi_{N}) B_{N-1}(z,l,\xi_{N},\bar\xi_{N-1}) \cdots B_{1}(z,l,\xi_{2},\bar\xi_{1})\nonumber\\ &=i^{N}t^{l}(x)e^{-\pi P},\end{aligned}$$ where $$P=\sum_{k=0}^{N}p_{k}$$ is the integral of motion, which commutes with $t^l(x)$. In the derivation of (104) we have moved all the factors $e^{-\pi p_{k}}$ to the right to form $e^{-\pi P}$. Gathering together (89), (95) and (104) we obtain the following functional relations: $$\begin{aligned} &Q^{(1)}(z-i(l+\frac{1}{2}))Q^{(2)}(z+i(l+\frac{1}{2}))- Q^{(1)}(z+i(l+\frac{1}{2}))Q^{(2)}(z-i(l+\frac{1}{2}))\nonumber\\ &=i^{N}t^{l}(x)e^{-\pi P}\end{aligned}$$ For $l=0$ the transfer matrix turns into $1$ and we have the simplest wronskian relation: $$\begin{aligned} Q^{(1)}(z-i/2)Q^{(2)}(z+i/2)- Q^{(1)}(z+i/2)Q^{(2)}(z-i/2)%\nonumber\\ =i^{N}e^{-\pi P}\end{aligned}$$ For the illustration of this identity the reader can use the $Q^{(i)}$-operators for one degree of freedom (47), (48). In this simplest case (107) reduces to the well-known identity for Bessel functions: $$I_{\nu}(z)I_{-\nu+1}(z)-I_{-\nu}(z)I_{\nu-1}(z)= -\frac{2\sin(\pi\nu)}{\pi z}$$ The general case (106) for one degree on freedom is related to Lommel polynomials [@Bateman]. The functional relations of the type (106) was first established for certain field theoretical model in [@BLZ]. In the recent paper of the author with Yu.Stroganov [@PS] we have discussed the analogues relation for the eigenvalues of $Q$-operators in the case of isotropic Heisenberg spin chain. Originally, since the Baxter paper[@Baxter] the existence of one $Q$-operator was considered as important alternative for Bethe ansatz. The relations (106) show the importance of the second $Q$-operator which together with the first one give rise to the numerous fusion relations (see e.g.[@BLZ],[@PS]). Discussion ========== The approach we have considered in the present paper could be applied also to the other with rational $R$-matrix – the discrete self-trapping (DST) model, considered in [@Sklyanin1]. The quantum determinant of Lax operator for this model is not unity and Baxter equation has the following form: $$t(x)Q(x)=(x-i/2)^N Q(x-i)+Q(x+i).$$ The general properties of the $Q$-operators for DST-model are similar to that of Toda system. The eigenvalues of one $Q$-operator are polynomial in spectrum parameter , while the eigenvalues of the second are meromorphic functions. In the case of Toda system the eigenvalues of $Q^{(1)}(x)$ are entire functions, the eigenvalues of $Q^{(2)}(x)$ are meromorphic. For the DST -model there also exist the functional relations similar to (106). The most interesting would be the application of the formalism to the case of XXX-spin chain. The situation here is the following. In [@Pronko] we have constructed the family of $Q(x,l)$-operators similar to (31). Moreover, from the results of [@PS] follows that for XXX-spin chain there exist the basic $Q$-operators. Making use of the formalism of the section 3, it is possible to the find the $M^{(i)}_{k}(x)$-operators for this case, but the trace of monodromies corresponding to $M^{(i)}_{k}(x)$ diverges. This puzzle deserves further investigation. Another interesting point we want to discuss is the relation of our $M^{(i)}_{k}(x)$-operators with quantum linear problem for Lax operator (5). In classical case the linear problem is the main ingredient of inverse scattering method, in the same time for the quantum theory it seems to be unnecessary (see for example excellent review on the subject [@Faddeev]). However, let us consider the following problem: $$\psi_{n+1}(x)=L_{n}(x)\psi_{n}(x),$$ where $L_{n}(x)$ is given in (5) and $\psi_{n}$ is two component quantum operator. From the multiplication rules (43) we obtain: $$\begin{aligned} \left(L_{n}(x)\right)_{ij}M_{n}^{(1)}(x)\left(\begin{array}{c} 1\\ \rho\end{array}\right)_{j}=\left(\begin{array}{c} 1\\ \rho\end{array}\right)_{i}M_{n}^{(1)}(x.i.).\end{aligned}$$ Now let us define the operator $$\begin{aligned} \left(\psi_{n}^{(1)}\right)_{i}(x)=Tr\left(\prod_{k=n}^{N}M_{k}^{(1)}(x) \left(\begin{array}{c} 1\\ \rho\end{array}\right)_{i}\prod_{k=1}^{n-1}M_{k}^{(1)}(x-i)\right),\end{aligned}$$ where the trace is taken over auxiliary space. Apparently, due to (111) the operator (112) does satisfy the equation (110). For $n=1$, the solution has the following form: $$\begin{aligned} \left(\psi_{1}^{(1)}\right)_{i}(x)=Tr\left(\prod_{k=1}^{N}M_{k}^{(1)}(x) \left(\begin{array}{c} 1\\ \rho\end{array}\right)_{i}\right)=Q^{(1)}(x)\left(\begin{array}{c} 1\\ ie^{q_{N}}\end{array}\right)_{i}, \end{aligned}$$ where on the last step we have used the explicit form of $Q^{(1)}(x)$-operator for the calculation of the trace. On the other hand the solution (113) translated to the period $N$ by the monodromy (8), due to (111) is given by $$\begin{aligned} \left(\psi_{N+1}^{(1)}\right)_{i}(x)=Tr\left( \left(\begin{array}{c} 1\\ \rho\end{array}\right)_{i}\prod_{k=1}^{N}M_{k}^{(1)}(x-i)\right)=Q^{(1)} (x-i)\left(\begin{array}{c} 1\\ ie^{q_{N}}\end{array}\right)_{i}.\end{aligned}$$ In other words we obtain $$T_{ij}(x)\left( \psi_{1}^{(1)}\right)_{j}(x)=\frac{Q^{(1)}(x-i)}{Q^{(1)}(x)}\left( \psi_{1}^{(1)}\right)_{i}(x).$$ This equation may be understood as quantum analogue of the property of Bloch solutions, which are the eigenvectors of the translation to the period. Similarly we can consider the second solution. Indeed, from the multiplications rules (43) for the $M_{n}^{(2)}(x)$ we obtain: $$\begin{aligned} \left(L_{n}(x)\right)_{ij}M_{n}^{(2)}(x)\left(\begin{array}{c} -\rho^+\\ 1 \end{array}\right)_{j}=\left(\begin{array}{c} -\rho^+\\ 1\end{array}\right)_{i}M_{n}^{(2)}(x-i).\end{aligned}$$ Therefore the operator $$\begin{aligned} \left(\psi_{n}^{(2)}\right)_{i}(x)=Tr\left(\prod_{k=n}^{N}M_{k}^{(2)}(x) \left(\begin{array}{c} -\rho^+\\ 1\end{array}\right)_{i}\prod_{k=1}^{n-1}M_{k}^{(2)}(x-i)\right),\end{aligned}$$ possesses the same properties as (112). The initial value of (117) is given by $$\begin{aligned} \left(\psi_{1}^{(2)}\right)_{i}(x)=Tr\left(\prod_{k=1}^{N}M_{k}^{(2)}(x) \left(\begin{array}{c} -\rho^+\\ 1\end{array}\right)_{i}\right)=Q^{(2)}(x)\left(\begin{array}{c}-ie^{q_{1}} \\ 1\end{array}\right)_{i}, \end{aligned}$$ where we again on the last step have used the explicit form of $M^{(2)}(x)$. As above we obtain $$T_{ij}(x)\left( \psi_{1}^{(2)}\right)_{j}(x)=\frac{Q^{(2)}(x-i)}{Q^{(2)}(x)}\left( \psi_{1}^{(2)}\right)_{i}(x).$$ In such a way using $M^{(i)}_{n}(x)$-operators we succeeded in the construction of the operators which may be interpreted as the quantum analogues of the Bloch functions of the corresponding linear problem. In the classical theory of finite-zone “potentials”, two Bloch solutions of the linear problem, as the functions of spectral parameter are actually the projections of the Backer-Akhiezer function, which is the single-valued meromorphic functions on an hyper-elliptic surface. In quantum case the Bloch functions (112) and (117) do not possess the branching points (in weak sense) which is the trace of the projection in the classical case, therefore their intimate relation is somehow hidden and it will be very interesting to uncover this relationship. Acknowledgments =============== The author is grateful to E.Skyanin, S.Sergeev, for their interest and discussions, This work was supported in part by ESPIRIT project NTCONGS and RFFI Grant 98-01-0070. [9]{} Baxter R.J.Stud.Appl.Math, L51-69, 1971; Ann.Phys. N.Y., v.70, 193-228, 1972; Ann.Phys. N.Y., v.76, 1-71, 1973. Bazhanov V.V., Lukyanov S.L., Zamolodchikov A.B. Commun. Math. Phys. v.190, 247-78, 1997; v.200, 297-324, 1998. Sklyanin E.K.,Kuznetsov V.B, Salerno M. solv-int/9908002 Pronko G.P. hep-th/ 9908179, to appear in Comm. Math. Phys. Sklyanin E.K. Lecture Notes in Physics v.226,196, 1985; Berezin F.A. The Method of Second Quantization. Academic Press, New York, 1966 Pasquer V., Gaudin M. J.Phys.A: Math.Gen. v.25, 5243-52, 1992. Erdelyi A. et al. Higher Transcemndental Functions NY:McGrow Hill, 1953. Pronko G.P., Stroganov Yu.G. J.Phys.A: Math.Gen. v.32, 2333-40, 1999. Faddeev L.D. UMANA v.40, 214, 1995 (hep-th/9605187)

[**MODEL CATEGORIES AND QUANTUM GRAVITY**]{} by: [**LOUIS CRANE; Math Department, KSU**]{} [**ABSTRACT:**]{} We propose a mathematically concrete way of modelling the suggestion that in quantum gravity the spacetime disappears, replacing it with a discrete approximation to the causal path space described as an object in a model category. One of the versions of our models appears as a thickening of spacetime, which we interpret as a formulation of relational geometry. Avenues toward constructing an actual quantum theory of gravity on our models are given a preliminary exploration. [**I. INTRODUCTION**]{} It has been shown recently [@Susskind] that a bounded region in a spacetime in general relativity can only transmit a finite amount of information to the exterior, proportional to its boundary area in Planck units. If we want to construct a quantum theory of GR which would describe the results of experiments which external observers perform on bounded regions of spacetime, then the description of spacetime as a point set continuum contains too much information. It was the view of Einstein [@Stachel] that the excess of information of a metric on a continuum spacetime is the source of the problems at the core of quantum field theory and quantum gravity. More recently [@AB] it has been proposed that the continuum disappears in the quantum theory of general relativity. There exists a model for quantum gravity, the BC model [@BC] which is finite on any finite four dimensional simplicial complex. This however leaves the problem of choosing the particular triangulation of spacetime. On the other hand there is the GFT model [@Rovelli], which sums over all triangulations. Unfortunately, this is an infinite sum. We have long believed that incorporating the finite information results for a bounded region into a more sophisticated model intermediate between BC and GFT, would be the resolution of this dilemma, but that that would need a more subtle type of topological structure. In this paper we search for models for the topology of a region of spacetime R which only contain information accessible to an external observer in general relativity. In order to do this, we make a physical conjecture that one type of information, namely the multiple images of a light source exterior to R as viewed by an observer through R, can give a complete description of the observable state of R. We call this the [**lensing hypothesis**]{}, and discuss its plausibility below. Assuming our conjecture, we show that well understood and very computable models for the topology of spacetime regions exist. There are several equivalent constructions, going under the names of Quillen [@Quillen] and Sullivan [@Sullivan] models. The mathematical reason we can do this is that some of the multiple images of the light source correspond to rational cohomology classes of the loop space of R via Morse theory. The rest of the images also fit into the Sullivan models in a natural way as contractible algebras. Perhaps the most interesting version of the model, in light of the developments of categorical state sums for TQFTs and quantum general relativity [@CY; @CF], is a differential graded Hopf algebra. Throughout this paper, we will assume that all the regions we study are simply connected. Mathematically, this is because the model category constructions we use only work in that case. The theorems we state would need to be weakened for multiply connected regions. Physically, we feel comfortable with this restriction because it is hard to imagine a multiply connected region on a macroscopic scale which could be treated in isolation from its surroundings. On a microscopic scale, we are effectively ruling out wormholes, which are incompatible with the positive energy condition. It would be possible, although more difficult, to weaken this assumption. The family of functors Quillen constructed allow us to reconvert the models into cellular and simplicial complexes in two different ways, raising the possibility of a BC type state sum on them. At the end of the paper, we begin a discussion of how to construct quantum theories over these models which could describe gravity and matter. This subject is still a work in progress, but there are several promising directions. Before we go on to explain the constructions of these models and how they relate to physical processes, we shall say a little about the philosophy which led us to them. In another famous quote, Einstein said that the experience of measuring the surface of the earth, which was the origin of geometry, led to the intuition that spacetime is a kind of substance, which he believed to be profoundly misleading. I think the idea of a fixed absolute spacetime point set, independent of the observer, is the result of such an intuition. The classical continuum, which we got from Euclid via Newton and Weierstrauss, is an extension of the results of everyday experience to a domain in which they do not apply. There is no good reason to suppose that a spacetime interval shorter than the Planck scale can be infinitely subdivided, and importing the assumption that it can via the trojan horse of background coordinates leads quantum theories of gravity into the swamp of the ultraviolet divergences. A very important motivation for this paper is the distinction between absolute and relational position. A region in spacetime thought of as a point set has its “position” determined by what set of points it is. This idea of position is unphysical. Rather, the foundation of a quantum theory of gravity should be [*relational*]{} position, by which I understand where some region appears to be to all external observers [@newcrane]. Now if we imagine two different spacetime metrics on a given bounded region, a subregion in one will have a different relational position from any subregion of the other. This is because the curvature of the metrics will bend light rays going in different directions by different amounts, so that a subregion that appeared to be in the same place as a subregion in a different metric to one outside observer would not appear so if observed by another observer from a different direction. Thus a region in a quantum state for the spacetime metric would have more subregions than a classical region. They would be arranged in layers, so that the quantum region would appear like a thickening of the classical region. I think of this as like a sheet of paper. At length scales large compared to its thickness, a sheet of paper can be described as a smooth surface, but at shorter scales it is a very complex web. If we are serious about abandoning the absolute point set background, we need to take the point of view that quantum gravity lives on some such thickening. For this reason, I think it is interesting that one of the collection of functors we will be studying, namely the geometric realisation of Sullivan algebras, produces precisely such a thickening. We will explore near the end of this paper the possibility of constructing a state sum model for general relativity on the geometric realisation of a Sullivan algebra. A region of spacetime which is to be treated as quantum cannot have classical observers inside it. Like any quantum process it can only be discussed after it is complete. We must not ask what it looks like on the inside, but only about regularities of relationships between external observations. Like any quantum system, GR must have observables that are complementary, that is cannot simultaneously possess sharp values. Already the development of the BC model [@BC] shows us that different geometrical data such as areas of faces cannot have simultaneous sharp values. This has implications for the topology appropriate to the descriptions of spacetime regions. Rather than a single topology, we should expect to find complementary descriptions of the topology appropriate to complementary ways of observing the geometry. This may seem hard to imagine within the framework of point set topology, but modern homotopy theory has many structures which can be considered to have a topology, and natural approaches exist connecting them to different ways of probing the spacetime geometry of a region. The different forms of the Quillen and Sullivan models we shall describe below suggest complementary descriptions of spacetime, as we shall indicate. The reason is that some of the models correspond to discretizations of the spacetime region itself, and some to discretizations of its loop space, or causal path space. [**II. SOME THOUGHTS ON THE LENSING HYPOTHESIS**]{} In a sense the lensing hypothesis is rather natural. If a region of spacetime is too small to enter by a classical observer, there isn’t much else to do except “hold it up to the light”. A classical observer inside a region of spacetime would decohere it, so unless we want to think about many worlds pictures, it is contradictory for a quantum region to have an internal observer. Also, since as we mentioned above, a bounded region can only transmit a finite amount of information to its exterior, and since most nontrivial topologies have an infinite number of images in a classical spacetime description of lensing, [@Perlick] there is certainly enough data. What we directly see when we observe a curved region of spacetime is only a discrete decomposition of its space of causal paths. Mathematicians have discovered that such decompositions have algebraic structure (inherited from the multiplication on loop space) which are handy descriptions of topological spaces. Such are the Quillen models which we are proposing to use in GR. A natural objection to the lensing hypothesis is that a large flat region of spacetime transmits only a single image of a light source. I believe the resolution of this is that a classical state of any quantum system emerges as in the decoherence picture as a superposition of many similar quantum states. We can then invoke the suggestion of Rovelli [@Rovelli2] that a particle travelling through a quantized spacetime would automatically appear quantized, since even classical GR imposes the equations of motion on any particle travelling through it. Combining the idea of many similar quantum states reinforcing with Rovelli’s suggestion, we end up with the picture that the single fuzzy quantum image of a source is the superposition of many multiple images in which the source is scattered by many smaller strongly curved quantum regions, obtaining the quantum propagator from a kind of random walk. In other words, the quantum mechanical uncertainty of the positions of optical images can be explained as a superposition of the multiple images of an ensemble of spacetime geometries, each highly curved near the Planck scale. If this picture could be precisely implemented, it would add strength to the lensing hypothesis. Finally, to be honest, we must invoke the streetlight principle. The mathematical picture which results from the hypothesis is rich and computationally accessible, so we investigate it because we can. It would be possible also to state a [**weak lensing hypothesis,**]{} namely that the locations and intensities of the multiple images define an interesting sector of quantum gravity in a region. We choose not to preface every result in the following with the phrase “in the lensing sector,” but the reader may do so. [**III. DESIDERATA FOR A DESCRIPTION OF QUANTUM REGIONS**]{} If a region of spacetime is not described mathematically by a point set, then some other mathematical structure is necessary. Now let us list the properties a mathematical description of a region of quantum spacetime should have. We shall include the lensing hypothesis in our assumptions. [*A model of quantum spacetime should:*]{} [*A. Assign a mathematical structure Q(R) to each region of spacetime.*]{} [*B. Assign to each mapping of regions $L : R \longrightarrow S$ a map of Q structures* ]{} [*which takes compositions to compositions.*]{} [*C. Have a class of homotopies on maps of Q structures so that given any two maps of one region to another which are homotopic (can be smoothly deformed into one another), there exists a homotopy of the corresponding maps on Q structures, again preserving composition.*]{} [*D. The structure Q(R) should have enough information to compute the probability amplitudes for the apparent positions and arrival times of the multiple images of light sources seen through R, and be reconstructable from this information.*]{} The physical motivation for this list is that subregions can be effectively included, and the principle of diffeomorphism invariance, so central to general relativity, can be expressed; together of course, with the lensing hypothesis. The mathematical motivation is that the construction be functorial or actually 2-functorial. That is the operation Q is actually a functor from the 2-category of regions, smooth maps and smooth homotopies to a category of models for quantum regions, also equipped with maps and homotopies. The central purpose of the rest of this paper is to show that the problem of constructing models satisfying our list is already solved in the context of rational homotopy theory by the theory of model categories. This is due to a string of mathematical coincidences which is actually rather obvious once one has assembled the necessary background. [**IV. MORSE THEORY, GEODESICS, AND LENSING**]{} Morse theory is a branch of smooth topology which obtains information about the topology of a smooth manifold by studying the singularities of generic smooth functions on it. A function is generic if it has only singularities of the lowest order, i.e. if the first derivatives only all vanish at isolated points, and the matrix of second derivatives at each such point is nonsingular. For a good introduction, see [@Milnor]. When one has a Morse function on a manifold, the gradient flow down to the singularities divides the manifold into standard structures called handles. Thus a Morse function is a snapshot of a handlebody decomposition of the manifold. Critically important for us are the Morse inequalities. These tell us that a Morse function on a manifold M must have at least as many singularities of degree d as the number of generators of the cohomology of M in dimension d with real coefficients. A perfect Morse function has just so many singularities; in general a Morse function may have more than these, but they come in cancelling pairs of adjacent dimensions. The reason we know this is a branch of mathematics called Cerf theory [@Cerf], which studies how we can move from one Morse function to another, moving through intermediate functions with only the simplest types of higher singularities. This gets us from one Morse function to another by a series of “moves,” and the only move which changes the number of singularities adds two cancelling handles in adjacent dimensions, not changing the topology. The singularities of Morse functions only correspond to the cohomology with real (or equivalently rational) coefficients; cocycles which are torsion (some multiple is zero) play no part. Cohomology with real coefficients is always a real vector space with the same dimension as the rational dimension of the rational cohomology. Mathematicians like to speak of rational cohomology, or sometimes cohomology with coefficients in an arbitrary field of characteristic zero. Physicists prefer to speak of real cohomology. I shall speak of them interchangeably. The classical Morse theory was for finite dimensional manifolds, but infinite dimensional Morse theory has been developed by many authors \[11, 12, 13\]. An important application was the study of closed geodesics on Riemannian manifolds. If we consider the space of smooth closed loops on M with a given basepoint $ \Omega (M)$, which is a smooth infinite dimensional manifold, the length, or better the energy (see [@Milnor]) of the path turns out to be a Morse function generically. Thus the problem of closed geodesics on M is related to finding the rational cohomology of $ \Omega (M)$. Now the space of paths in M with fixed endpoints p and q is very similar to the space of closed loops. This is because the smooth path space P(p,q,M) is of the same homotopy type as $ \Omega(M)$. Hence counting the geodesics with fixed endpoints is a very similar problem to the classical one of counting closed geodesics. This is another result from topology which will be important to us. Now let us turn to the problem of gravitational lensing, or describing the multiple images which an observer in a spacetime may see of a light source in its causal past. Fermat’s principle continues to hold in general relativity. Classical light rays correspond to null geodesics in the spacetime. Thus the problem of counting and describing the images an observer sees of a pointlike light source is closely analogous to finding the geodesics in a Riemannian manifold with fixed endpoints, but with a Lorentzian signature metric. The spacetime approach to gravitational lensing rests on an adaptation of Morse theory to the spacetime case [@Perlick; @Uhlenbeck]. One has to modify the application of Morse theory by considering the space of all smooth null paths which begin at a fixed event x in a spacetime S and end at some point on the worldline of an observer y(t). This is because different null geodesics have different arrival times. We denote this infinite dimensional space P(x,y,S). The arrival time acts as the Morse function. Once again, it is closely related to an infinite dimensional space, which in physically reasonable cases (i.e. globally hyperbolic spacetimes) is of the same homotopy type as the loop space of the spacetime [@Perlick]. If the time lapse happens to be a perfect Morse function, then the images will count a set of generators of the cohomology ring of the loop space $H_Q ( \Omega (S))$ for the reasons we have discussed. In general, this is not so. There can also be other images which come in pairs, with opposite orientations. These pairs of images disappear by meeting at “folds in the sky” as the position of the observer or source is varied. The folds in the sky appear as caustics in lensing images, and have been photographed by astronomers. Folds in the sky correspond to cancelling handles. [**EXAMPLE: SCHWARTZSCHILD BLACK HOLE**]{} It is known from an exact calculation [@K] that the Schwartzschild black hole has an infinite series of images as a gravitational lens. This reflects the fact that orbits in general relativity are not closed, and can circle the central mass any number of times before escaping to infinity. It would be a difficult problem to determine directly what any perturbation of the Schwartzschild solution might do to its lensing properties. It is therefore of interest that the infinite number of images can be predicted by a purely topological calculation [@Perlick]. This is possible because the Schwartzschild solution has the homotopy type of $S^2$. Thus the number of lensing images is at least the dimension of $H_Q ( \Omega (S^2))$, which is known to be infinite. As we shall see below, calculation of the rational cohomology of the loop groups of spheres was one of the seminal problems for rational homotopy theory, and can most easily be solved by means of model category theory. At this point we can already see that the lensing hypothesis leads us into a situation where the information we would observe about a region is closely related to rational cohomology of loop spaces and therefore to rational homotopy theory. [**V. RATIONAL HOMOTOPY THEORY, MODEL CATEGORIES AND LOOP SPACES.** ]{} If we were able to observe the lensing effect of a highly curved region of spacetime, the images that did not disappear in pairs as we rotated our position of observation around the region would correspond to a set of generators for the rational cohomology of the region. (It is also possible for a cancelling pair to appear and for one of the new handles to cancel an image formerly representing a cohomology cycle, leaving its partner to represent the cycle, but that doesnt change the point). This suggests that a quantum theory of the lensing sector of gravity could be constructed by replacing the region with its rational homotopy type. It will be seen that the images which cancel in pairs and therefore represent cancelling handles also fit into the models for rational homotopy theory as contractible models. We remind the reader that in this paper all regions are assumed to be simply connected. In fact the Quillen and Sullivan models are sophisticated descriptions of discrete decompositions of spacetime or its causal path space. They inherit their algebraic structure from the multiplication on the loop space or the multiplication of differential forms. The rational homotopy type of a space is much simpler than its general homotopy type. The rational homotopy and cohomology groups of simple spaces can be computed exactly, whereas the homotopy and cohomology groups including the torsion, are not all simultaneously known for any nontrivial space, one being hard whenever the other is easy. The calculations of rational homotopy and cohomology proceed by replacing the space with a model for its rational homotopy type. The calculations for loop spaces proceed by finding a model loop space for the model. There are several different constructions, which have been shown to be equivalent [@M]. In this section we give a nontechnical introduction to these mathematical ideas. Two maps $ L_1, L_2 : A \rightarrow B$ are homotopic if there exists a map $ L: A \times I \rightarrow B $ such that $L(a,0) = L_1 (a), L(a,1) = L_2 (a) $. L is called a homotopy. Two spaces A and B are said to have the same homotopy type if there are continuous maps $f: A \rightarrow B$ and $ g: B \rightarrow A $ such that both $ f \circ g$ and $g \circ f$ are homotopic to the identity. For most purposes, we can think of A and B as topologically the same in this case. A weak homotopy equivalence is a map between two spaces which induces an isomorphism on cohomology. In many cases this implies homotopy equivalence. Now we can define a rational homotopy equivalence as a map between two spaces which induces an isomorphism on the rational cohomology groups of the two spaces. The equivalence class of a space under this equivalence relation is its rational homotopy type. Note that the existence of a map is necessary in this definition. Two spaces whose rational cohomology groups are isomorphic as groups need not be rationally homotopy equivalent. If two spaces are rational homotopy equivalent, it does follow that their rational cohomology is isomorphic. Any space has a rationalisation, i.e. a map to a space whose homotopy and cohomology groups are rational (that is, torsion free), such that the map is a rational homotopy equivalence. For example, none of the spheres are rational, since they all have torsion in their higher homotopy groups. Even the complete homotopy groups of $S^2$ are unknown. By contrast the rational part of the homotopy groups of $S^2$ are generated by one generator in dimension 2 and one in dimension 3. The reason rational homotopy theory is so computable is that the rational homotopy category is equivalent to a number of model categories. Put more concretely, a space can be replaced, for the purposes of rational homotopy theory, by a mathematical structure which is simpler and can be computed with and constructed on as if it were geometrical. This can be done several ways; and the constructions have strong resonances with mathematical Physics. [**B. THE QUILLEN AND SULLIVAN MODELS**]{} The first models for rational homotopy types appear in Quillen [@Quillen] who had already formalized the notion of a model category in [@Quillen2]. (I reccommend that anybody not primarily grounded in homotopy theory learn about the examples first and the formalisation only as needed.) Quillen proved that the category of rational homotopy types of topological spaces is equivalent to several other categories. One is the category of complete cocommutative differential graded Hopf algebras (A graded algebra or coalgebra is graded (co)commutative if two odd indexed elements anti(co)commute, and any other combination (co)commutes). Quillen’s construction proceeds by a series of transformations of the topological space. Each transformation is shown to be a homotopy equivalence of categories. Each category in the chain is an abstract model category, which means that objects in it can be thought of as having topology, or differently put, that homotopy theory can be done inside the category. Each transformation is expressed by a pair of adjoint functors that go both ways between the two categories. This means that each transformation can be reversed, without changing the rational homotopy type. This possibility of going back and forth between different constructions is what leads me to believe that the Quillen mathematical setting is a flexible and powerful one. In particular, it is possible to change the algebraic models back into cellular or simplicial complexes, thus opening the possibility of putting categorical state sum models for gravity, such as the BC model, on them. In the interest of accessibility of exposition, we shall not discuss the formal definition of a model category; this can be found in [@Quillen2], or a good number of more recent books. Quillen originally did only the simply connected case, this can be generalized, but will be sufficient for our purposes. The first step is begun by replacing the space X with a simplicial complex. A physicist might think of this as triangulating the space, then describing the space only by the discrete combinatorial structure of the triangulation. Quillen actually replaces the space X with the complex C(X) of all singular simplices in X. (A singular simplex in X is a map of the standard simplex into X.) This is homotopy equivalent to the naive picture, but more flexible. As a downside, it produces an enormous space, which is difficult to compute with directly. The first step is concluded by replacing the simplicial complex by a 2-reduced simplicial complex, i.e. one in which there is only one vertex. This can always be done for a simply connected complex by contracting a maximal tree in its 1-skeleton. This last is not strictly necessary, but greatly simplifies the subsequent algebra. In the second step, the 2-reduced complex C(X) is replaced by the free group (group of strings) of simplices in C(X). This is denoted $C( \Omega (X))$. The reason for the notation is that $C( \Omega (X))$ is a complex for the loop space of X. (Think of a string of simplices as the family of paths in X which traverse those simplices in that order.) Strings are multiplied by concatenation, the inverse of a string traverses the simplices backwards and in the reverse order. $C( \Omega (X))$ is an example of a simplicial group, which is a simplicial complex where the simplices of each dimension form a group and the boundary and degeneracy operations are homomorphisms. The singular simplicial complex C(G) of a topological group G is a simplicial group under pointwise multiplication of simplices, and the simplicial group structure of $C( \Omega (X))$ comes from the natural group structure on $ \Omega (X)$. Quillen also proves that any reduced simplicial group comes from a space in this way. The third step is to replace the simplicial group with the differential graded Hopf algebra $GHA( \Omega (X))$ of rational functions on the set of its simplices, then completing it in a certain topology. A differential graded Hopf algebra is a differential graded vector space, (i.e. a sequence of spaces $V_i$ for each integer i with a map $d_i: V_i \rightarrow V_{i-1} $ with $d^2=0$) with a multiplication and comultiplication which are additive in degree and satisfy the axioms of a Hopf algebra. The multiplication in this Hopf algebra comes from composition in $\Omega(X)$; its comultiplication is the standard comultiplication on simplices, dual to the cup product of cohomology. In this section the definition of Hopf algebra we are using includes the assumption that it is graded cocommutative. In the section on the Sullivan model, we are in a dual picture, so Hopf algebras are graded commutative. Since this is in particular a category of graded vector spaces, it is no surprise that the torsion part of the homology has disappeared, so the ordinary homotopy theory of complete DG Hopf algebras corresponds to the rational homotopy theory of the original space. The definition of the intrinsic homotopy theory of this category is quite technical. As in the earlier steps, Quillen actually shows that the category of “complete” (I am suppressing that part of the definition for brevity) cocommutative DG-Hopf Algebras is homotopy equivalent to the earlier categories, in particular the category of rational homotopy types of simply connected spaces. After this, Quillen goes on to show that the class of Hopf algebras we are interested in are all universal enveloping algebras of differential graded Lie algebras, so that there are also differential graded Lie algebra models. The Lie algebras can be recovered as the primitive elements of the Hopf algebras, i.e. those which are not non-trivial products. The Lie algebra model for X has the property that its homology is a graded Lie algebra, which corresponds to the rational homotopy groups of X with the Whitehead product [@Book] for bracket. The rational homology of the loop space of X is the universal enveloping algebra of the Lie algebra of the rational homotopy groups of X We will not need here to examine Quillen’s last model, in which he replaces the DG Lie algebra by the DG coalgebra of its bar construction, or classifying space. Another transformation is possible which Quillen did not develop. We can replace the DG Hopf algebra by the category of its representations $Rep(HA( \Omega (X)))$. This will be worked out in detail in a subsequent paper. Given the analogous roles played by Hopf algebras and tensor categories of representations in the construction of TQFTs [@CY; @CF], this further transformation is very natural for our purposes. We will comment on the physical interest of this new variant of Quillen’s model below. Sullivan [@Sullivan] discovered a new approach to constructing models for rational homotopy theory. He began by constructing the graded algebra of forms on a simplicial complex. To each simplex S he associated the differential graded commutative algebra of differential forms $ \Lambda (S) $ whose coefficients were polynomial in the coordinates on the simplex. (A singular n-simplex in X is a map of the standard n-simplex in $R^{n+1} $ into X , so it inherits natural coordinates). He then made the natural boundary restrictions on forms, and defined the differential graded algebra on the simplicial complex X as the subset of the product over simplices S of X of $ \Lambda (S) $ where forms on each S agree with their restrictions to the boundary of S. The resulting graded commutative algebra is denoted A(X). In simpler words, A(X) is the differential graded algebra of forms on X which are polynomial in all the simplices of some triangulation of X. If X is a smooth manifold, then the resulting differential graded algebra is homotopy equivalent to the de Rham complex of smooth differential forms on X. Now since Sullivan has started with forms, which are contravariant, the new model is dual as a graded vector space to the Quillen models. If we ask for a complete commutative DG Hopf algebra which is homotopy equivalent to a Sullivan model, its Lie algebra of pure elements is dual to the rational homotopy groups of X. The Sullivan models admit transformations back to simplicial complexes by means of a functor called geometrical realization. It is constructed by assigning n-simplices to all homomorphisms of the DG algebra A(X) to the differential graded algebra of polynomial forms on a standard n-simplex, and identifying the faces with the homomorphisms given by restricting to the forms on a face of the standard n-simplex. The resulting complex is denoted $ <A(X)> $. If we begin with a simplicial comples X, form its Sullivan algebra A(X), and then form the geometrization $ <A(X)> $; we get a space which is of the same homotopy type as X but is a thickening of it, with many parallel copies of X glued together. We shall discuss the possibility of constructing a state sum model analogous to the BC model on it below. [**C. MODELS OF LOOP SPACES**]{} It is possible to assign to an object A of any of the above described model categories a loop space object $ \Omega (A)$ of the same category directly without reference to any geometric object. The basic tool is the path fibration. If X is a connected topological space, then the path space with any two fixed endpoints is homotopy equivalent to the loop space of X. Now if we pick any point $x \in X$ then the set of all paths starting at x with arbitrary endpoint, which we denote P(x,X), is contractible, since the paths can be uniformly shrunk back to x. On the other hand, mapping each path to its endpoint gives us a map $ P(x,X) \rightarrow X $, whose fiber over any point is $\Omega (X)$. It turns out that any fibration with contractible total space and base X is homotopy equivalent to this, so it is a homotopy path fibration, and its fiber is homotopic to $\Omega (X)$. We can imitate this in any model category by embedding any object A in a contractible space and then taking the quotient. This is one of the tricks which make model category theory so useful. [**VI. SULLIVAN ALGEBRAS, MINIMAL AND CONTRACTIBLE MODELS**]{} Up to this point we have constructed various models for a space as differential graded algebraic objects. Whatever the intellectual attractions of these models, they are huge objects, and therefore not terribly helpful in computation. Sullivan discovered a procedure to replace these models with homotopy equivalent ones which are remarkably simple, and therefore enable us to perform all sorts of calculations. Sullivan began by defining a special class of commutative DG algebras called Sullivan algebras. [**DEFINITION:**]{} A [*Sullivan algebra*]{} is an algebra of the form $( \Lambda V, d)$, where V is a graded vector space and $ \Lambda $ denotes the graded commutative free algebra, with the properties that V is the union of an upwards graded filtration $ V= \bigcup V_k $, such that $d:V_k \rightarrow \Lambda V_{k-1} $. This definition implies that as a graded algebra, a Sullivan algebra is the tensor product of a symmetric tensor algebra, generated by the even degree elements of V, with an alternating or exterior algebra generated by the odd elements. [**DEFINITION:**]{} A Sullivan algebra is [*minimal*]{} if the image of d is contained in $ \Lambda ^+ V \cdot \Lambda ^+ V $ where $ \Lambda ^+$ is the free algebra generated by elements of V with positive degree. In other words, all images of d are nontrivial products. The definition of minimal excludes [*contractible*]{} Sullivan algebras. [**DEFINITION:**]{} A Sullivan algebra is [*contractible*]{} if has a basis of the form $ \{ x_i, y_i \} $ where $d(x_i )= y_i, d(y_i ) =0 $. It is clear from the definition that a contractible Sullivan algebra has no cohomology. [**THEOREM:**]{} Every Sullivan algebra is the product of a minimal Sullivan algebra and a contractible one. Now Sullivan goes on to show that every differential graded cochain algebra (this means a graded algebra of only nonnegative degree, where d goes up one in degree like differential forms) is quasiisomorphic to a Sullivan model, and in fact to a minimal one. This proves that simply connected rational homotopy types of spaces correspond one to one with isomorphism classes of minimal Sullivan models, since quasiisomorphic minimal models are in fact isomorphic. His procedure is an inductive one reminiscent of the construction of Eilenberg-Maclane spaces. We first put in generators for the cohomology of our model, then add in new generators to kill any unwanted cohomology which has appeared. We then repeat the second step to remove any unwanted cohomology it has generated, and repeat, infinitely if necessary. In practice, this procedure terminates on quite simple DG algebraic structures. We can then perform the loop space construction above on the “minimal Sullivan Models, “ and obtain very simple models for loop spaces. Let us describe the examples of the spheres. [**Case 1. odd dimensional spheres**]{} The odd dimensional sphere $S^{2k-1} $ has a cohomology generator in d=2k-1. This means it has for Sullivan minimal model the graded algebra $ \Lambda (e) $ with d=0. Since e has odd degree, $e^2 =0$. We can construct an algebraic path fibration for it by embedding it in the contractible differential graded algebra $ \Lambda <e,u>, du=e$, where u has degree 2k-2. Now to construct its algebraic loop space we pass to the quotient algebra $ \Lambda <u>, du=0.$ Since u has even degree, we see that the rational cohomology of $ \Omega (S^{2k-1}) $ has precisely one generator in each dimension which is a multiple of 2k-2. [**Case 2, even dimension**]{} This case of $S^{2k}$ is slightly more complicated. We again start with $\Lambda <e>, d=0$, However, we get an infinite dimensional algebra, since e now has even grade. According to Sullivan’s construction, we need to kill off the higher cohomology by making it into coboundaries by adding in new generators. Fortunately, when we add in $e'$ with $ de' =e^2 $ with $e'$ of degree 4k-1, we find by a calculation that all the higher cycles are already killed off by monomials in e and $e'$. Thus the Sullivan minimal model is given by $ \Lambda <e,e'> : de'=e$. We now proceed to construct its algebraic path fibration, given by $\Lambda <e, e', u,u'> de'=e^2, du=e, du'=e'$. Once again we obtain the loop space by passing to the quotient $\Lambda <u,u'>$. We thus obtain, in particular the infinite set of images of the Schwartzschild black hole for k=1. [**B. FREE LIE ALGEBRA MODELS**]{} The subclass of differential graded Lie algebra models analogous to the Sullivan models is the free Lie algebra models. [**DEFINITION:**]{} A free differential graded Lie algebra is the minimal Lie subalgebra of the free tensor algebra of a differential graded vector space V which contains V. In other words, it is the DG Lie algebra generated by all multiple brackets of elements of V with no relations except the axioms of a graded Lie algebra, namely graded Jacobi identitiy and graded commutativity. [**Theorem:**]{} Every topological space has a free DG Lie algebra model. [**DEFINITION**]{} A Free DG Lie algebra is minimal if the image under d of any element of V has projection 0 onto V. Very similarly to the situation for Sullivan models, every free Lie algebra model is the product of a minimal one and a contractible one. Again, just as for Sullivan models, any two quasiisomorphic minimal free DG Lie algebras are isomorphic, so there is a 1-1 correspondence between minimal free DG Lie algebras and rational homotopy types of spaces. [**C. THE REDUCED BAR CONSTRUCTION**]{} As we have indicated, Quillen gave functors which established homotopy equivalences between his various model categories. The functor from DG Lie algebra models to DG algebra models is especially important for us. It is called the reduced bar construction [@Book]. The reduced bar construction has the property that if applied to a free DG Lie algebra model which is finitely generated, it produces a Sullivan model. Since physically interesting models will be finitely generated because of energy and information bounds, this connection will be very important for us. Let us briefly describe the reduced bar construction. Beginning with a free DG Lie algebra $(L_V, d_1) $, we first form the dual graded vector space ${L_V}^*$. We then shift its grading up by 1, and form the free graded commutative tensor algebra on it, denoting the result as $ \Lambda (T (s{L_V}^*) $. Note that the multiplication here is unrelated to the bracket on $L_V$. It is just a free product. On this new algebra, graded by length of strings, we put a new differential $d= d_1 + d_2 $, where $d_1$ is the dual of the differential on $L_V$ and $d_2$ is the dual of the bracket on $L_V$. The bracket goes from $L_V \otimes L_V $ to $L_V$, so its dual is quadratic on ${L_V}^* $. From our point of view, the reduced bar construction is an extremely convenient piece of magic. It converts a handlebody decomposition of the causal path space into an algebraic model for the differential forms on spacetime itself. The shift in grading reflects the transition from path space to spacetime, while the duality reflects the transition from cycles to forms. [**VII. QUANTUM GEOMETRY OF Q REGIONS**]{} [**A. QUANTUM TOPOLOGY**]{} We can see that there is a strong coincidence between the structure of Sullivan algebras and the apparent multiple images of a light source seen through a highly curved region R of spacetime. The minimal part corresponds to the topologically unavoidable images which correspond to the generators of the rational cohomology of R, the contractible part to the pairs corresponding to cancelling handles, which can be seen to join and disappear as the observation point shifts. Astrophysicists refer to these as folds in space. Furthermore, we would expect any physical region to be represented by a finitely generated model. We would also regard the free Lie algebra model as primary, since it relates directly to the path space of the region, and want to derive our model of the region itself from it. We therefore form the [**CONJECTURE:**]{} [*Quantum regions of spacetime are described by Sullivan algebras produced from finitely generated free Lie algebra models by the reduced bar construction.*]{} Note that the proposed models for quantum regions are richer than rational homotopy types, since they include the contractible algebras as well The structure we are proposing is a model for a handlebody decomposition of the path space of a region, not only its topology. We can consider models with only cancelling pairs of handles, only free generators, or combinations of both. Since Sullivan algebras are free algebras, they may be constructed as products of simple graded algebras with one generator each, and minimal contractible algebras $\Lambda <x,y>, dy=x$. We think of these two types of generators as quantum versions of black holes and “galaxies” , since a small weak gravitational source produces a fold in the sky [@Perlick]. Mathematically, rational homotopy types are as simple as they are because they admit free algebras as models. Since free algebras are simple products of parts with one generator each, this kind of decomposition is inherent in their structure. The d operation in the combined algebra could make topological links between the parts. It is related to the Whitehead product. As we have repeatedly emphasized, it is interesting from our point of view that our category of rational homotopy types has so many different but homotopy equivalent formulations. We conjecture that they correspond to complementary descriptions of the quantum spacetime geometry. In particular, the geometric realizations of Sullivan models is very suggestive. Recall that it is like a thickening of a region, where the different layers are labelled by different homomorphisms of the Sullivan algebra into the algebra of polynomial forms. In the case of a finitely generated free Sullivan algebra, the homomorphisms are determined by images of the generators, This seems like a natural setting for descriptions in which a subregion within our region observes different generators, and information about the geometry of the region is thereby measured. Such an description seems to be complementary to observing an external light source, just as a localized state for a particle is complementary to a beam. It is also possible to start from a free DG lie algebra model $L_V$, and directly produce a complex called the cellular model of the Lie algebra. This consists of one cell for each basis element of the DG vector space V which generates the Lie algebra L, with attachments determined by the d operation on L restricted to V. This is of the same rational homotopy type as the geometric realization of the Sullivan algebra associated to $L_V$ by the Quillen functor, but is much simpler. Thus more than one avenue toward a state sum model for quantum gravity over a Quillen model seems possible. [**B. QUANTUM GEOMETRY**]{} So far, we have been investigating a new type of topology which may be relevant in quantum gravity. Now we begin the investigation of how a quantum theory of general relativity might be constructed over a Sullivan model. There are two natural approaches, which we believe to be complementary. To recapitulate, since the geometrical version of spaces Quillen began with was combinatorial, namely simplicial complexes, one natural avenue is to put state sum models similar to the BC model on them. Thus we could take the geometrization of a minimal Sullivan model as a starting point, and put a categorical state sum on it. The original BC model came out of quantizing the geometry of a four dimensional simplicial complex by quantizing variables corresponding to the immersion of each simplex into Minkowski space [@BC]. We could attempt something similar, attaching geometric variables to represent quantum geometrical degrees of freedom. A paricularly interesting situation arises if we begin with a finitely generated free DG Lie algebra model, transform it into a Sullivan model by the reduced bar construction, and form its geometric realisation. The model contains forms giving the relational geometry between the generating “black holes” and “galaxies”, together with higher forms coming from the multiple brackets in the Lie algebra which would express higher quantum geometric correlations between them. If a state sum can be regularized to be finite over the “layers” of the Sullivan realisation, as we conjectured above, this would be an rich picture. We think that this has a good chance of working, because much of the information in the realisation groups together into families which are the same for any labelling of the boundary of any subcomplex. If such a state sum turns out to be possible, it will be an interesting departure, since the data on the simplices would not be local geometrical information, but only forms that indicated where the simplex “saw” the handles corresponding to the generators of the original Lie algebra model. Another approach might be to regard the minimal Hopf algebra model as an approximation to the loop space, and rewrite Einstein’s equations as conditions on the variation of the holonomy along the path under small perturbations. We could then write a topological version of this on the handlebody decompositions. of $\Omega(R)$ which correspond to the Sullivan model. Alternatively, we could write the Einstein-Hilbert Lagrangian on loop space by formulating the Riemann curvature in terms of variations of holonomy on the loop space, then approximate the path integral by a categorical state sum. [**C. MATTER FIELDS**]{} We can approach representation of matter fields over a Sullivan model via the replacement of the Hopf algebra models by their graded tensor categories of representations. A representation of the Hopf algebra model is an approximation to a representation of the loop group, which is a way of describing a connection (composition of paths would be represented by composition of holonomies). In the constructions of TQFTs and models for quantum gravity, the role of topology was played by simplicial complexes, and the fields came from Hopf algebras, tensor categories, and various categorifications thereof. There is something very suggestive about the topology of spacetime itself being represented by a Hopf algebra or tensor category. Various classes of functors are available for building blocks for a theory. There is also the possibility of investigating quantum deformations of the Hopf algebra models in the sense of quantum groups. Perhaps the resulting restriction of the representation theory could shed some light on what matter fields would appear. [**D. A CONJECTURE: CRYSTALS**]{} Since the Sullivan models can be constructed as products of two basic pieces, which we suggestively named above as black holes and galaxies, it would be a natural question to ask what a quantum theory of gravity would do on such an aggregate. Gravity being an attractive force, we would expect them to clump together into some sort of crystalline structure, black holes standing in for nuclei and galaxies for electrons. Could the spacetime continuum appear in the limit from such crystals? Would matter appear as phonons in our spacetime crystal? [**E. OUTLOOK**]{} The critical thing for this program is to develop one or more of the approaches to constructing a quantum theory of gravity on our proposed models. We should consider only models that are effectively four dimensional, i.e. only constructed from 2 and 3-handles as generators for a Sullivan algebra. Since we have constructed complexes which mirror the complexity of data which an external observer can see, topological state sum models on them would be plausible candidates for a quantum theory of gravity. Preliminary investigation of the Sullivan realisation geometric suggests that large “gauge” degrees of freedom exist, so that a state sum model has a good chance of being finite. This would base quantum gravity in a mathematically well defined version of relational geometry as we outlined in the introduction. The geometric realisation is an interesting candidate for the thickening of spacetime at the quantum level. If this is so, or if one of the other approaches to constructing a quantum theory outlined above proves tractible, then the new models proposed here could begin to be applied to the interesting problems of quantum gravity, and compared to other approaches. [**ACKNOWLEDGEMENTS:**]{} The author wishes to thank The Maths Department of Nottingham University for their hospitality while this paper was being written. He thanks David Yetter, John Barrett, and Catherine Meusburger for useful conversations. Special thanks to Dan Christensen who caught several mistakes in an earlier draft. The work was supported by a grant and a minigrant from FQXi. [99]{} L. Susskind, An introduction to black holes, information and the string theory revolution: the holographic universe, Singapore; Hackensack NJ; World Scientific 2005 Stachel, J.Einstein from “B” to “Z”. (English summary) Einstein Studies, 9. Birkhäuser Boston, Inc., Boston, MA, 2002. xii+556 pp. ISBN: 0-8176-4143-2 A. Ashtekar, M. Bojowald Black hole evaporation: A paradigm, Class.Quant. Grav. 22 (2005) 3349-3362 Barrett, John W.; Crane, Louis A Lorentzian signature model for quantum general relativity. Classical Quantum Gravity 17 (2000), no. 16, 3101–3118. De Pietri, Roberto; Freidel, Laurent; Krasnov, Kirill; Rovelli, Carlo Barrett-Crane model from a Boulatov-Ooguri field theory over a homogeneous space. Nuclear Phys. B 574 (2000), no. 3, 785–806 Quillen, D. Rational homotopy theory. Ann. of Math. (2) 90 1969 205-295 D. Sullivan, Infinitesimal calculations in topology, Pub. IHES 47 (1977)269-331. L. Crane, L.H. Kauffman, D.N. Yetter, State Sum Invariants of 4- Manifolds I, Arxiv preprint hep-th/9409167 Crane, Louis; Frenkel, Igor B. Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases. Topology and physics. J. Math. Phys. 35 (1994), no. 10, 5136–5154. L. Crane A Pointless Model for the Continuum as the Foundation for Quantum Gravity arXiv:0804.0030 V. Perleck, Gravitational lensing from a spacetime perspective, Living reviews in relativity, 2004-emis.ams.org J. Milnor Morse theory. Annals of Mathematics Studies, PUP 1969 K. Uhlenbeck, A morse theory for geodesics on a lorentzian manifold, Topology 14 (1975) 69-90. Cerf, Jean La stratification naturelle des espace des fonctions differentiables reeles et la theoreme de la pseudoisotopie, IHES publications math. No. 39 (1970) 5-173 S. Frittelli, C. N. Kozameh, E.T. Newman, C. Rovelli, R. S. Tate, Fuzzy spacetime fron a null-surface version of GR, Class. Quant. Grav. 14 (1997) A143-154 Quillen, D.. Homotopical algebra. Lecture Notes in Mathematics No. 43 Springer, N.Y.1967. Y. Felix, S. Halperin, J.-C. Thomas, Rational homotopy Theory, Springer 2000 M. Majewski, Rational homotopical models and uniqueness, AMS Providence RI, 2000. Frolov, V. P., and Novikov I. D., Black hole physics : basic concepts and new developments, Dordrecht ; Boston : Kluwer, 1998.

--- abstract: | The Mallows measure on the symmetric group $S_n$ is the probability measure such that each permutation has probability proportional to $q$ raised to the power of the number of inversions, where $q$ is a positive parameter and the number of inversions of $\pi$ is equal to the number of pairs $i<j$ such that $\pi_i > \pi_j$. We prove a weak law of large numbers for the length of the longest increasing subsequence for Mallows distributed random permutations, in the limit that $n \to \infty$ and $q \to 1$ in such a way that $n(1-q)$ has a limit in ${\mathbf{R}}$. [**Keywords: random permutations**]{} .2 cm [**MCS numbers: 335**]{} author: - Carl Mueller$^1$ and Shannon Starr$^2$ date: 'March 24, 2011' title: The Length of the Longest Increasing Subsequence of a Random Mallows Permutation --- Main Result {#sec:MainResult} =========== There is an extensive literature dealing with the longest increasing subsequence of a random permutation. Most of these papers deal with uniform random permutations. Our goal is to study the longest increasing subsequence under a different measure, the Mallows measure, which is motivated by statistics [@Mallows]. We begin by defining our terms and stating the main result, and then we give some historical perspective. The $\textrm{Mallows}(n,q)$ probability measure on permutations $S_n$ is given by $$\label{eq:DefMallowsMeasure} \mu_{n,q}(\{\pi \})\, =\, [Z(n,q)]^{-1} q^{{\operatorname{inv}}(\pi)}\, ,$$ where $\textrm{inv}(\pi)$ is the number of “inversions” of $\pi$, $${\operatorname{inv}}(\pi)\, =\, \#\{ (i,j) \in \{1,\dots,n\}^2\, :\, i<j\, ,\ \pi_i>\pi_j \}\, .$$ The normalization is $Z(n,q) = \sum_{\pi \in S_n} q^{{\operatorname{inv}}(\pi)}$. See [@DiaconisRam] for more background and interesting features of the Mallows measure. The measure is related to representations of the Iwahori-Hecke algebra as Diaconis and Ram explain. It is also related to a natural $q$-deformation of exchangeability which has been recently discovered and explained by Gnedin and Olshanski [@GnedinOlshanski; @GnedinOlshanski2]. We are interested in the length of the longest increasing subsequence in this distribution. The length of the longest increasing subsequence of a permutation $\pi \in S_n$ is $$\label{eq:DefLengthLIS} \ell(\pi)\, =\, \max \{k\leq n\, :\, \pi_{i_1} < \dots < \pi_{i_k}\ \textrm{ for some }\ i_1<\dots<i_k \} \, .$$ Our main result is the following. \[thm:main\] Suppose that $(q_n)_{n=1}^{\infty}$ is a sequence such that the limit $\beta = \lim_{n \to \infty} n(1-q_n)$ exists. Then $$\lim_{n \to \infty} \mu_{n,q_n}\left ( \left \{ \pi \in S_n \, : \, |n^{-1/2} \ell(\pi) - \mathcal{L}(\beta)| < \epsilon \right \} \right )\, =\, 1\, ,$$ for all $\epsilon>0$, where $$\label{eq:LDefinition} \mathcal{L}(\beta)\, =\, \begin{cases} 2 \beta^{-1/2} \sinh^{-1}(\sqrt{e^{\beta}-1}) & \text{ for $\beta>0$,}\\ 2 & \text{ for $\beta=0$,}\\ 2 |\beta|^{-1/2} \sin^{-1}(\sqrt{1-e^{\beta}})& \text{ for $\beta<0$.} \end{cases}$$ There is another formula for the function in \[eq:LDefinition\] as an integral in equation \[eq:integralformula\], below. In a recent paper [@BorodinDiaconisFulman] Borodin, Diaconis and Fulman asked about the Mallows measure, “Picking a permutation randomly from $P_{\theta}$ (their notation for the Mallows measure), what is the distribution of the cycle structure, longest increasing subsequence, . . . ?" We answer the question about the longest increasing subsequence at the level of the weak law of large numbers. Note that the Mallows measure for $q=1$ reduces to the uniform measure on $S_n$: $$\notag \mu_{n,1}(\pi)\, =\, \frac{1}{n!}\, ,$$ for all $\pi \in S_n$. For the uniform measure, Vershik and Kerov [@VershikKerov] and Logan and Shepp [@LoganShepp] already proved a weak law of large numbers for the length of the longest increasing subsequence. We will use their result in our proof, so we state it here: \[prop:VKLSPermutation\] $$\label{eq:PropositionVKLS} \lim_{n \to \infty} \mu_{n,1}\{ \pi \in S_n\, :\, |n^{-1/2} \ell(\pi) - 2 | > \epsilon \} \, =\, 0\, ,$$ for all $\epsilon>0$. The reader can find the proof of this proposition in [@VershikKerov] and [@LoganShepp]. The proof of Vershik, Kerov, Logan and Shepp involved a deep connection in combinatorics known as the Robinson-Schensted-Knuth algorithm. In the RSK algorithm, for each permutation $\pi \in S_n$, one associates a pair of Young tableaux, with the same shape: a Young diagram or partition, $\lambda \vdash n$. Referring to this as $\lambda = \operatorname{sh}(\pi)$, the first row, $\lambda_1$ is the length of the longest increasing subsequence of $\pi$. Vershik and Kerov considered the Plancherel measure on the set of partitions, where $\mu_n(\lambda)$ is proportional to the number of permutations $\pi$ such that $\operatorname{sh}(\pi) = \lambda$. Using the hook length formula of Frame, Robinson and Thrall for this probability measure, they defined the “hook integral” for limiting (rescaled) shapes of Young tableaux. They showed concentration of measure of the Plancherel measure, in the $N \to \infty$ limit, around the optimal shape of the hook integral. See, for example, [@Kerov] for many more details. The RSK approach was taken further by Baik, Deift and Johansson [@BaikDeiftJohansson]. In their seminal work, they gave a complete description of the fluctuations of $\lambda_1$, which are similar to fluctuations of the largest eigenvalue in random matrix theory [@TracyWidom]. For the Mallows measure, we do not believe that there is a straightforward analogue of the RSK algorithm. The reason for this is that the shape of the Young tableaux in the RSK algorithm does not determine the number of inversions. For progress on this difficult problem, see a result of Hohlweg [@Hohlweg]. On the other hand, long after the original proof of Vershik, Kerov, Logan and Shepp, there was a return to the original probabilistic approach of Hammersley [@Hammersley]. Hammersley had originally asked about the length of the longest increasing subsequence, as an example of an open problem with partial results, delived in an address directed primarily to graduate students. Using methods from interacting particle processes, Aldous and Diaconis established a hydrodynamic limit of Hammersley’s process, thereby giving an independent proof of Theorem \[prop:VKLSPermutation\]. This approach was also developed by Seppalainen [@Seppalainen], Cator and Groeneboom [@CatorGroeneboom], and others. This allows different generalizations than the original RSK proof of Vershik and Kerov. For instance, Deuschel and Zeitouni considered the length of the longest increasing subsequence for IID random points in the plane [@DeuschelZeitouni; @DeuschelZeitouni2]. For us, their results are key. On the other hand if, contrary to our expectations, there is a generalization of the RSK algorithm which applies to Mallows distributed random permutations, that would allow the discoverer to try to extend the powerful methods of Baik, Deift and Johansson. The rest of the paper is devoted to the proof of Theorem \[thm:main\]. We begin by stating the key ideas. This occupies Sections 2 through 6. Certain important technical assumptions will be stated as lemmas. These lemmas are independent of the main argument, although the main argument relies on the lemmas. The lemmas will be proved in Sections 7 and 8. A Boltzmann-Gibbs measure {#sec:BoltzmannGibbs} ========================= In a previous paper [@Starr] one of us proved the following result. \[lem:StarrOLD\] Suppose that the sequence $(q_n)_{n=1}^{\infty}$ has the limit $\beta = \lim_{n \to \infty} n (1-q_n)$. For $n \in {\mathbf{N}}$, let $\pi(\omega) \in S_n$ be a $\textrm{Mallows}(n,q_n)$ random permutation. For each $n \in {\mathbf{N}}$, consider the empirical measure $\tilde{\rho}_n(\cdot,\omega)$ on ${\mathbf{R}}^2$, such that $$\notag \tilde{\rho}_n(A,\omega)\, =\, \frac{1}{n} \sum_{k=1}^{n} {\mathbf{1}}\left\{ \left(\frac{k}{n} , \frac{\pi_k(\omega)}{n}\right) \in A \right \}\, ,$$ for each Borel set $A \subseteq {\mathbf{R}}^2$. Note that $\tilde{\rho}_n(\cdot,\omega)$ is a random measure. Define the non-random measure ${\rho}_{\beta}$ on ${\mathbf{R}}^2$ by the formula $$\label{eq:uDefin} d{\rho}_{\beta}(x,y)\, =\, \frac{(\beta/2) \sinh(\beta/2) {\mathbf{1}}_{[0,1]^2}(x,y)}{\big(e^{\beta/4} \cosh(\beta[x-y]/2) - e^{-\beta/4} \cosh(\beta[x+y-1]/2)\big)^2}\, dx\, dy\, .$$ Then the sequence of random measures $\tilde{\rho}_n(\cdot,\omega)$ converges in distribution to the non-random measure ${\rho}_{\beta}$, as $n \to \infty$, where the convergence is in distribution, relative to the weak topology on Borel probability measures. We will reformulate Lemma \[lem:StarrOLD\], using a Boltzmann-Gibbs measure for a classical spin system. The underlying spins take values in ${\mathbf{R}}^2$. We define a two body Hamiltonian interaction $h : {\mathbf{R}}^2 \to {\mathbf{R}}$ as $$h(x,y)\, =\, {\mathbf{1}}\{xy<0\}\, .$$ Then the $n$ particle Hamiltonian function is $H_n : ({\mathbf{R}}^2)^n \to {\mathbf{R}}$, $$H_n((x_1,y_1),\dots,(x_n,y_n))\, =\, \frac{1}{n-1} \sum_{i=1}^{n-1} \sum_{j=i+1}^{n} h(x_i-x_j,y_i-y_j)\, .$$ One also needs an [*a priori*]{} measure $\alpha$ which is a Borel probability measure on ${\mathbf{R}}^2$. Given all this, the Boltzmann-Gibbs measure on $({\mathbf{R}}^2)^n$ with “inverse-temperature” $\beta \in {\mathbf{R}}$ is defined as $\mu_{n,\alpha,\beta}$, $$d\mu_{n,\alpha,\beta}((x_1,y_1),\dots,(x_n,y_n))\, =\, \frac{\exp\big(-\beta H_n((x_1,y_1),\dots,(x_n,y_n))\big)\, \prod_{i=1}^{n} d\alpha(x_i,y_i)} {Z_n(\alpha,\beta)}$$ where the normalization, known as the “partition function” is $$Z_n(\alpha,\beta)\, =\, \int_{({\mathbf{R}}^2)^n} \exp\big(-\beta H_n((x_1,y_1),\dots,(x_n,y_n))\big)\, \prod_{i=1}^{n} d\alpha(x_i,y_i)\, .$$ Usually in statistical physics one only considers positive temperatures, corresponding to $\beta\geq 0$. But we will also consider $\beta \leq 0$, because it makes mathematical sense and is an interesting parameter range to study. A special situation arises when the [*a priori*]{} measure $\alpha$ is a product measure of two one-dimensional measures without atoms. If $\lambda$ and $\kappa$ are Borel probability measures on ${\mathbf{R}}$ without atoms, then $$\label{eq:equiv} \begin{split} \mu_{n,\lambda\times \kappa,\beta}\Big\{((x_1,y_1),\dots,(x_n,y_n)) \in &({\mathbf{R}}^2)^n\, : \, \exists i_1<\dots<i_k\, ,\\ &\hspace{-5mm} \text{ such that }\ (x_{i_j}-x_{i_\ell})(y_{i_j} - y_{i_\ell}) > 0\ \textrm{ for all $j\neq \ell$} \Big\} \\ &\hspace{5mm} =\, \mu_{n,\exp(-\beta/(n-1))}(\{ \pi \in S_n \, : \, \ell(\pi) \geq k \})\, , \end{split}$$ for each $k$. This follows from the definitions. In particular, the condition for an increasing subsequence of a permutation $i_1<\dots<i_k$ is that if $i_j < i_\ell$ then we must have $\pi_{i_j} < \pi_{i_\ell}$. For the variables $(x_1,y_1),\dots,(x_n,y_n)$ replacing the permutation, we obtain the condition listed above. We will also use results from [@DeuschelZeitouni] by Deuschel and Zeitouni. They define the record length of $n$ points in ${\mathbf{R}}^2$ as $$\label{eq:RecordDefinition} \ell((x_1,y_1),\dots,(x_n,y_n))\, =\, \max \{ k \, : \, \exists i_1<\dots<i_k\, ,\ (x_{i_j}-x_{i_\ell})(y_{i_j} - y_{i_\ell}) > 0\ \textrm{ for all $j<\ell$ } \}\, .$$ Equation (\[eq:equiv\]) says that the distribution of $\ell((X_1(\omega),Y_1(\omega),\dots,X_n(\omega),Y_n(\omega)))$ with respect to the Boltzmann-Gibbs measure $\mu_{n,\lambda \times \kappa,\beta}$ is equal to the distribution of $\ell(\pi(\omega))$ with respect to the $\operatorname{Mallows}(n,\exp(-\beta/(n-1)))$ measure $\mu_{n,\exp(-\beta/(n-1))}$. Using the equivalence and Lemma \[lem:StarrOLD\], we may also deduce a weak convergence result for the measures $\mu_{n,\lambda\times \kappa,\beta}$. In fact there is a special choice of measure for $\lambda$ and $\kappa$, depending on $\beta$, which makes the limit nice. For each $\beta \in {\mathbf{R}}\setminus \{0\}$ define $$L(\beta)\, =\, [(1-e^{-\beta})/\beta]^{1/2}\, ,$$ and define $L(0)=1$. Define the Borel probability $\lambda_{\beta}$ on ${\mathbf{R}}$ by the formula $$\notag d\lambda_{\beta}(x)\, =\, \frac{L(\beta) {\mathbf{1}}_{[0,L(\beta)]}(x)}{1-\beta L(\beta) x}\, dx\, ,$$ for $\beta\neq 0$, and $d\lambda_0(x) = {\mathbf{1}}_{[0,1]}(x)\, dx$. Also define a measure $\sigma_{\beta}$ on ${\mathbf{R}}^2$ by the formula $$d\sigma_{\beta}(x,y)\, =\, \frac{{\mathbf{1}}_{[0,L(\beta)]^2}(x,y)}{(1-\beta x y)^2}\, dx\, dy\, .$$ Both the $x$ and $y$ marginals of $\sigma_{\beta}$ are equal to the one-dimensional measure $\lambda_{\beta}$. Using this, the next lemma follows from Lemma \[lem:StarrOLD\] and the strong law of large numbers. In fact, the strong law implies that an empirical measure arising from i.i.d. samples always converges in distribution to the underlying measure, relative to the weak topology on measures. \[lem:StarrLimit\] For $n \in {\mathbf{N}}$, let $((X_{n,1}(\omega),Y_{n,k}(\omega)),\dots,(X_{n,n}(\omega),Y_{n,n}(\omega)))$ be distributed according to the Boltzmann-Gibbs measure $\mu_{n,\lambda_{\beta}\times \lambda_{\beta},\beta}$, where we used the special [*a priori*]{} measure just constructed. Define the random empirical measure $\tilde{\sigma}_n(\cdot,\omega)$ on ${\mathbf{R}}^2$, such that $$\tilde{\sigma}_{n}(A,\omega)\, :=\, \frac{1}{n} \sum_{i=1}^{n} {\mathbf{1}}\{ (X_{n,i}(\omega),Y_{n,i}(\omega)) \in A \} \, ,$$ for each Borel measurable set $A \subseteq {\mathbf{R}}^2$. Then the sequence of random measures $(\tilde{\sigma}_n(\cdot,\omega))_{n=1}^{\infty}$ converges in distribution to the non-random measure $\sigma_{\beta}$, in the limit $n \to \infty$, where the convergence in distribution is relative to the topology of weak convergence of Borel probability measures. We could have also chosen a different [*a priori*]{} measure to obtain convergence to the same measure $\rho_{\beta}$ from Lemma \[lem:StarrOLD\]. But we find the new measure $\sigma_{\beta}$ to be a nicer parametrization. We may re-parametrize the measures like this by changing the [*a priori*]{} measure. The ability to re-parametrize the measures will also be useful later. Deuschel and Zeitouni’s record lengths ====================================== In [@DeuschelZeitouni], Deuschel and Zeitouni proved the following result. We thank Janko Gravner for bringing this result to our attention. \[thm:DeuschelZeitouni\] Suppose that $u$ is a density on the box $[a_1,a_2]\times [b_1,b_2]$, i.e., $d\alpha(x,y) = u(x,y)\, dx\, dy$ is a probability measure on the box $[a_1,a_2]\times [b_1,b_2]$. Also suppose that $u$ is differentiable in $(a_1,a_2)\times (b_1,b_2)$ and the derivative is continuous up to the boundary. Finally, suppose there exists a constant $c>0$ such that $$u(x,y) \geq c\, ,$$ for all $(x,y) \in [a_1,a_2] \times [b_1,b_2]$. Let $(U_1,V_1),(U_2,V_2),\dots$ be i.i.d., $\alpha$-distributed random vectors in $[a_1,a_2]\times [b_1,b_2]$. Then the rescaled random record lengths, $$n^{-1/2} \ell((U_1,V_1),\dots,(U_n,V_n))\, ,$$ converge in distribution to a non-random number $\mathcal{J}^*(u)$ defined as follows. Let $\mathcal{C}^1_{\nearrow}([a_1,a_2]\times [b_1,b_2])$ be the set of all $\mathcal{C}^1$ curves from $(a_1,b_1)$ to $(a_2,b_2)$ whose tangent line has positive (and finite) slope at all points. For $\gamma \in \mathcal{C}^1_{\nearrow}([a_1,a_2]\times [b_1,b_2])$ and any $\mathcal{C}^1$ parametrization $(x(t),y(t))$, define $$\label{eq:JDefin} \mathcal{J}(u,\gamma)\, =\, 2 \int_{\gamma} \sqrt{u(x(t),y(t))\, x'(t)\, y'(t)}\, dt\, .$$ This is parametrization independent. Then $$\mathcal{J}^*(u)\, =\, \sup_{\gamma \in \mathcal{C}^1_{\nearrow}([a_1,a_2]\times [b_1,b_2])} \mathcal{J}(u,\gamma)\, .$$ This is Theorem 2 in Deuschel and Zeitouni’s paper. The fact that $\mathcal{J}(u,\gamma)$ is parametrization independent is useful. We generalize their definition of $\mathcal{J}(u,\gamma)$ a bit, attempting to mimic the definition of entropy made by Robinson and Ruelle in [@RobinsonRuelle]. This is useful for establishing continuity properties of $\mathcal{J}$ and it allows us to drop the assumption that $u$ is differentiable. Given a box $[a_1,a_2]\times [b_1,b_2]$, we define $\Pi_n([a_1,a_2]\times [b_1,b_2])$ to be the set of all $(n+1)$-tuples $\mathcal{P} = ((x_0,y_0),\dots,(x_n,y_n)) \in ({\mathbf{R}}^2)^{n+1}$ satisfying $$a_1 = x_0 \leq \dots \leq x_n = a_2 \quad \textrm{ and } \quad b_1 = y_0 \leq \dots \leq y_n = b_2 \, .$$ We define $$\label{eq:Jpart} \tilde{\mathcal{J}}(u,\mathcal{P})\, =\, 2 \sum_{k=0}^{n-1} \left(\int_{x_k}^{x_{k+1}} \int_{y_k}^{y_{k+1}} u(x,y)\, dx\, dy\right)^{1/2}\, .$$ For later reference, we note the following continuity property of $\tilde{\mathcal{J}}(u,\mathcal{P})$ as a function of $u$ for a fixed $\mathcal{P}$. Suppose that $u$ and $v$ are nonnegative functions in $\mathcal{C}([a_1,a_2] \times [b_1,b_2])$. Using the simple fact that $|a - b| \leq \sqrt{|a^2-b^2|}$, for all $a,b\geq 0$, we see that $$|\tilde{\mathcal{J}}(u,\mathcal{P}) - \tilde{\mathcal{J}}(v,\mathcal{P})| \, \leq \, 2 \sum_{k=0}^{n-1} \left(\int_{x_k}^{x_{k+1}} \int_{y_k}^{y_{k+1}} |u(x,y)-v(x,y)|\, dx\, dy\right)^{1/2}\, .$$ We define $\|u\|$ to be the supremum norm. Using this and the Cauchy inequality, $$\label{eq:Holder} \begin{split} |\tilde{\mathcal{J}}(u,\mathcal{P}) - \tilde{\mathcal{J}}(v,\mathcal{P})|\, &\leq \, 2 \|u-v\|^{1/2}\, \sum_{k=0}^{n-1} \sqrt{(x_{k+1}-x_k)(y_{k+1}-y_k)}\\ &\leq \, \|u-v\|^{1/2}\, \sum_{k=0}^{n-1} (x_{k+1}-x_k+y_{k+1}-y_k)\\ &=\, \|u-v\|^{1/2}\, (a_2-a_1+b_2-b_1)\, . \end{split}$$ Now we state a technical lemma. \[lem:USC\] Let $\mathcal{B}_{\nearrow}([a_1,a_2]\times [b_1,b_2])$ be the set of all connected sets $\Upsilon \subset [a_1,a_2]\times [b_1,b_2]$ containing $(a_1,b_1)$ and $(a_2,b_2)$, and having the property that $(x_1-x_2)(y_1-y_2) \geq 0$ for all $(x_1,y_1), (x_2,y_2) \in \Upsilon$. Define $\Pi_n(\Upsilon)$ to be the set of all $\mathcal{P} = ((x_0,y_0),\dots,(x_n,y_n))$ in $\Pi_n$ such that $(x_k,y_k) \in \Upsilon$ for each $k$, and let $\Pi(\Upsilon) = \bigcup_{n=1}^{\infty} \Pi_n(\Upsilon)$. Finally, define $$\tilde{\mathcal{J}}(u,\Upsilon)\, =\, \lim_{\epsilon \to 0}\, \inf \left \{ \tilde{\mathcal{J}}(u,\mathcal{P}) \, :\, \mathcal{P} \in \bigcup_{n=1}^{\infty} \Pi_n(\Upsilon)\, ,\ \|\mathcal{P}\|< \epsilon \right \} \, .$$ Then $\tilde{\mathcal{J}}(u,\cdot)$ is an upper semi-continuous function of $\mathcal{B}_{\nearrow}([a_1,a_2]\times [b_1,b_2])$, endowed with the Hausdorff metric. If $\Upsilon$ is the range of a curve $\gamma \in \mathcal{C}^1_{\nearrow}([a_1,a_2]\times [b_1,b_2])$, then $\tilde{\mathcal{J}}(u,\Upsilon) = \mathcal{J}(u,\gamma)$ because for each partition $\mathcal{P} \in \Pi(\Upsilon)$, the quantity $\tilde{\mathcal{J}}(u,\mathcal{P})$ just gives a Riemann sum approximation to the integral in $\mathcal{J}(u,\gamma)$. Now, let us denote the density of $\sigma_{\beta}$ as $$\label{eq:Definitionubeta} u_{\beta}(x,y)\, =\, \frac{{\mathbf{1}}_{[0,L(\beta)]^2]}(x,y)}{(1-\beta xy)^2}\, .$$ Then we may prove the following variational calculation. \[lem:variational\] For any $\Upsilon \in \mathcal{B}_{\nearrow}([0,L(\beta)]^2)$, $$\tilde{\mathcal{J}}(u_{\beta},\Upsilon)\, \leq\, \int_0^{L(\beta)} \frac{2}{1-\beta t^2}\, dt\, =\, \mathcal{L}(\beta)\, .$$ Let us quickly verify the lemma in the special case $\beta=0$. We have set $L(0)=1$ and we know that $u_0$ is identically 1 on the rectangle $[0,1]^2$. By equation (\[eq:Holder\]), we know that $$\tilde{\mathcal{J}}(u_0,\mathcal{P})\, \leq\, 2\, ,$$ by comparing $u=u_0$ with $v=0$. That means that $\tilde{\mathcal{J}}(u_0,\Upsilon) \leq 2$ for every choice of $\Upsilon$. It is easy to see that taking $\Upsilon = \{(t,t)\, :\, 0\leq t\leq 1\}$, which is the graph of the straight line curve $\gamma$ parametrized by $x(t)=y(t)=t$ for $0\leq t\leq 1$, $$\tilde{\mathcal{J}}(u_0,\Upsilon)\, =\, \mathcal{J}(u_0,\gamma)\, =\, 2 \int_0^1 \sqrt{u_0(x(t),y(t)) x'(t) y'(t)}\, dt\, =\, 2\, .$$ Therefore, using Deuschel and Zeitouni’s theorem, this shows that the straight line is the optimal path for the case of a constant density on a square. This lemma in general is proved using basic inequalities, as above, combined with the fact that $\mathcal{J}(u,\gamma)$ is parametrization independent, which allows us to reparametrize time for any curve $(x(t),y(t))$. As with the other lemmas, we prove this in Section 7 at the end of the paper. Coupling to IID point processes =============================== Now, suppose that $\beta$ is fixed, and consider a triangular array of random vectors in ${\mathbf{R}}^2$, $$((X_{n,k},Y_{n,k})\, :\, n \in {\mathbf{N}}\, ,\ 1\leq k\leq n)\\, ,$$ where for each $n \in {\mathbf{N}}$, the random variables $(X_{n,1},Y_{n,1}),\dots,(X_{n,n},Y_{n,n})$ are distributed according to the Boltzmann-Gibbs measure $\mu_{n,\lambda_{\beta}\times \lambda_{\beta},\beta}$. We know that $$\mu_{n,\exp(-\beta/(n-1))}\{\ell(\pi)=k\}\, =\, {\bf P} \{\ell((X_{n,1},Y_{n,1}),\dots,(X_{n,n},Y_{n,n})) = k\}\, ,$$ for each $k$. We also know that the empirical measure associated to $((X_{n,1},Y_{n,1}),\dots,(X_{n,n},Y_{n,n}))$ converges to the special measure $\sigma_{\beta}$. It is natural to try to apply Deuschel and Zeitouni’s Theorem \[thm:DeuschelZeitouni\], even though the points $(X_{n,1},Y_{n,1}),\dots,(X_{n,n},Y_{n,n})$ are not i.i.d., a requirement for the random variables $(U_1,V_1),\dots,(U_n,V_n)$ of their theorem. It is useful to generalize our perspective slightly. Let us suppose that $\lambda$ and $\kappa$ are general Borel probability measures on ${\mathbf{R}}$ without atoms, and let us consider a triangular array of random vectors in ${\mathbf{R}}^2$: $((X_{n,k},Y_{n,k})\, :\, n \in {\mathbf{N}}\, ,\ 1\leq k\leq n)$, where for each $n \in {\mathbf{N}}$, the random variables $(X_{n,1}(\omega),Y_{n,1}(\omega)),\dots,(X_{n,n}(\omega),Y_{n,n}(\omega))$ are distributed according to the Boltzmann-Gibbs measure $\mu_{n,\lambda\times \kappa,\beta}$. Let us define the random non-normalized, integer valued Borel measure $\xi_n(\cdot,\omega)$ on ${\mathbf{R}}^2$, by $$\label{eq:Definitionxin} \xi_n(A,\omega)\, =\, \sum_{i=1}^{n} {\mathbf{1}}\{ (X_{n,i}(\omega),Y_{n,i}(\omega)) \in A \}\, ,$$ This is a [*random point process*]{}. A general point process is a random, locally finite, nonnegative integer valued measure. We will restrict attention to finite point processes. Therefore, let $\mathcal{X}$ denote the set of all Borel measures $\xi$ on ${\mathbf{R}}^2$ such that $\xi(A) \in \{0,1,\dots \}$ for each Borel measurable set $A \subseteq {\mathbf{R}}^2$. Then, almost surely, $\xi_n(\cdot,\omega)$ is in $\mathcal{X}$. In fact $\xi_n({\mathbf{R}}^2,\omega)$ is a.s. just $n$. For a general random point process, the total number of points may be random. \[def:nu\] Let $\nu_{n,\lambda\times \kappa,\beta}$ be the Borel probability measure on $\mathcal{X}$ describing the distribution of the random element $\xi_n(\cdot,\omega) \in \mathcal{X}$ defined in (\[eq:Definitionxin\]), where $(X_{n,1}(\omega),Y_{n,1}(\omega)),\dots,(X_{n,n}(\omega),Y_{n,n}(\omega))$ are distributed according to the Boltzmann-Gibbs measure $\mu_{n,\lambda\times \kappa,\beta}$. Given a measure $\xi \in \mathcal{X}$, we extend the definition of the record length to $$\label{eq:DefinitionRecordPointProcess} \begin{split} \ell(\xi)\, =\, \max \{ k \, &: \, \exists (x_1,y_1),\dots,(x_k,y_k) \in {\mathbf{R}}^2\ \text { such that }\\ & \xi(\{(x_1,y_1),\dots,(x_k,y_k)\}) \geq k\ \text{ and } (x_i-x_j)(y_i-y_j) \geq 0 \text{ for all } i,j \}\, . \end{split}$$ With this definition, $$\label{eq:LBoldL} \ell(\xi_n(\cdot,\omega))\, =\, \ell((X_{n,1}(\omega),Y_{n,1}(\omega)),\dots,(X_{n,n}(\omega),Y_{n,n}(\omega)))\, ,$$ almost surely. There is a natural order on measures. If $\mu,\nu$ are two measures on ${\mathbf{R}}^2$, then let us say $\mu \leq \nu$ if $\mu(A) \leq \nu(A)$, for each Borel set $A \subseteq {\mathbf{R}}^2$. The function $\ell$ is monotone non-decreasing in the sense that if $\xi,\zeta$ are two measures in $\mathcal{X}$ then $\xi \leq \zeta \Rightarrow \ell(\xi) \leq \ell(\zeta)$. \[lem:CouplingAboveBelow\] Suppose that $\lambda$ and $\kappa$ each have no atoms. Then for each $n \in {\mathbf{N}}$, the following holds. - There exists a pair of random point processes $\eta_n,\xi_n$, defined on the same probability space, such that $\eta_n \leq \xi_n$, a.s., and satisfying these conditions: $\xi_n$ has distribution $\nu_{n,\lambda \times \kappa,\beta}$; there are i.i.d., Bernoulli-$p$ random variables $K_1,\dots,K_n$, for $p=\exp(-|\beta|)$, and i.i.d., $\lambda \times \kappa$-distributed points $(U_1,V_1),\dots,(U_{K_1+\dots+K_n},V_{K_1+\dots+K_n})$, such that $\eta_n(A) = \sum_{i=1}^{K_1+\dots+K_n} {\mathbf{1}}\{ (U_i,V_i) \in A \}$. - There exists a pair of random point processes $\xi_n,\zeta_n$, defined on the same probability space, such that $\xi_n \leq \zeta_n$, a.s., and satisfying these conditions: $\xi_n$ has distribution $\nu_{n,\lambda \times \kappa,\beta}$; there are i.i.d., geometric-$p$ random variables $N_1,\dots,N_n$, for $p=\exp(-|\beta|)$, and i.i.d., $\lambda \times \kappa$-distributed points $(U_1,V_1),\dots,(U_{N_1+\dots+N_n},V_{N_1+\dots+N_n})$, such that $\zeta_n(A) = \sum_{i=1}^{N_1+\dots+N_n} {\mathbf{1}}\{ (U_i,V_i) \in A \}$. We may combine this lemma with the weak law of large numbers and the Vershik and Kerov, Logan and Shepp theorem, to conclude the following: \[cor:ProbabilityBounds\] Suppose that $(q_n)_{n=1}^{\infty}$ is a sequence such that $\lim_{n \to \infty} n(1-q_n) = \beta \in {\mathbf{R}}$. Then, $$\notag \lim_{n \to \infty} \mu_{n,q_n} \{ \pi \in S_n \, : \, n^{-1/2} \ell(\pi) \in (2 e^{-|\beta|/2} - \epsilon,2e^{|\beta|/2}+\epsilon) \}\, =\, 1\, ,$$ for each $\epsilon>0$. Let us quickly prove this corollary, conditional on previously stated lemmas whose proofs will appear later. [**Proof of Corollary \[cor:ProbabilityBounds\]:**]{} Let $\beta_n$ be defined so that $\exp(-\beta_n/(n-1)) = q_n$. Let $\pi \in S_n$ be a random permutation, distributed according to $\mu_{n,q_n}$, and let $((X_{n,1},Y_{n,1}),\dots,(X_{n,n},Y_{n,n}))$ be distributed according to $\mu_{n,\lambda\times \kappa,\beta_n}$. We have the equality in distribution of the random variables $$\ell((X_{n,1},Y_{n,1}),\dots,(X_{n,n},Y_{n,n}))\, \stackrel{\mathcal{D}}{=} \, \ell(\pi)\, ,$$ as we noted in Section 2, before. Note $\lim_{n \to \infty} n (1-q_n) = \beta$, implies that $\lim_{n \to \infty} \beta_n = \beta$. For a fixed $n$, we apply Lemma \[lem:CouplingAboveBelow\], but with $\beta$ replaced by $\beta_n$, to conclude that there are random point processes $\eta_n(\cdot,\omega), \xi_n(\cdot,\omega) \in \mathcal{X}$ defined on the same probability space $\Omega$, and separately, there are random point processes $\xi_n(\cdot,\omega), \zeta_n(\cdot,\omega) \in \mathcal{X}$, defined on the same probability space, satisfying the conclusions of that lemma but with $\beta$ replaced by $\beta_n$. By (\[eq:LBoldL\]), we know that $$\ell(\pi)\, \stackrel{\mathcal{D}}{=} \, \ell(\xi)\, .$$ By monotonicity of $\ell$, and Lemma \[lem:CouplingAboveBelow\] we know that for each $k$ $$\label{eq:CouplingAboveBelowk} {\bf P}\{\ell(\eta) \geq k\}\, \leq\, {\bf P}\{\ell(\xi) \geq k\} \quad \text { and } \quad {\bf P}\{\ell(\xi) \geq k\} \, \leq \, {\bf P}\{\ell(\zeta) \geq k\}\, .$$ Using equations (\[eq:equiv\]) and (\[eq:LBoldL\]), this implies that for each $\epsilon>0$ $$\begin{gathered} \mu_{n,q_n}\{ \pi \in S_n\, :\, n^{-1/2} \ell(\pi) \leq 2 e^{|\beta|/2}+\epsilon \}\, \geq\, {\bf P}\{n^{-1/2} \ell(\zeta) \leq 2 e^{|\beta|/2}+\epsilon \}\, ,\\ \mu_{n,q_n}\{ \pi \in S_n\, :\, n^{-1/2} \ell(\pi) \geq 2 e^{-|\beta|/2}-\epsilon \}\, \geq\, {\bf P}\{n^{-1/2} \ell(\eta) \geq 2 e^{-|\beta|/2}-\epsilon \}\, .\end{gathered}$$ Since the $(U_i,V_i)$’s end at $i=K_1+\dots+K_n$ or $i=N_1+\dots+N_n$ in the two cases, let us also define new i.i.d., $\lambda\times\kappa$-distributed points $(U_i,V_i)$ for all greater values of $i$. We assume these are independent of everything else. Then all $(U_i,V_i)$ are i.i.d., $\lambda\times \kappa$ distributed. So, for any non-random number $m \in {\mathbf{N}}$, the induced permutation $\pi_m \in S_m$, corresponding to $((U_1,V_1),\dots,(U_m,V_m))$ is uniformly distributed. The random integers $K_1,\dots,K_n$ and $N_1,\dots,N_n$ from Lemma \[lem:CouplingAboveBelow\] are not independent of $(U_1,V_1),(U_2,V_2),\dots$. But, for instance, for any deterministic number $m$, conditioning on the event $\{\omega \in \Omega\, :\, K_1(\omega) + \dots + K_n(\omega) \leq m\}$, we have that $$\ell(\zeta)\, \leq\, \ell\big((U_1,V_1),\dots,(U_{m},V_{m})\big)\, ,$$ by using monotonicity of $\ell$ again. Therefore, for each $n \in {\mathbf{N}}$, and for any non-random number $M_n^+ \in {\mathbf{N}}$, we may bound $$\begin{aligned} {\bf P}\{n^{-1/2} \ell(\zeta) \leq 2 e^{|\beta|/2}+\epsilon \}\, &\geq\, \mu_{M_n^+,1}\{ \pi \in S_{M_n^+}\, :\, n^{-1/2} \ell(\pi) \leq 2 e^{|\beta|/2}+\epsilon \}\\ &\qquad - {\bf P}(\{\omega\in \Omega\, :\, K_1(\omega)+\dots+K_n(\omega) > M_n^+\})\, .\end{aligned}$$ Similarly, for any non-random number $M_n^-$, we may bound $$\begin{aligned} {\bf P}\{n^{-1/2} \ell(\zeta) \geq 2 e^{-|\beta|/2}-\epsilon \}\, &\geq\, \mu_{M_n^-,1}\{ \pi \in S_{M_n^-}\, :\, n^{-1/2} \ell(\pi) \geq 2 e^{-|\beta|/2}-\epsilon \}\\ &\qquad - {\bf P}(\{\omega \in \Omega\, :\, N_1(\omega)+\dots+N_n(\omega) < M_n^-\})\, .\end{aligned}$$ We choose $\delta$ such that $0<\delta<\epsilon$, and then we take sequences $M_n^+ = \lfloor n (e^{-|\beta|}+\delta) \rfloor$ and $N_n^- = \lceil n (e^{|\beta|} - \delta) \rceil$. Since $K_1,K_2,\dots$ are i.i.d., Bernoulli random variables with mean $e^{-|\beta|}$, and $N_1,N_2,\dots$ are i.i.d., geometric random variables with mean $e^{|\beta|}$, we may appeal to the weak law of large numbers to deduce $$\lim_{n \to \infty} {\bf P}(\{\omega\in \Omega\, :\, K_1(\omega)+\dots+K_n(\omega) > M_n^+\})\, =\, \lim_{n \to \infty} {\bf P}(\{\omega \in \Omega\, :\, N_1(\omega)+\dots+N_n(\omega) < M_n^-\})\, =\, 0\, .$$ Finally, by Proposition \[prop:VKLSPermutation\], we know that $$\begin{gathered} \liminf_{n \to \infty} \mu_{M_n^+,1}\{ \pi \in S_{M_n^+}\, :\, n^{-1/2} \ell(\pi) \leq 2 e^{|\beta|/2}+\epsilon \}\\ \geq\, \liminf_{n \to \infty} \mu_{M_n^+,1}\left\{ \pi \in S_{M_n^+}\, :\, (M_n^+)^{-1/2} \ell(\pi) \leq 2 \frac{e^{|\beta|/2}+\epsilon}{e^{|\beta|/2}+\delta}\right\} =\, 1\, ,\end{gathered}$$ and $$\begin{gathered} \liminf_{n \to \infty} \mu_{M_n^-,1}\{ \pi \in S_{M_n^-}\, :\, n^{-1/2} \ell(\pi) \geq 2 e^{-|\beta|/2}-\epsilon \}\\ \geq\, \liminf_{n \to \infty} \mu_{M_n^-,1}\left\{ \pi \in S_{M_n^-}\, :\, (M_n^-)^{-1/2} \ell(\pi) \geq 2 \frac{e^{-|\beta|/2}-\epsilon}{e^{-|\beta|/2}-\delta}\right\} =\, 1\, .\end{gathered}$$ The bounds in Corollary \[cor:ProbabilityBounds\] are useful for small values of $|\beta|$. For larger values of $\beta$, they are useful when combined with the following easy lemma: \[lem:TemperatureRenormalization\] Suppose $\lambda$ and $\kappa$ have no atoms, and let the random point process $\xi \in \mathcal{X}$ be distributed according to $\nu_{n,\lambda\times \kappa,\beta}$. Suppose that $R = [a_1,a_2] \times [b_1,b_2]$ is any rectangle. Let $\xi \restriction R$ denote the restriction of $\xi$ to this rectangle: i.e., $(\xi \restriction R)(A) = \xi(A \cap R)$. Note that this is still a random point process in $\mathcal{X}$ but one with a random total mass between $0$ and $n$. Then, for any $m \in \{1,\dots,n \}$, and any $k \in \{1,\dots,m\}$, we have $${\bf P}(\{\ell(\xi \restriction R) = k\}\, |\, \{\xi(R)=m\})\, =\, \mu_{m,q}\{ \pi \in S_m\, :\, \ell(\pi) = k\}\, ,$$ for $q = \exp(-\beta/(m-1))$. In order to use this lemma, we introduce an idea we call “paths of boxes.” Paths of boxes ============== We now introduce a method to derive Deuschel and Zeitouni’s Theorem \[thm:DeuschelZeitouni\] for our point process. For each $n$ we decompose the unit square $[0,1]^2$ into $n^2$ sub-boxes $$R_n(i,j)\, =\, \left[\frac{i-1}{n},\frac{i}{n}\right] \times \left[\frac{j-1}{n},\frac{j}{n}\right]\, .$$ We consider a basic path to be a sequence $(i_1,j_1),\dots,(i_{2n-1},j_{2n-1})$ such that $(i_1,j_1)=(1,1)$, $(i_{2n-1},j_{2n-1}) = (n,n)$ and $(i_{k+1}-i_k,j_{k+1}-j_k)$ equals $(1,0)$ or $(0,1)$ for each $k=1,\dots,2n-2$. In this case the basic path of boxes is the union $\bigcup_{k=1}^{2n-1} R_n(i_k,j_k)$. Note that $$\begin{gathered} (i_{k+1}-i_k,j_{k+1}-j_k)\, =\, (1,0)\quad \Rightarrow\quad R_n(i_k,j_k) \cap R_n(i_{k+1},j_{k+1})\, =\, \{i_k/n\} \times [(j_k-1)/n,j_k/n]\, ,\\ (i_{k+1}-i_k,j_{k+1}-j_k)\, =\, (0,1)\quad \Rightarrow\quad R_n(i_k,j_k) \cap R_n(i_{k+1},j_{k+1})\, =\,[(i_k-1)/n,i_k/n] \cap \{j_k/n\}\, .\end{gathered}$$ Now we consider a refined notion of path. We are motivated by the fact that Deuschel and Zeitouni’s $\mathcal{J}(u,\gamma)$ function does depend on the derivative of $\gamma$. To get reasonable error bounds we must allow for a choice of slope for each segment of the path. So, given $m \in {\mathbf{N}}$ and $n \in \{2,3,\dots \}$, we consider a set of “refined” paths $\Pi_{n,m}$ to be the set of all sequences $$\Gamma\, :=\, ((i_1,j_1),r_1,(i_2,j_2),r_2,(i_3,j_3),r_3,\dots,(i_{2n-2},j_{2n-2}),r_{2n-2},(i_{2n-1},j_{2n-1}))\, ,$$ where $((i_1,j_1),(i_2,j_2),\dots,(i_{2n-1},j_{2n-1})$ is a basic path, as described in the last paragraph, and $r_1,r_2,\dots,r_{2n-2}$ are integers in $\{1,\dots,m\}$ satisfying the additional condition: if $i_{k} = i_{k+1} = i_{k+2}$ or if $j_k=j_{k+1}=j_{k+2}$ then $r_{k+1} \geq r_k$, for each $k=1,\dots,2n-3$. We now explain the importance of this condition. Suppose that $R_n(i_k,j_k) \cap R_n(i_{k+1},j_{k+1}) = \{i_k/n\} \times [(j_k-1)/n,j_{k}/n]$. Then we decompose this interval into $m$ subintervals $$I_{n,m}^{(2)}(i_k;j_k,j_{k+1};r)\, =\, \left \{\frac{i_k}{n}\right\} \times \left[ \frac{j_k-1}{n} + \frac{r-1}{mn}, \frac{j_k-1}{n} + \frac{r}{m} \right]\, .$$ Similarly, if $R_n(i_k,j_k) \cap R_n(i_{k+1},j_{k+1}) = [(i_k-1)/n,i_{k}/n] \times \{j_k/n\}$, then we define $$I_{n,m}^{(1)}(i_k,i_{k+1};j_k;r)\, =\, \left[ \frac{i_k-1}{n} + \frac{r-1}{m}, \frac{i_k-1}{n} + \frac{r}{mn} \right] \times \left \{\frac{j_k}{n}\right\}\, .$$ In either case, the choice of $r_k$ is which subinterval the “path” passes through in going from $R_n(i_k,j_k)$ to $R_n(i_{k+1},j_{k+1})$. We define $I_k$ to be $I_{n,m}^{(2)}(i_k;j_k,j_{k+1};r_k)$ or $I_{n,m}^{(1)}(i_k,i_{k+1};j_k;r_k)$ depending on which case it is. We also define $(x_k,y_k)$ to be the center of the interval, either $$(x_k,y_k)\, =\, \left(\frac{i_k}{n},\frac{j_k-1}{n} + \frac{r-(1/2)}{mn}\right)\quad \textrm{or}\quad (x_k,y_k)\, =\, \left(\frac{i_k-1}{n} + \frac{r-(1/2)}{mn},\frac{j_k}{n}\right)\, .$$ The additional condition that we require for a refined path just guarantees that $x_{k+1} \geq x_k$ and $y_{k+1} \geq y_k$ for each $k$. We also define $(a_k,b_k) \in {\mathbf{R}}^2$ and $(c_k,d_k) \in {\mathbf{R}}^2$ to be the endpoints of the interval $I_k$. With these definitions, we may state our main result for paths of boxes. \[lem:PathOfBoxes\] Suppose that $\Gamma \in \Pi_{n,m}$ is a refined path. Also suppose that $\xi \in \mathcal{X}$ is a point process with support in $[0,1]^2$, such that no point lies on any line $\{(x,y)\, :\, x = i/n \}$ for $i \in {\mathbf{Z}}$ or any line $\{(x,y)\, :\, y=j/n \}$ for $j \in {\mathbf{Z}}$. Then $$\ell(\xi)\, \geq\, \sum_{k=1}^{2n-1} \ell(\xi \restriction [x_{k-1},x_k]\times [y_{k-1},y_k])\, ,$$ where we define $(x_0,y_0)=(0,0)$ and $(x_{2n-1},y_{2n-1})=(1,1)$. Also, $$\ell(\xi)\, \leq\, \max_{\Gamma \in \Pi_{n,m}} \sum_{k=1}^{2n-1} \ell(\xi \restriction [a_{k-1},c_k] \times [b_{k-1},d_k])\, ,$$ where we define $(a_0,b_0)=(0,0)$ and $(c_{2n-1},d_{2n-1}) = (1,1)$. We will prove this lemma in Section 8, after we have proved the other lemmas, since it requires several steps. Another useful lemma follows: \[lem:boxes\] Suppose that $u:[0,1]^2 \to {\mathbf{R}}$ is a probability density which is also continuous. Then, $$\max_{\Upsilon \in\mathcal{B}_{\nearrow}([0,1]^2)} \tilde{\mathcal{J}}(u,\Upsilon)\, =\, 2 \lim_{N \to \infty}\, \lim_{m \to \infty}\, \max_{\Gamma \in \Pi_{n,m}} \sum_{k=1}^{2N-1} \left(\int_{x_{k-1}}^{x_k} \int_{y_{k-1}}^{y_k} u(x,y)\, dx\, dy\right)^{1/2}\, .$$ We will prove this simple lemma in Section 7. With these preliminaries done, we may now complete the proof of the theorem. Completion of the Proof ======================= Suppose that $\beta \in {\mathbf{R}}$ is fixed. At first we will consider a fixed sequence $q_n = \exp(-\beta/(n-1))$, which does satisfy $n(1-q_n) \to \beta$ as $n \to \infty$. Define the triangular array of random vectors in ${\mathbf{R}}^2$: $((X_{n,k},Y_{n,k})\, :\, n \in {\mathbf{N}}\, ,\ 1\leq k\leq n)$, where for each $n \in {\mathbf{N}}$, the random variables $(X_{n,1},Y_{n,1}),\dots,(X_{n,n},Y_{n,n})$ are distributed according to the Boltzmann-Gibbs measure $\mu_{n,\lambda\times \kappa,\beta}$. Let $\xi_n \in \mathcal{X}$ be the random point process such that $$\xi_n(A)\, =\, \sum_{k=1}^{n} {\mathbf{1}}\{ (X_{n,k},Y_{n,k}) \in A \} \, ,$$ for each Borel measurable set $A \subseteq {\mathbf{R}}^2$. As we have noted before, we then have $$\begin{aligned} \notag \mu_{n,q_n}\{ \pi \in S_n \, : \, \ell(\pi) = k \} \, &=&\, P\{ \ell((X_{n,1},Y_{n,1}),\dots,(X_{n,n},Y_{n,n})) = k \} \\ \notag &=&\, P\{ \ell(\xi_n) = k \}\, ,\end{aligned}$$ for each $k$. Now suppose that $m, N \in {\mathbf{N}}$ are fixed. We consider “refined” paths in $\Pi_{N,m}$. By Lemma \[lem:PathOfBoxes\], which applies by first rescaling the unit square $[0,1]^2$ to $[0,L(\beta)]^2$, $$\label{eq:lower} \ell(\xi_n)\, \geq \, \max_{\Gamma \in \Pi_{N,m}} \sum_{k=1}^{2N-1} \ell(\xi_n \restriction [L(\beta) x_{k-1},L(\beta) x_k]\times [L(\beta) y_{k-1},L(\beta) y_k])\, .$$ The only difference is that we use the square $[0,L(\beta)]^2$ in place of $[0,1]^2$. Also, $$\label{eq:upper} \ell(\xi_n)\, \leq \, \max_{\Gamma \in \Pi_{N,m}} \sum_{k=1}^{2N-1} \ell(\xi_n \restriction [L(\beta) a_{k-1},L(\beta) c_k] \times [L(\beta) b_{k-1}, L(\beta) d_k])\, .$$ Now suppose that $\Gamma \in \Pi_{N,m}$ is fixed. Also consider a fixed sub-rectangle of $\Gamma$, $$R_k\, =\, [L(\beta) x_{k-1},L(\beta) x_k]\times [L(\beta) y_{k-1},L(\beta) y_k]\, .$$ By Lemma \[lem:StarrLimit\], we know that the random variables $\xi_n(R_k)/n$ converge in probability to the non-random limit $\sigma_{\beta}(R_k)$, as $n \to \infty$. Moreover, conditioning on the total number of points in the sub-rectangle $\xi_n(R_k)$, Lemma \[lem:TemperatureRenormalization\] tells us that $${\bf P}( \{\ell(\xi_n \restriction R_k) = \bullet\}\, |\, \{\xi_n(R_k) = r\})\, =\, \mu_{r,q_n}\{\pi \in S_r\, :\, \ell(\pi) = \bullet \}\, .$$ Note that the sequence of random variables $\xi_n(R_k) (1 - q_n)$ converges in probability to $\beta \sigma_{\beta}(R_k)$ as $n\to \infty$, because $$\xi_n(R_k) (1 - q_n)\, =\, n(1-q_n)\, \frac{\xi_n(R_k)}{n}\, ,$$ and $n(1-q_n) \to \beta$ as $n \to \infty$. Therefore, using Corollary \[cor:ProbabilityBounds\], this implies for each $\epsilon>0$ $$\lim_{n \to \infty} {\bf P}\left\{\xi_n(R_k)^{-1/2} \ell(\xi_n \restriction R) \in (2 e^{-\beta \sigma_{\beta}(R_k)/2} - \epsilon, 2 e^{\beta \sigma_{\beta}(R_k)/2} + \epsilon) \right\} \, =\, 1\, .$$ Since we have a limit in probability for $\xi_n(R_k)/n$, we may then conclude for each $\epsilon>0$ that $$\lim_{n \to \infty} {\bf P}\left\{ n^{-1/2} \ell(\xi_n \restriction R_k) \in (2 [\sigma_{\beta}(R_k)]^{1/2} e^{-\beta \sigma_{\beta}(R_k)/2} - \epsilon, 2 [\sigma_{\beta}(R_k)]^{1/2} e^{\beta \sigma_{\beta}(R_k)/2} + \epsilon) \right) \, =\, 1\, .$$ This is true for each sub-rectangle $R_k$ comprising $\Gamma$, and $\Gamma$ is in $\Pi_{N,m}$. But there are only finitely many sub-rectangles in $\Gamma$, and there are only finitely many possible choices of a refined path of boxes $\Gamma \in \Pi_{N,m}$, for $N$ and $m$ fixed. Combining this with (\[eq:lower\]) implies that for any $\epsilon>0$ we have $$\label{eq:FinalLowerBound} \lim_{n \to \infty} {\bf P}\left\{ n^{-1/2} \ell(\xi_n) \geq \max_{\Gamma \in \Pi_{m,n}} \sum_{k=1}^{2N-1} 2 [\sigma_{\beta}(R_k)]^{1/2} e^{-\beta \sigma_{\beta}(R_k)/2} - \epsilon \right\}\, =\, 1\, .$$ By exactly similar arguments and (\[eq:upper\]) we may also conclude that for each $\epsilon>0$ $$\label{eq:FinalUpperBound} \lim_{n \to \infty} {\bf P}\left\{ n^{-1/2} \ell(\xi_n) \leq \max_{\Gamma \in \Pi_{m,n}} \sum_{k=1}^{2N-1} 2 [\sigma_{\beta}(R^*_k)]^{1/2} e^{\beta \sigma_{\beta}(R^*_k)/2} + \epsilon \right\}\, =\, 1\, ,$$ where we define $$R_k^*\, =\, [L(\beta) a_{k-1},L(\beta) c_k],[L(\beta) b_{k-1},L(\beta) d_k]\, ,$$ for each $k = 1,\dots,2N-1$. We apply Lemma \[lem:boxes\] to $u_{\beta}$. For $N $ fixed, taking the limit $m\to \infty$, the area of the symmetric differences of the boxes $R_k^*$ and $R_k$ converges to zero, uniformly in $\Gamma \in \Pi_{N,m}$ for each $k=1,\dots,2N-1$. Since $\sigma_{\beta}$ has a density, the same is true replacing area by $\sigma_{\beta}$-measure. Moreover, $\exp(-\beta \sigma_{\beta}(R_k))$ and $\exp(\beta \sigma_{\beta}(R_k^*))$ converge to 1 uniformly as $N \to \infty$. Therefore, $$\label{eq:continuity} \begin{split} &\lim_{N \to \infty} \lim_{m \to \infty} \max_{\Gamma \in \Pi_{m,n}} \sum_{k=1}^{2N-1} 2 [\sigma_{\beta}(R_k)]^{1/2} e^{-\beta \sigma_{\beta}(R_k)/2}\\ &\hspace{1cm} =\, \lim_{N \to \infty} \lim_{m \to \infty} \max_{\Gamma \in \Pi_{m,n}} \sum_{k=1}^{2N-1} 2 [\sigma_{\beta}(R_k^*)]^{1/2} e^{-\beta \sigma_{\beta}(R_k^*)/2}\\ &\hspace{2cm} = \max_{\Upsilon \in \mathcal{B}_{\nearrow}([0,L(\beta)]^2)} \tilde{\mathcal{J}}(u_{\beta},\Upsilon)\, . \end{split}$$ Combined with (\[eq:FinalLowerBound\]) and (\[eq:FinalUpperBound\]), this implies that for each $\epsilon>0$, $$\lim_{n \to \infty} {\bf P} \left\{\left| n^{-1/2} \ell(\xi_n) - \max_{\Upsilon \in \mathcal{B}_{\nearrow}([0,L(\beta)]^2} \tilde{\mathcal{J}}(u_{\beta},\Upsilon)\right| < \epsilon \right\}\, =\, 1\, .$$ Finally, we use Lemma \[lem:variational\] to conclude that $$\max_{\Upsilon \in \mathcal{B}_{\nearrow}([0,L(\beta)]^2)} \tilde{\mathcal{J}}(u_{\beta},\Upsilon)\, \leq\, \mathcal{L}(\beta)\, .$$ But taking $\Upsilon = \{(t,t)\, :\, t \in [0,L(\beta)]\}$, which is the graph of the straight line curve $\gamma \in \mathcal{C}^1_{\nearrow}([0,L(\beta)]^2)$, gives $$\tilde{\mathcal{J}}(u_{\beta},\Upsilon)\, =\, \mathcal{J}(u_{\beta},\gamma)\, =\, 2 \int_0^{L(\beta)} \frac{1}{1-\beta t^2}\, dt\, .$$ This integral gives $\mathcal{L}(\beta)$. Thus, the proof is completed, for the special choice of $(q_n)$ equal to $(\exp(-\beta/(n-1)))$. Because the answer is continuous in $\beta$, if we consider any sequence $(q_n)$ satisfying $n (1-q_n) \to \beta$, then we get the same answer. All that is left is to prove all the lemmas. Proofs of Lemmas \[lem:USC\], \[lem:variational\], \[lem:TemperatureRenormalization\] and \[lem:boxes\] ======================================================================================================== We now prove the lemmas, in an order which is not necessarily the same as the order they were stated. This facilitates using arguments from one proof for the next one. [**Proof of Lemma \[lem:USC\].**]{} Define $$\tilde{\mathcal{J}}_{\epsilon}(u,\Upsilon)\, =\, \inf \{ \tilde{\mathcal{J}}(u,\mathcal{P})\, :\, \mathcal{P} \in \Pi(\Upsilon)\, ,\ \|\mathcal{P}\| < \epsilon \}$$ for each $\epsilon>0$. We first show that this function is upper semi-continuous. Let $\Pi_n$ denote $\Pi_n([a_1,a_2]\times [b_1,b_2])$. We remind the reader that this is the set of all $(n+1)$-tuples $\mathcal{P} = ((x_0,y_0),\dots,(x_n,y_n)) \in ({\mathbf{R}}^2)^{n+1}$ such that $a_1=x_0\leq \dots \leq x_n=a_n$ and $b_1=y_0\leq \dots \leq y_n=b_2$. For each $\mathcal{P} \in \Pi_n$, we have $$\tilde{\mathcal{J}}(u,\mathcal{P})\, =\, \sum_{k=0}^{n-1} \left(\int_{x_k}^{x_{k+1}} \int_{y_k}^{y_{k+1}} u(x,y)\, dx\, dy\right)^{1/2}\, .$$ Since $u$ is continuous, the mapping $\tilde{\mathcal{J}}(u,\cdot) : \Pi_n \to {\mathbf{R}}$ is continuous when $\Pi_n$ has its usual topology as a subset of $({\mathbf{R}}^2)^{n+1}$. Consider a fixed path $\Upsilon \in \mathcal{B}_{\nearrow}([a_1,a_2] \times [b_1,b_2])$ and a partition $\mathcal{P} \in \Pi(\Upsilon)$ such that $\|\mathcal{P}\| < \epsilon$. Note that there is some $n$ such that $\mathcal{P} \in \Pi_n(\Upsilon)$. Suppose that $(\Upsilon^{(k)})_{k=1}^{\infty}$ is a sequence in $\mathcal{B}_{\nearrow}([a_1,a_2] \times [b_1,b_2])$ converging to $\Upsilon$ in the Hausdorff metric. Then for each point $(x,y) \in \Upsilon$, there is a sequence of points $(x^{(k)},y^{(k)}) \in \Upsilon^{(k)}$ converging to $(x,y)$. Therefore, we may choose a sequence of partitions $\mathcal{P}^{(k)} \in \Pi_n(\Upsilon^{(k)})$ converging to $\mathcal{P}$ in $\Pi_n$. By the continuity mentioned above, $$\lim_{k \to \infty} \tilde{\mathcal{J}}(u,\mathcal{P}^{(k)}) \to \tilde{\mathcal{J}}(u,\mathcal{P})\, .$$ Also, $\|\mathcal{P}^{(k)}\|$ converges to $\|\mathcal{P}\|$ which is less than $\epsilon$. So, for large enough $k$, we have $\|\mathcal{P}^{(k)}\| < \epsilon$, and hence $$\tilde{\mathcal{J}}(u,\mathcal{P}^{(k)})\, \geq \, \tilde{\mathcal{J}}_{\epsilon}(u,\Upsilon^{(k)})\, ,$$ since the right hand side is the infimum. Therefore, we see that $$\limsup_{k \to \infty}\, \tilde{\mathcal{J}}_{\epsilon}(u,\Upsilon^{(k)})\, \leq \, \tilde{\mathcal{J}}(u,\mathcal{P}) \, .$$ Since this is true for all $\mathcal{P} \in \Pi(\Upsilon)$ with $\| \mathcal{P} \| < \epsilon$, taking the infimum we obtain $$\limsup_{k \to \infty}\, \tilde{\mathcal{J}}_{\epsilon}(u,\Upsilon^{(k)})\, \leq \, \tilde{\mathcal{J}}_{\epsilon}(u,\Upsilon)\, .$$ Since this is true for every $\Upsilon \in \mathcal{B}_{\nearrow}([a_1,a_2] \times [b_1,b_2])$ and every sequence $(\Upsilon^{(k)})$ converging to $\Upsilon$ in the Hausdorff metric, this proves that $\tilde{\mathcal{J}}_{\epsilon}(u,\cdot)$ is upper semi-continuous on $\mathcal{B}_{\nearrow}([a_1,a_2] \times [b_1,b_2])$. [**Proof of Lemma \[lem:boxes\]:**]{} The proof of this lemma is also used in the proof of Lemma \[lem:variational\]. This is the reason it appears first. Recall the definition of the basic boxes for $i,j \in \{1,\dots,N\}$, $$R_N(i,j)\, =\, \left[\frac{i-1}{N},\frac{i}{N}\right] \times \left[\frac{j-1}{N},\frac{j}{N}\right]\, .$$ Given $N \in {\mathbf{N}}$, let us define $u_N^+$ and $u_N^-$ so that $$\begin{gathered} u_N^+(x,y)\, =\, \sum_{i,j=1}^{N} \max_{(x',y') \in R_N(i,j)} u(x',y') \cdot {\bf 1}_{R_N(i,j)}(x,y)\, ,\\ u_N^-(x,y)\, =\, \sum_{i,j=1}^{N} \min_{(x',y') \in R_N(i,j)} u(x',y') \cdot {\bf 1}_{R_N(i,j)}(x,y)\, .\end{gathered}$$ By monotonicity, $\mathcal{J}(u_N^-,\Upsilon) \leq \mathcal{J}(u,\Upsilon) \leq \mathcal{J}(u_N^+,\Upsilon)$ for every $\Upsilon \in \mathcal{B}_{\nearrow}([0,1]^2)$. But since $u_N^-$ and $u_N^+$ are constant on squares, we know that the optimal $\Upsilon$’s for $u_N^-$ and $u_N^+$ are graphs of rectifiable curves $\gamma$ that are piecewise straight line curves on squares. This follows from the discussion immediately following the statement of Lemma \[lem:variational\], where we verified the special case of that lemma for constant densities. The only degrees of freedom for such curves are the slopes of each straight line, i.e., where they intersect the boundaries of each basic square. For $(x_k,y_k), (x_{k+1},y_{k+1}) \in R_N(i,j)$ representing two points on the boundary, such that $x_{k-1}\leq x_k$ and $y_{k-1}\leq y_k$, considering $\gamma_k$ to be the straight line joining these points, $$\int_{\gamma_k} \sqrt{u_N^+(x(t),y(t)) x'(t) y'(t)}\, dt\, =\, \sqrt{(x_k-x_{k-1})(y_k-y_{k-1})} \max_{(x,y) \in R_N(i,j)} \sqrt{u(x,y)}\, ,$$ with a similar formula for $u^-$. This is a continuous function of the endpoints. We may approximate the actual optimal piecewise straight line path by the “refined paths” of boxes in $\Pi_{N,m}$ if we take the limit $m \to \infty$ with $N$ fixed. Therefore, we find that $$\max_{\Upsilon \in \mathcal{B}_{\nearrow}([0,1]^2)} \tilde{\mathcal{J}}(u^\pm_N,\Upsilon)\, =\, \lim_{m \to \infty}\, \max_{\Gamma \in \Pi_{m,n}} \sum_{k=1}^{2N-1} \left(\int_{x_{k-1}}^{x_k} \int_{y_{k-1}}^{y_k} u^\pm_N(x,y)\, dx\, dy\right)^{1/2}\, .$$ Note that by upper semicontinuity, for each fixed $N$, the limit as $m \to \infty$ of the sequence $$\max_{\Gamma \in \Pi_{m,n}} \sum_{k=1}^{2N-1} \left(\int_{x_{k-1}}^{x_k} \int_{y_{k-1}}^{y_k} u(x,y)\, dx\, dy\right)^{1/2}$$ also exists, and is the supremum over $m \in {\mathbf{N}}$. Therefore, we conclude that for each fixed $N \in {\mathbf{N}}$, $$\begin{aligned} &\hspace{-1cm} \lim_{m \to \infty}\, \max_{\Gamma \in \Pi_{m,n}} \sum_{k=1}^{2N-1} \left(\int_{x_{k-1}}^{x_k} \int_{y_{k-1}}^{y_k} u(x,y)\, dx\, dy\right)^{1/2}\\ &\hspace{1cm} \in \left[\max_{\Upsilon \in \mathcal{B}_{\nearrow}([0,1]^2)} \tilde{\mathcal{J}}(u^-_N,\Upsilon)\, ,\ \max_{\Upsilon \in \mathcal{B}_{\nearrow}([0,1]^2)} \tilde{\mathcal{J}}(u^+_N,\Upsilon)\right]\, .\end{aligned}$$ But taking $N \to \infty$, we see that $u_N^+$ and $u_N^-$ converge to $u$, uniformly due to the continuity of $u$. Therefore, by the bound from equation (\[eq:Holder\]), the lemma follows. [**Proof of Lemma \[lem:variational\]:**]{} Suppose that $x(t)$, $y(t)$ is a $\mathcal{C}^1$ parametrization of a curve $\gamma \in \mathcal{C}^1_{\nearrow}([0,L(\beta)]^2)$. We may consider another time parametrization $x_1(t) = x(f(t))$ and $y_1(t) = y(f(t))$ for a $\mathcal{C}^1$ function $f(t)$ such that $$x_1(t) y_1(t)\, =\, t^2\, .$$ Indeed, we obtain $x(f(t)) y(f(t)) = t^2$. Setting $g(t) = x(t) y(t)$, our assumptions on $x(t)$ and $y(t)$ guarantee that $g$ is continuous and $g'(t)$ is strictly positive and finite for all $t$. We then take $f(t) = g^{-1}(t^2)$. Since a change of time parametrization does not affect $\mathcal{J}(u_{\beta},\gamma)$, we will simply assume that $x(t) y(t) = t^2$ is satisfied at the outset. Then we obtain $$\mathcal{J}(u_{\beta},\gamma)\, =\, \int_0^{L(\beta)} \frac{2 \sqrt{x'(t) y'(t)}}{1-\beta t^2}\, dt\, ,$$ due to the formula for $u_{\beta}$, and the fact that $x(t) y(t) = t^2 = L^2(\beta)$ at the endpoint of $\gamma$. Now since we have $x(t) y(t) = t^2$, that implies that $$\label{eq:constraint} x(t) y'(t) + y(t) x'(t)\, =\, 2 t\, .$$ We know that $x'(t)$ and $y'(t)$ are nonnegative. Therefore, we may use Cauchy’s inequality with $\epsilon$ $$\sqrt{x'(t) y'(t)}\, = \, [x'(t)]^{1/2} [y'(t)]^{1/2}\, \leq \, \frac{\epsilon}{2} x'(t) + \frac{1}{2\epsilon} y'(t)\, ,$$ for each $\epsilon \in (0,\infty)$. Taking $\epsilon = y(t)/t$ we get $\epsilon^{-1} = t/y(t)$ which is $x(t)/t$ since we chose the parametrization that $x(t) y(t) = t^2$. Therefore, we obtain $$\sqrt{x'(t) y'(t)}\, \leq \, \frac{ y(t) x'(t) + x(t) y'(t)}{2t}\, .$$ Taking into account our constraint (\[eq:constraint\]), this gives $$\sqrt{x'(t) y'(t)}\, \leq \, 1\, .$$ Since this is true at all $t \in [0,L(\beta)]$ this proves the desired inequality. But this upper bound gives the integral $$\label{eq:integralformula} \int_0^{L(\beta)} \frac{dt}{1-\beta t^2}\, =\, \int_0^{[(1-e^{-\beta})/\beta]^{1/2}} \frac{dt}{1-\beta t^2}\, ,$$ which equals the formula for $\mathcal{L}(\beta)$ from (\[eq:LDefinition\]). The argument works even if $\gamma$ is only piecewise $\mathcal{C}^1$, with finitely many pieces. Moreover, by the proof of Lemma \[lem:boxes\], we know that the maximum over all $\Upsilon$ is arbitrarily well approximated by optimizing over piecewise linear paths. So the inequality is true in general. [**Proof of Lemma \[lem:TemperatureRenormalization\]:**]{} This lemma is related to an important independence property of the Mallows measure. Gnedin and Olshanski prove this in Proposition 3.2 of [@GnedinOlshanski2], and they note that Mallows also stated a version in [@Mallows]. Our lemma is slightly different so we prove it here for completeness. Using Definition \[def:nu\], we can instead consider $(X_{1},Y_{1}),\dots,(X_{n},Y_{n})$ distributed according to $\mu_{n,\lambda\times \kappa,\beta}$ in place of $\xi$ distributed according to $\nu_{n,\lambda\times \kappa,\beta}$. Given $m\leq n$, we note that, conditioning on the positions of $(X_{m+1},Y_{m+1}),\dots,(X_n,Y_n)$, the conditional distribution of $(X_1,Y_1),\dots,(X_m,Y_m)$ is the same as $\mu_{m,\alpha,\beta'}$, where $\beta' = (m-1)\beta/(n-1)$ and where $\alpha$ is the random measure $$d\alpha(x,y)\, =\, \frac{1}{Z_1}\, \exp\left(-\frac{\beta}{n-1} \sum_{i=m+1}^n h(x-X_i,y-Y_i)\right)\, d\lambda(x)\, d\kappa(y)\, ,$$ where $Z_1$ is a random normalization constant. By finite exchangeability of $\mu_{n,\lambda\times \kappa,\beta}$ it does not matter which $m$ points we assume are in the square $[a_1,a_2]\times [b_1,b_2]$ which is why we just chose the first $m$. If we could rewrite $\alpha$ as a product of two measures $\lambda',\kappa'$ without atoms then we could appeal to (\[eq:equiv\]). By inspection $\alpha$ is not a product of two measures. However, if we condition on the event that there are exactly $m$ points in the square $[a_1,a_2]\times [b_1,b_2]$ then we can accomplish this goal. Let use define the event $A = \{(X_{m+1},Y_{m+1}),\dots,(X_n,Y_n) \not\in [a_1,a_2]\times [b_1,b_2]\}$. Then, given the event $A$, we can write $$\label{eq:cond} {\bf 1}_{[a_1,a_2]\times [b_1,b_2]}(x,y)\, d\alpha(x,y)\, =\, d\lambda'(x)\, d\kappa'(y)\, ,$$ where $\lambda'$ and $\kappa'$ are random measures $$d\lambda'(x)\, =\, \frac{1}{Z_2}\, e^{-\beta h_1(x)/(n-1)}\, d\lambda(x)\, ,\quad d\kappa'(y)\, =\, \frac{1}{Z_3}\, e^{-\beta h_2(y)/(n-1)}\, d\kappa(y)\, ,$$ with $Z_2$ and $Z_3$ normalization constants and random functions $$h_1(x)\, =\, \sum_{i=m+1}^{n} [{\bf 1}_{\{Y_i < b_1\}} {\bf 1}_{(X_i,\infty)}(x) + {\bf 1}_{\{Y_i>b_2\}} {\bf 1}_{(-\infty,X_i)}(x)]\, ,$$ and $$h_2(y)\, =\, \sum_{i=m+1}^{n} [{\bf 1}_{\{X_i < a_1\}} {\bf 1}_{(Y_i,\infty)}(y) + {\bf 1}_{\{X_i>a_2\}} {\bf 1}_{(-\infty,Y_i)}(x)]\, .$$ This may appear not to reproduce $\alpha$ exactly because it may seem that $h_1$ and $h_2$ double-count some terms which are only counted once in $\sum_{i=m+1}^{n} h(x-X_i,y-Y_i)$. But this is compensated by the normalization constants $Z_1$ and $Z_2$ as we now explain. Note that for each $i \in \{m+1,\dots,n\}$ since $(X_i,Y_i) \not\in [a_1,a_2] \times [b_1,b_2]$ we either have $Y_i<b_1$, $Y_i>b_2$, $X_i<a_1$ or $X_i>a_2$. These are not exclusive. But for instance, if $Y_i<b_1$ and $X_i<a_1$ then for every $(x,y) \in [a_1,a_2] \times [b_1,b_2]$, we have ${\bf 1}_{\{Y_i < b_1\}} {\bf 1}_{(X_i,\infty)}(x)=1$ and ${\bf 1}_{\{X_i < a_1\}} {\bf 1}_{(Y_i,\infty)}(y)=1$. Therefore, these terms are constant in the functions $h_1(x)$ and $h_2(y)$: they do not depend on the actual position of $(x,y)$ as long as $(x,y) \in [a_1,a_2] \times [b_1,b_2]$. Therefore, using the normalization constants $Z_1$ and $Z_2$, this double-counting may be compensated. Since we are conditioning on $\{(X_1,Y_1),\dots,(X_m,Y_m) \in [a_1,a_2] \times [b_1,b_2]\}$ and the event $A$, the conditional identity (\[eq:cond\]) suffices to prove the claim. Proof of Lemma \[lem:CouplingAboveBelow\] ========================================== This is the most involved lemma to prove. It follows from a coupling argument. In fact we use the most basic type of coupling for discrete random variables, based on the total variation distance. See the monograph [@LevinPeresWilmer] (Chapter 4) for a nice and elementary exposition. But we also combine this with the fact that we have a measure which may be derived from a statistical mechanical model of [*mean field*]{} type. Because the model is of mean field type, the correlations are weak and spread out. In principle, this allows one to approximate by a mixture of i.i.d., points as one sees in de Finetti’s theorem in probability, or the Kac-Lebowitz-Penrose limit in statistical physics. (See [@Aldous] for a reference on the former, and the appendix of [@Thompson] for the latter.) Given a probability measure $\alpha$ on ${\mathbf{R}}^2$, let $\theta_{1,\alpha}$ be the distribution on $\mathcal{X}$ associated to the random point process $$\xi_1(A,\omega)\, =\, {\bf 1}_A(X(\omega),Y(\omega))\, ,$$ assuming $(X(\omega),Y(\omega))$ is $\alpha$-distributed. \[lem:Coupling\] Suppose that $\alpha$ and $\tilde{\alpha}$ are two measures on ${\mathbf{R}}^2$ such that $\tilde{\alpha} \ll \alpha$, and suppose that for some $p \in (0,1]$ there are uniform bounds $$p\, \leq\, \frac{d\tilde{\alpha}}{d\alpha}\, \leq\, p^{-1}\, .$$ Then the following holds. - There exists a pair of random point processes $\eta_1, \xi_1$, defined on the same probability space, such that $\eta_1 \leq \xi_1$, a.s., and satisfying these properties: $\xi_1$ has distribution $\theta_{1,\tilde{\alpha}}$; there is an $\alpha$-distributed random point $(U_1,V_1)$, and independently there is a Bernoulli-$p$ random variable $K_1$, such that $\eta_1(A) = K_1 {\bf 1}_A(U_1,V_1)$. - There exists a pair of random point processes $\xi_1,\zeta_1$, defined on the same probability space, such that $\xi_1 \leq \zeta_1$, a.s., and satisfying these properties: $\xi_1$ has distribution $\theta_{1,\tilde{\alpha}}$; there is a sequence of i.i.d., $\alpha$-distributed points $(U_1,V_1),(U_2,V_2),\dots$ and a geometric-$p$ random variable $N_1$, such that $\zeta_1(A) = \sum_{i=1}^{N_1} {\bf 1}_A(U_i,V_i)$. Let $f = d\tilde{\alpha}/d\alpha$. We follow the standard approach, for example in Section 4.2 of [@LevinPeresWilmer]. We describe it here in detail, in order to be self-contained. Define $g(x) = (1-p)^{-1} [f(x)-p]$, which is nonnegative by assumption, and let $\hat{\alpha}$ be the probability measure such that $d\hat{\alpha}/d\alpha = g$. Note that $\tilde{\alpha}$ can be written as a mixture: $\tilde{\alpha} = p \alpha + (1-p) \hat{\alpha}$. Independently of one another, let $(U_1,V_1) \in {\mathbf{R}}^2$ be $\alpha$-distributed, and let $(W_1,Z_1) \in {\mathbf{R}}^2$ be $\hat{\alpha}$-distributed. Independently of all that, also let $K_1$ be Bernoulli-$q$. Then, taking $$(X_1,Y_1)\, =\, \begin{cases} (U_1,V_1) & \text { if $K_1 = 1$,}\\ (W_1,Z_1) & \text { otherwise,} \end{cases}$$ we see that $(X_1,Y_1)$ is $\tilde{\alpha}$-distributed. We let $\eta_1(A,\omega) = K_1(\omega) {\bf 1}_A(U_1(\omega),V_1(\omega))$. If $K_1(\omega)=1$ then $(U_1(\omega),V_1(\omega)) = (X_1(\omega),Y_1(\omega))$. Therefore taking $\xi_1(A,\Omega) = {\bf 1}_A(X_1(\omega),Y_1(\omega))$, we see that $\eta_1(\cdot,\omega) \leq \xi_1(\cdot,\omega)$, a.s. This proves (a). The proof for (b) is analogous. Let $h(x) = (p^{-1} - 1)^{-1}[p^{-1} - f(x)]$, which is nonnegative by hypothesis. Let $\check{\alpha}$ be the probability measure such that $d\check{\alpha}/d\alpha = h$. Then $\alpha$ can be written as the mixture: $\alpha = p \tilde{\alpha} + (1-p) \check{\alpha}$. Independently of each other, let $(X_1,Y_1),(X_2,Y_2),\dots$ be i.i.d., $\tilde{\alpha}$ distributed random variables, and let $(Z_1,W_1), (Z_2,W_2),\dots$ be i.i.d., $\check{\alpha}$ distributed random variables. Also, independently of all that, let $K_1,K_2,\dots$ be i.i.d., Bernoulli-$q$ random variables. For each $i$, we define $$(U_i,V_i)\, =\, \begin{cases} (X_i,Y_i) & \text { if $K_i=1$,}\\ (Z_i,W_i) & \text { otherwise.} \end{cases}$$ Then $(U_1,V_1),(U_2,V_2),\dots$ are i.i.d., $\alpha$-distributed random variables. Let $N_1 = \min\{n\, :\, K_n = 1\}$. We see that $(X_{N_1},Y_{N_1}) = (U_{N_1},V_{N_1})$. So clearly ${\bf 1}_A(X_{N_1},Y_{N_1}) \leq \sum_{k=1}^{N_1} {\bf 1}_A(U_k,V_k)$. Note that $K_1$ and $N_1$ are random variables which are dependent on $(U_1,V_1),(U_2,V_2),\dots$. But, for instance, conditioning on the event $\{N_1\geq i\}$, we do see that $(U_i,V_i)$ is $\alpha$-distributed. This is for the usual reason, as in Doob’s optional stopping theorem: the event $\{N_1 \geq i\}$ is measurable with respect to the $\sigma$-algebra generated by $K_1,\dots,K_{i-1}$, while the point $(U_i,V_i)$ is independent of that $\sigma$-algebra. This will be useful when we consider $n>1$, which is next. Resampling and Coupling for $n>1$ --------------------------------- In order to complete the proof of Lemma \[lem:CouplingAboveBelow\] we want to use Lemma \[lem:Coupling\]. More precisely we wish to iterate the bound for $n>1$. Suppose that $\tilde{\alpha}_n$ is a probability measure on $({\mathbf{R}}^2)^n$, and $\alpha$ is a probability measure on ${\mathbf{R}}^2$. Let $\theta_{n,\tilde{\alpha}_n}$ be the distribution on $\mathcal{X}$ associated to the random point process $$\xi_n(A,\omega)\, =\, \sum_{k=1}^{n} {\bf 1}_A(X_k(\omega),Y_k(\omega))\, ,$$ assuming $(X_1(\omega),Y_1(\omega)),\dots,(X_n(\omega),Y_n(\omega))$ are $\tilde{\alpha}_n$-distributed. If $\tilde{\alpha}_n$ was a product measure then it would be trivial to generalize Lemma \[lem:Coupling\] to compare it to the product measure $\alpha^n$. But there is another condition which makes it equally easy to generalize. Let $\mathcal{F}$ denote the Borel $\sigma$-algebra on ${\mathbf{R}}^2$. Let $\mathcal{F}^n$ denote the Borel $\sigma$-algebra on $({\mathbf{R}}^2)^n$. Let $\mathcal{F}^n_k$ denote the sub-$\sigma$-algebra of $\mathcal{F}^n$ generated by the maps $((x_1,y_1),\dots,(x_n,y_n)) \mapsto (x_j,y_j)$ for $j \in \{1,\dots,n\} \setminus \{k\}$. We suppose that there are regular conditional probability measures for each of these sub-$\sigma$-algebras. Let us make this precise: \[def:rcpd\] We say that $\tilde{\alpha}_{n,k} : \mathcal{F} \times ({\mathbf{R}}^2)^n \to {\mathbf{R}}$ is a regular conditional probability measure for $\tilde{\alpha}_n$, relative to the sub-$\sigma$-algebra $\mathcal{F}^n_k$ if the following three conditions are met: 1. For each $((x_1,y_1),\dots,(x_n,y_n)) \in ({\mathbf{R}}^2)^n$ the mapping $$A\mapsto \tilde{\alpha}_{n,k}\big(A;(x_1,y_1),\dots,(x_n,y_n)\big)$$ defines a probability measure on $\mathcal{F}$. 2. For each $A \in \mathcal{F}$, the mapping $$((x^1,y^2),\dots,(x^n,y^n)) \mapsto \tilde{\alpha}_{n,k}\big(A;(x_1,y_1),\dots,(x_n,y_n)\big)$$ is $\mathcal{F}^n$ measurable. 3. The measure $\tilde{\alpha}_{n,k}$ is a [*version*]{} of the conditional expectation ${\bf E}^{\tilde{\alpha}_n}[\cdot\, |\, \mathcal{F}^n_k]$. In this case this means precisely that for each $A_1,\dots,A_n \in \mathcal{F}$, $${\bf E}^{\tilde{\alpha}_n}\Bigg[\tilde{\alpha}_{n,k}\Big(A_k;(X_1,Y_1),\dots,(X_n,Y_n)\Big) \prod_{\substack{j=1\\ j\neq k}}^n {\bf 1}_{A_j}(X_j,Y_j)\Bigg]\, =\, \tilde{\alpha}_n(A_1\times \cdots \times A_n)\, .$$ For $p \in (0,1]$, we will say that $\tilde{\alpha}_n$ satisfies the $p$-resampling condition relative to $\alpha$ if the following conditions are satisfied: - There exist regular conditional probability distributions $\tilde{\alpha}_{n,k}$ relative to $\mathcal{F}^n_k$ for $k=1,\dots,n$. - For each $((x_1,y_1),\dots,(x_n,y_n)) \in {\mathbf{R}}^n$, and for each $k=1,\dots,n$, $$\tilde{\alpha}_{n,k}(\cdot;(x_1,y_1),\dots,(x_n,y_n))\, \ll\, \alpha\, .$$ - The following uniform bounds are satisfied for each $((x_1,y_1),\dots,(x_n,y_n)) \in {\mathbf{R}}^n$, and for each $k=1,\dots,n$: $$p\, \leq\, \frac{d\tilde{\alpha}_{n,k}(\cdot;(x_1,y_1),\dots,(x_n,y_n))}{d\alpha}\, \leq p^{-1}\, .$$ \[lem:CouplingGeneral\] Suppose that for some $p \in (0,1]$, the measure $\tilde{\alpha}_n$ satisfies the $p$-resampling condition relative to $\alpha$. Then the following holds. - There exists a pair of random point processes $\eta_n, \xi_n$, defined on the same probability space, such that $\eta_n \leq \xi_n$, a.s., and satisfying these properties: $\xi_n$ has distribution $\theta_{n,\tilde{\alpha}_n}$; there are i.i.d., $\alpha$-distributed points $\{(U^k_1,V^k_1)\}_{k=1}^n$, and independently there are i.i.d., Bernoulli-$p$ random variables $K_1,\dots,K_n$, such that $\eta_n(A) = \sum_{k=1}^{n} K_k {\bf 1}_A(U^k_1,V^k_1)$. - There exists a pair of random point processes $\xi_n,\zeta_n$, defined on the same probability space, such that $\xi_n \leq \zeta_n$, a.s., and satisfying these properties: $\xi_n$ has distribution $\theta_{n,\tilde{\alpha}_n}$; there are i.i.d., $\alpha$-distributed points $\{(U^k_i,V^k_i)\, :\, k=1,\dots,n\, ,\ i=1,2,\dots\}$, and i.i.d., geometric-$p$ random variables $N_1,\dots,N_n$, such that $\zeta_n(A) = \sum_{k=1}^{n} \sum_{i=1}^{N_k} {\bf 1}_A(U^k_i,V^k_i)$. We start with an $\tilde{\alpha}_n$-distributed random point $((X^k_1,Y^k_1),\dots,(X^k_n,Y^k_n))$. Then iteratively, for each $k=1,\dots,n$, we update this point as follows. Conditional on $$((X^{k-1}_1,Y^{k-1}_1),\dots,(X^{k-1}_n,Y^{k-1}_n))\, ,$$ we choose $(X^k_k,Y^k_k)$ randomly, according to the distribution $$\tilde{\alpha}_{n,k}\big(\cdot;(X^{k-1}_1,Y^{k-1}_1),\dots,(X^{k-1}_n,Y^{k-1}_n)\big)\, .$$ We let $(X^k_j,Y^k_j) = (X^{k-1}_j,Y^{k-1}_j)$ for each $j \in \{1,\dots,n\} \setminus \{k\}$. With this resampling rule, we can see that $((X^k_1,Y^k_1),\dots,(X^k_n,Y^k_n))$ is $\tilde{\alpha}_n$-distributed for each $k$. Also $(X^n_k,Y^n_k) = (X^k_k,Y^k_k)$. We apply Lemma \[lem:Coupling\] to each of the points $(X^k,Y^k)$, in turn. Since they all have distributions satisfying the hypotheses of the lemma, this may be done. Note that by our choices, the various $(U^k_i,V^k_i)$’s and $K_k$’s and $N_k$’s have distributions which are prescribed just in terms of $p$ and $\alpha$. Their distributions do not depend on the regular conditional probability distributions, as long as the hypotheses of the present lemma are satisfied. Therefore, they are independent of one another. Given the lemma, for part (a) we let $(U_1,V_1),\dots,(U_{K_1+\dots+K_n},V_{K_1+\dots+K_n})$ be equal to the points $(U^k_1,V^k_1)$ such that $K_k=1$, suitably relabeled, but keeping the relative order. By the idea, related to Doob’s stopping theorem, that we mentioned before, one can see that $$(U_1,V_1),\dots,(U_{K_1+\dots+K_n},V_{K_1+\dots+K_n})$$ are i.i.d., $\alpha$-distributed. We do similarly in case (b). This allows us to match up our notation with Lemma \[lem:CouplingAboveBelow\]. The only thing left is to check that “$p$-resampling condition” for the regular conditional probability distributions is satisfied for Boltzmann-Gibbs distributions. Regular conditional probability distributions for the Boltzmann-Gibbs measure ----------------------------------------------------------------------------- In Lemma \[lem:CouplingAboveBelow\], we assume that $((X_1,Y_1),\dots,(X_n,Y_n))$ are distributed according to the Boltzmann-Gibbs measure $\mu_{n,\lambda \times \kappa,\beta}$. Then we let $\nu_{n,\lambda\times \kappa,\beta}$ be the distribution of the random point process $\xi_n$, such that $$\xi_n(A)\, =\, \sum_{k=1}^{n} {\bf 1}_A(X_k,Y_k)\, .$$ In other words, the distribution $\mu_{n,\lambda\times\kappa,\beta}$ corresponds to the distribution we have denoted $\theta_{n,\tilde{\alpha}_n}$ if we let $\tilde{\alpha}_n = \mu_{n,\lambda \times \kappa,\beta}$. We take $\alpha = \lambda \times \kappa$. Now we want to verify the hypotheses of Lemma \[lem:CouplingGeneral\] for $p=e^{-|\beta|}$. Referring back to Section \[sec:BoltzmannGibbs\], we see that $\tilde{\alpha}_n$ is absolutely continuous with respect to the product measure $\alpha^n$. Moreover, $$\frac{d\tilde{\alpha}_n}{d\alpha^n}((x_1,y_1),\dots,(x_n,y_n))\, =\, \frac{1}{Z_n(\alpha,\beta)}\, \exp\Big(-\beta H_n((x_1,y_1),\dots,(x_n,y_n))\Big)\, .$$ Here the Hamiltonian is $$H_n((x_1,y_1),\dots,(x_n,y_n))\, =\, \frac{1}{n-1} \sum_{i=1}^{n-1} \sum_{j=i+1}^{n} h(x_i-x_j,y_i-y_j)\, .$$ This leads us to define a conditional Hamiltonian for the single point $(x,y)$ substituted in for $(x_k,y_k)$ in the configuration $((x_1,y_1),\dots,(x_n,y_n))$: $$H_{n,k}\big((x,y);(x_1,y_1),\dots,(x_n,y_n)\big)\, =\, \frac{1}{n-1} \sum_{\substack{j=1\\j\neq k}}^n h_n(x-x_j,y-y_j)\, .$$ With this, we define a measure $\tilde{\alpha}_{n,k}\big(\cdot;(x_1,y_1),\dots,(x_n,y_n)\big)$, which is absolutely continuous with respect to $\alpha$, and such that $$\frac{d\tilde{\alpha}_{n,k}\big(\cdot;(x_1,y_1),\dots,(x_n,y_n)\big)}{d\alpha}(x,y)\, =\, \frac{1} {Z_{n,k}\big(\alpha,\beta;(x_1,y_1),\dots,(x_n,y_n)\big)}e^{-\beta H_{n,k}((x,y);(x_1,y_1),\dots,(x_n,y_n))}\, .$$ The normalization is $$Z_{n,k}\big(\alpha,\beta;(x_1,y_1),\dots,(x_n,y_n)\big)\, =\, \int_{{\mathbf{R}}^2} e^{-\beta H_{n,k}((x,y);(x_1,y_1),\dots,(x_n,y_n))}\, d\alpha(x,y)\, .$$ To see that this is the desired regular conditional probability distribution, note that in the product $$\frac{d\tilde{\alpha}_{n,k}\big(\cdot;(x_1,y_1),\dots,(x_n,y_n)\big)}{d\alpha}(x,y)\, \frac{d\tilde{\alpha}_n}{d\alpha^n}((x_1,y_1),\dots,(x_n,y_n))$$ we have the product of two factors: $$\frac{1} {Z_{n,k}\big(\alpha,\beta;(x_1,y_1),\dots,(x_n,y_n)\big)}e^{-\beta H_{n,k}((x,y);(x_1,y_1),\dots,(x_n,y_n))}$$ and $$\frac{1}{Z_n(\alpha,\beta)}\, \exp\Big(-\beta H_n((x_1,y_1),\dots,(x_n,y_n))\Big)\, .$$ The first factor does not depend on $(x_k,y_k)$. The second factor does depend on it, but integrating against $d\alpha(x_k,y_k)$ gives, $$\int_{{\mathbf{R}}^2} \frac{ e^{-\beta H_n((x_1,y_1),\dots,(x_n,y_n))}}{Z_n(\alpha,\beta)}\, d\alpha(x_k,y_k)\, =\, \frac{Z_{n,k}\big(\alpha,\beta;(x_1,y_1),\dots,(x_n,y_n)}{Z_n(\alpha,\beta)} e^{-\beta H_{n,k}'((x_1,y_1),\dots,(x_n,y_n))}$$ where $$H_{n,k}'((x_1,y_1),\dots,(x_n,y_n))\, =\, \frac{1}{n-1} \sum_{\substack{i=1\\i\neq k}}^{n-1} \sum_{\substack{j=i+1\\j\neq k}}^{n} h(x_i-x_j,y_i-y_j)\, ,$$ and we have $$\begin{gathered} H_{n,k}'((x_1,y_1),\dots,(x_n,y_n)) + H_{n,k}\big((x,y);(x_1,y_1),\dots,(x_n,y_n)\big)\\ =\, H_{n}\big((x_1,y_1),\dots,(x_{k-1},y_{k-1}),(x,y),(x_{k+1},y_{k+1}),\dots,(x_n,y_n)\big)\, .\end{gathered}$$ Therefore, $$\int_{{\mathbf{R}}^2} \frac{d\tilde{\alpha}_{n,k}\big(\cdot;(x_1,y_1),\dots,(x_n,y_n)\big)}{d\alpha}(x,y)\, \frac{d\tilde{\alpha}_n}{d\alpha^n}((x_1,y_1),\dots,(x_n,y_n))\, d\alpha(x_k,y_k)$$ equals $$\frac{d\tilde{\alpha}_n}{d\alpha^n}\big((x_1,y_1),\dots,(x_{k-1},y_{k-1}),(x,y),(x_{k+1},y_{k+1}),\dots,(x_n,y_n)\big)\, .$$ This implies condition 3 in Definition \[def:rcpd\]. Conditions 1 and 2 are true because of the joint measurability of the density, which just depends on the Hamiltonian. Note that for any pair of points $(x,y)$, $(x',y')$, we have $$\label{ineq:variation} \left|H_{n,k}\big((x,y);(x_1,y_1),\dots,(x_n,y_n)\big) - H_{n,k}\big((x',y');(x_1,y_1),\dots,(x_n,y_n)\big)\right|\, \leq\, 1\, ,$$ because $|h(x-x_j,y-y_j) - h(x'-x_j,y'-y_j)|$ is either $0$ or $1$ for each $j$, and $H_{n,k}$ is a sum of $n-1$ such terms, then divided by $n-1$. We may write $$\left(\frac{d\tilde{\alpha}_{n,k}\big(\cdot;(x_1,y_1),\dots,(x_n,y_n)\big)}{d\alpha}(x,y)\right)^{-1}\, =\, Z_{n,k}\big(\alpha,\beta;(x_1,y_1),\dots,(x_n,y_n)\big)e^{\beta H_{n,k}((x,y);(x_1,y_1),\dots,(x_n,y_n))}$$ as an integral $$\int_{{\mathbf{R}}^2} e^{\beta \left[H_{n,k}\big((x,y);(x_1,y_1),\dots,(x_n,y_n)\big) - H_{n,k}\big((x',y');(x_1,y_1),\dots,(x_n,y_n)\big)\right]}\, d\alpha(x',y')\, .$$ Therefore, the inequality (\[ineq:variation\]) implies that $$e^{-|\beta|}\, \leq\, \left(\frac{d\tilde{\alpha}_{n,k}\big(\cdot;(x_1,y_1),\dots,(x_n,y_n)\big)}{d\alpha}(x,y)\right)^{-1}\, \leq\, e^{|\beta|}\, .$$ Of course, this implies the same bounds for the reciprocal. For all $(x,y) \in {\mathbf{R}}^2$, $$e^{-|\beta|}\, \leq\, \frac{d\tilde{\alpha}_{n,k}\big(\cdot;(x_1,y_1),\dots,(x_n,y_n)\big)}{d\alpha}(x,y)\, \leq\, e^{|\beta|}\, .$$ So, taking $p = e^{-|\beta|}$, this means that the hypotheses of Lemma \[lem:CouplingGeneral\] are satisfied: $\tilde{\alpha}_n$ has the “$p$-resampling” property relative to the measure $\alpha$. Hence, we conclude that Lemma \[lem:CouplingAboveBelow\] is true. Acknowledgements {#acknowledgements .unnumbered} ================ We are very grateful to Janko Gravner for helpful suggestions, including directing us to reference [@DeuschelZeitouni]. S.S is also grateful for advice from Bruno Nachtergaele, and he is grateful for the warm hospitality of the Erwin Schrödinger Institute where part of this research occurred. [10]{} D. J. Aldous. Exchangeability and related topics. In P.L. Hennequin (ed.), [*École d’été de probabilités de Saint-Flour, XII–1983, Lecture Notes in Math. v. 1117*]{}. 1985 Springer, Berlin, pp. 1–198. <http://www.stat.berkeley.edu/~aldous/Papers/me22.pdf>. David Aldous and Persi Diaconis. Hammersley’s Interacting Particle Process and Longest Increasing Subsequences. , [199–213]{} (1995). Jinho Baik, Percy Deift and Kurt Johansson. On the distribution of the length of the longest increasing subsequence of random permutations , [1119–1178]{} (1999). Alexei Borodin, Persi Diaconis and Jason Fulman. On adding a list of numbers (and other one-dependent determinantal processes). , 639-670 (2010). Eric Cator and Pietr Groeneboom. Second class particles and cube root asymptotics for Hammersley’s process. , 879–905 (2005). Jean-Dominique Deuschel and Ofer Zeitouni. Limiting Curves for I.I.D. Records. , 852–878 (1995). Jean-Dominique Deuschel and Ofer Zeitouni. On increasing subsequences of I.I.D. samples. , no. 3, 247–263 (1999). Persi Diaconis and Arun Ram. Analysis of Systematic Scan Metropolis Algorithms Using Iwahori-Hecke Algebra Techniques. , [157–190]{} (2000). Alexander Gnedin and Grigori Olshanski. A q-analogue of de Finetti’s Theorem. , 1, R78, (2009), <http://www.combinatorics.org/Volume_16/PDF/v16i1r78.pdf>. Alexander Gnedin and Grigori Olshanski. q-Exchangeability via quasi-invariance. , no. 6, 2103–2135 (2010). <http://projecteuclid.org/euclid.aop/1285334202>. John Michael Hammersley. A few seedlings of research. In [*Proc. Sixth Berkeley Symp. Math. Statist. Probab.*]{} [**v. 1**]{}, pp. 345–394. Univ. California Press, Berkeley, 1972. Christophe Hohlweg. Minimal and maximal elements in Kazhdan-Lusztig two-sided cells of $S_n$ and Robinson-Schensted correspondance. , no. 1, 79–87 (2005). S. V. Kerov. AMS, Providence, RI, 2003. David A. Levin, Yuval Peres and Elizabeth L. Wilmer. American Mathematical Society, Providence, RI, 2009. Benjamin F. Logan and Lawrence A. Shepp. A variational problem for random Young tableaux. , 206–222 (1977). C. L. Mallows. Non-null ranking models. I. , 114–130 (1957). <http://www.jstor.org/stable/2333244>. Derek W. Robinson and David Ruelle. Mean entropy of states in classical statistical mechanics. , 288–300 (1967). Timo Sepplinen. A Microscopic Model for the Burgers Equation and Longest Increasing Subsequences. , 1–51 (1994). Shannon Starr. Thermodynamic limit for the Mallows model on $S_n$. , 095208 (2009). C. J. Thompson. 1972 The Macmillan Company, New York. Craig Tracy and Harold Widom. Level-spacing distributions and the Airy kernel. , 151–174 (1994). Anatoly M. Vershik and Sergei V. Kerov. Asymptotics of the Plancherel measure of the symmetric group and the limiting form of Young tableaux. , 527–531 (1977). (Translation of [*Dokl. Acad. Nauk. SSR*]{} [**32**]{}, 1024–1027.)

--- abstract: 'We investigate a possibility of scale invariant but non-conformal supersymmetric field theories from a perturbative approach. The explicit existence of monotonically decreasing $a$-function that generates beta-functions as a gradient flow provides a strong obstruction for such a possibility at two-loop order. We comment on the “discovery" of scale invariant but non-conformal renormalization group trajectories via a “change of scheme" in $(4-\epsilon)$ dimension proposed in literatures.' author: - Yu Nakayama title: 'Comments on scale invariant but non-conformal supersymmetric field theories' --- Introduction ============ It is a long-standing problem whether scale invariant unitary relativistic quantum field theories in four dimension actually show conformal invariance. In two dimension, the equivalence was proved [@Zamolodchikov:1986gt][@Polchinski:1987dy][@Mack1], and in higher dimension with $d>4$, there is at least one explicit counterexample [@Jackiw:2011vz][@ElShowk:2011gz]. A search for counterexamples in $d = 4-\epsilon$ has been discussed [@Dorigoni:2009ra][@Fortin:2011ks][@Fortin:2011sz]. We also have a gravitational argument supporting the equivalence [@Nakayama:2009qu][@Nakayama:2009fe][@Nakayama:2010wx]. In this paper, we study a possibility to construct scale invariant but non-conformal field theories with a help of supersymmetry in four as well as $(4-\epsilon)$ dimension. The general structure was investigated in [@Antoniadis:2011gn], but without the detail of the beta-functions, it has been unknown whether the scale invariant supersymmetric field theories are superconformal or not. We show that the existence of monotonically decreasing $a$-function in supersymmetric field theories at two-loop order hinders scale invariant but non-conformal renormalization group trajectories within a perturbation theory. Thus, at two-loop order, we conclude that supersymmetric scale invariant field theories are all superconformal. We also comment on the “discovery" of scale invariant trajectories via a “change of scheme" in $(4-\epsilon)$ dimension proposed by [@Fortin:2011ks][@Fortin:2011sz]. We argue that the existence of the scale invariant but non-conformal trajectories as well as the monotonically decreasing $a$-function indeed depend on their “change of scheme" but such a change is not natural at least in supersymmetric field theories. This subtlety is intrinsic to the ambiguity in $\epsilon$ expansion and should be absent in strict four-dimensional theories. Formalism ========= In scale invariant relativistic quantum field theories in arbitrary dimension greater than two, trace of an energy-momentum tensor $T_{\mu\nu}$ is given by the divergence of a Virial current $V_\mu$ [@Coleman:1970je]: $$\begin{aligned} T^{\mu}_{\ \mu} = \partial^\mu V_\mu \ .\end{aligned}$$ The theory is conformal invariant if and only if we can improve the energy-momentum tensor so that it is traceless. In terms of the Virial current, if and only if the Virial current is a total derivative $$\begin{aligned} V_\mu = \partial^\nu L_{\nu \mu} + J_\mu\end{aligned}$$ up to a conserved current $J_\mu$ (i.e. $\partial^\mu J_\mu = 0$), we can improve the energy-momentum tensor so that the theory is conformal [@Polchinski:1987dy]. In this paper, we consider the supersymmetric field theories in $(4-\epsilon)$ dimension with R-symmetry that allow perturbative scale invariance. Such theories consist of vector multiplets whose field strength supermultiplets are $W_i$ with arbitrary gauge group and their coupling constants $g^2_{ij}$, and matter chiral multiplets $\Phi_a$ with arbitrary representation (modulo anomaly cancellation) and Yukawa-coupling constants $Y_{abc}$. The theory is completely specified by the superpotential $$\begin{aligned} W = \frac{1}{g_{ij}^2} W_i W_j + Y_{abc} \Phi^a \Phi^b \Phi^c \ .\end{aligned}$$ Note that Yukawa-coupling constants $Y_{abc}$ can be chosen to be completely symmetric in their indices. One may introduce $\theta$-parameters, but they will not affect the following perturbative analysis. It was shown [@Antoniadis:2011gn] that the structure of the Virial current supermultiplet $\mathcal{V}_\mu$, whose lowest component is $V_\mu$, is highly constrained from the unitarity in supersymmetric field theories with R-symmetry: $$\begin{aligned} \mathcal{V}_\mu = \partial_\mu U + \sigma_{\mu}^{\dot{\alpha} \beta} [\bar{D}_{\dot{\alpha}}, D_\beta] \mathcal{O} \ .\end{aligned}$$ Here $U$ and $\mathcal{O}$ are dimension-two singlet real scalar operators. The first term is irrelevant for our study because we can always improve the energy momentum tensor to eliminate it. To all orders in perturbation theory, $\mathcal{O}$ must take the following form: $$\begin{aligned} \mathcal{O} = Q_{b}^{\ a} \bar{\Phi}_a \Phi^b \ , \label{virial}\end{aligned}$$ where $Q_{b}^{\ a}$ is an Hermitian matrix. Actually, the same conclusion follows even if we did not assume the R-symmetry [^1]. The corresponding Virial current is $$\begin{aligned} V_\mu = iQ_{b}^{\ a} (\partial_\mu \bar{\phi}_a \phi^b - \bar{\phi}_a \partial_\mu \phi^b + \bar{\psi}_a \gamma_\mu \psi^b) . \end{aligned}$$ The application of the equation of motion gives the divergence of the Virial current $$\begin{aligned} &\partial^\mu V_\mu \cr &= i (Q_{a}^{\ d} Y_{d bc} + Q_{b}^{\ d}Y_{adc} + Q_{c}^{\ d} Y_{abd}) (\phi^{a} \psi^{b} \psi^c) + c.c. \cr &+ Q|Y|^2\phi^4 .\end{aligned}$$ The supersymmetry relates the Yukawa-interaction with the $\phi^4$ interaction schematically represented by $Q|Y|^2 \phi^4$, and the detail of the latter is not important in our study. Here it is important to note that we have implicitly assumed the existence of the renormalization scheme where the equation of motion holds in the perturbation theory. On the other hand, the trace of the energy-momentum tensor is given by the beta-functions of the coupling constants $g^2_{ij}$ and $Y^{abc}$. $$\begin{aligned} T^{\mu}_{\ \mu} &= \beta^{g^{2}_{ij}} F_{\mu \nu}^i F^{j \mu\nu} \cr &+\beta^{Y_{abc}} (\phi^{a } \psi^{b} \psi^c) + c.c. \cr &+ \cdots \ ,\end{aligned}$$ where $\cdots$ denotes the other terms that complete the supersymmetry such as $\phi^4$ interactions or those that involve gauginos. Due to the supersymmetry, these omitted terms will not add any new information in the following discussion. Scale invariance means that the trace of the energy-momentum tensor must be expressed as a divergence of the Virial current. First of all, when $\epsilon = 0$, it demands that the beta-functions for the gauge coupling constant must vanish $$\begin{aligned} \beta^{{g}^{2}_{ij}} = 0 \ . \label{beta1}\end{aligned}$$ We assume that these equations are solved. When $\epsilon = 0$, the additional condition from the Yukawa-coupling constants is $$\begin{aligned} &i (Q_{a}^{\ d} Y_{d bc} + Q_{b}^{\ d}Y_{adc} + Q_{c}^{\ d} Y_{abd}) \cr &= (\Gamma_{a}^{\ d} Y_{d bc} + \Gamma_{b}^{\ d}Y_{adc} + \Gamma_{c}^{\ d} Y_{abd}) \ . \label{beta2}\end{aligned}$$ Here, we have used the fact that the anomalous dimension $\Gamma_{a}^{\ b}$ will determine the beta-functions of the Yukawa-coupling constants in supersymmetric theories. A priori, it is not obvious whether all the solutions of and demand $QY \equiv (Q_{a}^{\ d} Y_{d bc} + Q_{b}^{\ d}Y_{adc} + Q_{c}^{\ d} Y_{abd}) = 0 $. In $\mathcal{N}=2$ supersymmetric case, means that the anomalous dimension matrix $\Gamma_{a}^{\ b}$ automatically vanish, so obviously they are superconformal. In generic $\mathcal{N}=1$ supersymmetric case, we need more detailed information of the beta-functions. One of the advantages of the supersymmetry, however, is that the stability of the potential is automatically guaranteed unlike the non-supersymmetric case studied in [@Fortin:2011ks][@Fortin:2011sz]. At one-loop order, $\Gamma_{a}^{\ d} = \frac{1}{2}Y_{abc} \bar{Y}^{dbc} + 3g^2 (t^2)^{\ d}_{a}$ [^2], so by contracting $QY$ on the both side of , and by observing that the left-hand side is a pure imaginary number while the right-hand side is a real number, we conclude $Q Y$ must vanish at this order (see [@Dorigoni:2009ra][@Fortin:2011ks] for a similar argument in non-supersymmetric case). At two-loop or higher, we have not been aware of a simple proof of vanishing of $QY$. From phenomenological studies with a small number of matters, however, all the solutions of seem to indicate $QY =0$ via analytical as well as numerical approach. In the next section, we will give an indirect evidence of the non-existence of scale invariant but non-conformal trajectories in these perturbative supersymmetric theories based on the explicit existence of the monotonically decreasing $a$-function that generates beta-functions as a gradient flow. Perturbative existence of $a$-function ====================================== In this section, we embody the idea to utilize the existence of monotonically decreasing $a$-function [@Freedman:1998rd] to show the non-existence of scale invariant but non-conformal field theories at two-loop order. Let us consider the most generic Wess-Zumino model in $(4-\epsilon)$ dimension, which is completely specified by the Yukawa-coupling constants $Y_{abc}$. We can directly show that the two-loop beta-function [@Jack:1996qq] $$\begin{aligned} \beta_{Y_{abc}} = -\epsilon Y_{abc} + (\Gamma_{a}^{\ d} Y_{d bc} + \Gamma_{b}^{\ d}Y_{adc} + \Gamma_{c}^{\ d} Y_{abd}) \label{betatwo} \end{aligned}$$ with the anomalous dimension matrix $$\begin{aligned} \Gamma_{a}^{\ b} = P_{a}^{\ b} -\bar{Y}^{b cd} Y_{a ce} P_d^{\ e} \ ,\end{aligned}$$ where $P_{a}^{\ b} = \frac{1}{2} \bar{Y}^{bcd} Y_{acd}$, can be generated by an “$a$-function" $$\begin{aligned} a &= a_0 - \frac{\epsilon}{3} \mathrm{tr}(\bar{Y}Y) \cr &+ (\frac{1}{4} + \frac{\epsilon}{8}) \mathrm{tr}((\bar{Y}Y)(\bar{Y}Y))- \bar{Y}^{acd}Y_{bce}P^{\ e}_{d}P^{\ b}_{a} \cr &- \frac{1}{24}\mathrm{tr}((\bar{Y}Y)(\bar{Y}Y)(\bar{Y}Y)) \ , \label{a}\end{aligned}$$ where $a_0$ is a constant and $(\bar{Y} Y)^{\ a}_{b} = \bar{Y}^{acd}Y_{bcd}$, with the “metric" of the coupling constant space (symbolically denoted by $g^I$) $$\begin{aligned} ds^2 &= G_{IJ} dg^{I}dg^{J} \cr &= \frac{2}{3} d\bar{Y}^{abc} dY_{abc} -\frac{1}{2} \mathrm{tr}((d\bar{Y} dY)(\bar{Y} Y) ) \ . \end{aligned}$$ as a gradient flow $$\begin{aligned} \partial_{Y_{abc}} a = G_{Y_{abc} \bar{Y}^{efg}}\beta^{\bar{Y}^{efg}} \ . \end{aligned}$$ We have added $\epsilon$ dependent terms to the result found in the literature [@Freedman:1998rd] in order to reproduce the classical part in the beta-function . Up to two-loop accuracy, we still have some freedom to modify the $a$-function as well as the metric [@Freedman:1998rd], but it is not relevant for our study of two-loop search of the scale invariant trajectories. We also note that our $a$ may be different from the actual “central charge" away from the conformal invariant fixed point. This gradient flow immediately implies that the $a$-function is monotonically decreasing $$\begin{aligned} \frac{da}{d\log \mu} = - \beta^{Y_{abc}} G_{Y_{abc} \bar{Y}^{efg}} \beta^{\bar{Y}^{efg}}\end{aligned}$$ for sufficiently small coupling constants $Y_{abc} \sim \sqrt{\epsilon} \ll 0$ near all one-loop conformal fixed points because the “metric" is positive definite when $\epsilon \ll 1$. We now argue that the existence of scale invariant but non-conformal trajectories at two-loop is inconsistent with the monotonically decreasing $a$-function explicitly constructed here. Suppose that there exists such a trajectory, then along the trajectory, $a$ continues to decrease: $$\begin{aligned} \frac{da}{d\log \mu} =- (\bar{Y} Q) G (Q Y) < 0 \ .\end{aligned}$$ At this point, we recall that the two-loop scale invariant trajectory should approximately lie on the one-loop conformal fixed points. The one-loop conformal fixed points form a trajectory generated by the action of $Q$ on a given reference one-loop conformal fixed point $Y_0$ (which is nothing but the field redefinition to generate new fixed points), and the two-loop scale invariant trajectory, if any, approximately moves along this trajectory generated by the same $Q$ with the speed of order $\epsilon^2$ for its one-loop consistency [^3]. In particular, after a sufficiently large time (at least of order $\epsilon^{-2}$), the two-loop scale invariant trajectory generated by non-zero beta-functions will return to the original point $Y_0$ (when the trajectory closes), or it comes back arbitrary close to the original point (when the trajectory is chaotic) [@Fortin:2011ks][@Fortin:2011sz], while $a$ decreases forever. However, we know that $a$ is a globally defined smooth function on the coupling constant space, which is explicitly given by , so it is impossible for $a$ to decrease perpetually along the closed or chaotic but bounded trajectory. Thus, the existence of the explicit $a$-function at two-loop order means that we cannot find scale invariant but non-conformal trajectories in $(4-\epsilon)$ dimensional Wess-Zumino model with the same accuracy of the loop approximation. Indeed analytic as well as numerical study of the model with small numbers of fields verify this statement. Note that our argument is quite generic so it applies not only to the Wess-Zumino models in $(4-\epsilon)$ dimension, but also to supersymmetric gauge theory with arbitrary matters and Yukawa-coupling constants in strict four-dimensional limit thanks to the existence of the explicit $a$-function [@Freedman:1998rd] that generates beta-functions as a gradient flow. Thus within a perturbatively accessible regime, where the $\epsilon$ parameter is replaced by a small control parameter such as $\frac{3N_c-N_f}{N_c}$ in SQCD, it is impossible to find scale invariant but non-conformal trajectories in supersymmetric gauge theories at two-loop order. The existence of the gradient flow is argued positively at higher-loop orders [@Freedman:1998rd][@Jack:1996qq][@Wallace:1974dy], and if this is true, we will never find a scale invariant but non-conformal trajectory in these models. Comment on [@Fortin:2011ks][@Fortin:2011sz] =========================================== Most of the discussion in this paper is actually applicable to non-supersymmetric field theories as well when we accept the existence of the monotonically decreasing “$a$-function" that generates the renormalization group as a gradient flow. The existence was claimed to all orders in perturbation theory in [@Wallace:1974dy][@Jack:1990eb][@Freedman:1998rd]. In particular, the scale invariant but non-conformal field theories cannot exist due to the same reason presented in the last section. How is this consistent with the “discovery" of the scale invariant but non-conformal renormalization group trajectory in non-supersymmetric field theories in ($4-\epsilon$) dimension proposed in [@Fortin:2011ks][@Fortin:2011sz] at the “two-loop" order? The authors are aware of the obstruction [@Wallace:1974dy][@Jack:1990eb] and claim that the gradient flow does not exist in their examples. Let us elaborate on this point. First, we emphasize that in the dimensional regularization scheme what they found actually is that all the theories under investigation are conformal invariant up to two loops where their approximation can be trusted. At this point, it must be consistent with the existence of $a$-function at two-loop order. Then, they proposed a most generic “change of scheme" to show the existence of scale invariant but non-conformal trajectories at two-loop order. We would like to clarify the nature of their “change of scheme" here: in particular, it is curious how the “change of scheme" can affect the physical conclusion whether the theory is conformal or only scale invariant. Indeed, if the “change of scheme" used in [@Fortin:2011ks][@Fortin:2011sz] were a mere reparametrization in the space of coupling constants, then the conclusion should not change. For instance, the condition of the conformal invariance $$\begin{aligned} \beta^I = \frac{d g^{I}}{d\log\mu} = 0 \end{aligned}$$ is invariant under the small reparametrization $ g^I \to g^I + A^{I}_{JK} g^J g^K$. So is the existence of the $a$-function: $$\begin{aligned} \partial_I a = g_{IJ} \beta^{J} \ , \ \ \frac{da}{d\log\mu} = -g_{IJ} \beta^I \beta^J \ .\end{aligned}$$ These equations are manifestly covariant under the reparametrization, and the physical conclusion should not depend on the change of scheme. The resolution of the apparent puzzle is that the “change of scheme" used in [@Fortin:2011ks][@Fortin:2011sz] in ($4-\epsilon$) dimension is [*not*]{} a simple reparametrization. Suppose that in one particular dimensional regularization scheme, we have obtained the beta-function $$\begin{aligned} \beta^I = -\epsilon g^I + \beta^{(1) I}_{KL} g^{K}g^{L} + \beta^{(2) I}_{KLM} g^{K} g^{L} g^{M} + \cdots \ . \label{genbeta}\end{aligned}$$ A small reparametrization $g^I \to g^I + A^{I}_{JK} g^{J}g^{K}$ will generically introduce not only the “two-loop" shift mentioned in [@Fortin:2011ks][@Fortin:2011sz] but also the “one-loop" term that explicitly depends on $\epsilon$: $$\begin{aligned} \delta \beta^{I} = +\epsilon A^{I}_{JK} g^{J} g^{K} \ . \label{epone}\end{aligned}$$ This term is crucial to maintain the covariance of the conformal invariant condition as well as the existence of the perturbative $a$-function in $(4-\epsilon)$ dimension under reparametrization. In [@Fortin:2011ks][@Fortin:2011sz], the explicit $\epsilon$ dependent term beyond the classical term such as is dropped by hand [*after*]{} the reparametrization, and this affects the structure of the conformal invariance at $O(\epsilon^2)$, which enabled them to “discover" a scale invariant but non-conformal trajectory via the “change of scheme". They could deliver the validity of the procedure by arguing that they could have done the change of scheme in the strict four-dimensional limit (i.e. $\epsilon = 0$), and then added the classical piece of the beta-function so that the $\epsilon$ dependent one-loop term such as would never appear. In other words, since the change of scheme and the introduction of non-zero $\epsilon$ do not commute, we should have known the regularization scheme where there is no other $\epsilon$ dependence in from the beginning to argue the existence of the $a$-function. Unless we give an independent argument which regularization is natural, we cannot conclude whether the lack of the integrability in beta-functions in $(4-\epsilon)$ dimension after the change of scheme is physical or not. In our supersymmetric case, on the other hand, there exists a manifestly supersymmetric regularization scheme where the holomorphic structure is maintained, and the random change of scheme would not be allowed. Therefore, the existence of the perturbative $a$-function in $(4-\epsilon)$ dimension seems rather robust. For instance, we could try their “change of scheme" in the general Wess-Zumino model studied in the last section. If we dropped the $\epsilon$ dependent one-loop term by hand subsequently after the small reparametrization of the Yukawa-coupling constants $Y \to A|Y|^2Y$, then we would observe that the existence of the monotonically decreasing $a$-function is generically lost, and we would be able to find a scale invariant but non-conformal trajectories that emerge as early as at two-loop order. An explicit example is in order. Consider the Wess-Zumino model with four chiral fields $\Phi_i$ $i=1,2,3,4$ with the superpotential $$\begin{aligned} W = (x_1 \Phi_1 + x_2 \Phi_2) \Phi_3 \Phi_3 + (y_1 \Phi_1 + y_2\Phi_2) \Phi_4\Phi_4 \ \end{aligned}$$ in $(4-\epsilon)$ dimension. We can easily show that to all orders in perturbation theory, fixed points are, if any, all conformal (e.g. $x_1 = x_2 \sim \sqrt{\epsilon}$, $y_1 = -y_2 \sim \sqrt{\epsilon}$) and the non-trivial Virial current vanishes. If we performed a “change of scheme" induced by the reparametrization $ x_i \to x_i + k (\bar{y}_{j}y_j) y_i$ and crucially discard the $\epsilon$ dependent one-loop part of the beta-functions by hand, then the scale invariant but non-conformal trajectories emerge, where the parameter $Q$ for the Virial current $i(\bar{\Phi}_{1} \Phi^2 - \bar{\Phi}_{2} \Phi^1)$ is proportional to $k$. This change of scheme is very artificial, however, because it does not preserve the holomorphic structure as well as the spurious charge assignment for $x_i$ and $y_i$. This example clearly illustrates the possibility that the “change of scheme" employed in [@Fortin:2011ks][@Fortin:2011sz] could transform even all-order conformal fixed points into a scale invariant but non-conformal trajectory at two-loop [^4]. Conclusion ========== We have investigated a possibility of scale invariant but non-conformal supersymmetric field theories from a perturbative approach. The explicit existence of monotonically decreasing $a$-function provides a strong obstruction for such a possibility at two-loop order. We have argued that the “change of scheme" employed in [@Fortin:2011ks][@Fortin:2011sz] is not covariant under reparametrization in $(4-\epsilon)$ dimension, and this is the reason why they circumvented the obstruction coming from the existence of $a$-function in a conventional dimensional regularization scheme. The rather artificial lack of covariance under the reparametrization discussed in the last section due to a subtlety or ambiguity in $\epsilon$ expansion disappears in the strict four-dimensional limit, and the existence of the $a$-function that generates beta-functions as a gradient flow should be reparametrization invariant or scheme independent. Thus, in the strict four-dimensional limit, we conclude that there indeed exists an obstruction to find a scale invariant but non-conformal field theories within a perturbative regime if the monotonically decreasing $a$-function is constructed in line with the claim in [@Jack:1990eb]. It would be interesting to see whether the non-perturbative reformulation of the $a$-function (see e.g. [@Barnes:2004jj][@Komargodski:2011vj] for recent discussions) would lead to a more comprehensive understanding of the relation between scale invariance and conformal invariance in four dimension. At the same time, it may be possible (beyond the perturbation theory) that the gradient flow may not exist and the scale invariant but non-conformal field theory is possible in four dimension unlike in two-dimension where the existence was proved. The most natural value of the anomalous dimension matrix that will lead to a conformal fixed point is given by the “$a$-maximization" procedure [@Intriligator:2003jj]. However, a priori, there is no guarantee that given a particular value of the anomalous dimension matrix via the $a$-maximization, we can solve the corresponding coupling constants from the actual beta-functions. One known possibility is that the corresponding value breaks the unitarity and the contribution from the accidental symmetry must be taken into account. The other logical possibility here, is that you may have to augment the Virial current contribution to solve them. Then, the (almost) derivation of the $a$-theorem would not hold. To understand the gradient flow of the $a$-function beyond the perturbation theory, the $a$-maximization with Lagrange multipliers (see e.g. [@Kutasov:2003ux][@Erkal:2010sh]) seemed promising, but we have to overcome this inverse problem in any way. Finally, it would be very interesting to see if the relation between the existence of the gradient flow and the scale invariant but non-conformal trajectories are clarified from the holographic viewpoint. In particular, we would like to shed more light on the role of the null energy condition in both sides (see e.g. [@Nakayama:2010wx][@Nakayama:2011zw]). This direction is under investigation. Acknowledgments {#acknowledgments .unnumbered} =============== We thank M. Buican, S. Rychkov and Y. Tachikawa for stimulating discussions. We are greatly indebted to J. -F. Fortin for clarification of the arguments in their papers. There is no need to say, of course, that all the possible misunderstandings of their claim is on us. The work is supported by the World Premier International Research Center Initiative of MEXT of Japan. [99]{} A. B. Zamolodchikov, JETP Lett.  [**43**]{} (1986) 730 \[Pisma Zh. Eksp. Teor. Fiz.  [**43**]{} (1986) 565\]. J. Polchinski, Nucl. Phys.  B [**303**]{}, 226 (1988). M. Lüscher and G. Mack, 1976, unpublished; G. Mack, “NONPERTURBATIVE QUANTUM FIELD THEORY. PROCEEDINGS, NATO ADVANCED STUDY INSTITUTE, CARGESE, FRANCE, JULY 16-30, 1987”. R. Jackiw, S. -Y. Pi, J. Phys. A [**A44**]{}, 223001 (2011). \[arXiv:1101.4886 \[math-ph\]\]. S. El-Showk, Y. Nakayama, S. Rychkov, Nucl. Phys.  [**B848**]{}, 578-593 (2011). \[arXiv:1101.5385 \[hep-th\]\]. D. Dorigoni, V. S. Rychkov, \[arXiv:0910.1087 \[hep-th\]\]. J. -F. Fortin, B. Grinstein, A. Stergiou, Phys. Lett.  [**B704**]{}, 74-80 (2011). \[arXiv:1106.2540 \[hep-th\]\]. J. -F. Fortin, B. Grinstein, A. Stergiou, \[arXiv:1107.3840 \[hep-th\]\]. Y. Nakayama, arXiv:0907.0227 \[hep-th\]. Y. Nakayama, JHEP [**1001**]{}, 030 (2010) \[arXiv:0909.4297 \[hep-th\]\]. Y. Nakayama, \[arXiv:1009.0491 \[hep-th\]\]. I. Antoniadis, M. Buican, Phys. Rev.  [**D83** ]{} (2011) 105011. \[arXiv:1102.2294 \[hep-th\]\]. S. R. Coleman and R. Jackiw, Annals Phys.  [**67**]{} (1971) 552. S. Zheng, Y. Yu, \[arXiv:1103.3948 \[hep-th\]\]. D. Z. Freedman, H. Osborn, Phys. Lett.  [**B432**]{}, 353-360 (1998). \[hep-th/9804101\]. I. Jack, D. R. T. Jones, C. G. North, Nucl. Phys.  [**B473**]{}, 308-322 (1996). \[hep-ph/9603386\]. D. J. Wallace, R. K. P. Zia, Annals Phys.  [**92** ]{} (1975) 142. I. Jack, H. Osborn, Nucl. Phys.  [**B343**]{}, 647-688 (1990). E. Barnes, K. A. Intriligator, B. Wecht, J. Wright, Nucl. Phys.  [**B702**]{}, 131-162 (2004). \[hep-th/0408156\]. Z. Komargodski, A. Schwimmer, \[arXiv:1107.3987 \[hep-th\]\]. K. A. Intriligator, B. Wecht, Nucl. Phys.  [**B667**]{}, 183-200 (2003). \[hep-th/0304128\]. D. Kutasov, \[hep-th/0312098\]. D. Erkal, D. Kutasov, \[arXiv:1007.2176 \[hep-th\]\]. Y. Nakayama, \[arXiv:1107.2928 \[hep-th\]\]. [^1]: The discussion in [@Zheng:2011bp] is incomplete, and the relation between the dilatation multiplet and the Virial current multiplet in non R-symmetric case must be modified with the additional $x^{\mu}(D^2 {X}+\bar{D}^2\bar{X})$ term in their formula. After the suitable modification, one can show that the Virial current multiplet is given by $\mathcal{V}_\mu = \sigma^{\alpha \dot{\alpha}}_\mu(\bar{D}_{\dot{\alpha}} \Gamma_{\alpha} - D_{\alpha} \bar{\Gamma}_{\dot{\alpha}})$. Within a perturbation theory, is anyway the only possibility with $\Gamma_{\alpha} = D_{\alpha}(\bar{\Phi} \Phi)$. Besides, the one-loop fixed point is superconformal, so the theory should possess R-symmetry. [^2]: In this paper, we use the convention that our coupling constants are divided by $(4\pi)$ compared with the standard textbook convention. [^3]: The one-loop fixed point would never nontrivially “split" into more than two two-loop trajectories with $Q=0$ and $Q\neq 0$. If this could happen, we would construct interpolating solutions between $Q=0$ and non-zero $Q$ at a fixed order of perturbative expansion by using the linearity of the perturbation equation. Since we could construct arbitrary large $Q$ solution in this way, it would be reasonable only when the Virial current corresponding to $Q$ is conserved. [^4]: For instance, in [@Fortin:2011ks], it was argued that when the Yukawa-coupling constants can be reduced to a matrix, the Virial current must be trivial to all orders in perturbation theory due to a particular structure of the (unrenormalized) Feynman diagrams. However, it is easy to break this structure, albeit artificial, by the “counter-terms" that are implied by the most generic “change of scheme" discussed here.

--- abstract: 'The [*Herschel Space Observatory*]{} has revealed a very different galaxyscape from that shown by optical surveys which presents a challenge for galaxy-evolution models. The [*Herschel*]{} surveys reveal (1) that there was rapid galaxy evolution in the very recent past and (2) that galaxies lie on a a single Galaxy Sequence (GS) rather than a star-forming ‘main sequence’ and a separate region of ‘passive’ or ‘red-and-dead’ galaxies. The form of the GS is now clearer because far-infrared surveys such as the Herschel ATLAS pick up a population of optically-red star-forming galaxies that would have been classified as passive using most optical criteria. The space-density of this population is at least as high as the traditional star-forming population. By stacking spectra of H-ATLAS galaxies over the redshift range $0.001 < z < 0.4$, we show that the galaxies responsible for the rapid low-redshift evolution have high stellar masses, high star-formation rates but, even several billion years in the past, old stellar populations—they are thus likely to be relatively recent ancestors of early-type galaxies in the Universe today. The form of the GS is inconsistent with rapid quenching models and neither the analytic bathtub model nor the hydrodynamical EAGLE simulation can reproduce the rapid cosmic evolution. We propose a new gentler model of galaxy evolution that can explain the new [*Herschel*]{} results and other key properties of the galaxy population.' author: - | Stephen Eales$^{1}$[^1], Dan Smith$^2$, Nathan Bourne$^3$, Jon Loveday$^4$, Kate Rowlands$^5$, Paul van der Werf$^6$, Simon Driver$^7$, Loretta Dunne$^{1,3}$, Simon Dye$^8$, Cristina Furlanetto$^{8}$, R.J. Ivison$^{9,3}$, Steve Maddox$^{1,3}$, Aaron Robotham$^7$, Matthew W.L. Smith$^1$, Edward N. Taylor$^{10}$, Elisabetta Valiante$^1$, Angus Wright$^{7,11}$, Philip Cigan$^1$, Gianfranco De Zotti$^{12,13}$, Matt J. Jarvis$^{14,15}$, Lucia Marchetti$^{16}$, Micha[ł]{} J. Micha[ł]{}owski$^{17}$, Steven Phillipps$^{18}$, Sebastien Viaene$^{19}$and Catherine Vlahakis$^{20}$\ $^{1}$School of Physics and Astronomy, Cardiff University, The Parade, Cardiff CF24 3AA, UK\ $^2$Centre for Astrophysics Research, School of Physics, Astronomy and Mathematics, University of Hertfordshire,\ College Lane, Hatfield, AL10 9AB, UK\ $^3$Institute for Astronomy, The University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ, UK\ $^4$Astronomy Centre, University of Sussex, Falmer, Brighton BN1 9QH, UK\ $^5$Department of Physics and Astronomy, JHU, Bloomberg Center, 3400 N. Charles St., Baltimore, MD 21218, USA\ $^6$ Leiden Observatory, PO Box 9513, 2300 RA Leiden, the Netherlands\ $^7$International Centre for Radio Astronomy Research, 7 Fairway, The University of Western\ Australia, Crawley, Perth, WA 6009, Australia\ $^8$School of Physics and Astronomy, University of Nottingham, University Park, Nottingham NG7 2RD, UK\ $^9$European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748, Garching, Germany\ $^{10}$ Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Hawthorn 3122, Australia\ $^{11}$ Argelander-Institut fur Astronomie, Auf dem Hugel 71, D-53121 Bonn, Germany\ $^{12}$ INAF-Osservatorio Astronomico di Padova, Vicolo Osservatorio 5, I-35122 Padova, Italy\ $^{13}$ SISSA, Via Bonomea 265, I-34136 Trieste, Italy\ $^{14}$ Astrophysics, Department of Physics, Keble Road, Oxford, OX1 3RH, UK\ $^{15}$ Physics and Astronomy Department, University of the Western Cape, Bellville 7535, South Africa\ $^{16}$ Department of Physical Sciences, The Open University, Milton Keynes, MK7 6AA, UK\ $^{17}$ Astronomical Observatory Institute, Faculty of Physics, Adam Mickiewicz University, ul. S[ł]{}oneczna 36, 60-286 Pozna[ń]{}, Poland\ $^{18}$ Astrophysics Group, Department of Physics, University of Bristol, Tyndall Avenue, Bristol BS8 1TL\ $^{19}$ Sterrenkundig Observatorium,Universiteit Gent, Krijgslaan 281 S9, B-9000 Gent, Belgium\ $^{20}$ North American ALMA Science Center, National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22901, USA title: The New Galaxy Evolution Paradigm Revealed by the Herschel Surveys --- \[firstpage\] galaxies: evolution Introduction ============ Over the last decade a simple phenomenological model of galaxy evolution has been become widely used by astronomers to interpret observations. In this model, star-forming galaxies lie on the ‘Galaxy Main Sequence’ (henceforth GMS), a distinct region in a plot of star-formation rate versus galaxy stellar mass (e.g Noeske et al. 2007; Daddi et al. 2007; Elbaz et al. 2007; Peng et al. 2010; Rodighiero et al. 2011; Whitaker et al. 2012; Lee et al. 2015). Over cosmic time, the GMS gradually moves downwards in star-formation rate, which decreases by a factor of $\simeq$20 from a redshift of 2 to the current epoch (Daddi et al. 2007). Observations of the molecular gas and dust in galaxies show that the principal cause of this evolution is that high-redshift galaxies contained more gas and therefore formed stars at a faster rate (Tacconi et al. 2010; Dunne et al. 2011; Genzel et al. 2015; Scoville et al. 2016). In this phenomenological paradigm an individual galaxy moves along the GMS until some process quenches the star formation in the galaxy, which then moves rapidly (in cosmic terms) across the diagram to the region occupied by ‘red and dead’ or ‘passive’ galaxies. Possible quenching processes include galaxy merging (Toomre 1977), with a starburst triggered by the merger rapidly using up the available gas; the expulsion of gas by a wind from an active galactic nucleus (Cicone et al. 2014); the rapid motion of star-forming clumps towards the centre of the galaxy (Noguchi 1999; Bournaud et al. 2007; Genzel et al. 2011, 2014), leading to the formation of a stellar bulge, which then stabilizes the star-forming gas disk and reduces the rate at which the gas collapses to form stars (Martig et al. 2009); and a variety of environmental processes which either reduce the rate at which gas is supplied to a galaxy or which drive out most of the existing gas (Boselli and Gavazzi 2006). As can be seen from the long list of possible quenching mechanisms, the physics underlying this paradigm is unknown. Peng et al. (2010) have shown that many statistical properties of star-forming and passive galaxies can be explained by a model in which both the star-formation rate and the probability of quenching are proportional to the galaxy’s stellar mass, but the physics behind both proportionalities is unknown. Although it is clear that the increased star-formation rates in high-redshift galaxies are largely due to their increased gas content, there is also evidence that the star-formation efficiency is increasing with redshift (Rowlands et al. 2014; Santini et al. 2014; Genzel et al. 2015; Scoville et al. 2016); so either the physics of star formation or the properties of the interstellar gas (Papadopoulos and Geach 2012) must also be changing with redshift in some way. Implicit in this paradigm is the assumption that there are two physically-distinct classes of galaxy. These two classes of galaxy are variously called ‘star-forming’ and ‘passive’, ‘star-forming’ and ‘red-and-dead’ or ‘star-forming’ and ‘quenched’. The most visually striking evidence that there are two separate classes is that on plots of optical colour versus optical absolute magnitude, galaxies fall in two distinct areas: a ‘blue cloud’ of star-forming galaxies and a tight ‘red sequence’ representing the passive galaxies (e.g. Bell et al. 2004). However, in an earlier paper (Eales et al. 2017) we argued that the the tight red sequence is the result of optical colour depending only very weakly on specific star-formation rate (star-formation rate divided by galaxy stellar mass, henceforth SSFR) for $\rm SSFR < 5 \times 10^{-12}\ year^{-1}$; the red sequence is therefore better thought of as the accumulated number of galaxies that have passed below this critical SSFR, all of which pile up at the same colour, rather than representing a distinct class of galaxy. The two classes are largely the same as the morphological classes of early-type and late-type galaxies (henceforth ETGs and LTGs). Although there is plenty of evidence for a gradual change in the properties of galaxies along the morphological Hubble sequence (e.g. Kennicutt 1998), there is now little evidence for a clear dichotomy between the two broad morphological classes (Section 5.5 of this paper). The launch in 2009 of the [*Herschel Space Observatory*]{} (Pilbratt et al. 2010) gave a different view of the galaxy population from the one given by optical surveys. Apart from the interest of this new galaxyscape, produced by the different selection effects operating on optical and far-infrared surveys (§2), [*Herschel’s*]{} launch provided two immediate practical benefits for astronomers studying galaxy evolution. The first was that [*Herschel*]{} made it possible for astronomers to directly measure the part of the energy output of stars that is hidden by dust. For example, by using [*Herschel*]{} and other telescopes to measure the bolometric luminosity of different galaxy classes, it is possible to measure the size of the morphological transformation that has occurred in the galaxy population in the last eight billion years (Eales et al. 2015). The second benefit is that with [*Herschel*]{} submillimetre photometry, which now exists for $\sim10^6$ galaxies, it is possible to estimate the mass of a galaxy’s ISM from its dust emission (Eales et al. 2012; Scoville et al. 2014; Groves et al. 2015), a technique that has since been profitably extended to ALMA observations (Scoville et al. 2016). In this paper, we investigate this new galaxyscape. The paper is based on two [*Herschel*]{} surveys: the [*Herschel*]{} Reference Survey (henceforth HRS) and the [*Herschel*]{} Astrophysical Terrahertz Large Area Survey (henceforth H-ATLAS). We describe these surveys in more detail in Section 2 but, briefly, the HRS is a volume-limited sample of 323 galaxies, designed before launch to be a complete as possible census of the stellar mass in the Universe today; each galaxy was then individually observed with [*Herschel*]{} (Boselli et al. 2010; Smith et al. 2012b; Cortese et al. 2012; Ciesla et al. 2012; Eales et al. 2017). The H-ATLAS was the largest (in sky area, 660 square degrees) [*Herschel*]{} extragalactic survey, consisting of imaging at 100, 160, 250, 350 and 500 $\mu$m of five fields, two large fields at the North and South Galactic Poles and three smaller fields on the celestial equator (Eales et al. 2010). There is already one important advance in our knowlege of galaxy evolution provided by the [*Herschel*]{} surveys, although we suspect this has yet been absorbed by the wider astronomical community. In an early H-ATLAS paper (Dye et al. 2010), we showed that the 250-$\mu$m luminosity function evolves remarkably rapidly, even showing significant evolution by a redshift of 0.1, which we have confirmed recently with a much larger dataset (Wang et al. 2016a). We have also shown (Dunne et al. 2011; Bourne et al. 2012) that there is rapid evolution in the masses of dust in galaxies, and therefore in the mass of the ISM. Using radio continuum emission to trace the star formation, we found that there is also rapid evolution at low redshift in the star-formation rates of galaxies (Hardcastle et al. 2016). Marchetti et al. (2016) reached the same conclusion from a different [*Herschel*]{} survey and using a different method of estimating the star-formation rate (from the bolometric dust luminosity). This rapid low-redshift evolution is important because, as we show in this paper, it is not predicted by important galaxy-evolution models. A note on nomenclature: in this paper, we generally use the term ‘Galaxy Sequence’ rather than ‘Galaxy Main Sequence’. The latter term is implicitly based on the phenomenological paradigm, in which galaxies, like stars, spend most of their life in an active phase, which then comes to a definite end. We prefer the empirical term ‘Galaxy Sequence’, which is free of theoretical assumptions. Our definition of the term is that it refers to the distribution of galaxies in a plot of specific star-formation rate versus stellar mass that contains most of the stellar mass in a given volume of space. The arrangement of this paper is as follows. In Section 2 we describe the two [*Herschel*]{} surveys in more detail. Section 3 and 4 describes results from the two surveys that have implications for galaxy evolution. In Section 5, we discuss the implications of these results for the phenomenological paradigm and propose an alternative model for galaxy evolution that is in better agreement with these results. We suggest that readers not interested in the details of the [*Herschel*]{} results but interested in their implications skip to Section 5, which we start with a summary of the main observational results. A summary of the main results of this paper is given in Section 6. We assume a Hubble constant of 67.3 $\rm km\ s^{-1}\ Mpc^{-1}$ and the other [*Planck*]{} cosmological parameters (Planck Collaboration 2014). The Herschel Surveys ==================== The HRS consists of 323 galaxies with distances between 15 and 25 Mpc and with a near-infrared K-band limit of $K < 8.7$ for early-type galaxies (ETGs) and $K < 12$ for late-type galaxies (LTGs, Boselli et al. 2010). The sample was designed to be a volume-limited sample of galaxies selected on the basis of stellar mass. Eales et al. (2017) estimate that within the HRS volume the survey is complete for LTGs with stellar masses above $\simeq 8 \times 10^8\ M_{\odot}$ and for ETGs with stellar masses above $\simeq 2 \times 10^{10}\ M_{\odot}$. The survey therefore misses low-mass ETGs, but Eales et al. (2017) show that there is very little mass contained in these objects: $\simeq$90% of the total stellar mass in ETGs with masses $>10^8\ M_{\odot}$ in the HRS volume is contained in the galaxies in the sample. As a result of the [*Herschel*]{} photometry (Ciesla et al. 2012; Smith et al. 2012; Cortese et al. 2014) and the proximity of the galaxies, there are extremely sensitive measurements of the dust continuum emission in five far-infrared bands for each of the HRS galaxies. Even though ETGs are often assumed to contain very little dust, Smith et al. (2012b) detected continuum dust emission from 31 of the 62 HRS ETGs and obtained very tight limits on the amount of dust in the remainder. High-quality photometry in 21 photometric bands, from the $ UV$ to the far-infared, makes the HRS ideal for the application of galaxy modelling programs such as MAGPHYS (Da Cunha et al. 2008). De Vis et al. (2017) applied MAGPHYS to the HRS galaxies, obtaining estimates of key galaxy properties such as star-formation rate and stellar mass. Eales et al. (2017) used these results to look at the relationship between specific star-formation rate and stellar mass in the HRS volume (Figure 1), finding that galaxies follow a smooth curved Galaxy Sequence (GS), with a gradual change in galaxy morphology along the sequence and no abrupt change between LTGs and ETGs. They showed that the location and shape of the GS in Figure 1 is consistent with other recent attempts either to plot the entire GS (Gavazzi et al. 2016) or to plot the part of the GS classified as star-forming (Renzini and Peng 2015). Oemler et al. (2017, O17) have recently carried out a reanalysis of the SDSS galaxy survey, taking account of several selection effects, and have found a galaxy distribution very like that in Figure 1. Since Figure 1 contains all the LTGs in the HRS volume with masses $\succeq 8 \times 10^8\ M_{\odot}$, and since there is very little stellar mass in the ETGs in its bottom-left-hand corner, the diagram should be a good representation of where the stars in the Universe are today, after 12 billion years of galaxy evolution. ![specific star-formation rate versus stellar mass for the galaxies in the [*Herschel*]{} Reference Survey, a volume-limited sample designed to contain most of the stellar mass in the survey volume, reproduced from Eales et al. (2017 - see that paper for more details). The colours show the morphologies of the galaxies: maroon - E and E/S0; red - S0; orange - S0a and Sa; yellow - Sab and Sb; green - Sbc; cyan - Sc and Scd; blue - Sd, Sdm; purple - I, I0, Sm and Im. The coloured ellipses show the $1\sigma$ error region on the mean position for each morphological class, with the colours being the same as for the individual galaxies. The dashed line shows the results of fitting a second-order polynomial to the H-ATLAS galaxies (Section 3), using a method that corrects for the effect of Malmquist bias. Note the consistency in where the GS lies, whether its location is obtained from a volume-limited survey such as the HRS or a far-infrared survey such as H-ATLAS. ](Figure1_revised.eps){width="49.00000%"} While the HRS was a sample of galaxies selected in the near-infrared, which were then observed in the far-infared with [*Herschel*]{}, the H-ATLAS was a ‘blind’ far-infrared survey in which the galaxies were selected based on their far-infrared flux density. In its five fields the H-ATLAS detected $\sim500,000$ sources. The fields we use in this paper are the three small fields on the celestial equator, which cover a total area of 161.6 square degrees and were the same fields surveyed in the Galaxy and Mass Assembly project (henceforth GAMA). GAMA was a deep spectroscopic survey (Driver et al. 2009; Liske et al. 2015) complemented with matched-aperture photometry throughout the $UV$, optical and IR wavebands (Driver et al. 2016). We used the optical SDSS images to find the galaxies producing the Herschel sources and then the GAMA data to provide redshifts and matched-aperture photometry for these galaxies. We have recently released our final images and catalogues for the GAMA fields (Valiante at al. 2016; Bourne et al. 2016[^2]). The catalogue of [*Herschel*]{} sources (Valiante et al. 2016) contains 120,230 sources detected at $>4\sigma$ at 250, 350 or 500 $\mu$m, of which 113,995 were detected above this signal-to-noise at 250 $\mu$m, our most sensitive wavelength, which corresponds to a flux-density limit of $\simeq$30 mJy. We have also released a catalogue of 44,835 galaxies which are the probable sites of the [*Herschel*]{} sources (Bourne et al. 2016). We found these galaxies by looking for galaxies on the the r-band SDSS images close to the positions of the [*Herschel*]{} sources; we then used the magnitude of the galaxy and its distance from the [*Herschel*]{} source to estimate a Bayesian probability (the ‘reliability’ in our nomenclature) that the galaxy is producing the far-infrared emission. Our base sample in this paper are the 19556 galaxies in this catalogue detected at $>4 \sigma$ at 250 $\mu$m, with matched-aperture multi-wavelength photometry and spectroscopic redshifts in the redshift range $0.001 < z < 0.4$, the lower redshift limit chosen to minimize the effect of galaxy peculiar motions. The redshift distribution of this sample is shown in Table 1. There are several possible sources of error that we need to consider. The first is the possibility that we have incorrectly associated a [*Herschel*]{} source and an SDSS galaxy. We can estimate the number of sources that may have been misidentified in this way by adding up $1-R$ for all the sources, where $R$ is the reliability. We calculate that of our base sample, 435 sources (2.2%) have been incorrectly associated with SDSS galaxies. The second is that there are an additional 819 galaxies which satisfy the other criteria above but which do not have multi-wavelength aperture-matched photometry, since their redshifts were measured after the completion of the GAMA photometry program. These are essentially random omissions from the sample and are thus very unlikely to have any effect on the results in sections 3 and 4 (these objects are included in the investigation of the evolution of the luminosity function in Section 4.3). A more important issue is the possibility that we have missed associations. Bourne et al. (2016) have estimated, as a function of source redshift, the probability that we will have found the galaxy producing the [*Herschel*]{} source. Their estimates are shown in Table 1, which range from 91.3% in the redshift range $0.001 < z < 0.1$ to 72.2% in the highest redshift bin, $0.3 < z < 0.4$. The final source of error is the requirement that the galaxy has a spectroscopic redshift, which we have made so as not to introduce any additional errors into our spectral fits (Section 3). We can estimate the overall completeness of the base sample by finding the additional galaxies that have photometric redshifts in the redshift range $0.001 < z < 0.4$ but which do not have spectroscopic redshifts. There are 5701 of these galaxies, which implies the spectroscopic sample is 78% complete. However, we expect the completeness of the base sample to be a strong function of redshift. We investigated how the completeness varies with redshift using the following method. The magnitude limit of the GAMA spectroscopic survey was $r=19.8$ (Liske et al. 2015). We can guage the possibility that we have missed galaxies because they are fainter than this limit by counting the number of galaxies in the base sample that fall in the half magnitude [*brighter*]{} than the spectroscopic limit: $19.3 < r < 19.8$. If this number is small we would not expect incompleteness from this effect to be an issue. We list these percentages in Table 1. In the lowest redshift bin, the percentage is very small (0.7%), but it is very high in the two highest redshift bins. Therefore, the lowest-redshift bin should not be significantly affected but the two highest-redshift bins will be significantly incomplete. The incompleteness will be most severe for galaxies with low stellar masses. In summary, there are number of sources of systematic error associated with the method used to find the galaxies producing the [*Herschel*]{} sources. The numbers in Table 1 show that these errors are likely to be quite small for the lowest-redshift bin but large for the two highest-redshift bins, making the base sample highly incomplete at $z > 0.2$. ------------------- ------- ------------- --------- ------- -------- Redshift No. ID fraction Last bad GAMA range 0.5 mag fits $0.001 < z < 0.1$ 3,456 91.3% 0.7% 12.1% 17,768 $0.1 < z < 0.2$ 7,096 87.7% 5.1% 7.1% 38,768 $0.2 < z < 0.3$ 5,400 80.4% 22.3% 9.2% 21,323 $0.3 < z < 0.4$ 3,604 72.2% 39.5% 12.8% 7,289 ------------------- ------- ------------- --------- ------- -------- Notes: Col. 1 - redshift range; col. 2 - No. of galaxies in base sample - these are the galaxies used in the anlysis in this paper; col. 3 - Estimated percentage of H-ATLAS sources in this redshift range for which our search procedure should have found the galaxies producing the submillimetre emission (Bourne et al. 2016); col. 4 - percentage of galaxies in column 2 with r-band magnitude in the range $19.3 < r < 19.8$ (see text for significance); col. 5 - Percentage that were excluded from the base sample because there was $<$1% probability that the best-fit MAGPHYS SED was a good fit to the multi-wavelength photometry; col. 6 - No. of galaxies from the GAMA survey in this redshift range (Driver et al. 2009; Liske et al. 2015). On top of these errors, there is is the unavoidable selection effect found in all flux-density-limited surveys: Malmquist bias. The H-ATLAS is biased towards galaxies with high 250-$\mu$m luminosities in the same way that an optical survey such as the SDSS is biased towards galaxies with high optical luminosities. Thus galaxies with low masses of interstellar dust, such as ETGs, will be under-represented in H-ATLAS. For example, the ETGs in the HRS have a mean dust mass of $\rm \sim 10^5\ M_{\odot}$ (Smith et al. 2012b), but there are only 22 galaxies in our base sample with dust-mass estimates (§3) less than $10^{5.5}\ M_{\odot}$. In the next section we will make an attempt to correct H-ATLAS for the effect of Malmquist bias. The result of selection effects is that a very different galaxy population is found in a submillimetre survey such as H-ATLAS from an optical survey such as the SDSS. In optical colour-versus-absolute-magnitude diagrams, galaxies detected in optical surveys lie in a ‘blue cloud’ or on a ‘red sequence’ with a ‘green valley’ in between. We show in an accompanying paper (Eales et al. in preparation) that the distribution of H-ATLAS galaxies on the same diagram is almost the opposite of this, with the far-infared-selected galaxies forming a ‘green mountain’. We show in the accompanying paper that both distributions are the natural result of selection effects operating on the smooth GS shown in Figure 1. In the next section, we start from our biased sample of galaxies detected in H-ATLAS[^3] and investigate whether the GS we obtain after correcting for selection effects is consistent with the GS we see in Figure 1. The H-ATLAS Galaxy Sequence =========================== Method ------ The main purpose of the work described in this section was to investigate whether the low-redshift GS derived from H-ATLAS, after correcting for selection effects, is consistent with the GS derived from the [*Herschel*]{} Reference Survey that is shown in Figure 1. As for the HRS, we used MAGPHYS (Da Cunha, Charlot and Elbaz 2008) to estimate the specific star-formation rates and stellar masses of the galaxies in the base sample, which all have high-quality matched-aperture photometry in 21 bands from the ultraviolet, measured with the [*Galaxy Evolution Explorer*]{}, to the five [*Herschel*]{} measurements in the far infrared. For the reader that is not familiar with the model, we give here very brief details of the model and our application of it. MAGPHYS is a model of a galaxy based on the model of the ISM of Charlot and Fall (2000), who investigated the effects on a galaxy’s spectrum and SED of the newly-formed stars being more obscured by dust than the older stellar population, because they are still surrounded by the dust in their natal giant molecular clouds. MAGPHYS generates 50,000 possible models of the SED of an unobscured stellar population, ultimately based on the stellar synthesis models of Bruzual and Charlot (2003), and 50,000 models of the dust emission from the interstellar medium. By linking the two sets of models using a dust obscuration model that balances the radiation absorbed at the shorter wavelengths with the energy emitted in the infrared, the program generates templates which are then fitted to the galaxy photometry. From the quality of the fits between the templates and the measurements, the program produces probability distributions for important global properties of each galaxy. An advantage of the model is that the large number of templates make it possible to generate a probability distribution for each galaxy property. MAGPHYS uses the stellar initial mass function from Chabrier (2003). Our detailed procedure was the same, with the exceptions listed below, as that described by Smith et al. (2012a), who applied MAGPHYS to the galaxies in the small H-ATLAS field observed as part of the [*Herschel*]{} Science Demonstration Phase. As in the earlier paper, the value we use for each galaxy property is the median value from the probability distribution returned by MAGPHYS, since this is likely to be the most robust estimate (Smith et al. 2012a). The star-formation rate we use is the average star-formation rate over the last 0.1 Gyr. The biggest change from the earlier work was that we replaced the UKIRT near-infrared photometry and IRAS photometry with the photometry in five near-infrared bands ($z$, $Y$, $J$, $H$ and $K_s$) from the VISTA Kilo-Degree Infrared Galaxy Survey (VIKING, Edge et al. 2013) and in the four bands measured with the [*Wide-Field Infrared Survey Exlorer*]{}. A minor change was in the calibration errors used for photometry measured with the two cameras on [*Herschel*]{}, PACS and SPIRE, which we reduced from the values used in our earlier paper to 5.5% for SPIRE and 7% for PACS, the values recommended by Valiante et al. (2016). We excluded galaxies from the base sample for which there was $<$1% probability that the best-fit MAGPHYS SED was a good fit to the multi-waveband photometry. The number of these objects is shown in Table 1. Smith et al. (2012a) did a detailed examination of these objects and concluded that the vast majority are due to serious problems with the aperture-matched photometry, probably due to neighbouring objects within the aperture. We have checked that none of the results in this paper is spuriously generated by the exclusion of these object by repeating the analysis with them included, obtaining similar results. We have assumed that the SEDs are dominated by emission that is directly or indirectly from stars. This assumption is supported by the results of Marchetti et al. (2016), who, using [*Spitzer*]{} data, concluded that only $\simeq$3% of the galaxies at $z < 0.5$ in the HerMES Wide sample, a [*Herschel*]{} sample with a similar depth to our own, have an SED dominated by emission from an AGN. Many of the H-ATLAS galaxies which do have an AGN-dominated SED will anyway have been eliminated by the requirement that MAGPHYS produces a good fit to the measured SED. The results from MAGPHYS have been checked in a number of ways. Eales et al. (2017) showed that the MAGPHYS stellar mass estimates for the HRS galaxies agree well with the estimates of Cortese et al. (2012), who estimated stellar masses from i-band luminosities and a relation between mass-to-light ratio and g-i colour from Zibetti et al. (2009). In the same way, the MAGPHYS estimates of stellar mass for the H-ATLAS galaxies agree well with estimates from the optical spectral energy distributions (Taylor et al. 2011). We note, however, that these comparisons have been made with studies that are ultimately based on the stellar synthesis models and initial mass function on which MAGPHYS is based. We do not have independent measurements of the star-formation rate with which to test the MAGPHYS estimates. However, in a comparison of 12 different methods of estimating star-formation rates, Davies et al. (2016) showed that the use of MAGPHYS estimates does not lead to any biases in the relationship between SSFR and galaxy stellar mass (see their Fig. 10). Finally Hayward and Smith (2015) have demonstrated, using simulated galaxies, that MAGPHYS is very reliable for estimating galaxy stellar masses and star-formation rates, irrespective of star-formation history, viewing angle, black hole activity etc. Results ------- Figure 2 shows specific star-formation rate (SSFR) plotted against galaxy stellar mass for the H-ATLAS galaxies in four redshift bins. The errors in the estimates of the logarithm of SSFR are typically 0.2 but can be much larger for galaxies at the bottom of the diagram. A simple argument shows one of the effects of Malmquist bias. The H-ATLAS galaxies are selected based on their continuum dust emission and thus the sample will be biased towards galaxies with a large mass of dust, and thus consequently a large ISM mass and a high star-formation rate. Since lines of constant star-formation rate run roughly parallel to the galaxy distributions in Figure 2, the absence of galaxies to the bottom left of each panel may well be the result of Malmquist bias. Conversely, however, the upper envelope of each distribution, and its negative gradient, should not be significantly affected by this. The distribution in the lowest redshift bin appears curved. To assess whether this is statistically significant, we fitted both a straight line and a polynomial to the distribution, minimising the sum of $\chi^2$ in the SSFR direction. The polynomial had the form: $$\begin{aligned} log_{10}(SSFR) = a + & b \times (log_{10}M_* - 10.0)\nonumber\\ & + c \times (log_{10}M_*-10.0)^2\end{aligned}$$ The best-fit polynomial is shown in Figure 3. The reduction in the total value of $\chi^2$ obtained from using a polynomical rather than a straight line is 336. Since the expected reduction in $\chi^2$ when fitting a function with one additional parameter is itself distributed as $\chi^2$ with one degree of freedom, the reduction in $\chi^2$, and thus the curvature in the low-redshift GS, is highly significant. This adds to the other recent evidence that the GS is curved, whether only star-forming galaxies are plotted (Whitaker et al. 2014; Lee et al. 2015; Schreiber et al. 2016; Tomczak et al. 2016) or all galaxies are plotted (Gavazzi et al. 2015; Oemler et al. 2017 - henceforth O17). {width="64mm"} We used the following method to correct for the effect of Malmquest bias in Figure 3. We divided the SSFR-versus-stellar-mass diagram into rectangular bins and calculated the following quantity in each bin: $$\begin{aligned} N(SSFR,M_*) = \sum_i {1 \over V_{acc,i}}\end{aligned}$$ where the sum is over all the galaxies in that bin and $V_{acc,i}$ is the accessible volume of the i’th galaxy, the volume in which that galaxy could still have been detected above the 250-$\mu$m flux limit. This is given by: $$\begin{aligned} V_{acc,i} = \int_{z_{min}}^{z_{max}} dV\end{aligned}$$ In this equation, $z_{min}$ is the lower redshift limit (0.001) and $z_{max}$ is the lower of (a) the upper redshift limit of the redshift bin and (b) the redshift at which the flux density of the galaxy would equal the 250-$\mu$m flux limit of the sample. This technique is the standard technique for correcting for the effect of accessible volume. It will produce an unbiased estimate of $N(SSFR,M_*)$ as long as there are representatives of all kinds of galaxy in the sample. We have applied it to the lowest redshift bin because the low lower redshift limit (0.001) means there are galaxies with very low dust masses in the sample, which is not the case for the higher redshift bins. However, even if the answer is unbiased, it will be very noisy if there are only a few representatives of a class, which is the case for galaxies with the dust masses typical of ETGs (§2). This indeed is what we see in Figure 3, where $N(SSFR,M_*)$ is shown as a grayscale. The distribution, after it has been corrected for Malmquist bias, is clearly very noisy but lies, as expected, well below the observed distribution. Note that this reconstructed GS is particularly noisy at the lower right-hand end because of the small number of ETGs detected in H-ATLAS (Section 2). We fitted the polynomial in equation 1 to the datapoints again, this time weighting each point by $1/V_{acc}$ in order to correct for the effect of Malmquist bias. The dashed black line in Figure 3 shows the best-fitting polynomial. As expected, it lies well below the polynomial that is the best fit to the unscaled datapoints. We have also plotted this Malmquist-bias-corrected line onto Figure 1, which shows the GS derived from the HRS. Without being a particularly good fit, the line passes through the middle of the HRS points, showing that the GS derived from a volume-limited survey and the GS derived from correcting a far-infrared survey for Malmquist bias are consistent. In an earlier paper we showed that the GS from the HRS is consistent with the low-redshift GS derived using other methods (Eales et al. 2017), and the results from this section show that the GS derived from two very different Herschel surveys are also in reasonable agreement. {width="49.00000%"} In the remainder of this section we describe some analysis whose original goal was to test our method for correcting Malmquist bias but which turned out to have an unexpected and interesting result. We originally decided to test the method by using it to estimate the stellar mass function for star-forming galaxies, which we could then compare with the same stellar mass function derived from optical surveys. We estimated the galaxy mass function from the H-ATLAS galaxies in each redshift bin using the following formula: $$\begin{aligned} \phi(M_*) dM = \sum_i { 1 \over V_{acc,i}}\end{aligned}$$ In this formula, the sum is over all galaxies with $M_* < M_i < M_* + dM$, and $V_{acc,i}$ is the accessible volume of each galaxy (equation 2). After estimating the galaxy mass function from the H-ATLAS data, we then calculated: $$\begin{aligned} f = { \left( \phi(M_*) \right)_{submm} \over \left( \phi(M_*) \right)_{optical} }\end{aligned}$$ in which the numerator is the stellar mass function derived from the H-ATLAS galaxies using equation (4) and the denominator is the galaxy stellar mass function for star-forming galaxies derived from optical samples; for the latter, at $z < 0.2$ we used the mass function from Baldry et al. (2012) and at $z>0.2$ we used the mass function for $0.2 < z < 0.5$ derived by Ilbert et al. (2013). Figure 4 shows $f$ plotted against galaxy stellar mass for the four redshift bins. For the two highest redshift bins, the value of $f$ is much less than one, showing that at $z>0.2$, even after correcting for accessible volume, we are missing a large fraction of star-forming galaxies. This result is not unexpected because we showed in Section 2 that the base sample is seriously incomplete in these bins. In the second lowest redshift bin ($0.1 < z <0.2$), the H-ATLAS mass function is incomplete at stellar masses of $<10^{10}\ M_{\odot}$, and in the lowest redshift bin it is incomplete at stellar masses of $<10^9\ M_{\odot}$. Above these stellar masses, however, $f$ reaches values that are much greater than 1, reaching values of 3-5 at the highest stellar masses. At first sight, this result suggests that a far-infrared survey is actually much better at finding star-forming galaxies than an optical survey, with optical surveys missing a population of star-forming galaxies. We will investigate this result further in the following section. {width="49.00000%"} Red and Blue Galaxies as seen by Herschel ========================================= The Galaxy Sequence ------------------- In the comparison of the stellar mass functions at the end of the previous section, we implicitly assumed that the galaxies detected by H-ATLAS are star-forming galaxies. However, there is intriguing evidence that [*Herschel*]{} surveys do also detect a population of galaxies that have red colours (Dariush et al. 2011, 2016; Rowlands et al. 2012; Agius et al. 2013). These red colours might indicate a galaxy with an old stellar population or a star-forming galaxy with colours reddened by dust. In this section and the next one, we step back from our previous assumption about the kind of galaxy that should be detected by a far-infrared survey; instead we use the optical colours and spectra of the galaxies to determine empirically what kinds of galaxy are actually detected. {width="73mm"} {width="73mm"} {width="73mm"} In our investigation we have used the optical colours to separate the H-ATLAS galaxies into two classes using two alternative criteria from Baldry et al. (2012). Baldry et al. called these classes ‘star-forming’ and ‘passive’, but we will call them ‘blue’ and ‘red’, since the former nomenclature makes the assumption that red galaxies are not forming stars. As the first criterion we use the rest-frame $g-r$ colour of the galaxy to classify it as red or blue using the dividing line on the colour versus absolute magnitude diagram: $$\begin{aligned} g-r = -0.0311 M_r + 0.0344\end{aligned}$$ As the second criterion we use the rest-frame $u-r$ colour with the dividing line on the colour versus absolute magnitude diagram being: $$\begin{aligned} u-r = 2.06 - 0.244 {\rm tanh}\left( {M_r + 20.07 \over 1.09} \right)\end{aligned}$$ We calculated the rest-frame colours by calculating individual k-corrections for each galaxy by applying $KCORRECT\ v4\_2$ (Blanton et al. 2003; Blanton and Roweis 2007). In brief, this package finds the linear combination of five template spectra that gives the best fit to the five SDSS magnitudes for each galaxy and then uses this model to calculate the K-correction for the galaxy (Blanton and Roweis 2007). Some additional details of the implementation of the code are given in Loveday et al. (2012). Equations 6 and 7 were determined by Baldry et al. (2012) from the low-redshift galaxy population. However, even a galaxy today in which no stars have formed for the last 10 billion years will have had bluer colours in the past because of the evolution in the turnoff mass on the stellar main sequence. We therefore investigated the effect of adding a small correction to these equations to model the expected evolution in the colours of a very old stellar population. Our model of this effect was based on a model of a single stellar population from Bruzual and Charlot (2003) with a Salpeter initial mass function and solar metallicty. We assumed that the galaxy started forming 12 Gyr ago with the star-formation rate proportional to $exp(-t/\tau)$ and $\tau = 1\ Gyr$. Table 2 gives the percentages of red galaxies in the different redshift bins for the two colour criteria and also shows the effect of adding the evolutionary correction. Rather surprisingly, even without making the evolutionary correction, $\simeq$15-30% of the H-ATLAS galaxies are red galaxies. This is higher than the value of $\simeq4.2\%$ found by Dariush et al. (2016) for H-ATLAS galaxies at $z<0.2$. We suspect that the difference arises because Dariush et al. used optical-UV colours rather than optical colours, a suspicion we will explain in the next section (§4.2). Figure 5 shows the GS again, but this time with the points colour-coded to show which galaxies are red and blue according to equation 6 (equation 7 produces a very similar figure). The significant fraction of red galaxies explains the values of $f>$1 in Fig. 4, because these galaxies would have been classified as passive galaxies using optical criteria and so would have been omitted from the mass function for star-forming galaxies derived from optical surveys. However, these red galaxies still have significant reservoirs of interstellar gas (after all, they are detected because of the continuum emission from interstellar dust) and the MAGPHYS results imply they are still forming stars. We will investigate further the properties of this interesting population in the next section. Figures 5 is a striking demonstration of why the GMS produced from a subset of galaxies classified as star-forming will generally be flatter than the GS we have derived from the two [*Herschel*]{} surveys. Imagine removing all the red galaxies in Figures 5; the GMS would then have a much flatter slope. Although there are fewer optically-red galaxies than optically-blue galaxies in H-ATLAS, a simple argument shows that optically-red star-forming galaxies are not a peripheral population. The optically-red galaxies are under-represented in H-ATLAS because of Malmquist bias. At a given stellar mass, Figure 5 shows that optically-red galaxies generally have lower values of SSFR than optically-blue galaxies, which in turn means a lower star-formation rate and, through the Kennicutt-Schmidt law, a lower gas and dust mass - leading to a smaller accessible volume and Malmquist bias. Figure 4, where we have attempted to correct for Malmquist bias, shows this quantitatively. The implication of the figure is that, at a given stellar mass, the space-density of optically-red star-forming galaxies is at least as high as that of optically-blue star-forming galaxies. Optical investigators have generally missed this population because they classify these galaxies as passive. However, they have the following excuse. A comparison of the stellar mass functions given by Baldry et al. (2012; their Figure 15) for star-forming galaxies and passive galaxies (precisely equivalent to our optically-blue and optically-red classes) shows that the space density of the two galaxy types is the same at a stellar mass of $\simeq10^{10}\ M_{\odot}$, but that at higher stellar masses the space-density of passive galaxies is higher, with a maximum difference of a factor of $\simeq5$ at a stellar mass of $\simeq 4 \times 10^{10}\ M_{\odot}$. Given the much larger number of passive galaxies, even a small change in how one divides galaxies into the passive and star-forming classes will have a large effect on estimates of the space-density of star-forming galaxies. Not all optical investigators, however, have missed this population. In their elegant reanalysis of the SDSS galaxy sample, O17 showed there is an intermediate population of galaxies between those that are rapidly forming stars and passive galaxies. The galaxies in this intermediate class are still forming stars and seem identical to our optically-red star-forming galaxies. O17 conclude that at a given stellar mass the number of galaxies in this intermediate class is roughly the same as the number in the rapidly star-forming class, thus reaching exactly the same conclusion as we do but starting from a traditional optical survey. ------------------- ------- ------- ---------------- ---------------- Redshift (g-r) (u-r) (g-r) (u-r) plus evolution plus evolution $0.001 < z < 0.1$ 27% 16% 29% 18% $0.1 < z < 0.2$ 26% 15% 31% 18% $0.2 < z < 0.3$ 24% 18% 33% 23% $0.3 < z < 0.4$ 21% 21% 35% 28% ------------------- ------- ------- ---------------- ---------------- : Percentages of red galaxies in H-ATLAS The percentage of H-ATLAS galaxies in different redshift ranges classified as red using equation 6 (columns 2 and 4) and equation 7 (columns 3 and 5). In columns 4 and 5 we add a correction to equations 6 and 7 to allow for the expected evolution of a very old stellar population (see text). Stacking spectra - the nature of the red and blue galaxies ---------------------------------------------------------- The colours of the optically-red H-ATLAS galaxies might indicate an old stellar population or alternatively a star-forming galaxy whose colours are reddened by dust. We attempted to distinguish between these possibilities using the galaxies’ spectra. The spectra come from the GAMA and SDSS projects, with most of the spectra coming from the former. Hopkins et al. (2013) describe the calibration and other technical details of the GAMA spectra. The GAMA project used the AAOmega spectrograph on the AAT, which has fibres with an angular diameter on the sky of 2 arcsec. This corresponds to a physical size at redshifts of 0.1, 0.2, 0.3 and 0.4 of 3.6 kpc, 6.6 kpc, 8.9 kpc and 10.7 kpc, respectively. We divided the SSFR versus stellar mass diagrams for the H-ATLAS galaxies with $0.001 < z < 0.1$ and with $0.3<z<0.4$ each into five boxes, which are shown in the bottom and top panels of Figure 5. We then calculated the median rest-frame spectrum of all the GAMA and SDSS spectra in each box (Figures 6 and 7). We measured the equivalent width of the H$\alpha$ line from each spectrum, using the wavelength ranges $6555.5 < \lambda < 6574.9\AA$ to measure the flux in the line and the wavelength ranges $6602.5 < \lambda < 6622.5\AA$ and $6509.5 < \lambda < 6529.5\AA$ to estimate the mean value of the continuum at the line wavelength. The H$\alpha$ equivalent width, the ranges of stellar mass and SSFR for each box, and the number of galaxies in the box are shown by the side of each spectrum in Figures 6 and 7. First, let us consider the stacked spectra for the low-redshift galaxies (Figure 6). The galaxies in the bottom box in Figure 5 almost all have red optical colours. The stacked spectrum for this box, which is shown in the lowest panel in Figure 6, is strongly characteristic of an old stellar population. The red colours are therefore generally the result of the age of the stellar population rather than dust reddening. This figure gives further insights into why different studies of the GMS can find very different results. As we move down the panels in Figure 6, the appearance of the stacked spectra gradually changes, with the equivalent width of the H$\alpha$ line, the brightest emission line in the spectra, steadily decreasing. This is not surprising (although it is reassuring) because the luminosity of the H$\alpha$ line is often used to estimate a galaxy’s star-formation rate (Davies et al. 2016; Wang et al. 2016b). When the H$\alpha$ line is used in studies of the GMS to separate star-forming from passive galaxies, the dividing line is usually an H$\alpha$ equivalent width in the range $3\AA < EW < 10\AA$ (Bauer et al. 2013; Casado et al. 2015). The equivalent width of H$\alpha$ in the three lowest boxes in Figure 6 is, in order of increasing SSFR, 0.3, 5.9 and 14.0Å. Therefore, when this method is used to separate star-forming and passive galaxies, the shape of the GMS that is found depends critically on the exact value of the equivalent width used to divide the galaxies. These results also suggest something more fundamental which we will return to later. Whereas the optical view of the galaxy population is that there are two distinct classes of galaxy (§1), the [*Herschel*]{} results show more continuity. The red galaxies have colours and stacked spectra that imply they have old stellar populations but their detection by [*Herschel*]{} shows they contain a substantial ISM - and both our MAGPHYS results and the H$\alpha$ equivalent widths imply they are still forming stars. Furthermore, the overlap of red and blue galaxies in Figure 5 also implies that the two classes are not clearly physically distinct. The blurring between the two classes is even more evident when we turn to the high-redshift population. In Figure 7 we show the result of stacking the spectra of the galaxies in the redshift range $\rm 0.3 < z < 0.4$ and $\rm log_{10}(M/M_{\odot}) > 10.5$. The base sample is highly incomplete in this redshift range, although the incompleteness is most severe at lower stellar masses (Section 2). The stacked spectra for this high-redshift bin are visually quite startling because they all show clear evidence of both an old and a young stellar population. In all the stacked spectra, there is a clear 4000 Å break, evidence of an old stellar population. Since the SDSS u-band is centred at $\simeq$3500 Å, the existence of the 4000 Å break immediately explains why so many of the galaxies fall into the optically-red class. However, in all the stacked spectra there are also clear signs of a high star-formation rate, including strong emission lines, in particular H$\alpha$, and a $UV$ upturn. We don’t know whether this $UV$ upturn is present in the galaxies in the low-redshift bin because our spectra for this bin (Figure 6) do not extend to a low enough rest-frame wavelength, but a $UV$ upturn would explain why Dariush et al. (2016) found a much smaller fraction of red galaxies when using the $UV$-optical colours to classify the galaxies. Figure 7 shows that the rapid low-redshift evolution that we have seen in previous H-ATLAS studies (§1) is caused by galaxies with high stellar masses and a large population of old stars. In the Universe today, galaxies like this are forming stars at a very low rate but four billion years back in time they were clearly forming stars at a much faster rate. The evolution of the red and blue galaxies ------------------------------------------ In earlier papers (Dye et al. 2010; Wang et al. 2016a), we showed that the H-ATLAS 250-$\mu$m luminosity function shows rapid evolution over the redshift range $0 < z < 0.4$ with significant evolution even by a redshift of 0.1. Marchetti et al. (2016) found similar results from an analysis of the results of the other large [*Herschel*]{} extragalactic survey, HerMES. In this section we consider the evolution of the 250-$\mu$m luminosity function separately for red and blue galaxies. In this case we started with [*all*]{} the galaxies from the GAMA fields detected at $>4\sigma$ at 250 $\mu$m with spectroscopic (by preference) or photometric redshifts in the range $0.001 < z < 0.4$ (Valiante et al. 2016; Bourne et al. 2016). Of the 25,973 H-ATLAS galaxie in this sample, 20,012 have spectroscopic redshifts. We used the $u-r$ colour criterion (equation 7) to divide the H-ATLAS galaxies into red and blue galaxies, although the results using the $g-r$ colour criterion (equation 6) were very similar. We made the small correction to equation 7 that allows for the fact that the colours of even a very old stellar population must have been redder in the past (Section 4.1), although this actually makes very little difference to the results. To calculate the luminosity function for each class, we used the estimator invented by Page and Carerra (2000), since this has some advantages for submillimetre surveys (Eales et al. 2009): $$\begin{aligned} \phi(L_1<L<L_2,z_1<z<z_2)\Delta log_{10}L\Delta z = n/V\end{aligned}$$ in which $n$ is the number of galaxies in this bin of luminosity and redshift and $V$ is the accessible volume averaged over the luminosity range of this bin. We multiplied the luminosity function in each redshift bin by $ 1/C$, where $C$ is the estimated efficiency of our method for finding the galaxies producing the far-infrared emission (Bourne et al. 2016; column 3 of Table 1). Figure 8 shows the luminosity function for the optically-red and optically-blue galaxies. Strong evolution is seen in the luminosity function for both populations, with the evolution in the red population looking slightly stronger. We quantified the evolution by fitting a Schechter function to each empirical luminosity function. For the lowest-redshift luminosity function, we allowed all three parameters of the Schechter function - $\alpha$, $L_*$ and $\phi_*$ - to vary. For the other luminosity functions, which have a smaller range of luminosity, we only allowed $L_*$ and $\phi_*$ to vary, using the value of $\alpha$ from the low-redshift bin. We then used the estimates of the parameters at each redshift to investigate the evolution of $\phi_*$ and $L_*$. We assumed that the evolution has the form $\phi_* = \phi_{*0} (1+z)^n$ and $L_* = L_{*0} (1+z)^m$. We found $n=0.24\pm0.04$ and $m=3.69\pm0.01$ for the optically-blue galaxies and $n=3.66\pm0.06$ and $m=1.86\pm0.01$ for the optically-red galaxies. Therefore, there is strong evolution in $L_*$ and $\phi_*$ for the red galaxies and strong evolution in $L_*$ for the blue galaxies. Bourne et al. (2012) also found evidence for evolution in submillimetre luminosity and dust mass for both red and blue galaxies, with marginal evidence of stronger evolution for the red galaxies. The stacking analysis of Bourne et al. was carried out on an optically-selected sample of galaxies, and is therefore evidence that the strong evolution we see for optically-red galaxies is not just a phenomenon associated with an interesting, but ultimately unimportant population detected by [*Herschel*]{} but applies to the whole galaxy population. {width="74mm"} How star-formation efficiency varies along the GS ------------------------------------------------- Results from the [*Herschel*]{} Reference Survey show that galaxy morphology changes gradually along the GS, the morphologies moving to earlier types on the Hubble sequence as one moves down the GS (Fig. 1; Eales et al. 2017). This progression implies an increase in the bulge-to-disk ratio, which Martig et al. (2009) have argued should lead to a decrease in star-formation efficiency (SFE, star-formation rate divided by ISM mass). In this section we test this hypothesis by investigating whether SFE varies along the GS. There is already some evidence from other surveys that SFE and SSFR are correlated (Saintonge et al. 2012; Genzel et al. 2015). We have restricted our analysis to the H-ATLAS galaxies in the redshift range $0.001 < z < 0.1$. In our analysis we use the MAGPHYS estimates of the star-formation rate and the dust mass, using the dust mass of each galaxy to estimate the mass of the ISM. Many authors (Eales et al. 2012; Scoville et al. 2014; Groves et al. 2015; Genzel et al. 2015) have argued this is a better way of estimating the ISM mass than the standard method of using the 21-cm and CO lines, because of the many problems with CO, in particular the evidence that one third of the molecular gas in the Galaxy contains no CO (Abdo et al. 2010; Planck Collaboration 2011; Pineda et al. 2013), which is probably because of photodisintegration of the CO molecule. Dust grains, on the other hand, are quite robust, and the main problem with the dust method is the fact that the dust-to-gas ratio is likely to depend on metallicity, a problem of course that is shared by the CO method. There is a lot of evidence that above a transition metallicity ($\rm 12+log(O/H) \simeq 8.0$) the dust-to-gas ratio is proportional to the metallicity (James et al. 2002; Draine et al. 2007; Bendo et al. 2010; Smith et al. 2012c; Sandstrom et al. 2013; Rémy-Ruyer et al. 2014). In order to test how robust our results are to the metallicity correction, we have used three different methods for doing this correction. In the first method we make no correction for metallicity and assume that each galaxy has a dust-to-gas ratio of 0.01. In the second method we estimate the metallity of each galaxy from its stellar mass using the relationship found by Tremonti et al. (2004): $$\begin{aligned} 12 + log_{10}(O/H) = -1.492 & + 1.847(log_{10}M_*) \nonumber\\ & - 0.08026(log_{10}M_*)^2\end{aligned}$$ We then assume that the dust-to-gas ratio is proportional to the metallicity and that a galaxy with solar metallicity has a dust-to-gas ratio of 0.01. The third method is the same as the second except that we use the relationship found by Hughes et al. (2013) from their study of HRS galaxies: $$\begin{aligned} 12 + & log_{10}(O/H) = 22.8 - 4.821(log_{10}M_*) \nonumber\\ & +0.519(log_{10}M_*)^2 - 0.018(log_{10}M_*)^3\end{aligned}$$ The obvious statistical test is to see whether there is any correlation between SFE and SSFR. However, this is dangerous because both quantities are ratios with the star-formation rate in the numerator; thus errors in the star-formation-rate estimates may lead to a spurious correlation. Instead, we have compared the SFE of the optically-red and optically-blue galaxies, since this classification was done without using any of the MAGPHYS estimates and yet we know from Figure 5 that optically-red galaxies have lower SSFR values than optically-blue galaxies. Figure 9 shows the distributions of SFE for the optically-red and optically-blue galaxies when the metallicity correction is made using equation 9. The figures produced using the two other methods look very similar. The mean values of SFE for the optically-red and optically-blue galaxies are given in Table 3, for all three metallicity-correction methods and for both methods of classifying the galaxies as red or blue. In each case, we have compared the two distributions using the two-sample Kolmogorov-Smirnov test, testing the null hypothesis that the two samples are drawn from the same population. The values of the KS statistic and the probabilities that the two samples are drawn from the same population are given in Table 3. -------- ------------- ----------- ------------ ---------------- ---------------- ------ ---------- Colour metallicity $n_{red}$ $n_{blue}$ $<SFE_{red}>$ $<SFE_{blue}>$ KS Prob. correction $g-r$ None 885 2502 -9.59$\pm$0.01 -9.19$\pm$0.01 0.40 $<<$0.1% $g-r$ Tremonti 885 2502 -9.24$\pm$0.01 -8.95$\pm$0.01 0.31 $<<$0.1% $g-r$ Hughes 885 2502 -9.74$\pm$0.01 -9.41$\pm$0.01 0.35 $<<$0.1% $u-r$ None 534 2853 -9.68$\pm$0.02 -9.22$\pm$0.01 0.43 $<<$0.1% $u-r$ Tremonti 534 2853 -9.33$\pm$0.02 -8.97$\pm$0.01 0.38 $<<$0.1% $u-r$ Hughes 534 2853 -9.84$\pm$0.02 -9.43$\pm$0.01 0.42 $<<$0.1% -------- ------------- ----------- ------------ ---------------- ---------------- ------ ---------- The columns are as follows: Col. 1–the colour used to divide galaxies into optically-red and optically-blue galaxies; col. 2–the method used to correct the dust-to-gas ratio for the effect of metallicity (see text for details); col. 3–the number of optically-red galaxies; col. 4–the number of optically-blue galaxies; col. 5–the mean value of the logarithm of star-formation efficiency for the optically-red galaxies; col. 6–the mean value of the logarithm of star-formation efficiency for the optically-blue galaxies; col. 7–the value of the two-sample Kolmogorov-Smirnov statistic used to compare the SFE distributions of the optically-red and optically-blue galaxies; col. 8–the probability that such a high value of the KS statistic would be obtained if the two distributions were drawn from the same population. The KS test shows that for all six variations on the basic method the probability that the star-formation efficiency of the optically-red and optically-blue galaxies is the same is $<<0.1\%$. Table 3 shows that the mean SFE of the optically-red galaxies is lower by a factor of $\simeq1.9-2.9$ than the optically-blue galaxies. Saintonge et al. (2012) found that as the SSFR decreases by a factor of $\simeq$50, SFE decreases by a factor of $\simeq$4 (see their Fig. 2). The shift in SSFR between the optically-blue and optically-red galaxies in Figure 5 is a little less than this and so our result seems to be in reasonable agreement with their result. ![[Histograms of star-formation efficiency for the optically-red galaxies (red line) and optically-blue galaxies (blue line) for the H-ATLAS galaxies in the redshift range $0.001 < z < 0.1$. In this diagram, we have used equation 6 ($g-r$) to split the galaxies into the two classes and the metallicity relationship of Tremonti et al. (2004) to correct for the metallicity effect (see text). The histograms have been normalised so that the areas under both are the same.]{} ](Figure9_revised.eps){width="74mm"} Discussion ========== What have we learned from Herschel? ----------------------------------- We have learned three main things from the [*Herschel*]{} extragalactic surveys that need to be explained by any comprehensive theory of galaxy evolution. We have learned first that the galaxy population shows rapid evolution at a surprisingly low redshift. This is evident in the submillimetre luminosity function (Dye et al. 2010; Wang et al. 2016a), the dust masses of galaxies (Dunne et al. 2011; Bourne et al. 2012) and the star-formation rates, whether estimated from radio continuum observations (Hardcastle et al. 2016) or from bolometric dust emssion (Marchetti et al. 2016). The rapid evolution in dust mass and star-formation rate are of course connected, since stars form out of gas and the dust traces the interstellar gas reservoir. Several recent studies have found that the GS is curved, whether only star-forming galaxies are plotted (Whitaker et al. 2014; Lee et al. 2015; Schreiber et al. 2016; Tomczak et al. 2016) or all galaxies are plotted (Gavazzi et al. 2015; O17). The results from the two [*Herschel*]{} surveys confirm and extend this result. Both the [*Herschel*]{} surveys, selected in very different ways, show that galaxies lie on an extended curved GS rather than a star-forming GMS and a separate region of ‘passive’ or ‘red-and-dead’ galaxies (Figures 1 and 3). The GS shown in Figures 1 and 3 extends down to lower values of SSFR than most of the other studies because we have made no attempt to remove ‘passive’ galaxies, a distinction which we argue below is anyway rather meaningless. Third, we have learned that the operational division often used in optical studies between red and blue galaxies looks distinctly arbitrary when viewed from a submillimetre perspective. 15-30% of the H-ATLAS galaxies fall into the red category, but these galaxies clearly are not quenched or passive galaxies, since they still have large reservoirs of interstellar material and are still forming stars. The distinction looks even less clearcut at $z\simeq0.3-0.4$, where the spectra of both red and blue galaxies are qualitatively very similar (Fig. 7). After correcting for the effect of Malmquist bias in H-ATLAS, we find that, at a constant stellar mass the space-density of these optically-red star-forming galaxies, which are missing from most optical studies, is at least as high as the space-density of the optically-blue star-forming galaxies that are included in optical studies. O17 reached a similar conclusion from their reanalysis of the optically-selected SDSS galaxy sample. The combination of the second and third results point to a very different picture of the galaxy population than the standard division into two dichotomous classes - whether one calls these classes ‘late-type’ and ‘early-type’, ‘star-forming’ and ‘passive’, ‘star-forming’ and ‘red-and-dead’, or ‘star-forming’ and ‘quenched’. The smooth appearance of the GS when all galaxies are plotted rather than just galaxies classified as ‘star-forming’ (Figs 1 and 3), the gradual change in galaxy morphology along the GS (Fig. 1), the fact that the average galaxy spectra change gradually along the GS (Figs 6 and 7), the strong evolution shown by both optically-red and optically-blue galaxies - all of these suggest that galaxies are better treated as a unitary population rather than two dichotomous classes. As we discuss in §5.5, there is actually a a lot of recent evidence from observations in other wavebands that leads to the same conclusion. If our conclusion is correct, that galaxies form a unitary population rather than two dichotomous classes, much of the rationale for rapid quenching models (§1) vanishes. Finally, in this section, we consider the identity of the galaxies producing the rapid low-redshift evolution. The smoking gun is possibly Figure 7. The stacked spectra in this figure show that the high-redshift H-ATLAS galaxies both contain an old stellar population, shown by the significant 4000 Å spectral break (and the red optical colours), but are also forming stars at a high rate, shown by the ultraviolet emission and strong emission lines. The galaxies used to produce these average spectra all have stellar masses $\rm >10^{10.5}\ M_{\odot}$. Today most galaxies with stellar masses above this limit are ETGs (Fig. 1). Therefore it seems likely that the H-ATLAS galaxies are the fairly recent ancestors of some part of the ETG population in the Universe today. Arguments based on chemical abundances imply that at least 50% of the stellar mass of ETGs formed over $\simeq$8 billion years in the past (Thomas et al. 2005), but our results and those of Bourne et al. (2012) suggest that even looking back a few billion years is enough to see significantly enhanced star-formation rates in the ETG population. Comparision with theoretical models ----------------------------------- The recent large-area galaxy surveys, H-ATLAS in the far-infrared and GAMA in the optical, have made it possible to observe galaxy evolution, with good time resolution, over the last few billion years. The models have not kept pace with the observations and most theoretical papers make predictions over a longer time period with much coarser time resolution. It is therefore not possible to make a definitive comparision of our new results with the results from numerical galaxy simulations such as Illustris and EAGLE, although we do make a comparision with the predictions in a very recent EAGLE paper at the end of this section. We have therefore mostly used an analytic galaxy-evolution model to try to reproduce our results, which does have the advantage that it is generally easier to diagnose why an analytic model does not match the results than is the case for a numerical model. Given the clear evidence that the increased star-formation rate in galaxies at high redshift is because they contain more gas, both evidence from our results (Dunne et al. 2011) and from those of others (Genzel et al. 2015; Scoville et al. 2016), it is reasonable to assume that, to first order, galaxy evolution is governed by the supply of gas. A useful model based on this assumption, which we will use in this section, is the ‘bathtub model’ or ‘gas regulator model’ (Bouché et al. 2010; Lilly et al. 2013; Peng and Maiolino 2014). We will use the analytic version of the bathtub model described by Peng and Maiolino (2014; henceforth PM). The PM model is based on three assumptions: (a) the star formation rate is proportional to the mass of gas in the galaxy; (b) the rate at which gas is flowing out of the galaxy is proportional to the star-formation rate; (c) the rate at which gas is flowing into the galaxy is proportional to the growth rate of the dark-matter halo in which the galaxy is located. The final assumption can be represented by the following equation: $$\begin{aligned} \Phi \propto {dM_{halo} \over dt}\end{aligned}$$ in which $\Phi$ is the gas inflow rate. In their model, PM used the specific mass-increase rate for a halo found in the hydrodynamic simulations of Faucher-Giguere et al. (2011): $$\begin{aligned} <{1 \over M_{halo}} {dM_{halo} \over dt}> = & 0.0336(1+0.91z) \left( {M_{halo} \over 10^{12} M_{\odot}} \right)^{0.06}\nonumber\\ & \times \sqrt{ \Omega_M (1+z)^3 + \Omega_{\Lambda}}\ Gyr^{-1}\end{aligned}$$ We will examine whether this model can explain two of the key [*Herschel*]{} results. First, let us consider the shape of the GS. In their model, PM show that the equilbrium value of the SSFR of a galaxy is given by: $$\begin{aligned} SSFR = {1 \over t - t_{eq}}\end{aligned}$$ in which $t_{eq}$ depends on parameters such as the mass-loading factor for outflows, the fraction of the mass of newly formed stars that is eventually returned to the ISM, and the star-formation efficiency. With the possible exception of the last (§4.4), there is no obvious reason why any of these should depend on stellar mass, and so the model predicts that the SSFR should also be independent of stellar mass. Although this is consistent with the relatively flat GMS found in some optical studies (e.g. Peng et al. 2010), it is clearly completely inconsistent with our results and the other recent findings that the GMS is strongly curved (Whitaker et al. 2014; Lee et al. 2015; Gavazzi et al. 2015; Schreiber et al. 2016; Tomczak et al. 2016; O17). Now let us try to reproduce the rapid low-redshift evolution. We will try to reproduce both the evolution in the position of the GS seen in Figure 2 and the evolution in the mass of the ISM in galaxies found by Dunne et al. (2011). Although the two highest redshift bins in Figure 2 are highly incomplete, our completeness analysis (§3.2) showed that at $z < 0.2$ H-ATLAS is fairly complete for stellar masses $\rm >10^{10}\ M_{\odot}$ once a correction has been made for Malmquist bias (Figure 4). We therefore fitted a second-order polynomial to the datapoints in the two lowest redshift bins in Figure 2 for galaxies with masses $\rm >10^{10}\ M_{\odot}$. We found that the mean value of SSFR increases by a factor of $\simeq$2.7 between the two bins if no correction is made for Malmquist bias and $\simeq$3.0 if a correction is made. We can use equation 13 to predict how rapidly the mean SSFR should change between bins according to the PM model. If we assume that the galaxy formed 12 Gyr ago and that $t_{eq}$ is insignificant, the predicted change in SSFR between a redshift of 0.05 and 0.15 is only 1.12, much less than the observed evolution. Now let us consider the change in the mass of the ISM. Dunne et al. (2011) found that by a redshift of $\simeq0.45$ galaxies contain roughly 5 times as much dust (and therefore gas[^4]) as in the Universe today. In the PM model, the equibrium gas mass is proportional to the gas inflow rate. On this assumption and using equations 11 and 12, we calculate that the gas mass should increase by a factor of $\simeq1.3$ from a redshift of 0.05 to 0.45. In both cases, the PM model predicts much weaker evolution than we observe. The physical reason for this is the assumption that the rate of increase of gas flow into a galaxy is proportional to the rate of increase of the mass of the surrounding dark-matter halo (equation 11), since the growth in the masses of dark-matter halos is very slow at low redshift (equation 12). This will be a problem for any model, analytic or otherwise, in which the gas flow into a galaxy is proportional to the growth rate of the surrounding halo. Although it is not yet possible to make a definitive comparision with the predictions of of the numerical galaxy simulations, we will make a first attempt to see whether these might reproduce the rapid low-redshift evolution using the recent predictions for the star-formation rate function (SFRF; the space-density of galaxies as a function of star-formation rate) by the EAGLE team (Katsianis et al. 2017). Katsianis et al. show predictions for the SFRF at two redshifts in the redshift range of interest: $z = 0.1$ and $z = 0.4$. Inspection of their Figure 2 shows that the SFRF has a similar shape at the two redshifts but is shifted upwards in star-formation rate by a factor of $\simeq$1.55 from the lower to the higher redshift. This is quite similar to the way the submillimetre luminosity function evolves, because while its shape stays roughly the same it moves gradually to higher luminosity as the redshift increases (Figure 8 of this paper; Dye et al. 2010; Wang et al. 2016a; Marchetti et al. 2016). We can make a rough comparison of the observations and the model if we make the assumption that the characteristic luminosity of the best-fit Schechter function ($L_*$) is proportional to the star-formation rate. Wang et al. (2016a) and Marchetti et al. (2016) find, respectively, that $L*$ at 250 $\mu$m increases by a factor of 3.25 and 3.59 over this redshift range. This is much faster than the evolution predicted by the model, although this difference needs to be confirmed by an analysis in which the observed and predicted quantities are more obviously comparable. One possible way of strengthening the evolution in the simulation would to reduce the strength of the feedback in massive galaxies (Katsianis private communication). A new model - the flakey faucet model ------------------------------------- In this section we present a heuristic model that can reproduce the shape of the GS and the rapid low-redshift evolution. It is based on a model proposed by Peng et al. (2010) to explain the stellar mass functions of red and blue galaxies which we have modified to explain the new observations. Peng et al. (2010) showed that the difference in the stellar mass functions of the red and blue galaxies can be explained by a simple quenching model. In their model a galaxy evolves along the GMS with its star-formation rate being proportional to its stellar mass, thus producing a horizontal GMS on a plot of SSFR versus stellar mass, until a catastrophic quenching event occurs; the galaxy then moves rapidly to the ‘red and dead’ region of the diagram. The difference between the stellar mass functions of red and blue galaxies (e.g. Baldry et al. 2012) can be explained almost exactly if the probability of quenching is also proportional to stellar mass. However, this model doesn’t explain the curved GS and the rapid low-redshift evolution. The difference between our model and that of Peng et al. is that we assume something milder occurs when the galaxy is quenched. We assume that the quenching simply consists of the gas supply to the galaxy being turned off. We discuss possible physical explanations of this stochastic disruption of the gas supply in §5.4. After the gas supply is turned off, we model the evolution of the galaxy over the SSFR-versus-stellar-mass diagram; it is this post-quenching evolution that produces the curved GS and the rapid low-redshift evolution. Many of the details of the models are the same. We assume that as long as gas is being supplied to a galaxy, its star-formation rate is proportional to its stellar mass. We also assume like Peng et al. that the mean SSFR of the galaxies to which gas is still being supplied decreases with cosmic time. Using a slight modification of equation 1 of Peng et al., the SSFR of an indvidual galaxy, as long as gas is being supplied to it, is given by: $$\begin{aligned} {SFR \over M_*} = SSFR = 2.5 \left({t \over 3.5}\right)^{-2.2}\ Gyr^{-1} + k\end{aligned}$$ The constant $k$ represents the position of a galaxy relative to the mean SSFR at that time. We assume $k$ remains a constant as long as gas is being supplied to the galaxy. We also assume, like Peng et al., that the quenching probability is proportional to the galaxy’s stellar mass: $$\begin{aligned} prob_{quenching} = c \times M_*\end{aligned}$$ The constant $c$ is the first of the parameters of our model. To model the evolution of the galaxy after the gas supply is turned off, we need to know the gas mass at that time, which is given by: $$\begin{aligned} M_{gas} = SFR / \epsilon\end{aligned}$$ where $SFR$ is the star-formation rate and $\epsilon$ is the star-formation efficiency, which is the second of the two parameters in our model. With this equation and the following two equations, we can follow a galaxy’s evolution over the SSFR-versus-stellar-mass diagram once the galaxy’s gas supply has been turned off: $$\begin{aligned} \Delta M_* = SFR \Delta t\end{aligned}$$ $$\begin{aligned} \Delta M_{gas} = -SFR \Delta t\end{aligned}$$ in which $\Delta t$ is an interval of cosmic time. There are some implicit assumptions behind these equations. We are assuming, as we did in §4.4, that the star-formation rate is proportional to the gas mass rather than to a different power of the gas mass (there is evidence for both in the literature - Kennicutt and Evans 2012). Also, despite the evidence in this paper (§4.4) and elsewhere that the star-formation efficiency falls with decreasing SSFR, for simplicity we assume it is constant. We created a realisation of this model by continuously injecting stochastically galaxies onto the diagram with a stellar mass of $10^8 M_{\odot}$ and an SSFR given by equation 14, starting at a redshift of 4.0 and continuing to the present time. In giving a value of $k$ to each galaxy, we assumed that the distribution of $k$ is a Gaussian distribution with a standard deviation of 0.2 in $\rm log_{10}(SSFR)$. Once the galaxy is injected, we follow its motion accross the diagram, as long as the gas supply is switched on, using equations 14 and 17. In each time step, we use equation 15 and a random number generator to determine whether to switch off the gas supply; once the gas is switched off, we follow the evolution of the galaxy using equations 16-18. We assumed a star-formation efficiency ($\epsilon$ in equation 16) of $\rm 10^{-9}\ year^{-1}$, a typical value for optically-blue galaxies (§4.4, Figure 9). The only other parameter in this model is the quenching probability ($c$ in equation 15), which we adjusted until we got a galaxy distribution in the Universe today that looked like the observed GS. We found that we got reasonable agreement with the observations if $$\begin{aligned} prob_{quenching}(M_*) = \left( {M_* \over 3.3 \times 10^{9}} \right)\ Gyr^{-1}\end{aligned}$$ Figure 10 shows the distribution of the galaxies in a plot of SSFR versus stellar mass at a redshift of 0 for one realization of this model. The dashed lines show the region in the bottom left of the diagram where the HRS is incomplete (Section 2). If one mentally excludes the galaxies in Figure 10 that fall in this region, the figure’s resemblance to Figure 1, the GS as seen by the HRS, is quite good. The model explains the curvature of the GS as the result of the curved path an individual galaxy follows in this diagram once its gas supply has been cut off. We emphasise that the only thing we have done to get this agreement between the model and the observations is to adjust one parameter: the normalization in the quenching equation (equation 19). The simplicity of the model means that it is very easy to see what would happen if we change some of its features. In the model, there are no outflows once the gas supply has been turned off. If there continued to be outflows, galaxies would evolve down the diagram more rapidly, reducing the density of points in the figure. On the other hand, if star-formation efficiency decreases with decreasing SSFR, galaxies would move more slowly down the diagram, leading to a greater number of galaxies at intermediate values of SSFR. Since Fig. 10 shows that most of the galaxies with stellar masses $>10^{10}\ M_{\odot}$ in the Universe today are now evolving as closed boxes, the model will naturally lead to strong low-redshift evolution. Dunne et al. (2011) found that by a redshift of $\simeq0.45$ galaxies contain roughly five times more dust (and therefore gas) than today. The simplicity of the model makes a quantitative comparison with the observations pointless, but a rough calculation shows that the model should give approximately the right amount of evolution. If $\epsilon$ is the star-formation efficiency, the ratio of the gas mass at time $t$ to the gas mass at $t=0$ is given by $M_g/M_0 = e^{-\epsilon t}$. An increase of a factor of 5 in the gas mass by $z = 0.45$ requires a star-formation efficiency of $\rm \epsilon = 3.5 \times 10^{-10}\ year^{-1}$. This is close to the centre of the histogram of star-formation efficiencies in Figure 9. As long as most galaxies are now evolving as closed boxes, the observed evolution in gas mass is easy to explain. Although this model is outwardly not very different from the model of Peng et al. (2010), its assumptions about the galaxy population are very different. In our model, there is no longer a star-forming GMS and an area of ‘red-and-dead’ galaxies. Galaxies are still forming stars all the way down the Galaxy Sequence. The only physical distinction is between galaxies which are still being supplied by gas (the horizontal stub of galaxies in Figure 10) and the galaxies to which the gas supply has been shut off but which are still forming stars. Where’s the physics? -------------------- The heuristic model described in the last section was successful in reproducing in a qualitative, and at least in a semi-quantitative way some of the key properties of the galaxy population. What physical process could lie behind it? Any process must pass three tests. First, it must be a ‘weak quenching’ process, stopping any further supply of gas to the galaxy but not removing the gas that is already in the galaxy. Second, it must work in all environments, since the shape of the GS is similar in the field and in clusters (Eales et al. 2017; this paper; O17) and the rapid low-redshift evolution is also a feature of the overall galaxy population not just the galaxies in clusters[^5]. Third, it must have the stochastic element necessary to explain the different stellar mass functions of early- and late-type galaxies. Many processes suggested as quenching processes fail one or more of these tests. Ram-pressure stripping of gas by an intracluster medium (Gunn and Gott 1972) fails the first and second. Removal of the gas in a galaxy by wind or jet from an AGN or starburst - ‘feedback’ - fails the first. Note that we do not claim these processes are not occurring - there is plenty of evidence for feedback, for example (e.g. Cicone et al. 2014) - merely that they are not the processes responsible for switching off the future gas supply to galaxies. Galactic ‘strangulation’ or ‘starvation’ (Larson et al. 1980), in which the extended gaseous envelope around a galaxy is stripped away, is the kind of weak quenching process we require, since once the envelope is removed there will be no further inflow of gas onto the galaxy, but it was orginally suggested as a process that would take place in dense environments, thus failing test 2. In the remainder of this section, we describe one process that potentially passes all three tests, although at least part of this is speculation and we do not claim there are no other possibilities. The mass in the quenching equation (equation 19) is not a characteristic mass because changing the time unit, from Gyr to years for example, changes its value. However, we can derive a characteristic mass using the natural timescale for the growth of a galaxy, $SSFR^{-1}$. Rewriting equation 19 as the probability of quenching in this time period rather than per Gyr, the characteristic stellar mass is $\rm \simeq4 \times 10^8\ M_{\odot}$, $\rm 3 \times 10^9\ M_{\odot}$ and $\rm 9 \times 10^9\ M_{\odot}$ at $z=0$, $z=1$ and $z=2$, respectively. These correspond to masses of the surrounding dark-matter halos of $\rm \simeq 6 \times 10^{10}\ M_{\odot}$, $\rm 4\times10^{11}\ M_{\odot}$ and $\rm 8 \times 10^{11}\ M_{\odot}$, respectively (Moster, Naab and White 2013). Ideally, the physical process we require should pass both the three tests above and also explain these characteristic masses. Our speculative process was inspired by numerical simulations of the way gas cools onto galaxies. These simulations show that gas accretes on to galaxies in two ways: a ‘hot mode’ in which shocks are set up in the gas, heating it to the virial temperature, from which it slowly cools and is accreted by the galaxy; a ‘cold mode’ in which gas falls freely onto a galaxy along cold flows (Katz et al. 2003; Keres et al. 2005, 2009; Nelson et al. 2013). The simulations imply the transition halo mass below which gas can be supplied efficiently via cold flows is $\rm 10^{11}-10^{11.5}\ M_{\odot}$, fairly similar to the values we require. This process therefore provides a possible explanation of the characterisitic mass that we observe, although it is worth noting that (a) the transition from cold mode to hot mode may actually be rather gentle, occurring over a range of halo mass (Nelson et al. 2013), and (b) our attempt to use lensed [*Herschel*]{} sources to estimate the mass above which baryons cannot easily accrete on to galaxies gives a value $\simeq$10 times higher than the predictions of the numerical simulations (Amvrosiadis et al. 2017). The other requirement is stochasticity. Dekel and Birnboim (2006) show that cold flows can still sometimes penetrate to the central galaxy even in halos above the transition mass. Since the number of cold flows in an individual halo is small, we speculate that whether gas can be still supplied onto a galaxy in a halo above the transition mass may depend on the particular substructure in the individual halo, thus introducing the element of stochasticity that we need. Other evidence for a unitary population --------------------------------------- There is other recent evidence that instead of two separate classes of galaxies, there is actually a unitary population. The first evidence comes from the ATLAS$^{3D}$ survey of 260 ETGs. The survey has shown that 86% of ETGs have the velocity field expected for a rotating disk (Emsellem et al. 2011); for 92% of these ‘fast rotators’ there is also photometric evidence for a stellar disk (Krajnovic et al. 2013). Cappellari et al. (2013) used the ATLAS$^{3D}$ results to propose that there is a gradual change in galaxy properties from LTGs to ETGs rather than a dichotomy at the ETG/LTG boundary. The only exception are the 14% of the ETGs that are ‘slow-rotators’, for which there is generally (but not always) no photometric evidence for a stellar disk (Krajnovic et al. 2013). Very recently, Cortese et al. (2016), using integral-field spectroscopy from the SAMI survey, have shown that in terms of their kinematic properties LTGs and fast-rotator ETGs form a continuous class of objects. {width="74mm"} Other evidence comes from surveys of the ISM in ETGs. The best evidence for the existence of a cool ISM in many ETGs comes from the [*Herschel*]{} observations of the ETGs in the [*Herschel*]{} Reference Survey. Smith et al. (2012b) detected dust emission from 50% of the HRS ETGs; [*Herschel*]{} observations of a much larger sample of ETGs drawn from the ATLAS$^{3D}$ survey have detected a similar percentage (Smith et al. in preparation). The big increase in detection rate from ground-based CO observations of ETGs (e.g. Young et al. 2011) to [*Herschel*]{} observations suggests that the common assumption that ETGs do not contain a cool ISM is largely a function of instrumental sensitivity - if we had more sensitive instruments we would find an even higher fraction with a cool ISM. Interferometric observations of the molecular gas in ETGs generally show a rotating disk similar to what is seen in LTGs (Davis et al. 2013). Thus these recent results are consistent with our argument that there is a single population of galaxies (with the possible exception of the slow-rotating ETGs - §5.6). In our heuristic model (§5.3), the dust and gas detected in ETGs is the residual ISM left at the end of their evolution down the GS - these galaxies are ones in which the gas supply was shut off early and have been evolving as closed boxes ever since. A common counter-argument used against this idea is that the kinematics of the gas in ETGs are characteristic of gas acquired as the result of recent mergers. However, the fact that very few ETGs have a gas rotation axis pointing in the opposite direction to the stellar rotation axis, and in the vast majority of the cases the two rotation axes are broadly in the same direction, is strong evidence that most of the gas is this residual ISM (see Eales et al. (2017) for this argument in more detail). Steps to a new paradigm ----------------------- The main argument we have tried to make in this paper and a previous paper (Eales et al. 2017) is that the galaxyscale revealed by [*Herschel*]{} does not look much like the predictions of the current paradigm, in which there are two distinct classes of galaxy and a violent quenching process moves a galaxy from one class to the other. The curved Galaxy Sequence shown in Figure 1, with the gradual change in galaxy morphology along it, seems to us to require a gentler quenching process. The one we have suggested in Section 5.3 is the stochastic removal of the gas supply (a possible physical process is suggested in §5.4). If we are correct, the motion of galaxies along the Galaxy Sequence, as well as the rapid low-redshift evolution, is caused, not by the actual removal of the gas supply, but by the gradual evolution that occurs once the gas is turned off. A number of other groups have also recently investigated quenching processes in the galaxy population. Cassado et al. (2015) found evidence for a weak quenching process in low-density environments but for strong quenching in clusters, whereas Schawinski et al. (2014) found evidence for weak quenching in late-type galaxies but strong quenching in early-type galaxies. Peng, Maiolino and Cochrane (2015) used the metallicity distributions of star-forming and passive galaxies to argue that the evolution from one population to the other must have occurred over $\simeq$4 billion years - evidence for weak quenching. O17 reached a similar conclusion to us about the distribution of galaxies in the plot of SSFR versus stellar mass, but reached a slightly different conclusion about its significance. They concluded that there are actually three types of galaxy: star-forming galaxies, genuinely passive galaxies with no ISM, and an intermediate class in which galaxies are still forming stars but with a lower efficiency than in the first class (essentially our optically-red class). The gradual change in morphology along the GS seen in Figure 1 seems to us evidence against this idea. Like us, however, O17 concluded that the evolution from the star-forming class to the intermediate class occurs as the result of the consumption of gas by star formation. Very recently, Bremer et al. (in preparation) have looked at the SEDs and structures of galaxies in the intermediate region of the SSFR-versus-stellar-mass plot, concluding like us that the evolution accross this region is the result of the gradual consumption of gas by the formation of stars in the disks of galaxies. Although there is accumulating evidence for this gentler version of galaxy evolution, there are some unanswered questions. First, is there a subset of ETGs that are genuinely passive galaxies formed by some violent evolutionary process? One possible class are the 14% of ETGs that are slow rotators (§5.5). One way to test this possibility would be to measure the star-formation rate and map the ISM in [*all*]{} ETGs, in order to look for the residual star formation and disks that should be present if the only evolutionary process is the weak quenching process we have proposed. Unfortunately, the required observations get extremely challenging for galaxies with $log_{10}(SSFR) < -11.5$. The second bigger question, for which there is also not a definitive answer in the current paradigm, is what causes the morphological variation along the GS seen in Figure 1? We don’t have an answer but a possible clue is that the GS is also a sequence of time: the epoch during which most of the galaxy’s stars formed moves earlier as we move down the GS. One idea for forming a galaxy’s bulge is that it formed as the result of the rapid motion of star-forming clumps towards the galaxy’s centre (Noguchi et al. 1999; Bournaud et al. 2007; Genzel et al. 2011, 2014). If this process worked better at earlier times, this might explain the morphological change along the GS. Summary ======= The [*Herschel*]{} surveys gave a radically different view of galaxy evolution from optical surveys. We first list our basic observational results. 1. Rather than a star-forming Galaxy Main Sequence (GMS) and a separate region of red-and-dead galaxies, two [*Herschel*]{} surveys, selected in completely different ways, reveal a single, curved Galaxy Sequence (GS) extending to low values of specific star-formation rate (Figures 1 and 3). The curvature of the GS confirms other recent results (Whitaker et al. 2014; Gavazzi et al. 2015; Lee et al. 2015; Schreiber et al. 2016; Tomczak et al. 2016; O17) and we show in §5.5 that there is plenty of evidence from other wavebands for this picture of a unitary galaxy population rather than two populations. 2. The star-formation efficiency (star-formation rate divided by gas mass) falls as one moves down the GS (§4.4). 3. Galaxy morphology gradually changes as one moves down the GS rather than there being a jump from early-type to late-type galaxies (Fig. 1) 4. 15-30% of the galaxies in the far-infrared [*Herschel*]{} ATLAS would have been classified as passive galaxies based on their optical colours. We use stacked spectra to show that these red colours are caused by an old stellar population rather than reddening by dust. Nevertheless, these galaxies still contain significant reservoirs of interstellar gas, and the stacked spectra confirm that they are still forming stars. They are red but not dead. After correcting for Malmquist bias, we find that the space-density of optically-red star-forming galaxies, which are missed by optical studies, is at least as high as the space-density of optically-blue star-forming galaxies of the same stellar mass. 5. The 250-$\mu$m luminosity function of both optically-blue and optically-red galaxies shows rapid evolution at low redshift, with the evolution appearing stronger for the red galaxies (§4.3). 6. We use stacked optical spectra of the H-ATLAS galaxies at $0.3 < z < 0.4$ to show that the galaxies responsible for this rapid evolution have a significant spectral break at 4000 Å explaining the red optical colours, but also bright emission lines and a $UV$ upturn, implying a high star-formation rate. These galaxies are red but very lively. It seems likely that these galaxies, which have high stellar masses, high star-formation rates and, even a few billion years in the past, an old stellar population, are the relatively recent ancestors of the ETGs in the Universe today. We have explored whether existing galaxy-evolution models can explain these results. Our concluson that galaxies are a unitary population lying on a single GS removes the need for a rapid quenching process. We show that the popular gas regulator, or bathtub, model cannot explain the curved GS and the strong low-redshift evolution, and the EAGLE numerical galaxy simulation does not reproduce the strong low-redshift evolution. We propose an alternative model in which the gas supply to galaxies is stochastically cut off (the flaky faucet model). We show that this model provides a natural explanation of the curved GS, while still explaining earlier results such as the difference between the stellar mass functions of optically-red and optically-blue galaxies. The model naturally explains the rapid low-redshift evolution because most massive galaxies are no longer being supplied by gas and are now evolving as closed boxes. Acknowledgments {#acknowledgments .unnumbered} =============== We are grateful to the many scientists who have contributed to the success of the [*Herschel*]{} ATLAS and the [*Herschel*]{} Reference Survey. We thank Antonio Kitsianis for a useful e-mail interchange about his paper on the EAGLE predictions and for supplying a digital version of one of his figures. We also thank an anonymous referee for comments that significantly improved the paper, in particular its readability outside the far-infrared club. EV and SAE acknowledge funding from the UK Science and Technology Facilities Council consolidated grant ST/K000926/1. MS and SAE have received funding from the European Union Seventh Framework Programme (\[FP7/2007-2013\] \[FP7/2007-2011\]) under grant agreement No. 607254. PC, LD and SM acknowledge support from the European Research Council (ERC) in the form of Consolidator Grant [CosmicDust]{} (ERC-2014-CoG-647939, PI HLGomez). SJM LD and RJI acknowledge support from the ERC in the form of the Advanced Investi- gator Program, COSMICISM (ERC-2012-ADG 20120216, PI R.J.Ivison). GDZ acknowledges financial support from ASI/INAF agreement n.2014-024-R.0. M.J.M. acknowledges the support of the National Science Centre, Poland through the POLONEZ grant 2015/19/P/ST9/04010; this project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk[ł]{}odowska-Curie grant agreement No. 665778. [99]{} Abdo, A. et al. 2010, 710, 133 Agius, N. et al. 2013, MNRAS, 431, 1929 Amvrosiadis, A. et al. 2017, MNRAS, submitted Baldry, I.K. et al. (2012), MNRAS, 441, 2440 Bauer, A. et al. 2013, MNRAS, 434, 209 Bell, E.F. et al. 2004, ApJ, 608, 752 Bell, E.F. and de Jong, R.S. 2001, ApJ, 550, 212 Bendo, G.J. et al. 2010, MNRAS, 402, 1409 Blanton, M.R. et al. (2003), AJ, 125, 2348 Blanton, M.R. and Roweis, S. 2007, AJ, 133, 734 Boselli, A. & Gavazz, G. 2006, PASP, 118, 517 Boselli, A. et al. 2010, PASP, 122, 261 Bournaud, F., Elmgreen, B.G. & Elmgreen, D.M. 2007, ApJ, 670, 237 Bouché, N. et al. 2010, ApJ, 718, 1001 Bourne, N. et al. 2012, MNRAS, 421, 3027 Bourne, N. et al. 2016, MNRAS, 462, 1714 Bruzual, G. & Charlot, S. 2003, MNRAS, 344, 1000 Butcher, H. & Oemler, A. 1978, ApJ, 219, 18 Cappellari, M. et al. 2013, MNRAS, 432, 1862 Casado, J., Acasibar, Y., Gavilan, M., Terlevich, R., Terlevich, E., Hoyos, C. & Diaz, A.I. 2015, MNRAS, 451, 888 Chabrier, G. 2003, PASP, 115, 763 Charlot, S. & Fall, S.M. 2000, ApJ, 539, 718 Cicone, C. et al. 2014, A&A, 562, 21 Ciesla, L. et al. 2012, A&A, 543, 161 Cortese, L. et al. 2012, A&A, 540, 52 Cortese, L. et al. 2014, MNRAS, 440, 942 Cortese, L. et al. 2016, MNRAS, 463, 170 Da Cunha, E., Charlot, S. and Elbaz, D. 2008, MNRAS, 388, 1595 Daddi, E. et al. 2007, ApJ, 670, 156 Dariush, A. et al. 2011, MNRAS, 418, 64 Dariush, A. et al. 2016, MNRAS, 456, 2221 Davies, L.J.M. et al. 2016, MNRAS, 461, 458 Davis, T.A. et al. 2013, MNRAS, 429, 534 De Vis, P. et al. 2017, MNRAS, 464, 4680 Dekel, A. and Birnboim, Y. 2006, MNRAS, 368, 2 Draine, B.T. et al. 2007, ApJ, 663, 866 Driver, S. et al. 2009, Astron. Geophys., 50, 12 Driver, S. et al. 2012, ApJ, 827, 108 Dunne, L. et al. 2011, MNRAS, 417, 1510 Dye, S. et al. 2010, AA, 518, L10 Eales, S. et al. 2009, ApJ, 707, 1779 Eales, S. et al. 2010, PASP, 122, 499 Eales, S. et al. 2012, ApJ, 761, 168 Eales, S.A. et al. 2015, MNRAS, 452, 3489 Eales, S.A. et al. 2017, MNRAS, 465, 3125 Edge, A., Sutherland, W., Kuijken, K., Driver. S., McMahon, R., Eales, S. & Emerson, J.P. 2013, The Messenger, 154, 32 Elbaz, D. et al. 2007, A&A, 468, 33 Emsellem, E. et al. 2011, MNRAS, 414, 888 Faucher-Giguère, C.A., Keres, D. & Ma, , C.-P. 2011, MNRAS, 417, 2982 Gavazzi, G. et al. 2015, A&A, 580, 116 Genzel, R. et al. 2011, ApJ, 733, 30 Genzel, R. et al. 2014, ApJ, 785, 75 Genzel, R. et al. 2015, ApJ, 800, 20 Groves, B. et al. 2015, ApJ, 799, 96 Gunn, J.E. & Gott, J.R. 1972, ApJ, 176, 1 Hardcastle, M. et al. 2016, MNRAS, 462, 1910 Hayward, C. and Smith, D.J.B. 2015, MNRAS, 446, 1512 Hopkins, A. et al. 2013, MNRAS, 430, 2047 Hughes, T.M., Cortese, L., Boselli, A., Gavazzi, G. & Davies, J.I. 2013, A&A, 550, 115 Ilbert, O. et al. 2013, A&A, 556, 545 James, A., Dunne, L., Eales, S. and Edmunds, M. 2002, MNRAS, 335, 753 Katsianis, A. et al. 2017, MNRAS, in press (arXiv: 1708.01913) Katz, N., Keres, D., Davé, R. & Weinberg, D.H. 2003 in J.L. Rosenberg & M.E. Putman ed., The IGM/Galaxy Connection. The Distribution of Baryons at $z=0$, Vol. 281 of Astrophysics and Space Science Library, How Do Galaxies Get Their Gas? p185 Kennicutt, R.C. 1998, ARAA, 36, 189 Kennicutt, R.C. & Evans, N.J. 2012, ARAA, 50, 531 Keres, D., Katz, N., Weinberg, D.H. & Davé, R. 2005, MNRAS, 363, 2 Keres, N. & Hernquist, L. 2009, ApJ, 700, L1 Krajnovic, D. et al. 2013, MNRAS, 432, 1768 Larson, R.B., Tinsley, B.M. & Caldwell, C.N. 1980, ApJ 237, 692 Lee, N. et al. 2015, ApJ, 801, 80 Lilly, S.J., Carollo, M.C., Pipino, A., Renzini, A. & Peng, Y. 2013, ApJ, 772, 119 Liske, J. et al. 2015, MNRAS, 452, 2087 Loveday, J. et al. (2012), MNRAS, 420, 1239 Marchetti, L. et al. 2016, MNRAS, 456, 1999 Martig, M., Bournaud, F., Teyssier, R. & Dekel, A. 2009, ApJ, 707, 250 Moster, B.J., Naab, T. and White, S.D.M. 2013, MNRAS, 428, 3121 Nelson, D., Vogelsberger, M., Genel, S., Sijacki, D., Keres, D., Springel, V. & Hernquist, L. 2013, MNRAS, 429, 3353 Noguchi, M. 1999, ApJ, 514, 77 Noeske, K.G. et al. 2007, ApJ, 660, L43 Oemler, A., Abramson, L.E., Gladders, M.D., Dressler, A., Poggianti, B.M. & Vulcani, B. 2017, ApJ, 844, 45 (O17) Page, M. & Carerra, F. 2000, MNRAS, 311, 433 Papadopoulos, P. and Geach, J.E. 2012, ApJ, 757, 157 Peng, Y.-J. et al. 2010, ApJ, 721, 193 Peng, Y.-J. and Maiolino, R. 2014, MNRAS, 443, 3643 Peng, Y., Maiolino, R. & Cochrane, R. 2015, Nature, 521, 192 Pilbratt, G. et al. 2010, A&A, 518, L1 Pineda, J.L. et al. 2013, A&A, 554, 103 Planck Collaboration 2011, A&A, 536, 19 Planck Collaboration 2014, A&A, 571, 26 Rémy-Ruyer, A. et al. 2014, A&A, 563, 31 Renzini, A. & Peng, Y. 2015, ApJ, 801, L29 Rodighiero, G. et al. 2011, ApJ, 739, 40 Rowlands, K. et al. 2012, MNRAS, 419, 2545 Rowlands, K. et al. 2014, MNRAS, 441, 1017 Saintonge, A. et al. 2012, 758, 73 Sandstrom, K. et al. 2013, ApJ, 777, 5 Santini, P. et al. 2014, A&A, 562, 30 Schawinski, K. et al. 2014, MNRAS, 440, 889 Schreiber, C. et al. 2016, A&A, 589, 35 Scoville, N.Z. et al. 2014, 2014, ApJ, 783, 84 Scoville, N. et al. 2016, ApJ, 820, 83 Speagle, J.S., Steinhardt, C.L., Capak, P.L. & Silverman, J.D. 2014, ApJS, 214, 15 Smith, D.J.B. et al. 2011, MNRAS, 416, 857 Smith, D.J.B. et al. 2012a, MNRAS, 427, 703 Smith, M.W.L. et al. 2012b, ApJ, 748, 123 Smith, M.W.L. et al. 2012c, ApJ, 756, 40 Smith, M.W.L. et al. in preparation Tacconi, L. et al. 2010, Nature, 463, 781 Taylor, E. et al. 2011, MNRAS, 418, 1587 Thomas, D., Maraston, C., Bender, R. and Mendes de Oliveira, C. 2005, ApJ, 621, 673 Toomre, A. 1977, [*Evolution of Galaxies and Stellar Populations, Proceedings of a Conference at Yale University*]{}, edited by B.M. Tinsley and R.B. Larson (Yale University Observatory, New Haven, Conn.), 401 Tomczak, A. et al. 2016, ApJ, 817, 118 Tremonti, C. et al. 2004, ApJ, 613, 898 Wang, L. et al. 2016, MNRAS, submitted Wang, L. et al. 2016a, A&A, 592, L5 Wang, L. et al. 2016b, MNRAS, 461, 1719 Whitaker, K. et al. 2012, ApJ, 754, 29 Whitaker, K. et al. 2014, ApJ, 795, 104 Valiante, E. et al. 2016, MNRAS, 562, 3146 Young, L.M. et al. 2011, MNRAS, 414, 940 Zibetti, S., Charlot, S. & Rix, H.-W. 2009, MNRAS, 400, 1181 \[lastpage\] [^1]: E-mail: sae@astro.cf.ac.uk [^2]: This dataset can be obtained from h-atlas.org [^3]: No more biased, of course, than an optical survey. [^4]: We assume that the gas-to-dust ratio does not evolve significantly. [^5]: The Butcher-Oemler effect in clusters may have been an early example of this rapid low-redshift evolution (Butcher and Oemler 1978).

--- abstract: 'We consider the leptonic asymmetry generation in the $\nu MSM$ via hadronic decays of sterile neutrinos at $T\ll T_{EW}$, when the masses of two heavier sterile neutrinos are between $m_\pi$ and 2 GeV. The choice of upper mass bound is motivated by absence of direct experimental searches for singlet fermions with greater mass. We carried out computations at zero temperature and ignored the background effects. Combining constraints of sufficient value of the leptonic asymmetry for production of dark matter particles, condition for sterile neutrino to be out of thermal equilibrium and existing experimental data we conclude that it can be satisfied only for mass of heavier sterile neutrino in the range $1.4$ GeV $\lesssim M <2$ Gev and only for the case of normal hierarchy for active neutrino mass.' author: - | Volodymyr M. Gorkavenko[^1], Igor Rudenok[^2],\ and Stanislav I. Vilchynskiy[^3]\ *Department of Physics, Taras Shevchenko National University of Kyiv,\ *64 Volodymyrs’ka St., Kyiv, 01601, Ukraine** title: 'Leptonic asymmetry of the sterile neutrino hadronic decays in the $\nu MSM$ ' --- Introduction {#intro} ============ The Standard Model (SM) is minimal relativistic field theory, which is able to explain almost all particle physics experimental data [@SM]. However, there are several observable facts, that cannot be explained in the SM frame. Firstly, the neutrinos of SM are strictly massless, that contradict to the experimental fact of the neutrinos oscillations [@PG; @Strumia]. The second problem is the impossibility to explain the baryon asymmetry of the Universe (BAU) within the SM. Finally, the SM does not provide the dark matter (DM) candidate. Also the SM can not solve the strong CP problem in particle physics, the primordial perturbations problem and the horizon problem in cosmology, etc. The solutions of the above mentioned problems of the SM require some new physics between the electroweak and the Planck scales. An important challenge for the theoretical physics is to see if it is possible to solve them using only the extensions of the SM below the electroweak scale [@last]. The Neutrino Minimal Standard Model ($\nu MSM$) is an extension of the SM by three massive right-handed neutrinos (sterile neutrinos), which do not take part in the gauge interactions of the SM [^4]. The model was suggested by M. Shaposhnikov and T. Asaka [@Shap1; @Shap2]. The masses of sterile neutrinos are predicted to be smaller than electroweak scale, and thus there is no new energy scale introduced in the theory. The parameters of the $\nu MSM$ can be chosen in order to explain simultaneously the masses of active neutrinos, the nature of DM, and BAU. The lightest sterile neutrino (the mass is expected to be in the KeV range [@last]) can be intensively produced in the early Universe and have cosmologically long life-time. So, it might be a viable DM candidate. The sufficient amount of this neutrinos can be generated through an efficient resonant mechanism proposed by Shi and Fuller [@Shi]. In the $\nu MSM$ the required amount of the leptonic asymmetry (in accordance with Shi and Fuller mechanism) can be created due to decays of the two heavier sterile neutrinos. This particles are generated at temperature $T> T_{EW}$ and their masses are expected to be in range $m_\pi<M_I<T_{EW}$ [@Sh3], where $m_\pi$ is the pion mass and $M_I$ is the mass of $I$-sterile neutrino. The leptonic asymmetry at the temperature of the sphaleron freeze-out ($T\sim T_{EW}$) is related to the baryon asymmetry of the Universe. At temperature $T< T_{EW}$ the leptonic asymmetry from decays of heavier sterile neutrinos can not convert into the baryon asymmetry and is accumulated. As it was shown in [@last; @Sh1] the required amount of the leptonic asymmetry $\Delta=\Delta L/L=(n_L-n_{\bar L})/(n_L+n_{\bar L})$: $$\label{asymmetry} 10^{-3}<\Delta<2/11$$ has to already exist in the Universe at the moment of the beginning of production of the DM particles (takes place at the temperature around 0.1 GeV). We consider here the leptonic asymmetry generation at $T\ll T_{EW}$, when the masses of two heavier sterile neutrinos are between $m_\pi$ and 2 GeV. The motivation is following. The mass of heavier sterile neutrino can not be less then $m_\pi$ (the constraint is coming from accelerator experiments combined with Big Bang Nucleosynthesis (BBN) bounds [@Gorbunov; @Kusenko]) and there is no direct experimental searches for singlet fermions with mass more then 2 Gev [@Gorbunov]. Since the masses of active neutrinos in the $\nu MSM$ are produced by the “see-saw” mechanism [@2] some constraints on the parameters of the $\nu MSM$ come from active neutrino parameters that can be found from the experiments on the neutrino oscillations. Namely, these are the mass squared differences of active neutrinos and the mixing angles. Until recently the mixing angle $\theta_{13}$ was supposed to have a close to zero value. But new observations indicated its essential difference from zero [@T2Ktheta13]. The aim of this work is to obtain constraints on the parameters of the $\nu MSM$ from the required amount of the leptonic asymmetry and cosmology conditions. Also we want to investigate the influence of non-zero mixing angle $\theta_{13}$ on space of the allowed parameters of the $\nu MSM$. We do it following [@Tibor] using a simple model: we ignore the background effects and do computations at zero temperature. The paper is organized as follows. In Section \[rozdil2\] we present the Lagrangian of the $\nu MSM$, make its convenient parametrization and present the Yukawa couplings in terms of active neutrinos mass matrix parameters. In Section \[rozdil3\] we derive the expression for the leptonic asymmetry. The limitations on the $\nu MSM$ parameters are imposed in Section \[rozdil4\]. Section \[rozdil5\] is devoted to the analysis and conclusions. Basic formalism of the $\nu MSM$ {#rozdil2} ================================ In the $\nu MSM$ [@Shap1; @Shap2] the following terms are added to the Lagrangian of the SM (without taking into account the kinetic terms): $$\label{lagdiag0} \mathcal L^{ad}=-F_{\alpha I}\bar L_\alpha \tilde\Phi \nu_{IR}-\frac{M_{IJ}}{2}\bar \nu_{IR}^c\nu_{JR}+h.c.,$$ where index $\alpha=e,\mu,\tau$ corresponds to the active neutrino flavors, indices $I,J$ run from $1$ to $3$, $L_{\alpha}$ is for the lepton doublet of the left-handed particles, $\nu_{IR}$ is for the field functions of the sterile right-handed neutrinos, the superscript $\raisebox{-0.8em}{$"$}\!c"$ means charge conjugation, $F_{\alpha I}$ is for the new (neutrino) matrix of the Yukawa constants, $M_{IJ}$ is for the Majorana mass matrix of the right-handed neutrinos, $\Phi$ is for the field of the Higgs doublet, ${\tilde\Phi}=i\sigma_2\Phi^*$. After the spontaneous symmetry breaking the field of the Higgs doublet in unitary gauge is $$\Phi=\left(\begin{array}{c}0\\ \frac{v+h}{\sqrt{2}}\end{array}\right),$$ where $h$ is the neutral Higgs field and the parameter $v$ determines minimum of the Higgs field potential $(v\cong247\,\,\mbox{GeV})$. In this case Lagrangian acquires the Dirac-Majorana neutrino mass terms: $$\label{2} \mathcal L^{DM}=-\frac{v}{\sqrt{2}}F_{\alpha I}\bar{\nu}_\alpha\nu_{IR}-\frac{M_{IJ}}{2}\bar \nu_{IR}^c\nu_{JR}+h.c.,$$ or in conventional form [@Belenki] $$\label{dm1} \mathcal L^{DM}=-\left(\overline{(N_L)^c} \,\frac{M^{DM}}2\,N_L+h.c.\right),$$ where $$\label{dm2} N_L=\left(% \begin{array}{c} \nu_L \\ \nu_R^c \\ \end{array} \right)\!;\,\, N_L^c=\left( \begin{array}{c} \nu_L^c \\ \nu_R \\ \end{array} \right)\!;\,\, M^{DM}=\left(% \begin{array}{cc} M_L&M_D{}^T \\ M_D&M_R \\ \end{array} \right)\!$$ and $$\label{dm1a} M_L=0,\quad M_D=F^+\frac{v}{\sqrt2}\,,\quad M_R=M^*,$$ where $M,F$ are square matrix of the third order with elements $F_{\alpha I}$ and $M_{IJ}$. In zero approximation the $\nu MSM$ Lagrangian is assumed to be invariant under $U(1)_e\times U(1)_\mu\times U(1)_\tau$ transformations, that provides preservation of the $e, \mu, \tau$ lepton numbers separately. It is also assumed that two heavier sterile neutrinos interact with the active neutrinos, but the third (lightest) sterile neutrino does not interact[^5]. This assumption can be realized by following matrix $M^{DM}$[@sb]: $$M_R^{(0)}=\left(\begin{array}{ccc}0&0&0\\0&0&M\\0&M&0\end{array}\right),\qquad M_D^{(0)+}=\frac{v}{\sqrt{2}}\left(\begin{array}{ccc}0&h_{12}&0\\0&h_{22}&0\\0&h_{32}&0\end{array}\right),\qquad M_L^{(0)}=0$$ In this approximation we have two massive sterile neutrinos with equal mass $M$, the third neutrino is massless, and all active neutrinos have zero mass. It contradicts observable data [@PG; @Strumia]. To adjust it next small terms are added to the matrix $M^{DM}$ [@sb]: $$\begin{gathered} M_R^{(1)}=\Delta M=\left(\begin{array}{ccc}m_{11}e^{-i\alpha}&m_{12}&m_{13}\\m_{12}&m_{22}e^{-i\beta}&0\\m_{13}&0&m_{33}e^{-i\gamma}\end{array}\right),\\ M_D^{(1)+}=\frac{v}{\sqrt{2}}\left(\begin{array}{ccc}h_{11}&0&h_{13}\\h_{21}&0&h_{23}\\h_{31}&0&h_{33}\end{array}\right), M_L^{(1)}=0\end{gathered}$$ This correction violates $U(1)_e\times U(1)_\mu\times U(1)_\tau$ symmetry, leads to the appearance of the mass of the third sterile neutrino and takes off the mass degeneracy for two heavier sterile neutrinos. It’s also leads to the appearance of the extra small masses of the active neutrinos and nonzero mixing angles among them. In the terms of the introduced corrections Lagrangian is $$\label{nongiag1} \mathcal{L}^{ad}=-h_{\alpha I}\bar{L}_\alpha\tilde{N_I}\tilde{\Phi}-M\bar{\tilde{N_2^c}}\tilde{N_3}-\frac{\Delta M_{IJ}}{2}\bar{\tilde{N_I^c}}\tilde{N_J}+h.c.,$$ where $\tilde{N_I}$ are right-handed neutrinos in the gauge basis. In order to find the masses of the active neutrino one has to make the diagonalization of the matrix $M^{DM}$. The diagonalization undergoes in two steps. Firstly, $M^{DM}$ matrix is reduced to the block-diagonal form via the unitary transformation [@see-saw] in the “see-saw” approach: $$M_{block}=W^TM^{DM}W=\left(\begin{array}{cc}-(M_D)^T (M_R)^{-1} M_D&0\\0&M_R\end{array}\right)=\left(\begin{array}{cc}M_{light}&0\\0&M_{heavy}\end{array}\right),$$ where $$\label{ss3} W=\left( \begin{array}{cc} 1-\frac{1}{2}\varepsilon^+\varepsilon & \varepsilon^+ \\ -\varepsilon & 1-\frac{1}{2}\varepsilon\varepsilon^+ \\ \end{array}% \right),\quad \varepsilon=M_R^{-1}M_D\ll1$$ and $M_{light}=-(M_D)^T (M_R)^{-1} M_D$, $M_{heavy}=M_R$ are the mass matrix of the active and sterile neutrinos respectively. Now each block of the matrix $M^{DM}$ may be diagonalized independently by the matrix $$\label{ss4a} U=\left(% \begin{array}{cc} U_1 & 0 \\ 0 & U_2 \\ \end{array}% \right).$$ The mass matrix of the active and sterile neutrinos is diagonalized by unitary transformation $U_{1(2)}$: $$U_1^TM_{light}U_1=diag(m_1,m_2,m_3),\quad U_2^TM_{heavy}U_2=diag(M_1,M_2,M_3).$$ There is a standard parametrization [@PG] for $U_{1(2)}$: $$\begin{gathered} \label{ss8} U_{1(2)}=\left(\!\! \begin{array}{ccc} c_{12}c_{13} & c_{13}s_{12} & s_{13}e^{-i \delta} \\ -s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i \delta} & c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i \delta} & s_{23}c_{13} \\ s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i \delta} & -c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i \delta} & c_{23}c_{13} \\ \end{array}% \!\!\right)\times\\\times\left(\!\! \begin{array}{ccc} e^{i \alpha_1 /2} & 0 & 0 \\ 0 & e^{i \alpha_2 /2} & 0 \\ 0 & 0 & 1 \\ \end{array}% \!\!\right)\!\!,\end{gathered}$$ where $c_{ij}=\cos \theta_{ij}$, $s_{ij}=\sin \theta_{ij}$, $\theta_{12},\theta_{13},\theta_{23}$ are the three mixing angles; $\delta$ is the Dirac phase, and $\alpha_1, \alpha_2$ are the Majorana phases. The angles $\theta_{ij}$ can be in the region $0\leq\theta_{ij}\leq\pi/2$, phases $\delta,\alpha_1, \alpha_2$ vary from $0$ to $2\pi$. Each of the matrices $U_{1}$ and $U_{2}$ contains its own, independent angles and phases. Then the elements of the $M_{light}$ can be defined by masses and elements of mixing matrix $U$ of the active neutrinos: $$[M_{light}]_{\alpha\beta}=m_1U^*_{\alpha1}U^*_{\beta1}+m_2U^*_{\alpha2}U^*_{\beta2}+m_3U^*_{\alpha3}U^*_{\beta3}.$$ The data that come from the neutrino oscillation experiments are presented in Tab.1: ----------------------------------------------------------------------------------------------- central value 99$\%$ confidence interval -------------------------------------------------------------- -------------------------------- $\Delta m^2_{21} = (7.58\pm0.21)\cdot10^{-5}\,eV^2$ $(7.1 - 8.1)\cdot10^{-5} eV^2$ $|\Delta m^2_{23}| = (2.40 \pm 0.15) \cdot 10^{-3} eV^2$ $(2.1 - 2.8) 10^{-3} eV^2$ $tan^2 \theta_{12} = 0.484 \pm 0.048$ $31^0 < \theta_{12} < 39^0$ $sin^2 2\theta_{23} = 1.02 \pm 0.04$ $ 37^0 < \theta_{23} < 53^0$ $^*\quad$ $\sin^2 2\theta_{13} = 0.11$ ($ \theta_{13}=10^0$) ----------------------------------------------------------------------------------------------- \[Tab\] [Table 1. Experimental constraints on the parameters of active neutrinos [@Strumia], $^*$ — results of T2K Collaboration [@T2Ktheta13]: $0.03<\sin^2 2\theta_{13}<0.28$ in the case of the normal hierarchy and $0.04<\sin^2 2\theta_{13}<0.34$ in the case of the inverted hierarchy.]{} On the other hand, from the “see-saw” formula (in the approximation when the elements of the first column of the Yukawa matrix are neglected and $M\gg m_{ij}$) one can immediately obtain, that the mass of the lightest sterile neutrino is zero and the mass matrix of the active neutrinos has the form [@sb] $$\label{system} [M_{light}]_{\alpha\beta}=-\frac{v^2}{2M}(h_{\alpha 2}h_{\beta 3}+h_{\alpha 3}h_{\beta 2}),$$ and its eigenvalues is $$\label{m2m3} m_a=0,\quad m_{\tiny \left(\!\!\!\begin{array}{c} b\\c\end{array}\!\!\!\right)}=\frac{v^2(F_2F_3\mp |h^+ h|_{23})}{2M},$$ where $F_I^2=(h^+ h)_{II}$, $m_a$ is the mass of the lightest active neutrino, $m_c$ is the mass of the heaviest active neutrino. The sum over the neutrino masses is given by $$\label{sum} \frac{v^2F_2F_3}{M}=\sum_{i=1}^3m_{i}.$$ The system has infinite number of solutions. Indeed, the replacement $h_{\alpha2}\rightarrow zh_{\alpha2}$, $h_{\alpha3}\rightarrow h_{\alpha3}/{z}$ ($z$ is an arbitrary complex number) does not change the system. Then one can define the real quantity $\varepsilon$ $$\label{epsilondef} \varepsilon={F_3}/{F_2},\quad\varepsilon=|z|.$$ as an independent parameter of the model. As it was shown in [@GorkVil], the system has good solutions for ratios of the elements of second column of the Yukawa matrix: $$\label{solution} \left\{\begin{array}{c} \vspace{0.5em}A_{12}=\displaystyle{\frac{M_{12}}{M_{22}}\left(1\pm\sqrt{ 1-\frac{M_{11}M_{22}}{M_{12}{}^{2}} } \right)}\\ \vspace{0.5em}A_{13}=\displaystyle{\frac{M_{13}}{M_{33}}\left(1\pm\sqrt{1-\frac{M_{11}M_{33}}{M_{13}{}^{2}} } \right)}\\ A_{23}=\displaystyle{\frac{M_{23}}{M_{33}}\left(1\pm\sqrt{1-\frac{M_{22}M_{33}}{M_{23}{}^{2}}}\right)} \end{array}\right.$$ where $A_{12}={h_{12}}/{h_{22}}$, $A_{13}={h_{12}}/{h_{32}}$, $A_{23}={h_{22}}/{h_{32}}$ and $M_{IJ}$ is elements of $M_{light}$ matrix. The ratios of the third column elements of the Yukawa matrix are expressed through the $A_{ij}$ elements: $$\label{dop} \frac{h_{23}}{h_{13}}=A_{12}\frac{M_{22}}{M_{11}};\quad \frac{h_{33}}{h_{13}}=A_{13}\frac{M_{33}}{M_{11}}.$$ Though formally there are eight different choices for the solutions , only four are independent. For example, if we fix the sign before the square roots in the expressions for $A_{12}$ and $A_{13}$ then $A_{23}$ is unambiguously determined by the relation $$\label{a10} A_{23}=A_{13}/A_{12}.$$ The solutions allow one to find the ratios of the elements of Yukawa matrix [@GorkVil]: $$\label{haI1} \frac{\left(h_{12};h_{22};h_{32}\right)}{F_2}=\frac{e^{iarg(h_{12})}}{\sqrt{1+|A_{12}|^{-2}+|A_{13}|^{-2}}}\left(1;A_{12}^{-1};A_{13}^{-1}\right)$$ $$\label{haI2} \frac{\left(h_{13};h_{23};h_{33}\right)}{F_3}=\frac{e^{iarg(h_{13})}}{\sqrt{1+|A_{12}\frac{M_{22}}{M_{11}}|^{2}+|A_{13}\frac{M_{33}}{M_{11}}|^{2}}}\left(1;A_{12}\frac{M_{22}}{M_{11}};A_{13}\frac{M_{33}}{M_{11}}\right),$$ where phases of $h_{12}$, $ h_{13}$ are connected by condition $$arg(h_{12})+arg( h_{13})=arg(M_{11}).$$ This is the exact solution of that definitely expresses ratio of the elements of the Yukawa matrix via parameters of the active neutrino mass matrix. For fixed values of the active neutrino parameters there are only two choices for placing of the signs in the expressions for $A_{12},A_{13},A_{23}$ which are not inconsistent with condition . These two variants are distinguished from each other by simultaneous replacement of the sign in front of square roots in the expressions for $A_{12},A_{13}, A_{23}$. It can be shown that such replacement of the signs leads to interchanging and conjugating of the ratios of elements of the second and the third columns of the Yukawa matrix, notably $h_{22}/h_{12}\leftrightarrow h^*_{23}/h^*_{13}$, $h_{32}/h_{12}\leftrightarrow h^*_{33}/h^*_{13}$ [@GorkVil]. As it was announced in Introduction, only the two heavier sterile neutrino take part in the production of the leptonic asymmetry. Therefore we will exclude the lightest sterile neutrino from consideration, so hereinafter indexes $I,J$ take the value 2 or 3 referring to the two heavy sterile neutrinos. In this case there are 11 additional parameters in the $\nu MSM$ as compared with SM. Seven of them we will identify with the elements of the active neutrino mass matrix ($m_2$, $m_3$, $\theta_{12}$, $\theta_{13}$, $\theta_{23}$, $\delta$, $\alpha_2$). The other 4 we will define as follows: the average mass of two heavier sterile neutrinos $M=\frac{M_2+M_3}{2}$, their mass splitting $\Delta M=\frac{M_3-M_2}{2}$, the parameter $\varepsilon$ and the phase $\xi=arg(h_{12})$. Thus, we can parameterize the Lagrangian in the following way: $$\begin{gathered} \label{lagparam} \mathcal L=\left(\frac{M\sum m_{\nu_i}}{v^2}\right)^{\frac{1}{2}}\left[\frac{1}{F_2\sqrt{\varepsilon }}h_{\alpha 2}\bar L_{\alpha}\tilde N_\alpha+\frac{\sqrt{\varepsilon }}{F_3}h_{\alpha 3}\bar L_{\alpha}\tilde N_3\right]\tilde\Phi-\\-M\bar{\tilde N}_2^c\tilde N_3-\frac{1}{2}\Delta M\left(\bar{\tilde{ N_2^c}}\tilde N_2+\bar{\tilde{ N_3^c}}\tilde N_3\right)+h.c.,\end{gathered}$$ where $a_{\alpha I}={h_{\alpha I}}/{F_I}$ are defined by equations and . Lagrangian can be written in another basis, namely when the mass matrix of sterile right-handed neutrinos is diagonal. In this case the Lagrangian is $$\label{diag} \mathcal{L}^{ad}=-g_{\alpha I}\bar{L}_\alpha {N'_I}\tilde {\Phi}-\frac{M_I}{2}\bar{N}'{}_I^cN'_I+h.c.,$$ where ${N'_I}$ are right-handed neutrinos and $g_{\alpha I}$ are elements of the Yukawa matrix in this basis. Transition from presentation of Lagrangian in gauge and mass basis can be made with unitary transformation that transfers mass matrix of right-handed neutrino to diagonal form [@Tibor; @sb]: $$\label{transform} V^*\left(\begin{array}{cc}\Delta M&M\\M&\Delta M\end{array}\right)V=\left(\begin{array}{cc}M-\Delta M&0\\0&M+\Delta M\end{array}\right);\quad V=\frac1{\sqrt{2}}\left(\begin{array}{cc}-i&i\\1&1\end{array}\right).$$ So, the transition can be made by $$\label{transition} %\left(\begin{array}{c}N_2\\N_3\end{array}\right)=V\left(\begin{array}{c}\tilde N_2\\ %\tilde N_3\end{array}\right) \tilde N_I=V_{IJ} N'{}_J,\quad g_{\alpha I}=h_{\alpha J} V_{JI}.$$ With help of this relations it will be useful to express Lagrangian in terms of right-handed neutrino functions of Lagrangian $$\begin{gathered} \label{lagdiag1} \mathcal{L}^{ad}=-\left(\frac{M\sum m_{\nu_i}}{2v^2}\right)^{\frac{1}{2}}\left[\left(\frac{ia_{\alpha 2}}{\sqrt{\varepsilon }}-i\sqrt{\varepsilon }a_{\alpha 3}\right)\bar L_{\alpha} N'{}_2+\left(\frac{a_{\alpha 2}}{\sqrt{\varepsilon }}+\sqrt{\varepsilon }a_{\alpha 3}\right)\bar L_{\alpha} N'{}_3\right]\tilde\Phi-\\-\frac{1}{2}\left(M-\Delta M\right)\bar{ {N'{}_2^c}} N'{}_2-\frac{1}{2}(M+\Delta M)\bar{ {N'{}_3^c}} N'{}_3.\end{gathered}$$ After comparing and one can express Yukawa couplings in different presentations $$\begin{aligned} &g_{\alpha 2}=\left(\frac{M\sum_i m_{\nu_i}}{2v^2}\right)^{\frac{1}{2}}\left(\frac{ia_{\alpha 2}}{\sqrt{\varepsilon }}-i\sqrt{\varepsilon }a_{\alpha 3}\right),\\ &g_{\alpha 3}=\left(\frac{M\sum_i m_{\nu_i}}{2v^2}\right)^{\frac{1}{2}}\left(\frac{a_{\alpha 2}}{\sqrt{\varepsilon }}+\sqrt{\varepsilon }a_{\alpha 3}\right).\end{aligned}$$ The mass eigenstates neutrinos for Lagrangian with the mass matrix $M^{DM}$ can be easily expressed through the states of neutrino of Lagrangian , particularly: $$\label{through} N^c=\left(1-\frac{1}{2}\varepsilon\varepsilon^+\right)N'{}^c+\varepsilon\nu_{L}\simeq N'{}^c+\varepsilon\nu_{L},$$ where $N$ are mass eigenstates of the right-handed neutrinos in which they are produced and decay, $\nu_{L}$ are the active neutrinos of the SM in flavor basis, $$\label{theta} \varepsilon_{\alpha I}\equiv\Theta_{\alpha I}=\frac{v}{\sqrt{2}}\frac{g_{\alpha I}}{M_I}$$ is the mixing angle ($\varepsilon_{\alpha I} \ll 1$). The computation of the leptonic asymmetry. ========================================== $\qquad$ $\qquad$ \[rozdil3\] As it was pointed in Section \[intro\], leptonic asymmetry in the $\nu MSM$ is generated due to decays of the heavier sterile neutrinos on SM particles. At temperature $T\ll T_{EW}$ the interaction of the sterile neutrinos with SM particles via neutral Higgs field can be neglected. The only possible way of interaction of the sterile neutrino with matter is through the mixing with active neutrinos . For the sterile neutrino with the mass $m_\pi<M_I<$ 2 GeV the channels for the decay into two-body final state are: $$\label{decay} N_I\rightarrow \pi^0\nu_\alpha,\pi^+e_\alpha^-,\pi^-e_\alpha^+, K^+e_\alpha^-, K^-e_\alpha^+,\eta\nu_\alpha,\eta '\nu_\alpha,\rho^0\nu_\alpha,\rho^+e_\alpha^-,\rho^-e_\alpha^+.$$ The channel of decay $N_{2,{3}}\rightarrow N_1+...$ is strongly suppressed because of the small Yukawa coupling constants of $N_1$. The decay of the sterile neutrino into the $K^0$ state is forbidden, because the composition of $K^0$ ($d\bar s$) can not be obtained by decay of $Z$-boson. The three-body final state can be safely neglected and also the many hadron final state [@Gorbunov]. This last decay channels contribute for less than 10% for $M_I < 2$ GeV. For $m_\pi < M_I < 2$ GeV the decays into D-meson can also be neglected because its mass is not much smaller than 2 GeV. Let us consider the decay of the sterile neutrino in the $\nu MSM$. Sterile neutrino oscillates into active neutrino that decay into $Z$-bozon and active neutrino (or $W^\pm$-boson and charged lepton) in accordance with the SM. $Z$-boson (or $W^\pm$-boson) hereafter decays into quark-antiquark pair, see Fig.1. Since kinetic energy of this quarks are small enough the quark pair will form a bound state. Since $M_I<2$ GeV $\ll M_{Z(W)}$ we can use low energy Fermi theory and shrink the heavy boson propagator into an effective vertex and use for final state a meson, see Fig.2. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- {width="150mm"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------  ---------------------------------------------------------------------------------------------------------- The process of sterile neutrino decay into charged lepton and charged meson through $W^\pm$-boson is described by charged current interaction $$\label{epi1} \mathcal{L}_C=\frac{G_F}{\sqrt{2}}\left(j_\nu^{CC}\right)^+j^{\nu\,CC},$$ where $j_\nu^{CC}=j_\nu^{l\,CC}+j_\nu^{h\,CC}$ is charged lepton and hadron current, $$\label{epi2} j_\nu^{l\,CC}=\sum_\alpha {\bar e}_\alpha\gamma_\nu(1-\gamma^5)\nu_\alpha, \quad j_\nu^{h\,CC}=\sum_{n,m} V_{n,m}^*{\bar d}_m\gamma_\nu(1-\gamma^5)u_n.$$ The indices $m,n$ run over the quark generation, $\alpha=e,\mu,\tau$ and $V$ is Kabbibo-Kobayashi-Maskawa (CKM) matrix. Similarly, the process of sterile neutrino decay into active neutrino and neutral meson through $Z$-boson is described by neutral current interaction $$\label{epi4} \mathcal{L}_N=\sqrt{2}{G_F}\left(j_\nu^{NC}\right)^+j^{\nu\,NC},$$ where $j_\nu^{NC}=j_\nu^{l\,NC}+j_\nu^{h\,NC}$ is active neutrino and hadron current, $$\label{epi5} j_\nu^{l\,NC}=\sum_\alpha {\bar \nu}_\alpha\gamma_\nu\frac{1-\gamma^5}2\nu_\alpha, \quad j_\nu^{h\,NC}=\sum_{f} {\bar f}\gamma_\nu\left(t_3^f(1-\gamma^5)-2q_f\sin^2\theta_W\right)f,$$ where sum over $f$ means sum over all quarks, $t^f_3$ – is the weak isospin of the quark, $q_f$ — is the electric charge of quark in proton charge units, notably $t^f_3=1/2, q_f=+2/3$ for $u,c,t$ and $t^f_3=-1/2, q_f=-1/3$ for $d,s,b$ quarks. The matrix element corresponding to Feynman diagram of sterile neutrino decay (see, e.g., Fig.1,2) can be obtained from the interactive effective Lagrangian [@Tibor]. For example, effective Lagrangian of decay of $I$ sterile neutrino into the $\pi^\pm,\pi^0$ final states is: $$\begin{gathered} \label{Lpi} \mathcal L_{eff}^{\pi}=\frac{G_F}{2}M_If_{\pi}\Theta_{\alpha I}\bar \nu_{\alpha}(1+\gamma_5)N_I \pi^0+\\ +\left[\frac{G_F}{\sqrt{2}}M_If_{\pi}V_{ud}\Theta_{\alpha I}\bar e_{\alpha}\left((1+\gamma_5)-\frac{m_{\alpha}}{M_I}(1-\gamma_5)\right)N_I \pi^-+h.c.\right],%\\f_\pi=0.131\mbox{GeV},\quad|V_{ud}|=0.97\end{gathered}$$ where $G_F$ is Fermi coupling constant, $M_I$ is the mass of $I$-sterile neutrino, $m_\alpha$ is the mass of the charged lepton of $\alpha$ generation, $f_\pi$ is the $\pi$-meson decay constant that is defined as $$\label{constf} \langle\pi^+|\bar u(1+\gamma^5)\gamma_\nu d|0\rangle=-f_\pi\cdot (p_\pi)_\mu,$$ where $p_\pi$ is the pion 4-momentum. The leptonic asymmetry $\epsilon$ can be defined as $$\label{epsilon} \epsilon=\frac{\Gamma_{N\rightarrow l}-\Gamma_{N\rightarrow \bar l}}{\Gamma_{N\rightarrow l}+\Gamma_{N\rightarrow \bar l}}\,,$$ where $\Gamma_{N\rightarrow l}$ is the total decay rate of sterile neutrinos into leptons and $\Gamma_{N\rightarrow \bar l}$ is the total decay rate of sterile neutrinos into antileptons. At tree level the decay rates of the sterile neutrinos into leptons and antileptons are equal. Therefore we must compute the one loop diagrams, see Fig.3. In the case of nearly degenerated sterile neutrinos the contribution from the diagrams presented at Fig.3*b*) can be neglected as compared with diagrams presented at Fig.3*a*). Indeed the propagator of the sterile neutrino in the diagrams *a)* type is proportional to $1/\Delta M$ in the center of mass frame. The leading order contribution to the leptonic asymmetry comes from interference between one-loop diagrams and tree-level diagrams [@Davidson]. In this case $\Gamma_{N\rightarrow l}-\Gamma_{N\rightarrow \bar l}\sim \Theta^4$ and $\Gamma_{N\rightarrow l}+\Gamma_{N\rightarrow \bar l}\sim \Theta^2$, the leptonic asymmetry is suppressed. ------------------------------------------------------------- {width="90mm"} ------------------------------------------------------------- In our case, when the mass splitting between the two heavier sterile neutrinos is very small and it is of the same order as their decay rate (we obviously will see it later), the oscillations between $N_I$ and $N_J$ are important, see Fig.4. So, the corresponding mass eigenfunctions are no longer the $N_I$ states, but a mixture of them, namely $\psi_I$ [@Flanz; @Tibor]. It is these physical eigenstates which evolve in time with a definite frequency. The subsequent decay of these fields will produce the desired lepton asymmetry $$\label{Delta} \Delta=\frac{\Gamma_{\psi\rightarrow l}-\Gamma_{\psi\rightarrow \bar l}}{\Gamma_{\psi\rightarrow l}+\Gamma_{\psi\rightarrow \bar l}}\,,$$ where $\Gamma_{\psi\rightarrow l}$ and $\Gamma_{\psi\rightarrow \bar l}$ are the total decay rates of the sterile neutrino mass eigenfunctions $\psi_I$ into leptons and antileptons correspondingly. In this case the leading order contribution to the leptonic asymmetry comes from tree-level diagrams. --------------------------------------------------------------------------- {width="90mm"} --------------------------------------------------------------------------- In general case the correct description of the processes can be made in frame of the density matrix formalism, see, e.g., [@Shap1]. We will follow a simpler way by considering a non-hermitian Hamiltonian. The effective Hamiltonian in the basis of $\{N_2,N_3\}$ is the $H=H_0+\Delta H$, where $H_0$ is the diagonal Hamiltonian of equal mass particle $$\label{H0} H_0=\left(\begin{array}{cc}M&0\\0&M\end{array}\right).$$ The corrections to this Hamiltonian are given by the one-loop diagrams, see Fig.4: $$\label{DeltaH} \Delta H=\left(\begin{array}{cc}-\Delta M-\frac{i}{2}\Gamma_2&-\frac{i}{2}\Gamma_{23}\\-\frac{i}{2}\Gamma_{23}&\Delta M-\frac{i}{2}\Gamma_3\end{array}\right).$$ The dispersive part of these diagrams can be absorbed in the mass renormalization of the fields [@Flanz] and it brings to appearance of the mass splitting $\Delta M$. The absorptive part of the diagrams will define total decay rates of the sterile neutrino $\Gamma_I$ and the rate of oscillation between sterile neutrinos $\Gamma_{23}$. Total rates of $I$-sterile neutrino decays into charged mesons and leptons of $\alpha$-generation are $$\begin{gathered} \label{GP+-} \Gamma_I^{\,\alpha\pi^\pm}=\Gamma(N_I\rightarrow \pi^\pm+l_\alpha^{\mp})=\\=\frac{G_F^2f_\pi^2|V_{ud}|^2M^3}{8\pi}|\Theta_{\alpha I}|^2S(M,m_\alpha,m_\pi)\left[\left(1-\frac{m_\alpha^2}{M^2}\right)^2-\frac{m_\pi^2}{M^2}\left(1+\frac{m_\alpha^2}{M^2}\right)\right],\end{gathered}$$ $$\begin{gathered} \label{GK} \Gamma_I^{\,\alpha K}=\Gamma(N_I\rightarrow K^\pm+l_\alpha^{\mp})=\\=\frac{G_F^2f_K^2|V_{us}|^2M^3}{8\pi}|\Theta_{\alpha I}|^2S(M,m_\alpha,m_K)\left[\left(1-\frac{m_\alpha^2}{M^2}\right)^2-\frac{m_K^2}{M^2}\left(1+\frac{m_\alpha^2}{M^2}\right)\right],\end{gathered}$$ $$\begin{gathered} \Gamma_I^{\,\alpha\rho^\pm}=\Gamma(N_I\rightarrow \rho^\pm+l^\mp_{\alpha})=\\=\frac{G_F^2g_\rho^2|V_{ud}|^2M^3}{4\pi m_\rho^2}|\Theta_{\alpha I}|^2S(M,m_\alpha,m_\rho)\left[\left(1-\frac{m_\alpha^2}{M^2}\right)^2+\frac{m_\rho^2}{M^2}\left(1+\frac{m_\alpha^2-2m_\rho^2}{M^2}\right)\right],\end{gathered}$$ where $$\label{S} S(M_I,m_\alpha,m)=\sqrt{\left(1-\frac{(m-m_\alpha)^2}{M_I^2}\right)\left(1-\frac{(m+m_\alpha)^2}{M_I^2}\right)},$$ and values of decay constants and elements of CKM matrix are given in [@PG]: $f_\pi=0.131$ GeV, $f_K=0.16$ GeV, $g_\rho=0.102$ GeV$^2$, $|V_{ud}|=0.97$, $|V_{us}|=0.23$. Total rates of $I$-sterile neutrino decays into neutral mesons and active neutrinos are $$\label{GP0} \Gamma_I^{\,\alpha\pi^0}=\Gamma(N_I\rightarrow \pi^0+\nu_\alpha)= \frac{G_F^2f_\pi^2M^3}{16\pi}|\Theta_{\alpha I}|^2\left(1-\frac{m_\pi^2}{M^2}\right)^2,$$ $$\Gamma^{\,\alpha\rho^0}_I=\Gamma(N_I\rightarrow \rho_0+\nu_{\alpha})=\frac{G_F^2g_\rho^2M^3}{8\pi m_\rho^2}|\Theta_{\alpha I}|^2\left(1+2\frac{m_\rho^2}{M^2}\right)\left(1-\frac{m_\rho^2}{M^2}\right)^2,$$ $$\Gamma_I^{\,\alpha\eta}=\Gamma(N_I\rightarrow \eta+\nu_{\alpha})=\frac{G_F^2f_{\eta}^2M^3}{16\pi}|\Theta_{\alpha I}|^2\left(1-\frac{m_{\eta}^2}{M^2}\right)^2,$$ $$\label{eta'} \Gamma_I^{\,\alpha\eta'}=\Gamma(N_I\rightarrow \eta'+\nu_{\alpha})=\frac{G_F^2f_{\eta'}^2M^3}{16\pi}|\Theta_{\alpha I}|^2\left(1-\frac{m_{\eta'}^2}{M^2}\right)^2,$$ where $f_\eta=0.156$ GeV, $f_\eta'=-0.058$ GeV [@PG]. As one can see the decay rates into $\rho^\pm,\rho^0$ mesons are slightly different because they are vector mesons. The adduced decay rates – were obtained in [@Johnson; @Gorbunov]. The total decay rate of sterile neutrino decay into mesons and leptons is sum of the rates over all decay channels $\Lambda$ and over leptonic generation: $$\label{totalrate} \Gamma_{I}=\sum_{\alpha,\Lambda}\Gamma_I^{\,\alpha\Lambda}\,\Theta(y_{\alpha\Lambda}),$$ where $y_{\Lambda}$ is the difference of the $I$ sterile neutrino mass and total mass of all final particles of the decay channel $\Lambda$; $\Theta(x)$ is the usual Heaviside function. The rate of oscillation between $I$ and $J$ sterile neutrinos ($\Gamma_{IJ}$) can be expressed through the decay rates $$\label{IJ} \Gamma_{IJ}=\sum_{\alpha,\Lambda} \frac{{\rm Re}(\Theta_{\alpha I}\Theta_{\alpha J}^*)}{|\Theta_{\alpha I}|^2}\,\Gamma_I^{\,\alpha\Lambda}\,\Theta(y_{\alpha\Lambda}).$$ The eigenvalues and corresponding eigenfunctions of the non-hermitian Hamiltonian $H=H_0+\Delta H$ are given by $$\begin{aligned} \label{evalues} & \omega_2=M-\frac{i}{4}(\Gamma_2+\Gamma_3)-\frac{1}{4}c,\qquad\psi_2=\frac{1}{\sqrt{N}}\left(\begin{array}{c}B\\1\end{array}\right),\\ &\omega_3=M-\frac{i}{4}(\Gamma_2+\Gamma_3)+\frac{1}{4}c,\qquad \psi_3=\frac{1}{\sqrt{N}}\left(\begin{array}{c}1\\-B\end{array}\right),\end{aligned}$$ where $N$ is a normalization factor and $$c=\sqrt{(4\Delta M-i(\Gamma_3-\Gamma_2)^2-4(\Gamma_{23})^2},\quad B=(4i\Delta M+(\Gamma_3-\Gamma_2)+ic)/({2\Gamma_{23}}).$$ It should be noted the sterile neutrinos are not initially in the state $\psi_2$ and $\psi_3$, but in the state $N_2$ and $N_3$. The fact is that sterile neutrino where in thermal equilibrium before they propagated freely. The equilibrium was maintained by the weak interaction between the sterile neutrinos and particles in the background. The weak interaction eigenstates are $N_2$ and $N_3$, therefore at the beginning the sterile neutrinos are in the state $N_2$ or $N_3$. In general the initial state of sterile neutrino is the superposition of $N_2$ and $N_3$ states and can be described by a density matrix: $$\label{roini} \hat\rho_{initial}=\hat\rho(t=0)=\sum_{I=2,3}\alpha_I|N_I(0)\rangle\langle N_I(0)|,$$ where $\alpha_2+\alpha_3=1$. It was shown in [@Tibor] that leptonic asymmetry dependence on parameter $\alpha_I$ can be neglected. We confirmed this statement and, hereafter, we will consider the symmetric initial state $\alpha_2=\alpha_3={1}/{2}$. The time evolution of the density matrix can be obtain in a simple way. Since $$\label{uperetv} |\psi_I\rangle=U_{IJ}|N_J\rangle,$$ where $$\label{u} U=\frac1{\sqrt{N}}\left(\begin{array}{cc}B&1\\1&-B\end{array}\right),$$ the time evolution of $|N_I\rangle$ state is known $$\label{evolstates} |N_I(t)\rangle=U^{-1}_{IK}e^{-i\omega_Kt}|\psi_K(0)\rangle=U^{-1}_{IK}e^{-i\omega_Kt}U_{KJ}|N_J(0)\rangle=R_{IJ}|N_J(0)\rangle.$$ Thus $$\label{roT} \hat\rho(t)=\frac{1}{2}\sum_{I,J,K=2}^3R_{IK}(t)^\ast R_{IJ}(t)|N_J(0)\rangle\langle N_K(0)|=\frac{1}{2}\sum_{J,K=2}^3(R^\dag R)_{KJ}|N_J(0)\rangle\langle N_K(0)|.$$ The average production rate of leptons is given by $$\begin{gathered} \Gamma=\!\!\int_0^\infty \!\!\!\!\!dt\!\!\int \!\!\!d\Pi_2\sum_lTr\left[|l\rangle\langle l|\hat\rho(t)\right]=\frac{1}{2}\int_0^\infty \!\!\!\!\!dt\!\!\int \!\!\!d\Pi_2\, Tr\left[\sum_{l,K,J}(R^\dag R)_{KJ}\langle l|N_J(0)\rangle\langle N_K(0)|l\rangle\right]\!=\\=\frac{1}{2}\int_0^\infty \!\!\!\!\!dt\!\!\int \!\!\!d\Pi_2\sum_{l,J,K}(R^\dag R)_{KJ}A_{Jl}A_{Kl}^\ast,\end{gathered}$$ where sum over $l$ means sum over all leptons generations and include charged leptons and active neutrinos, $\langle l|N_J(0)\rangle=A_{Jl}$ is the transition amplitude of the decay of $I$ sterile neutrino into a lepton at tree level that includes all possible channels of reaction, and $d \Pi_2$ is the differential 2-body phase space $$d\Pi_2=\frac{d^3q}{(2\pi)^32E_q}\frac{d^3k}{(2\pi)^32E_k}(2\pi)^4\delta^4(p-q-k),$$ where $p,q,k$ are 4-momentums of initial and final particles in decay. Similarly the production rate of antileptons is $$\bar \Gamma=\frac{1}{2}\int_0^\infty \!\!\!\!\!dt\!\!\int \!\!\!d\Pi_2\sum_{l,J,K}(R^\dag R)_{KJ}A_{Jl}^\ast A_{KJ}.$$ The measure of the leptonic asymmetry is given by $$\begin{gathered} \!\Delta\!=\!\frac{\Gamma\!-\!\bar \Gamma}{\Gamma\!+\!\bar \Gamma}\!=\! \frac{\int dt\int d\Pi_2Im((R^\dag R)_{32})Im(A_{2l}^\ast A_{3l})}{\int dt\int d\Pi_2((R^\dag R)_{22}|A_2l|^2\!+\!(R^\dag R)_{33}|A_3l|^2\!+\!2Re(A_{2l}^\ast A_{3l})Re(R^\dag R)_{23})}\end{gathered}$$ The integration over $d\Pi_2$ gives [@Tibor]: $$\label{assym} \Delta=\frac{\int dt Im((R^\dag R)_{32})\sum_{\alpha}Im(\Theta_{\alpha 2}^\ast \Theta_{\alpha 3})V_{\alpha}}{\int dt\sum_{\alpha}((R^\dag R)_{22}|\Theta_{\alpha 2}|^2+(R^\dag R)_{33}|\Theta_{\alpha 3}|^2+2Re(\Theta_{\alpha 2}^\ast \Theta_{\alpha 3})Re(R^\dag R)_{23})V_{\alpha}},$$ where $V_\alpha$ is defined via sum over all possible channels of sterile neutrino decays into leptons of generation $\alpha$ $$\label{v} V_\alpha=\sum_{\Lambda} \frac{\Gamma_I^{\,\alpha\Lambda}}{|\Theta_{\alpha I}|^2}\,\Theta(y_{\alpha\Lambda}).$$ The restrictions on the parameters of the $\nu MSM$ =================================================== \[rozdil4\] As it was pointed in Section \[intro\], the leptonic asymmetry of the Universe has to be constrained by condition at the moment of the beginning of the DM particles production. It allows us to constrain parameters of the $\nu MSM$. To do it, we can construct the leptonic asymmetry as function of only three parameters of $\nu MSM$: $M$, $\Delta M$, $\varepsilon$. We do it in the following way. Leptonic asymmetry function is maximized over phases $\delta$, $\alpha_2$, $\xi$ (and $\alpha_1$ in case of the inverted hierarchy) and is taken at central value of active neutrino mass matrix parameters[^6], see Tab.1. This function contains dependence on ratios of the Yukawa matrix elements in mixing angle $\Theta_{\alpha I}$ that can be expressed through solutions with two possible choice of sign consistent with condition . So far as the relation for leptonic asymmetry has no symmetry for interchanging and conjugating of the ratios of elements of the second and the third columns of the Yukawa matrix we have to consider two variants of the solutions. For fixed values of the mixing angles and phases we will designate allowed solution of with 2 or more sign $(+)$ as solution of A type, and, vice-versa, the solution with 2 or more sign $(-)$ we will designate as solution of B type. It should be noted that our results , for B type of solution coincide with results of [@Sh3] where the ratios of the elements were obtained in the particular case $\theta_{13}\rightarrow 0$, $\theta_{23}\rightarrow\pi/4$. We separately consider the case of $\theta_{13}=0$ and $\theta_{13}=10^{\rm o}$ also. Thereby we construct allowed regions ($\Delta>10^{-3}$) in plane of parameters $\Delta M$ and $\varepsilon$ at fixed values of $M$. For the case of the normal hierarchy the deference between the case of $\theta_{13}=0$ or $\theta_{13}=10^{\rm o}$ and the case of solution of A or B types is not essential, so we illustrate allowed regions with help of only one figure on Fig.5. For the case of the inverted hierarchy, the difference between the case of solution of A or B types is not essential, but the cases of $\theta_{13}=0$ and $\theta_{13}=10^{\rm o}$ are substantially different. So we illustrate allowed regions with help of two figures on Fig.6. It should be noted that we investigated form of the allowed regions not only for the central value of $\theta_{13}$ angle, but for range given by data of [@T2Ktheta13]. We conclude that in case of the normal hierarchy the regions are almost not sensitive to value of $\theta_{13}$ in range $0<\theta_{13}<16^{\rm o}$. In case of the inverted hierarchy it is true for the regions on Fig6 *b*) and $\theta_{13}<18^{\rm o}$, but for $\theta_{13}=0$ the allowed regions are appreciably different. Also we illustrate regions where maximum of $\Delta$ can be more then $2/11$ on Fig.7 (white inner figures) for the case of both hierarchies. We do it only for the mass $M=1$ GeV because this regions are at small values of $\varepsilon$ and it will not intersect with other subsequent constrains. Moreover, at some values of phases leptonic asymmetry in this region can be less then $2/11$ and so we can not exclude this region ultimately. By way of example, we present possible values of $I$ sterile neutrino decay rate $\Gamma_I$ and rate of oscillations between $I$ and $J$ sterile neutrinos $\Gamma_{IJ}$ for $M=1$ GeV and $\theta_{13}=10^{\rm o}$ on Fig.8. As one can see the values of $\Gamma_I$, $\Gamma_{IJ}$ are really of the same order as $\Delta M$. It confirms previous assumption about necessity of taking into consideration oscillations between sterile neutrinos. {width="80mm"} {width="162mm"} In order to create a leptonic asymmetry the sterile neutrinos should be out of thermal equilibrium. That means that $$\label{out} \Gamma_2\lesssim H,$$ where $H$ is Hubble parameter that determines the expansion rate of the Universe. In the radiative dominated epoch Hubble parameter is given by $$\label{Hubble} H=\frac{T^2}{M_{PL}^\ast}$$ where $M_{PL}^\ast=\sqrt{\frac{90}{8\pi^3g^\ast(T)}}M_{PL}$, $M_{PL}=1.22\cdot10^{19}$ GeV is the Planck mass, $g^\ast(T)$ is the internal degrees of freedom [@kolb]. At temperature $T\sim1$ GeV we can take $g^*\simeq65$. So we get condition $$\sqrt{M_{PL}^*\Gamma_2}\lesssim T.$$ The out-of-equilibrium condition means that sterile neutrinos should decay at a temperature smaller than their mass ($T\lesssim M$). Moreover the sterile neutrinos should decay before the creation of DM so that the leptonic asymmetry enhances the DM production. The DM is created at $T\sim 0.1$ GeV. Therefore, $$\label{dmcon} 0.1\lesssim\frac{\sqrt{M_{PL^*}\Gamma_2}}{1{\rm GeV}}\lesssim \frac{M} {1\rm GeV}.$$ {width="162mm"} {width="155mm"} {width="85mm"} We illustrate on Fig.9 the region of values of $M$ and $\varepsilon$ where condition is satisfied for the case of the normal hierarchy. At scale of parameters presented on Fig.9 the difference between the case of $\theta_{13}=0$ or $\theta_{13}=10^{\rm o}$ and between the case of solution of A or B types is small, so we present only one figure. It is not true for the case of the inverted hierarchy, see Fig.10. It should be noted that region on Fig.9 is almost not sensitive to value of $\theta_{13}$ at range $0<\theta_{13}<16^{\rm o}$. In case of the inverted hierarchy it is true for the regions on Fig.10 *b*) and $\theta_{13}<18^{\rm o}$, but for the case of $\theta_{13}=0$ the regions are appreciably different. As one can see there are regions on Fig.9 (red) and Fig.10 *a*) (red and blue) where conditions and are satisfied simultaneously. This region of parameters is suitable for DM production in the $\nu MSM$. For the case of inverted hierarchy and nonzero value of $\theta_{13}$ we have no region that is suitable for DM production. So, in $\nu$ MSM for physical nonzero value of $\theta_{13}$ and mass of sterile neutrino $m_\pi<M<2$ GeV DM production can be realized only in case of normal hierarchy of active neutrino mass. The region suitable for DM production (the case of the normal hierarchy and nonzero $\theta_{13}$) can be used to obtain constraints for mass splitting of the sterile neutrino. Fixing mass of the sterile neutrino one obtains possible values of $\varepsilon$ (see Fig.9) and using Fig.5 one can obtain possible values of the mass splitting for the sterile neutrino with mass $M$. If mass of the sterile neutrino is on lower boundary of the allowed mass range ($M\simeq 1.4$ GeV) than value of $\Delta M$ is exactly known ($\Delta M\approx 5\cdot10^{-21}$ GeV). If mass of the sterile neutrino is on upper bound of the allowed mass range ($M=2$ GeV) than $\Delta M$ can possess the values from the range $10^{-21}\lesssim\Delta M/1 {\rm GeV}\lesssim10^{-20}$. {width="160mm"} Some existing experimental data restrict the area of parameters of $\nu MSM$. For $M < 0.45$ GeV the best constraints come from the CERN PS191 experiment. For $0.45 < M < 2$ GeV the constraints come from the NuTeV, CHARM and BEBC experiments. The range of parameters admitted by these experimental data is summarized in [@ShProg]. These parameters are the mixing angle $(\Theta^+\Theta)_{22}$ (it defines the range of reactions with sterile neutrino) and the mass of the heavier sterile neutrino $M$. To compare obtained in the present paper constraints on the $\nu MSM$ parameters (see Fig.9 and Fig.10) with constraints summarized in [@ShProg] one has to rebuild allowed regions in the space of parameters $M$ and $\theta^2_{\nu N_2}=(\Theta^+\Theta)_{22}$. In general case the relation between $(\Theta^+\Theta)_{22}$ and $\varepsilon$ is quite difficult. Really, in accordance with and we have $$\label{Theta22} (\Theta^+\Theta)_{22}=\frac{\nu^2}{2M^2} (V^+h^+hV)_{22}=\frac{\nu^2}{2M^2}(F_2^2+F_3^2-2 |h^+h|_{23}\sin\chi),$$ where $\chi=arg[(h^+h)_{23}]$. Using – we get $$\label{F2F3} F_2^2=\frac{M}{\nu^2\varepsilon}(m_c+m_b),\quad F_3=\varepsilon F_2,\quad |h^+h|_{23}=\frac{M}{\nu^2}(m_c-m_b)$$ and $$\label{Theta22F} (\Theta^+\Theta)_{22}=\frac{m_c+m_b}{2M\varepsilon }\left(1+\varepsilon^2-2\frac{m_c-m_b}{m_c+m_b}\varepsilon \sin\chi\right).$$ The problem is in parameter $\chi$ that is a complicated function of many parameters. But for our case (see Fig.9 and Fig.10, $\epsilon<0.16$) we can use approximate relation $$\label{Theta22Final} (\Theta^+\Theta)_{22}=\frac{m_c+m_b}{2M\varepsilon}.$$ The imposition of our constraints are presented on Fig.9 and Fig.10 for nonzero value of $\theta_{13}$ and summarized constraints from [@ShProg] is presented on Fig.11. Above the line marked “BAU”, baryogenesis is not possible: here sterile neutrinos come to thermal equilibrium above the $T_{EW}$ temperature. Below the line marked “See-saw”, the data on neutrino masses and mixing cannot be explained using “see-saw” mechanism. The region noted as “BBN” is disfavoured by the considerations of Big Bang Nucleosynthesis. The region marked ”Experiment” shows the part of the parameter space excluded by direct searches for singlet fermions. The regions market “Cos”, “$\Delta$” and “DM” were builded in this paper. The grey and blue region “Cos” shows the parametric space allowed by cosmological constraint (grey region corresponds to A and B type of solution, blue region corresponds to B type of solution), the dashed region market “$\Delta$” shows the parametric space allowed by constraint , the red region marked “DM” shows the parametric space where constraints and are noncontradictory. The last region is preferred for DM production according to calculations of the present paper. ![The imposition of our constraints and summarized constraints from [@ShProg]: *a*) the case of the normal hierarchy, *b*) the case of the inverted hierarchy.](Fig11){width="165mm"} The red region marked “DM” is shown on Fig.12 in the scaled-up form. The difference between the case of $\theta_{13}=0^{\rm 0}$ or $\theta_{13}=10^{\rm o}$, and between type of A or B solutions is illustrated. As one see the choice of solutions of A or B type makes greater change in the allowed region then the choice of $\theta_{13}=0^{\rm 0}$ or $\theta_{13}=10^{\rm 0}$. Conclusion {#rozdil5} ========== In the present paper we consider the leptonic asymmetry generation at $T\ll T_{EW}$ when the masses of two heavier sterile neutrinos are between $m_\pi$ and 2 GeV. We conclude that oscillations and decays of sterile neutrinos can produce a leptonic asymmetry that is large enough to enhance the DM production sufficiently to explain the observed DM in the Universe, but only for the case of the normal hierarchy of the active neutrino mass. The allowed range of parameters is narrow and it is presented on Fig.11 and Fig.12. It should be noted that allowed mass range for heavier sterile neutrino is $1.42 (1.55)\lesssim M<2$ GeV for B (A) type of solutions and the mixing angle between active and sterile neutrino is $-7.91(-7.98)\lesssim{\rm Log}_{10}(\Theta^+\Theta)_{22}\lesssim-8.41(-8.35)$ for B (A) type of solutions. If mass of the sterile neutrino is on lower boundary of the allowed mass range than value of $\Delta M$ is exactly known ($\Delta M\approx 5\cdot10^{-21}$ GeV). If mass of the sterile neutrino is on upper bound of the allowed mass range than $\Delta M$ can possess the values from the range $10^{-21}\lesssim\Delta M/1 {\rm GeV}\lesssim10^{-20}$. For the case of the inverted hierarchy there is no region suitable for DM production. The big range of parameters of the $\nu MSM$ is not forbidden by the existing experimental data, see Fig.11. Combining of this range with our constraints (red region “DM” on Fig.11) leads to conclusion that improvement of previous experiments, as NuTeV or CHARM, of one or two order of magnitude can exclude the $\nu MSM$ with $M < 2$ GeV or detect the right-handed neutrinos. It should be noted that our constraints are quite a rough and can be used only for estimation. Really, the form of red region “DM” is very sensitive to cosmological constraints. Applied condition $ 0.1 {\rm GeV} <\sqrt{M_{PL^*}\Gamma_2}<M $ is very approximate. The correct description of the processes can be made in frame of the density matrix formalism or Boltzmann equations. Our computation is not valid for $M > 2$ GeV. However, the extrapolation of our result, see Fig.11, suggests that the range of admitted parameters for the case of the normal hierarchy becomes bigger for masses above 2 GeV. We expect that for masses above 2 GeV DM production can be realized for the case of inverted hierarchy too. {width="115mm"} During computation we used two types (A or B) of solutions . This is due to the fact that ratios of the Yukawa matrix elements (enter into the expression for the mixing angle $\Theta_{\alpha I}$) can be expressed through solutions with two possible choice of sign consistent with condition . It is closely related to the symmetry of under replacing the elements of the second column of the Yukawa matrix by elements of the third column. This two variants are equal in rights. The computation of the leptonic asymmetry in the applied simple model allows us to make some conclusions that, seemingly, will be correct and under more rigorous consideration. Namely, the initial state of the right-handed neutrino in form are not important for lepton asymmetry generation (the final results are not sensitive to values of the constants $\alpha_I$). For the case of normal hierarchy the deviation of the mixing angle $\theta_{13}$ from its zero value (up to value $16^{\rm 0}$) almost does not change the region suitable for DM production. For the case of inverted hierarchy results are different for $\theta_{13}=0$ and $\theta_{13}\neq0$. Our calculations indicates that case of $\theta_{13}=0$ leads to existing of region suitable for DM production, but at nonzero values of $\theta_{13}$ this region does not exist. Values of $\theta_{13}$ in range $\theta_{13}<18^{\rm o}$ ($\theta_{13}\neq0$) almost does not change the region suitable for DM production. It’s essential to note that during computations we have used functions maximized over unknown parameters of the model (phases $\delta,\xi,\alpha_{2},\alpha_{1}$). If the maximization procedure was not performed the final functions are sensitive to values of mentioned phases. So, the obtained results are very optimistic. But if the proposed on Fig.11 region of parameters “DM” will be forbidden by experiment data it will mean that mass of heavier sterile neutrinos must be lager 2 Gev. An essential assumption we have made is that the background effects are negligible. We do not have justify that it can be neglected in the thermal bath of the universe. For simplicity the computations were made at zero temperature. A rigorous justification of this assumptions is needed. It should be noted that region suitable for DM production in $\nu MSM$ was recently calculated in frame of more general formalism in [@Drewes]. Certainly, results of [@Drewes] somewhat differ from our simple calculations. Acknowledgments {#acknowledgments .unnumbered} =============== We would like to thank Marco Drewes and Tibor Frossard for the idea of treating this subject, and for useful comments and discussions. This work has been supported by the Swiss Science Foundation (grant SCOPES 2010-2012, No. IZ73Z0\_128040). [200]{} S. Weinberg, Phys. Rev. Lett. **19**, 1264 (1967); S. L. Glashow, Nucl. Phys. **22**, 579 (1961); A. Salam, Proceedings of The Nobel Symposium Held 1968 At Lerum, Sweden, Stockholm 1968, 367-377. Particle Data Group, http://pdg.lbl.gov A. Strumia and F. Vissani, arXiv: hep-ph/0606054. A. Boyarsky, O. Ruchayskiy, and M. Shaposhnikov, Ann. Rev. Nucl. Part. Sci. **59**, 191 (2009). T. Asaka and M. Shaposhnikov, Phys. Let. B **620**, 17 (2005). T. Asaka, S. Blanchet, and M. Shaposhnikov, Phys. Let. B **631**, 151 (2005). X.-d. Shi and G. M. Fuller, Phys. Rev. Lett. **82**, 2832 (1999). M. Shaposhnikov, JHEP 0808:008,2008. M. Laine, M. Shaposhnikov, JCAP 0806:031,2008. D. Gorbunov and M. Shaposhnikov, JHEP 0710:015,2007. A. Kusenko, S. Pascoli and D. Semikoz, JHEP 0511:028,2005. R.N. Mohapatra and A.Y. Smirnov, Ann. Rev. Nucl. Sci. **56**, 569 (2006). T2K Collaboration, Phys. Rev. Lett. **107**, 041801 (2011). T. Frossard, Leptonic Asymmetry in the $\nu MSM$ — 2010. S. Bilenky and S. Petcov, Rev. Mod. Phys. **59** No. 3, Part 1 (1987). M. Shaposhnikov, Nucl. Phys. B **763**, 49 (2007). S. Bilenky, C. Giunti, W. Grimus, Prog. Part. Nucl. Phys. **43**, 1 (1999). V. Gorkavenko, S. Vilchynskiy, Eur. Phys. J. C **70**, 1091 (2010). S. Davidson, E. Nardi and Y. Nir, Phys. Rept. **466**, 105 (2008). M. Flanz, E. Paschos, U. Sarkar and J. Weiss, Phys. Lett. B **389**, 693 (1996). L. M. Johnson, D. W. McKay and T. Bolton, Phys. Rev. D **56**, 2970 (1997). E.W. Kolb , M.S. Turner “The early Universe”, Addison-Wesley Pub. Company, 1989. M. Shaposhnikov, Prog. Theor. Phys. **122** , 185 (2009). L. Canetti, M. Drewes and M. Shaposhnikov, arXiv:1204.3902, 2012. [^1]: E-mail: gorka@univ.kiev.ua [^2]: E-mail:igrudenok@gmail.com [^3]: E-mail: sivil@univ.kiev.ua [^4]: This is why these neutrinos are called sterile neutrinos. The left-handed neutrinos of the SM are called active neutrinos. [^5]: Therefore the lightest sterile neutrino in the $\nu MSM$ is a candidate for the DM particle. [^6]: In case of the normal hierarchy we have $m_1=0$, $m_2=\sqrt{\Delta m_{21}^2}=0.009\,eV$, $m_3=\sqrt{|\Delta m^2_{23}|+\Delta m_{21}^2}=0.05\,eV$. In case of inverted hierarchy we have $m_1=\sqrt{|\Delta m^2_{23}|-\Delta m_{21}^2}=0.048\,eV$, $m_2=\sqrt{|\Delta m^2_{23}|}=0.049\,eV$, $m_3=0$.

--- abstract: 'This paper is concerned with optimal switching over multiple modes in continuous time and on a finite horizon. The performance index includes a running reward, terminal reward and switching costs that can belong to a large class of stochastic processes. Particularly, the switching costs are modelled by right-continuous with left-limits processes that are quasi-left-continuous and can take both positive and negative values. We provide sufficient conditions leading to a well known probabilistic representation of the value function for the switching problem in terms of interconnected Snell envelopes. We also prove the existence of an optimal strategy within a suitable class of admissible controls, defined iteratively in terms of the Snell envelope processes.' author: - 'Randall Martyr[^1]' title: ' **Finite-horizon optimal multiple switching with signed switching costs**[^2]' --- [**MSC2010 Classification:** 93E20, 60G40, 91B99, 62P20.]{} [**Key words:** optimal switching, real options, stopping times, optimal stopping problems, Snell envelope.]{} Introduction. ============= The recent paper by Guo and Tomecek [@Guo2008b] showed a connection between Dynkin games and optimal switching problems with signed (positive and negative) switching costs. The results were obtained for a model in which the cost/reward processes were merely required to be adapted and satisfy mild integrability conditions. However, there are few theoretical results on the existence of optimal switching control policies under such general conditions. Optimal switching for models driven by discontinuous stochastic processes has been studied previously in papers such as [@Hamadene2007a; @Morimoto1987]. The paper [@Morimoto1987] used optimal stopping theory to study the optimal switching problem on an infinite time horizon with multiple modes. The model described in [@Morimoto1987] has bounded and non-negative running rewards which are driven by right-continuous processes, and switching costs that are strictly positive and constant. The paper [@Hamadene2007a] studied the finite time horizon optimal switching problem with two modes. The model has running rewards adapted to a filtration generated by a Brownian motion and an independent Poisson random measure, but excludes terminal data and assumes switching costs that are strictly positive and constant. The more recent paper [@Djehiche2009] has a model similar to [@Hamadene2007a] with switching costs assumed to be continuous stochastic processes and filtration generated by a Brownian motion. Nevertheless, the authors stated ([@Djehiche2009 p. 2753]) that their results can be adapted to a more general setup, possibly by using the same approach as in [@Hamadene2007a]. Most of the literature on optimal switching assumes non-negative switching costs. However, signed switching costs are important in models where the controller can (partially) recover its investment, or receive a subsidy/grant for investing in a new technology such as renewable (green) energy production [@Guo2008b; @Lumley2001; @LyVath2008]. The preprint [@ElAsri2012] sought to generalise the results of [@Djehiche2009] by permitting signed switching costs, but at the expense of limiting the total number of switches incurring negative costs. This limitation is absent in papers such as [@Bouchard2009; @Lundstrom2013a] where the optimal switching problem was studied within a Markovian setting. There are, however, other structural conditions and hypotheses made in [@Bouchard2009; @Lundstrom2013a]. For example, there is an assumption in [@Lundstrom2013a p. 1221] that the terminal reward is the same for all modes (which also implies the terminal values of the switching costs are non-negative). The Snell envelope approach, also known as the *method of essential supremum* [@Peskir2006], is a general approach to optimal stopping problems which does not require Markovian assumptions on the data. It was used in the aforementioned papers [@Hamadene2007a; @Djehiche2009] and the paper [@Bayraktar2010] on optimal switching problems for one-dimensional diffusions (albeit in a slightly different manner). In this paper we use the theory of Snell envelopes to extend Theorems 1 (verification) and 2 (existence) of [@Djehiche2009]. Our model allows for non-zero terminal data and switching costs which are real-valued stochastic processes with paths that are right-continuous with left limits and *quasi-left-continuous*. In contrast to [@ElAsri2012], our results do not presuppose a limit on the total number of switches incurring negative costs. We do, however, require that a certain “martingale hypothesis” $\textbf{M}$ on the switching costs be satisfied (see Section \[Section:ArbitraryNumberOfSwitches\] below). This hypothesis can be verified in many cases of interest, including the case of two modes, and does not require the terminal reward to be the same for all modes. We also assume that the filtration, in addition to satisfying the usual conditions of right-continuity and completeness, is *quasi-left-continuous*. This property, which generalises the assumption made in [@Hamadene2007a], is satisfied in many applications. For example, it holds when the filtration is the natural (completed) filtration of a Lévy process. Such models have wide ranging applications in finance, insurance and control theory [@Oksendal2007]. The layout of paper is as follows. Section \[Section:Optimal-Switching-Definitions\] introduces the probabilistic model and optimal switching problem. Preliminary concepts from the general theory of stochastic processes and optimal stopping are recalled in Section \[Section:Optimal-Switching-Preliminaries\]. The modelling assumptions for the optimal switching problem are given in Section \[Section:Optimal-Switching-Assumptions\]. A verification theorem establishing the relationship between the optimal switching problem and iterative optimal stopping is given in Section \[Section:Optimal-Switching-VerificationTheorem\]. Sufficient conditions for validating the verification theorem’s hypotheses are discussed in Section \[Section:Optimal-Switching-Existence-Proof\]. The conclusion, appendix, acknowledgements and references then follow. Definitions. {#Section:Optimal-Switching-Definitions} ============ Probabilistic setup. -------------------- We work on a time horizon $[0,T]$, where $0 < T < \infty$. It is assumed that a complete filtered probability space, $\left(\Omega,\mathcal{F},\mathbb{F},\mathsf{P}\right)$, has been given and the filtration $\mathbb{F} = \left(\mathcal{F}_{t}\right)_{0 \le t \le T}$ satisfies the usual conditions of right-continuity and augmentation by the $\mathsf{P}$-null sets. Let $\mathsf{E}$ denote the corresponding expectation operator. We use $\mathbf{1}_{A}$ to represent the indicator function of a set (event) $A$. The shorthand notation a.s. means “almost surely”. Let $\mathcal{T}$ denote the set of $\mathbb{F}$-stopping times $\nu$ which satisfy $0 \le \nu \le T$ $\mathsf{P}$-a.s. For a given $S \in \mathcal{T}$, write $\mathcal{T}_{S} = \{\nu \in \mathcal{T} \colon \nu \ge S \enskip \mathsf{P}-a.s.\}$. Unless otherwise stated, a stopping time is assumed to be defined with respect to $\mathbb{F}$. For notational convenience the dependence on $\omega \in \Omega$ is often suppressed. Problem definition. ------------------- The controller in an optimal switching problem influences a dynamical system over the horizon $[0,T]$ by choosing operating modes from a finite set $\mathbb{I} = \lbrace 1,\ldots,m \rbrace$ with $m \ge 2$. The instantaneous profit in mode $i \in \mathbb{I}$ is a mapping $\psi_{i} \colon \Omega \times [0,T] \to \mathbb{R}$. There is a cost for switching from mode $i$ to $j$ which is given by $\gamma_{i,j} \colon \Omega \times [0,T] \to \mathbb{R}$. There is also a reward for being in mode $i \in \mathbb{I}$ at time $T$, denoted by $\Gamma_{i}$, which is a real-valued random variable. The assumptions on these costs / rewards are discussed below in Section \[Section:Optimal-Switching-Assumptions\]. \[Definition:Optimal-Switching-AdmissibleStrategies\] Let $t \in [0,T]$ and $i \in \mathbb{I}$ be given. An admissible switching control strategy starting from $(t,i)$ is a double sequence $\alpha = \left(\tau_{n},\iota_{n}\right)_{n \ge 0}$ of stopping times $\tau_{n} \in \mathcal{T}_{t}$ and mode indicators $\iota_{n}$ such that: 1. $\tau_{0} = t$ and the sequence $\lbrace \tau_{n} \rbrace_{n \ge 0}$ is non-decreasing; 2. Each $\iota_{n} \colon \Omega \to \mathbb{I}$ is $\mathcal{F}_{\tau_{n}}$-measurable; $\iota_{0} = i$ and $\iota_{n} \neq \iota_{n+1}$ for $n \ge 0$; 3. Only a finite number of switching decisions can be made before the terminal time $T$: $$\label{eq:Optimal-Switching-FiniteStrategy} \mathsf{P}\left(\{\tau_{n} < T,\hspace{1bp} \forall n \ge 0\}\right) = 0.$$ 4. The family of random variables $\{ C^{\alpha}_{n}\}_{n \ge 1}$, where $C^{\alpha}_{n}$ is the total cost of the first $n \ge 1$ switches $$\label{eq:Optimal-Switching-CumulativeSwitchingCost} C^{\alpha}_{n} \coloneqq \sum\limits_{k = 1}^{n}\gamma_{\iota_{k-1},\iota_{k}}(\tau_{k})\mathbf{1}_{\{\tau_{k} < T\}}$$ satisfies $$\label{eq:Optimal-Switching-UISupremumClosureSwitchingCosts} \mathsf{E}\big[\sup_{n}\big|C^{\alpha}_{n}\big|\big] < \infty.$$ Let $\mathcal{A}_{t,i}$ denote the set of admissible switching control strategies (henceforth, just strategies). We write $\mathcal{A}_{i}$ when $t = 0$ and drop the subscript $i$ if it is not important for the discussion. \[Remark:Optimal-Switching-SubscriptModeIndicators\] Processes or functions with super(sub)-scripts in terms of the random mode indicators $\iota_{n}$ are interpreted in the following way: $$\begin{aligned} Y^{\iota_{n}} & = \sum\limits_{j \in \mathbb{I}}\mathbf{1}_{\lbrace \iota_{n} = j \rbrace} Y^{j}, \quad n\ge 0 \\ \gamma_{\iota_{n-1},\iota_{n}}\left(\cdot\right) & = \sum\limits_{j \in \mathbb{I}}\sum\limits_{k \in \mathbb{I}}\mathbf{1}_{\lbrace \iota_{n-1} = j \rbrace}\mathbf{1}_{\lbrace \iota_{n} = k \rbrace} \gamma_{j,k}\left(\cdot\right), \quad n \ge 1. \end{aligned}$$ Note that the summations are finite. We shall frequently use the notation $N(\alpha)$ to denote the (random) number of switches before $T$ under strategy $\alpha$: $$\label{eq:Optimal-Switching-RandomNumberOfSwitches} N(\alpha) = \sum_{n \ge 1}\mathbf{1}_{\{\tau_{n} < T\}}, \quad \alpha \in \mathcal{A}.$$ Associated with each strategy $\alpha \in \mathcal{A}$ is a mode indicator function $\mathbf{u} \colon \Omega \times [0,T] \to \mathbb{I}$ that gives the active mode at each time [@Djehiche2009; @Guo2008b]: $$\label{eq:Optimal-Switching-ModeIndicatorContinuousTime} \mathbf{u}_{t} \coloneqq \iota_{0}\mathbf{1}_{[\tau_{0},\tau_{1}]}(t) + \sum\limits_{n \ge 1}\iota_{n}\mathbf{1}_{(\tau_{n},\tau_{n+1}]}(t),\hspace{1em} t \in [0,T].$$ For a fixed time $t \in [0,T]$ and given mode $i \in \mathbb{I}$, the performance index for the optimal switching problem starting at $t$ in mode $i$ is given by: $$\label{eq:Optimal-Switching-DynamicPerformanceIndex} J(\alpha;t,i) = \mathsf{E}\left[\int_{t}^{T}\psi_{\mathbf{u}_{s}}(s){d}s + \Gamma_{\mathbf{u}_{T}} - \sum_{n \ge 1}\gamma_{\iota_{n-1},\iota_{n}}(\tau_{n})\mathbf{1}_{\{ \tau_{n} < T \}} \biggm \vert \mathcal{F}_{t}\right],\hspace{1em} \alpha \in \mathcal{A}_{t,i}.$$ The goal is to find a strategy $\alpha^{*} \in \mathcal{A}_{t,i}$ that maximises the performance index: $$\label{eq:Optimal-Switching-ValueFunction} J(\alpha^{*};t,i) = \operatorname*{ess\,sup}\limits_{\alpha \in \mathcal{A}_{t,i}}J\left(\alpha;t,i\right) \eqqcolon V(t,i).$$ The random function $V(t,i)$ is called the *value function* for the optimal switching problem. \[Note:IntegrabilityConcerns\] For $\alpha \in \mathcal{A}$, define $C^{\alpha}$ to be the total switching cost under $\alpha$: $$C^{\alpha} \coloneqq \sum\limits_{n \ge 1}\gamma_{\iota_{n-1},\iota_{n}}(\tau_{n})\mathbf{1}_{\{\tau_{n} < T\}}$$ By the finiteness condition , we have $\forall \alpha \in \mathcal{A}$: $$C^{\alpha} = \lim_{n \to \infty} C^{\alpha}_{n} \enskip \mathsf{P}-\text{a.s.}$$ Furthermore, by using condition , we have the following dominated convergence property: $$\label{eq:Optimal-Switching-ConvergenceOfSwitchingCost} \forall \alpha \in \mathcal{A}: \quad \lim_{n \to \infty}\mathsf{E}\left[C^{\alpha}_{n}\vert \mathcal{B}\right] = \mathsf{E}\left[C^{\alpha}\vert \mathcal{B}\right] \hspace{1em} \text{a.s. for every } \sigma\text{-algebra} \hspace{5bp} \mathcal{B} \subset \mathcal{F}.$$ Preliminaries. {#Section:Optimal-Switching-Preliminaries} ============== Some results from the general theory of stochastic processes. ------------------------------------------------------------- We need to recall a few results from the general theory of stochastic processes that are essential to this paper. For more details the reader is kindly referred to the references [@Elliott1982; @Jacod2003; @Rogers2000b]. ### Right-continuous with left-limits processes. An adapted process $X = \left(X_{t}\right)_{0 \le t \le T}$ is said to be càdlàg if it is right-continuous and admits left limits. The left-limits process associated with a càdlàg process $X$ is denoted by $X_{-} = \left(X_{t^{-}}\right)_{0 < t \le T}$ (here we follow the convention of [@Rogers2000b]). We also define the process $\triangle X$ by $\triangle X \coloneqq X - X_{-}$ and let $\triangle_{t}X \coloneqq X_{t} - X_{t^{-}}$ denote the size of the jump in $X$ at $t \in (0,T]$. ### Predictable random times. A random time $S$ is an $\mathcal{F}$-measurable mapping $S \colon \Omega \to [0,T]$. For two random times $\rho$ and $\tau$, the stochastic interval $[\rho,\tau]$ is defined as: $$[\rho,\tau] = \left\lbrace (\omega,t) \in \Omega \times [0,T] \colon \rho(\omega) \le t \le \tau(\omega) \right\rbrace.$$ Stochastic intervals $(\rho,\tau]$, $[\rho,\tau)$, $(\rho,\tau)$ are defined analogously. A random time $S > 0$ is said to be predictable if the stochastic interval $[0,S)$ is measurable with respect to the predictable $\sigma$-algebra (the $\sigma$-algebra on $\Omega \times (0,T]$ generated by the adapted processes with paths that are left-continuous with right-limits on $(0,T]$). Note that every predictable time is a stopping time [@Jacod2003 p. 17]. By Meyer’s previsibility (predictability) theorem ([@Rogers2000b], Theorem .12.6), a stopping time $S > 0$ is predictable if and only if it is announceable in the following sense: there exists a sequence of stopping times $\lbrace S_{n} \rbrace_{n \ge 0}$ satisfying $S_{n}(\omega) \le S_{n+1}(\omega) < S(\omega)$ for all $n$ and $\lim_{n}S_{n}(\omega) = S(\omega)$. ### Quasi-left-continuous processes and filtrations. A càdlàg process $X$ is called quasi-left-continuous if $\triangle_{S} X = 0$ a.s. for every predictable time $S$ (Definition .2.25 of [@Jacod2003]). The strict pre-$S$ $\sigma$-algebra associated with a random time $S > 0$, $\mathcal{F}_{S^{-}}$, is defined as [@Rogers2000b p. 345]: $$\mathcal{F}_{S^{-}} = \sigma\left(\lbrace A \cap \lbrace S > u \rbrace \colon 0 \le u \le T, A \in \mathcal{F}_{u} \rbrace\right).$$ According to [@Rogers2000b p. 346], a filtration $\mathbb{F} = (\mathcal{F}_{t})_{0 \le t \le T}$ which satisfies the usual conditions is said to be *quasi-left-continuous* if $\mathcal{F}_{S} = \mathcal{F}_{S^{-}}$ for every predictable time $S$. We have the following equivalence result for quasi-left-continuous filtrations (see [@Rogers2000b], Theorem .18.1 and [@Elliott1982], Theorem 5.36). \[Proposition:Optimal-Switching-qlcFiltrations\] The following statements are equivalent: 1. $\mathbb{F}$ satisfies the usual conditions (right-continuous and $\mathsf{P}$-complete) and is quasi-left-continuous; 2. For every bounded (and then for every uniformly integrable) càdlàg martingale $M$ and every predictable time $S$, we have $\triangle_{S} M = 0$ a.s.; 3. For every increasing sequence of stopping times $\lbrace S_{n} \rbrace$ with limit, $\lim_{n}S_{n} = S$, we have $$\mathcal{F}_{S} = \bigvee_{n} \mathcal{F}_{S_{n}}.$$ Some notation. {#Section:Continuous-Time-Optimal-Switching-Notation} -------------- Let us now define some notation that is frequently used below. 1. For $1 \le p < \infty$, we write $L^{p}$ to denote the set of random variables $Z$ satisfying $\mathsf{E}\left[|Z|^{p}\right] < \infty$. 2. Let $\mathcal{Q}$ denote the set of adapted, càdlàg processes which are quasi-left-continuous. 3. Let $\mathcal{M}^{2}$ denote the set of progressively measurable processes $X = \left(X_{t}\right)_{0 \le t \le T}$ satisfying, $$\mathsf{E}\left[\int_{0}^{T}|X_{t}|^{2}{d}t\right] < \infty.$$ 4. Let $\mathcal{S}^{2}$ denote the set of adapted, càdlàg processes $X$ satisfying: $$\mathsf{E}\left[\left(\sup\nolimits_{0 \le t \le T}\left|X_{t}\right|\right)^{2}\right] < \infty.$$ Properties of Snell envelopes. {#Section:Optimal-Switching-SnellEnvelopes} ------------------------------ The following properties of Snell envelopes are also essential for our results. Recall that a progressively measurable process $X$ is said to belong to class $[D]$ if the set of random variables $\lbrace X_{\tau}, \tau \in \mathcal{T} \rbrace$ is uniformly integrable. \[Proposition:Optimal-Switching-SnellEnvelopeProperties\] Let $U = (U_{t})_{0 \le t \le T}$ be an adapted, $\mathbb{R}$-valued, càdlàg process that belongs to class $[D]$. Then there exists a unique (up to indistinguishability), adapted $\mathbb{R}$-valued càdlàg process $Z = (Z_{t})_{0 \le t \le T}$ such that $Z$ is the smallest supermartingale which dominates $U$. The process $Z$ is called the Snell envelope of $U$ and it enjoys the following properties. 1. For any stopping time $\theta$ we have: $$Z_{\theta} = \operatorname*{ess\,sup}_{\tau \in \mathcal{T}_{\theta}}\mathsf{E}\left[U_{\tau} \vert \mathcal{F}_{\theta}\right],\text{ and therefore }Z_{T} = U_{T}.$$ 2. Meyer decomposition: There exist a uniformly integrable right-continuous martingale $M$ and two non-decreasing, adapted, predictable and integrable processes $A$ and $B$, with $A$ continuous and $B$ purely discontinuous, such that for all $0 \le t \le T$, $$\label{SnellEnvelope:MeyerDecomposition} Z_{t} = M_{t} - A_{t} - B_{t}, \hspace{1em} A_{0} = B_{0} = 0.$$ Furthermore, the jumps of $B$ satisfy $\lbrace \triangle B > 0 \rbrace \subset \lbrace Z_{-} = U_{-} \rbrace$. 3. Let a stopping time $\theta$ be given and let $\{\tau_{n}\}_{n \ge 0}$ be an increasing sequence of stopping times tending to a limit $\tau$ such that each $\tau_{n} \in \mathcal{T}_{\theta}$ and satisfies $\mathsf{E}\left[U^{-}_{\tau_{n}}\right] < \infty$. Suppose the following condition is satisfied for any such sequence, $$\label{eq:Optimal-Switching-LeftUpperSemicontinuity} \limsup_{n \to \infty} U_{\tau_{n}} \le U_{\tau}$$ Then the stopping time $\tau^{*}_{\theta}$ defined by $$\label{eq:Optimal-Switching-DebutTime} \tau^{*}_{\theta} = \inf\lbrace t \ge \theta \colon Z_{t} = U_{t} \rbrace \wedge T$$ is optimal after $\theta$ in the sense that: $$\label{eq:Optimal-Switching-OptimalityOfDebutTime} Z_{\theta} = \mathsf{E}\left[Z_{\tau^{*}_{\theta}} \vert \mathcal{F}_{\theta}\right] = \mathsf{E}\left[U_{\tau^{*}_{\theta}} \vert \mathcal{F}_{\theta}\right] = \operatorname*{ess\,sup}_{\tau \in \mathcal{T}_{\theta}}\mathsf{E}\left[U_{\tau} \vert \mathcal{F}_{\theta}\right].$$ 4. For every $\theta \in \mathcal{T}$, if $\tau^{*}_{\theta}$ is the stopping time defined in equation , then the stopped process $\left(Z_{t \wedge \tau^{*}_{\theta}}\right)_{\theta \le t \le T}$ is a (uniformly integrable) càdlàg martingale. 5. Let $\lbrace U^{n} \rbrace_{n \ge 0}$ and $U$ be adapted, càdlàg and of class $[D]$ and let $Z^{U^{n}}$ and $Z$ denote the Snell envelopes of $U^{n}$ and $U$ respectively. If the sequence $\lbrace U^{n} \rbrace_{n \ge 0}$ is increasing and converges pointwise to $U$, then the sequence $\lbrace Z^{U^{n}} \rbrace_{n \ge 0}$ is also increasing and converges pointwise to $Z$. Furthermore, if $U \in \mathcal{S}^{2}$ then $Z \in \mathcal{S}^{2}$. References for these properties can be found in the appendix of [@Hamadene2002] and other references such as [@ElKaroui1981; @Morimoto1982; @Peskir2006]. Proof of the fifth property can be found in Proposition 2 of [@Djehiche2009]. We also have the following result concerning integrability of the components in the Doob-Meyer decomposition. \[Proposition:Optimal-Switching-SquareIntegrableMartingale\] For $0 \le t \le T$, let $Z_{t} = M_{t} - A_{t}$ where 1. the process $Z = (Z_{t})_{0 \le t \le T}$ is in $\mathcal{S}^{2}$; 2. the process $M = (M_{t})_{0 \le t \le T}$ is a càdlàg, quasi-left-continuous martingale with respect to $\mathbb{F}$; 3. the process $A = (A_{t})_{0 \le t \le T}$ is an $\mathbb{F}$-adapted càdlàg increasing process. Then $A$ (and therefore $M$) is also in $\mathcal{S}^{2}$. The proof essentially uses an integration by parts formula on $(A_{T})^{2}$ and the decomposition $Z = M - A$ in the hypothesis. See Proposition A.5 of [@Hamadene2002] for further details, noting that the same proof works for quasi-left-continuous $M$. Assumptions {#Section:Optimal-Switching-Assumptions} =========== \[Assumption:QLCFiltration\] The filtration $\mathbb{F}$ satisfies the usual conditions of right-continuity and $\mathsf{P}$-completeness and is also quasi-left-continuous. \[Assumption:ProfitandCost\] For every $i,j \in \mathbb{I}$ we suppose: 1. the instantaneous profit satisfies $\psi_{i} \in \mathcal{M}^{2}$; 2. the switching cost satisfies $\gamma_{i,j} \in \mathcal{Q} \cap \mathcal{S}^{2}$. 3. the terminal data $\Gamma_{i} \in L^{2}$ and is $\mathcal{F}_{T}$-measurable. \[Assumption:SwitchingCosts\] For every $i,j,k \in \mathbb{I}$ and $\forall t \in [0,T]$, we have a.s.: $$\label{eq:Optimal-Switching-NoArbitrage} \begin{cases} \gamma_{i,i}\left(t\right) = 0 \\ \gamma_{i,k}\left(t\right) < \gamma_{i,j}\left(t\right) + \gamma_{j,k}\left(t\right),\hspace{1em} \text{ if } i \neq j \text{ and } j \neq k,\\ \Gamma_{i} \ge \max\limits_{j \neq i}\left\lbrace \Gamma_{j} -\gamma_{i,j}(T) \right\rbrace. \end{cases}$$ \[Remark:SubOptimalSwitchTwice\] The first line in condition  shows there is no cost for staying in the same mode. The other two rule out possible arbitrage opportunities (also see [@Guo2008b; @Hamadene2012]). In particular, we can always restrict our attention to those strategies $\alpha = \left(\tau_{n},\iota_{n}\right)_{n \ge 0} \in \mathcal{A}$ such that $\mathsf{P}\big( \lbrace \tau_{n} = \tau_{n+1}, \tau_{n} < T \rbrace) = 0$ for $n \ge 1$. Indeed, if $H_{n} \coloneqq \lbrace \tau_{n} = \tau_{n+1}, \tau_{n} < T \rbrace$ satisfies $\mathsf{P}(H_{n}) > 0$ for some $n \ge 1$, then by the second line in condition  we get $$\begin{aligned} \left(\gamma_{\iota_{n-1},\iota_{n}}(\tau_{n}) + \gamma_{\iota_{n},\iota_{n+1}}(\tau_{n+1})\right)\mathbf{1}_{H_{n}} & = \left(\gamma_{\iota_{n-1},\iota_{n}}(\tau_{n}) + \gamma_{\iota_{n},\iota_{n+1}}(\tau_{n})\right)\mathbf{1}_{H_{n}} \\ & > \left(\gamma_{\iota_{n-1},\iota_{n+1}}(\tau_{n})\right)\mathbf{1}_{H_{n}} \end{aligned}$$ which shows it is suboptimal to switch twice at the same time. A verification theorem. {#Section:Optimal-Switching-VerificationTheorem} ======================= Throughout this section, we suppose that there exist processes $Y^{1},\ldots,Y^{m}$ in $\mathcal{Q} \cap \mathcal{S}^{2}$ defined by $$\label{eq:Optimal-Switching-OptimalSwitchingProcessesDefinition} \begin{split} Y^{i}_{t} & = \operatorname*{ess\,sup}\limits_{\tau \in \mathcal{T}_{t}}\mathsf{E}\left[\int_{t}^{\tau}\psi_{i}(s){d}s + \Gamma_{i}\mathbf{1}_{\{\tau = T \}} + \max\limits_{j \neq i}\left\lbrace Y^{j}_{\tau} -\gamma_{i,j}(\tau) \right\rbrace\mathbf{1}_{\{\tau < T \}}\biggm\vert \mathcal{F}_{t}\right],\\ Y^{i}_{T} & = \Gamma_{i}. \end{split}$$ Sufficient conditions ensuring the existence of $Y^{1},\ldots,Y^{m}$ with these properties are given in Section \[Section:Optimal-Switching-Existence-Proof\]. Theorem \[Theorem:Optimal-Switching-VerificationPartial\] below verifies that the solution to the optimal switching problem  can be written in terms of these $m$ stochastic processes. In preparation of this *verification theorem*, we need a few preliminary results. Let $U^{i} = \left(U^{i}_{t}\right)_{0 \le t \le T}$, $i \in \mathbb{I}$, be a càdlàg process defined by: $$\label{eq:Optimal-Switching-InterconnectedObstacleProcess} U^{i}_{t} \coloneqq \Gamma_{i}\mathbf{1}_{\{ t = T \}} + \max\limits_{j \neq i} \left\lbrace Y^{j}_{t} - \gamma_{i,j}\left(t\right)\right\rbrace\mathbf{1}_{\{ t < T \}}, \hspace{1em} 0 \le t \le T.$$ Recall that for every $i,j \in \mathbb{I}$ we have $\gamma_{i,j}, Y^{i} \in \mathcal{Q} \cap \mathcal{S}^{2}$, $\Gamma_{i} \in L^{2}$ by assumption. Hence the process $U^{i} \in \mathcal{S}^{2}$ and is therefore of class $[D]$. Recalling Proposition \[Proposition:Optimal-Switching-SnellEnvelopeProperties\] and rewriting equation  for $Y^{i}_{t}$ as follows, $$\begin{aligned} \label{eq:Optimal-Switching-SnellEnvelopeCharacterisation} Y^{i}_{t} & = \operatorname*{ess\,sup}\limits_{\tau \in \mathcal{T}_{t}}\mathsf{E}\left[\int_{t}^{\tau}\psi_{i}(s){d}s + U^{i}_{\tau}\biggm\vert \mathcal{F}_{t}\right] \nonumber \\ & = \operatorname*{ess\,sup}\limits_{\tau \in \mathcal{T}_{t}}\mathsf{E}\left[\int_{0}^{\tau}\psi_{i}(s){d}s + U^{i}_{\tau}\biggm\vert \mathcal{F}_{t}\right] - \int_{0}^{t}\psi_{i}(s){d}s,\hspace{1em}\mathsf{P}-\text{a.s.}\end{aligned}$$ we can verify that $\left(Y^{i}_{t} + \int_{0}^{t} \psi_{i}(s){d}s \right)_{0 \le t \le T}$ is the Snell envelope of $\left(U^{i}_{t} + \int_{0}^{t} \psi_{i}(s){d}s\right)_{0 \le t \le T}$. \[Lemma:Optimal-Switching-VerificationLemma\] Suppose that $Y^{1},\ldots,Y^{m}$ defined in are in $\mathcal{Q} \cap \mathcal{S}^{2}$. For each $i \in \mathbb{I}$, let $U^{i}$ be defined as in equation . Then for every $\tau_{n} \in \mathcal{T}$ and $\mathcal{F}_{\tau_{n}}$-measurable $\iota_{n} \colon \Omega \to \mathbb{I}$, we have $$\label{eq:Optimal-Switching-VerificationLemmaSnellEnvelope} Y^{\iota_{n}}_{t} = \operatorname*{ess\,sup}\limits_{\tau \in \mathcal{T}_{t}}\mathsf{E}\left[\int_{t}^{\tau}\psi_{\iota_{n}}(s){d}s + U^{\iota_{n}}_{\tau} \biggm \vert \mathcal{F}_{t}\right], \hspace{1em} \mathsf{P}-\text{a.s. }\forall \hspace{2bp} \tau_{n} \le t \le T.$$ Furthermore, there exist a uniformly integrable càdlàg martingale $M^{\iota_{n}} = \left(M^{\iota_{n}}_{t}\right)_{\tau_{n} \le t \le T}$ and a predictable, continuous, increasing process $A^{\iota_{n}} = \left(A^{\iota_{n}}_{t}\right)_{\tau_{n} \le t \le T}$ such that $$\label{eq:Optimal-Switching-VerificationLemmaMeyerDecomposition} Y^{\iota_{n}}_{t} + \int_{0}^{t}\psi_{\iota_{n}}(s){d}s = M^{\iota_{n}}_{t} - A^{\iota_{n}}_{t},\hspace{1em}\mathsf{P}-\text{a.s. }\forall \hspace{2bp} \tau_{n} \le t \le T.$$ The claim  is established in the same way as the first few lines of Theorem 1 in [@Djehiche2009] so the proof is sketched. We need to show that $Y^{\iota_{n}}_{t} + \int_{0}^{t}\psi_{\iota_{n}}(s){d}s$ is the Snell envelope of $U^{\iota_{n}}_{t} + \int_{0}^{t}\psi_{\iota_{n}}(s){d}s$ for $\tau_{n} \le t \le T$. Our previous discussion established under the current hypotheses that, for every $i \in \mathbb{I}$, $Y^{i}_{t} + \int_{0}^{t}\psi_{i}(s){d}s$ is the Snell envelope of $U^{i}_{t} + \int_{0}^{t}\psi_{i}(s){d}s$ on $[0,T]$. Since $\mathbf{1}_{\lbrace \iota_{n} = i \rbrace}$ is non-negative and $\mathcal{F}_{\tau_{n}}$-measurable, we can show that $\left(Y^{i}_{t} + \int_{0}^{t}\psi_{i}(s){d}s\right)\mathbf{1}_{\lbrace \iota_{n} = i \rbrace}$ is the smallest càdlàg supermartingale dominating $\left(U^{i}_{t} + \int_{0}^{t}\psi_{i}(s){d}s\right)\mathbf{1}_{\lbrace \iota_{n} = i \rbrace}$ on $[\tau_{n},T]$. By summing over $i \in \mathbb{I}$ (recall $\mathbb{I}$ is finite), we have $\left(Y^{\iota_{n}}_{t} + \int_{0}^{t}\psi_{\iota_{n}}(s){d}s\right)$ is the smallest càdlàg supermartingale dominating $\left(U^{\iota_{n}}_{t} + \int_{0}^{t}\psi_{\iota_{n}}(s){d}s\right)$ for $\tau_{n} \le t \le T$. In particular, we have $$Y^{\iota_{n}}_{t} + \int_{0}^{t}\psi_{\iota_{n}}(s){d}s = \operatorname*{ess\,sup}\limits_{\tau \in \mathcal{T}_{t}}\mathsf{E}\left[\int_{0}^{\tau}\psi_{\iota_{n}}(s){d}s + U^{\iota_{n}}_{\tau}\biggm\vert \mathcal{F}_{t}\right],\hspace{1em}\mathsf{P}-\text{a.s. }\forall \hspace{1bp} t \le \tau_{n} \le T,$$ and equation  follows by $\mathcal{F}_{t}$-measurability of the integral term for $t \ge \tau_{n}$. For the second part of the claim, we use the unique Meyer decomposition of the Snell envelope (property 2 of Proposition \[Proposition:Optimal-Switching-SnellEnvelopeProperties\]) to show that for every $i \in \mathbb{I}$, $$Y^{i}_{t} + \int_{0}^{t}\psi_{i}(s){d}s = M^{i}_{t} - A^{i}_{t}\hspace{1em} \text{for } t \in [0,T],$$ where $M^{i} = \left(M^{i}_{t}\right)_{0 \le t \le T}$ is a càdlàg uniformly integrable martingale and $A^{i} = \left(A^{i}_{t}\right)_{0 \le t \le T}$ is a predictable, increasing process. The Snell envelope $\left(Y^{i}_{t} + \int_{0}^{t}\psi_{i}(s){d}s\right)_{0 \le t \le T}$ is in $\mathcal{Q} \cap \mathcal{S}^{2}$ since $Y^{i} \in \mathcal{Q} \cap \mathcal{S}^{2}$ and $\psi_{i} \in \mathcal{M}^{2}$. This means the Snell envelope is a regular supermartingale of class $[D]$ and Theorem .10 of [@Dellacherie1982] asserts that its compensator, $A^{i}$, is continuous. Using the Meyer decomposition, we see that $$\label{eq:Optimal-Switching-VerificationMeyerDecompositionFull} Y^{\iota_{n}}_{t} + \int_{0}^{t}\psi_{\iota_{n}}(s){d}s \coloneqq \sum_{i \in \mathbb{I}}\left(Y^{i}_{t} + \int_{0}^{t}\psi_{i}(s){d}s\right)\mathbf{1}_{\{\iota_{n} = i\}} = \sum_{i \in \mathbb{I}}\left(M^{i}_{t} - A^{i}_{t}\right)\mathbf{1}_{\{\iota_{n} = i\}}.$$ Now, using $\mathbf{1}_{\lbrace \iota_{n} = i \rbrace}$ is non-negative and $\mathcal{F}_{t}$-measurable for $t \ge \tau_{n}$, we see that $M^{\iota_{n}}$ defined on $[\tau_{n},T]$ by $$\label{eq:Optimal-Switching-VerificationMeyerDecompositionMartingale} M^{\iota_{n}(\omega)}_{t}(\omega) \coloneqq \sum_{i \in \mathbb{I}}M^{i}_{t}(\omega)\mathbf{1}_{\{\iota_{n} = i\}}(\omega),\hspace{1em} \forall\hspace{1bp} (\omega,t) \in [\tau_{n},T]$$ is a uniformly integrable càdlàg martingale $\mathsf{P}$-a.s. for every $\tau_{n} \le t \le T$. Likewise, $A^{\iota_{n}}$ defined on $[\tau_{n},T]$ by $$\label{eq:Optimal-Switching-VerificationMeyerDecompositionCompensator} A^{\iota_{n}(\omega)}_{t}(\omega) \coloneqq \sum_{i \in \mathbb{I}}A^{i}_{t}(\omega)\mathbf{1}_{\{\iota_{n} = i\}}(\omega),\hspace{1em} \forall\hspace{1bp} (\omega,t) \in [\tau_{n},T]$$ is a continuous, predictable increasing process $\mathsf{P}$-a.s. for every $\tau_{n} \le t \le T$. By equation , $M^{\iota_{n}}_{t}$ and $A^{\iota_{n}}_{t}$ provide the (unique) Meyer decomposition of $Y^{\iota_{n}}_{t}$ $\mathsf{P}$-a.s. for every $\tau_{n} \le t \le T$. \[Theorem:Optimal-Switching-VerificationPartial\] Suppose there exist $m$ unique processes $Y^{1},\ldots,Y^{m}$ in $\mathcal{Q} \cap \mathcal{S}^{2}$ which satisfy equation . Define a sequence of times $\left\lbrace\tau^{*}_{n}\right\rbrace_{n \ge 0}$ and mode indicators $\left\lbrace\iota^{*}_{n}\right\rbrace_{n \ge 0}$ as follows: $$\begin{gathered} \label{eq:Optimal-Switching-OptimalStoppingStrategy} \tau^{*}_{0} = t,\hspace{1em} \iota^{*}_{0} = i, \\ \begin{cases} \tau^{*}_{n} = \inf\left\lbrace s \ge \tau^{*}_{n-1} \colon Y^{\iota^{*}_{n-1}}_{s} = \max\limits_{j \neq \iota^{*}_{n-1}} \left(Y^{j}_{s} - \gamma_{\iota^{*}_{n-1},j}\left(s\right) \right) \right\rbrace \wedge T,\\ \iota^{*}_{n} = \sum\limits_{j \in \mathbb{I}} j \mathbf{1}_{F^{\iota^{*}_{n-1}}_{j}} \end{cases} \nonumber \\ \text{for } n \ge 1, \text{ where } F^{\iota^{*}_{n-1}}_{j} \text{ is the event} : \nonumber \\ F^{\iota^{*}_{n-1}}_{j} \coloneqq \left\lbrace Y^{j}_{\tau^{*}_{n}} -\gamma_{\iota^{*}_{n-1},j}\left(\tau^{*}_{n}\right) = \max\limits_{k \neq \iota^{*}_{n-1}}\left( Y^{k}_{\tau^{*}_{n}} -\gamma_{\iota^{*}_{n-1},k}\left(\tau^{*}_{n}\right) \right) \right\rbrace. \nonumber \end{gathered}$$ Then the sequence $\alpha^{*} = \left(\tau^{*}_{n},\iota^{*}_{n}\right)_{n \ge 0} \in \mathcal{A}_{t,i}$ and satisfies $$\label{eq:Optimal-Switching-UpperBoundOnThePerformanceIndex} Y^{i}_{t} = J(\alpha^{*};t,i) = \operatorname*{ess\,sup}\limits_{\alpha \in \mathcal{A}_{t,i}}J(\alpha;t,i) \hspace{1em} \mathsf{P}-\text{a.s.}$$ Standard arguments can be used to verify that $\tau^{*}_{n}$ is a stopping time and each $\iota^{*}_{n}$ is $\mathcal{F}_{\tau^{*}_{n}}$-measurable. The appendix confirms that $\alpha^{*} \in \mathcal{A}_{t,i}$. As for the claim , it holds trivially for $t = T$ since $Y^{i}_{T} = \Gamma_{i} = V(t,i)$ a.s. for every $i \in \mathbb{I}$. Henceforth, we assume that $t \in [0,T)$. Recall the process $U^{i} = \left(U^{i}_{t}\right)_{0 \le t \le T}$ defined in equation . By our assumptions on $Y^{i},\psi_{i},\Gamma_{i}$ and $\gamma_{i,j}$ for every $i,j \in \mathbb{I}$, we have $U^{i} \in \mathcal{S}^{2}$ and we assert that $\left(Y^{i}_{t} + \int_{0}^{t} \psi_{i}(s){d}s \right)_{0 \le t \le T}$ is the Snell envelope of $\left(U^{i}_{t} + \int_{0}^{t} \psi_{i}(s){d}s \right)_{0 \le t \le T}$. For $i,j \in \mathbb{I}$, using $Y^{j}_{T} = \Gamma_{j}$ $\mathsf{P}$-a.s., quasi-left-continuity of $Y^{j}$ and $\gamma_{i,j}$, and Assumption \[Assumption:SwitchingCosts\] on the terminal condition for the switching costs, we have $$\lim_{t \uparrow T}\left(\max_{j \neq i}\left\lbrace Y^{j}_{t} -\gamma_{i,j}(t) \right\rbrace\right) = \max_{j \neq i}\left\lbrace \Gamma_{j} - \gamma_{i,j}(T) \right\rbrace \le \Gamma_{i} \hspace{1em} \mathsf{P}-\text{a.s.}$$ Therefore, $U^{i}$ is quasi-left-continuous on $[0,T)$ and $\lim_{t \uparrow T}U^{i}_{t} \le U^{i}_{T}$ $\mathsf{P}$-a.s. Combining this with the continuity of the integral, we see that $\left(U^{i}_{t} + \int_{0}^{t} \psi_{i}(s){d}s \right)_{0 \le t \le T}$ satisfies the hypotheses of property 3 in Proposition \[Proposition:Optimal-Switching-SnellEnvelopeProperties\]. Let $\left(\tau_{n}^{*},\iota_{n}^{*}\right)_{n \ge 0}$ be the pair of random times and mode indicators in the statement of the theorem and $\mathbf{u}^{*}$ be the associated mode indicator function. In conjunction with Lemma \[Lemma:Optimal-Switching-VerificationLemma\], $\lbrace \tau_{n}^{*}\rbrace$ defines a sequence of stopping times where, for $n \ge 1$, $\tau^{*}_{n}$ is optimal for an appropriately defined optimal stopping problem. The remaining arguments, which are similar to those establishing Theorem 1 in [@Djehiche2009], are only sketched here. The main idea is as follows: starting from an initial mode $i \in \mathbb{I}$ at time $t \in [0,T]$, iteratively solve the optimal stopping problem on the right-hand-side of using the theory of Snell envelopes and Lemma \[Lemma:Optimal-Switching-VerificationLemma\]. The minimal optimal stopping times characterise the switching times whilst the maximising modes are paired with them to give the switching strategy. This characterisation will eventually lead to: $$\label{eq:Optimal-Switching-VerificationProof5} \begin{split} \forall N \ge 1,\hspace{1em} Y^{i}_{t} = {} & \mathsf{E}\left[\int_{t}^{\tau^{*}_{N}}\psi_{\mathbf{u}^{*}_{s}}(s){d}s + \sum_{n = 1}^{N}\Gamma_{\iota^{*}_{n-1}}\mathbf{1}_{\lbrace \tau^{*}_{n-1} < T \rbrace}\mathbf{1}_{\lbrace \tau^{*}_{n} = T \rbrace} - \sum_{n = 1}^{N}\gamma_{\iota^{*}_{n-1},\iota^{*}_{n}}\left(\tau^{*}_{n}\right)\mathbf{1}_{\lbrace \tau^{*}_{n} < T \rbrace} \biggm \vert \mathcal{F}_{t} \right] \\ & + \mathsf{E}\left[Y^{\iota^{*}_{N}}_{\tau^{*}_{N}} \mathbf{1}_{\lbrace \tau^{*}_{N} < T \rbrace} \biggm \vert \mathcal{F}_{t} \right] \end{split}$$ By Lemma \[Lemma:Optimal-Switching-FiniteStrategyCandidateOptimal\] and Theorem \[Theorem:Optimal-Switching-SquareIntegrableSwitchingCosts\] in the appendix respectively, the times $\left\lbrace\tau_{n}^{*}\right\rbrace_{n \ge 0}$ satisfy the finiteness condition  and $\mathsf{E}\big[\sup_{n}\big|C^{\alpha^{*}}_{n}\big|\big] < \infty$ holds for the cumulative switching costs. Appealing also to the conditional dominated convergence theorem (cf. ), we may take the limit as $N \to \infty$ in equation  and use the definition of $\mathbf{u}^{*}$ to get: $$\label{eq:Optimal-Switching-VerificationProof6} Y^{i}_{t} = \mathsf{E}\left[\int_{t}^{T}\psi_{\mathbf{u}^{*}_{s}}(s){d}s + \Gamma_{\mathbf{u}^{*}_{T}} - \sum_{n \ge 1}\gamma_{\iota^{*}_{n-1},\iota^{*}_{n}}\left(\tau^{*}_{n}\right)\mathbf{1}_{\lbrace \tau^{*}_{n} < T \rbrace} \biggm \vert \mathcal{F}_{t}\right] = J(\alpha^{*};t,i).$$ Now, take any arbitrary admissible strategy $\alpha = \left(\tau_{n},\iota_{n}\right)_{n \ge 0} \in \mathcal{A}_{t,i}$. Since the sequence $(\tau_{n},\iota_{n})_{n \ge 1}$, does not necessarily achieve the essential suprema / maxima in the iterated optimal stopping problems, we have for all $N \ge 1$: $$\begin{split} Y^{i}_{t} \ge {} & \mathsf{E}\left[\int_{t}^{\tau_{N}}\psi_{\mathbf{u}_{s}}(s){d}s + \sum_{n = 1}^{N}\Gamma_{\iota_{n-1}}\mathbf{1}_{\lbrace \tau_{n-1} < T \rbrace}\mathbf{1}_{\lbrace \tau_{n} = T \rbrace} - \sum_{n = 1}^{N}\gamma_{\iota_{n-1},\iota_{n}}\left(\tau_{n}\right)\mathbf{1}_{\lbrace \tau_{n} < T \rbrace} \biggm \vert \mathcal{F}_{t} \right] \\ & + \mathsf{E}\left[Y^{\iota_{N}}_{\tau_{N}} \mathbf{1}_{\lbrace \tau_{N} < T \rbrace} \biggm \vert \mathcal{F}_{t} \right] \end{split}$$ Passing to the limit $N \to \infty$ and using the conditional dominated convergence theorem, we obtain $$J(\alpha^{*};t,i) = Y^{i}_{t} \ge \mathsf{E}\left[\int_{t}^{T}\psi_{\mathbf{u}_{s}}(s){d}s + \Gamma_{\mathbf{u}_{T}} - \sum_{n \ge 1}\gamma_{\iota_{n-1},\iota_{n}}\left(\tau_{n}\right)\mathbf{1}_{\lbrace \tau_{n} < T \rbrace} \biggm \vert \mathcal{F}_{t}\right] = J(\alpha;t,i).$$ Since $\alpha \in \mathcal{A}_{t,i}$ was arbitrary we have just proved . Existence of the candidate optimal processes. {#Section:Optimal-Switching-Existence-Proof} ============================================= The existence of the processes $Y^{1},\ldots,Y^{m}$ which satisfy Theorem \[Theorem:Optimal-Switching-VerificationPartial\] is proved in this section following the arguments of [@Djehiche2009]. The interested reader may also compare the proof to that of Lemma 2.1 and Corollary 2.1 in [@Bayraktar2010]. The case of at most $n \ge 0$ switches. --------------------------------------- For each $n \ge 0$, define process $Y^{1,n},\ldots,Y^{m,n}$ recursively as follows: for $i \in \mathbb{I}$ and for any $0 \le t \le T$, first set $$\label{eq:Optimal-Switching-OptimalProcess0Switches} Y^{i,0}_{t} = \mathsf{E}\left[\int_{t}^{T}\psi_{i}(s){d}s + \Gamma_{i} \biggm\vert \mathcal{F}_{t}\right],$$ and for $n \ge 1$, $$\label{eq:Optimal-Switching-OptimalProcessnSwitches} Y^{i,n}_{t} = \operatorname*{ess\,sup}\limits_{\tau \in \mathcal{T}_{t}}\mathsf{E}\left[\int_{t}^{\tau}\psi_{i}(s){d}s + \Gamma_{i}\mathbf{1}_{\{\tau = T \}} + \max\limits_{j \neq i}\left\lbrace Y^{j,n-1}_{\tau} - \gamma_{i,j}(\tau) \right\rbrace\mathbf{1}_{\{\tau < T \}}\biggm\vert \mathcal{F}_{t}\right].$$ Define another process $\hat{U}^{i,n} = (\hat{U}^{i,n}_{t})_{0 \le t \le T}$ by: $$\hat{U}^{i,n}_{t} \coloneqq \int_{0}^{t}\psi_{i}(s){d}s + \Gamma_{i}\mathbf{1}_{\{ t = T \}} + \max_{j \neq i}\left\lbrace Y^{j,n-1}_{t} - \gamma_{i,j}(t) \right\rbrace\mathbf{1}_{\{ t < T \}}$$ If $\hat{U}^{i,n}$ is of class $[D]$, then by Proposition \[Proposition:Optimal-Switching-SnellEnvelopeProperties\] its Snell envelope exists and is defined by $$\operatorname*{ess\,sup}\limits_{\tau \in \mathcal{T}_{t}}\mathsf{E}\bigl[\hat{U}^{i,n}_{\tau}\big \vert \mathcal{F}_{t}\bigr] = Y^{i,n}_{t} + \int_{0}^{t} \psi_{i}(s){d}s.$$ Some properties of $Y^{i,n}$ which verify this are proved in the following lemma. In order to simplify some expressions in the proof, introduce a new process $\hat{Y}^{i,n} = (\hat{Y}^{i,n}_{t})_{0 \le t \le T}$ which is defined by: $$\label{eq:Optimal-Switching-YHatProcess} \hat{Y}^{i,n}_{t} \coloneqq Y^{i,n}_{t} + \int_{0}^{t}\psi_{i}(s){d}s.$$ \[Lemma:Optimal-Switching-FinitelyManySwitchesQLC\_S2\] For all $n \ge 0$, the processes $Y^{1,n},\ldots,Y^{m,n}$ defined by and are in $\mathcal{Q} \cap \mathcal{S}^{2}$. The proof is similar to the one in [@Djehiche2009]. By $\mathcal{F}_{t}$-measurability of the integral term, we have $$\hat{Y}^{i,0}_{t} \coloneqq Y^{i,0}_{t} + \int_{0}^{t}\psi_{i}(s){d}s = \mathsf{E}\left[\int_{0}^{T}\psi_{i}(s){d}s + \Gamma_{i} \biggm\vert \mathcal{F}_{t}\right].$$ Since $\psi_{i} \in \mathcal{M}^{2}$ and $\Gamma_{i} \in L^{2}$, the conditional expectation is well-defined and $\hat{Y}^{i,0}$ is a uniformly integrable martingale which we can take to be càdlàg (Section .67 of [@Rogers2000a]). By Doob’s maximal inequality it follows that $\hat{Y}^{i,0} \in \mathcal{S}^{2}$ and therefore $Y^{i,0}$. Since the filtration is assumed to be quasi-left-continuous, Proposition \[Proposition:Optimal-Switching-qlcFiltrations\] verifies that $\hat{Y}^{i,0} \in \mathcal{Q}$ and therefore $Y^{i,0} \in \mathcal{Q}$. Therefore, $Y^{i,n} \in \mathcal{Q} \cap \mathcal{S}^{2}$ for every $i \in \mathbb{I}$ when $n = 0$. Now, suppose by an induction hypothesis on $n \ge 0$ that for all $i \in \mathbb{I}$, $Y^{i,n} \in \mathcal{Q} \cap \mathcal{S}^{2}$. We first show that $Y^{i,n+1} \in \mathcal{S}^{2}$. By the induction hypothesis on $Y^{i,n}$ and since $\gamma_{i,j} \in \mathcal{Q} \cap \mathcal{S}^{2}$ and $\psi_{i} \in \mathcal{M}^{2}$, we verify that $\hat{U}^{i,n+1} \in \mathcal{S}^{2}$. Therefore, by Proposition \[Proposition:Optimal-Switching-SnellEnvelopeProperties\], $\hat{Y}^{i,n+1}$ is the Snell envelope of $\hat{U}^{i,n+1}$. It is then not difficult to show that $\hat{U}^{i,n+1} \in \mathcal{S}^{2} \implies \hat{Y}^{i,n+1} \in \mathcal{S}^{2}$ (also property 5 of Proposition \[Proposition:Optimal-Switching-SnellEnvelopeProperties\]). Since $\psi_{i} \in \mathcal{M}^{2}$ we conclude that $Y^{i,n+1} \in \mathcal{S}^{2}$. We now show that $Y^{i,n+1} \in \mathcal{Q}$ by arguing similarly as in the proof of Proposition 1.4a in [@Hamadene2003]. First, recall that $Y^{i,n}$ is in $\mathcal{Q} \cap \mathcal{S}^{2}$ for every $i \in \mathbb{I}$ by the induction hypothesis, and $\gamma_{ij} \in \mathcal{Q} \cap \mathcal{S}^{2}$ for every $i,j \in \mathbb{I}$. This means the process $\left(\max\nolimits_{j \neq i}\left\lbrace -\gamma_{i,j}(t) + Y^{j,n}_{t} \right\rbrace\right)_{0 \le t \le T}$ is also in $\mathcal{Q} \cap \mathcal{S}^{2}$. Using $Y^{j,n}_{T} = \Gamma_{j}$, $\mathsf{P}$-a.s. and Assumption \[Assumption:SwitchingCosts\] on the switching costs, we also have $$\lim_{t \uparrow T}\left(\max_{j \neq i}\left\lbrace Y^{j,n}_{t} - \gamma_{i,j}(t) \right\rbrace\right) = \max_{j \neq i}\left\lbrace \Gamma_{j} -\gamma_{i,j}(T) \right\rbrace \le \Gamma_{i}.$$ Thus $\hat{U}^{i,n+1}$ is quasi-left-continuous on $[0,T)$ and has a possible positive jump at time $T$. Next, by Proposition \[Proposition:Optimal-Switching-SnellEnvelopeProperties\], $\hat{Y}^{i,n+1}$ has a unique Meyer decomposition: $$\hat{Y}^{i,n+1} = M - A - B,$$ where $M$ is a right-continuous, uniformly integrable martingale, and $A$ and $B$ are predictable, non-decreasing processes which are continuous and purely discontinuous respectively. Let $\tau \in \mathcal{T}$ be any predictable time. The process $A$ is continuous so $A_{\tau^{-}} = A_{\tau}$ holds almost surely. Moreover, the martingale $M$ also satisfies $M_{\tau} = M_{\tau^{-}}$ a.s. since, by Proposition \[Proposition:Optimal-Switching-qlcFiltrations\], it is quasi-left-continuous. Predictable jumps in $\hat{Y}^{i,n+1}$ therefore come from $B$, and we need only consider the two events $\lbrace \triangle_{\tau}B > 0 \rbrace$ and $\lbrace \triangle_{\tau}B = 0 \rbrace$ since $B$ is non-decreasing. By property 2 of Proposition \[Proposition:Optimal-Switching-SnellEnvelopeProperties\], we have $$\lbrace \triangle_{\tau}B > 0 \rbrace \subset \lbrace \hat{Y}^{i,n+1}_{\tau^{-}} = \hat{U}^{i,n+1}_{\tau^{-}} \rbrace$$ and, using the dominating property of $\hat{Y}^{i,n+1}$ and non-negativity of the predictable jumps of $\hat{U}^{i,n+1}$, this gives $$\label{eq:Optimal-Switching-JumpAtPredictableTime1} \mathsf{E}\left[\hat{Y}^{i,n+1}_{\tau^{-}}\mathbf{1}_{\lbrace \triangle_{\tau}B > 0 \rbrace}\right] = \mathsf{E}\left[\hat{U}^{i,n+1}_{\tau^{-}}\mathbf{1}_{\lbrace \triangle_{\tau}B > 0 \rbrace}\right] \le \mathsf{E}\left[\hat{U}^{i,n+1}_{\tau}\mathbf{1}_{\lbrace \triangle_{\tau}B > 0 \rbrace}\right] \le \mathsf{E}\left[\hat{Y}^{i,n+1}_{\tau}\mathbf{1}_{\lbrace \triangle_{\tau}B > 0 \rbrace}\right]$$ On the other hand, the Meyer decomposition of $\hat{Y}^{i,n+1}$ and the almost sure continuity of $M$ and $A$ at $\tau$ yield the following: $$\begin{aligned} \label{eq:Optimal-Switching-JumpAtPredictableTime2} \mathsf{E}\left[\hat{Y}^{i,n+1}_{\tau^{-}}\mathbf{1}_{\lbrace \triangle_{\tau}B = 0 \rbrace}\right] & = \mathsf{E}\left[\left(M_{\tau^{-}} - A_{\tau^{-}} - B_{\tau^{-}}\right)\mathbf{1}_{\lbrace \triangle_{\tau}B = 0 \rbrace}\right] \nonumber \\ & = \mathsf{E}\left[\left(M_{\tau} - A_{\tau} - B_{\tau}\right)\mathbf{1}_{\lbrace \triangle_{\tau}B = 0 \rbrace}\right] \nonumber \\ & = \mathsf{E}\left[\hat{Y}^{i,n+1}_{\tau}\mathbf{1}_{\lbrace \triangle_{\tau}B = 0 \rbrace}\right]\end{aligned}$$ From and we get the inequality, $\mathsf{E}\big[\hat{Y}^{i,n+1}_{\tau^{-}}\big] \le \mathsf{E}\big[\hat{Y}^{i,n+1}_{\tau}\big]$. However, $\mathsf{E}\big[\hat{Y}^{i,n+1}_{\tau^{-}}\big] \ge \mathsf{E}\big[\hat{Y}^{i,n+1}_{\tau}\big]$ since $\hat{Y}^{i,n+1}$ is a right-continuous supermartingale (in $\mathcal{S}^{2}$) and $\tau$ is predictable (Theorem .14 of [@Dellacherie1982]). Thus $\mathsf{E}\big[\hat{Y}^{i,n+1}_{\tau^{-}}\big] = \mathsf{E}\big[\hat{Y}^{i,n+1}_{\tau}\big]$ for every predictable time $\tau$. This means $Y^{i,n+1}$ is a regular supermartingale (of class $[D]$) and, by Theorem .10 of [@Dellacherie1982], the predictable non-decreasing component of the Meyer decomposition of $Y^{i,n+1}$ must be continuous. Therefore, $B \equiv 0$ and $Y^{i,n+1} \in \mathcal{Q}$ since the only jumps it experiences are those from the quasi-left-continuous martingale $M$. \[Lemma:OptimalSwitching-ConvergenceOfTheOptimalProcesses\] For every $i \in \mathbb{I}$, the process $Y^{i,n}$ solves the optimal switching problem with at most $n \ge 0$ switches: $$\label{eq:Optimal-Switching-AtMostnSwitches} Y^{i,n}_{t} = \operatorname*{ess\,sup}\limits_{\alpha \in \mathcal{A}^{n}_{t,i}}\mathsf{E}\left[\int_{t}^{T}\psi_{\mathbf{u}_{s}}(s){d}s + \Gamma_{\mathbf{u}_{T}} - \sum_{j = 1}^{n}\gamma_{\iota_{j-1},\iota_{j}}(\tau_{j})\mathbf{1}_{\{ \tau_{j} < T \}}\biggm\vert \mathcal{F}_{t}\right],\hspace{1em} t \in [0,T].$$ Moreover, the sequence $\left\lbrace Y^{i,n}\right\rbrace_{n \ge 0}$ is increasing and converges pointwise $\mathsf{P}$-a.s. for any $0 \le t \le T$ to a càdlàg process $\tilde{Y}^{i}$ satisfying: $\forall t \in [0,T]$, $$\label{eq:Optimal-Switching-ConvergenceOfSwitchingProcess} \tilde{Y}^{i}_{t} = \operatorname*{ess\,sup}\limits_{\alpha \in \mathcal{A}_{t,i}}J\left(\alpha;t,i\right) \eqqcolon V(t,i) \quad \text{a.s.}$$ Let $t \in [0,T]$, $i \in \mathbb{I}$ be given and for $n \ge 0$ define $\mathcal{A}^{n}_{t,i}$ as the subset of admissible strategies with at most $n$ switches: $$\mathcal{A}^{n}_{t,i} = \left\lbrace \alpha \in \mathcal{A}_{t,i} \colon \tau_{n+1} = T,\hspace{2bp} \mathsf{P}-a.s. \right\rbrace$$ Define a double sequence $\hat{\alpha}^{(n)} = \left(\hat{\tau}_{k},\hat{\iota}_{k}\right)^{n+1}_{k = 0}$ as follows $$\begin{gathered} \label{eq:Optimal-Switching-Optimal-Strategy-n-switches} \hat{\tau}_{0} = t,\hspace{1em} \hat{\iota}_{0} = i, \nonumber \\ \begin{cases} \hat{\tau}_{k} = \inf\left\lbrace s \ge \hat{\tau}_{k-1} \colon Y^{\hat{\iota}_{k-1},n-(k-1)}_{s} = \max\limits_{j \neq \hat{\iota}_{k-1}} \left(Y^{j,n-k}_{s} - \gamma_{\hat{\iota}_{k-1},j}\left(s\right) \right) \right\rbrace \wedge T,\nonumber \\ \hat{\iota}_{k} = \sum\limits_{j \in \mathbb{I}} j \mathbf{1}_{F^{\hat{\iota}_{k-1}}_{j}} \end{cases} \\ \text{for } k = 1,\ldots,n \text{ where } F^{\hat{\iota}_{k-1}}_{j} \text{ is the event} : \\ F^{\hat{\iota}_{k-1}}_{j} \coloneqq \left\lbrace Y^{j,n-k}_{\hat{\tau}_{k}} -\gamma_{\hat{\iota}_{k-1},j}\left(\hat{\tau}_{k}\right) = \max\limits_{\ell \neq \hat{\iota}_{k-1}}\left( Y^{\ell,n-k}_{\hat{\tau}_{k}} -\gamma_{\hat{\iota}_{k-1},\ell}\left(\hat{\tau}_{k}\right) \right) \right\rbrace, \nonumber\end{gathered}$$ and set $\hat{\tau}_{n+1} = T,~\hat{\iota}_{n+1}(\omega) = j \in \mathbb{I}$ with $j \neq \hat{\iota}_{n}(\omega)$. Since $Y^{i,n} \in \mathcal{Q} \cap \mathcal{S}^{2}$, one verifies that $\hat{\alpha}^{(n)} \in \mathcal{A}^{n}_{t,i}$ and, using the arguments of Theorem \[Theorem:Optimal-Switching-VerificationPartial\], that $Y^{i,n}_{t} = J(\hat{\alpha}^{(n)};t,i)$ and has the representation . Furthermore, since $\mathcal{A}^{n}_{t,i} \subset \mathcal{A}^{n+1}_{t,i} \subset \mathcal{A}_{t,i}$, it follows that $Y^{i,n}_{t}$ is non-decreasing in $n$ for all $t \in [0,T]$ and $Y^{i,n}_{t} \le Y^{i,n+1}_{t} \le V(t,i)$ almost surely. Recalling also the processes $\hat{U}^{i,n}$ and $\hat{Y}^{i,n}$ from Lemma \[Lemma:Optimal-Switching-FinitelyManySwitchesQLC\_S2\] and that $\hat{Y}^{i,n}$ is the Snell envelope of $\hat{U}^{i,n}$ for each $n \ge 0$, we deduce $\lbrace \hat{Y}^{i,n} \rbrace_{n \ge 0}$ is an increasing sequence of càdlàg supermartingales. Theorem .18 of [@Dellacherie1982] shows that this sequence converges to a limit $\hat{Y}^{i}$ defined pointwise on $[0,T]$ by $$\label{eq:Continuous-Time-Optimal-Switching-Limiting-Snell-Envelope} \hat{Y}^{i}_{t} \coloneqq \sup_{n}\hat{Y}^{i,n}_{t} = \sup_{n}\left(Y^{i,n}_{t} + \int_{0}^{t}\psi_{i}(s){d}s\right).$$ This random function $\hat{Y}^{i} = (\hat{Y}^{i}_{t})_{0 \le t \le T}$ is indistinguishable from a càdlàg process, but is not necessarily a supermartingale since we have not established its integrability. Nevertheless, the sequence $\lbrace Y^{i,n} \rbrace_{n \ge 0}$ converges pointwise on $[0,T]$ to a limit $\tilde{Y}^{i}$ which, modulo indistinguishability, is a càdlàg process given by $$\label{eq:Continuous-Time-Optimal-Switching-LimitingProcess} \tilde{Y}^{i}_{t} = \sup_{n}Y^{i,n}_{t} = \hat{Y}^{i}_{t} - \int_{0}^{t}\psi_{i}(s){d}s.$$ Next, let $\alpha = (\tau_{k},\iota_{k})_{k \ge 0} \in \mathcal{A}_{t,i}$ be arbitrary. By Remark \[Remark:SubOptimalSwitchTwice\], we can restrict our attention to those strategies such that $\mathsf{P}\big( \lbrace \tau_{k} = \tau_{k+1}, \tau_{k} < T \rbrace) = 0$ for $k \ge 1$. Define $\alpha^{n} = (\tau^{n}_{k},\iota^{n}_{k})_{k \ge 0}$ to be the strategy obtained from $\alpha$ when only the first $n$ switches are kept: $$\begin{cases} (\tau^{n}_{k},\iota^{n}_{k}) = (\tau_{k},\iota_{k}),\hspace{1em} k \le n, \\ \tau^{n}_{k} = T,\hspace{1em} k > n. \end{cases}$$ The difference between the performance indices under $\alpha$ and $\alpha^{n}$ is: $$\begin{aligned} J(\alpha;t,i) - J(\alpha^{n};t,i) & = \mathsf{E}\left[\int_{\tau_{n}}^{T}\left(\psi_{\mathbf{u}_{s}}(s) - \psi_{\iota^{n}_{n}}(s)\right){d}s + \Gamma_{\mathbf{u}_{T}} - \Gamma_{\iota^{n}_{n}} - \sum_{k > n}\gamma_{\iota_{k-1},\iota_{k}}(\tau_{k})\mathbf{1}_{\{\tau_{k} < T\}} \biggm\vert \mathcal{F}_{t}\right] \nonumber \\ & = \mathsf{E}\left[\int_{\tau_{n}}^{T}\left(\psi_{\mathbf{u}_{s}}(s) - \psi_{\iota^{n}_{n}}(s)\right){d}s + \Gamma_{\mathbf{u}_{T}} - \Gamma_{\iota^{n}_{n}} - (C^{\alpha} - C^{\alpha}_{n}) \biggm\vert \mathcal{F}_{t}\right]\end{aligned}$$ where $\mathbf{u}$ is the mode indicator function associated with $\alpha$ and $\iota^{n}_{n} = \iota_{n \wedge N(\alpha)}$ is the last mode switched to before $T$ under $\alpha^{n}$. Since $\alpha \in \mathcal{A}_{t,i}$, $\psi_{i} \in \mathcal{M}^{2}$ and $\Gamma_{i} \in L^{2}$ for every $i \in \mathbb{I}$, the conditional expectation above is well-defined for every $n \ge 1$. This also leads to an integrable upper bound for $J(\alpha;t,i)$, $$\label{eq:Optimal-Switching-DifferenceBetweenPerformanceIndices2} \begin{split} J(\alpha;t,i) \le {} & \mathsf{E}\left[\left(\int_{\tau_{n}}^{T}\left|\psi_{\mathbf{u}_{s}}(s) - \psi_{\iota^{n}_{n}}(s)\right|{d}s + |\Gamma_{\mathbf{u}_{T}} - \Gamma_{\iota^{n}_{n}}| + \left|C^{\alpha} - C^{\alpha}_{n}\right|\right)\mathbf{1}_{\lbrace N(\alpha) > n \rbrace} \biggm\vert \mathcal{F}_{t}\right]\\ & + J(\alpha^{n};t,i) \end{split}$$ Using these integrability conditions again together with the observation that $N(\alpha) < \infty$ $\mathsf{P}$-a.s. and $\lbrace \tau_{k} \rbrace$ is (strictly) increasing towards $T$, we may pass to the limit $n \to \infty$ in equation  to get, $$\label{eq:Optimal-Switching-DifferenceBetweenPerformanceIndices3} J(\alpha;t,i) \le \lim\limits_{n \to \infty}J(\alpha^{n};t,i) \quad \text{a.s.}$$ However, as $\alpha^{n} \in \mathcal{A}^{n}_{t,i}$ for each $n \ge 0$, from and we get for every $t \in [0,T]$: $$J(\alpha;t,i) \le \lim\limits_{n \to \infty}J(\alpha^{n};t,i) \le \lim\limits_{n \to \infty}Y^{i,n}_{t} = \tilde{Y}^{i}_{t} \quad \text{a.s.}$$ Since $\alpha \in \mathcal{A}_{t,i}$ was arbitrary, we have just shown for every $t \in [0,T]$ $$V(t,i) \coloneqq \operatorname*{ess\,sup}\limits_{\alpha \in \mathcal{A}_{t,i}}J\left(\alpha;t,i\right) \le \tilde{Y}^{i}_{t}\quad \text{a.s.}$$ The reverse inequality holds since $Y^{i,n}_{t} = J(\hat{\alpha}^{(n)};t,i) \le V(t,i)$ almost surely for $n \ge 0$ (cf. ) and $\tilde{Y}^{i}$ is the pointwise supremum of the sequence $\{Y^{i,n}\}_{n \ge 0}$. The case of an arbitrary number of switches. {#Section:ArbitraryNumberOfSwitches} -------------------------------------------- This section gives sufficient conditions under which the limiting processes $\tilde{Y}^{1},\ldots,\tilde{Y}^{m}$ satisfy the verification theorem \[Theorem:Optimal-Switching-VerificationPartial\]. The main difficulty is in proving that $\tilde{Y}^{i} \in \mathcal{S}^{2}$, and in order to achieve this we make the following hypothesis. > 1em **(M)** There exists a family of martingales $\{M_{ij} = (M_{ij})_{0 \le t \le T} \colon i,j \in \mathbb{I}\}$ such that for every $i,j,k \in \mathbb{I}$: $$\begin{aligned} > i. \quad & M_{i,j} \in \mathcal{S}^{2} \nonumber \\ > ii. \quad & -\gamma_{i,j}(\cdot) \le M_{i,j}(\cdot), \enskip \mathsf{P}-\text{a.s.} \enskip \text{ if } i \neq j \nonumber \\ > iii. \quad & M_{i,j}(\cdot) + M_{j,k}(\cdot) \le M_{i,k}(\cdot), \enskip \mathsf{P}-\text{a.s.} \enskip \text{ if } i \neq j \text{ and } j \neq k. > \end{aligned}$$ This hypothesis can be verified in the following cases: - The switching costs are martingales – since we can set $M_{i,j} = -\gamma_{i,j}$ (with strict inequality in property iii. above). This includes the case $\gamma_{i,j}(t) = \gamma_{i,j}$, $t \in [0,T]$, with $\gamma_{i,j} \in L^{2}$ and $\mathcal{F}_{0}$-measurable; - The switching costs are non-negative – since we can set $M_{i,j} \equiv 0$ for $i,j \in \mathbb{I}$; - There are two modes ($\mathbb{I} = \{0,1\}$ as per convention). For $i \in \{0,1\}$ and $j = 1-i$, let $Z_{i,j}$ denote the Snell envelope of $(-\gamma_{i,j}(t))_{0 \le t \le T}$, which exists and is in $\mathcal{S}^{2}$ since $\gamma_{i,j} \in \mathcal{S}^{2}$ (see Proposition \[Proposition:Optimal-Switching-SnellEnvelopeProperties\]). We may then take $M_{i,j}$ to be the martingale component in the Doob-Meyer decomposition of $Z_{i,j}$, and set $M_{0,0} = M_{1,1} = M_{0,1} + M_{1,0}$. This case includes many examples of *Dynkin games* (see [@Martyr2014c]). \[Lemma:Martingale-Hypothesis-Lemma\] Assume Hypothesis **(M)**, then $\forall \alpha = (\tau_{n},\iota_{n})_{n \ge 0} \in \mathcal{A}_{t,i}$, $(t,i) \in [0,T] \times \mathbb{I}$: $$\label{eq:Optimal-Switching-Constant-Upper-Limit} \forall N \ge 1, \quad \mathsf{E}\left[-\sum_{n = 1}^{N}\gamma_{\iota_{n-1},\iota_{n}}(\tau_{n}) \biggm\vert \mathcal{F}_{t}\right] \le \mathsf{E}\left[\max_{j_{1},j_{2} \in \mathbb{I}}|M_{j_{1},j_{2}}(T)| \biggm \vert \mathcal{F}_{t}\right] \hspace{1em}\mathsf{P}-\text{a.s.}$$ Let $\alpha = (\tau_{n},\iota_{n})_{n \ge 0} \in \mathcal{A}_{t,i}$ be arbitrary. For $n \ge 1$ and $i,j \in \mathbb{I}$ we have $\tau_{n} \le T$, $\mathbf{1}_{\{\iota_{n-1} = i\}}\mathbf{1}_{\{\iota_{n} = j\}}$ is non-negative and $\mathcal{F}_{\tau_{n}}$-measurable, $M_{ij} \in \mathcal{S}^{2}$ is a martingale with $-\gamma_{i,j}(\cdot) \le M_{i,j}(\cdot)$ for $i \neq j$. We can therefore show for $N \ge 1$: $$\mathsf{E}\left[-\sum_{n = 1}^{N}\gamma_{\iota_{n-1},\iota_{n}}(\tau_{n}) \biggm\vert \mathcal{F}_{t}\right] \le \mathsf{E}\left[\sum_{n = 1}^{N}M_{\iota_{n-1},\iota_{n}}(\tau_{n}) \biggm\vert \mathcal{F}_{t}\right] = \mathsf{E}\left[\sum_{n = 1}^{N} M_{\iota_{n-1},\iota_{n}}(T) \biggm\vert \mathcal{F}_{t}\right]$$ The proof can be completed by showing $$\label{eq:Optimal-Switching-Constant-Upper-Limit-Proof} \forall N \ge 1, \quad \sum_{n = 1}^{N} M_{\iota_{n-1},\iota_{n}}(T) \le \max_{j_{1},j_{2} \in \mathbb{I}}|M_{j_{1},j_{2}}(T)| \hspace{1em}\mathsf{P}-\text{a.s.}$$ and concluding by arbitrariness of $\alpha$. The inequality shall be proved via induction similarly to [@LyVath2007 p. 399]. First note that is true for $N = 1$. Now, suppose that is satisfied for $N \ge 1$. Since $M_{\iota_{N-1},\iota_{N}}(T) + M_{\iota_{N},\iota_{N+1}}(T) \le M_{\iota_{N-1},\iota_{N+1}}(T)$ a.s. we have $$\sum_{n = 1}^{N+1}M_{\iota_{n-1},\iota_{n}}(T) \le \sum_{n = 1}^{N-1}M_{\iota_{n-1},\iota_{n}}(T) + M_{\iota_{N-1},\iota_{N+1}}(T) \enskip\mathsf{P}-\text{a.s.}$$ Define a new strategy $\tilde{\alpha} = (\tilde{\tau}_{n},\tilde{\iota}_{n})_{n \ge 0} \in \mathcal{A}_{t,i}$ by $(\tilde{\tau}_{n},\tilde{\iota}_{n}) = (\tau_{n},\iota_{n})$ for $n = 1,\ldots,N-1$ and $(\tilde{\tau}_{n},\tilde{\iota}_{n}) = (\tau_{n+1},\iota_{n+1})$ for $n \ge N$. Then, using the induction hypothesis on $\tilde{\alpha}$, one gets $$\sum_{n = 1}^{N+1}M_{\iota_{n-1},\iota_{n}}(T) \le \sum_{n = 1}^{N}M_{\tilde{\iota}_{n-1},\tilde{\iota}_{n}}(T) \le \max_{j_{1},j_{2} \in \mathbb{I}}|M_{j_{1},j_{2}}(T)| \enskip\mathsf{P}-\text{a.s.}$$ \[Theorem:Optimal-Switching-ExistenceVerification\] Suppose Hypothesis **(M)**. Then the limit processes $\tilde{Y}^{1},\ldots,\tilde{Y}^{m}$ of Lemma \[Lemma:OptimalSwitching-ConvergenceOfTheOptimalProcesses\] satisfy the following: for $i \in \mathbb{I}$, 1. $\tilde{Y}^{i} \in \mathcal{Q} \cap \mathcal{S}^{2}$. 2. For any $0 \le t \le T$, $$\label{eq:Optimal-Switching-LimitingSnellEnvelope} \begin{split} \tilde{Y}^{i}_{t} & = \operatorname*{ess\,sup}\limits_{\tau \ge t}\mathsf{E}\left[\int_{t}^{\tau}\psi_{i}(s){d}s + \Gamma_{i}\mathbf{1}_{\{\tau = T \}} + \max_{j \neq i}\left\lbrace \tilde{Y}^{j}_{\tau} - \gamma_{i,j}(\tau) \right\rbrace\mathbf{1}_{\{\tau < T \}}\biggm\vert \mathcal{F}_{t}\right],\\ \tilde{Y}^{i}_{T} & = \Gamma_{i}. \end{split}$$ In particular, $\tilde{Y}^{1},\ldots,\tilde{Y}^{m}$ are unique and satisfy the verification theorem. Recall the limit processes $\hat{Y}^{1},\ldots,\hat{Y}^{m}$ and $\tilde{Y}^{1},\ldots,\tilde{Y}^{m}$ from Lemma \[Lemma:Optimal-Switching-FinitelyManySwitchesQLC\_S2\], equation . Under Hypothesis **(M)** one verifies directly using Lemma \[Lemma:Martingale-Hypothesis-Lemma\] and the arguments in Lemma \[Lemma:OptimalSwitching-ConvergenceOfTheOptimalProcesses\] that the $\mathbb{F}$-martingale $\zeta = (\zeta_{t})_{0 \le t \le T}$ defined by $$\label{eq:Optimal-Switching-BoundOnMartingale} \zeta_{t} \coloneqq \mathsf{E}\left[\int_{0}^{T}\max_{j \in \mathbb{I}}\left|\psi_{j}(s)\right|{d}s + \max_{j \in \mathbb{I}}|\Gamma_{j}| + \max_{j_{1},j_{2} \in \mathbb{I}}|M_{j_{1},j_{2}}(T)| \biggm \vert \mathcal{F}_{t}\right]$$ satisfies $\zeta \in \mathcal{S}^{2}$ and $\forall n \ge 0$, $|\hat{Y}^{i,n}_{t}| \le \zeta_{t}$ $\mathsf{P}$-a.s. for every $t \in [0,T]$. Moreover, since $\hat{Y}^{i}$ is the pointwise supremum of $\lbrace \hat{Y}^{i,n} \rbrace_{n \ge 0}$ we also have $\hat{Y}^{i}_{t} \le \zeta_{t}$ for each $t \in [0,T]$. These observations give $-\zeta_{t} \le \hat{Y}^{i}_{t} \le \zeta_{t}$ $\mathsf{P}-\text{a.s. } \forall~ t \in [0,T]$. Since $\zeta \in \mathcal{S}^{2}$, it follows that $\hat{Y}^{i} \in \mathcal{S}^{2}$ and also $\tilde{Y}^{i} \in \mathcal{S}^{2}$ since $\psi_{i} \in \mathcal{M}^{2}$. Now define a process $\hat{U}^{i} = (\hat{U}^{i}_{t})_{0 \le t \le T}$ for $i = 1,\ldots,m$ similarly to $\hat{U}^{i,n}$ used in Lemma \[Lemma:Optimal-Switching-FinitelyManySwitchesQLC\_S2\]: $$\begin{aligned} \hat{U}^{i}_{t} \coloneqq \int_{0}^{t}\psi_{i}(s){d}s + \Gamma_{i}\mathbf{1}_{\{ t = T \}} + \max_{j \neq i}\left\lbrace \tilde{Y}^{j}_{t} - \gamma_{i,j}(t) \right\rbrace\mathbf{1}_{\{ t < T \}}\end{aligned}$$ The $\mathcal{S}^{2}$ processes $\hat{Y}^{i}$ and $\hat{U}^{i}$ are the respective limits of the increasing sequences of càdlàg $\mathcal{S}^{2}$ processes $\lbrace \hat{Y}^{i,n} \rbrace_{n \ge 0}$ and $\lbrace \hat{U}^{i,n} \rbrace_{n \ge 0}$. Since $\hat{Y}^{i,n}$ is also the Snell envelope of $\hat{U}^{i,n}$, property 5 of Proposition \[Proposition:Optimal-Switching-SnellEnvelopeProperties\] verifies that $\hat{Y}^{i}$ is the Snell envelope of $\hat{U}^{i}$. This leads to equation  for $\tilde{Y}^{i}$ and the uniqueness claim. The final part is to show that $\tilde{Y}^{i} \in \mathcal{Q}$. Let $\tau \in \mathcal{T}$ be any predictable time. Since $\hat{Y}^{i}$ is the Snell envelope of $\hat{U}^{i}$, it has a Meyer decomposition (cf. Proposition \[Proposition:Optimal-Switching-SnellEnvelopeProperties\]) $$\hat{Y}^{i} = M - A - B,$$ where $M$ is a uniformly integrable càdlàg martingale and $A$ (resp. $B$) is non-decreasing, predictable and continuous (resp. discontinuous). Remember that $M$ is also quasi-left-continuous due to Assumption \[Assumption:QLCFiltration\] and Proposition \[Proposition:Optimal-Switching-qlcFiltrations\]. We therefore have $$\triangle_{\tau}\hat{Y}^{i} = \left(M_{\tau} - A_{\tau} - B_{\tau}\right) - \left(M_{\tau^{-}} - A_{\tau^{-}} - B_{\tau^{-}}\right) = -\triangle_{\tau}B \quad \text{a.s.}$$ By property 2 of Proposition \[Proposition:Optimal-Switching-SnellEnvelopeProperties\] concerning the jumps of $\hat{Y}^{i}$ (and therefore $\tilde{Y}^{i}$), we have $$\lbrace \triangle_{\tau}B > 0 \rbrace \subset \lbrace \hat{Y}^{i}_{\tau^{-}} = \hat{U}^{i}_{\tau^{-}} \rbrace$$ and by using the definitions of $\hat{Y}^{i}$ and $\hat{U}^{i}$ we get: $$\label{eq:Optimal-Switching-JumpInIndexi} \tilde{Y}^{i}_{\tau} < \tilde{Y}^{i}_{\tau^{-}} = \max_{j \neq i}\left\lbrace \tilde{Y}^{j}_{\tau^{-}} - \gamma_{i,j}(\tau^{-}) \right\rbrace \text{ on } \lbrace \triangle_{\tau}B > 0 \rbrace.$$ Since $\mathbb{I}$ is finite, implies that there exists an $\mathbb{I}$-valued random variable $j^{*}$, $j^{*} \neq i$, such that $$\label{eq:Optimal-Switching-ContradictionAtJump1} \tilde{Y}^{i}_{\tau^{-}} = \tilde{Y}^{j^{*}}_{\tau^{-}} - \gamma_{i,j^{*}}(\tau^{-}) = \max_{j \neq i}\left\lbrace \tilde{Y}^{j}_{\tau^{-}} -\gamma_{i,j}(\tau^{-}) \right\rbrace \text{ on } \lbrace \triangle_{\tau}B > 0 \rbrace.$$ However, also implies that the process $\left(\max_{j \neq i}\left\lbrace \tilde{Y}^{j}_{t} - \gamma_{i,j}(t)\right\rbrace\right)_{0 \le t \le T}$ jumps at time $\tau$ (since it is dominated by $\tilde{Y}^{i}$). As the switching costs are quasi-left-continuous, we conclude that $\tilde{Y}^{j^{*}}$ jumps at time $\tau$. Using the Meyer decomposition of $\hat{Y}^{j^{*}}$ and the properties of the jumps as before, this leads to $$\tilde{Y}^{j^{*}}_{\tau^{-}} = \max_{l \neq j^{*}}\left\lbrace \tilde{Y}^{l}_{\tau^{-}} - \gamma_{j^{*},l}(\tau^{-}) \right\rbrace \text{ on } \lbrace \triangle_{\tau}B > 0 \rbrace$$ and there exists an $\mathbb{I}$-valued random variable $l^{*}$, $l^{*} \neq j^{*}$, such that $$\label{eq:Optimal-Switching-ContradictionAtJump2} \tilde{Y}^{j^{*}}_{\tau^{-}} = \tilde{Y}^{l^{*}}_{\tau^{-}} - \gamma_{j^{*},l^{*}}(\tau^{-}) = \max_{l \neq j^{*}}\left\lbrace \tilde{Y}^{l}_{\tau^{-}} - \gamma_{j^{*},l}(\tau^{-}) \right\rbrace \text{ on } \lbrace \triangle_{\tau}B > 0 \rbrace$$ Putting and together, then using the quasi-left-continuity of the switching costs and Assumption \[Assumption:SwitchingCosts\], the following (almost sure) inequality and contradiction to the optimality of $j^{*}$ is obtained: $$\begin{aligned} \tilde{Y}^{i}_{\tau^{-}} = -\gamma_{i,j^{*}}(\tau^{-}) + \tilde{Y}^{j^{*}}_{\tau^{-}} & = -\gamma_{i,j^{*}}(\tau^{-}) -\gamma_{j^{*},l^{*}}(\tau^{-}) + \tilde{Y}^{l^{*}}_{\tau^{-}} \\ & = -\gamma_{i,j^{*}}(\tau) -\gamma_{j^{*},l^{*}}(\tau) + \tilde{Y}^{l^{*}}_{\tau^{-}} \\ & < -\gamma_{i,l^{*}}(\tau) + \tilde{Y}^{l^{*}}_{\tau^{-}} \\ & < -\gamma_{i,l^{*}}(\tau^{-}) + \tilde{Y}^{l^{*}}_{\tau^{-}} \enskip \text{on} \enskip \lbrace \triangle_{\tau}B > 0 \rbrace.\end{aligned}$$ This means $\triangle_{\tau}B = 0$ a.s. for every predictable time $\tau$, and $\tilde{Y}^{i} \in \mathcal{Q}$ for every $i \in \mathbb{I}$. Conclusion. {#Section:Optimal-Switching-Continuous-Time-Conclusion} =========== This paper extended the study of the multiple modes optimal switching problem in [@Djehiche2009] to account for 1. non-zero, possibly different terminal rewards; 2. signed switching costs modelled by càdlàg, quasi-left-continuous processes; 3. filtrations which are only assumed to satisfy the usual conditions and quasi-left-continuity. Just as in Theorem 1 of [@Djehiche2009], it was shown that the value function of the optimal switching problem can be defined stochastically in terms of interconnected Snell envelope-like processes. The existence of these processes was proved in a manner similar to Theorem 2 of [@Djehiche2009], by a limiting argument for sequences of processes solving the optimal switching problem with at most $n \ge 0$ switches. The limits of these sequences are right-continuous processes, but may not satisfy the integrability assumptions of the Snell envelope representation in general. Sufficient conditions for this representation were obtained by further hypothesizing the existence of a family of martingales satisfying particular relations among themselves and the switching costs. We explained that this “martingale hypothesis” can be verified quite easily in the following cases: - the switching costs are martingales; - the switching costs are non-negative; - the case of two modes (starting and stopping problem). Admissibility of the candidate optimal strategy. {#Section:Optimal-Switching-IntegrabilityOfTheSwitchingCost} ================================================ Let $\alpha^{*} = \left(\tau^{*}_{n},\iota^{*}_{n}\right)_{n \ge 0}$ be the sequence of times and random mode indicators defined in equation  of Theorem \[Theorem:Optimal-Switching-VerificationPartial\]. In this section we prove that $\alpha^{*} \in \mathcal{A}_{t,i}$ (cf. Definition \[Definition:Optimal-Switching-AdmissibleStrategies\]). One readily verifies (by right-continuity) that $\left\lbrace\tau^{*}_{n}\right\rbrace_{n \ge 0} \subset \mathcal{T}$ is non-decreasing with $\tau^{*}_{0} = t$, and each $\iota^{*}_{n}$ is an $\mathcal{F}_{\tau^{*}_{n}}$-measurable $\mathbb{I}$-valued random variable with $\iota^{*}_{0} = i$ and $\iota^{*}_{n} \neq \iota^{*}_{n+1}$ for $n \ge 0$. The remaining properties are established in a number of steps, beginning with the following lemma on the switching times. \[Lemma:Optimal-Switching-FiniteStrategyCandidateOptimal\] Let $\left\lbrace\tau^{*}_{n}\right\rbrace_{n \ge 0}$ be the switching times defined in equation  of Theorem \[Theorem:Optimal-Switching-VerificationPartial\]. Then these times satisfy $$\label{eq:FiniteAndStrictlyIncreasingSwitchingTimes} \begin{split} i) & \quad \mathsf{P}(\{\tau^{*}_{n} = \tau^{*}_{n+1}, \tau^{*}_{n} < T\}) = 0,\quad n \ge 1 \\ ii) & \quad \mathsf{P}\left(\{\tau^{*}_{n} < T,\hspace{1bp} \forall n \ge 0\}\right) = 0 \end{split}$$ Condition -i) can be proved via contradiction using Assumption \[Assumption:SwitchingCosts\] (recall Remark \[Remark:SubOptimalSwitchTwice\]). Condition -ii) can also be proved by contradiction using Assumption \[Assumption:SwitchingCosts\] and the same arguments of [@Hamadene2012 pp. 192–193] (since the switching costs are quasi-left-continuous). The details are therefore omitted. The rest of this section is devoted to verifying condition  for the strategy $\alpha^{*}$. Recall that the cumulative cost of switching $n \ge 1$ times is given by, $$C^{\alpha^{*}}_{n} = \sum\limits_{k=1}^{n}\gamma_{\iota^{*}_{k-1},\iota^{*}_{k}}(\tau^{*}_{k})\mathbf{1}_{\{\tau^{*}_{k} < T\}}$$ Since the switching costs satisfy $\gamma_{i,j} \in \mathcal{S}^{2}$ for every $i,j$ in the finite set $\mathbb{I}$, $C^{\alpha^{*}}_{n} \in L^{2}$ for every $n \ge 1$. We define a sequence $$N^{*}_{n} \coloneqq \sum_{k=1}^{n}\mathbf{1}_{\{\tau^{*}_{k} < T\}},\quad n = 1,2,\ldots$$ which we use to rewrite the expression for $C^{\alpha^{*}}_{n}$ as follows: $$\label{eq:Optimal-Switching-CostOfSwitchingFinitelyManyTimes2} C^{\alpha^{*}}_{n} = \sum\limits_{k=1}^{N^{*}_{n}}\gamma_{\iota^{*}_{k-1},\iota^{*}_{k}}(\tau^{*}_{k}).$$ The following proposition gives an alternative representation of $C^{\alpha^{*}}_{n}$ in terms of the processes $Y^{1},\ldots,Y^{m}$ and their Meyer decomposition with random superscripts (cf. Lemma \[Lemma:Optimal-Switching-VerificationLemma\]). \[Proposition:Optimal-Switching-AdmissibilityCandidateOptimalStrategy\] Let $\alpha^{*} = \left(\tau^{*}_{n},\iota^{*}_{n}\right)_{n \ge 0} \in \mathcal{A}_{t,i}$ be the switching control strategy defined in equation  of Theorem \[Theorem:Optimal-Switching-VerificationPartial\] and let $\mathbf{u}^{*}$ be the associated mode indicator function. Then $C^{\alpha^{*}}_{n}$, the cumulative cost of switching $n \ge 1$ times under $\alpha^{*}$, satisfies $$\label{eq:Optimal-Switching-CumulativeCostOfSwitchingMeyerDecompositionFinal} C^{\alpha^{*}}_{n} = Y^{\iota^{*}_{N^{*}_{n}}}_{\tau^{*}_{N^{*}_{n}}} - Y^{\iota^{*}_{0}}_{\tau^{*}_{0}} + \int_{\tau^{*}_{0}}^{\tau^{*}_{N^{*}_{n}}}\psi_{\mathbf{u}^{*}_{s}}(s){d}s - \sum_{k = 1}^{N^{*}_{n}}\left(M^{\iota^{*}_{k-1}}_{\tau^{*}_{k}} - M^{\iota^{*}_{k-1}}_{\tau^{*}_{k-1}}\right) \hspace{1em} \mathsf{P}-\text{a.s.}$$ where $M^{\iota^{*}_{k}}$, $k \ge 0$, is the martingale component of the Meyer decomposition in Lemma \[Lemma:Optimal-Switching-VerificationLemma\]. By definition of the strategy $\alpha^{*}$ (cf. ), optimality of the time $\tau^{*}_{n}$ and the definition of $\iota^{*}_{n}$, for $n \ge 1$ the cost of switching at $\tau^{*}_{n}$ is, $$\label{eq:Optimal-Switching-CostOfSwitching} \gamma_{\iota^{*}_{n-1},\iota^{*}_{n}}(\tau^{*}_{n})\mathbf{1}_{\{\tau^{*}_{n} < T\}} = \left(Y^{\iota^{*}_{n}}_{\tau^{*}_{n}} - Y^{\iota^{*}_{n-1}}_{\tau^{*}_{n}}\right)\mathbf{1}_{\{\tau^{*}_{n} < T\}}\hspace{1em}\mathsf{P}-\text{a.s.}$$ Therefore, from equation  and the cost of the first $n$ switches can be rewritten as, $$\label{eq:Optimal-Switching-CumulativeCostOfSwitching} C^{\alpha^{*}}_{n} = \sum_{k = 1}^{N^{*}_{n}}\left(Y^{\iota^{*}_{k}}_{\tau^{*}_{k}} - Y^{\iota^{*}_{k-1}}_{\tau^{*}_{k}}\right)\hspace{1em}\mathsf{P}-\text{a.s.}$$ Now, Lemma \[Lemma:Optimal-Switching-VerificationLemma\] proved that the following Meyer decomposition holds for $k \ge 0$ (cf. equation ): $$\label{eq:Optimal-Switching-OptimalProcessMeyerDecomposition} Y^{\iota^{*}_{k}}_{t} + \int_{0}^{t}\psi_{\iota^{*}_{k}}(s){d}s = M^{\iota^{*}_{k}}_{t} - A^{\iota^{*}_{k}}_{t},\hspace{1em}\mathsf{P}-\text{a.s. }\forall \hspace{2bp} \tau^{*}_{k} \le t \le T.$$ where, on $[\tau^{*}_{k},T]$, $M^{\iota^{*}_{k}}$ is a uniformly integrable càdlàg martingale and $A^{\iota^{*}_{k}}$ is a predictable, continuous and increasing process. The Meyer decomposition is used to rewrite equation  for the cumulative switching costs as follows: $\mathsf{P}$-a.s., $$\label{eq:Optimal-Switching-CumulativeCostOfSwitchingMeyerDecomposition} \begin{split} C^{\alpha^{*}}_{n} = {} & \sum_{k = 1}^{N^{*}_{n}}\left(M^{\iota^{*}_{k}}_{\tau^{*}_{k}} - M^{\iota^{*}_{k-1}}_{\tau^{*}_{k}}\right) - \sum_{k = 1}^{N^{*}_{n}}\left(A^{\iota^{*}_{k}}_{\tau^{*}_{k}} - A^{\iota^{*}_{k-1}}_{\tau^{*}_{k}}\right) \\ & \qquad - \sum_{k = 1}^{N^{*}_{n}}\left(\int_{0}^{\tau^{*}_{k}}\psi_{\iota^{*}_{k}}(s){d}s - \int_{0}^{\tau^{*}_{k}}\psi_{\iota^{*}_{k-1}}(s){d}s \right). \end{split}$$ The first summation term in equation  can be rewritten as: $$\label{eq:Optimal-Switching-SimplificationOfASummationTerm1} \sum_{k = 1}^{N^{*}_{n}}\left(M^{\iota^{*}_{k}}_{\tau^{*}_{k}} - M^{\iota^{*}_{k-1}}_{\tau^{*}_{k}}\right) = M^{\iota^{*}_{N^{*}_{n}}}_{\tau^{*}_{N^{*}_{n}}} - M^{\iota^{*}_{0}}_{\tau^{*}_{0}} - \sum_{k = 1}^{N^{*}_{n}}\left(M^{\iota^{*}_{k-1}}_{\tau^{*}_{k}} - M^{\iota^{*}_{k-1}}_{\tau^{*}_{k-1}}\right)$$ For every $k \ge 0$, by the definition of $\tau^{*}_{k+1}$ and property 4 of Proposition \[Proposition:Optimal-Switching-SnellEnvelopeProperties\], we know that $\left(Y^{\iota^{*}_{k}}_{t} + \int_{0}^{t}\psi_{\iota^{*}_{k}}(s){d}s\right)$ is a martingale $\mathsf{P}$-a.s. for every $\tau^{*}_{k} \le t \le \tau^{*}_{k+1}$. By using the Meyer decomposition , we therefore observe that $\forall k \ge 0$, $A^{\iota^{*}_{k}}_{t}$ *is constant* $\mathsf{P}$-a.s. $\forall \hspace{2bp} \tau^{*}_{k} \le t \le \tau^{*}_{k+1}$. The summation term in with respect to $A^{\iota^{*}_{k-1}}$ can then be simplified as follows, $$\label{eq:Optimal-Switching-SimplificationOfASummationTerm2} \sum_{k = 1}^{N^{*}_{n}}\left(A^{\iota^{*}_{k}}_{\tau^{*}_{k}} - A^{\iota^{*}_{k-1}}_{\tau^{*}_{k}}\right) = \sum_{k = 1}^{N^{*}_{n}}\left(A^{\iota^{*}_{k}}_{\tau^{*}_{k}} - A^{\iota^{*}_{k-1}}_{\tau^{*}_{k-1}}\right) = A^{\iota^{*}_{N^{*}_{n}}}_{\tau^{*}_{N^{*}_{n}}} - A^{\iota^{*}_{0}}_{\tau^{*}_{0}} \hspace{1em} \mathsf{P}-\text{a.s.}$$ By writing out the terms and using the definition of the mode indicator function $\mathbf{u}^{*}$, the third summation term in is simplified as follows: $\mathsf{P}$-a.s., $$\begin{aligned} \label{eq:Optimal-Switching-SimplificationOfASummationTerm3} {} & -\sum_{k = 1}^{N^{*}_{n}}\left(\int_{0}^{\tau^{*}_{k}}\psi_{\iota^{*}_{k}}(s){d}s - \int_{0}^{\tau^{*}_{k}}\psi_{\iota^{*}_{k-1}}(s){d}s \right) \nonumber \\ = \quad & \int_{0}^{\tau^{*}_{1}}\psi_{\iota^{*}_{0}}(s){d}s + \sum_{k = 1}^{N^{*}_{n}-1}\int_{\tau^{*}_{k}}^{\tau^{*}_{k+1}}\psi_{\iota^{*}_{k}}(s){d}s - \int_{0}^{\tau^{*}_{N^{*}_{n}}}\psi_{\iota^{*}_{N^{*}_{n}}}(s){d}s \nonumber \\ = \quad & \int_{0}^{\tau^{*}_{1}}\psi_{\iota^{*}_{0}}(s){d}s + \int_{\tau^{*}_{1}}^{\tau^{*}_{N^{*}_{n}}}\psi_{\mathbf{u}^{*}_{s}}(s){d}s - \int_{0}^{\tau^{*}_{N^{*}_{n}}}\psi_{\iota^{*}_{N^{*}_{n}}}(s){d}s\end{aligned}$$ Substitute equations , , and into equation  for the cumulative switching cost, then use the Meyer decomposition  and the definition of $\mathbf{u}^{*}$ to get, $$\begin{aligned} C^{\alpha^{*}}_{n} = {} & M^{\iota^{*}_{N^{*}_{n}}}_{\tau^{*}_{N^{*}_{n}}} - M^{\iota^{*}_{0}}_{\tau^{*}_{0}} - \sum_{k = 1}^{N^{*}_{n}}\left(M^{\iota^{*}_{k-1}}_{\tau^{*}_{k}} - M^{\iota^{*}_{k-1}}_{\tau^{*}_{k-1}}\right) - A^{\iota^{*}_{N^{*}_{n}}}_{\tau^{*}_{N^{*}_{n}}} + A^{\iota^{*}_{0}}_{\tau^{*}_{0}} \nonumber \\ & + \int_{0}^{\tau^{*}_{1}}\psi_{\iota^{*}_{0}}(s){d}s + \int_{\tau^{*}_{1}}^{\tau^{*}_{N^{*}_{n}}}\psi_{\mathbf{u}^{*}_{s}}(s){d}s - \int_{0}^{\tau^{*}_{N^{*}_{n}}}\psi_{\iota^{*}_{N^{*}_{n}}}(s){d}s \nonumber \\ = {} & Y^{\iota^{*}_{N^{*}_{n}}}_{\tau^{*}_{N^{*}_{n}}} - \left(Y^{\iota^{*}_{0}}_{\tau^{*}_{0}} + \int_{0}^{\tau^{*}_{0}}\psi_{\iota^{*}_{0}}(s){d}s\right) + \int_{0}^{\tau^{*}_{1}}\psi_{\iota^{*}_{0}}(s){d}s + \int_{\tau^{*}_{1}}^{\tau^{*}_{N^{*}_{n}}}\psi_{\mathbf{u}^{*}_{s}}(s){d}s \nonumber \\ & - \sum_{k = 1}^{N^{*}_{n}}\left(M^{\iota^{*}_{k-1}}_{\tau^{*}_{k}} - M^{\iota^{*}_{k-1}}_{\tau^{*}_{k-1}}\right) \nonumber \\ = {} & Y^{\iota^{*}_{N^{*}_{n}}}_{\tau^{*}_{N^{*}_{n}}} - Y^{\iota^{*}_{0}}_{\tau^{*}_{0}} + \int_{\tau^{*}_{0}}^{\tau^{*}_{N^{*}_{n}}}\psi_{\mathbf{u}^{*}_{s}}(s){d}s - \sum_{k = 1}^{N^{*}_{n}}\left(M^{\iota^{*}_{k-1}}_{\tau^{*}_{k}} - M^{\iota^{*}_{k-1}}_{\tau^{*}_{k-1}}\right) \hspace{1em} \mathsf{P}-\text{a.s.}\end{aligned}$$ Convergence of the family of cumulative switching costs. {#Section:Optimal-Switching-ConvergenceOfSwitchingCostsInL2} -------------------------------------------------------- ### A discrete-parameter martingale. For $k \ge 0$, define an $\mathcal{F}_{\tau^{*}_{k}}$-measurable random variable $\xi_{k}$ by, $$\label{eq:Optimal-Switching-MartingaleIncrements} \xi_{k} \coloneqq \begin{cases} M^{\iota^{*}_{k-1}}_{\tau^{*}_{k}} - M^{\iota^{*}_{k-1}}_{\tau^{*}_{k-1}} & \hspace{1em}\text{on } k \ge 1 \text{ and } \lbrace \tau^{*}_{k} < T \rbrace ,\\ 0 & \hspace{1em}\text{otherwise.} \end{cases}$$ Note that the limit $\xi_{\infty}$ is a well-defined $\mathcal{F}_{T}$-measurable random variable which satisfies $$\xi_{\infty} \coloneqq \lim_{k}\xi_{k} = \begin{cases} 0, \hspace{1em} \text{on }\hspace{2bp} \lbrace N(\alpha^{*}) < \infty \rbrace,\\ 0, \hspace{1em}\text{a.s. on }\hspace{2bp} \lbrace N(\alpha^{*}) = \infty \rbrace. \end{cases}$$ where the second line holds since $M^{i}$, $i \in \mathbb{I}$, is quasi-left-continuous, and the switching times $\lbrace \tau^{*}_{k} \rbrace_{k \ge 1}$ announce $T$ on $\lbrace N(\alpha^{*}) = \infty \rbrace$ (cf. Lemma \[Lemma:Optimal-Switching-FiniteStrategyCandidateOptimal\]). In this case set $\iota^{*}_{\infty} \coloneqq \mathbf{u}^{*}_{T}$. Since $M^{i} \in \mathcal{S}^{2}$ for $i \in \mathbb{I}$ (cf. Proposition \[Proposition:Optimal-Switching-SquareIntegrableMartingale\]) and the set $\mathbb{I}$ is finite, the sequence $\lbrace \xi_{k} \rbrace_{k \ge 0}$ is in $L^{2}$. Properties of square-integrable martingales and conditional expectations can be used to show: $$\begin{aligned} \label{eq:Optimal-Switching-UniformBoundOnTheSquaredDifferences} \forall n \ge 1, \quad \mathsf{E}\left[\sum_{k = 1}^{n} (\xi_{k})^{2}\right] & = \mathsf{E}\left[\sum_{k = 1}^{n}(M^{\iota^{*}_{k-1}}_{\tau^{*}_{k}} - M^{\iota^{*}_{k-1}}_{\tau^{*}_{k-1}})^{2}\mathbf{1}_{\{\tau^{*}_{k} < T\}}\right] \nonumber \\ & \le \sum_{i = 1}^{m}\sum_{k = 1}^{n}\mathsf{E}\left[\left((M^{i}_{\tau^{*}_{k}})^{2} - 2 \cdot M^{i}_{\tau^{*}_{k-1}} \cdot \mathsf{E}\big[M^{i}_{\tau^{*}_{k}} \bigm\vert \mathcal{F}_{\tau^{*}_{k-1}}\big] + (M^{i}_{\tau^{*}_{k-1}})^{2}\right)\right] \nonumber \\ & \le \sum_{i = 1}^{m}\mathsf{E}\left[(\sup\nolimits_{0 \le s \le T}|M^{i}_{s}|)^{2}\right] \nonumber \\ & \le 4 \cdot m \cdot \max_{i \in \mathbb{I}}\mathsf{E}\left[(M^{i}_{T})^{2}\right]\end{aligned}$$ Finally, almost surely for $1 \le k \le N^{*}_{n}$, $$\mathsf{E}\bigl[\xi_{k} \bigm\vert \mathcal{F}_{\tau^{*}_{k-1}}\bigr] = \mathsf{E}\bigl[M^{\iota^{*}_{k-1}}_{\tau^{*}_{k}} - M^{\iota^{*}_{k-1}}_{\tau^{*}_{k-1}} \bigm\vert \mathcal{F}_{\tau^{*}_{k-1}}\bigr] = \sum_{i \in \mathbb{I}}\mathbf{1}_{\{\iota^{*}_{k-1} = i\}}\mathsf{E}\bigl[M^{i}_{\tau^{*}_{k}} - M^{i}_{\tau^{*}_{k-1}} \bigm\vert \mathcal{F}_{\tau^{*}_{k-1}}\bigr] = 0$$ and letting $n \to \infty$ shows that $\mathsf{E}\big[\xi_{k} \bigm\vert \mathcal{F}_{\tau^{*}_{k-1}}\big] = 0$ for $k \ge 1$. Now define an increasing family of sub-$\sigma$-algebras of $\mathcal{F}$, $\mathbb{G} = \left(\mathcal{G}_{n}\right)_{n \ge 0}$, by $\mathcal{G}_{n} \coloneqq \mathcal{F}_{\tau^{*}_{n}}$. Applying Lemma \[Lemma:Optimal-Switching-FiniteStrategyCandidateOptimal\] and Proposition \[Proposition:Optimal-Switching-qlcFiltrations\] shows that $$\mathcal{G}_{\infty} \coloneqq \bigvee_{n}\mathcal{G}_{n} = \bigvee_{n}\mathcal{F}_{\tau^{*}_{n}} = \mathcal{F}_{T}.$$ The sequence $\left(X_{n},\mathcal{G}_{n}\right)_{n \ge 0}$ with $X_{n}$ defined by $$\label{eq:Optimal-Switching-MartingaleFromMartingaleDifference} X_{n} \coloneqq \sum_{k=0}^{n}\xi_{k}$$ is a discrete-parameter martingale in $L^{2}$. The probability space $\left(\Omega,\mathcal{F},\mathsf{P}\right)$ with filtration $\mathbb{G} = \left(\mathcal{G}_{n}\right)_{n \ge 0}$ will be used to discuss convergence and integrability properties of $\left(X_{n}\right)_{n \ge 0}$. ### Convergence of the discrete-parameter martingale. As discussed previously, the $\mathbb{G}$-martingale $\left(X_{n}\right)_{n \ge 0}$ is in $L^{2}$. It is not hard to verify, by the conditional Jensen inequality for instance, that the sequence $\left(X^{2}_{n}\right)_{n \ge 0}$ is a positive $\mathbb{G}$-submartingale. By Doob’s Decomposition (Proposition -1-2 of [@Neveu1975]), $\left(X^{2}_{n}\right)_{n \ge 0}$ can be decomposed uniquely as $$\label{eq:Optimal-Switching-DoobDecomposition} X_{n}^{2} = Q_{n} + R_{n}$$ where $\left(Q_{n}\right)_{n \ge 0}$ is an integrable $\mathbb{G}$-martingale and $\left(R_{n}\right)_{n \ge 0}$ is an increasing process (starting from $0$) with respect to $\mathbb{G}$. Convergence of $\left(X_{n}\right)_{n \ge 0}$ depends on the properties of the compensator $\left(R_{n}\right)_{n \ge 0}$, and this is made more precise by the following proposition. \[Proposition:Optimal-Switching-ConvergenceOfL2DiscreteParameterMartingales\] Let $\left(X_{n}\right)_{n \ge 0}$ be a square-integrable $\mathbb{G}$-martingale such that (without loss of generality) $X_{0} = 0$, and $\left(R_{n}\right)_{n \ge 0}$ denote the increasing process associated with the $\mathbb{G}$-submartingale $\left(X^{2}_{n}\right)_{n \ge 0}$ by the Doob decomposition . Then if $\mathsf{E}[R_{\infty}]< \infty$, the martingale $\left(X_{n}\right)_{n \ge 0}$ converges in $L^{2}$; furthermore, $\mathsf{E}[(\sup_{n \ge 0}|X_{n}|)^{2}] \le 4\mathsf{E}[R_{\infty}]$. We can now prove the main result. \[Theorem:Optimal-Switching-SquareIntegrableSwitchingCosts\] The sequence $\{ C^{\alpha^{*}}_{n}\}_{n \ge 1}$ converges in $L^{2}$ and also satisfies $\mathsf{E}\big[(\sup_{n}\big|C^{\alpha^{*}}_{n}\big|)^{2}\big] < \infty$. Proposition \[Proposition:Optimal-Switching-AdmissibilityCandidateOptimalStrategy\] gave the following representation for the switching cost sum: $$\begin{aligned} \label{eq:Optimal-Switching-CumulativeSwitchingCostUniformIntegrabilityCheck} C^{\alpha^{*}}_{n} & = Y^{\iota^{*}_{N^{*}_{n}}}_{\tau^{*}_{N^{*}_{n}}} - Y^{\iota^{*}_{0}}_{\tau^{*}_{0}} + \int_{\tau^{*}_{0}}^{\tau^{*}_{N^{*}_{n}}}\psi_{\mathbf{u}^{*}_{s}}(s){d}s - \sum_{k = 1}^{N^{*}_{n}}\left(M^{\iota^{*}_{k-1}}_{\tau^{*}_{k}} - M^{\iota^{*}_{k-1}}_{\tau^{*}_{k-1}}\right) \nonumber \\ & = Y^{\iota^{*}_{N^{*}_{n}}}_{\tau^{*}_{N^{*}_{n}}} - Y^{\iota^{*}_{0}}_{\tau^{*}_{0}} + \int_{\tau^{*}_{0}}^{\tau^{*}_{N^{*}_{n}}}\psi_{\mathbf{u}^{*}_{s}}(s){d}s - X_{N^{*}_{n}} \hspace{1em} \mathsf{P}-\text{a.s.}\end{aligned}$$ Since $N(\alpha^{*}) < \infty$ almost surely, the sequences $\{\tau^{*}_{N^{*}_{n}}\}_{n \ge 1}$ and $\{\iota^{*}_{N^{*}_{n}}\}_{n \ge 1}$ converge almost surely to $\tau^{*}_{N(\alpha^{*})} \le T$ and $\iota^{*}_{N(\alpha^{*})} = \mathbf{u}^{*}_{T}$ respectively. Noting that $Y^{i} \in \mathcal{S}^{2}$ and $\psi_{i} \in \mathcal{M}^{2}$ for every $i \in \mathbb{I}$, we can prove the claim by showing that the martingale $\left(X_{n}\right)_{n \ge 0}$ converges in $L^{2}$ and $\mathsf{E}[(\sup_{n \ge 0}|X_{n}|)^{2}] < \infty$. For this it suffices to prove the hypothesis of Proposition \[Proposition:Optimal-Switching-ConvergenceOfL2DiscreteParameterMartingales\]. Towards this end, we apply Fatou’s Lemma to the increasing process $\left(R_{n}\right)_{n \ge 0}$ associated with the $\mathbb{G}$-submartingale $\left(X^{2}_{n}\right)_{n \ge 0}$ to get $$\label{eq:Optimal-Switching-FatouLemmaCompensator} \mathsf{E}\left[R_{\infty}\right] \le \liminf_{n \to \infty}\mathsf{E}\left[R_{n}\right].$$ For $n \ge 1$, the random variable $R_{n}$ can be decomposed as follows [@Neveu1975 p. 148]: $$\label{eq:Optimal-Switching-CompensatorDecomposition} R_{n} = \sum_{k = 0}^{n-1}R_{k+1} - R_{k} = \sum_{k = 0}^{n-1}\mathsf{E}\left[\left(X_{k+1}-X_{k}\right)^{2} \bigm \vert \mathcal{G}_{k}\right] = \sum_{k = 0}^{n-1}\mathsf{E}\left[\left(\xi_{k+1}\right)^{2} \bigm \vert \mathcal{G}_{k}\right].$$ Using equation  in and applying the tower property of conditional expectations leads to $$\label{eq:Optimal-Switching-FatouLemmaAndCompensatorDecomposition} \mathsf{E}\left[R_{\infty}\right] \le \liminf_{n \to \infty}\mathsf{E}\left[\sum_{k = 0}^{n-1}\left(\xi_{k+1}\right)^{2}\right].$$ The inequalities leading up to  above show that the right-hand side of is finite and we conclude by applying Proposition \[Proposition:Optimal-Switching-ConvergenceOfL2DiscreteParameterMartingales\]. Acknowledgments {#acknowledgments .unnumbered} =============== This research was partially supported by EPSRC grant EP/K00557X/1. The author would like to thank his PhD supervisor J. Moriarty and colleague T. De Angelis for their feedback which led to an improved draft of the paper. The author also expresses his gratitude to others who commented on a previous version of the manuscript. [10]{} Erhan Bayraktar and Masahiko Egami. . , 35(1):140–159, 2010. Bruno Bouchard. . , 81(2):171–197, 2009. Claude Dellacherie and Paul-Andr[é]{} Meyer. , volume 72 of [*North-Holland Mathematics Studies*]{}. Elsevier, Amsterdam, 1982. Boualem Djehiche, Said Hamad[è]{}ne, and Alexandre Popier. . , 48(4):2751–2770, 2009. Brahim [El Asri]{} and Imade Fakhouri. , 2012, arXiv:1204.1683. Nicole [El Karoui]{}. . , 1981. Robert J. Elliott. . Springer-Verlag, New York, 1982. Xin Guo and Pascal Tomecek. . , 47(1):421–443, 2008. Said Hamad[è]{}ne. . , 74(3):571–596, 2002. Said Hamad[è]{}ne and I Hdhiri. . , 7(1):1081803–1081804, 2007. Said Hamad[è]{}ne and M. A. Morlais. . , 67(2):163–196, 2013. Said Hamad[è]{}ne and Youssef Ouknine. . , 8(2000):1–20, feb 2003. Jean Jacod and Albert N. Shiryaev. , volume 288 of [ *Grundlehren der mathematischen Wissenschaften*]{}. Springer Berlin Heidelberg, Berlin, Heidelberg, 2003. Richard R. Lumley and Mihail Zervos. . , 26(4):637–653, 2001. Niklas L. P. Lundstr[ö]{}m, Kaj Nystr[ö]{}m, and Marcus Olofsson. . , 193(4):1213–1247, 2014. Vathana [Ly Vath]{} and Huy[ê]{}n Pham. . , 46(2):395–426, 2007. Vathana [Ly Vath]{}, Huy[ê]{}n Pham, and St[é]{}phane Villeneuve. . , 18(3):1164–1200, 2008. Randall Martyr. , 2014, arXiv:1411.4438. Hiroaki Morimoto. . , 34(3):407–416, 1982. Hiroaki Morimoto. . , 16(1):1–17, 1987. Jacques Neveu. . North-Holland, Amsterdam, 1975. Bernt [Ø]{}ksendal and Agn[è]{}s Sulem. . Springer Berlin Heidelberg, Berlin, Heidelberg, 2007. Goran Peskir and Albert N. Shiryaev. . Lectures in Mathematics. ETH Z[ü]{}rich. Birkh[ä]{}user Basel, 2006. L. C. G. Rogers and David Williams. . Cambridge University Press, Cambridge, 2nd edition, 2000. L. C. G. Rogers and David Williams. . Cambridge University Press, Cambridge, 2nd edition, 2000. [^1]: School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom. email: `randall.martyr@postgrad.manchester.ac.uk` [^2]: This research was partially supported by EPSRC grant EP/K00557X/1.

--- abstract: | A new stochastic theory of a foreign exchange markets dynamics is developed. As a result we have the new probability distribution which well describes statistical and scaling dependencies ”experimentally” observed in foreign exchange markets in recent years. The developed dynamic theory is compared with well-known phenomenological Levy distribution approach which is widely applied to this problem. It is shown that the developed stochastic dynamics and phenomenological approach based on the Levy distribution give the same statistical and scaling dependencies. [*PACS* ]{}number(s): 02.50 Ey, 05.40. Fb author: - 'Nikolai Laskin[^1]' date: | Isotrace Laboratory, University of Toronto\ 60 St. George Street, Toronto, ON M5S 1A7\ Canada title: '[**Stochastic Theory of  Foreign Exchange Market Dynamics** ]{}' --- tcilatex Introduction ============ Statistical behavior of the foreign exchange (FX) markets and price fluctuations in currency have been the subject of studies in recent years [@Mantega], [@Dodge]. The high-frequency data for financial markets has made it possible to investigate market dynamics on timescales as short as 1 min, a value close to the minimum time needed to perform transaction in the market. It was observed that the short-term price fluctuations in FX market, for example, between US dollar and German mark, has the same statistical behavior as the velocity differences in hydrodynamic turbulence [@Dodge]. Probability distributions in turbulence well fit the experimental data by superposition of the Gaussians with log-normal distributions of its variances [@Dodge],[@Castaing]. The convergence of the velocity differences distributions toward a Gaussian shape corresponds to a decrease of the log-normal variance for increasing of spatial distances $\Delta r$. In the FX fluctuation dynamics the statistical distributions of the price difference separated by time $\Delta t$ was elaborated by the theoretical model of a Levy walk or Levy flight [@Mantega],[@Levy]. I have developed a new stochastic dynamic theory for the FX market. The Langevin type stochastic differential equation is proposed. I introduce the random force with some general characteristics in order to fit the theoretical predictions of my model with observed statistical dependencies. It is shown that the developed model describes exactly the ”experimental” data of the dynamics of a price index of the New York Stock Exchange. The probability distributions of the Standard & Poor’s 500 index differences $% \Delta x$ are reproduced exactly by the $P_{Laskin}(\Delta x,\Delta t)$. The new model describes well the scaling behavior of the ”probability of return” $P_{\Delta t}(0)$ as a function of $\Delta t$ (for definitions, see [@Mantega]). Dynamic model of price changes ============================== We will describe the dynamics of the FX price $x(t)$ by the stochastic differential equation $$\stackrel{\cdot }{x}(t)=F(t), \label{eq1}$$ where $F(t)$ is the random force. The quantity which were interested about is the price differences separated by the time scale $\Delta t$, $$\Delta x=x(t+\Delta t)-x(t), \label{eq2}$$ The distribution function of stochastic process $\Delta x$ is defined as follow $$P(\Delta x,\Delta t)=<\delta (\Delta x-\int\limits_t^{t+\Delta t}d\tau F(\tau ))>, \label{eq3}$$ where ... means the averaging over the all possible realizations of the random force $F(t)$. Let us construct the general stochastic force $F(t)$ by the following way $$F(t)=\sum\limits_{k=1}^na_k\varphi (t-t_k), \label{eq4}$$ here $a_k$ are the random amplitudes, $\varphi (t)$ is the response function, $t_k$ are the homogeneously distributed (on time interval $[0,T]$) moments of time, the number $n$ of which obeys the Poisson law. Thus, the averaging includes three statistically independent averaging procedures: 1\. Averaging over random amplitudes $a_k$, $<...>_{a_k}$, $$<...>_{a_k}=\int da_1...da_nP(a_1,...,a_n)..., \label{eq5}$$ where $P(a_1,...,a_n)$ is the probability distribution of amplitudes $a_k$. 2\. Averaging over $t_k$ on time interval $T$, $$<...>_T=\frac 1T\int\limits_0^Tdt_1...\frac 1T\int\limits_0^Tdt_n.... \label{eq6}$$ 3\. Averaging over random numbers $n$ of time moments $t_k$, $$<...>_n=\sum\limits_{n=0}^n\frac{\overline{n}^n}{n!}e^{-\overline{n}}..., \label{eq7}$$ where $\overline{n}=\nu T$ and $\nu $ is the density of points $t_k$ on time interval $T$. Taking into account the definition Eq.(\[eq3\]) and performing the averaging in accordance with Eqs.(\[eq5\])-(\[eq7\]) we will have $$P_{Laskin}(\Delta x,\Delta t)=\frac 1\pi \int\limits_0^\infty d\xi \cos (\xi \cdot \Delta x)e^{-L(\xi ,\Delta t)}, \label{eq8}$$ where we put notation $$L(\xi ,\Delta t)=\nu \delta \int\limits_0^{\frac{\sigma \delta }{\sqrt{2}}% (1-e^{-\Delta t/\delta })}\frac{du}u(1-e^{-u^2}). \label{eq9}$$ For simplicity the Eq.(\[eq8\]) is considered for the market currency situation when $P(a_1,...,a_n)$ is factorized as follow $$P(a_1,...,a_n)=\prod\limits_{k=1}^nP_1(a_k), \label{eq10}$$ where the Gaussian distribution $P_1(a)$ is given by $$P_1(a)=\frac 1{\sigma \sqrt{2\pi }}\exp (-\frac{a^2}{2\sigma ^2}). \label{eq11}$$ We also choose the response function in the form $$\varphi (t)=e^{-|t|/\delta }. \label{eq12}$$ The response function describes the influence of a piece of information which has become available at the delay time $t$ on the decision of a trader to propose or accept a price change. The parameter $\delta $ is the character scale of the time delay. We also have considered the response function in the form $$\varphi (t)=\frac 1{1+(|t|/\delta )^\beta }, \label{eq13}$$ where $\beta $ is a new parameter. For example, for $\beta =1$ we have $$P_{Laskin}(\Delta x,\Delta t;\beta =1)=\frac 1\pi \int\limits_0^\infty d\xi \cos (\xi \cdot \Delta x)e^{-L_\beta (\xi ,\Delta t;\beta =1)}, \label{eq14}$$ where $$L_\beta (\xi ,\Delta t;\beta =1)=\nu \Delta t\int\limits_0^{\ln \frac{\Delta t+\delta }\delta }\frac{dze^z}{(e^z-1)^2}(1-e^{-\frac{\sigma ^2\delta ^2\xi ^2}2z^2}). \label{eq15}$$ Thus, we derive the new $P_{Laskin}(\Delta x,\Delta t)$ distribution (see Eqs. (\[eq8\]), (\[eq9\])) starting from the differential stochastic equation Eq.(\[eq1\]). As it was mentioned the Levy stable distribution is widely applied to this problem. The comparison between $P_{Laskin}(\Delta x,\Delta t)$ transformed to the form $$P_{Laskin}(\Delta x,\Delta t)=\frac 1\pi \int\limits_0^\infty dy\cos (y\cdot \Delta x)\times , \label{eq16}$$ $$\times \exp \left\{ -\frac 12D\int\limits_0^{\tau ^2(\Delta t)y^2}\frac{dz}% z(1-e^{-z})\right\} ,$$ the Levy stable distribution [@Mantega], [@Levy], [@Feller] $$P_{Levy}(\Delta x,\Delta t)=\frac 1\pi \int\limits_0^\infty dy\cos (y\cdot \Delta x)e^{-\gamma \Delta ty^\alpha }, \label{eq17}$$ and the Gauss distribution $$P_{Gauss}(\Delta x)=\frac 1{\sqrt{2\pi }\sigma }\exp \{-x^2/2\sigma ^2\}. \label{eq18}$$ is shown in Fig.1. Figure 2 is a comparison of scaling properties of $P_{Laskin}(0,\Delta t)$, and $P_{Levy}(0,\Delta t).$ It is interesting that the [*Laskin*]{} stochastic dynamics and phenomenological approach based on the [*Levy*]{} distribution give the same scaling for the probability of return $P_{\Delta t}(0)$ of the S&P 500 index variations as a function of the $\Delta t$ [@Mantega]. General FX rate dynamic model ============================= The main goal of the developed theory is to predict the behavior of currency exchange market in the future and based on these prediction to develop the efficient currency market strategy. As we will see further it is useful to generalize the Eqs.(\[eq1\]), (\[eq4\]) in following way $$\stackrel{\cdot }{x}(t)=-\lambda x+\sum\limits_{k=1}^na_k\varphi (t-t_k) \label{eq19}$$ where $\lambda $ is a ”price dissipative coefficient” the financial mean of which will be discussed further. Let us define the new PDF $P_{Laskin}(x,t)$ as follow $$P_{Laskin}(x,t)=<\delta (x-x(t,x_0))>_{a,T,n} \label{eq20}$$ where $x(t,x_0)$ is the formal solution of the Eq.(\[eq19\]). and $$<...>_{a,T,n}=\int da_1...da_nP(a_1,...,a_n)\sum\limits_{n=0}^n\frac{% \overline{n}^n}{n!}e^{-\overline{n}}\frac 1T\int\limits_0^Tdt_1...\frac 1T\int\limits_0^Tdt_n...$$ Using the Eqs.(\[eq19\]), (\[eq20\]) it is easy to obtain the evolution equation for PDF $P_{Laskin}(x,t)$ $$\frac{\partial P_{Laskin}(x,t)}{\partial t}=\lambda \frac \partial {\partial x}xP_{Laskin}(x,t)+ \label{eq21}$$ $$+\nu \int daP_1(a)\int\limits_{t_0}^tdt^{\prime }\varphi (t-t^{\prime })\left\{ e^{-a\int\limits_{t_0}^td\tau e^{\lambda (t-\tau )}\varphi (\tau -t^{\prime })\frac \partial {\partial x}}-1\right\} P_{Laskin}(x,t)$$ The initial condition for this equation has the form $$P_{Laskin}(x,t)=<\delta (x-x(t,x_0))>_{a,T,n} \label{eq22}$$ The Eq.(\[eq21\]), (\[eq22\])will serve as the main equations to predict the behavior of FX market in the future. It is easy to see that the solution of the problem Eqs.(\[eq21\]) and (\[eq22\]) can be written as $$P_{Laskin}(x,t)=\frac 1{2\pi }\int\limits_{-\infty }^\infty d\xi e^{i\xi (x-e^{\lambda (t-t_0)}x_0)}\times \label{eq23}$$ $$\times \exp \left\{ -\nu \int\limits_{t_0}^tdt^{\prime }\left( 1-W(\xi \int\limits_{t_0}^td\tau e^{\lambda (t-\tau )}\varphi (\tau -t^{\prime }))\right) \right\}$$ where $W(\xi )$ is characteristic function of the random amplitude $a$. It is well-known [@Feller] that the PDF $P_1(a)$ and characteristic function $W(\xi )$ are connected each other $$P_1(a)=\frac 1{2\pi }\int\limits_{-\infty }^\infty d\xi e^{i\xi a}W(\xi )$$ and $$W(\xi )=\int\limits_{-\infty }^\infty dae^{-i\xi a}P_1(a)$$ The Eqs.(\[eq21\]) and (\[eq22\]) allow one to study the evolution problems in FX currency market. Some analytically solvable evolution problems will be demonstrated in the next publication. Conclusions =========== The new dynamic stochastic theory of FX currency market dynamics is developed. We have established the new [*Laskin*]{} distribution which well describes the observed statistical dependencies of the price difference separated by time scale $\Delta t$ and the [*Laskin*]{} PDF which allows elaborate evolution of the currency market and predict behavior of market dynamics. The theoretical predictions based on the [*Laskin*]{} distribution are compared with phenomenological the [*Levy*]{} distribution based approach and observed statistical dependencies. It is interesting that the [*Laskin*]{} stochastic dynamics and phenomenological approach based on the [*Levy*]{} distribution give the same statistical and scaling dependencies. [1]{} R.N. Mantega and H.E. Stanley, Scaling behaviour in the dynamics of economic index, Nature, v.[**376**]{} 1995, p.46-49. S. Ghashghaie, W.Breymann, J. Peinke, P.Talkner, Y.Dodge, Turbulent cascades in foreign exchange markets, Nature, v.[**381**]{}, 1996, p.767-770. B. Castaing, Y. Gagne and E. Hopfinger, Velocity probability density functions of high Reynolds number turbulence, Physica [**D46**]{}, 1990, p.177-200 P. Levy, [*Theorie de l’addition des variables aleatoires*]{}, Gauthiere-Villars, Paris, 1937 W. Feller, [*An Introduction to Probability Theory and its Applications*]{} (Wiley, New York, 1971) Figure captions =============== Fig.1. Comparison between $P_{Laskin}(\Delta x,\Delta t)$, $P_{Levy}(\Delta x,\Delta t)$ and $P_{Gauss}(\Delta x)$ (see the definitions Eqs.(\[eq16\]), (\[eq17\]), (\[eq18\])). For $P_{Laskin}(\Delta x,\Delta t)$ the parameters are $\Delta t=0.5$, $D=1.4$, $S=0.16$ and $\tau (\Delta t)=S\cdot (1-e^{-\Delta t})$. For $P_{Levy}(\Delta x,\Delta t)$ the parameters are $% \alpha =0.00375$ and $\gamma =1.4$. For $P_{Gauss}(\Delta x)$, $\sigma =0.0508$ (see, [@Mantega]). Fig.2 Scaling dependencies of $P_{Laskin}(0,\Delta t)$ and $% P_{Levy}(0,\Delta t)$ ”probabilities of return”. All parameters are the same as for Fig.1. [^1]: E-mail: nlaskin@rocketmail.com

--- abstract: 'We study the quantization problem for certain types of jump processes. The probabilities for the number of jumps are assumed to be bounded by Poisson weights. Otherwise, jump positions and increments can be rather generally distributed and correlated. We show in particular that in many cases entropy coding error and quantization error have distinct rates. Finally, we investigate the quantization problem for the special case of $\R^d$-valued compound Poisson processes.' address: 'Technische Universität Berlin, Institut für Mathematik, Sekr. MA 7-5, Straße des 17. Juni 136, 10623 Berlin, Germany.' author: - Frank Aurzada - Steffen Dereich - Michael Scheutzow - Christian Vormoor title: High resolution quantization and entropy coding of jump processes --- , , , and High resolution quantization; entropy coding; complexity; jump process; compound Poisson process; Lévy process: metric entropy Introduction and results ======================== Statement of the problem ------------------------ In this article, we study the quantization- and entropy coding problem for certain types of jump processes. Given a random variable $X$, the aim is to find a good approximation $\hat X$ to $X$ that satisfies a particular *complexity constraint*. Let $s>0$, $X$ be a random variable in a measurable space $(E,\EE)$, $\rho$ a distortion measure on $E$ (i.e. a measurable, symmetric function $\rho : E\times E \to \R_{\geq 0}$ with $\rho(x,y)=0$ iff $x=y$), and $r\geq 0$. Then we define the *quantization error* as follows: $$D^{(q)}( r ~|~ X,\rho,s) := \inf\left\lbrace \left( \E \min_{a\in \CC} \rho(X,a)^s\right)^{1/s}\quad : \quad \log \# \CC\leq r\right\rbrace.$$ The number $D^{(q)}$ represents the best-achievable average error when encoding the signal $X$ with $r$ nats. The term ‘nats’ is used instead of ‘bits’, since we calculate the amount of information using the natural logarithm. Further, we investigate the *entropy coding error*, which can be understood as the average error when encoding the signal $X$ using – on average – $r$ nats: $$D^{(e)}( r ~|~ X,\rho,s) := \inf\left\lbrace \left( \E \rho(X,\hat{X})^s\right)^{1/s} ~:~\text{$\hat{X}$ random var.\ with $H(\hat{X})\leq r$}\right\rbrace,$$ where $H$ is the (discrete) entropy of a random variable: $$H(X):= \begin{cases} - \sum_{x} {\P\left(X=x\right)} \log {\P\left(X=x\right)}& \text{$X$ discrete,} \\ \infty & \text{otherwise.} \end{cases}$$ In slight misuse of notation we also write $D^{(q)}( r ~|~ X,{\left\|.\right\|},s)$ if $\rho(x,y)={\left\|x-y\right\|}$ for a norm distortion ${\left\|.\right\|}$. Analogously, we deal with the entropy coding error. We recall that $D^{(e)}( r ~|~ X,\rho,s) \leq D^{(q)}( r ~|~ X,\rho,s)$. The problems described above arise naturally in coding theory, where for instance, the complexity of a signal has to be reduced due to capacity restrictions of a channel or simply (lossy) data compression is considered (see for instance [@CoTho91] for a general account on coding theory and [@Kol68] for a historic outline of the information constraints). Beyond these information-theoretic applications, the quantization error is tightly related to certain quadrature problems: the quantization error can be defined equivalently as the worst-case error of a particular quadrature problem. Moreover, further quadrature problems are linked to the quantization problem via estimates involving both quantities. Recent results in that direction can be found in [@CDMR08] (see also [@PaPri05] for earlier results). The analysis of the quantization- and entropy coding error started in the 40s of the 20th century. At that time research was mainly focused on finite-dimensional signals; and the numerous publications mainly appeared in the engineering literature. A mathematical account of the results for finite-dimensional signals is provided by [@grafluschgy]. Since about 2000 researchers are attracted by the problem in the case where the original signal is infinite-dimensional. A series of articles followed on (infinite-dimensional) random vectors $X$ that are Gaussian (see for instance [@DFMS03], [@LuPa04], [@dereichscheutzow]), diffusions ([@LuPa04b], [@Der06a], [@Der06b]), and Lévy processes ([@luschgypages06], [@aurzadadereich]). In this article, we provide asymptotic estimates for the quantization- and entropy coding error for certain jump processes. The results are shown to be sharp in several cases. In contrast to the (infinite-dimensional) settings studied before, there is a qualitative difference in the (best-achievable) approximation error induced by the two constraints. Some notation and the model --------------------------- Let us now introduce the jump processes that we investigate in this article. We define the space $\DD([0,1[,E)$ to be the space of all functions $f:[0,1[\to E$ that are piecewise constant and possess a finite number of jumps, where if $f$ has a jump at $t$ from the value $a\in E$ to $b\in E$, then $f(t)=b$. We endow $\DD([0,1[,E)$ with the $\sigma$-field induced by the projections. In the sequel, $X=(X(t))_{t\in[0,1[}$ denotes a $\DD([0,1[,E)$-valued random vector. We denote by $N_X$ the random number of jumps of $X$, let $0<Y_1<\dots<Y_{N_X}<0$ be the *jump positions* of $X$, and set $Y_0=0$ and $Y_{N_X+1}=1$. Moreover, we denote by $$Z_i:=\rho(X(Y_{i-1}),X(Y_i))$$ the *moduli of the increments* and, in the case where $E$ is a linear space, we denote by $$Z^{(i)}:=X(Y_i)-X(Y_{i-1})$$ the *increments*. As distortion measure on $\DD([0,1[,E)$ we consider $$\rho_\DD(f,g) := \int_0^1 \rho(f(t),g(t))\, \d t,\qquad f,g \in \DD([0,1[,E), \label{eqn:distm}$$ where $\rho$ is a distortion measure on $E$. It is straightforward to extend the results of this paper to the distortion measure $\rho_\DD^p(f,g) = ( \int_0^1 \rho(f(t),g(t))^p \, \d t)^{1/p}$, with $1\leq p<\infty$. Our lower bounds require that the jump positions constitute a *Poisson point process* with intensity $\lambda$. The upper bounds on the complexity are proven under weaker assumptions on $X$. Here, we only assume that the total number of jumps can be estimated against the probability weights of a Poisson random variable: $${\P\left(N_X = k\right)} \leq \frac{\lambda^k}{k!}\, e^{-\lambda} K,\qquad k\geq 0, \label{decrcond}$$ where $\lambda>0$ and $K\ge1$ are some fixed parameters. In particular, one can choose $K=1$, if the jump positions are induced by a Poisson point process. Sometimes we shall also impose the following condition: 1. The jump positions are independent of the jump destinations, which means that, given the event $\{N_X=k\}$, the vector $(Y_1, \ldots, Y_k)$ is independent of the vector $(X(Y_0), \ldots, X(Y_k))$. Let us introduce some more notation. Firstly, we make use of the concept of metric entropy. If $\rho$ is a distortion measure on $E$ we define its covering numbers by $$N(E,\rho,{\varepsilon}) := \min\lbrace n \in \N \,:\, \exists x_1, \ldots, x_n \in E~\forall x\in E~ \exists i~:~ \rho(x,x_i)\leq {\varepsilon}\rbrace.$$ A set $\{ x_1, \ldots, x_n \}$ for which the defining property of $N$ holds is called an ${\varepsilon}$-net of $E$. Note that in general one has to assume that $N(E,\rho,{\varepsilon})$ is well-defined, i.e. that for all ${\varepsilon}>0$ there is an ${\varepsilon}$-net of $E$. This is ensured if, for example, $(E,\rho)$ is a precompact metric space. We also introduce the inverse concept of $D^{(q)}$, which we call $d^{(q)}$, given by $$d^{(q)}({\varepsilon}\,|\,X,\rho,s) := \inf\left\lbrace n\geq 1, n\in\N \,:\, D^{(q)}( \log n ~|~ X,\rho, s) \leq {\varepsilon}\right\rbrace.$$ In other words, $d^{(q)}$ is the number of points needed to quantize with error at most ${\varepsilon}$, i.e. roughly it is the inverse function of $D^{(q)}(\log (.))$. We shall also need the notation of strong and weak asymptotics. Namely, we write $f\lesssim g$, if $\limsup f/g \leq 1$. Analogously, $f\gtrsim g$ is defined. Furthermore, $f\sim g$ means $\lim f/g = 1$. We also use $f\approx g$ if $0<\liminf f/g \leq \limsup f/g < \infty$. Finally, throughout the article $\lambda_d$ denotes the $d$-dimensional Lebesgue measure. The paper is organized as follows. In the rest of this section we state the main results. In Section \[sec:iupper\], we state the upper bounds for both quantities under various additional assumptions. In Section \[sec:ilower\], the upper bounds are complemented by corresponding lower bounds. In particular, we obtain that the upper and lower bounds are tight in many cases. Finally, Section \[sec:cpp\] is devoted to the particular setting where $X$ is a compound Poisson process. The proofs for the upper bounds can be found in Section \[sec:pg\]. There, explicit coding strategies are constructed. The lower bounds are proven in Sections \[sec:lb\] and \[sec:ece\] for quantization- and entropy coding, respectively. The proofs for the lower bounds of the quantization error rely on a small ball argument, whereas the lower bounds for the entropy coding error are derived using the Shannon lower bound for a related problem. Upper bounds {#sec:iupper} ------------ Our first result concerns the case where the space $E$ has finite covering numbers. In the case where $(E,\rho)$ is a metric space, this corresponds to the assumption that $E$ is precompact. Assume that $w:=\sup_{x,y\in E} \rho(x,y)<\infty$ and that the upper box dimension $$\gamma:=\limsup_{\ep\to 0} \frac{\log N(E,\rho,{\varepsilon})}{\log 1/{\varepsilon}} \label{eqn:assbox}$$ is finite. Then $$\label{eqn:q} - \log D^{(q)}( r ~|~ X,\rho_\DD,s) \gtrsim \sqrt{\frac{2}{s (1+\gamma)}\, r \log r}.$$ \[thm:q\] Assume that $N(E,\rho,{\varepsilon})<\infty$ for all ${\varepsilon}>0$ and that $w:=\sup_{x,y\in E} \rho(x,y)<\infty$. 1. For all $r>r_0=r_0(\lambda)$, $$\label{eqn:e} D^{(e)}\left( K \left( \lambda r + (\lambda+1) \log N(E,\rho,e^{-r}) \right) ~|~ X,\rho_\DD,s\right) \leq C_{s} (w+1) K^{1/s} e^{- r},$$ where the constant $C_s$ depends on $s$ only, and $K$ and $\lambda$ are the constants from (\[decrcond\]). 2. In particular, if the jump positions are distributed according to a Poisson point process with rate $\lambda$, we obtain for $r>r_0=r_0(\lambda)$ $$\label{eqn:epoisson} D^{(e)}( \lambda r + (\lambda+1) \log N(E,\rho,e^{-r}) ~|~ X,\rho_\DD,s) \leq C_{s} (w+1) e^{- r}.$$ 3. In the case of a discrete space $E=\{x_1,\ldots, x_q\}$ we even have the more precise estimate $$D^{(e)}( K \left( \lambda r + (\lambda+1) \log q \right) ~|~ X,\rho_\DD,s) \leq 4 K^{1/s} w \min(1,\lambda)^{1/s}\, e^{- r},\label{eqn:view2}$$ for $r>r_0=r_0(\lambda)$. \[thm:e\] Theorem \[thm:e\] can be interpreted in the following way. In order to quantize with error $e^{-r}$ one needs, on average, $\lambda r$ nats to encode the jump positions, $\lambda \log N(E,\rho,e^{-r})$ nats in order to encode the increments, and another $\log N(E,\rho,e^{-r})$ nats in order to encode the initial position $X(0)$. In particular, the same result can be proved without the $\log N(E,\rho,e^{-r})$ term if the initial value of the process is deterministic. Let us compare Theorem \[thm:q\] and Theorem \[thm:e\] in the case where $N(E,\rho,{\varepsilon})\leq q {\varepsilon}^{-\gamma}$. We point out that the asserted rate of the quantization error is different to the one of the entropy coding error. As we will see below neither the quantization error bounds nor the entropy coding error bounds can be improved significantly. Finally note that $N(E,\rho,{\varepsilon})<\infty$ for all ${\varepsilon}>0$ does not necessarily imply that $w:=\sup_{x,y\in E} \rho(x,y)<\infty$ if $\rho$ does not satisfy the triangle inequality. For the remainder of this subsection, let us assume that $(E,{\left\|.\right\|})$ is a normed linear space with distortion measure $\rho(x,y)={\left\|x-y\right\|}$. We assume that the jump destinations of $X$ (and thus the increments $Z^{(i)}$) are independent of the jump positions ([condition (\*)]{}). Furthermore, assume that the increments, conditioned upon $N_X=k$, are identically distributed (not necessarily independent among each other) with the same law as the $E$-valued random variable, say, $Z^{(1)}$. Furthermore we assume that $X(0)$ is deterministic, i.e. that for some $x_0\in E$ $X(0)=x_0$ a.s. Under the above assumtions the following statements are true. 1. If $$\gamma:=\limsup_{{\varepsilon}\to 0} \frac{\log d^{(q)}({\varepsilon}\,|\,Z^{(1)},{\left\|.\right\|},s)}{\log 1/{\varepsilon}} \in [0,\infty[,$$ then (\[eqn:q\]) is valid for the newly defined $\gamma$. 2. If $d^{(q)}({\varepsilon}\,|\,Z^{(1)},{\left\|.\right\|},s) <\infty$ for all ${\varepsilon}>0$, then $$D^{(e)}\left( K\left( \lambda r + \lambda \log d^{(q)}(e^{-r}\,|\,Z^{(1)},{\left\|.\right\|},s)\right) ~|~ X,\rho_\DD,s\right) \leq C\, e^{- r}.$$ holds with some constant $C>0$ depending on the parameters $s,K,\lambda$, and $\E||Z^{(1)}||^s$. \[thm:assumed\] Theorem \[thm:assumed\] relates the complexity of coding $X$ to that of coding the increments. If the assumptions of both Theorems \[thm:e\] and \[thm:assumed\] are satisfied, then the bounds of the latter theorem provide a better estimate since in general $d^{(q)}({\varepsilon})\leq N({\varepsilon})$ (see Lemma \[lem:quantentropy\]). However, note that in contrast to Theorems \[thm:q\] and \[thm:e\], Theorem \[thm:assumed\] requires that the increments are identically distributed and independent of the jump positions. In case of Theorems \[thm:q\] and \[thm:e\], this is not necessary since, by assumption, the space $E$ is sufficiently well-structured (in the sense of small metric entropy $N$). Let us remark that the assumption in Theorem \[thm:assumed\] that $X(0)$ be deterministic is for simplicity only. If instead $X(0)$ is a random variable in $E$, one has to add $d^{(q)}({\varepsilon}\,|\,X(0),\rho,s)$ to the average number of nats needed to encode $X$ conditioned upon $X(0)$. Finally we mention that one can also prove counterparts to assertions (b) and (c) of Theorem \[thm:e\] in the setting of Theorem \[thm:assumed\]. Lower bounds {#sec:ilower} ------------ As an illustration consider the case of a discrete space $E$, namely let $E=\{x_1, \ldots, x_q\}$, which was first studied in [@Vor07]. Then $N(E,\rho,{\varepsilon})\leq q$, and we thus obtain from Theorems \[thm:q\] and \[thm:e\]: $$- \log D^{(q)}( r ~|~ X,\rho_\DD,s) \gtrsim \sqrt{\frac{2}{s}\, r \log r}\quad\text{and}\quad -\log D^{(e)}( K r ~|~ X,\rho_\DD,s) \gtrsim \frac{r}{\lambda}.\label{eqn:dissresult}$$ Now we ask for lower bounds. Clearly, one cannot expect a non-trivial lower bound when only assuming (\[decrcond\]). Thus, let us assume in this subsection that the jump positions constitute a Poisson point process and that [condition (\*)]{} holds. In this case, we show in Theorem \[thm:onoff\] that the order of $D^{(q)}$ in (\[eqn:dissresult\]) is in fact the true order on this scale. Below, in Theorem \[thm:steffenmod\], we show that the order of $D^{(e)}$ is the correct one, too. We consider a more general situation than a finite, discrete space. We only have to assume that there is sufficient uncertainty in the model in order to ensure that every jump indeed has to be encoded. Concretely, assume that [condition (\*)]{} holds and that the jump positions form a Poisson point process. Furthermore, we assume that, given the event that $k$ jumps occur ($\{N_X=k\}$), the moduli of the increments $Z_1, \ldots, Z_k$ are such that there are ${\varepsilon}_0>0$ and $\delta_0>0$ (independent of $k$) such that for all $i=1,\ldots, k$, ${\P\left(Z_i>{\varepsilon}_0\,|\,N_X=k\right)} \geq \delta_0$. Additionally, we now impose that $(E,\rho)$ is a [*metric*]{} space. Under the above assumptions, $$-\log D^{(q)}( r ~|~ X,\rho_\DD,s) \lesssim \sqrt{\frac{2}{s}\, r \log r}.$$ In particular, for a discrete metric space $E=\{x_1, \ldots, x_q\}$, $$-\log D^{(q)}( r ~|~ X,\rho_\DD,s) \sim \sqrt{\frac{2}{s}\, r \log r}.$$ \[thm:onoff\] Note that in view of (\[eqn:view2\]) the rates for quantization error and entropy coding error must be different in case of a discrete metric space $E=\{x_1, \ldots, x_q\}$. Moreover, the order of convergence of the quantization error depends strongly on the moment $s$. In particular, one has for two distinct moments $0<s<s'$ that $$\lim_{r\to\infty} \frac{D^{(q)}( r ~|~ X,\rho_\DD,s)}{D^{(q)}( r ~|~ X,\rho_\DD,s')} =0.$$ This contrasts earlier results on quantization where the same order of convergence is obtained for all moments $s>0$. Let us consider a simple example. Let $X$ be an alternating Poisson process, i.e. $$X(t)=\sum_{i=1}^{N(t)}(-1)^{i-1}, \label{eqn:alternpp}$$ where $(N(t))_{t\in[0,1]}$ is a Poisson (counting) process with rate $\lambda$ (cf. Section \[sec:cpp\]) with the natural metric $|.|$ on $E=\{0,1\}$. Then Theorem \[thm:onoff\] yields $$-\log D^{(q)}( r ~|~ X,\rho_\DD,s) \sim \sqrt{\frac{2}{s}\, r \log r}.$$ Recall that the assertions of the upper bounds, do not require that $(E,\rho)$ is a metric space. The statement is valid for any distortion measure $\rho$. However, the lower bound from Theorem \[thm:onoff\] fails for general distortion measures. This can be seen from the following simple example. Let $E=\{ 0,1\}\cup \{ 2^{-n}, n\geq 1\}$ and $\rho(0,1)=1$, $\rho(0,2^{-n})=\rho(1,2^{-n})=2^{-n}$, $\rho(2^{-n},2^{-m})=1$ for $n\neq m$ and $n,m\geq 1$. Note that this is not a metric space. Consider the alternating Poisson process, i.e. the model from (\[eqn:alternpp\]). Then $X$ satisfies the assumptions of Theorem \[thm:onoff\], in particular those for the moduli of the increments (since $Z_i=1$), but $D^{(q)}( r ~|~ X,\rho_\DD,s)=0$, for all $r\geq 0$. Next, we will prove a lower bound for the entropy coding error. \[thm:steffenmod\] Let $X$ be a jump process satisfying condition (\*). We assume that he jump positions $(Y_i)$ form a Poisson point process with rate $\lambda$. Moreover, we suppose that $\rho$ defines a metric on $E$ and that a.s. the moduli of the jumps are bounded from below by ${\varepsilon}_0>0$. Then for all $s\geq 1$ and all sufficiently large $r$ $$D^{(e)}( r ~|~ X,\rho_\DD,s) \geq {\varepsilon}_0 C \min(1,\lambda) \, e^{-r/\lambda},$$ where $C>0$ is an absolute constant. The lower bound in Theorem \[thm:steffenmod\] is actually shown to hold for the *distortion rate function* $D(r~|~X,\rho,s)$ defined in Section \[sec:ece\]. We obtain the following corollary as a special case. \[cor:steffen\] Let $X$ satisfy the conditions of Theorem \[thm:steffenmod\]. Assume additionally that $X(0)$ is deterministic and consider the case of a discrete metric space $E=\{x_1,\ldots, x_q\}$ with $w := \max_{x,y\in E} \rho(x,y)$. Then for $s\ge 1$ $$C_1 {\varepsilon}_0 \min(1,\lambda) e^{-r/\lambda}\leq D^{(e)}( r ~|~ X,\rho_\DD,s) \leq C_2 \,q \, w\, \min(1,\lambda)^{1/s} \, e^{-r/\lambda},$$ for large enough $r$ and absolute constants $C_1, C_2>0$. The corollary follows immediately from part (c) of Theorem \[thm:e\] and the remark after it and Theorem \[thm:steffenmod\]. This result shows that the bounds for the entropy coding error in Theorems \[thm:e\] and \[thm:assumed\] are tight. Consider again the alternating Poisson process from (\[eqn:alternpp\]) with the natural metric $|.|$. Then Corollary \[cor:steffen\] yields, for all $s\geq 1$, $$C_1 \min(1,\lambda) e^{-r/\lambda} \leq D^{(e)}( r ~|~ X,\rho_\DD,s) \leq C_2 \min(1,\lambda)^{1/s} \, e^{-r/\lambda},$$ for large enough $r$ and absolute constants $C_1, C_2>0$. Let us illustrate the influence of a random initial position on our estimates. For this purpose, consider an alternating Poisson process with random initial position, i.e.  $$X(t)=X(0) + \sum_{i=1}^{N(t)}(-1)^{i-1+X(0)},$$ where $X(0)$ equals $0$ and $1$ with probability $1/2$, respectively, cf. [@Vor07]. Our Theorem \[thm:e\], part (c), and Theorem \[thm:steffenmod\] yield $$C_1 \min(1,\lambda) e^{-r/\lambda} \leq D^{(e)}( r ~|~ X,\rho_\DD,s) \leq C_2 \min(1,\lambda)^{1/s} 2^{1/\lambda} \, e^{-r/\lambda},$$ for all $s\geq 1$ and all large enough $r$ and absolute constants $C_1, C_2>0$. Application to compound Poisson processes in $\R^d$ {#sec:cpp} --------------------------------------------------- As an application of our results, let us determine the coding complexity of $\R^d$-valued compound Poisson processes. Recall that a Lévy process with finite Lévy measure is a compound Poisson process with the following structure, cf. e.g. [@bertoin]. Let $(N(t))_{t\geq 0}$ be a Poisson (counting) process with intensity $\lambda>0$, i.e. let $N(t):=\max\{ n \geq 0 \,:\,\sum_{i=1}^n e_j\leq \lambda t\}$, where $(e_j)$ are i.i.d. standard exponential random variables. Consider $$X(t) = \sum_{i=1}^{N(t)} Z^{(i)},\qquad t\in[0,1[, \label{eqn:cpp}$$ where the $Z^{(i)}$, $i=1,2, \ldots$, are i.i.d. and distributed according to any probability distribution in $\R^d$ with ${\P\left(Z^{(1)}=0\right)}=0$. Note that this notation is consistent with the one employed above for the increments. Note furthermore that for compound Poisson processes [condition (\*)]{} is satisfied. We consider the distortion measure $${\left\|X\right\|}_1 := \int_0^1 {\left\|X(t)\right\|}_\infty \, \d t,$$ which of course coincides with $\rho_\DD$ for $\rho={\left\|.\right\|}_\infty$, where as usual ${\left\|x\right\|}_\infty:=\max_{i=1,\ldots, d} |x_i|$. However, one can replace ${\left\|.\right\|}_\infty$ by any norm on $\R^d$, which would change only the constants. Theorem \[thm:assumed\] yields the following corollary. Let $X$ be a compound Poisson process as defined in (\[eqn:cpp\]) and $s>0$. 1. Assume that $$\gamma:=\limsup_{{\varepsilon}\to 0} \frac{\log d^{(q)}({\varepsilon}\,|\,Z^{(1)},{\left\|.\right\|}_\infty,s)}{\log 1/{\varepsilon}} \label{eqn:dbox}$$ is finite. Then $$- \log D^{(q)}( r ~|~ X,\rho_\DD,s) \gtrsim \sqrt{\frac{2}{s (1+\gamma)}\, r \log r}.$$ 2. Let $d^{(q)}({\varepsilon}\,|\,Z^{(1)},\rho,s) <\infty$ for all ${\varepsilon}>0$. Then, for $r\geq r_0$ and a constant $C=C(s,\lambda,\E || Z^{(1)}||_\infty^s)$, we have $$D^{(e)}( \lambda r + (\lambda+1) d^{(q)}(e^{-r}\,|\,Z^{(1)},\rho,s) ~|~ X,\rho_\DD,s) \leq C e^{- r}.$$ \[cor:cpp\] Alternatively, one can study the consequences of Theorems \[thm:q\] and \[thm:e\] if one has additional information on the range of $X$. As for lower bounds we can apply Theorem \[thm:onoff\], which gives the following. Let $X$ be a compound Poisson process as defined in (\[eqn:cpp\]) and $s>0$. Then $$- \log D^{(q)}( r ~|~ X,\rho_\DD,s) \lesssim \sqrt{\frac{2}{s}\, r \log r}.$$ If additionally (\[eqn:dbox\]) holds with $\gamma=0$, then $$- \log D^{(q)}( r ~|~ X,\rho_\DD,s) \sim \sqrt{\frac{2}{s}\, r \log r}.$$ We obtain a similar result in the case that the distribution of the increments has an absolutely continuous component. Let $X$ be a compound Poisson process as defined in (\[eqn:cpp\]) and $s>0$. Assume that the distribution of $Z^{(1)}$ has an absolutely continuous component. Then $$-\log D^{(q)}( r ~|~ X,\rho_\DD,s) \lesssim \sqrt{\frac{2}{s(1+d)}\, r \log r}.$$ If additionally (\[eqn:dbox\]) holds with $\gamma=d$ then $$- \log D^{(q)}( r ~|~ X,\rho_\DD,s) \sim \sqrt{\frac{2}{s(1+d)}\, r \log r}.$$ \[thm:lowerac\] Theorems \[thm:onoff\] and \[thm:lowerac\] show that the upper bound for the quantization rate in Theorems \[thm:q\] and \[thm:assumed\] (and thus Corollary \[cor:cpp\]) cannot be improved in general (for all $\gamma\in\N$). Let us finally list a corollary of Theorem \[thm:steffenmod\]. Let $X$ be a compound Poisson process as defined in (\[eqn:cpp\]). Assume that ${\left\|Z^{(1)}\right\|}_\infty>{\varepsilon}_0$ a.s. Then, for all $s\geq 1$, $$D^{(e)}( r ~|~ X,\rho_\DD,s) \geq {\varepsilon}_0 C \min(1,\lambda) \, e^{-r/\lambda},$$ for $r>r_0$ and $C>0$ an absolute constant. The most instructive examples of the application of the results of this subsection are given now. Consider a Poisson (counting) process with intensity $\lambda$, i.e. let $Z^{(1)}=1$. Then $$-\log D^{(q)}( r ~|~ X,\rho_\DD,s) \sim \sqrt{\frac{2}{s}\, r \log r}$$ and $$C_1 \min(1,\lambda) e^{-r/\lambda}\leq D^{(e)}( r ~|~ X,\rho_\DD,s) \leq C_2 e^{-r/\lambda},$$ for $s\geq 1$, $r>r_0$, where $C_1>0$ is an absolute constant, and $C_2>0$ depends on $s$ and $\lambda$. \[example:PP\] Let $Z^{(1)}$ be uniformly distributed in $[0,1]^d$. Then $$-\log D^{(q)}( r ~|~ X,\rho_\DD,s) \sim \sqrt{\frac{2}{s(1+d)}\, r \log r}$$ and $$D^{(e)}( r ~|~ X,\rho_\DD,s) \leq C_2 e^{-r/((1+d)\lambda)},$$ for $s\geq 1$, $r>r_0$, where $C_2>0$ depends on $s$ and $\lambda$. We conjecture that the order on the right-hand side is the correct one. Let $Z^{(1)}$ be uniformly distributed in $C$, where $C$ is the Cantor set in $[0,1]$. Set $\gamma:=\log 2/\log 3$. Then $$\sqrt{\frac{1}{s}\, r \log r} \gtrsim -\log D^{(q)}( r ~|~ X,\rho_\DD,s) \gtrsim \sqrt{\frac{2}{s(1+\gamma)}\, r \log r}$$ and $$D^{(e)}( r ~|~ X,\rho_\DD,s) \leq C_2 e^{-r/((1+\gamma)\lambda)},$$ for $s\geq 1$,s $r>r_0$, where $C_2>0$ depends on $s$ and $\lambda$. We conjecture that the orders on the right-hand side, respectively, are the correct ones. The theorems and examples presented in this subsection complement results from [@aurzadadereich], where general real-valued Lévy processes are studied. The main result for compound Poisson processes in that paper states that, for any compound Poisson process with $\E \log \max(|Z^{(1)}|,1) < \infty$ and all $s\geq 1$, $$\log D^{(e)} (r~|~ X,\rho_\DD,s) \approx - r.$$ No result on the quantization error for compound Poisson processes is obtained in [@aurzadadereich]. Our findings also improve the results in [@luschgypages06], where an upper bound for the quantization error of real-valued compound Poisson processes is obtained. In particular, it is shown that for the Poisson (counting) process and all $s\geq 1$, $$-\log D^{(q)}( r ~|~ X,\rho_\DD,s) \gtrsim \sqrt{\frac{1}{s}\, r \log r}.$$ The correct rate on this scale is given in Example \[example:PP\]. Upper bounds {#sec:pg} ============ In this section, we provide the proofs of the upper bounds for the quantization error and the entropy coding error stated in Theorems \[thm:q\], \[thm:e\], and \[thm:assumed\], respectively. In the proofs, the following four technical lemmas are needed. First we prove a result on the asymptotic behaviour of a certain sum occurring in the calculations. Let $c>0$. Then $$\log \left( \sum_{k=0}^\infty \frac{c^k}{k!}\, e^{-c} e^{- r / (k+1)} \right)~ \sim~ - \sqrt{2 r \log r},\qquad \text{as $r\to \infty$.}$$ \[lem:crucialorder\] Let $V$ be a random variable that is Poisson distributed with mean $c$. Then the term in question equals $$\log \E e^{- r / (V+1)}.$$ By the so-called de Bruijn Tauberian theorem (cf. [@bgt], Theorem 4.12.9), considering the Laplace transform is equivalent to considering the lower tail of $(V+1)^{-1}$. Thus, consider $$\log {\P\left(\frac{1}{V+1} <{\varepsilon}\right)} = \log {\P\left(V > \frac{1}{{\varepsilon}} -1 \right)} = \log \sum_{\{ k> \frac{1}{{\varepsilon}}-1\}} \frac{c^k}{k!}\, e^{-c} \sim -\frac{1}{{\varepsilon}} \log \frac{1}{{\varepsilon}},$$ where we used Stirling’s Formula in the last step. Using the above-mentioned Tauberian theorem returns the asserted order of the Laplace transform, including the constant. Secondly, we prove a quantization result for random variables in a space $E$ with known metric entropy. This is needed in order to encode the increments of the process $X$. Let $X$ be any random variable on a space $E$. Then, for all $s>0$ and all ${\varepsilon}>0$, $$D^{(q)}( \log N(E,\rho,{\varepsilon}) ~|~ X,\rho,s) \leq {\varepsilon}.$$ In other words, $d^{(q)}( {\varepsilon}~|~ X,\rho,s) \leq N(E,\rho,{\varepsilon})$. \[lem:quantentropy\] For given ${\varepsilon}>0$ let $\CC$ be an ${\varepsilon}$-net of $(E,\rho)$. By the definition of the covering numbers, $\CC$ can be chosen to contain only $N(E,\rho,{\varepsilon})$ elements. Thus $$D^{(q)}( \log N(E,\rho,{\varepsilon}) ~|~ X,d,s) \leq \left( \E \min_{a\in\CC} \rho(X,a)^s \right)^{1/s} \leq {\varepsilon}.$$ By using product quantization, it is clear that for a random variable $X$ in $E^k:=E\times \ldots \times E$ with $\rho^k(x,y) := \max_{i=1,\ldots, k} \rho(x_i,y_i)$ we have $$D^{(q)}( k \log N(E,\rho,{\varepsilon}) ~|~ X,\rho^k,s) \leq {\varepsilon}.$$ \[rem:pqe\] Essentially the same technique is applied in the proof of the next lemma. The result is comparable, but slightly more precise. This version is used to encode the jump positions. Let $Y$ be any random variable in $[0,1]^k$. Then, for all $s>0$, $r\geq 0$, $$D^{(q)}( r ~|~ Y,{\left\|.\right\|}_\infty,s) \leq e^{-r/k}.$$ If $Y$ is such that $Y_1\leq \ldots \leq Y_k$ almost surely then we can restrict ourselves to codebooks $\CC$ with $\hat Y_1\leq \ldots \leq \hat Y_k$ for all $\hat Y \in \CC$.\[lem:unifde\] Note that this may be a fairly weak estimate in concrete cases; however, it holds for all $k\geq 1$ and all $r\geq 0$. If more is known about the distribution of $Y$, much better (asymptotic) estimates are available, cf. [@grafluschgy], e.g. Theorem 6.2. Let us first consider the case $e^r=n=((m+1)/2)^k$ with $m\geq 1$. Then we can use a simple product quantizer. Namely, we set $$\CC:= \{ (y_1/m,\ldots,y_k/m)\in[0,1]^k ~:~ y_i\in\{1,3,5,\ldots\}, i=1,\ldots, k\}.$$ Then $\# \CC \leq ((m+1)/2)^k$ and thus $$D^{(q)}( \log ((m+1)/2)^k ~|~ Y,{\left\|.\right\|}_\infty,s) \leq \left(\E \min_{a\in \CC} {\left\|Y-a\right\|}_\infty^s\right)^{1/s} \leq m^{-1}.$$ For any $r>0$ with $e^r\geq 2^k$, there exists an $m\geq 1$ such that $(\frac{m+1}{2})^k \leq e^r < (\frac{m+2}{2})^k$. Then $$\begin{gathered} D^{(q)}( r ~|~ Y,{\left\|.\right\|}_\infty,s) \leq D^{(q)}( \log ((m+1)/2)^k ~|~ Y,{\left\|.\right\|}_\infty,s) \\ \leq m^{-1} \leq (2 e^{r/k}-2)^{-1} \leq e^{-r/k},\end{gathered}$$ where we used $e^r\geq 2^k$ in the last step. Finally, for $1\leq e^r\leq 2^k$, $$D^{(q)}( r ~|~ Y,{\left\|.\right\|}_\infty,s) \leq \left(\E {\left\|Y-(1/2, \ldots, 1/2)\right\|}_\infty^s\right)^{1/s}\leq 1/2 \leq e^{-r/k}.$$ The last lemma can be strengthened if it is known that the random vector $Y=(Y_1, \ldots , Y_k)$ satisfies $Y_1\leq \ldots \leq Y_k$. There are absolute constants $c^*, \kappa>0$ such that, for any random variable $Y$ in $[0,1]^k$ such that almost surely $Y_1\leq \ldots \leq Y_k$, we have, for all $s>0$, $$D^{(q)}( r ~|~ Y,{\left\|.\right\|}_\infty,s) \leq \frac{\kappa}{k}\, e^{-r/k},\qquad \text{for all $r\geq c^* k$.} \label{eqn:save1}$$ \[lem:improvedlemorder\] Let $m\geq k$ and consider $$\CC := \left\lbrace \left(\frac{y_1}{m}, \ldots, \frac{y_k}{m}\right) ~:~ 1\leq y_1\leq y_2\leq \ldots \leq y_k\leq m, y_i\in\{ 1,\ldots, m\}\right\rbrace.$$ Clearly, $$\# \CC=\binom{m+k-1}{k}.$$ Note that for any $y\in[0,1]^k$ with $y_1\leq\ldots\leq y_k$ we have $\min_{a\in\CC} {\left\|y-a\right\|}_\infty \leq 1/m$. Thus, $$D^{(q)}(\log (\#\CC)~|~ Y,{\left\|.\right\|}_\infty,s) \leq \frac{1}{m},$$ for any random variable $Y$ that satisfies the assumption of the lemma. Note that, by Stirling’s Formula, for some absolute constants $C_1,C_2,c^*>0$, $$\binom{k+1+k-1}{k} =\frac{(2k)!}{(k!)^2}\leq C_1 \frac{ 2^{2k}k^{2k}e^{-2k}\sqrt{2 \pi 2 k}}{k^{2k}e^{-2k}2\pi k} \leq C_2 2^{2k}\leq e^{c^* k}. \label{eqn:lqstar}$$ Let $r\geq c^* k$. Then there is an $m\geq k$ such that $$\binom{m+k-1}{k} < e^r \leq \binom{m+1+k-1}{k}, \label{eqn:lqplus}$$ because, as seen in (\[eqn:lqstar\]), $$\min_{m\geq k} \binom{m+k-1}{k} = \binom{k+k-1}{k} < \binom{k+1+k-1}{k} \leq e^{c^* k} \leq e^r.$$ Thus, $$D^{(q)}(r~|~ Y,{\left\|.\right\|}_\infty,s)\leq D^{(q)}\left(\left.\log \binom{m+k-1}{k}~\right|~ Y,{\left\|.\right\|}_\infty,s\right) \leq \frac{1}{m}.\label{eqn:lqtristar}$$ By (\[eqn:lqplus\]) and Stirling’s Formula, for some absolute constants $C_3,C_4>0$, $$\begin{gathered} e^r\leq \binom{m+1+k-1}{k}= \frac{(m+k)!}{m! k!} \\ \leq C_3 \, \frac{(m+k)^m(m+k)^k e^{-m-k} \sqrt{2\pi (m+k)}}{m^m e^{-m}\sqrt{2 \pi m}\, k^k e^{-k}\sqrt{2 \pi k}} \\ \leq C_4\, \left(1+\frac{k}{m}\right)^m \frac{(2 m)^k}{k^k} \, \sqrt{\frac{m+k}{m k}}.\label{eqn:lqdoustar}\end{gathered}$$ Observe that $\left(1+\frac{k}{m}\right)^m\leq e^k$ and that $\frac{m+k}{m k}=\frac{1}{k}+\frac{1}{m}\leq 2$, for all $m$ and $k$. Therefore, the term in (\[eqn:lqdoustar\]) can be estimated by $$C_5 (2 e)^k \,\frac{m^k}{k^k}\leq \kappa^k\, \frac{m^k}{k^k},$$ where $\kappa$ is an absolute constant. This implies $k e^{r/k}\leq \kappa m$ or $1/m\leq \kappa e^{-r/k}/k$. We deduce from (\[eqn:lqtristar\]) that for any $r\geq c^* k$ (\[eqn:save1\]) holds, as asserted. Now we can proceed with the proof of our first main result. Let $X_k$ be a random variable that has the distribution of $X$ conditioned upon the event that $N_X=k$, i.e. $X$ that has $k$ jumps. Let $Y$ be the vector in $[0,1]^k$ with the jump positions of $X_k$ (in increasing order) and $Z$ be the $E^{k+1}$-vector containing values of the process $X_k$ between the jumps (in the order corresponding to when they occur), i.e. the initial value and the $k$ jump destinations. Note that we can reconstruct $X_k$ completely from the vectors $Y$ and $Z$. Thus, it is sufficient to find good codebooks for $Z$ and $Y$. Let $\delta>0$. By assumption, there is an ${\varepsilon}_0={\varepsilon}_0(\delta)\in]0,1[$ such that for all $0<{\varepsilon}\leq{\varepsilon}_0$, $$\log N(E,\rho,{\varepsilon}) \leq (\gamma+\delta) \log 1/{\varepsilon}.\label{eqn:boxasmpt}$$ Let $r\geq \log 1/{\varepsilon}_0$. For $0\leq k \leq k_0:=k_0(\delta,r):=r \min(1, (\log 1/{\varepsilon}_0(\delta))^{-1})-1$, let $\CC_k''$ be a codebook for $Z$ in $(E^{k+1},\rho^{k+1})$ with $$\left( \E \min_{\hat{Z}\in \CC_k''} \rho^{k+1}(Z,\hat{Z})^s\right)^{1/s} \leq 2 e^{-r/(k+1)}. \label{eqn:gets4}$$ By Remark \[rem:pqe\], $\CC_k''$ can be chosen such that $$\log \#\CC_k'' \leq (k+1) \log N(E,\rho,e^{-r/(k+1)} ) \leq (k+1) (\gamma+\delta) \log\left( e^{r/(k+1)} \right)= (\gamma +\delta) r,\label{eqn:repl1}$$ where we used (\[eqn:boxasmpt\]) and the choice of $k_0$. For $1\leq k \leq k_0$, let $\CC_k'$ be a codebook for $Y$ in $(\R^k,{\left\|.\right\|}_\infty)$ with $$\left( \E \min_{\hat{Y}\in \CC_k'} {\left\|Y-\hat{Y}\right\|}_\infty^s\right)^{1/s} \leq 2 \left( e^{r-k}\right)^{-1/k}. \label{eqn:gets2}$$ By Lemma \[lem:unifde\], $\CC_k'$ can be chosen such that $\log \#\CC_k' \leq r-k$. Define $\CC_0:=\CC_0''$. For $k\neq 0$, let $\CC_k$ be the Cartesian product of the codebooks $\CC_k'$ and $\CC_k''$. Then $\log \#\CC_k \leq r-k + (\gamma +\delta)r$ for all $0\leq k\leq k_0$. Let us define the following notation: for any $\hat{Y}\in\CC_k'$, we set $$F:=\bigcup_{i=1}^k \left[\hat{Y}_i,Y_i\right[ \cup \left[Y_i,\hat{Y}_i\right[ \subseteq [0,1[. \label{eqn:defnff}$$ Note that on $[0,1[\setminus F$, $X$ can be reconstructed up to the error given in (\[eqn:gets4\]). Furthermore, note that the Lebesgue measure of $F$ is less than $k\,{\left\| Y-\hat{Y}\right\|}_\infty$. With the help of this information, we can estimate the error of approximating by $\CC_k$ when $k\neq 0$: $$\begin{aligned} \E \min_{a\in \CC_k} \rho_\DD(X_k,a)^s \notag &=& \E \min_{a\in \CC_k} \left( \int_0^1 \rho(X_k(t),a(t)) \, \d t\right)^s \notag \\ &\leq & C_s \E \min_{a\in \CC_k} \left( \left( \int_{F} \ldots \, \d t \right)^s + \left( \int_{[0,1[\setminus F} \ldots\, \d t \right)^s\right)\notag \\ &\leq & C_s \E \min_{\hat{Y}\in \CC_k'} \min_{\hat{Z}\in \CC_k''} \left( \left( w k {\left\|Y-\hat{Y}\right\|}_\infty \right)^s + \left( \rho^{k+1}(Z,\hat{Z}) \right)^s\right)\label{eqn:crucialchange} \\ &=& C_s \left( (k w)^s \E \min_{\hat{Y}\in \CC_k'} {\left\|Y-\hat{Y}\right\|}_\infty^s + \E \min_{\hat{Z}\in \CC_k''} \rho^{k+1}(Z,\hat{Z})^s \right)\notag \\ &\leq& C_s \left( (2 k w)^s \left( e^{r-k}\right)^{-s/k} + 2^s e^{-s r/(k+1)}\right) \notag \\ & \leq& D k^s e^{-r s /(k+1)}, \label{eqn:gets1}\end{aligned}$$ having used (\[eqn:gets2\]) and (\[eqn:gets4\]) in the last but one step, where $D:=C_s 2^s ((e w)^s +1)$. We define the codebook $\CC := \bigcup_{0\leq k \leq k_0} \CC_k$. Then $$\#\CC\leq \sum_{0\leq k\leq k_0} e^{r-k+(\gamma+\delta) r} \leq e^{r+(\gamma+\delta) r} \sum_{k=0}^{\infty} e^{-k} \leq e^{r+(\gamma+\delta) r +1}.$$ Thus, $$\begin{gathered} D^{(q)}( (1+\gamma+\delta) r +1 ~|~ X,\rho_\DD,s)^s \leq \E \min_{a\in \CC} \rho_\DD(X,a)^s \\ \leq \sum_{0\leq k \leq k_0} {\P\left(N_X=k\right)} \E \min_{a\in \CC_k} \rho_\DD(X_k,a)^s + \sum_{k > k_0} {\P\left(N_X=k\right)} \E \min_{a\in \CC_0} \rho_\DD(X_k,a)^s.\label{eqn:reas2}\end{gathered}$$ Using (\[decrcond\]), (\[eqn:gets1\]), and the trivial fact that $\rho_\DD(X_k,a)\leq w$, the last expression is seen to be less than $$\begin{aligned} &&K e^{-\lambda}\left( 2^s e^{-s r} + \sum_{1\leq k\leq k_0} \frac{\lambda^k}{k!}\, D k^s e^{-rs/(k+1)} + \sum_{k> k_0} \frac{\lambda^k}{k!}\, w^s\right) \notag \\ &=& K e^{-\lambda}\left(2^s e^{-s r} + D \sum_{1\leq k\leq k_0} \frac{\lambda^k}{k!}\, k^s e^{-r s / (k+1)} + w^s \sum_{k> k_0} \frac{\lambda^k}{k!}\, e^{-rs/(k+1)} e^{rs/(k+1)}\right) \notag \\ &\leq& K e^{-\lambda}\left( 2^s e^{-s r} + D \sum_{1\leq k\leq k_0} \frac{\lambda^k}{k!}\, e^{ks} e^{-r s / (k+1)} + \left(\frac{w}{{\varepsilon}_0}\right)^s \sum_{k> k_0} \frac{\lambda^k}{k!}\, e^{-r s / (k+1)} \right) \notag \\ &\leq& K e^{-\lambda} 2^s e^{-s r} + C_{K,s,w,\lambda,{\varepsilon}_0(\delta)}\sum_{k=0}^{\infty} \frac{(e^s \lambda)^k}{k!}\, e^{-e^s \lambda} e^{-r s /(k+1)} \label{eqn:save2}.\end{aligned}$$ Recall from Lemma \[lem:crucialorder\] that the exponential order of the sum, when $r\to\infty$, is $$- \sqrt{2 r s \log (r s)} \sim - \sqrt{2 r s \log r}$$ and that the constant in front of it does not depend on $r$. The first term in (\[eqn:save2\]) also has no influence. Thus, for any $\delta>0$, $$\limsup_{r\to \infty} \frac{\log D^{(q)}( (1+\gamma+\delta)r+1 ~|~ X,\rho_\DD,s)}{\sqrt{r \log r}} \leq - \sqrt{\frac{2}{s}}.$$ Therefore $$\limsup_{r\to \infty} \frac{\log D^{(q)}( r ~|~ X,\rho_\DD,s)}{\sqrt{r \log r}} \leq - \sqrt{\frac{2}{s(1+\gamma+\delta)}},$$ which holds for any $\delta>0$. Letting $\delta$ tend to $0$ gives the assertion. First we treat part (a). Again we condition upon the event that $k$ jumps occur. Let $X_k$ be a random variable that has the distribution of $X$ conditioned upon the event that $N_X=k$, i.e. that $X$ has $k$ jumps. Let, as above, $Y$ be the vector in $[0,1]^k$ with the jump positions of $X_k$ and $Z$ be the $E^{k+1}$-vector containing the values of the process $X_k$ between the jumps. Recall that one can reconstruct $X_k$ from $Y$ and $Z$, so it suffices to find good codebooks for $Y$ and $Z$. Let $c^*,\kappa>0$ be the absolute constants from Lemma \[lem:improvedlemorder\]. Let $r\geq \lambda c^*$. Fix $k\geq 1$. Let $\CC_k'$ be a codebook for $Y$ in $(\R^k,{\left\|.\right\|}_\infty)$ with $$\left( \E \min_{\hat{Y}\in \CC_k'} {\left\|Y-\hat{Y}\right\|}_\infty^s\right)^{1/s} \leq \frac{2\kappa}{k}\, \left( e^{r k/\lambda}\right)^{-1/k} = \frac{2\kappa}{k}\, e^{-r/\lambda}.$$ By Lemma \[lem:improvedlemorder\] and the fact that $Y_1\leq \ldots\leq Y_k$, $\CC_k'$ can be chosen such that $\log \#\CC_k' \leq k r/\lambda$. Furthermore, for $k\geq 0$, let $\CC_k''$ be a codebook for $Z$ in $(E^{k+1},\rho^{k+1})$ with $$\left( \E \min_{\hat{Z}\in \CC_k''} \rho^{k+1}(Z,\hat{Z})^s\right)^{1/s} \leq 2 e^{-r/\lambda}. \label{eqn:gets4be}$$ By Remark \[rem:pqe\], $\CC_k''$ can be chosen such that $$\log \#\CC_k'' \leq (k+1) \log N(E,\rho, e^{-r/\lambda} ). \label{eqn:repl2}$$ Let $\CC_k$ be the Cartesian product of the codebooks $\CC_k'$ and $\CC_k''$. Then $\log \#\CC_k\leq k r /\lambda + (k+1) \log N(E,\rho, e^{-r/\lambda} )$. Let $F$ be defined as in (\[eqn:defnff\]). In case $k$ jumps occur ($k\neq 0$) we approximate $X$ by a function from $\CC_k$, which gives an error of at most $$\begin{aligned} \E \min_{a\in \CC_k} \rho_\DD(X_k,a)^s \notag &=& \E \min_{a\in \CC_k} \left( \int_0^1 \rho(X_k(t),a(t)) \, \d t\right)^s \notag \\ &\leq & C_s \E \min_{a\in \CC_k} \left( \left( \int_{F} \ldots \, \d t \right)^s + \left( \int_{[0,1[\setminus F} \ldots\, \d t \right)^s\right)\notag \\ &\leq & C_s \E \min_{\hat{Y}\in \CC_k'} \min_{\hat{Z}\in \CC_k''} \left( \left( w k {\left\|Y-\hat{Y}\right\|}_\infty \right)^s + \left( \rho^{k+1}(Z,\hat{Z}) \right)^s\right) \label{eqn:crucialchange2} \\ &=& C_s \left( (k w)^s \E \min_{\hat{Y}\in \CC_k'} {\left\|Y-\hat{Y}\right\|}_\infty^s + \E \min_{\hat{Z}\in \CC_k''} \rho^{k+1}(Z,\hat{Z})^s \right)\notag \\ &\leq& C_s 2^s \left( (w \kappa)^s e^{-rs/\lambda} + e^{-s r/\lambda}\right)\notag \\ & \leq& C_s 2^s ((w \kappa)^s+1) e^{-r s /\lambda}. \label{eqn:lsmodif} \end{aligned}$$ For $k=0$, set $\CC_0:=\CC_0''$. Then the error is less than $2e^{-rs/\lambda}$, by (\[eqn:gets4be\]). On the other hand, this procedure has an expected nat length of at most $$\begin{gathered} K e^{-\lambda} \sum_{k=0}^\infty \frac{\lambda^k}{k!}\, \log \left( \#\CC_k \right) = K \sum_{k=0}^\infty \frac{\lambda^k}{k!}\, e^{-\lambda} \left( k r /\lambda + (k+1) \log N(E,\rho, e^{-r/\lambda} )\right) \\ = K \left( r + (\lambda+1) \log N(E,\rho, e^{-r/\lambda} )\right).\end{gathered}$$ Therefore, similarly to (\[eqn:reas2\]), $$\begin{gathered} D^{(e)}( K \left( r + (\lambda+1) \log N(E,\rho, e^{-r/\lambda} )\right) ~|~ X,\rho_\DD,s)^s \\ \leq K e^{-\lambda}\left( 2e^{-rs/\lambda} + \sum_{k=1}^\infty \frac{\lambda^k}{k!}\, C_s 2^s ((w \kappa)^s+1) e^{-rs/\lambda} \right) \\ = K \left( 2e^{-rs/\lambda} e^{-\lambda} + C_s 2^s ((w \kappa)^s+1) e^{-rs/\lambda} \sum_{k=1}^\infty \frac{\lambda^k}{k!}\, e^{-\lambda}\right) \leq K C_s' (w^s+1) e^{-rs/\lambda}. \label{eqn:modifdisc}\end{gathered}$$ where $C_s'$ only depends on $s$. This yields the assertion (a). To see (b) one only has to recall that in case the jump positions are distributed as a Poisson point process we can choose $K=1$ in (\[decrcond\]). Let us finally show (c). In the case of a discrete space $E=\{x_1,\ldots, x_q\}$ with $w= \max_{x,y\in E} \rho(x,y)$, we can choose $\log \# \CC_k''=(k+1)\log q$. Thus, on $[0,1[\setminus F$, no error arises. This allows to replace the right-hand side in (\[eqn:lsmodif\]) by $(2 w \kappa)^s e^{-r s /\lambda}$. Therefore, the upper bound in (\[eqn:modifdisc\]) becomes $K (2 w \kappa)^s (1-e^{-\lambda}) e^{-rs/\lambda}$, where $\kappa>0$ is the absolute constant from Lemma \[lem:improvedlemorder\]. This finishes the proof of (c). \[rem:modifindiscretecase\] Note that no assumption is necessary on the correlation of the jump positions and increments. Let us now indicate the changes that are necessary to prove Theorem \[thm:assumed\]. The proof carries over almost literally from Theorems \[thm:q\] and \[thm:e\], respectively. The only differences concern the assumption on $d^{(q)}$ instead of the metric entropy $N$, the fixed initial position, and the possibly unbounded jumps. In this case, we encode the increments instead of the jump destinations. Let $Y$ be as above, but $Z$ denote the $E^{k}$ vector with the increments, i.e. $Z:=(Z^{(1)},\ldots,Z^{(k)})$ with $Z^{(i)} := X(Y_i) - X(Y_i-)$. Note that we can reconstruct $X$ from $Y$ and $Z$, since we asssumed $X(0)$ to be deterministic. The first change is to replace (\[eqn:repl1\]) by $$\log \#\CC_k'' \leq k \log d^{(q)}(e^{-r/k} ~|~ Z^{(1)},\rho,s ) \leq k (\gamma+\delta)\log e^{r/k} = (\gamma+\delta) r$$ in the proof for the quantization error. For the entropy coding error one has to replace (\[eqn:repl2\]) by $$\log \#\CC_k'' \leq k \log d^{(q)}(e^{-r/\lambda} ~|~ Z^{(1)},\rho,s ).$$ The second issue concerns a certain refinement in order to deal with the possibly unbounded jumps. Here, we need that we deal with a normed space. We will show that, on average, the high jumps do not have any influence on the rate. In fact, the only modification affects (\[eqn:crucialchange\]), where we estimate by $$C_s \E \min_{\hat{Y}\in \CC_k'} \min_{\hat{Z}\in \CC_k''} \left[ \left(\max_{1\leq n,m\leq k}{\left\|\sum_{i=1}^n Z^{(i)} - \sum_{i=1}^{m} \hat{Z}^{(i)}\right\|} k {\left\|Y-\hat{Y}\right\|}_\infty \right)^s + \left( k \rho^{k}(Z,\hat{Z}) \right)^s\right],$$ which is required due to the fact that we cannot estimate by a finite diameter $w$ (modification in the first term) and the errors may add up over all the jumps, since we encode the increments and not the absolute positions (modification in the second term). The first term can be estimated by $$\begin{gathered} C_s \E \min_{\hat{Y}\in \CC_k'} \min_{\hat{Z}\in \CC_k''} \left(\sum_{i=1}^k {\left\| Z^{(i)} - \hat{Z}^{(i)}\right\|} + \sum_{i=1}^k {\left\|Z^{(i)}\right\|} \right)^s k^{s} {\left\|Y-\hat{Y}\right\|}_\infty^s \\ \leq C_s^2 \left(k^s \E \min_{\hat{Z}\in \CC_k''} \rho^{k}\left( Z, \hat{Z}\right)^s + k^{s+1} \E {\left\|Z^{(1)}\right\|}^s \right) k^s \E \min_{\hat{Y}\in \CC_k'}{\left\|Y-\hat{Y}\right\|}_\infty^s \\ \leq C_s^2(2^s+ \E{\left\|Z^{(1)}\right\|}^s) k^{2s+1} \E \min_{\hat{Y}\in \CC_k'}{\left\|Y-\hat{Y}\right\|}_\infty^s, \end{gathered}$$ where the last step comes from (\[eqn:gets4\]). This leads to an additional factor $C k^{s+1}$ in (\[eqn:gets1\]) which has no influence on the order. Note furthermore that this argument needs that the jump positions and the increments are independent (in order to separate the expectations) and that the increments are identically distributed (as $Z^{(1)}$). It is not needed that the increments are independent among each other. Analogously, for the proof of the entropy coding error, (\[eqn:crucialchange2\]) is modified, which leads to an additional factor of $C k^{s+1}$ in (\[eqn:lsmodif\]), which leaves the resulting order unchanged, but which *does* change the constant. Lower bound for the quantization error {#sec:lb} ====================================== In this section, we prove the lower bounds for the quantization error. Essentially we employ a small ball argument, i.e. we construct an event of not too small probability that still leaves sufficient uncertainty for the error to be large. First we prove Theorem \[thm:onoff\]. Let us fix $k>0$ and $\delta>0$ (to be chosen later) and define intervals $I_j:=\left[\frac{j-1}{k}+\frac{1}{4k},\frac{j}{k}-\frac{1}{4k}\right]$, $j=1,\ldots, k$. Note that $\lambda_1(I_j)=1/(2k)$. Let $A$ be the event that $X$ has exactly $k$ jumps at $Y_1, \ldots, Y_k$, such that $Y_j \in I_j$, for all $j=1, \ldots, k$, and that the moduli of the increments are all greater than ${\varepsilon}_0$. Since the $Y_i$ and $Z_i$ are independent (by [condition (\*)]{}) and the $Y_i$ are distributed according to a Poisson point process, we have $$\begin{aligned} {\P\left(A\right)} &=&\prod_{j=1}^k {\P\left(\text{exactly one jump in $I_j$, $Z_j>{\varepsilon}_0$}\right)} \cdot {\P\left(\text{no jump in $\left[(j-1)/k,j/k\right]\setminus I_j$}\right)} \\ &\geq& \prod_{j=1}^k \left(\frac{\lambda}{2k} e^{-\lambda/(2k)} \delta_0 \cdot e^{-\lambda/(2k)} \right)= \left(\frac{\delta_0 \lambda}{2k}\right)^k e^{-\lambda}.\end{aligned}$$ [*Step 1:*]{} Let $X_A$ be a random variable with the distribution of $X$ conditioned upon the event $A$. Then $$\begin{aligned} D^{(q)}( r ~|~ X_A,\rho_\DD,s)^s \notag &=& \inf_{\log (\#\CC)\leq r} \E_{X_A} \min_{f\in\CC} \rho_\DD(X_A,f)^s \notag \\ &\geq& \inf_{\log (\#\CC)\leq r} \delta^s {\P\left( \forall f\in\CC \,:\, \rho_\DD(X_A,f) \geq \delta \right)} \notag \\ &=& \inf_{\log (\#\CC)\leq r} \delta^s \left(1 - {\P\left( \exists f\in\CC \,:\, \rho_\DD(X_A,f) < \delta \right)}\right) \notag \\ &\geq& \inf_{\log( \#\CC)\leq r} \delta^s \left(1 - (\#\CC) \sup_f {\P\left( \rho_\DD(X_A,f) < \delta \right)}\right) \notag \\ &\geq& \delta^s \left(1 - e^r \sup_{f} {\P\left( \rho_\DD(X_A,f) < \delta \right)}\right),\label{smallballe}\end{aligned}$$ where the supremum is taken over all functions $f$ in $\DD([0,1[,E)$. For such $f$, we have $$\begin{gathered} {\P\left( \rho_\DD(X_A,f) < \delta \right)} = {\P\left( \int_0^1 \rho( X_A(t),f(t)) \, \d t < \delta \right)} \\ \leq {\P\left( \bigcap_{j=1}^k \left\lbrace\int_{I_j} \rho( X_A(t),f(t)) \, \d t < \delta \right\rbrace\right)} = \E \,{\P\left( \left. \bigcap_{j=1}^k \left\lbrace\int_{I_j} \rho( X_A(t),f(t)) \, \d t < \delta \right\rbrace \right| Z\right)} , \label{eqn:lacr1} \end{gathered}$$ where $Z=(X_A(0), X_A(Y_1), \ldots, X_A(Y_k))$ is the vector with the jump destinations. By [condition (\*)]{}, we have that, conditioned upon $Z$, the events $$\left( \left\lbrace\int_{I_j} \rho( X_A(t),f(t)) \, \d t < \delta \right\rbrace \right)_{j=1}^k$$ are independent, since each of them only depends on the jump position in the respective interval. This together with (\[eqn:lacr1\]) shows $$\label{eqn:lblaste} \sup_{f} {\P\left( \rho_\DD(X_A,f) < \delta \right)} \leq \sup_{f} \E \prod_{j=1}^k{\P\left( \left. \int_{I_j} \rho( X_A(t),f(t)) \, \d t < \delta \right| Z\right)} .$$ [*Step 2:*]{} Now we estimate each term in the product separately. Fix $j\in\{ 1,\ldots, k\}$. Define $l_j:=\frac{j-1}{k}+\frac{1}{4k}$, i.e. the left end point of the interval $I_j$. Furthermore, we define $$B_j:=\left\lbrace t\in I_j : \rho(X_A(l_j),f(t)) < {\varepsilon}_0/2\right\rbrace\qquad\text{and}\qquad C_j:= \left\lbrace t\in I_j : X_A(t)=X_A(l_j)\right\rbrace.$$ Then we show that $$\int_{I_j} \rho(X_A(t),f(t))\, \d t < \delta \qquad \Rightarrow \qquad \lambda_1( B_j \Delta C_j ) < \frac{2\delta}{{\varepsilon}_0}, \label{eqn:mimic01}$$ where $B_j \Delta C_j := (B_j^c \cap C_j) \cup (B_j \cap C_j^c)$. Indeed, assume that we had $\lambda_1( B_j \Delta C_j ) \geq 2\delta/{\varepsilon}_0$. Then $$\begin{gathered} \int_{I_j} \rho(X_A(t),f(t))\, \d t \\ \geq \int_{B_j^c\cap C_j} \rho(X_A(t),f(t))\, \d t + \int_{B_j\cap C_j^c} \rho(X_A(t),f(t))\, \d t \geq \frac{{\varepsilon}_0}{2}\, \lambda_1( B_j \Delta C_j ) \geq \delta,\end{gathered}$$ where we used the triangle inequality in the last but one step. This shows (\[eqn:mimic01\]); and we thus have $${\P\left(\left. \int_{I_j} \rho( X_A(t),f(t)) \, \d t < \delta\right|Z\right)} \leq {\P\left( \left. | \lambda_1(B_j) - \lambda_1(C_j) | < \frac{2\delta}{{\varepsilon}_0} \right| Z\right)} .$$ Note that, conditioned upon $Z$, $\lambda_1(B_j)$ is a deterministic value (depending on $X_A(l_j)$ and $f$), whereas $\lambda_1(C_j)$ is a random variable that is uniformly distributed in $[0,1/(2k)]$, since the point in $I_j$ where the jump of $X_A$ occurs is uniformly distributed in $I_j$. Therefore, $${\P\left( \left. | \lambda_1(B_j) - \lambda_1(C_j) | < \frac{2\delta}{{\varepsilon}_0} \right| Z\right)} \leq \frac{8 \delta k}{{\varepsilon}_0}.$$ [*Step 3:*]{} This shows, continuing (\[eqn:lblaste\]), that $\sup_{f} {\P\left( \rho_\DD(X_A,f) < \delta \right)} \leq (8 k \delta / {\varepsilon}_0)^k$. Substituting this estimate back into (\[smallballe\]), we obtain $$D^{(q)}( r ~|~ X_A,\rho_\DD,s)^s \geq \delta^s \left(1 - e^r (8 \delta k / {\varepsilon}_0)^k\right).$$ Therefore, $$D^{(q)}( r ~|~ X,\rho_\DD,s)^s \geq {\P\left(A\right)} \cdot D^{(q)}( r ~|~ X_A,\rho_\DD,s)^s \geq \left(\frac{\delta_0 \lambda}{2 k}\right)^k e^{-\lambda} \delta^s \left(1 - e^r \left(\frac{8 k\delta}{{\varepsilon}_0}\right)^k\right).$$ Now we can optimize $k\geq 1$ and $\delta>0$ to obtain the largest possible lower bound. We set $$\delta:=\frac{{\varepsilon}_0}{8k}\, \left( \frac{1}{2}\, e^{-r} \right)^{1/k}.$$ Then the last estimate becomes $$D^{(q)}( r ~|~ X,\rho_\DD,s) \geq \left(\frac{\delta_0 \lambda}{2k}\right)^{k/s} e^{-\lambda/s} \delta \, 2^{-1/s}.$$ We set $$k := \lfloor \sqrt{ 2 s r / \log r} \rfloor \sim \sqrt{ 2 s r / \log r}.$$ Taking logarithms of the last estimate shows that $$-\log D^{(q)}( r ~|~ X,\rho_\DD,s) \lesssim \frac{k}{s} \log k + r/k \sim \sqrt{ \frac{2}{s}\, r \log r},$$ as asserted. The proof of Theorem \[thm:lowerac\] contains the same idea as the one of Theorem \[thm:onoff\] and carries over almost literally. Therefore, we only indicate the necessary changes. By assumption, $Z^{(1)}$ has an absolutely continuous component. Let $S\subseteq \R^d$ be a measurable set with $\lambda_d(S)>0$ on which $Z^{(1)}$ has a positive bounded density w.r.t. the Lebesgue measure and such that $0\notin S$. Define ${\varepsilon}_0:={\operatorname*{dist}}(S,0)/2>0$. This time, $A$ is defined as follows: let $A$ be the event that $X$ has exactly $k$ jumps at $Y_1, \ldots, Y_k$, such that $Y_j \in I_j$, for all $j=1, \ldots, k$, and that the corresponding increments (i.e. $Z^{(j)}=X(Y_j)-X(Y_j-)$) are of a height in $S$. Due to the Poissonian nature of the point process and since increments and positions are independent, we have $$\begin{aligned} {\P\left(A\right)}&=&\prod_{j=1}^k {\P\left(\text{exactly one jump in $I_j$}\right)} \cdot {\P\left(X(Y_j)-X(Y_j-)\in S\right)}\cdot \\ &&\qquad\qquad \cdot {\P\left(\text{no jump in $\left[(j-1)/k,j/k\right]\setminus I_j$}\right)} \\ &=& \prod_{j=1}^k \left(\frac{\lambda}{2k} e^{-\lambda/(2k)}\cdot q_S \cdot e^{-\lambda/(2k)} \right)= \left(\frac{\lambda q_S}{2k}\right)^k e^{-\lambda},\end{aligned}$$ where $q_S:={\P\left(Z^{(1)}\in S\right)}>0$. Regarding (\[smallballe\]), the proof is analogous to that of Theorem \[thm:onoff\]. We set $Z=(X_A(Y_1), \ldots, X_A(Y_k))$ for the vector with the jump destinations. In (\[eqn:lacr1\]) and (\[eqn:lblaste\]) we estimate a bit more carefully and obtain: $$\begin{gathered} {\P\left( \rho_\DD(X_A,f) < \delta \right)} \\ \leq \E \prod_{j=1}^k {\P\left( \left. \int_{I_j} \rho(X_A(t),f(t)) \, \d t < \delta, \int_{(4j-1)/(4k)}^{j/k} \rho(X_A(t),f(t)) \, \d t < \delta \right|Z\right)}.\end{gathered}$$ As in the proof of Theorem \[thm:onoff\], the sets $B_j$ and $C_j$ are introduced and (\[eqn:mimic01\]) is established. Let $r_j':=j/k$ and $r_j:=r_j'-1/(4k)$. Because of (\[eqn:mimic01\]) and since $X_A(t)=X_A(r_j')=X_A(r_j)$ on $[r_j,r_j']$, the last expression is less than $$\E \prod_{j=1}^k {\P\left( \left.\lambda_d(B_j\Delta C_j) < \delta, \int_{r_j}^{r_j'} \rho(X_A(r_j),f(t)) \, \d t < \delta\right| Z\right)}.$$ Note that, conditioned upon $Z$, the events $\lambda_d(B_j\Delta C_j) < \delta$ and $\int_{r_j}^{r_j'} \rho(X_A(r_j),f(t)) \, \d t < \delta$ are independent, since the second event only depends on $Z$, i.e. it is deterministic. Thus the last expression equals $$\E \prod_{j=1}^k {\P\left( \left. \lambda_d(B\Delta C) < \delta \right| Z\right)} \, {\P\left(\left. \int_{r_j}^{r_j'} \rho(X_A(r_j),f(t)) \, \d t < \delta\right|Z\right)}.$$ The first term can be estimated as in the proof of Theorem \[thm:onoff\] by $8 \delta k/{\varepsilon}_0$, which allows to estimate the last expression by $$\left(\frac{8 \delta k}{{\varepsilon}_0}\right)^k\, \E \prod_{j=1}^k {\P\left(\left. \int_{r_j}^{r_j'} \rho(X_A(r_j),f(t)) \, \d t < \delta\right|Z\right)}.$$ In order to treat the second term, note that it equals $$\begin{gathered} {\P\left(\int_{r_j}^{r_j'} \rho(X_A(r_j),f(t)) \, \d t < \delta,j=1,\ldots, k\right)} \\ = \E {\P\left(\left.\int_{r_j}^{r_j'} \rho(X_A(r_j),f(t)) \, \d t < \delta,j=1,\ldots, k\right| Z^{(1)}, \ldots, Z^{(k-1)}\right)}. \label{eqn:verbr}\end{gathered}$$ Note that the last condition (for $j=k$) is the only non-deterministic condition in the probability. It depends on $Z^{(k)}$, which is an $\R^d$-valued random variable distributed as $Z^{(1)}$. By the definition of the event $A$, $Z^{(k)}$ attains values in $S$. Thus, $$\begin{gathered} {\P\left(\left.\int_{r_k}^{r_k'} \rho(X_A(r_k),f(t)) \, \d t < \delta\right| Z^{(1)}, \ldots, Z^{(k-1)}\right)} \\ = {\P\left(\left.\int_{r_k}^{r_k'} {\left\| \sum_{j=1}^k Z^{(j)} - f(t)\right\|}_\infty \, \d t < \delta\right| Z^{(1)}, \ldots, Z^{(k-1)}\right)} \\ \leq{\P\left(\left.{\left\| \frac{Z^{(k)}}{4k} + \int_{r_k}^{r_k'} \sum_{j=1}^{k-1} Z^{(j)} - f(t) \, \d t \right\|}_\infty < \delta\right| Z^{(1)}, \ldots, Z^{(k-1)}\right)},\end{gathered}$$ where the integral is to be understood componentwise. Note that $\int_{r_k}^{r_k'} \sum_{j=1}^{k-1} Z^{(j)} - f(t) \, \d t$ is a deterministic value in $\R^d$, conditioned upon $(Z^{(1)}, \ldots, Z^{(k-1)})$. Thus, the last term is bounded from above by ${\varepsilon}_0' (8 k \delta)^d$, where ${\varepsilon}_0'$ is the supremum of the density of $Z^{(k)}{\stackrel{d}{=}}Z^{(1)}$ in $S$. In the same way, successively the other terms can be reduced; and the expression in (\[eqn:verbr\]) can be estimated by ${\varepsilon}_0'^k (8 k \delta)^{d k}$. Therefore, $$\sup_{f} {\P\left( \rho_\DD(X_A,f) < \delta \right)} \leq ( k \delta {\varepsilon}_0'')^{k(1+d)},$$ where ${\varepsilon}_0''=8\min(1/{\varepsilon}_0,{\varepsilon}_0')$. Continuing as in Step 3 of the proof of Theorem \[thm:onoff\] shows $$D^{(q)}( r ~|~ X,\rho_\DD,s)^s \geq {\P\left(A\right)} \cdot D^{(q)}( r ~|~ X_A,\rho_\DD,s)^s \geq \left(\frac{\lambda q_S}{2k}\right)^k e^{-\lambda} \delta^s \left(1 - e^r \left(k\delta {\varepsilon}_0''\right)^{k(1+d)}\right).$$ This time we set $$\delta:=\frac{1}{{\varepsilon}_0'' k}\, \left( \frac{1}{2}\, e^{-r} \right)^{1/(k(1+d))}.$$ Then again $$D^{(q)}( r ~|~ X,\rho_\DD,s) \geq \left(\frac{\lambda q_S}{2k}\right)^{k/s} e^{-\lambda/s} \delta \, 2^{-1/s},$$ where this time we set $$k := \left\lfloor \sqrt{ \frac{2 s}{1+d}\, \frac{r}{\log r}} \right\rfloor \sim \sqrt{ \frac{2 s}{1+d}\, \frac{r}{\log r}}.$$ This eventually leads to $$-\log D^{(q)}( r ~|~ X,\rho_\DD,s) \lesssim \frac{k}{s} \log k + r/(k(1+d)) \sim \sqrt{ \frac{2}{s(1+d)}\, r \log r},$$ as asserted. Lower bound for the entropy coding error {#sec:ece} ======================================== In this section we prove a corresponding lower bound for the entropy coding error (in fact, for the distortion rate function) for a jump process where the underlying point process is Poissonian. We use the notation from [@Iha93], in particular, for the distortion rate function $$D(r~|~X,\rho_D,s):=\inf\left\lbrace \left( \E \rho_\DD(X,\hat X)^s \right)^{1/s} : I(X;\hat X)\leq r\right\rbrace,$$ and the notion of mutual information: $$I(X;\hat X)= \begin{cases} \int \log \frac{\d\P_{X,\hat X}}{\d\P_X\otimes \P_{\hat X}} \,\d\P_{X,\hat X} & \text{ if } \P_{X,\hat X}\ll \P_X\otimes \P_{\hat X}\\ \infty & \text{ otherwise.} \end{cases}$$ We recall that $D(r~|~X,\rho,s)\leq D^{(e)}(r~|~X,\rho,s)\leq D^{(q)}(r~|~X,\rho,s)$ for any random variable, all moments and any distortion measure. Therefore, a lower bound for $D$ immediately translates into a lower bound for $D^{(e)}$. Let us state the assumptions of the main result of this section. We shall require that $X$ is a jump process (on the index set $[0,1[$) whose jumps form a Poisson process of intensity $\lambda>0$. Furthermore, we assume that $\rho$ defines a metric on $E$ and that the moduli of the jumps of $X$ are a.s. bounded from below by a constant ${\varepsilon}_0>0$. As before, we denote by $(Y_i)$ the jump times of the process $X$ and by $N_X$ the random number of jumps of $X$; we set $Y_0=0$. Moreover, we assume that conditioned upon $N_X=k$ the random vector $(A_i):=(X(Y_i))_{i=0}^k$ and the jump times $(Y_i)$ are independent ([condition (\*)]{}). In the rest of this section, we prove the following stronger version of Theorem \[thm:steffenmod\]. \[th0930-1\] Under the above assumptions one has $$D( r ~|~ X,\rho_\DD,1) \geq {\varepsilon}_0 C \min(1,\lambda) \, e^{-r/\lambda},$$ where $C>0$ is an absolute constant. Let us shortly describe the idea of the proof. We relate the coding complexity of the jump process to that of the random jump times. Controlling the complexity of the jump times by using Shannon’s lower bound then leads to a lower bound in terms of a variational problem. The proof is based on several lemmas and a particular random partition of $[0,1[$. We denote by ${{\mathcal{D}}_m}$ the dyadic subintervals of $[0,1[$ of the $m$-th level, that is $${\mathcal{D}}_m =\{[(j-1)2^{-m}, j2^{-m}):j=1,\dots,2^m\}$$ We construct for any collection $t_1,\dots,t_k$ of distinct points in $[0,1[$ a finite binary tree as follows. Let $\mu=\sum_{i=1}^k \delta_{t_k}$ with $\delta_z$ denoting the Dirac mass in $z$. The root of the tree will be associated with the interval $[0,1[$ and it will be marked by the number $\mu([0,1[)=k$. If $k\in\{0,1\}$, then the construction ends and the root is also a leaf of the tree. If $k\geq 2$, the root of the tree is attached two children namely the two dyadic intervals of ${\mathcal{D}}_1$ that are contained in $[0,1[$: $[0,1/2[$ and $[1/2,1[$. Again we mark each of the nodes with their corresponding masses. Each node that has mass $0$ or $1$ becomes a leaf of the tree, and for each node with mass greater than $1$ we attach the two dyadic intervals of the next level that are contained in the interval and we continue in analogy to above. By the construction, each leaf contains either one or no point. We shall denote by $\pi_k(t_1,\dots,t_k):=(I_1,\dots, I_k)$ the $k$-intervals associated to the leaves with positive mass. In order to make the definition unique we arrange the intervals in their natural order. \[le0823-1\] Let $k\geq 1$ and $(I_1,\dots,I_k)\in \mathrm{im}(\pi_k)$. Conditioned upon the event $\{N_X=k, \pi_k(Y_1,\dots,Y_k)=(I_1,\dots,I_k)\}$ we have that $(Y_1,\dots,Y_k) {\stackrel{d}{=}}(U_1,\dots,U_k)$, where $U_i$ are independent random variables that are uniformly distributed on $I_i$, respectively. First note that for any collection of distinct points $t_1,\dots,t_k\in [0,1[$ such that $\sum_{j=1}^k {1\hspace{-0.098cm}\mathrm{l}}_{I_i}(t_j)=1$ for all $i=1,\dots,k$ one retrieves $\pi_k(t_1,\dots,t_k)=(I_1,\dots,I_k)$. On the other hand, any collection of points which yields $\sum_{j=1}^k {1\hspace{-0.098cm}\mathrm{l}}_{I_i}({t_j})\not =1$ for one $i$, induces a different tree and $\pi_k(t_1,\dots,t_k)\not=(I_1,\dots,I_k)$. Therefore, the following two events coincide $$\{N_X=k, \pi_k(Y_1,\dots,Y_k)=(I_1,\dots,I_k)\}=\Bigl\{N_X=k, \sum_{j=1}^k{1\hspace{-0.098cm}\mathrm{l}}_{I_i}({Y_j})=1 \text{ for } i=1,\dots,k\Bigr\}.$$ Recall that the times $(Y_i)$ form a Poisson process on $[0,1[$ so that conditioned on $\{N_X=k, \pi(Y_1,\dots,Y_k)=(I_1,\dots,I_k)\}$ one has $(Y_1,\dots,Y_k){\stackrel{d}{=}}(U_{1},\dots,U_{k})$, where $U_i$ are independent random variables uniformly distributed on $I_i$. \[le1002-1\] Fix $k\geq 1$, $(I_1,\dots,I_k)\in \mathrm{im}(\pi_k)$ and distinct points $a_0,\dots,a_k\in E$ with $|a_{i}-a_{i-1}|\geq {\varepsilon}_0$ for $i=1,\dots,k$. Moreover, let $\mu$ denote the distribution of a process in $\DD([0,1[,E)$ that has jump positions at $k$ uniformly distributed times in the intervals $I_1,\dots,I_k$ and that attains the values $a_0,\dots,a_k$ in the given order. Then $$D(r~|~\mu,\rho_\DD,1)\geq {\varepsilon}_0 \frac{k}{2e} \Bigl(\prod_{i=1}^k |I_i|\Bigr)^{1/k} \, e^{-r/k}.$$ \[lem:steffen1\] With slight abuse of notation we shall denote by $X=(X(t))_{t\in[0,1[}$ a $\mu$-distributed process and we let $Y_1,\dots,Y_k$ denote the ordered $k$ jump positions of $X$. Due to Lemma \[le0823-1\] the times $Y_1,\dots,Y_k$ are independent and each $Y_i$ is uniformly distributed on $I_i$. Now let $\hat X=(\hat X(t))_{t\in[0,1[}$ denote a $\DD([0,1[,E)$-valued reconstruction with $I(X;\hat X)\leq r$. We define $X^i_t = a_{i-1}$ for $t<Y_i$ and $X^i_t=a_i$ for $t\geq Y_i$. Also we set $\hat X^i_t=\hat X(t)$ for $t\in I_i$ and $\hat X^i_t=X^i_t$ otherwise. Then clearly $$\rho_\DD (X,\hat X)\geq \sum_{i=1}^k \rho_\DD (X^i,\hat X^i).$$ Next, we will provide a lower bound for the right hand side in the latter inequality. For each fixed $i=1,\dots,k$ we define $\nu_i$ to be the probability kernel of the regular conditional probability $\P(Y_i\in\cdot |\hat X=\cdot)$. Next we choose $\hat Y_i=\hat Y_i(\hat X)$ to be the first time $t\in[0,1[$ for which the probability $\nu_i(\hat X, [0,t])$ is greater or equal to the threshold $1/2$. We observe that for $t\in [0,1[$ $$\begin{gathered} \E[ \rho( X^i_t,\hat X^i_t) |\hat X] \geq {\P\left(X^i_t=a_{i-1}|\hat X\right)}\wedge {\P\left(X^i_t=a_{i}|\hat X\right)}\left[ \rho( a_{i-1},\hat X^i_t)+\rho( a_{i},\hat X^i_t)\right]\\ \geq {\P\left( X^i_t=a_{i-1}|\hat X\right)}\wedge {\P\left( X^i_t=a_{i}|\hat X\right)} \ \rho(a_{i-1},a_i)\\ ={\varepsilon}_0\ {\P\left( Y_i<t|\hat X\right)} \wedge {\P\left( Y_i\geq t|\hat X\right)}.\end{gathered}$$ Consequently, the approximation error satisfies $$\begin{aligned} \E[\rho_\DD (X^i,\hat X^i) |\hat X] & \geq {\varepsilon}_0 \int_0^1 {\P\left( Y_i<t|\hat X\right)} \wedge {\P\left( Y_i\geq t | \hat X\right)} \,\d t\\ &={\varepsilon}_0 \left[ \int_0^{\hat Y_i} \E [{1\hspace{-0.098cm}\mathrm{l}}_{\{Y_i<t\}}|\hat X] \,\d t + \int_{\hat Y_i}^1 \E [{1\hspace{-0.098cm}\mathrm{l}}_{\{Y_i\geq t\}}|\hat X] \,\d t\right]\\ &={\varepsilon}_0 \, \E\left(|Y_i-\hat Y_i| \ |\hat X\right)\end{aligned}$$ and one gets $$\begin{aligned} \E[\rho_\DD (X,\hat X) ]\geq {\varepsilon}_0 \sum_{i=1}^k \E |Y_i-\hat Y_i|. \label{eqn:abslusssatz}\end{aligned}$$ We shall now use the Shannon lower bound to derive a lower bound for the right hand side of the latter equation. For ease of notation we write shortly $Y=(Y_1,\dots,Y_k)$ and $\hat Y=(\hat Y_1,\dots,\hat Y_k)$. We need the notation for the continuous entropy and its conditional counterpart: for $\R^k$-valued random vectors $Z$ and $\hat Z$ we denote $$h(Z):= - \int \log \frac{\d \P_Z}{\d \lambda_k}\, \d \P_Z \ \text{ and } \ h(Z|\hat Z):= - \int \log \frac{\d \P_{Z|\hat Z}}{\d \lambda_k}\, \d \P_{Z,\hat Z},$$ provided the Radon-Nikodym derivatives exist and the integrals are well-defined. Since $\hat Y$ is $\sigma(\hat X)$-measurable we have $I(Y;\hat Y)\leq I(X;\hat X)\leq r$; so that by the Shannon lower bound $$\begin{aligned} r&\geq I(Y;\hat Y)=h(Y)-h(Y|\hat Y)\\ & = h(Y)-h(Y- \hat Y|\hat Y) \geq h(Y)-h(Y- \hat Y).\end{aligned}$$ In particular, $Y-\hat Y$ is absolutely continuous and its differential entropy is well-defined. Next, we set $d:=\E ||Y-\hat Y||_{\ell_1^k}$ and estimate the term $h(Y-\hat Y)$ from above by $$\phi(d)= {\sup_{\substack{\text{$Z$, $\P_Z\ll \lambda_k$} \\ \E{\left\|Z\right\|}_{\ell_1^k}\leq d}}} h(Z).$$ Using Lemma 6.4 from [@dereichankirchnerimkeller] (which is based on ideas from [@csiszar]) one can easily show that $$\sup_{\text{$Z$, $\P_Z\ll \lambda_k$}, \sum_{i=1}^k \E |Z_i|\leq d} h(Z) = \sum_{i=1}^k \log\left( \frac{2 d e}{k}\right).$$ Consequently, $r\geq h(Y)- k \log(2ed/k)$ or, equivalently, $$d\geq \frac{k}{2e} \,e^{h(Y)/k}\, e^{-r/k}.$$ Moreover, the entropy of $Y$ satisfies $$h(Y)=\sum_{i=1}^k h(Y_i)= \sum_{i=1}^k \log \frac1{|I_i|}.$$ and we conclude that $$d\geq \frac{k}{2e} (\prod_{i=1}^k |I_i|)^{1/k} \, e^{-r/k},$$ which together with (\[eqn:abslusssatz\]) shows the assertion. A crucial quantity in the latter lower bound for the distortion rate function is the length of the intervals $I_i$. Later we will use the following estimate: \[le1002-2\] Let $t_1,\dots,t_k\in[0,1[$ denote $k$ distinct points ordered by their size and let $(I_1,\dots,I_k)=\pi_k(t_1,\dots,t_k)$. With $t_0=-\infty$ and $t_{k+1}=\infty$ we get for each $i=1,\dots,k$ that $$|I_i|\geq \frac12 (t_{i}-t_{i-1})\wedge (t_{i+1}-t_i).$$\[lem:steffen2\] By definition $I_i$ is the largest dyadic interval that only contains the point $t_i$ and the assertion follows since all half-open intervals of length $(t_{i}-t_{i-1})\wedge (t_{i+1}-t_i)$ that contain $t_i$ do not contain any of the other points. \[le1003-2\] There exists a universal constant $\alpha_1\in\R$ and a function $\alpha_2 : \N \to \R$ such that for any $k\geq 1$ and $i\in\{1,\dots,k\}$ $$\E[\log [(Y_i-Y_{i-1})\wedge (Y_{i+1}-Y_i)]|N_X=k]= \alpha_1 - \alpha_2(k).$$ Let $\tilde Y_1,\dots, \tilde Y_k$ denote the order statistics of $k$ independent $[0,1[$-uniformly distributed random variables, and let $(\bar Y_i)$ denote the random jump positions of a Poisson process of intensity $1$ on $[0,\infty[$. First let $i\in \{1,\dots,k-1\}$ $$\begin{aligned} \E[\log (Y_i-Y_{i-1})\wedge (Y_{i+1}-Y_i)|N_X=k]&= \E \log \bigl((\tilde Y_i-\tilde Y_{i-1})\wedge (\tilde Y_{i+1}-\tilde Y_i)\bigr)\\ &= \E\log\bigl( \frac{ \bar Y_i- \bar Y_{i-1}}{\bar Y_{k+1}} \wedge \frac{ \bar Y_{i+1}- \bar Y_i}{\bar Y_{k+1}}\bigr)\\ &= \E \log \bigl( ( \bar Y_1- \bar Y_{0}) \wedge (\bar Y_{2}-\bar Y_1)\bigr)- \E[\log \bar Y_{k+1}].\end{aligned}$$ For the second equality see e.g. [@breiman], Proposition 13.15. Setting $\alpha_1:=\E \log (( \bar Y_1- \bar Y_{0}) \wedge ( \bar Y_{2}-\bar Y_1))$ and $\alpha_2(k):=\E[\log \bar Y_{k+1}]$ finishes the proof in this case. The statement follows analogously for $i=k$. Furthermore, we will need asymptotic estimates for $$A:=N_X\Bigl(\prod_{i=1}^{N_X} (Y_{i}-Y_{i-1})\wedge (Y_{i+1}-Y_i)\Bigr)^{1/{N_X}}$$ and $$R_\beta:={N_X} \log_+ \beta (\prod_{i=1}^{N_X} (Y_i-Y_{i-1})\wedge (Y_{i+1}-Y_i))^{1/{N_X}}= \log_+ \prod_{i=1}^{N_X} \beta (Y_i-Y_{i-1})\wedge (Y_{i+1}-Y_i),$$ where $\beta>0$. \[le1003-1\] One has $$\E R_\beta \geq \lambda \log \beta +c$$ for the constant $c=c(\lambda)=\lambda \alpha_1- \E[N_X \alpha_2(N_X)]\in\R$, where $\alpha_1$ and $\alpha_2$ are as in the previous lemma. Moreover, $$\lim_{\beta\to\infty} \beta \, \E\left(\frac {N_X}\beta \wedge A\right) = \lambda.$$ Applying Lemma \[le1003-2\] we get $$\begin{aligned} \E R_\beta &= \E[\E \bigl[R_\beta|{N_X}]]\geq \E \sum_{i=1}^{N_X} \Bigl[\log \beta +\E\bigl[\log (Y_i-Y_{i-1})\wedge (Y_{i+1}-Y_i)|{N_X}\bigr]\Bigr]\\ &= \E {N_X} \Bigl[ \log \beta +\alpha_1- \alpha_2({N_X})\Bigr]= \lambda \log \beta +c.\end{aligned}$$ The second statement is an immediate consequence of the monotone convergence theorem: since $A>0$ a.s. one has $$\beta \, \E\Bigl[ \frac {N_X}\beta \wedge A\Bigr]= \E [{N_X} \wedge \beta A]\to \E {N_X}=\lambda.$$ We are now in the position to prove Theorem \[th0930-1\]. Let $\hat X$ be $\DD([0,1[,E)$-valued reconstruction with $I(X;\hat X)\leq r$ for some fixed $r\geq0$. Furthermore, we denote by $$\begin{gathered} G(k, (I_1,\dots, I_k),(a_0,\dots,a_k)) \\ = I(X;\hat X| {N_X}=k , \pi_k(Y)=(I_1,\dots,I_k),(A_0,\dots,A_k)=(a_0,\dots,a_k))\end{gathered}$$ the conditional mutual information of $X$ and $\hat X$ given ${N_X}$, $\pi_{N_X}(Y)$, and $(A_0,\dots, A_{N_X})$. We consider the non-negative random variable $R= G({N_X},\pi_{N_X}(Y),(A_0,\dots,A_k))$. Since $({N_X},\pi_{N_X}(X),(A_0,\dots,A_k))$ is $\sigma(X)$-measurable one has $$r\geq I(X;\hat X)\geq I(X;\hat X| {N_X}, \pi_{N_X}(Y),(A_0,\dots,A_k))= \E R.$$ Moreover, Lemma \[le1002-1\] together with Lemma \[le1002-2\] implies that $$\E \rho_\DD (X,\hat X)\geq \frac{{\varepsilon}_0}{4e} \E \Bigl[{N_X} \Bigl(\prod_{i=1}^{N_X} (Y_i-Y_{i-1})\wedge (Y_{i+1}-Y_i) \Bigr)^{1/{N_X}} \, e^{-R/{N_X}}\Bigr].$$ In order to get a lower bound for the coding error we next analyze the minimization problem $$\E \left[ {N_X} \left(\prod_{i=1}^{N_X} (Y_i-Y_{i-1})\wedge (Y_{i+1}-Y_i) \right)^{1/{N_X}} \, e^{-\bar R/{N_X}}\right]=\min !$$ where the infimum is taken over all non-negative random variables $\bar R$ satisfying $\E \bar R\leq r$. We let again $A={N_X} (\prod_{i=1}^{N_X} (Y_i-Y_{i-1})\wedge (Y_{i+1}-Y_i) )^{1/{N_X}}$. Using Lagrange multipliers one gets that for every $\beta>0$ $$R_\beta = {N_X} \log_+ \frac{\beta A}{{N_X}}={N_X} \log_+ \beta \Bigl(\prod_{i=1}^{N_X} (Y_i-Y_{i-1})\wedge (Y_{i+1}-Y_i) )^{1/{N_X}}\Bigr),$$ is a minimizer when $r=r_\beta:=\E R_\beta=\E[{N_X} \log_+ \frac{\beta A}{{N_X}}]$. Moreover, elementary computations give that the corresponding minimal value in the minimization problem is $$d_\beta:=\E \left[A \exp \left(- \log_+ \frac{\beta A}{{N_X}}\right)\right]=\E\left(\frac {N_X} \beta\wedge A\right).$$ For given $r\geq 0$ we now choose $\beta=\beta(r)= \exp((r-c)/\lambda)$ where $c$ is as in Lemma \[le1003-1\]. Then $ r=\lambda \log \beta+c\leq \E R_\beta $ and due to the variational formula above one has $$D(r~|~X,\rho_\DD,1) \geq \frac{{\varepsilon}_0}{4e} \E \left(\frac{{N_X}}{\beta(r)} \wedge A\right).$$ Thus letting $r$ tend to infinity we get $$D(r~|~X,\rho_\DD,1) \gtrsim \frac{{\varepsilon}_0}{4e}\,\frac \lambda {\beta(r)} =\frac{{\varepsilon}_0}{4e} \lambda \exp(-(r-c)/\lambda).$$ Thus, one has for all sufficiently large $r$ that $$D(r~|~X,\rho_\DD,1) \ge \frac{1}{8e} \lambda e^{-c/\lambda} {\varepsilon}_0 e^{-r/\lambda}= C_\lambda {\varepsilon}_0 e^{-r/\lambda},$$ where $$C_\lambda=\frac{\lambda}{8 e}\, e^{c/\lambda}\text{ and } c=\lambda \alpha_1 - \E [ N_X \,\alpha_2(N_X)].$$ Moreover, $\alpha_1$ and $\alpha_2$ can be expressed in terms of i.i.d. standard exponential random variables $(e_i)$ as $\alpha_1=\E \log (e_1\wedge e_2)$ and $\alpha_2(n)=\E \log \sum_{i=1}^{n+1} e_i$, cf. the proof of Lemma \[le1003-2\]. After some calculations (using Mathematica) one obtains $$\alpha_1 =\int_0^\infty (\log x) 2 e^{-2 x} \, \d x = -\gamma - \log 2,\qquad \alpha_2(n)=\frac{\Gamma'(n+1)}{\Gamma(n+1)},$$ where $\gamma=0.57721\ldots$ is the Euler–Mascheroni constant and $\Gamma$ is the Gamma function. Some more calculations show that $$c/\lambda =- \gamma - \log 2 - \log \lambda - \int_{\lambda}^\infty x^{-1} e^{-x}\, \d x - \frac{1-e^{-\lambda}}{\lambda}.$$ Closer analysis of this term shows that $$\lim_{\lambda\to \infty} C_\lambda = \frac{1}{8 e}\, e^{-\gamma},\qquad \lim_{\lambda\to 0} C_\lambda/\lambda = \frac{1}{8 e^2},$$ which altogether shows that $C_\lambda$ can be estimated from below by $D \min(1,\lambda)$ with some absolute constant $D>0$. [**Acknowledgements.**]{} The research of Frank Aurzada was supported by the DFG Research Center <span style="font-variant:small-caps;">Matheon</span> “Mathematics for key technologies” in Berlin. Christian Vormoor was supported by the DFG Graduiertenkolleg 251. [99]{} S. Ankirchner, S. Dereich, and P. Imkeller, The Shannon information of filtrations and the additional logarithmic utility of insiders, Ann. Probab. 34 (2006) 743-778. F. Aurzada and S. Dereich, The coding complexity of Lévy processes, Preprint (2007), available from: http://arxiv.org/abs/0707.3040 J. Bertoin, Lévy processes, volume 121 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, UK, 1996. N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, volume 27 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, UK, 1989. L. Breiman, Probability, volume 7 of Classics in Applied Mathematics, SIAM, Philadelphia, USA, 1993. T. M. Cover and J. A. Thomas, Elements of information theory, Wiley Series in Telecommunications, John Wiley & Sons, Inc., New York, USA, 1991. J. Creutzig, S. Dereich, Th. Müller-Gronbach, and K. Ritter, Infinite-dimensional quadrature and approximation of distributions, Preprint (2007). I. Csiszar, $I$-divergence geometry of probability distributions and minimization problems, Ann. Probab. 3 (1975) 146–158. S. Dereich, The coding complexity of diffusion processes under supremum norm distortion, to appear in: Stochastic Process. Appl., available from: http://dx.doi.org/ 10.1016/j.spa.2007.07.003, 2007. S. Dereich, The coding complexity of diffusion processes under ${L}^p[0,1]$-norm distortion, to appear in: Stochastic Process. Appl., available from: http://dx.doi.org/ 10.1016/j.spa.2007.07.002, 2007. S. Dereich, F. Fehringer, A. Matoussi and M. Scheutzow, On the link between small ball probabilities and the quantization problem for Gaussian measures on Banach spaces, J. Theoret. Probab. 16 (2003) 249–265. S. Dereich and M. Scheutzow, High resolution quantization and entropy coding for fractional Brownian motion, Electronic J. Prob. 11 (2006) 700–722. S. Graf and H. Luschgy, Foundations of [Q]{}uantization for [P]{}robability [D]{}istributions, volume 1730 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2000. S. Ihara, Information theory for continuous systems, World Scientific, Singapore, 1993. A. N. Kolmogorov, Three approaches to the quantitative definition of information, Internat. J. Comput. Math. 2 (1968) 157–168. H. Luschgy and G. Pagès, Sharp asymptotics of the functional quantization problem for [G]{}aussian processes, Ann. Probab. 32 (2004) 1574–1599. H. Luschgy and G. Pagès, Functional quantization of a class of [B]{}rownian diffusions: a constructive approach, Stochastic Process. Appl. 116 (2006) 310–336. H. Luschgy and G. Pagès, Functional quantization rate and mean pathwise regularity of processes with an application to Lévy processes, to appear in: Ann. Appl. Probab. 2007. G. Pagès and J. Printems, Functional quantization for pricing derivatives, Université de Paris VI, LPMA no. 930, Preprint, 2004. C. Vormoor, High resolution coding of point processes and the Boolean model, PhD thesis, Technische Universität Berlin, 2007.

--- abstract: 'We introduce a concept of causality in the framework of generalized pseudo-Riemannian Geometry in the sense of J.F. Colombeau and establish the inverse Cauchy-Schwarz inequality in this context. As an application, we prove a dominant energy condition for some energy tensors as put forward in Hawking and Ellis’s book The large scale structure of space-time. Our work is based on a new characterization of free elements in finite dimensional modules over the ring of generalized numbers.' address: 'University of Vienna, Faculty of Mathematics, Nordbergstrasse 15, 1090 Vienna, Austria' author: - Eberhard Mayerhofer title: On Lorentz geometry in algebras of generalized functions --- [^1] Introduction ============ The theory of distributions is an indispensable tool for investigating linear partial differential equations. As an example we mention the theorem of Malgrange-Ehrenpreis which asserts that every linear PDE with constant coefficients has a fundamental solution in $\mathcal D'$. However, there are natural limitations in its applicability to non-linear problems. Concerning the analysis of PDEs with non-constant coefficients, the desire to solve a differential equation in all of $\mathcal D'$ soon requires the definition of products of distributions. Such definitions, however, are usually restricted to specific subspaces of $\mathcal D'$ (e.g. Sobolev spaces) or fail to display certain algebraic properties of a product. More explicitly, we mention - (lack of consistency) The definitions vary from application to application, e.g., the definition $H\delta=c\delta$ may be reasonable for every complex number $c$ (cf. [@Bible], Examples 1.1.1). - (product properties) The product in general lacks nice properties, such as commutativity or associativity. Indeed, assuming we are given an associative product $\circ$ on $\mathcal D'$ and let ${\mathop{\mathrm{vp}}}(1/x)$ denote the principal value of $1/x$. Then we would have $$\delta=\delta\circ(x \circ {\mathop{\mathrm{vp}}}(1/x))=(\delta\circ x)\circ {\mathop{\mathrm{vp}}}(1/x)=0,$$ which is impossible, since $\delta\neq 0$. For a more detailed study we refer to ([@MObook]). The need for defining an unrestricted multiplication of elements of $\mathcal D'$ therefore motivates the search for non-linear extension of the space of distribution. More precisely it is desirable to have an associative, commutative algebra $({\mathcal G},+,\circ)$ such that: 1. \[eig1alg\] There exists a linear embedding $\iota:{\mathcal D}' \hookrightarrow {\mathcal G}$ such that $\iota(1)$ is the unit in ${\mathcal G}$. 2. There exist derivation operators $D_i: {\mathcal G} \to {\mathcal G}$ ($1\leq i \leq s$), which are linear and satisfy the Leibniz-rule. 3. $D_i \mid_{{\mathcal D}'} = \frac{\partial}{\partial x_i}$ ($1\le i \le s$), that is the derivation operators restricted to $\mathcal D'$ are the usual partial derivations. 4. \[propfour\] $\circ \mid_{{\mathcal C}^\infty\times {\mathcal C}^\infty}$ is the point-wise product of functions. Item (\[propfour\]) corresponds to the natural requirement that the new product should coincide with the usual point-wise product on a reasonable subspace of $\mathcal D'$. Schwartz’s celebrated impossibility result ([@Schw1]) states that such an algebra does not exist if (\[propfour\]) is weakened to the respective requirement on $C^k$ functions (the space of $k$-times differentiable functions). The construction of a differential algebra $(\mathcal G,+,\circ)$ which satisfies (\[eig1alg\])–(\[propfour\]) was achieved by J.F. Colombeau ([@Colombeau; @C]). The key idea of his construction is regularization of distributions. Generalized functions are basically described by nets of smooth functions parametrized by the smoothing parameter and satisfying a specific asymptotic growth property with respect to the latter. Now there are a number of such algebras of generalized functions. For a general construction scheme, cf. [@Bible]. A non-linear theory of generalized functions in a geometric setting has been developed by Kunzinger and Steinbauer ([@PenroseKS; @KS1], cf. also [@Bible] and section \[sec2\]). This approach allows for mathematically rigorous investigations of distributional geometries. In the context of general relativity this theory has proved valuable for formulating and solving problems, e.g. concerning weak singularities such as cosmic strings and impulsive gravitational waves (cf. [@CVW; @HS]). These are singularities which admit a locally bounded metric, but the curvature has to be calculated on the distributional level. Since the curvature tensor is a non-linear function of the metric tensor and its first two derivatives, its calculation in general involves ill-defined products of distributions, unless one deviates from the distributional framework. This paper is a result of recent research on the intersection of general relativity and the theory of generalized function algebras. Related work concerns, for instance, classifying singularities of space-times following a concept of C.J.S. Clarke: Singularities in space-times are considered essential if they disrupt the evolution of the wave-equation (generalized hyperbolicity , cf. [@Clarke; @VW]). It turned out that for a deeper understanding of singular space-times as modelled in algebras of generalized functions it is indispensable to reinterpret the notion of causality in this framework. The present article meets this requirement by contributing some algebraic foundations for ongoing research in this field. In addition, we hope that the results laid out in this paper may also be of independent interest to the field of nonlinear generalized functions. Program of the paper {#program-of-the-paper .unnumbered} -------------------- In sections \[sec2\] and \[secin\] we recall constructions in generalized pseudo-Riemannian geometry and we revisit invertibility and positivity issues in the special algebra. Sections \[sec4\], \[sec5\] and \[sec6\] form the core of the paper. Section \[sec4\] deals with symmetric generalized matrices, introducing a notion of generalized eigenvalues of the latter. By means of the positivity concept revisited in section \[secin\] we introduce in section \[sec5\] a generalized concept of causality. Furthermore, the inverse Cauchy-Schwarz inequality is proved in this context. As an application a dominant energy condition for a class of generalized Energy tensors is established. The final section \[sec6\] presents generalized point value characterizations of generalized pseudo-Riemannian metrics and of causality of generalized vector fields by means of the theory developed in the preceding two sections. The paper ends with an appendix on further algebraic properties of finite dimensional modules over the ring of generalized numbers. Preliminaries {#sec2} ============= The ring of generalized numbers and a partial order --------------------------------------------------- Throughout the symbol $\mathbb K$ denotes $\mathbb R$ resp. $\mathbb C$. In what follows we use the index set $I:=(0,1]\subset \mathbb R$. We shall construct generalized numbers based on nets of real or complex number  $(a_\varepsilon)_\varepsilon$ indexed by $\varepsilon\in I$, i.e., on elements of $\mathbb K^I$. The ring of generalized numbers over $\mathbb K$ is constructed in the following way: Given the ring of moderate nets of numbers $$\mathcal E_M:=\{(x_{\varepsilon})_{\varepsilon}\in\mathbb K^I \mid \exists\; m:\vert x_{\varepsilon}\vert=O(\varepsilon^m)\,(\varepsilon\rightarrow 0)\}$$ and, similarly, the ideal of negligible nets in $\mathcal E(\mathbb K)$ which are of the form $$\mathcal N:=\{(x_{\varepsilon})_{\varepsilon}\in\mathbb K^I\mid \forall\; m:\vert x_{\varepsilon}\vert=O(\varepsilon^m)\,(\varepsilon\rightarrow 0)\},$$ we may define the generalized numbers as the factor ring $$\widetilde{\mathbb K}:=\mathcal E_M/\mathcal N.$$ Given a moderate net $(a_\varepsilon)_\varepsilon\in\mathcal E_M$, we denote by $[(a_\varepsilon)_\varepsilon]$ its class in $\widetilde{\mathbb K}$. Next we show how a partial order $\leq$ can be introduced on $\widetilde{\mathbb R}$ (cf. [@MOHor; @PSMO]). For $a\,,b\in\widetilde{\mathbb R}$, we say $a\leq b$ if and only if there exist representatives $(a_\varepsilon)_\varepsilon$, $(b_\varepsilon)_\varepsilon$ of $a,b$ such that for each $\varepsilon>0$ we have $a_\varepsilon\leq b_\varepsilon$, in the usual order on the real line. This is equivalent to saying that for arbitrary representatives $(\bar a_\varepsilon)_\varepsilon$, $(\bar b_\varepsilon)_\varepsilon$ there is a negligible number $(n_\varepsilon)_\varepsilon$ such that $$(\forall \varepsilon>0)(\bar a_\varepsilon\leq \bar b_\varepsilon+n_\varepsilon).$$ $(\widetilde{\mathbb R},\, \leq)$ is a partially ordered ring (cf. [@Bible], Proposition 1.2.36), however $\leq$ is not a total order on $\widetilde{\mathbb R}$. As an example for a pair of numbers which are not comparable with respect to this order, we define $c,\,d\in\widetilde{\mathbb R}$ on the level of representatives by $$c_\varepsilon:=\begin{cases} 1, \textit{ if } \varepsilon =1/n\; (n\in\mathbb N)\\ 0, \textit{ otherwise}\end{cases},\;\;\; d_\varepsilon:=1-c_\varepsilon,\; (\varepsilon\in I).$$ Another difference to the situation on the real numbers is the following. Suppose we are given a number $a\in\widetilde{\mathbb R}$, with a representatives $(a_\varepsilon)_\varepsilon$ satisfying $$\label{posdef} \forall \varepsilon>0, a_\varepsilon>0.$$ This does not imply that $a$ is invertible: Note that even $0$ admits positive representatives, for instance $n_\varepsilon:=\exp(-1/\varepsilon)$. Indeed, $(n_\varepsilon)_\varepsilon$ tends to zero faster than any power of $\varepsilon$, for $\varepsilon\rightarrow 0$, hence is a representative of $0$. The above example motivates us to introduce, apart from $\geq 0$, a further order. We will call an element $a\in\widetilde{\mathbb R}$ strictly positive if and only if $a$ admits a representative $(a_{\varepsilon})_{\varepsilon}$ such that $$\label{strposdef} (\exists\, m\geq 0)(\exists\, \varepsilon_0)(\forall\,\varepsilon<\varepsilon_0,\, a_{\varepsilon}\geq \varepsilon^m)$$ In this case we shall write $a>0$. Contrary to the above situation (\[posdef\]) where $a\geq 0$, strict positivity implies invertibility. The main reason for this difference is that property (\[strposdef\]) is stable under a change of representatives, whereas (\[posdef\]) is not. For more information on positivity as well as on invertibility we refer to section \[secin\]. Let $A\subset I$, then the characteristic function $\chi_A\in\widetilde{\mathbb R}$ is given by the class of $(\chi_{\varepsilon})_{\varepsilon}$, where $$\chi_{\varepsilon}:=\begin{cases} 1,\qquad\mbox{if}\qquad \varepsilon\in A\\ 0, \qquad\mbox{otherwise}\end{cases}.$$ $\widetilde{\mathbb R}^n$ shall be considered as an $\widetilde{\mathbb R}$–module of dimension $n\geq 1$. Clearly the latter can also be constructed by a quotient of moderate nets of vectors by negligible nets of vectors. The special Colombeau algebra on manifolds ------------------------------------------ This section is devoted to introducing the special algebra on manifolds in a coordinate independent way as in [@K]. A translation into coordinate expressions of the respective objects is given in the end of this section. The material presented until the end of section \[sec2\] stems from the original sources [@K; @KS]. For a comprehensive presentation we refer to the–meanwhile standard reference on generalized function algebras – [@Bible]. Moreover, for further works in geometry based on Colombeau’s ideas we refer to ([@GlobTh; @KKo1; @KU2; @3MikesV; @KS; @KSV; @GenConKSV]). In this paper, $X$ shall denote a paracompact, smooth Hausdorff manifold of dimension $n$ and by $\mathcal P(X)$ we denote the space of linear differential operators on $X$. $K\subset\subset X$ denotes a set $K$ compactly contained in $X$. The special algebra of generalized functions on $X$ is constructed as the quotient $\mathcal G(X):=\mathcal E_M(X)/\mathcal N(X)$, where the ring of moderate (resp. negligible) nets of smooth functions is given by $$\begin{aligned} \nonumber \mathcal E_M(X):=\{(u_{\varepsilon})_{\varepsilon}\in (C^{\infty}(X))^I\mid\forall\;K\subset\subset X\;\forall\;P\in\mathcal P(X)\;\exists\;N\in\mathbb N:\\\sup_{x\in K}\vert Pu_{\varepsilon}\vert=O(\varepsilon^{-N})\,(\varepsilon\rightarrow 0)\}\end{aligned}$$ resp.$$\begin{aligned} \nonumber \mathcal N(X):=\{(u_{\varepsilon})_{\varepsilon}\in (C^{\infty}(X))^I\mid\forall\;K\subset\subset X\;\forall\;P\in\mathcal P(X)\;\forall\;m\in\mathbb N:\\\sup_{x\in K}\vert Pu_{\varepsilon}\vert=O(\varepsilon^m)\,(\varepsilon\rightarrow 0)\}.\end{aligned}$$ Given a moderate net $(u_\varepsilon)_\varepsilon\in \mathcal E_M(X)$ we dennote by $[(u_\varepsilon)_\varepsilon]$ its class in $\mathcal G(X)$. The $C^{\infty}$-sections of a vector bundle $(E,X,\pi)$ with base space $X$ we denote by $\Gamma(X, E)$. Moreover, let $\mathcal P(X,E)$ be the space of linear partial differential operators acting on $\Gamma(X,E)$. The $\mathcal G(X)$-module of generalized sections $\Gamma_{\mathcal G}(X,E)$ of a vector bundle $(E,X,\pi)$ on $X$ is defined similarly as (the algebra of generalized functions on $X$) above, in that we use asymptotic estimates with respect to the norm induced on the respective fibers by some arbitrary Riemannian metric. That is, we define the quotient $$\Gamma_{\mathcal G}(X,E):=\Gamma_{\mathcal E_M}(X,E)/\Gamma_{\mathcal N}(X,E),$$ where the module of moderate (resp. negligible) nets of sections is given by $$\begin{aligned} \nonumber \Gamma_{\mathcal E_M}(X,E):=\{(u_{\varepsilon})_{\varepsilon}\in (\Gamma(X,E))^I\mid\forall\;K\subset\subset X\;\forall\;P\in\mathcal P(X, E)\;\exists\;N\in\mathbb N:\\\sup_{x\in K}\|Pu_{\varepsilon}\|=O(\varepsilon^N)\,(\varepsilon\rightarrow 0)\}\end{aligned}$$ resp.$$\begin{aligned} \nonumber \Gamma_{\mathcal N}(X,E):=\{(u_{\varepsilon})_{\varepsilon}\in (\Gamma(X,E))^I\mid\forall\;K\subset\subset X\;\forall\;P\in\mathcal P(X, E)\;\forall\;m\in\mathbb N:\\\sup_{x\in K}\|Pu_{\varepsilon}\|=O(\varepsilon^m)\,(\varepsilon\rightarrow 0)\}.\end{aligned}$$ In this article we shall deal with generalized sections of the tensor bundle $\mathcal T^{r}_{s}(X)$ over $X$, which we denote by $$\mathcal G^{r}_{s}(X):=\Gamma_{\mathcal G}(X,\mathcal T^{r}_{s}(X)).$$ We call elements of $\mathcal G^{r}_{s}(X)$ [*generalized tensors of type $(r,s)$*]{}. We end this section by translating the global description of generalized vector bundles into coordinate expressions. Following the notation of [@KS], we denote by $(V,\Psi)$ a vector bundle chart over a chart $(V,\psi)$ of the base $X$. With $\mathbb R^{n'}$, the typical fibre, we can write: $$\Psi:\pi^{-1}(V)\rightarrow\psi(V)\times \mathbb R^{n'},$$ $$z\mapsto(\psi(p),\psi^1(z),\dots,\psi^ {n'}(z)).$$ Let now $s\in\Gamma_{\mathcal G}(X,E)$. Then the local expressions of $s$, $s^i=\Psi^i\circ s\circ \psi^{-1}$ lie in $\mathcal G(\psi(V))$. An equivalent local definition  of generalized vector bundles can be achieved by defining moderate nets $(s_\varepsilon)_\varepsilon$ of smooth sections $s_\varepsilon$ to be such for which the local expressions $s_\varepsilon^i=\Psi^i\circ s_\varepsilon\circ \psi^{-1}$ are moderate, that is $(s_\varepsilon^i)_\varepsilon\in\mathcal E_M(\psi(V))$ (the notion negligible is defined completely analogously). This follows from the fact that every linear differential operator can be localized (cf. [@Bible], p. 289). Uniqueness in $\mathcal G(X)$ {#uniqueness} ----------------------------- A function $f\in\mathcal G(X)$ can be evaluated on standard points $x\in X$. To be more precise, let $(f_\varepsilon)_\varepsilon$ be a representative of $f$. Then the mapping $$\label{eqxx} f: X\rightarrow \widetilde{\mathbb R},\qquad x\mapsto f(x):=(f_\varepsilon(x))_\varepsilon+\mathcal N$$ is well defined (cf. [@MO1]). It is customary to call $f(x)$ the point value of $f$ at $x$. Note that the above constitutes a slight abuse of notation: On the one hand, $f$ is a generalized function and on the other hand, $f$ denotes the evaluation mapping (\[eqxx\]). Generalized functions are [*not*]{} uniquely determined by evaluation on standard points ([@Eyb4; @MO1]). To illustrate this important feature of generalized function algebras we recall Example 2.1 from [@MO1]: Take some $\varphi\geq 0\in\mathcal D(\mathbb R)$ with ${\mathop{\mathrm{supp}}}\varphi\in[-1,1]$ and $\int \varphi=1$ and set $u_\varepsilon:=\varphi_\varepsilon(x-\varepsilon)$, where $\varphi_\varepsilon(y):=\frac{1}{\varepsilon}\varphi(\frac{y}{\varepsilon})$. Then $(u_\varepsilon)_\varepsilon\in\mathcal E_M(\mathbb R)$, so $u:=[(u_\varepsilon)_\varepsilon]\in\mathcal G(\mathbb R)$. One can easily see that for all $x\in\mathbb R$, $u_\varepsilon(x)=0$, whenever $\varepsilon$ is sufficiently large. Hence, $u(x)=0$ in $\widetilde{\mathbb R}$. But $u\neq 0$. However, if we allow the point $x$ to vary with $\varepsilon$ (on the level of representatives this means inserting a net $(x_\varepsilon)_\varepsilon$ into $(f_\varepsilon)_\varepsilon$ instead of standard points only as in eq. (\[eqxx\])), we can uniquely determine generalized functions by evaluation. More precisely, the following holds ([@MO1], Theorem 2.4 and [@KS], Theorem 1): Let $f\in\mathcal G(X)$. The following are equivalent: 1. $f=0$ in $\mathcal G(X)$, 2. $f(x_c)=0$ in $\widetilde{\mathbb R}$ for each $x_c\in\widetilde{X}_c$. Here $\widetilde X_c$ denotes the class of nets $(x_\varepsilon)_\varepsilon$ of compactly supported points factored by the equivalence relation $\sim$ given by $$(x_\varepsilon)_\varepsilon\sim (y_\varepsilon)_\varepsilon\Leftrightarrow \forall m\geq 0:\; d(x_\varepsilon,y_\varepsilon)=O (\varepsilon^m), \;\textit{whenever }\;(\varepsilon\rightarrow 0)$$ where $d$ is the distance function induced by an arbitrary Riemannian metric. Completely analogous to (\[eqxx\]), the evaluation of $f$ at points in $\widetilde{X}_c$ is well defined. Generalized pseudo-Riemannian metrics {#introducerepseudoriemannereetconnexione} ------------------------------------- We begin by recalling the following characterization of non-degenerateness of symmetric (generalized) tensor fields of type (0,2) on $X$ ([@KS1], Theorem 3.1) \[chartens02\] Let $g\in \mathcal G^0_2(X)$. The following are equivalent: 1. \[chartens021\] For each chart $(V_{\alpha},\psi_{\alpha})$ and each $\widetilde x\in (\psi_{\alpha}(V_{\alpha}))^{\sim}_c$ the map\ $g_{\alpha}(\widetilde x): \widetilde{\mathbb R}^n\times \widetilde{\mathbb R}^n\rightarrow \widetilde{\mathbb R}$ is symmetric and non-degenerate. 2. $g: \mathcal G^0_1(X)\times \mathcal G^0_1(X)\rightarrow \mathcal G(X)$ is symmetric and for each chart $(V_{\alpha},\psi_{\alpha})$, $\det g_\alpha$ is invertible in $\mathcal G(\psi_\alpha(V_\alpha))$. 3. \[chartens023\] For each chart $(V_{\alpha},\psi_{\alpha})$, $\det g_\alpha$ is invertible in $\mathcal G(\psi_\alpha(V_\alpha))$ and for each relatively compact open set $V\subset X$ there exists a representative $(g_{\varepsilon})_{\varepsilon}$ of $g$ and $\varepsilon_0>0$ such that $g_{\varepsilon}\mid_V$ is a smooth pseudo-Riemannian metric for all $\varepsilon<\varepsilon_0$. Furthermore, the index of $g\in \mathcal G^0_2(X)$ is introduced in the following well defined way (cf. Definition 3.2 and Proposition 3.3 in [@KS1]): \[defpseud\] Let $g\in \mathcal G^0_2(X)$ satisfy one (hence all) of the equivalent conditions in Theorem \[chartens02\]. If there exists some $j\in\mathbb N$ with the property that for each relatively compact open set $V\subset X$ there exists a representative $(g_{\varepsilon})_{\varepsilon}$ of $g$ as in Theorem \[chartens02\] (\[chartens023\]) such for each $\varepsilon<\varepsilon_0$ the index of $g_{\varepsilon}$ is equals $j$ we say $g$ has index $j$. Such symmetric 2-forms we call generalized pseudo-Riemannian metrics on $X$. The field of generalized pseudo–Riemannian geometry deals with pairs $(X, g)$, where $g$ is a pseudo–Riemannian metrics on $X$ with index $\nu$. Invertibility and strict positivity in generalized function algebras revisited {#secin} ============================================================================== This section is devoted to elaborating a new characterization of invertibility as well as of strict positivity of generalized numbers resp. functions. The first investigation on which many works in this field are based was done by M. Kunzinger and R. Steinbauer in [@KS1]; the authors of the latter work established the fact that invertible generalized numbers are precisely such for which the modulus of any representative is bounded from below by a fixed power of the smoothing parameter (cf. the proposition below). It is, however, noteworthy that component-wise invertibility on the level of representatives describes invertibility of generalized numbers entirely: \[genpointinv\] Let $\gamma\in\widetilde{\mathbb R}$. The following are equivalent: 1. \[inv1\] $\gamma$ is invertible. 2. \[inv2\] $\gamma$ is strictly nonzero, that is: for some (hence any) representative $(\gamma_{\varepsilon})_{\varepsilon}$ of $\gamma$ there exists an $m_0$ and an $\varepsilon_0\in I$ such that for each $\varepsilon<\varepsilon_0$ we have $\vert \gamma_{\varepsilon}\vert>\varepsilon^{m_0}$. 3. \[inv3\] For each representative $(\gamma_{\varepsilon})_{\varepsilon}$ of $\gamma$ there exists some $\varepsilon_0\in I$ such that for all $\varepsilon<\varepsilon_0$ we have $\gamma_{\varepsilon}\neq 0$. 4. \[inv4\] $\vert \gamma\vert$ is strictly positive. Since (\[inv1\]) $\Leftrightarrow$ (\[inv2\]) by ([@KS1], Theorem 1.2.38) and (\[inv1\]) $\Leftrightarrow $ (\[inv4\]) follows from the definition of strict positivity, we only need to establish the equivalence (\[inv2\]) $\Leftrightarrow$ (\[inv3\]) in order to complete proof. As the reader can easily verify, the definition of strictly non-zero is independent of the representative, that is for each representative $(\gamma_{\varepsilon})_{\varepsilon}$ of $\gamma$ we have some $m_0$ and some $\varepsilon_0$ such that for all $\varepsilon<\varepsilon_0$ we have $\vert\gamma_{\varepsilon}\vert>\varepsilon^{m_0}$. By this consideration (\[inv3\]) follows from (\[inv2\]). In order to show the converse direction, we proceed by an indirect argument. Assume there exists a representative $(\gamma_{\varepsilon})_{\varepsilon}$ of $\gamma$ such that for some zero sequence $\varepsilon_k\rightarrow 0$ ($k\rightarrow \infty$) we have $\vert\gamma_{\varepsilon_k}\vert<\varepsilon_k^k$ for each $k>0$. Define a moderate net $(\hat{\gamma}_{\varepsilon})_{\varepsilon}$ in the following way: $$\hat{\gamma}_{\varepsilon}:=\begin{cases} 0\qquad\mbox{if}\qquad \varepsilon=\varepsilon_k\\ \gamma_{\varepsilon} \qquad\mbox{otherwise} \end{cases}.$$ It can then easily be seen that $(\hat{\gamma}_{\varepsilon})_{\varepsilon}-(\gamma_{\varepsilon})_{\varepsilon}\in \mathcal N(\mathbb R)$ which means that $(\hat{\gamma}_{\varepsilon})_{\varepsilon}$ is a representative of $\gamma$ as well. However the latter violates (\[inv3\]) and we are done. We can characterize the strict order relation on the ring of generalized real numbers in a similar manner: \[secinverweis\] Let $\gamma\in\widetilde{\mathbb R}$. The following are equivalent: 1. \[inv11\] $\gamma$ is strictly positive, that is: for some (hence any) representative $(\gamma_{\varepsilon})_{\varepsilon}$ of $\gamma$ there exists an $m_0$ and an $\varepsilon_0\in I$ such that for each $\varepsilon<\varepsilon_0$ we have $\gamma_{\varepsilon}>\varepsilon^{m_0}$. 2. \[inv21\] $\gamma$ is strictly nonzero and has a representative $(\gamma_{\varepsilon})_{\varepsilon}$ which is positive for each index $\varepsilon>0$. 3. \[inv31\] For each representative $(\gamma_{\varepsilon})_{\varepsilon}$ of $\gamma$ there exists some $\varepsilon_0\in I$ such that for all $\varepsilon<\varepsilon_0$ we have $\gamma_{\varepsilon}> 0$. The statement can be shown similarly to the preceding one. Next, we draw our attention to the question of invertibility and strict positivity of generalized functions. We start with the definition of the latter: A function $f\in\mathcal G(X)$ is called strictly positive in $\mathcal G(X)$, if $f$ is invertible and if for each compact subset $K\subset X$ there exists a representative $(f_{\varepsilon})_{\varepsilon}$ of $f$ which is non-negative on $K$. We shall write $f>0$. $f\in\mathcal G(X)$ is called strictly negative in $\mathcal G(X)$, if $-f>0$ on $X$. Next, we show that Propositions \[genpointinv\] and \[secinverweis\] have immediate generalizations to generalized functions on $X$ : \[downhilliseasier\] Let $u\in\mathcal G(X)$. The following are equivalent: 1. \[invf1\] $u$ is invertible (resp. strictly positive). 2. \[invf1extra\] For each compactly supported point $x_c\in \widetilde{X}_c$, $u(x_c)$ is an invertible element of $\widetilde{\mathbb R}$. 3. \[invf2\] For each representative $(u_{\varepsilon})_{\varepsilon}$ of $u$ and each compact set $K$ in $X$ there exists some $\varepsilon_0\in I$ and some $m_0$ such that for all $\varepsilon<\varepsilon_0$ we have $\inf_{x\in K} \vert u_{\varepsilon}(x)\vert>\varepsilon^{m_0}$ (resp. $\inf_{x\in K} u_{\varepsilon}(x)>\varepsilon^{m_0}$). 4. \[invf3\] For each representative $(u_{\varepsilon})_{\varepsilon}$ of $u$ and each compact set $K$ in $X$ there exists some $\varepsilon_0\in I$ such that $\forall\; x\in K\;\forall\; \varepsilon<\varepsilon_0: u_{\varepsilon}(x)\neq 0$ (resp. $u_{\varepsilon}(x)>0$). We only show that the characterization of invertibility holds, the rest of the statement is then clear. (\[invf1\])$\Leftrightarrow$(\[invf2\]) hold according to ([@KS1], Proposition 2.1). Furthermore the equivalence (\[invf1\]) $\Leftrightarrow$ (\[invf1extra\]) can be easily elaborated by modifying suitably the proofs of Theorem 2.4 in ([@MO1]) resp. of Proposition 3.4 in ([@PSMO]). It is therefore sufficient to establish the equivalence of the latter two statements. Since (\[invf2\])$\Rightarrow$(\[invf3\]) is evident, we finish the proof by showing the converse direction. Assume (\[invf2\]) does not hold, then there exists a compactly supported sequence $(x_k)_k\in X^{\mathbb N}$ such that for some representative $(u_{\varepsilon})_{\varepsilon}$ of $u$ we have $\vert u_{\varepsilon_k}(x_k)\vert<\varepsilon_k^k$ for each $k$. Similarly to the proof of Proposition \[genpointinv\] we observe that $(\hat u_{\varepsilon})_{\varepsilon}$ defined by $$\hat{u}_{\varepsilon}:=\begin{cases} u_{\varepsilon}-u_{\varepsilon}(x_k),\qquad\mbox{if}\qquad \varepsilon=\varepsilon_k\\ u_{\varepsilon}, \qquad\mbox{otherwise}\end{cases}$$ yields another representative of $u$ which, however, violates (\[invf3\]) and we are done. We will frequently employ the notion of positivity characterized here (e.g. in Definition \[causaldef1\] of causality). However, positivity in the generalized sense is a fundamental property which has proved useful in other contexts as well. We refer here to papers by Oberguggenberger et al concerning positivity and positive definiteness in generalized function algebras ([@MOHor]) and also on elliptic regularity for partial differential equations with generalized coefficients ([@PSMO]). Matrices over $\widetilde{\mathbb R}$ {#sec4} ===================================== We denote by $\widetilde{\mathbb R}^{n^2}:=\mathcal M_n(\widetilde{\mathbb R})$ the ring of $n\times n$ matrices over $\widetilde{\mathbb R}$. A matrix $A$ is called orthogonal, if $UU^t=\mathbb I$ in $\widetilde{\mathbb R}^{n^2}$ and $\det U=1$ in $\widetilde{\mathbb R}$. Clearly, there are two different ways to introduce $\widetilde{\mathbb R}^{n^2}$: Denote by $\mathcal E_M(\mathcal M_n(\mathbb R))$ the ring of moderate nets of $n\times n$ matrices over $\mathbb R$, a subring of $\mathcal M_n(\mathbb R)^I$. Similarly let $\mathcal N(\mathcal M_n(\mathbb R))$ denote the ideal of negligible nets of real $n\times n$ matrices. There is a ring isomorphism $\varphi: \widetilde{\mathbb R}^{n^2}\rightarrow \mathcal E_M(\mathcal M_n(\mathbb R))/\mathcal N(\mathcal M_n(\mathbb R))$. For the convenience of the reader we repeat Lemma 2.6 from [@KS1]: \[nondeg\] Let $A\in\widetilde{\mathbb R}^{n^2}$. The following are equivalent: 1. \[charnondeg1\] $A$ is non-degenerate, that is, $\xi \in \widetilde{\mathbb R}^n,\;\xi^t A\eta=0$ for each $\eta\in\widetilde{\mathbb R}^n$ implies $\xi=0$. 2. \[charnondeg2\] $A: \widetilde{\mathbb R}^n\rightarrow \widetilde{\mathbb R}^n$ is injective. 3. \[charnondeg3\] $A: \widetilde{\mathbb R}^n\rightarrow \widetilde{\mathbb R}^n$ is bijective. 4. \[charnondeg4\] $\det A$ is invertible in $\widetilde{\mathbb R}$. Note that the equivalence of (\[charnondeg1\])–(\[charnondeg3\]) and (\[charnondeg4\]) results from the fact that in $\widetilde{\mathbb R}$ any nonzero non-invertible element is a zero-divisor. Since we deal with symmetric matrices throughout, we start by giving a basic characterization of symmetry of generalized matrices: \[symmetry\] Let $A\in\widetilde{\mathbb R}^{n^2}$. The following are equivalent: 1. \[asymmetry\] $A$ is symmetric, that is $A=A^t$ in $\widetilde{\mathbb R}^{n^2}$. 2. \[bsymmetry\] There exists a symmetric representative $(A_{\varepsilon})_{\varepsilon}:=((a_{ij}^{\varepsilon})_{ij})_{\varepsilon}$ of $A$. Since (\[bsymmetry\]) $\Rightarrow$ (\[asymmetry\]) is clear, we only need to show (\[asymmetry\]) $\Rightarrow$ (\[bsymmetry\]). Let $((\bar a_{ij}^{\varepsilon})_{ij})_{\varepsilon}$ a representative of $A$. Symmetrizing yields the desired representative $$(a_{ij}^{\varepsilon})_{\varepsilon}:=\frac{ (\bar a_{ij}^{\varepsilon})_{\varepsilon}+(\bar a_{ji}^{\varepsilon})_{\varepsilon} } {2}$$ of $A$. This follows from the fact that for each pair $(i,j)\in\{1,\dots,n\}^2$ of indices one has $(\bar a_{ij}^{\varepsilon})_{\varepsilon}-(\bar a_{ji}^{\varepsilon})_{\varepsilon} \in \mathcal N(\mathbb R)$ due to the symmetry of $A$. Denote by $\|\,\|_F$ the Frobenius norm on $\mathcal M_n(\mathbb C)$. In order to prepare a notion of eigenvalues for symmetric matrices, we repeat a numeric result given in [@SJ] (Theorem 5. 2): \[perturbation\] Let $A\in\mathcal M_n(\mathbb C)$ be a Hermitian matrix with eigenvalues $\lambda_1\geq\dots\geq\lambda_n$. Denote by $\widetilde A$ a non-Hermitian perturbation of $A$, i. e., $E=\widetilde A- A$ is not Hermitian. We further call the eigenvalues of $\widetilde A$ (which might be complex) $\mu_k+i\nu_k\;(1\leq k\leq n)$ where $\mu_1\geq\dots\geq\mu_n$. In this notation, we have $$\sqrt{ \sum_{k=1}^n\vert(\mu_k+i\nu_k)-\lambda_k \vert^2 }\leq\sqrt 2\| E\|_F.$$ \[eigenvalues\] Let $A\in\widetilde{\mathbb R}^{n^2}$ be a symmetric matrix and let $(A_{\varepsilon})_{\varepsilon}$ be an arbitrary representative of $A$. Let for any $\varepsilon\in I$, $\theta_{k,\varepsilon}:=\mu_{k,\varepsilon}+i\nu_{k,\varepsilon}\;(1\leq k\leq n)$ be the eigenvalues of $A_{\varepsilon}$ ordered by the size of the real parts, i. e., $\mu_{1,\varepsilon}\geq\dots\geq\mu_{n,\varepsilon}$. The generalized eigenvalues $\theta_k\in\widetilde{\mathbb C}\;(1\leq k\leq n)$ of $A$ are defined as the classes $(\theta_{k,\varepsilon})_{\varepsilon}+\mathcal N(\mathbb C)$. \[schur\] Let $A\in\widetilde{\mathbb R}^{n^2}$ be a symmetric matrix. Then the eigenvalues $\lambda_k\;(1\leq k\leq n)$ of $A$ as introduced in Definition \[eigenvalues\] are well defined elements of $\widetilde{\mathbb R}$. Furthermore, there exists an orthogonal $U\in \widetilde{\mathbb R}^{n^2}$ such that $$\label{eqdecschur} U A U^t={\mathop{\mathrm{diag}}}(\lambda_1,\dots,\lambda_n).$$ We call $\lambda_i\;(1\leq i\leq n)$ the eigenvalues of $A$. $A$ is non-degenerate if and only if all generalized eigenvalues are invertible. Before we prove the lemma, we note that throughout this paper we shall omit the term generalized  (eigenvalues) and we shall call the generalized numbers constructed in the above way simply eigenvalues (of a generalized symmetric matrix). Due to Lemma \[symmetry\] we may choose a symmetric representative $(A_{\varepsilon})_{\varepsilon}=((a_{ij}^{\varepsilon})_{ij})_{\varepsilon}\in\mathcal E_M(\mathcal M_n(\mathbb R))$ of $A$ . For any $\varepsilon$, denote by $\lambda_{1,\varepsilon}\geq\dots\geq\lambda_{n,\varepsilon}$ the resp. (real) eigenvalues of $(a_{ij}^{\varepsilon})_{ij}$ ordered by size. For any $i\in\{1,\dots,n\}$, define $\lambda_i:=(\lambda_{i,\varepsilon})_\varepsilon+\mathcal N(\mathbb R)\in\widetilde{\mathbb R}$. For the well-definedness of the eigenvalues of $A$, we only need to show that for any other (not necessarily symmetric) representative of $A$, the resp. net of eigenvalues lies in the same class of $\mathcal E_M(\mathbb C)$; note that the use of complex numbers is indispensable here. Let $ (\widetilde A_{\varepsilon})_{\varepsilon}=((\widetilde a_{ij}^{\varepsilon})_{ij})_{\varepsilon}$ be another representative of $A$. Denote by $\mu_{k,\varepsilon}+i\nu_{k+\varepsilon}$ the eigenvalues of $\widetilde A_{\varepsilon}$ for any $\varepsilon\in I$ such that the real parts are ordered by size, i. e., $\mu_{1,\varepsilon}\geq\dots\geq \mu_{n,\varepsilon}$. Denote by $(E_{\varepsilon})_{\varepsilon}:=(\widetilde A_{\varepsilon})_{\varepsilon}-(A_{\varepsilon})_{\varepsilon}$. Due to Theorem \[perturbation\] we have for each $\varepsilon\in I$: $$\label{nullboundsE} \sqrt{\sum_{k=1}^n\vert(\mu_{k,\varepsilon}+i\nu_{k,\varepsilon})-\lambda_{k,\varepsilon}\vert^2 }\leq\sqrt 2\|E_{\varepsilon}\|_F.$$ Since $(E_{\varepsilon})_{\varepsilon}\in\mathcal N(\mathcal M_n(\mathbb R))$, (\[nullboundsE\]) implies for any $k\in\{1,\dots,n\}$ and any $m$, $$\vert(\mu_{k,\varepsilon}+i\nu_{k,\varepsilon})-\lambda_{k,\varepsilon}\vert=O(\varepsilon^m)\;(\varepsilon\rightarrow 0)$$ which means that the resp. eigenvalues of $(A_\varepsilon)_\varepsilon$ and of $(\widetilde A_\varepsilon)_\varepsilon$ in the above order belong to the same class in $\mathcal E_M(\mathbb C)$. In particular they yield the same elements of $\widetilde{\mathbb R}$. The preceding argument and Lemma \[symmetry\] show that without loss of generality we may construct the eigenvalues of $A$ by means of a symmetric representative $(A_{\varepsilon})_{\varepsilon}=((a_{ij}^{\varepsilon})_{ij})_{\varepsilon}\in\mathcal E_M(\mathcal M_n(\mathbb R))$. For such a choice we have for any $\varepsilon$ an orthogonal matrix $U_{\varepsilon}$ such that $$U_{\varepsilon}A_{\varepsilon}U_{\varepsilon}^t={\mathop{\mathrm{diag}}}(\lambda_{1,\varepsilon},\dots,\lambda_{n,\varepsilon}),\;\lambda_{1,\varepsilon}\geq\dots\geq\lambda_{n,\varepsilon}.$$ Declaring $U$ as the class of $(U_{\varepsilon})_{\varepsilon}\in\mathcal E_M(\mathcal M_n(\mathbb R))$ yields the proof of the second claim, since orthogonality for any $U_{\varepsilon}$ implies orthogonality of $U$ in $\mathcal M_n(\widetilde{\mathbb R})$. Finally, decomposition (\[eqdecschur\]) gives, by applying the multiplication theorem for determinants and the orthogonality of $U$, $\det A=\prod_{i=1}^n \lambda_i$. This shows in conjunction with Lemma \[nondeg\] that invertibility of all eigenvalues is a sufficient and necessary condition for the non-degenerateness of $A$ and we are done. A remark on the notion eigenvalue of a generalized symmetric matrix $A\in\widetilde{\mathbb R}^{n^2}$ is in order: Since for any eigenvalue $\lambda$ of $A$ we have $\det (A-\lambda \mathbb I)=\det (U(A-\lambda \mathbb I)U^t)=\det ((UAU^t)-\lambda \mathbb I)=0$, Lemma \[charnondeg\] implies that $A-\lambda \mathbb I: \widetilde{\mathbb R}^n\rightarrow\widetilde{\mathbb R}^n$ is not injective. However, again by the same lemma, $\det (A-\lambda \mathbb I)=0$ is not necessary for $A-\lambda \mathbb I$ to be not injective, and a $\theta\in\widetilde{\mathbb R}$ for which $A-\theta I$ is not injective need not be an eigenvalue of $A$. More explicitly, we give two examples of possible scenarios here: 1. Let $\forall\;i\in\{1,\dots,n\}: \lambda_i\neq 0$ and for some $i$ let $\lambda_i$ be a zero divisor. Then besides $A-\lambda_i\;(i=1,\dots,n)$, also $ A: \widetilde{\mathbb R}^n\rightarrow\widetilde{\mathbb R}^n$ fails to be injective. 2. Mixing representatives of $\lambda_i,\lambda_j\; (i\neq j)$ might give rise to generalized numbers $\theta\in\widetilde{\mathbb R}, \,\theta\neq \lambda_j\,\forall j\in\{1,\dots,n\}$ for which $A-\theta \mathbb I$ is not injective as well. Consider for the sake of simplicity the matrix $D:={\mathop{\mathrm{diag}}}(1,-1)\in \mathcal M_2(\mathbb R)$. A rotation $U_{\varphi}:=\left(\begin{array}{cc} \cos(\varphi)& \sin(\varphi)\\-\sin(\varphi)&\cos(\varphi) \end{array}\right)$ yields by matrix multiplication $$U_{\varphi} D U_{\varphi}^t=\left(\begin{array}{cc} \cos(2\varphi)& -\sin(2\varphi)\\-\sin(2\varphi)&-\cos(2\varphi) \end{array}\right).$$ The choice of $\varphi=\pi/2$ therefore switches the order of the entries of $D$, that is $U_{\pi/2}DU_{\pi/2}^t={\mathop{\mathrm{diag}}}(-1,1)$. Define $U,\lambda$ as the classes of $(U_{\varepsilon})_{\varepsilon}, (\lambda_{\varepsilon})_{\varepsilon}$ defined by $$U_{\varepsilon}:=\begin{cases} I:\;\varepsilon\in I\cap\mathbb Q\\ U_{\pi/2}:\;\mbox{else} \end{cases},$$ $$\lambda_{\varepsilon}:=\begin{cases} 1:\;\varepsilon\in I\cap\mathbb Q\\ -1\;\mbox{else} \end{cases},$$ further define $\mu\in\widetilde{\mathbb R} $ by $\mu+\lambda=0$. Then we have for $A:= [(D)_{\varepsilon}]$: $$UDU^t={\mathop{\mathrm{diag}}}(\lambda,\mu).$$ Therefore as shown above, $D-\lambda \mathbb I,\; D-\mu \mathbb I$ are not injective considered as maps $\widetilde{\mathbb R}^n\rightarrow\widetilde{\mathbb R}^n$. But neither $\lambda$, nor $\mu$ are eigenvalues of $D$. \[indexmatrixdef\] Let $A\in\widetilde{\mathbb R}^{n^2}$. We denote by $\nu_{+}(A)$ (resp. $\nu_{-}(A)$) the number of strictly positive (resp. strictly negative) eigenvalues, counting multiplicity. Furthermore, if $\nu_{+}(A)+\nu_{-}(A)=n$, we simply write $\nu(A):=\nu_{-}(A)$. If $A$ is symmetric and $\nu(A)=0$, we call $A$ a positive definite symmetric matrix. If $A$ is symmetric and $\nu_{+}(A)+\nu_{-}(A)=n$ and $\nu(A)=1$, we say $A$ is a symmetric $L$-matrix. The following corollary shows that for a symmetric non-degenerate matrix in $\widetilde{\mathbb R}^{n^2}$ counting $n$ strictly positive resp. negative eigenvalues is equivalent to having a (symmetric) representative for which any $\varepsilon$-component has the same number (total $n$) of positive resp. negative real eigenvalues. The proof can be obtained by using ideas of the proof of Proposition \[secinverweis\]. \[charindexmatrix\] Let $A\in\widetilde{\mathbb R}^{n^2}$ be symmetric and non-degenerate and $j\in\{1,\dots,n\}$. The following are equivalent: 1. $\nu_{+}(A)+\nu_{-}(A)=n$, $\nu(A)=j$. 2. \[charindexmatrix2\] For each symmetric representative $(A_{\varepsilon})_{\varepsilon}$ of $A$ there exists some $\varepsilon_0 \in I$ such that for any $\varepsilon<\varepsilon_0$ we have for the eigenvalues $\lambda_{1,\varepsilon}\geq\dots\geq\lambda_{n,\varepsilon}$ of $A_{\varepsilon}$: $$\lambda_{1,\varepsilon},\dots,\lambda_{n-j,\varepsilon}>0,\;\;\lambda_{n-j+1,\varepsilon},\dots,\lambda_{n,\varepsilon}<0.$$ Causality and the inverse Cauchy-Schwarz inequality {#sec5} =================================================== In a free module over a commutative ring $R\neq \{0\}$, any two bases have the same cardinality. Therefore, any free module $\mathfrak M_n$ of dimension $n\geq 1$ (i. e., with a basis having $n$ elements) is isomorphic to $R^n$ considered as module over $R$ (which is free, since it has the canonical basis). As a consequence we may confine ourselves to considering the module $\widetilde{\mathbb R}^n$ over $\widetilde{\mathbb R}$ and its submodules. We further assume that from now on $n$, the dimension of $\widetilde{\mathbb R}^n$, is greater than $1$. It is quite natural to start with an appropriate version of the Steinitz exchange lemma: \[steinitzprop\] Let $\mathcal B=\{v_1,\dots,v_n\}$ be a basis for $\widetilde{\mathbb R}^n$. Let $w=\lambda_1 v_1+\dots+\lambda_nv_n\in \widetilde{\mathbb R}^n$ such that for some $j\;(1\leq j\leq n)$, $\lambda_j$ is strictly nonzero. Then, also $\mathcal B':=\{v_1,\dots,v_{j-1},w,v_{j+1},\dots,v_n\}$ is a basis for $\widetilde{\mathbb R}^n$. Since every strictly nonzero number is invertible, one can prove the claim by using the proof of the well known one in the vector space setting. \[bilformdef\] Let $b:\widetilde{\mathbb R}^n\times\widetilde{\mathbb R}^n\rightarrow \widetilde{\mathbb R}$ be a symmetric bilinear form. Suppose there exists a number $j\in\mathbb N_0$ such that for some basis $\mathcal B:=\{e_1,\dots,e_n\}$ of $\widetilde{\mathbb R}^n$ we have $\nu((b(e_i,e_j))_{ij})=j$. Then we call $j$ the index of $b$. If $j=0$ we say that $b$ is positive definite and if $j=1$ we call $b$ a symmetric bilinear form of Lorentzian signature. Note that as in the classical setting, there is no notion of ’eigenvalues’ of a symmetric bilinear form, since a change of coordinates that is not induced by an orthogonal matrix need not conserve the eigenvalues of the original coefficient matrix. We are obliged to show that the notion above is well defined. The main argument is Sylvester’s inertia law (cf. [@GF], pp. 306): The index of a bilinear form $b$ on $\widetilde{\mathbb R}^n$ as introduced in Definition \[bilformdef\] is well defined. Let $\mathcal B$, $\mathcal B'$ be bases of $\widetilde{\mathbb R}^n$ and let $A$ be a matrix describing a linear map which maps $\mathcal B$ onto $\mathcal B'$ (this map is uniquely determined in the sense that it only depends on the order of the basis vectors of the resp. bases). Let $B$ be the coefficient matrix of the given bilinear form $b$ and let further $k:=\nu(B)$. The change of bases results in a ’generalized’ equivalence transformation of the form $$B\mapsto T:=A^tBA,$$ $T$ being the coefficient matrix of $h$ with respect to $\mathcal B'$. We only need to show that $\nu(B)=\nu(T)$. Since the index of a matrix is well defined (and this again follows from Lemma \[schur\], where it is proved that the eigenvalues of a symmetric generalized matrix are well defined), it is sufficient to show that for one any symmetric representative $(T_{\varepsilon})_{\varepsilon}$ of $T$ there exists an $\varepsilon_0\in I$ such that for each $\varepsilon<\varepsilon_0$ we have $$\lambda_{1,\varepsilon}>0,\dots,\lambda_{n-k,\varepsilon}>0,\lambda_{n-k+1,\varepsilon}<0,\dots,\lambda_{n-k,\varepsilon}<0,$$ where $(\lambda_{i,\varepsilon})_{\varepsilon}$ ($i=1,\dots,n$) are the ordered eigenvalues of $(T_{\varepsilon})_{\varepsilon}$. To this end, let $(B_{\varepsilon})_{\varepsilon}$ be a symmetric representative of $B$, and define by $(T_{\varepsilon})_{\varepsilon}$ a representative of $T$ component-wise via $$T_{\varepsilon}:=A_{\varepsilon}^tB_{\varepsilon}A_{\varepsilon}.$$ Clearly $(T_{\varepsilon})_{\varepsilon}$ is symmetric. For each $\varepsilon$ let $\lambda_{1,\varepsilon}\geq\dots\geq\lambda_{n,\varepsilon}$ be the ordered eigenvalues of $T_{\varepsilon}$ and let $\mu_{1,\varepsilon}\geq\dots\geq\mu_{n,\varepsilon}$ be the ordered eigenvalues of $B_{\varepsilon}$. Since $A$ and $B$ are non-degenerate, there exists some $\varepsilon_0\in I$ and an integer $m_0$ such that for each $\varepsilon<\varepsilon_0$ and for each $i=1,\dots,n$ we have $$\vert \lambda_{i,\varepsilon}\vert\geq \varepsilon^{m_0}\qquad\mbox{and}\qquad \vert \mu_{i,\varepsilon}\vert\geq \varepsilon^{m_0}.$$ Furthermore due to our assumption $k=\nu(B)$, therefore taking into account the component-wise order of the eigenvalues $\mu_{i,\varepsilon}$, for each $\varepsilon<\varepsilon_0$ we have: $$\mu_{i,\varepsilon}\geq \varepsilon^{m_0}\;\;(i=1,\dots,n-k)\qquad \mbox{and}\qquad \mu_{i,\varepsilon}\leq -\varepsilon^{m_0}\;\;(i=n-k+1,\dots,n).$$ As a consequence of Sylvester’s inertia law we therefore have for each $\varepsilon<\varepsilon_0$: $$\lambda_{i,\varepsilon}\geq \varepsilon^{m_0}\;\; (i=1,\dots,n-k)\qquad \mbox{and}\quad \lambda_{i,\varepsilon}\leq -\varepsilon^{m_0}\;\; (i=n-k+1,\dots,n),$$ since for each $\varepsilon<\varepsilon_0$ the number of positive resp. negative eigenvalues of $B_{\varepsilon}$ resp. $T_{\varepsilon}$ coincides. We have thereby shown that $\nu(T)=k$ and we are done. Let $b:\widetilde{\mathbb R}^n\times\widetilde{\mathbb R}^n\rightarrow \widetilde{\mathbb R}$ be a symmetric bilinear form. A basis $\mathcal B:=\{e_1,\dots,e_k\}$ of $\widetilde{\mathbb R}^n$ is called an orthogonal basis with respect to $b$, if $b(e_i,e_j)=0$ whenever $i\neq j$. \[exorthbasis\] Any symmetric bilinear form $b$ on $\widetilde{\mathbb R}^n$ admits an orthogonal basis. Let $\mathcal B:=\{v_1,\dots,v_n\}$ be some basis of $\widetilde{\mathbb R}^n$, then the coefficient matrix $A:=(b(v_i,v_j))_{ij}\in\widetilde{\mathbb R}^{n^2}$ is symmetric. Due to Lemma \[schur\], there is an orthogonal matrix $U\in\widetilde{\mathbb R}^{n^2}$ and generalized numbers $\theta_i\;(1\leq i\leq n)$ (the so-called eigenvalues) such that $UAU^t={\mathop{\mathrm{diag}}}( \theta_1,\dots,\theta_n)$. Therefore the (clearly non-degenerate) matrix $U$ induces a mapping $\widetilde{\mathbb R}^n\rightarrow\widetilde{\mathbb R}^n$ which maps $\mathcal B$ onto some basis $\mathcal B'$ which is orthogonal. Let $\lambda_1,\dots,\lambda_k\in\widetilde{\mathbb R}$ ($k\geq 1$). Then the span of $\lambda_i$, $(1\leq i\leq k)$ in $\widetilde{\mathbb R}^n$ is denoted by $\langle\{\lambda_1,\dots,\lambda_n\}\rangle$. We now introduce a notion of causality in our framework: \[causaldef1\] Let $g$ be a symmetric bilinear form of Lorentzian signature on $\widetilde{\mathbb R}^n$. Then we call $u\in\widetilde{\mathbb R}^n$ 1. time-like, if $g(u,u)<0$, 2. null, if $u=0$ or $u$ is free and $g(u,u)=0$, 3. space-like, if $g(u,u)>0$. Furthermore, we say two time-like vectors $u,v$ have the same time-orientation whenever $g(u,v)<0$. Note that there exist elements in $\widetilde{\mathbb R}^n$ which are neither time-like, nor null, nor space-like. The next statement provides a crucial characterization of free elements in $\widetilde{\mathbb R}^n$. We shall repeatedly make use of it in the sequel. \[freechar\] Let $v$ be an element of $\widetilde{\mathbb R}^n$. Then the following are equivalent: 1. \[freechar1\] For any positive definite symmetric bilinear form $h$ on $\widetilde{\mathbb R}^n$ we have $$h(v,v)>0$$ 2. \[freechar2\] The coefficients of $v$ with respect to some (hence any) basis span $\widetilde{\mathbb R}$. 3. \[freechar3\]$v$ is free. 4. \[freechar4\] The coefficients $v^i$ ($i=1,\dots,n$) of $v$ with respect to some (hence any) basis of $\widetilde{\mathbb R}^n$ satisfy the following: For any choice of representatives $(v^i_{\varepsilon})_{\varepsilon}\;(1\leq i\leq n)$ of $v^i$ there exists some $\varepsilon_0\in I$ such that for each $\varepsilon<\varepsilon_0$ we have $$\max_{i=1,\dots,n} \vert v^i_{\varepsilon}\vert>0.$$ 5. \[freeloader6\] For each representative $(v_{\varepsilon})_{\varepsilon}\in\mathcal E_M(\mathbb R^n)$ of $v$ there exists some $\varepsilon_0\in I$ such that for each $\varepsilon<\varepsilon_0$ we have $v_{\varepsilon}\neq 0$ in $\mathbb R^n$. 6. \[freeloader7\] There exists a basis of $\widetilde{\mathbb R}^n$ such that the first coefficient $v^i$ of $v$ is strictly non-zero. 7. \[freeloader8\] $v$ can be extended to a basis of $\widetilde{\mathbb R}^n$. 8. \[freechar5\] Let $v^i$ ($i=1,\dots,n$) denote the coefficients of $v$ with respect to an arbitrary basis of $\widetilde{\mathbb R}^n$. Then we have $$\| v\widetilde{\|}:=\left(\sum_{i=1}^n (v^i)^2\right)^{1/2}>0.$$ The equivalences of (\[freechar1\]) $\Leftrightarrow$ (\[freechar5\]) as well as (\[freechar4\]) $\Leftrightarrow$ (\[freeloader6\]) are evident. We start by establishing the implications (\[freechar1\]) $\Rightarrow$ (\[freechar2\]) $\Rightarrow$ (\[freechar3\]) $\Rightarrow$ (\[freechar1\]) and the equivalence (\[freechar4\]) $\Leftrightarrow$ (\[freechar5\]). We end the proof by showing (\[freechar4\]) $\Rightarrow$ (\[freeloader7\]) $\Rightarrow$ (\[freeloader8\]) $\Rightarrow$ (\[freechar4\]).\ If $v=0$ the equivalences are trivial. We shall therefore assume $v\neq 0$.\ (\[freechar1\]) $\Rightarrow$ (\[freechar2\]): Let $(h_{ij})_{ij}$ be the coefficient matrix of $h$ with respect to some fixed basis $\mathcal B$ of $\widetilde{\mathbb R}^n$. Then $\lambda:=\sum_{1\leq i,j\leq n}h_{ij}v^iv^j=h(v,v)>0$, in particular $\lambda$ is invertible and $\sum_j(\sum_i \frac{h_{ij}v^i}{\lambda})v^j=1$ which shows that $\langle\{ v^1,\dots,v^n\}\rangle=\widetilde{\mathbb R}$. Since the choice of the basis was arbitrary, (\[freechar2\]) is shown.\ (\[freechar2\]) $\Rightarrow$ (\[freechar3\]): We assume $\langle\{ v^1,\dots,v^n\}\rangle=\widetilde{\mathbb R}$ but that there exists some $\lambda\neq 0: \lambda v=0$, that is, $\forall\;i: 1\leq i\leq n:\,\lambda v^i=0$. Since the coefficients of $v$ span $\widetilde{\mathbb R}$, there exist $\mu_1,\dots,\mu_n$ such that $\lambda=\sum_{i=1}^n\mu_iv^i$. It follows that $\lambda^2=\sum_{i=1}^n\mu_i(\lambda v^i)=0$ but this is impossible, since $\widetilde{\mathbb R}$ contains no nilpotent elements.\ (\[freechar3\]) $\Rightarrow$ (\[freechar1\]): Due to Lemma \[schur\] we may assume that we have chosen a basis such that the coefficient matrix with respect to the latter is in diagonal form, i. e., $(h_{ij})_{ij}={\mathop{\mathrm{diag}}}(\lambda_1,\dots,\lambda_n)$ with $\lambda_i>0\; (1\leq i\leq n)$. We have to show that $h(v,v)=\sum_{i=1}^n\lambda_i (v^i)^2>0$. Since there exists $\varepsilon_0\in I$ such that for all representatives of $\lambda_1,\dots,\lambda_n, v^1,\dots, v^n$ we have for $\varepsilon<\varepsilon_0$ that $\gamma_{\varepsilon}:=\lambda_{1\varepsilon} (v^1_{\varepsilon})^2+\dots+\lambda_{n\varepsilon} (v^n_{\varepsilon})^2\geq 0$, $h(v,v)\not> 0$ would imply that there exists a zero sequence $\varepsilon_k\rightarrow 0$ ($k\rightarrow \infty$) such that $\gamma_{\varepsilon_k}<\varepsilon^k$. This implies that $h(v,v)$ is a zero divisor and it means that all summands share a simultaneous zero divisor, i. e., $\exists\;\mu\neq 0\,\forall\;i\in\{1,\dots,n\}:\,\mu \lambda_i (v^i)^2=0$. Since $v$ was free, this is a contradiction and we have shown that (\[freechar1\]) holds.\ We proceed by establishing the equivalence (\[freechar4\]) $\Leftrightarrow$ (\[freechar5\]). First, assume (\[freechar5\]) holds, and let $(v^i_{\varepsilon})_{\varepsilon}\;(1\leq i\leq n)$ be arbitrary representatives of $v^i\;(i=1,\dots,n)$. Then $$\left(\sum_{i=1}^n (v^i_{\varepsilon})^2\right)_{\varepsilon}$$ is a representative of $(\|v\widetilde{\|})^2$ as well, and since $\|v\widetilde{\|}$ is strictly positive, there exists some $m_0$ and some $\varepsilon_0\in I$ such that $$\forall\;\varepsilon<\varepsilon_0:\sum_{i=1}^n (v^i_{\varepsilon})^2>\varepsilon^{m_0}.$$ This immediately implies (\[freechar4\]). In order to see the converse direction, we proceed indirectly. Assume (\[freechar5\]) does not hold, that is, we assume there exist representatives $(v_{\varepsilon}^i)_{\varepsilon}$ of $v^i$ for $i=1,\dots,n$ such that for some sequence $\varepsilon_k\rightarrow 0$ ($k\rightarrow\infty$) we have for each $k>0$ that $$\sum_{i=1}^n (v_{\varepsilon_k}^i)^2<\varepsilon_k^k.$$ Therefore one may even construct representatives $(\widetilde v_{\varepsilon}^i)_{\varepsilon}$ for $v^i$ ($i=1,\dots,n$) such that for each $k>0$ and each $i\in\{1,\dots,n\}$ we have $\widetilde v_{\varepsilon_k}^i=0$. It is now evident that $(\widetilde v_{\varepsilon}^i)_{\varepsilon}$ violate condition (\[freechar4\]) and we are done with (\[freechar4\]) $\Leftrightarrow$ (\[freechar5\]). Finally we prove the chain of implications (\[freechar4\]) $\Rightarrow$ (\[freeloader7\]) $\Rightarrow$ (\[freeloader8\]) $\Rightarrow$ (\[freechar4\]). Clearly (\[freeloader8\]) $\Rightarrow$ (\[freechar4\]). To see (\[freechar4\]) $\Rightarrow$ (\[freeloader7\]), let $v_i$ ($i=1,\dots,n$) be the coefficients of $v$ with respect to some basis of $\widetilde{\mathbb R}^n$. We first observe that condition (\[freechar4\]) implies that there exists some $m_0$ such that for suitable representatives $(v_{\varepsilon}^i)_{\varepsilon}$ of $v^i$ ($i=1,\dots,n$) we have for each $\varepsilon\in I$ $\max_{i=1,\dots,n}\vert v_{\varepsilon}^i\vert>\varepsilon^{m_0}$, i. e., $$\forall\;\varepsilon\in I\;\exists\; i(\varepsilon)\in\{1,\dots,n\}:\vert v_{\varepsilon}^{i(\varepsilon)}\vert>\varepsilon^{m_0}.$$ We may view $(v_{\varepsilon})_{\varepsilon}:=((v_{\varepsilon}^1,\dots,v_{\varepsilon}^n)^t)_{\varepsilon}\in\mathcal E_M(\mathbb R^n)$ as a representative of $v$ in $\mathcal E_M(\mathbb R^n)/\mathcal N(\mathbb R^n)$. Denote for each $\varepsilon\in I$ by $A_{\varepsilon}$ the representing matrix of the linear map $\mathbb R^n\rightarrow\mathbb R^n$ that merely permutes the $i(\varepsilon)$ th. canonical coordinate of $\mathbb R^n$ with the first one. Define $A:\widetilde{\mathbb R}^n\rightarrow\widetilde{\mathbb R}^n$ the bijective linear map with representing matrix $$A:=(A_{\varepsilon})_{\varepsilon}+\mathcal E_M(\mathcal M_n(\mathbb R)).$$ What is evident now from our construction, is: The first coefficient of $$\widetilde v:=Av=(\mathcal A_{\varepsilon}v_{\varepsilon})_{\varepsilon}+\mathcal E_M(\mathbb R^n)$$ is strictly nonzero and we have shown (\[freeloader7\]). Finally we verify (\[freeloader7\]) $\Rightarrow$ (\[freeloader8\]). Let $\{e_i\mid 1\leq i\leq n\}$ denote the canonical basis of $\widetilde{\mathbb R}^n$. Point (\[freeloader7\]) ensures the existence of a bijective linear map $A$ on $\widetilde{\mathbb R}^n$ such that the first coefficient $\bar v^1$ of $\bar v=(\bar v^1,\dots,\bar v^n)^t:=Av$ is strictly non-zero; applying Proposition \[steinitzprop\] yields another basis $\{\bar v,e_2,\dots,e_n\}$ of $\widetilde{\mathbb R}^n$. Since $A$ is bijective, $\{v=A^{-1}\bar v,A^{-1}e_2,\dots,A^{-1}e_n\}$ is a basis of $\widetilde{\mathbb R}^n$ as well and we are done. We may add a non-trivial example of a free vector to the above characterization: For $n>1$, let $\lambda_i\in\widetilde{\mathbb R}\; (1\leq i\leq n)$ have the following properties: 1. \[ex1\] $\lambda_i^2=\lambda_i\;\forall\; i\in\{1,\dots,n\}$ 2. \[ex2\] $\lambda_i\lambda_j=0\;\forall\;i\neq j$ 3. \[ex3\] $\langle\{\lambda_1,\dots,\lambda_n\}\rangle=\widetilde{\mathbb R}$ This choice of zero divisors in $\widetilde{\mathbb R}$ is possible (idempotent elements in $\widetilde{\mathbb R}$ are thoroughly discussed in [@A1], pp. 2221–2224). Now, let $\mathcal B=\{e_1,\dots,e_n\}$ be the canonical basis of $\widetilde{\mathbb R}^n$. Set $v:=\sum_{i=1}^n (-1)^{(i+1)(n+1)}\lambda_i e_i$ satisfies Theorem \[freechar\] (\[freechar3\]). Thus $v$ is free. Furthermore let $\gamma\in\Sigma_n$ be the cyclic permutation which sends $\{1,\dots,n\}$ to $\{n,1,\dots,n-1\}$. Clearly the sign of $\gamma$ is positive if and only if $n$ is odd. Define $n$ vectors $v_j\;(1\leq j\leq n)$ by $v_1:=v$, and such that $v_j$ is given by $v_j:=\sum_{k=1}^n \lambda_{\gamma^{j-1}(k)}e_k$ whenever $j>1$. Let $A$ be the matrix having the $v_j$’s as column vectors. Then $$\det A=\sum_{l=1}^n\lambda_l^n=\sum_{l=1}^n\lambda_l.$$ By properties (\[ex1\]) and (\[ex3\]), $\det A$ is invertible. Therefore, $\mathcal B':=\{v,v_2,\dots,v_n\}$ is a basis of $\widetilde{\mathbb R}^n$, too. The reader is invited to check further equivalent properties of $v$ according to Theorem \[freechar\]. Since any symmetric bilinear form admits an orthogonal basis due to Corollary \[exorthbasis\] we further conclude by means of Theorem \[freechar\]: \[corbilpos\] Let $b$ be a symmetric bilinear form on $\widetilde{\mathbb R}^n$. Then the following are equivalent: 1. \[corbilpos1\] For any free $v\in\widetilde{\mathbb R}^n$, $b(v,v)>0$. 2. $b$ is positive definite. For showing further algebraic properties of $\widetilde{\mathbb R}^n$ (cf. section \[semi\]), also the following lemma will be crucial: \[posprop\] Let $h$ be a positive definite symmetric bilinear form. Then we have the following: 1. \[h1\] $\forall\; v\in\widetilde{\mathbb R}^n: h(v,v)\geq 0$ and $h(v,v)=0\Leftrightarrow v=0$. 2. \[h2\]Let $\mathfrak m$ be a free submodule of $\widetilde{\mathbb R}^n$. Then $h$ is a positive definite symmetric bilinear form on $\mathfrak m$. First, we verify (\[h1\]): Let $v^i\;(1\leq i\leq n)$ be the coefficients of $v$ with respect to some orthogonal basis $\mathcal B$ for $h$. Then we can write $h(v,v)=\sum_{i=1}^n\lambda_i(v^i)^2$ with $\lambda_i$ strictly positive for each $i\in\{1,\dots,n\}$. Thus $h(v,v)\geq 0$, and $h(v,v)=0$ implies $\forall\; i\in\{1\dots n\}:v^i=0$, i. e., $v=0$. This finishes the proof of part (\[h1\]). In order to show (\[h2\]) we first notice that by definition, any free submodule admits a basis. Let $\mathcal B_{\mathfrak m}:=\{w_1,\dots,w_k\}$ be such for $\mathfrak m$ and denote by $h_{\mathfrak m}$ the restriction of $h$ to $\mathfrak m$. Then, due to Theorem \[freechar\] (\[freechar1\]), we have for all $1\leq i\leq k$, $h_{\mathfrak m}(w_i,w_i)>0$. Let $A:=(h_{\mathfrak m}(w_i,w_j))_{ij}$ be the coefficient matrix of $h_{\mathfrak m}$ with respect to $\mathcal B_{\mathfrak m}$. Since $h_{\mathfrak m}$ is symmetric, so is the matrix $A$ and thus, due to Lemma \[schur\] there is an orthogonal matrix $U\in\widetilde{\mathbb R}^{k^2}$ and there are generalized numbers $\lambda_i\;(1\leq i\leq k)$ such that $UAU^t={\mathop{\mathrm{diag}}}(\lambda_1,\dots,\lambda_k)$ which implies that the (orthogonal, thus non-degenerate) $U$ maps $\mathcal B_{\mathfrak m}$ on an orthogonal basis $\mathcal B:=\{e_1,\dots,e_k\}$ of $\mathfrak m$ with respect to $h_{\mathfrak m}$ and again by Theorem \[freechar\] (\[freechar1\]) we have $\lambda_i>0\;(1 \leq i\leq k)$. By Definition \[bilformdef\], $h_{\mathfrak m}$ is also positive definite on $\mathfrak m$ and we are done. Since any time-like or space-like vector is free, we further have as a consequence of Theorem \[freechar\]: \[coraustausch\] Suppose we are given a bilinear form of Lorentzian signature on $\widetilde{\mathbb R}^n$ and let $u\in\widetilde{\mathbb R}^n\setminus \{0\}$ be time-like, null or space-like. Then $u$ can be extended to a basis of $\widetilde{\mathbb R}^n$. In the case of a time-like vector we know a specific basis in which the first coordinate is invertible: Suppose we are given a bilinear form $b$ of Lorentzian signature on $\widetilde{\mathbb R}^n$, let $u$ be a time-like vector. Due to the definition of $g$ we may suppose that we have a basis so that the scalar product of $u$ takes the form $$g(u,u)=-\lambda_1 (u^1)^2+\lambda_2 (u^2)^2\dots+\lambda_n (u^n)^2.$$ with $\lambda_i$ strictly positive for each $i=1,\dots,n$. Since $g(u,u)<0$, we see that the first coordinate $u^1$ of $u$ must be strictly non-zero. It is worth mentioning that an analogue of the well known criterion of positive definiteness of matrices in $\mathcal M_n(\mathbb R)$ holds in our setting: \[criterion\] Let $A\in\widetilde{\mathbb R}^{n^2}$ be symmetric. If the determinants of all principal subminors of $A$ (that are the submatrices $A^{(k)}:=(a_{ij})_{1\leq i,j\leq k}\;(1\leq k\leq n)$) are strictly positive, then $A$ is positive definite. Choose a symmetric representative $(A_{\varepsilon})_{\varepsilon}$ of $A$ (cf.  Lemma \[symmetry\]). Clearly the assumption $\det A^{(k)}>0\; (1\leq k\leq n)$ implies that $\exists\; \varepsilon_0\;\exists\; m\;\forall\; k: 1\leq k\leq n\; \forall\; \varepsilon<\varepsilon_0:\det A^{(k)}_{\varepsilon}\geq \varepsilon^m$, that is, for each sufficiently small $\varepsilon$, $A_{\varepsilon}$ is a positive definite symmetric matrix due to a well known criterion in linear algebra. Furthermore $\det A^{(n)}=\det A>0$ implies $A$ is non-degenerate which finally shows that $A$ is positive definite. Before we go on we note that type changing of tensors on $\widetilde{\mathbb R}^n$ by means of a non-degenerate symmetric bilinear form $g$ clearly is possible. Moreover, given a (generalized) metric $g\in\mathcal G^0_2(X)$ on a manifold $X$ (cf. section \[introducerepseudoriemannereetconnexione\]), lowering (resp. raising) indices of generalized tensor fields on $X$ (resp. tensors on $\widetilde{\mathbb R}^n$) is compatible with evaluation on compactly supported generalized points (which actually yields the resp.  object on $\widetilde{\mathbb R}^n$). This basically follows from Proposition 3.9 ([@KS1]) combined with Theorem 3.1 ([@KS1]). As usual we write the covector associated to $\xi\in\widetilde{\mathbb R}^n$ in abstract index notation as $\xi_a:=g_{ab}\xi^b$. We call $\xi_i\;(i=1,\dots,n)$ the covariant components of $\xi$.\ The following technical lemma is required in the sequel: \[uvfree\] Let $u,v\in\widetilde{\mathbb R}^n$ such that $u$ is free and $u^tv=0$. Then for each representative $(u_\varepsilon)_\varepsilon$ of $u$ there exists a representative $(v_\varepsilon)_\varepsilon$ of $v$ such that for each $\varepsilon \in I$ we have $u^t_\varepsilon v_\varepsilon=0$. Let $(u_\varepsilon)_\varepsilon$, $(\hat v_\varepsilon)_\varepsilon$ be representatives of $u,v$ respectively. Then there exists $(n_\varepsilon)_\varepsilon\in\mathcal N$ such that $$(u_\varepsilon^t)_\varepsilon(\hat v_\varepsilon)_\varepsilon=(n_\varepsilon)_\varepsilon.$$ By Theorem \[freechar\] (\[freechar4\]) we conclude $$\exists\; \varepsilon_0\;\exists\; m_0\;\forall\;\varepsilon<\varepsilon_0\;\exists\; j(\varepsilon):\;\vert u_\varepsilon^{j(\varepsilon)}\vert\geq \varepsilon^{m_0}.$$ Therefore we may define a new representative $(v_\varepsilon)_\varepsilon$ of $v$ in the following way: For $\varepsilon\geq\varepsilon_0$ we set $v_\varepsilon:=0$, otherwise we define $$v_\varepsilon:=\begin{cases}\hat v_\varepsilon^{j}, \quad j\neq j(\varepsilon)\\\hat v_{\varepsilon}^{j(\varepsilon)}-\frac{n_{\varepsilon}}{u_\varepsilon^{j(\varepsilon)}}\quad \mbox{otherwise} \end{cases}$$ and clearly we have $u^t_\varepsilon v_\varepsilon=0$ for each $\varepsilon\in I$. The following result in the style of [@FL1] (Lemma 3.1.1, p. 74) prepares the inverse Cauchy-Schwarz inequality in our framework. We follow the book of Friedlander which helps us to calculate the determinant of the coefficient matrix of a symmetric bilinear form, which then turns out to be strictly positive, thus invertible. This is equivalent to non-degenerateness of the bilinear form (cf. Lemma \[nondeg\]): \[procs\] Let $g$ be a symmetric bilinear form of Lorentzian signature. If $u\in\widetilde{\mathbb R}^n$ is time-like, then $u^{\perp}$ is an $n$$-$$1$ dimensional submodule of $\widetilde{\mathbb R}^n$ and $g\mid_{u^{\perp}\times u^{\perp}}$ is positive definite. Due to Proposition \[coraustausch\] we can choose a basis of $\widetilde{\mathbb R}^n$ such that $\Pi:=\langle\{u\}\rangle$ is spanned by the first vector, i. e., $$\Pi=\{\xi\in\widetilde{\mathbb R}^n\vert \xi^A=0, A=2,\dots,n\}.$$ Consequently we have $$\langle \xi,\xi\rangle\vert_{\Pi\times\Pi}=g_{11}(\xi^1)^2,$$ and $g_{11}=\langle u,u\rangle<0$. If $\eta\in \Pi':=u^{\perp}$, then $\langle \xi,\eta\rangle=\xi^i\eta_i$, hence the covariant component $\eta_1$ must vanish (set $\xi:=u$, i. e., $\langle \xi,\eta\rangle=\langle u,\eta\rangle=\eta_1=0$). Therefore we have $$\label{uperp} \langle \eta ,\theta\rangle\vert_{\Pi'\times\Pi'}=g^{AB}\eta_A\theta_B.$$ Our first observation is that $u^{\perp}$ is a free ($n-1$ dimensional) submodule with the basis $\xi_{(2)},\dots,\xi_{(n)}$ given in terms of the chosen coordinates above via $$\xi_{(k)}^j:=g^{ij}\delta_i^k,\quad k=2,\dots,n$$ (cf. (\[matmulti\]) below, these are precisely the $n-1$ row vectors there!) Due to the matrix multiplication $$\label{matmulti} \left (\begin{array}{cccc} 1 & 0 &\dots &0 \\ g^{21}& g^{22}&\dots&g^{2n}\\ \dots &\dots&\dots&\dots\\g^{n1}& g^{n2}&\dots&g^{nn}\end{array}\right)(g_{ij})=\left(\begin{array}{cc} g_{11}& *\\ 0&\mathbb I_{n-1}\end{array} \right)$$ evaluation of the determinants yields $$\det g^{AB}\det g_{ij}=g_{11}.$$ And it follows from $\det g_{ij}<0, g_{11}<0$ that $\det g^{AB}>0$ which in particular shows that $g^{AB}$ is a non-degenerate symmetric matrix, $g\mid_{u^{\perp}\times u^{\perp}}$ therefore being a non-degenerate symmetric bilinear form on an $n-1$ dimensional free submodule. What is left to prove is positive definiteness of $g^{AB}$. We claim that for each $u\in v^{\perp}$, $g(v,v)\geq 0$. In conjunction with the fact that $g\mid_{u^{\perp}\times u^\perp}$ is non-degenerate, it follows that $g(v,v)>0$ for any free $v\in u^{\perp}$ (this can be seen by using a suitable basis for $u^{\perp}$ which diagonalizes $g\mid_{u^{\perp}\times u^{\perp}}$, cf. Corollary \[corbilpos\]) and we are done. To show the subclaim we have to undergo an $\varepsilon$-wise argument. Let $(u_{\varepsilon})_{\varepsilon}\in\mathcal E_M(\mathbb R^n)$ be a representative of $u$ and let $((g^{\varepsilon}_{ij})_{ij})_{\varepsilon}\in\mathcal E_M(\mathcal M_n(\mathbb R))$ be a symmetric representatives of $(g_{ij})_{ij}$, where $(g_{ij})_{ij}$ is the coefficient matrix of $g$ with respect to the canonical basis of $\widetilde{\mathbb R}^n$. For each $\varepsilon$ we denote by $g_{\varepsilon}$ the symmetric bilinear form induced by $(g^{\varepsilon}_{ij})_{ij}$, that is, the latter shall be the coefficient matrix of $g_{\varepsilon}$ with respect to the canonical basis of $\mathbb R^n$. First we show that $$\label{identitynormalspaces} u^{\perp}=\{(v_{\varepsilon})_{\varepsilon}\in\mathcal E_M(\mathbb R^n):\; \forall\; \varepsilon>0: v_{\varepsilon}\in u_{\varepsilon}^{\perp}\}+\mathcal N(\mathbb R^n),$$ Since the inclusion relation $\supseteq$ is clear, we only need to show that $\subseteq$ holds. To this end, pick $v\in u^{\perp}$. Then $g(u,v)=g_{ij}u^iv^j=0$ and the latter implies that for each representative $(\hat v_{\varepsilon})_{\varepsilon}$ of $v$ there exists $(n_{\varepsilon})_{\varepsilon}\in\mathcal N$ such that $$(g_{ij}^{\varepsilon}u_{\varepsilon}^i\hat v_{\varepsilon}^j)_{\varepsilon}=(n_{\varepsilon})_{\varepsilon}.$$ We may interpret $(g_{ij}^{\varepsilon}u_{\varepsilon}^i) (j=1,\dots,n)$ as the representatives of the coefficients of a vector $w$ with coordinates $w_j:=g_{ij}u^i$, and $w$ is free, since $u$ is free and $g$ is non-degenerate. Therefore we may employ Lemma \[uvfree\] which yields a representative $(v_{\varepsilon}^j)_{\varepsilon}$ of $v$ such that $$(g_{ij}^{\varepsilon}u_{\varepsilon}^i v_{\varepsilon}^j)_{\varepsilon}=0.$$ This precisely means that there exists a representative $(v_{\varepsilon})_\varepsilon$ of $v$ such that for each $\varepsilon$ we have $v_\varepsilon\in u_\varepsilon^\perp$. We have thus finished the proof of identity (\[identitynormalspaces\]). To finish the proof of the claim, that is $g(v,v)\geq0$, we pick a representative $(v_{\varepsilon})_{\varepsilon}$ of $v$ and an $\varepsilon_0\in I$ such that for each $\varepsilon<\varepsilon_0$ we have 1. each $g_{\varepsilon}$ is of Lorentzian signature 2. $u_{\varepsilon}$ is time-like 3. $v_{\varepsilon}\in u_{\varepsilon}^\perp$. Note that this choice is possible due to (\[identitynormalspaces\]). Further, by the resp. classic result of Lorentz geometry (cf. [@FL1], Lemma 3. 1. 1) we have $g_{\varepsilon}(v_{\varepsilon},v_\varepsilon)\geq 0$ unless $v_{\varepsilon}=0$. Since $(g_{ij}^{\varepsilon}v_\varepsilon^i v_\varepsilon^j)_\varepsilon$ is a representative of $g(v,v)$ we have achieved the subclaim. \[dirsum1\] Let $u\in\widetilde{\mathbb R}^n$ be time-like. Then $u^{\perp}:=\{v\in \widetilde{\mathbb R}^n:\langle u,v\rangle=0\}$ is a submodule of $\widetilde{\mathbb R}^n$ and $\widetilde{\mathbb R}^n=\langle \{u\}\rangle \oplus u^{\perp}$. The first statement is obvious. For $v\in\widetilde{\mathbb R}^n$, define the orthogonal projection of $v$ onto $\langle\{u\}\rangle$ as $P_u(v):=\frac{\langle u, v\rangle}{\langle u,u\rangle}u$. Then one sees that $v=P_u(v)+(v-P_u(v))\in \langle \{u\}\rangle+u^{\perp}$. Finally, assume $\widetilde{\mathbb R}^n\neq \langle \{u\}\rangle \oplus u^{\perp}$, i. e., $\exists\; \xi\neq 0, \xi\in \langle \{u\}\rangle \cap u^{\perp} $. It follows $\langle \xi,\xi\rangle\leq 0$ and due to the preceding proposition $\xi\in u^{\perp}$ implies $\langle \xi,\xi\rangle \geq 0$. Since we have a partial ordering $\leq $, this is impossible unless $\langle \xi,\xi\rangle=0$. However by Lemma \[posprop\] (\[h1\]) we have $\xi=0$. This contradicts our assumption and proves that $\widetilde{\mathbb R}^n$ is the direct sum of $u$ and its orthogonal complement. The following statement on the Cauchy–Schwarz inequality is a crucial result in generalized Lorentz Geometry. It slightly differs from the classical result as is shown in Example \[csex\]. However it seems to coincide with the classical inequality in physically relevant cases, since algebraic complications which mainly arise from the existence of zero divisor in our scalar ring of generalized numbers, presumably are not inherent in the latter. Our proof follows the lines of the proof of the analogous classic statement in O’Neill’s book ([@ON], chapter 5, Proposition 30, pp. 144): (Inverse Cauchy–Schwarz inequality)\[cs\] Let $u,\;v \in \widetilde{\mathbb R}^n$ be time-like vectors. Then 1. \[cs1\] $\langle u,v\rangle^2\geq \langle u,u\rangle \langle v,v\rangle$, and 2. \[cs2\] equality in (\[cs1\]) holds if $u,v$ are linearly dependent over $\widetilde{\mathbb R}^*$, the units in $\widetilde{\mathbb R}$. 3. \[cs3\] If $u,v$ are linearly independent, then $\langle u,v\rangle^2>\langle u,u\rangle \langle v,v\rangle$. In what follows, we keep the notation of the preceding corollary. Due to Corollary \[dirsum1\], we may decompose $u$ in a unique way $v=a u+w$ with $a\in\widetilde{\mathbb R},\, w\in u^{\perp}$. Since $u$ is time-like, $$\langle v,v\rangle=a^2 \langle u,u\rangle+\langle w,w\rangle<0.$$ Then $$\label{eqcs} \langle u,v\rangle^2=a^2\\\langle u,u\rangle^2=(\langle v,v\rangle-\langle w,w\rangle)\langle u,u\rangle\geq \langle u,u\rangle \langle v,v\rangle$$ since $\langle w,w\rangle\geq 0$ and this proves (\[cs1\]).\ In order to prove (\[cs2\]), assume $u,v$ are linearly dependent over $\widetilde{\mathbb R}^*$, that is, there exist $\lambda,\,\mu$, both units in $\widetilde{\mathbb R}$ such that $\lambda u+\mu v=0$. Then $u=-\frac{\mu}{\lambda} v$ and equality in (\[cs2\]) follows.\ Proof of (\[cs3\]): Assume now, that $u,v$ are linearly independent. We show that this implies that $w$ is free. For the sake of simplicity we assume without loss of generality that $\langle u,u\rangle=\langle v,v\rangle=-1$ and we choose a basis $\mathcal B=\{e_1,\dots,e_n\}$ with $e_1=u$ due to Proposition \[coraustausch\]. Then with respect to the new basis we can write $u=(1,0,\dots,0)^t$, $v=(v^1,\dots,v^n)^t$, $w=v-P_u(v)=(v^1-(-g(v,e_1)), v^2,\dots,v^n)^t=(0,w^2,\dots,w^n)^t$. Assume $\exists\; \lambda\neq0: \lambda w=0$, then $$(\lambda v^1)u+\lambda v=\lambda v^1 e_1-\lambda g(v,e_1) e_1=\lambda v^1 e_1-\lambda v^1 e_1=0$$ which implies that $u,v$ are linearly dependent. This contradicts the assumption in (\[cs3\]). Thus $w$ indeed is free. Applying Theorem \[freechar\] yields $\langle w,w\rangle>0$. A glance at (\[eqcs\]) shows that the proof of (\[cs3\]) is finished. The following example indicates what happens when in \[cs\] (\[cs2\]) linear dependence over the units in $\widetilde{\mathbb R}$ is replaced by linear dependence over $\widetilde{\mathbb R}$: \[csex\] Let $\lambda\in\widetilde{\mathbb R}$ be an idempotent zero divisor, and write $\alpha:=[(\varepsilon)_{\varepsilon}]$. Let $\eta={\mathop{\mathrm{diag}}}(-1,1\dots,1)$ be the Minkowski metric. Define $u=(1,0,\dots,0)^t,v=(1,\lambda\alpha,0,\dots,0)^t$. Clearly $\langle u,u\rangle=-1,\langle v,v\rangle=-1+\lambda^2\alpha^2<0$ But $$\langle u,v\rangle^2=1\neq \langle u,u\rangle\langle v,v\rangle=-(-1+\lambda^2\alpha^2)=1-\lambda^2\alpha^2.$$ However, also the strict relation fails, i. e., $\langle u,v\rangle^2 \not> \langle u,u\rangle\langle v,v\rangle$, since $\lambda$ is a zero divisor. Applications {#energygeneralizedsection} ------------ In this subsection we establish a generalized dominant energy condition as an application of the Cauchy-Schwarz inequality. This generalizes known results in Relativity by Hawking and Ellis ([@HE]) in the context of the special algebra. Throughout this subsection $g$ denotes a symmetric bilinear form of Lorentz signature on $\widetilde{\mathbb R}^n$, and for $u,v\in\widetilde{\mathbb R}^n$ we write $\langle u,v\rangle:=g(u,v)$. We introduce the notion of a (generalized) Lorentz transformation: We call a linear map $L:\widetilde{\mathbb R}^n\rightarrow\widetilde{\mathbb R}^n$ a Lorentz transformation, if it preserves the metric, that is $$\forall \xi \in \widetilde{\mathbb R}^n:\;\langle L\xi,L\eta\rangle=\langle \xi,\eta\rangle$$ or equivalently, $$L^{\mu}_{\lambda}L^{\nu}_{\rho}g_{\mu\nu}=g_{\lambda\rho}.$$ The following statement is straightforward (cf. [@RB]): \[lorentz\] Let $\xi, \eta\in \widetilde{\mathbb R}^n$ be time-like unit vectors with the same time-orientation. Then $$L^{\mu}_{\lambda}:=\delta^{\mu}_{\lambda}-2\eta^{\mu}\xi_{\lambda}+\frac{(\xi^{\mu}+\eta^{\mu})(\xi_{\lambda}+\eta_{\lambda})}{1-\langle \xi,\eta\rangle}$$ is a Lorentz transformation with the property $L\xi=\eta$. The following proposition is a crucial ingredient in the subsequent proof of the (generalized) dominant energy condition for certain energy tensors of this section: \[metrconstr\] Let $u,v \in \widetilde{\mathbb R}^n$ be time-like vectors such that $\langle u,v\rangle<0$. Then $$h_{\mu\nu}:=u_{(\mu}v_{\nu)}-\frac{1}{2}\langle u,v\rangle g_{\mu\nu}$$ is a positive definite symmetric bilinear form on $\widetilde{\mathbb R}^n$. Symmetry and bilinearity of $h$ are clear. What would be left is to show that the coefficient matrix of $h$ with respect to an arbitrary basis is invertible. However, determining the determinant of $h$ is nontrivial. So we proceed by showing that for any free $w\in\widetilde{\mathbb R}^n$, $h(w,w)$ is strictly positive (thus also deriving the classic statement). We may assume $\langle u ,u\rangle=\langle v,v\rangle=-1$; this can be achieved by scaling $u,v$ (note that this is due to the fact that for a time-like (resp. space-like) vector $u$, $\langle u, u\rangle$ is strictly non-zero, thus invertible in $\widetilde{\mathbb R}$). We may assume we have chosen an orthogonal basis $\mathcal B=\{e_1,\dots,e_n\}$ of $\widetilde{\mathbb R}^n$ with respect to $g$, i. e., $g(e_i,e_j)=\varepsilon_{ij}\lambda_i$, where $\lambda_1\leq \dots\leq \lambda_n$ are the eigenvalues of $(g(e_i,e_j))_{ij}$. Due to Lemma \[lorentz\] we can treat $u,v$ by means of generalized Lorentz transformations such that both vectors appear in the form $u=(\frac{1}{\lambda_1},0,0,0)$, $v=\gamma(v)(\frac{1}{\lambda_1},\frac{V}{\lambda_2},0,0)$, where $\gamma(v)=\sqrt{-g(v,v)}=\sqrt{1-V^2} >0$ (therefore $\vert V\vert <1$). Let $w=(w^1,w^2,w^3,w^4)\in \widetilde{\mathbb R}^n$ be free (in particular $w\neq 0$). Then $$\label{beig1} h(w,w):=h_{ab}w^aw^b= \langle u,w\rangle \langle v,w\rangle-\frac{1}{2}\langle w,w\rangle \langle u,v\rangle.$$ Obviously, $\langle u,w \rangle=-w^1,\langle v,w\rangle=\gamma(v)(-w^1+Vw^2),\langle u,v\rangle=-\gamma(v)$. Thus $$\begin{gathered} \nonumber h(w,w)=\gamma(v)(-w^1)(-w^1+Vw^2)+\frac{\gamma(v)}{2}(-(w^1)^2+(w^2)^2+(w^3)^2+(w^4)^2)=\\= -\gamma(v) V w^1w^2+\frac{1}{2}\gamma(v) (+(w^1)^2+(w^2)^2+(w^3)^2+(w^4)^2).\end{gathered}$$ If $Vw^1w^2\leq 0$, we are done. If not, replace $V$ by $\vert V\vert$ ($-V\geq-\vert V \vert$) and rewrite the last formula in the following form : $$\label{estw} h(w,w)\geq\frac{\gamma(v)}{2}\left ( (\vert V \vert(w^1-w^2)^2 +(1-\vert V \vert)(w^1)^2+(1-\vert V \vert)(w^2)^2+(w^3)^2+(w^4)^2\right).$$ Clearly for the first term on the right side of (\[estw\]) we have $\vert V\vert(w^1-w^2)^2\geq 0$. From $v$ is time-like we further deduce $1-\vert V \vert=\frac{1-V^2}{1+\vert V \vert}>0$. Since $w$ is free we may apply Theorem \[freechar\], which yields $(1-\vert V\vert)(w^1)^2+(1-\vert V\vert)(w^2)^2+(w^3)^2+(w^4)^2>0$ and thus $h(w,w)>0$ due to equation (\[estw\] and we are done. Finally we are prepared to show a dominant energy condition in the style of Hawking and Ellis ([@HE], pp. 91–93) for a generalized energy tensor. In what follows, we use abstract index notation. \[dec\] For $\theta\in\widetilde{\mathbb R}^n$ the energy tensor $E^{ab}(\theta):=(g^{ac}g^{bd}-\frac{1}{2}g^{ab}g^{cd})\theta_c\theta_d$ has the following properties 1. \[energy1\] If $\xi,\eta\in\widetilde{\mathbb R}^n$ are time-like vectors with the same orientation, then we have for any free $\theta$, $E^{ab}(\theta)\xi_a\eta_b>0$. 2. \[energy2\] Suppose $\langle \theta,\theta\rangle$ is invertible in $\widetilde{\mathbb R}$. If $\xi\in\widetilde{\mathbb R}^n$ is time-like, then $\eta^b:=E^{ab}(\theta)\xi_a$ is time-like and $\eta^a\xi_a>0$, i. e., $\eta$ is past-oriented with respect to $\xi$. Conversely, if $\langle \theta,\theta\rangle$ is a zero divisor, then $\eta$ fails to be time-like. (\[energy1\]): Define a symmetric bilinear form $h^{ab}:=(g^{(ac}g^{b)d}-\frac{1}{2}g^{ab}g^{cd})\xi_c\eta_d$. Due to our assumptions on $\xi$ and $\eta$, Proposition \[metrconstr\] yields that $h^{ab}$ is a positive definite symmetric bilinear form. By Theorem \[freechar\] we conclude that for any free $\theta\in\widetilde{\mathbb R}^n$, $h_{ab}\theta^a\theta^b>0$. It is not hard to check that $E^{ab}(\theta)\xi_a\eta_b=h^{ab}\theta_a\theta_b$ and therefore we have proved (\[energy1\]).\ (\[energy2\]): To start with, assume $\eta$ is time-like. Then $g(\xi,\eta)=g_{ab}\xi^a\eta^b=g_{ab}\xi^aE(\theta)^{ac}\xi_c=E^{ab}(\theta)\xi_a\xi_b$. That this expression is strictly greater than zero follows from (\[energy1\]), i. e., $E^{ab}(\theta)\xi_a$ is past-directed with respect to $\xi$ whenever $\langle\theta,\theta\rangle$ is invertible, since the latter implies $\theta$ is free. It remains to prove that $\langle \eta,\eta\rangle<0$. A straightforward calculation yields $$\langle \eta,\eta\rangle=\langle E(\theta)\xi,E(\theta)\xi\rangle=\frac{1}{4}\langle \theta,\theta\rangle^2\langle \xi,\xi\rangle.$$ Since $\langle \theta,\theta\rangle$ is invertible and $\xi$ is time-like, we conclude that $\eta$ is time-like as well. Conversely, if $\langle \theta,\theta\rangle$ is a zero-divisor, also $\langle E(\theta)\xi,E(\theta)\xi\rangle$ clearly is one. Therefore, $\eta=E(\theta)\xi$ cannot be time-like, and we are done. A remark on this statement is in order. A comparison with ([@HE], pp. 91–93) shows, that our dominant energy condition on $T^{ab}$ is stronger, since the vectors $\xi,\eta$ in (\[energy1\]) need not coincide. Furthermore, if in (\[energy2\]) the condition $\langle \theta,\theta\rangle$ is invertible was dropped, then (as in the classical (smooth ) theory) we could conclude that $\eta$ was not space-like, however, unlike in the smooth theory, this does not imply $\eta$ to be time-like or null (cf. the short note after Definition \[causaldef1\]). Generalized point value characterizations of generalized pseudo-Riemannian metrics and of causality of generalized vector fields {#sec6} ================================================================================================================================ The first goal of this section is to characterize generalized pseudo-Riemannian metrics through evaluation on generalized points. Then we describe causality of generalized vector fields on $X$ by means of causality in $\widetilde{\mathbb R}^n$. The importance of the latter comes from the fact that generalized functions are not uniquely determined by evaluation on standard points (cf. the discussion in subsection \[uniqueness\]). We start by establishing a point-value characterization of generalized pseudo-Riemannian metrics with respect to their index: \[charindexg\] Let $g\in \mathcal G^0_2(X)$ satisfy one (hence all) of the equivalent statements of Theorem \[chartens02\], $j\in\mathbb N_0$. The following are equivalent: 1. \[charindexg1\] $g$ has (constant) index $j$. 2. \[charindexg2\] For each chart $(V_{\alpha},\psi_{\alpha})$ and each $\widetilde x \in (\psi_{\alpha}(V_{\alpha}))_c^{\sim}$, $g_{\alpha}(\widetilde x)$ is a symmetric bilinear form on $\widetilde{\mathbb R}^n$ with index $j$. (\[charindexg1\])$\Rightarrow$(\[charindexg2\]): Let $\widetilde x\in\psi_{\alpha}(V_{\alpha})_c^{\sim}$ be supported in $K\subset\subset \psi_{\alpha}(V_{\alpha})$ and choose a representative $(g_{\varepsilon})_{\varepsilon}$ of $g$ as in Theorem \[chartens02\] (\[chartens023\]) and Definition \[defpseud\]. According to Theorem \[chartens02\] (\[chartens021\]), $g_{\alpha}(\widetilde x):\widetilde{\mathbb R}^n\times\widetilde{\mathbb R}^n\rightarrow\widetilde{\mathbb R}$ is symmetric and non-degenerate. So it merely remains to prove that the index of $g_\alpha(\widetilde x)$ coincides with the index of $g$. Since $\widetilde x$ is compactly supported, we may shrink $V_{\alpha}$ to $U_\alpha$ such that the latter is an open relatively compact subset of $X$ and $\widetilde x\in \psi_\alpha(U_\alpha)$. By Definition \[defpseud\] there exists a symmetric representative $(g_\varepsilon)_\varepsilon$ of $g$ on $U_\alpha$ and an $\varepsilon_0$ such that for all $\varepsilon<\varepsilon_0$, $g_\varepsilon$ is a pseudo-Riemannian metric on $U_\alpha$ with constant index $\nu$. Let $(\widetilde x_\varepsilon)_\varepsilon$ be a representative of $\widetilde x$ lying in $U_\alpha$ for each $\varepsilon<\varepsilon_0$. Let $g_{\alpha,\,ij}^\varepsilon$ be the coordinate expression of $g_\varepsilon$ with respect to the chart $(U_\alpha,\psi_\alpha)$. Then for each $\varepsilon<\varepsilon_0$, $g_{\alpha,\,ij}^\varepsilon(\widetilde x_\varepsilon)$ has precisely $\nu$ negative and $n-\nu$ positive eigenvalues, therefore due to Definition \[indexmatrixdef\], the class $g_{ij}:=[(g_{\alpha,\,ij}^\varepsilon(\widetilde x_\varepsilon))_\varepsilon]\in\mathcal M_n(\widetilde {\mathbb R})$ has index $\nu$. By Definition \[bilformdef\] it follows that the respective bilinear form $g_{\alpha}(\widetilde x)$ induced by $(g_{ij})_{ij}$ with respect to the canonical basis of $\widetilde{\mathbb R}$ has index $\nu$ and we are done.\ To show the converse direction, one may proceed by an indirect proof. Assume the contrary to (\[charindexg1\]), that is, $g$ has non-constant index $\nu$. In view of Definition \[defpseud\] there exists an open, relatively compact chart $(V_\alpha, \psi_\alpha)$, a symmetric representative $(g_\varepsilon)_\varepsilon$ of $g$ on $V_\alpha$ and a zero sequence $\varepsilon_k$ in $I$ such that the sequence $(\nu_k)_k$ of indices $\nu_k$ of $g_{\varepsilon_k}\mid_{V_\alpha}$ has at least two accumulation points, say $\alpha\neq\beta$. Let $(x_\varepsilon)_\varepsilon$ lie in $\psi_\alpha(V_\alpha)$ for each $\varepsilon$. Therefore the number of negative eigenvalues of $(g_{ij})_{ij}:=(g_{\alpha,ij}^\varepsilon(x_\varepsilon))_{ij}$ is not constant for sufficiently small $\varepsilon$, and therefore for $\widetilde x:=[(x_\varepsilon)_\varepsilon]$, the respective bilinear form $g_{\alpha}(\widetilde x)$ induced by $(g_{ij})_{ij}$ with respect to the canonical basis of $\widetilde{\mathbb R}$ has no index and we are done. Theorem \[downhilliseasier\] provides the appropriate machinery to characterize causality of generalized vector fields: \[characterizationcausality\] Let ${\xi}\in\mathcal G^1_0(X)$, $ g\in \mathcal G^0_2(X)$ be a Lorentzian metric. The following are equivalent: 1. \[causalitychar1\] For each chart $(V_{\alpha},\psi_{\alpha})$ and each $\widetilde x \in (\psi_{\alpha}(V_{\alpha}))_c^{\sim}$, ${\xi}_{\alpha}(\widetilde x)\in\widetilde{\mathbb R}^n$ is time-like (resp. space-like, resp. null) with respect to $ g_{\alpha}(\widetilde x)$ (a symmetric bilinear form on $\widetilde{\mathbb R}^n$ of Lorentz signature). 2. \[causalitychar2\] $ g({\xi},\xi)< 0$ (resp. $>0$, resp. $=0$) in $\mathcal G(X)$. (\[causalitychar2\])$\Leftrightarrow$ $\forall\;\widetilde x\in X_c^{\sim}: g(\xi,\xi)(\widetilde x)<0$ (due to Theorem \[downhilliseasier\], (\[invf1extra\])) $\Leftrightarrow$ for each chart $(V_{\alpha},\psi_{\alpha})$ and for all $\widetilde x_c\in \psi_{\alpha}(V_{\alpha})_c^{\sim}$ we have $g_{\alpha}(\widetilde x)(\xi_{\alpha}(\widetilde x),\xi_{\alpha}(\widetilde x))<0$ in $\widetilde{\mathbb R}$ $\Leftrightarrow$ (\[causalitychar1\]). The preceding theorem gives rise to the following definition: \[defcausalityglobal\] A generalized vector field $ \xi\in \mathcal G^1_0(X)$ is called time-like (resp. space-like, resp. null) if it satisfies one of the respective equivalent statements of Theorem \[characterizationcausality\]. Moreover, two time-like vector fields $\xi,\eta$ are said to have the same time orientation, if $\langle \xi,\eta\rangle<0$. Due to the above, this notion is consistent with the point-wise one given in \[causaldef1\]. We conclude this section by harvesting constructions of generalized pseudo-\ Riemannian metrics by means of point-wise results of the preceding section in conjunction with the point-wise characterizations of the global objects of this paper: \[genRiemannmetricconstrglobal\] Let $g$ be a generalized Lorentzian metric and let $\xi,\eta\in \mathcal G ^1_0(X)$ be time-like vector fields with the same time orientation. Then $$h_{ab}:=\xi_{(a}\eta_{b)}-\frac{1}{2}\langle \xi,\eta\rangle g_{ab}$$ is a generalized Riemannian metric. Use Proposition \[metrconstr\] together with Theorem \[characterizationcausality\] and Theorem \[charindexg\]. A final remark on this section is in order. We based our initial considerations in this paper on Theorem \[chartens02\], a characerization of generalized pseudo-Riemannian metrics. Point (\[chartens021\]) motivated us to study bilinear forms $b$ on $\widetilde{\mathbb R}^n$ in section \[sec5\] by introducing the index of $b$. We described successfully free vectors in $\widetilde{\mathbb R}^n$, established elementary (and expected) facts in generalized Lorentz geometry, for instance the Cauchy-Schwarz inequality. It was then quite natural to return to the global objects of Theorem \[chartens02\] we had started with and to characterize them in terms of the machinery we had developed in previous sections. This section shows that the chosen notion in linear algebra on $\widetilde{\mathbb R}^n$ matchs perfectly this setting. Appendix. Further algebraic properties of finite dimensional modules over the ring of generalized numbers {#appendix.-further-algebraic-properties-of-finite-dimensional-modules-over-the-ring-of-generalized-numbers .unnumbered} ========================================================================================================= This section is devoted to a discussion of direct summands of submodules inside $\widetilde{\mathbb R}^n$. The question first involves free submodules of arbitrary dimension. However, we establish a generalization of Theorem \[freechar\] (\[freeloader8\]) not only with respect to the dimension of the submodule; the direct summand we construct is also an orthogonal complement with respect to a given positive definite symmetric bilinear form. Having established this in \[semi\], we subsequently show that $\widetilde{\mathbb R}^n$ is not semisimple, i. e., non-free submodules in our module do not admit direct summands. Direct summands of free submodules {#semi} ---------------------------------- The existence of positive bilinear forms on $\widetilde{\mathbb R}^n$ ensures the existence of direct summands of free submodules of $\widetilde{\mathbb R}^n$: Any free submodule $\mathfrak m$ of $\widetilde{\mathbb R}^n$ has a direct summand. Denote by $\mathfrak m$ the free submodule in question with $\dim \mathfrak m=k$, let $h$ be a positive definite symmetric bilinear form on $\mathfrak m$ and $h_{\mathfrak m}$ its restriction to $\mathfrak m$. Now, due to Lemma \[posprop\] (\[h2\]), $h_{\mathfrak m}$ is a positive definite symmetric bilinear form. In particular, there exists an orthogonal basis $\mathcal B_{\mathfrak m}:=\{e_1,\dots,e_k\}$ of $\mathfrak m$ with respect to $h_{\mathfrak m}$. We further may assume that the latter one is orthonormal. Denote by $P_{\mathfrak m}$ the orthogonal projection on $\mathfrak m$ which due to the orthogonality of $\mathcal B_{\mathfrak m}$ may be written in the form $$P_{\mathfrak m}:\;\widetilde{\mathbb R}^n\rightarrow \mathfrak m,\; v\mapsto \sum_{i=1}^k\langle v,e_i\rangle e_i.$$ Finally, we show ${\mathfrak m}^{\perp}=\ker P_{\mathfrak m}$: $$\begin{aligned} \nonumber {\mathfrak m}^{\perp}&=&\{v\in\widetilde{\mathbb R}^n\mid \forall\; u\in \mathfrak m: h(v,u)=0\}=\\\nonumber &=&\{v\in\widetilde{\mathbb R}^n\mid \forall\; i=1,\dots,k: h(v,e_i)=0\}=\\\nonumber &=&\{v\in \widetilde{\mathbb R}^n\mid P_{\mathfrak m}(v)=0\}=\ker P_{\mathfrak m}.\end{aligned}$$ Where both of the last equalities are due to the definition of $P_{\mathfrak m}$ and the fact that $B_{\mathfrak m}$ is a basis of $\mathfrak m$. As always in modules, ${\mathfrak m}^{\perp}=\ker P_{\mathfrak m}\Leftrightarrow {\mathfrak m}^{\perp}$ is a direct summand and we are done. An alternative end of this proof is provided by Lemma \[posprop\]: Since we have $\mathfrak m+\mathfrak m^{\perp}=\widetilde{\mathbb R}^n$, we only need to show that this sum is a direct one. But Lemma \[posprop\] (\[h1\]) shows that $0 \neq u \in \mathfrak m\cap {\mathfrak m}^{\perp}$ is absurd, since $h$ is positive definite. We thus have also shown (cf. Theorem \[freechar\]): \[orthdecriem\] Let $w\in\widetilde{\mathbb R}^n$ be free and let $h$ be a positive definite symmetric bilinear form. Then $\widetilde{\mathbb R}^n=\langle \{w\}\rangle \oplus w^{\perp}$. We therefore have added a further equivalent property to Theorem \[freechar\]. $\widetilde{\mathbb R}^n$ is not semisimple {#secsemi} ------------------------------------------- In this section we show that $\widetilde{\mathbb R}^n$ is not semisimple. Recall that a module $B$ over a ring $R$ is called simple, if $RA\neq \{0\}$ and if $A$ contains no non-trivial strict submodules. For the convenience of the reader, we recall the following fact on modules (e. g., see [@Hungerford], p. 417): \[charsemisimple\] The following conditions on a nonzero module $A$ over a ring $R$ are equivalent: 1. \[charsemisimple1\] $A$ is the sum of a family of simple submodules. 2. \[charsemisimple2\] $A$ is the direct sum of a family of simple submodules. 3. For every nonzero element a of $A$, $Ra\neq 0$; and every submodule $B$ of $A$ is a direct summand (that is, $A=B\oplus C$ for some submodule $C$. Such a module is called semisimple. However, property (\[charsemisimple1\]) is violated in $\widetilde{\mathbb R}^n$ $(n\geq 1)$: Every submodule $A\neq \{0\}$ in $\widetilde{\mathbb R}^n$ contains a strict submodule. Let $u\in A$, $u\neq 0$. We may write $u$ in terms of the canonical basis $e_i\;(i=1,\dots,n)$, $u=\sum_{i=1}^n \lambda_ie_i$ and without loss of generality we may assume $\lambda_1\neq 0$. Denote a representative of $\lambda_1$ by $(\lambda_1^{\varepsilon})_{\varepsilon}$. $\lambda_1\neq 0$ in particular ensures the existence of a zero sequence $\varepsilon_k \searrow 0$ in $I$ and an $m>0$ such that for all $k\geq 1$, $\vert \lambda_1^{\varepsilon_k}\vert\geq \varepsilon_k^m$. Define $D:=\{\varepsilon_k\mid k\geq 1\}\subset I$, let $\chi_D\in\widetilde{\mathbb R}$ be the characteristic function on $D$. Clearly, $\chi_D u\in A$, furthermore, if the submodule generated by $\chi_D u$ is not a strict submodule of $A$, one may replace $D$ by $\bar D:=\{\varepsilon_{2k}\mid k\geq 1\}$ to achieve one in the same way, which however is a strict submodule of $A$ and we are done. The preceding proposition in conjunction with Theorem \[charsemisimple\] gives rise to the following conclusion: $\widetilde{\mathbb R}^n$ is not semisimple. Our discussion on algebraic properties of the finite dimensional module $\widetilde{\mathbb R}^n$ in this paper lets us draw the following important conclusion. Though there are obvious differences to linear algebra in $\widetilde{\mathbb R}^n$, important facts still hold in the generalized setting. In particular, what is said in this section indicates that properties for linear subspaces of $\mathbb R^n$, have appropriate counterparts for [*free*]{} submodules of $\widetilde{\mathbb R}^n$. Acknowledgement {#acknowledgement .unnumbered} =============== I want to express my gratitude to my supervisors Michael Kunzinger and Roland Steinbauer for the great research environment they have offered me inside the DIANA research group at the University of Vienna during the last 3 years. I am further indebted to Professor Robert Beig for discussions on section \[sec5\], particularly on generalized Energy tensors. [10]{} , [*Some structural properties of the topological ring of [C]{}olombeau’s generalized numbers*]{}, Comm. Algebra, 29 (2001), pp. 2201–2230. , [*Lecture notes on special and general relativity, unpublished*]{}, University of Vienna, Physics Institute, (2004). , [*Generalized hyperbolicity in singular spacetimes*]{}, Class. Quantum Grav., 15 (1998), pp. 975–984. , [*Generalized functions and distributional curvature of cosmic strings*]{}, Class. Quantum Grav., 13 (1996), pp. 2485–2498. , [*New generalized functions and multiplication of distributions*]{}, vol. 84 of North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam, 1984. Notas de Matemática \[Mathematical Notes\], 90. height 2pt depth -1.6pt width 23pt, [*Elementary introduction to new generalized functions*]{}, vol. 113 of North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam, 1985. Notes on Pure Mathematics, 103. , [*Lineare [A]{}lgebra*]{}, vol. 17 of Grundkurs Mathematik, Friedr. Vieweg & Sohn, Braunschweig, fifth ed., 1979. In collaboration with Richard Schimpl. , [*The wave equation on a curved space-time*]{}, Cambridge University Press, Cambridge, 1975. Cambridge Monographs on Mathematical Physics, No. 2. , [ *Geometric theory of generalized functions with applications to general relativity*]{}, vol. 537 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 2001. , [*A global theory of algebras of generalized functions*]{}, Adv. Math., 166 (2002), pp. 50–72. , [*The large scale structure of space-time*]{}, Cambridge University Press, London, 1973. Cambridge Monographs on Mathematical Physics, No. 1. , [*Remarks on the distributional [S]{}chwarzschild geometry*]{}, J. Math. Phys., 43 (2002), pp. 1493–1508. , [*Elliptic regularity and solvability for partial differential equations with [C]{}olombeau coefficients*]{}, Electr. Jour. Diff. Equ.,, (2004), pp. 1–30. , [*Algebra*]{}, Hgolt, Rinehart and Winston, Inc., New York, 1974. , [*Generalized group actions in a global setting*]{}, J. Math. Anal. Appl., 322 (2006), pp. 420–436. , [*Generalized functions valued in a smooth manifold*]{}, Monatsh. Math., 137 (2002), pp. 31–49. height 2pt depth -1.6pt width 23pt, [*Nonsmooth differential geometry and algebras of generalized functions*]{}, J. Math. Anal. Appl., 297 (2004), pp. 456–471. Special issue dedicated to John Horváth. , [ *Generalized flows and singular [ODE]{}s on differentiable manifolds*]{}, Acta Appl. Math., 80 (2004), pp. 221–241. , [*A note on the [P]{}enrose junction conditions*]{}, Classical Quantum Gravity, 16 (1999), pp. 1255–1264. height 2pt depth -1.6pt width 23pt, [*Foundations of a nonlinear distributional geometry*]{}, Acta Appl. Math., 71 (2002), pp. 179–206. height 2pt depth -1.6pt width 23pt, [*Generalized pseudo-[R]{}iemannian geometry*]{}, Trans. Amer. Math. Soc., 354 (2002), pp. 4179–4199 (electronic). , [*Intrinsic characterization of manifold-valued generalized functions*]{}, Proc. London Math. Soc. (3), 87 (2003), pp. 451–470. height 2pt depth -1.6pt width 23pt, [*Generalised connections and curvature*]{}, Math. Proc. Cambridge Philos. Soc., 139 (2005), pp. 497–521. , [*On the characterization of p-adic [C]{}olombeau-[E]{}gorov generalized functions by their point values*]{}, Math. Nachr., 280 (2007), pp. 1297-1301. , [*Multiplication of distributions and applications to partial differential equations*]{}, vol. 259 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, 1992. , [*Characterization of [C]{}olombeau generalized functions by their pointvalues*]{}, Math. Nachr., 203 (1999), pp. 147–157. , [ *Positivity and positive definiteness in generalized function algebras*]{}, J. Math. Anal. Appl., 328 (2007), pp. 1321–1335. , [*Semi-[R]{}iemannian geometry*]{}, vol. 103 of Pure and Applied Mathematics, Academic Press Inc., New York, 1983. With applications to relativity. , [*Sur l’impossibilité de la multiplication des distributions*]{}, C. R. Acad. Sci. Paris, 239 (1954), pp. 847–848. , [*Matrix perturbation theory*]{}, Computer Science and Scientific Computing, Academic Press Inc., Boston, MA, 1990. , [*Generalized hyperbolicity in conical spacetimes*]{}, Class. Quantum Grav., 17 (2000), pp. 1333–1260. [^1]: Work supported by FWF research grants P16742-N04 and Y237-N13

--- author: - | D. E. Wolf[^1], T. Scheffler and J. Schäfer[^2]\ Theoretische Physik, Gerhard-Mercator-Universität Duisburg\ D-47048 Duisburg, Germany title: 'Granular Flow, Collisional Cooling and Charged Grains' --- Abstract {#abstract .unnumbered} ======== The non-Newtonian character of granular flow in a vertical pipe is analyzed. The time evolution of the flow velocity and the velocity fluctuations (or granular temperature) is derived. The steady state velocity has a power law dependence on the pipe width with an exponent between $3/4$ and $3/2$. The flow becomes faster the more efficient the collisional cooling is, provided the density remains low enough. The dependence of collisional cooling on the solid fraction, the restitution coefficient and a possible electric charge of all grains is discussed in detail. Introduction ============ Granular materials like dry sand are classical many particle systems which differ significantly from solids and liquids in their dynamical behavior. They seem to have some properties in common with solids, like the ability of densely packed grains to sustain shear. Other properties, like the ability to flow through a hopper, remind of a fluid. Upon looking more closely, however, it turns out that such similarities are only superficial: Force localization or arching in granular packings and their plastic yield behaviour are distictly different from solids, cluster instabilities and nonlinear internal friction make granular flow very different from that of ordinary, Newtonian fluids. These differences manifest themselves often in extraordinarily strong fluctuations, which may cause accidents, when ignored in technological applications. This is why understanding the statistical physics of granular materials is important and has been very fruitful (see e.g. [@reviews; @Wolf]). In this paper a few selected examples will be presented, which on the one hand document the progress in understanding brought about by applying concepts from statistical physics, and on the other hand point out some areas where important and difficult questions invite future research. One of the simplest geometries displaying the non-Newtonian character of granular flow is an evacuated vertical tube through which the grains fall. The experimental investigation is difficult, however, because the flow depends sensitively on electric charging and humidity [@Raafat]. Nevertheless the ideal uncharged dry granular medium falling in vacuum through a vertical pipe is an important reference case and a natural starting point for computer simulations. Using this idealized example we shall show that the properties of granular flow can be explained, if two essential physical ingredients are understood: The interaction of the granular flow with the container walls and the phenomenon of collisional cooling. This technical term draws an analogy between the disordered relative motion of the agitated grains and the thermal motion of gas molecules. In a granular “gas”, differently from a molecular gas, the relative motion of the grains is reduced in every collision due to the irreversible loss of kinetic energy to the internal degrees of freedom of the grains. This is called collisional cooling. Agitated dry grains are usually electrically charged due to contact electrification. Its effect on the dynamical behavior of granular materials has hardly been studied so far. The reason is certainly not lack of interest, as intentional charging is the basis of several modern applications of granular materials in industrial processes. The reason is that well controlled experiments with electrically charged grains are difficult, as is the theory, because of the long range nature of the Coulomb interaction. One such application is the electrostatic separation of scrap plastics into the raw materials for recycling. A similar technique is used to separate Potassium salts, a raw material for fertilizers, from rock salt. In a “conditioning process” chemically different grains get charged oppositely. Then they fall through a condenser tower, where they are deflected in opposite directions and hence separated. Such a dry separation has the advantage of avoiding the environmental damage, the old fashioned chemical separation method would cause. Another application is powder varnishing. In order to avoid the harmful fumes of ordinary paints, the dry pigment powder is charged monopolarly and attracted to the grounded piece of metal to be varnished. Once covered with the powder, the metal is heated so that the powder melts and forms a continuous film. Recently a rather complete understanding has been reached, how monopolar charging affects collisional cooling [@Scheffler1; @Scheffler2]. However, little is known about the influence of the charges on the grain-wall interaction statistics. The monopolar case is much simpler than the bipolar one: If all grains repell each other, collisional cooling cannot lead to the clustering instability observed for neutral grains [@Goldhirsch; @McNamara]. The case, where the grains carry charges of either sign, is much more difficult, because the clustering instability might even be enhanced. It has not been investigated yet. Why the laws of Hagen, Poisseuille and Bagnold fail for granular pipe flow ========================================================================== Since the flow through a vertical pipe is such a basic example, it has been addressed many times, including some classical work from the last century. Here we remind the reader of some of the most elementary ideas and results concerning pipe flow, and at the same time show, where they fail. The general situation is much more complex, as we are going to point out in the subsequent sections. Force balance requires that the divergence of the stress tensor $\sigma_{ij}$ compensates the weight per unit volume in the steady state of the flowing material: $$\partial_x \sigma_{zx} + \partial_z \sigma_{zz} = - mgn , \label{force_balance}$$ where $m$ denotes the molecular or grain mass, $n$ the number density of molecules or grains and $g$ the gravitational acceleration. The partial derivative in the vertical (the $z$-) direction vanishes because of translational invariance along the pipe. $\partial_x$ denotes the partial derivative in the transversal direction.[^3] In Newtonian or simple liquids the stress tensor is assumed to be proportional to the shear rate $$\sigma_{zx} = \eta \partial_x v_z. \label{newton}$$ The proportionality constant $\eta$ is the viscosity. Inserting this into (\[force\_balance\]) immediately gives the parabolic velocity profile of Hagen-Poisseuille flow, $v_z = v_{\rm max} - (mgn/2\eta) x^2$. No-slip boundary conditions then imply that the flow velocity averaged over the cross section of the pipe scales like ${\bar v}\propto W^2$. According to kinetic theory the viscosity $\eta$ is proportional to the thermal velocity. In lowest order the thermal motion of liquid molecules is independent of the average flow velocity. It is given by the coupling of the liquid to a heat bath. For a granular gas, the thermal velocity must be replaced by a typical relative velocity ${\delta v}$ of the grains. Due to collisional cooling ${\delta v}$ would drop to zero, if there was no flow. This is the most important difference between liquid and granular flow. It shows, that for a granular gas the collision rate between the grains, and hence the viscosity cannot be regarded as independent of the average flow velocity in lowest order. Bagnold [@Bagnold] argued that the typical relative motion should be proportional to the absolute value of the shear rate, $\eta \propto \delta v \propto |\partial_x v_z|$, so that $$\sigma_{zx} \propto |\partial_x v_z| \partial_x v_z.$$ Inserting this into (\[force\_balance\]) leads to the result, that the average flow velocity must scale with the pipe diameter as ${\bar v}\propto W^{3/2}$. However, Bagnold’s argument ignores, that there is a second characteristic velocity in the system, which is $\sqrt{gd}$, where $d$ is the diameter of the grains. It enters due to the nonlinear coupling between the flow velocity and the irregular grain motion, as we are going to point out in the next section. Hence, for granular flow through a vertical pipe, the viscosity is a function of both the average flow velocity and $\sqrt{gd}$. This will change the scaling of ${\bar v}$ with the diameter of the pipe, of course. Very little is known about the flow of dry granular materials at high solid fractions, where the picture of gas-like dynamics, which we employed so far, no longer applies. Hagen studied the discharge from a silo [@Hagen] and postulated, that the flow rate is not limited by plastic deformations inside the packing but by arching at the outlet. He assumes that the only dimensionful relevant parameters for outlets much larger than the grain size are $g$ and the width $W$ of the outlet, for which we use the same notation as for the pipe diameter. Therefore, he concludes, that up to dimensionless prefactors $${\bar v}\propto \sqrt{gW}. \label{Eq:Hagen}$$ He confirmed this experimentally for the silo geometry, where the outlet is smaller than the diameter of the container. It is tempting to expect that this holds also for pipe flow at high solid fractions. However, in our computer simulations we never observed such a behaviour, although we studied volume fractions, which were so high, that the addition of a single particle would block the pipe completely. Without investigating dense granular flow any further in this paper, we just want to point out that Hagen’s dimensional argument seems less plausible for a pipe than for a silo, because important arching now occurs at any place simultaneously along the pipe, and the dynamics of decompaction waves [@Luding] and plastic deformations far from the lower end of the pipe may well depend on the dimensionless ratio $W/d$, for instance. This spoils the argument leading to (\[Eq:Hagen\]), of course. General equations of momentum and energy balance ================================================ A vertical pipe can be viewed essentially as a one-dimensional system, if one averages all dynamical quantities over the cross section. In the following we derive the time evolution of such cross sectional averages of the velocity and velocity fluctuations, assuming they are constant along the pipe. This assumption ignores the spontaneous formation of density waves, which is legitimate if the pipe is sufficiently short. Then a homogeneous state can be maintained. It needs not be stationary, though, and the equations we shall derive describe its temporal evolution. The physical significance of this study is based on the assumption that short sections of a long pipe are locally homogeneous and close to the corresponding steady state. The translational invariance along the pipe implies that the average velocity only has an axial component. Its time evolution is given by the competition of a gain term, which is the gravitational acceleration $g$, and a loss term due to the momentum transfer to the pipe wall. Here we focus on the behavior at low enough densities, where the dynamics are dominated by collisions rather than frictional contacts. Then the momentum transfer to the pipe wall is proportional to the number of grain-wall collisions, $\dot N_{\rm w}$. In each such collision the axial velocity of the colliding particle changes by an average value $\Delta {\bar v}$. All grains are assumed to be equal for simplicity. Hence the average axial velocity changes by $\Delta {\bar v}/N$ in a wall collision. The momentum balance then reads: $$\dot {\bar v} = g - \dot N_{\rm w} \Delta {\bar v}/N . \label{E_v}$$ More subtle is the energy balance which gives rise to an equation for the root mean square fluctuation of the velocity, ${\delta v} = \sqrt{\langle \vec{v}^2 \rangle - \langle \vec{v} \rangle ^2}$. This can be regarded as the typical absolute value of relative velocities. The rates of energy dissipation $\dot E_{\rm diss}$ and of change of kinetic and potential energy, $\dot E_{\rm kin}$ and $\dot E_{\rm pot}$ must add up to zero due to energy conservation, $$0=\dot E_{\rm diss} + \dot E_{\rm kin} + \dot E_{\rm pot}. \label{E_energy_balance}$$ The change in kinetic energy per unit time is $$\dot E_{\rm kin} = N m ({\bar v} \dot {\bar v} + {\delta v} \dot {\delta v} ) , \label{E_E_kin}$$ where $N$ is the total number of particles in the pipe and $m$ their mass. The potential energy (in the absence of Coulomb interactions between the grains) changes at a rate $$\dot E_{\rm pot} = - N m g {\bar v} . \label{E_E_pot}$$ If only the irreversible nature of binary grain collisions is taken into account the energy dissipation rate is proportional to the number of binary collisions per unit time, $\dot N_{\rm g}$, times the loss of kinetic energy in the relative motion of the collision partners, $$\dot E_{\rm diss} = \dot N_{\rm g} \Delta E, \label{E_E_diss}$$ with $$\Delta E = \Delta (m {\delta v}^2/2) = m {\delta v} \Delta({\delta v}). \label{E_DeltaE}$$ Solving (\[E\_energy\_balance\]) for $\dot {\delta v}$ and replacing $\dot {\bar v} - g$ using (\[E\_v\]) gives $$\dot {\delta v} = ({\bar v}/{\delta v}) \dot N_{\rm w} \Delta {\bar v}/N - \dot N_{\rm g} \Delta({\delta v})/N . \label{E_sigma}$$ As for the average velocity, (\[E\_v\]), the typical relative velocity has a gain and a loss term. The gain term has a remarkable symmetry to the loss term in (\[E\_v\]), which is completely general. Only the second term in (\[E\_sigma\]) may be different, if additional modes of energy dissipation like collisions with the walls or friction are included. The gain term in (\[E\_sigma\]) subsumes also the production of granular temperature in the interior of the pipe due to the finite shear rate, which is remarkable, as it expresses everything in terms of physics at the wall. Once the loss terms of the balance equations, (\[E\_v\]) and (\[E\_sigma\]), are known, the time evolution of the average velocity and the velocity fluctuations can be calculated, because the gain terms are given. In this sense, it is sufficient to have a statistical description of collisional cooling (which gives the loss term in (\[E\_sigma\])) and of the momentum transfer of the granular gas to a wall (which gives the loss term in (\[E\_v\])) in order to describe granular flow in a vertical pipe. It turns out, that collisional cooling is easier, because it cannot depend on the average velocity due to the Galilei invariance of the grain-grain-interactions, whereas the momentum transfer to the walls depends on both, ${\bar v}$ and $\delta v$. Collisional cooling =================== We shall now specify $\dot N_{\rm g}$ and $\Delta({\delta v})$. The time between two subsequent collisions of a particle can be estimated by the mean free path, $\lambda$, divided by a typical relative velocity, ${\delta v}$. Hence the number of binary collisions per unit time is proportional to $$\dot N_{\rm g} \propto N {\delta v}/ \lambda . \label{E_coll.rate0}$$ Here we assumed that the flow is sufficiently homogeneous, that the local variations of $\lambda$ and ${\delta v}$ are unimportant. In each collision the relative normal velocity gets reduced by a factor, the restitution coefficient $e_{\rm n}<1$. For simplicity we assume that the restitution coefficient is a constant. Correspondingly a fraction of the kinetic energy of relative motion is dissipated in each collision. $$\Delta E = (1-e_{\rm n}^2){m\over2}{\delta v}^2 \label{E_restitution}$$ with the grain mass $m$. According to (\[E\_DeltaE\]) $\Delta({\delta v}) = (1-e_{\rm n}^2) {\delta v}/2$. Putting this together, the dissipation rate (\[E\_E\_diss\]) is [@Haff] $$\dot E_{\rm diss} = k_{\rm g} N {m\over d} {\delta v}^3. \label{E_E_diss1}$$ The dimensionless proportionality constant $k_{\rm g}$ contains the dependence on the solid fraction $\nu \propto d/\lambda$ and the restitution coefficient $e_{\rm n}$ and can be calculated analytically, if one assumes that the probability distribution of the particles is Gaussian [@Lun]. From these considerations one can draw a very general conclusion for the steady state values of ${\bar v}$ and ${\delta v}$. In the steady state the kinetic energy is constant, so that (\[E\_energy\_balance\]) together with (\[E\_E\_pot\]) and (\[E\_E\_diss1\]) implies $${{{\bar v}_{\rm s}}\over {{\delta v}_{\rm s}^3}} = {{k_{\rm g}}\over {gd}} %\approx {\nu(1-e_{\rm n}^2)\over gd}. \label{scaling1}$$ Whenever the dissipation is dominated by irreversible binary collisions and the flow is sufficiently homogeneous, the steady flow velocity in a vertical pipe should be proportional to the velocity fluctuation to the power $3/2$. The proportionality constant does not depend on the width of the pipe. We tested this relation by two dimensional event driven molecular dynamics simulations [@Wolf]. The agreement is surprisingly good, given the simple arguments above, even quantitatively. However, it turns out that the proportionality constant in (\[scaling1\]) has a weak dependence on the width of the pipe, which can be traced back to deviations of the velocity distribution from an isotropic Gaussian: The vertical velocity component has a skewed distribution with enhanced tail towards zero velocity [@Schaefer]. Interaction of the granular flow with the wall ============================================== The collision rate $\dot N_{\rm w}$ with the walls of the vertical pipe can be determined by noting that the number of particles colliding with a unit area of the wall per unit time for low density $n$ is given by $|v_{\perp}| n$. As the typical velocity perpendicular to the pipe wall, $|v_{\perp}|$ is proportional to $\delta v$, one obtains $$\dot N_{\rm w} \propto N {\delta v}/W.$$ This is the place where the pipe width $W$ enters into the flow dynamics. To specify, by how much the vertical velocity of a grain changes on average, when it collides with the wall, is much more difficult, as it depends on the local geometry of the wall. In our simulations the wall consisted of a dense array of circular particles of equal size. When a grain is reflected from such a wall particle, a fraction of the vertical component of its velocity will be reversed. Instead of averaging this over all collision geometries, we give some general arguments narrowing down the possible functional form of $\Delta {\bar v}$. If we assume that the velocity distribution is Gaussian, all moments of any velocity component must be functions of ${\bar v}$ and ${\delta v}$. This must be true for $\Delta {\bar v}$, as well. For dimensional reasons it must be of the form $$\Delta {\bar v} = {\bar v} f\left({{\delta v}\over {\bar v}}\right) \label{E_DeltaV}$$ with a dimensionless function $f$. The physical interpretation of this is the following: The loss term in the momentum balance can be viewed as an effective wall friction. As long as the granular flow in the vertical pipe approaches a steady state, the friction force must depend on the velocity ${\bar v}$. The ratio ${\delta v}/{\bar v}$ can be viewed as a characteristic impact angle, so that the function $f$ contains the information about the average local collision geometry at the wall. In principle all dimensionless parameters characterizing the system may enter the funcion $f$, that is, apart from $\nu$ also the restitution coefficient $e_{\rm n}$ and the ratios $W/d$ and $gd/{\bar v}^2$. However, it is hard to imagine, that the width $W$ of the pipe or the gravitational acceleration $g$ influences the local collision geometry. Therefore we shall assume that $f$ does not depend on $W/d$ or $gd/{\bar v}^2$. On the other hand, it is plausible, that the restitution coefficient enters $f$. It will influence the spatial distribution of particles and also accounts for the correlation of the velocities, if some particle is scattered back and forth between the wall and neighboring particles inside the pipe, and hence hits the wall twice or more times without a real randomization of its velocity. Due to positional correlations among the particles, $f$ should also depend on the solid fraction $\nu$: One can easily imagine, that the average collision geometry is different in dense and in dilute systems. Lacking a more precise understanding of the function $f$ we make a simple power law ansatz for it and write the loss term of (\[E\_v\]) as $$\begin{aligned} {{\dot N_{\rm w}}\over N} \Delta {\bar v} &=& {1\over W}\, {\delta v} {\bar v} \, k_{\rm w}\left({{\delta v}\over {\bar v}}\right)^{\beta}\nonumber \\ &=& k_{\rm w}\, W^{-1} {\delta v}^{1+\beta} {\bar v}^{1-\beta}.\end{aligned}$$ The dimensionless parameters $k_{\rm w}$ and $\beta$ will be functions of $\nu$ and $e_{\rm n}$. Time evolution and steady state =============================== With these assumptions, the equations of motion (\[E\_v\]) and (\[E\_sigma\]) for granular flow through a vertical pipe become $$\begin{aligned} \label{dot_v} \dot {\bar v} &=& g - k_{\rm w}\, W^{-1} {\delta v}^{1+\beta} {\bar v}^{1-\beta},\\ \label{dot_sigma} \dot {\delta v} &=& k_{\rm w}\, W^{-1} {\delta v}^{\beta} {\bar v}^{2-\beta} - k_{\rm g}\, d^{-1} {\delta v}^2.\end{aligned}$$ As the time evolution should not be singular for ${\bar v}=0$ or ${\delta v} = 0$, the values of $\beta$ are restricted to the interval $$0\leq \beta \leq 1 . \label{E_Einschraenkung}$$ The meaning of the exponent $\beta$ becomes clear, if we calculate the steady state velocity from (\[dot\_v\]) and (\[scaling1\]). The result is $${\bar v}_{\rm s} = \sqrt{gd}\, k_{\rm g}^{\gamma -1/2} k_{\rm w}^{-\gamma}\, \left({W\over d}\right)^{\gamma}. \label{v_steady}$$ The exponent $\gamma$, which determines the dependence of the average flow velocity on the pipe diameter, is related to $\beta$ by $$\gamma = {3\over 2(2-\beta)}.$$ Due to (\[E\_Einschraenkung\]) we predict that in granular pipe flow $$3/4 \leq \gamma \leq 3/2,$$ as long as the flow is sufficiently homogeneous and the main dissipation mechanism are binary collisions. Note, that the exponent is always smaller than 2, which would be its value for Hagen-Poisseuille flow of a Newtonian fluid. $\gamma=3/2$ is the prediction of Bagnold’s theory, but in our simulations we found also values as small as 1, depending on the values of the solid fraction and the restitution coefficient [@Schaefer]. The stationary value ${\delta v}_{\rm s}$ directly follows from (\[v\_steady\]) and (\[scaling1\]). One obtains the same formula as (\[v\_steady\]) with $\gamma$ replaced by $\gamma/3$. Collisional cooling for monopolar charged grains ================================================ In this section we summarize our recent results [@Scheffler1; @Scheffler2], how the dissipation rate (\[E\_E\_diss1\]) is changed if all grains carry the same electrical charge $q$ (besides having the same mass $m$, radius $r$ and restitution coefficient $e_{\rm n}$). For simplicity we assume that the charges are located in the middle of the insulating particles. The results are valid for grains in a three dimensional space, $D=3$. Whereas the hard sphere gas has no characteristic energy scale, the Coulomb repulsion introduces such a scale, $$E_{\rm q} = q^2/d.$$ It is the energy barrier that two collision partners have to overcome, when approaching each other from infinity. It has to be compared to the typical kinetic energy stored in the relative motion of the particles, which by analogy with molecular gases is usually expressed in terms of the “granular temperature” $$T = \delta v^2/D. \label{eq:T}$$ If $E_{\rm q} \ll m\,T$ one expects that the charges have negligible effect on the dissipation rate. Using (\[eq:T\]) and the expression $$\nu = \frac{\pi}{6} n d^3 \quad {\rm with}\quad n=N/V$$ for the three dimensional solid fraction, the dissipation rate (\[E\_E\_diss1\]) can be written in the form $$\dot E_{\rm diss}/V = k \, n^2 d^2 m T^{3/2} \label{eq:E_diss2}$$ with the dimensionless prefactor $$k = k_{\rm g} \pi \sqrt{3}/2\nu. \label{eq:k}$$ The advantage of writing it this way is that the leading $n$- or $\nu$-dependence is explicitely given: In the dilute limit $\nu \rightarrow 0$ the dissipation rate should be proportional to $n^2$, i.e. to the probability that two particles meet in an ideal gas. Since the remaining factors in (\[eq:E\_diss2\]) are uniquely determined by the dimension of the dissipation rate, this equation must hold for charged particles as well. However, in this case the prefactor $k$ will not only depend on $e_{\rm n}$ and $\nu$, but also on the dimensionless energy ratio $E_{\rm q}/mT$. We found [@Scheffler1; @Scheffler2] that the following factorization holds $$k = k_0 (e_{\rm n}) g_{\rm chs}(\nu, E_{\rm q}/m T),$$ where $$k_0 = 2\sqrt{\pi}(1-e_{\rm n}^2) \label{eq:k0}$$ is the value of $k$ for $\nu=E_{\rm q}/mT=0$. $g_{\rm chs}$ denotes the radial distribution function for charged hard spheres (chs) at contact, normalized by the one for the uncharged ideal gas. For $\nu < 0.2$ and $E_{\rm q}/mT < 8$ our computer simulations show that $$g_{\rm chs}\left(\nu,\frac{E_{\rm q}}{m T}\right) \approx g_{\rm hs}(\nu) \exp\left(-\frac{E_{\rm q}}{m T}f(\nu)\right) \label{eq:g}$$ is a very good approximation. Here, $$g_{\rm hs} = \frac{2-\nu}{2(1-\nu)^3} \geq 1$$ is the well-known Enskog correction for the radial distribution function of (uncharged) hard spheres (hs) [@CarStar]. This factor describes that the probability that two particles collide is enhanced due to the excluded volume of all the remaining particles. The second, Boltzmann-like factor describes that the Coulomb repulsion suppresses collisions. The effective energy barrier $E_{\rm q} f(\nu)$ decreases with increasing solid fraction, because two particles which are about to collide not only repel each other but are also pushed together by being repelled from all the other charged particles in the system. A two parameter fit gives $$f(\nu) \approx 1 - c_0 \, \nu^{1/3} + c_1 \, \nu^{2/3}$$ with $$c_0 = 2.40 \pm 0.15, \quad {\rm and} \quad c_1 = 1.44 \pm 0.15 . \label{eq:fit}$$ Very general arguments [@Scheffler1] lead to the prediction that $c_1 = (c_0/2)^2$, which is confirmed by (\[eq:fit\]). We expect deviations from (\[eq:g\]) for larger $\nu$ and $E_{\rm q}/mT$, because the uncharged hard sphere system has a fluid-solid transition close to $\nu \approx 0.5$, and the charged system may become a Wigner crystal for any solid fraction, provided the temperature gets low enough. Conclusion ========== We presented four main results: The steady state velocity of granular flow in a vertical pipe should have a power law dependence on the diameter $W$ of the pipe with an exponent $\gamma$ ranging between 3/4 and 3/2, depending on the solid fraction and the restitution coefficient of the grains. This result was derived ignoring possible electric charges of the grains and assuming that the flow is sufficiently homogeneous and the main dissipation mechanism are binary collisions. This illustrates the genuinely non-Newtonian character of granular flow. Second, the dependence of the steady state velocity on the solid fraction $\nu$, the restitution coefficient $e_{\rm n}$ and – in the case of monopolar charging – the ratio between Coulomb barrier and kinetic energy, $E_{\rm q}/mT$, is contained in the factor $k_{\rm g}^{\gamma -1/2} k_{\rm w}^{-\gamma}$ in (\[v\_steady\]). In the dilute limit $\nu \rightarrow 0$ as well as in the limit of nearly elastic particles $e_{\rm n} \rightarrow 1$ the coefficient $k_{\rm w}$, which describes how sensitive the momentum transfer to the wall depends on the local collision geometry, should remain finite, whereas $k_{\rm g}$ vanishes like $\nu (1-e_{\rm n}^2)$ according to (\[eq:k\]), (\[eq:k0\]). As $\gamma-1/2 > 0$, this implies that the flow through a vertical pipe becomes faster the higher the solid fraction and the less elastic the collisions between the grains are (in the limit of low density and nearly elastic collisions). The physical reason for this is that in denser and more dissipative systems collisional cooling is more efficient, reducing the collisions with the walls and hence their braking effect. This remarkable behaviour has been confirmed in computer simulations [@Schaefer]. Third, monopolar charging leads to a Boltzmann-like factor in $k_{\rm g}$ or the dissipation rate, respectively, which means that for low granular temperature the dissipation rate becomes exponentially weak. The higher the density the less pronounced is this effect, because the effective Coulomb barrier $E_{\rm q} f(\nu)$ hindering the collisions becomes weaker. Finally, we derived the evolution equations for the flow velocity and the velocity fluctuation for granular flow through a vertical pipe, (\[dot\_v\]) and (\[dot\_sigma\]). These equations apply to the situation of homogeneous flow, which can only be realized in computer simulations of a sufficiently short pipe with periodic boundary conditions. In order to generalize these equations for flow that is inhomogeneous along the pipe one should replace the time derivatives by $\partial_t + \bar{v}(z,t) \partial_z$. In addition a third equation, the continuity equation, is needed to describe the time evolution of the density of grains along the pipe. Such equations have been proposed previously [@Lee; @Valance; @Riethmuller] in order to study the kinetic waves spontaneously forming in granular pipe flow. Our equations are different. Acknowledgements {#acknowledgements .unnumbered} ================ We thank the Deutsche Forschungsgemeinschaft for supporting this research through grant no. Wo 577/1-3. The computer simulations supporting the theory presented in this paper were partly done at the John von Neumann-Institut für Computing (NIC) in Jülich. [99]{} H. J. Herrmann, J.-P. Hovi and S. Luding (eds.): [*Physics of Dry Granular Media*]{} (Kluwer, Dordrecht 1998); R. P. Behringer and J. T. Jenkins (eds.): [*Powders and Grains ’97*]{} (A. A. Balkema, Rotterdam 1997); H. M. Jaeger, S. R. Nagel and R. P. Behringer, Rev. Mod. Phys. [ **68**]{}, 1259 (1996) D. E. Wolf, in: [*Computational Physics: Selected Methods, Simple Exercises, Serious Applications*]{}, eds. K. H. Hoffmann and M. Schreiber (Springer, Berlin 1996) pp. 64 – 95 T. Raafat, J. P. Hulin and H. J. Herrmann, Phys. Rev. E [**53**]{}, 4345 (1996) T. Scheffler and D. E. Wolf, submitted to Phys. Rev. E; T. Scheffler, J. Werth and D. E. Wolf, in: Structure and Dynamics of Heterogeneous Systems, eds. P. Entel and D. E. Wolf (World Scientific, Singapore, 1999) I. Goldhirsch and G. Zanetti, Phys. Rev. Lett. [**72**]{}, 1619 (1993) S. McNamara, W. R. Young, Phys. Fluids A [**4**]{}, 496 (1992); Phys. Rev. E [**53**]{}, 5089 (1996) R. A. Bagnold, Proc. Roy. Soc. A [**225**]{}, 49 (1954) G. Hagen, Monatsberichte Preu[ß]{}. Akad. d. Wiss. (Berlin, 1852), pp. 35 – 42 J. Duran, T. Mazozi, S. Luding. E. Clément and J. Raichenbach, Phys. Rev. E [**53**]{}, 1923 (1996) P. K. Haff, J. Fluid Mech. [**134**]{}, 401 (1983) C. K. L. Lun, S. B. Savage, D. J. Jeffrey and N. Chepurniy, J. Fluid Mech. [**140**]{}, 223 (1984) T. Scheffler, J. Schaefer, D. E. Wolf, to be published N. F. Carnahan and K. E. Starling, J. Chem .Phys. [**51**]{}, 635 (1969) J. Lee and M. Leibig, J. Phys. I France [**4**]{}, 507 (1994) A. Valance and T. Le Pennec, Eur. Phys. J. B [**5**]{}, 223 (1998) T. Riethmüller, L. Schimansky-Geier, D. Rosenkranz and T. Pöschel, J. Stat. Phys. [**86**]{}, 421 (1997) [^1]: d.wolf@uni-duisburg.de [^2]: Present address: Procter & Gamble European Services GmbH, D-65823 Schwalbach am Taunus, Germany [^3]: For the sake of transparency the equations are given for the two dimensional case in this section.

--- abstract: 'Passive shims are often used to reduce the size and cost of room-temperature magnetic dipoles. In this paper we revisit an analytic approach to the problem of optimum shim design, and we extend it by taking into consideration the effect of magnetic saturation. We derive an abacus curve to determine optimum shim dimensions as a function of the desired dipole nominal field. We show that, for nominal fields below 1.2T, a pole with such shims can be made at least one half gap height narrower than a pole without. We discuss the range of validity of this approach and verify its predictions using 2 and 3-dimensional finite-element calculations.' address: - 'TRIUMF, 4004 Wesbrook Mall, Vancouver, BC, Canada V6T 2A3' - 'University of British Columbia Okanagan, 3333 University Way, Kelowna, BC, Canada V1V 1V7' author: - 'Thomas Planche[^1], Paul M. Jung, Suresh Saminathan, Rick Baartman' - 'Matthew J. Basso[^2]' bibliography: - 'Mybib.bib' - 'AllDN.bib' title: Conformal Mapping Approach to Dipole Shim Design --- Magnet design ,shim ,conformal mapping Introduction ============ In the context of accelerator magnet design, a shim is a device used to provide fine adjustments of the magnetic field profile. There are two types: active shims, which are extra coils mounted on the poles; and passive shims, which are localized modifications of the pole shape. Active shims are often used in NMR and MRI instruments [@anderson1961electrical; @hoult1994accurate; @terada2011development] and superconducting particle accelerator magnets [@ferracin2014magnet]. Passive shims are often used in room temperature magnets for particle accelerators [@rose1938magnetic; @danby1999precision], and are generally designed based on empirical rules and iterative processes involving 2 and/or 3-dimensional finite element calculations [@russenschuck2011shim]. To facilitate this design process it is useful to start from a ‘good guess’. Such initial guess can be chosen among the set of optimum shim dimensions analytically derived by Rose [@rose1938magnetic] using conformal maps. The issue is that the solution derived by Rose is not unique, leaving some room for arbitrariness in the choice of the initial guess. In this paper, after a brief review of Rose’s work, we attempt to complete his work by taking into consideration the effect of magnetic saturation. We show that, for a given nominal field, the effect of magnetic saturation reduces the set of optimum “Rose” shims to a narrow range of optimum solutions. We provide an abacus curve to determine optimum shim dimensions as a function of the desired dipole nominal field, and we test the validity of our approach using 2 and 3-dimensional finite-element calculations. Magnetic Dipole Shim {#sec:2} ==================== The relative permeability of ferromagnetic materials, which is much larger than 1 at low excitation, converges asymptotically to 1 at high excitation: this is what is referred to as “magnetic saturation”. As a starting point we make the assumption that the pole surface is not saturated, i.e. the pole surface is equipotential. We will discuss the range of validity of this assumption in \[sec:saturation\]. We also assume that we work with dipoles which are long compared to their gap height. Under these assumptions, our problem reduces to solving Laplace’s equation in 2 dimensions. We also assume that we work with dipoles that are wide compared to their gap height, which allows us to consider only one pole edge at a time. We will show in \[sec:narrowpole\] that in fact this approximation remains valid for pole widths down to about only 1 gap height. The last assumption we make is to neglect the direct contribution from the coil to the magnetic field distribution. Note that the direct contribution from the coil could be taken into account following Ref. [@walstrom2012dipole]. But for the sake of simplicity, and as we concern ourselves with optimizing field flatness well inside the magnet gap, we choose to neglect this effect. Conformal Mapping ----------------- Conformal mapping can be used to find analytical 2-dimensional solutions to Laplace’s equation. The technique depends on constructing a transformation which preserves angles locally, from a geometry for which a solution is known, to the geometry of interest. Detailed introductions to the technique can be found in several text books (see for instance [@weber1950electromagnetic]). All the geometries presented in this paper are solved starting from the solution of Laplace’s equation for a flat condenser (see \[fig:flatcondenser-equipot\]): $$\label{eq:pot} \phi(u,v)=\frac{\Delta V }{\pi }\arctan\left(\frac{u}{v}\right) +V_0\,,$$ where $u$ and $v$ are real. One can verify that \[eq:pot\] satisfies $\nabla^2 \phi=0$, and that the boundary condition: $$\begin{aligned} \phi(u,v=0) & = & \left\{ \begin{array}{lc} V_0+\frac{\Delta V }{2 }, & u>0,\\[0.5em] V_0-\frac{\Delta V }{2 }, & u<0,\\[0.5em] V_0, & u=0. \end{array}\right. \end{aligned}$$ is satisfied. ![Schematic representation of a flat condenser. The red and blue lines represent constant potential boundaries; they extend to $u=-\infty$, and $u=+\infty$, respectively. A few more equipotential are shown as thin grey lines.[]{data-label="fig:flatcondenser-equipot"}](flatcondenser-equipot){width="61.00000%"} In the electro-static case $\Delta V$ is the difference of potential between the plates of the condenser. In the magnetic case $\Delta V=\mu_0 NI$, with $NI$ the amount of current flowing perpendicularly to the ($u,v$) plane along (0,0). Let $x$ and $y$ be the Cartesian coordinates of the geometry we are trying to solve for. If we construct two complex variables: $t=u+iv$ and $z=x+iy$, the relation between them is given by the Schwarz-Christoffel formula: $$\label{eq:Schwarz} \frac{{\rm d}z(t)}{{\rm d}t}=C\prod_n(t-t_n)^{\alpha_n/\pi-1}\,,$$ where $\alpha_n$ is the internal angle of the $n^{{\rm th}}$ corner of the studied geometry, and $t_n$ is the value of $t$ associated with this corner in the in complex t-plane. The value of the constant $C$ is determined according to the specific scale of the problem. The magnetic field distribution is obtained from: $$\label{eq:field} H(t)=H_x-iH_y=-\frac{{\rm d}\phi}{{\rm d}z}=-\frac{{\rm d}\phi}{{\rm d}t}\left(\frac{{\rm d}z}{{\rm d}t}\right)^{-1}\,.$$ This way, as noted by Rose [@rose1938magnetic], the field distribution on the midplane can be obtained as a function of $t$ without analytic integration of \[eq:Schwarz\] for complex values of $t$ and $z$. Rose’s Shim {#sec:Rose} ----------- In this section we revisit Rose’s work on rectangular shim design [@rose1938magnetic]. Unlike Rose, we choose to take advantage of the up-down symmetry by setting a fixed potential along the dipole mid-plane, see \[fig:Rose\]. This geometry possesses four corners (see \[tab:Rose\]) leading to: $$\label{eq:zRose} \frac{{\rm d}z}{{\rm d}t}=C\frac{ \sqrt{t-1} \sqrt{t-t_1}}{t \sqrt{t-t_2}}\,.$$ $n$ $t_n$ $\alpha_n$ $z$ ----- ------- ------------------ ------------ 0 1 $\frac{3\pi}{2}$ $i(1-b)$ 1 $t_1$ $\frac{3\pi}{2}$ $a+i(1-b)$ 2 $t_2$ $\frac{\pi}{2}$ $a+i$ 3 0 0 $+\infty$ : Parameters of the four corners of the Rose shim geometry.[]{data-label="tab:Rose"} ![Dipole edge with a rectangular shim. The blue line is the equipotential that goes along the pole surface; the red line is the equipoltential that goes along the magnet mid-plane. A few more equipotentials, obtained from numerical integration of \[eq:zRose\] along the equipotential lines of \[fig:flatcondenser-equipot\], are shown as thin grey lines.[]{data-label="fig:Rose"}](Rose){width="61.00000%"} We set the values of $\Delta V$ and $C$ such that both the half-gap height and the magnitude of the field at $t=0$ are equal to 1, leading to: $\Delta V = -1$, $C =\frac{i \sqrt{t_2}}{\pi \sqrt{t_1}}$, and $$\label{eq:RoseH} H(t)=\frac{\sqrt{t_1 (t-t_2)}}{\sqrt{t-1} \sqrt{t_2 (t-t_1)}}\,.$$ At this point the geometry possesses two degrees of freedom: the value of $t_1$ and $t_2$ which determine the value of $a$ and $b$ through: $$\label{eq:ab} a=\int_1^{t_1} \frac{{\rm d}z}{{\rm d}t}{\rm d}t\,, \quad b=\int_{t_1}^{t_2} \frac{{\rm d}z}{{\rm d}t}{\rm d}t\,.$$ The problem consists in finding the particular values of $a$ and $b$ that lead to optimal field flatness inside the magnet. To solve this problem Rose [@rose1938magnetic] expands \[eq:RoseH\] around $t=0$: $$\label{eq:Hexp} i\,H(t)=1+\frac{t}{2} \left(\frac{1}{t_1}-\frac{1}{t_2}+1\right)+O\left(t^2\right)\,.$$ Expanding \[eq:zRose\] and solving this differential equation for $t$ close to 0 leads to: $$\label{eq:tapprox} t(z)\approx \left(i\sin(\pi y)-\cos(\pi y)\right)e^{-\pi x}\,.$$ Substituting in \[eq:Hexp\] leads to an approximate expression of the vertical field along the magnet mid-plane: $$\label{eq:Hyexp} H_y(x)\approx 1-\frac{e^{-\pi x}}{2} \left(\frac{1}{t_1}-\frac{1}{t_2}+1\right)+O\left(e^{-2\pi x}\right)\,.$$ To make the field “optimally” homogeneous Rose chooses the value of $t_2$ such that the first order term in $e^{-\pi x}$ vanishes: $$\label{eq:t2} t_2 = \frac{t_1}{1 + t_1}\,.$$ With this additional constraint the geometry has only one degree of freedom left. This last degree of freedom cannot be used to cancel the next order term as \[eq:Hyexp\] now reads: $$H_y(x)\approx 1-\frac{e^{-2\pi x}}{2 t_1} +O\left(e^{-3\pi x}\right)\,,$$ and $0<t_1<1$. The contribution from the term in $e^{-2\pi x}$ is minimized for shims that are relatively narrow ($a$ small) and tall ($b$ large), see \[fig:shimeff\]. The magnetic field distribution along the magnet mid-plane is obtained as a function of the (real) negative parameter $u$ : $$\label{eq:Hy(x)} \left\{\begin{array}{rl} H_y(u)&=\Im\{H(u)\}=-\sqrt{\frac{u^2}{(u-1) (t_1-u)}+1}\\ x(u)&=x_0+\int_{-1}^u \frac{{\rm d}z}{{\rm d}t} {\rm d}t\\ \end{array}\right. \,,$$ where: $$\label{eq:x0} x_0=z(t=-1)=\Re\left\{\int_{1}^{-1} \frac{{\rm d}z}{{\rm d}t} {\rm d}t\right\} \,.$$ ![Field distribution along the magnet mid-plane for Rose shims of various heights $b$. Lines are calculated using \[eq:Hy(x)\]; round dots are obtained from [Opera-2d]{} calculations for the same geometry, assuming linear steel properties.[]{data-label="fig:shimeff"}](shim-2d){width="70.00000%"} Mid-plane field profile obtained from \[eq:Hy(x)\] and from a 2D finite element calculation using [Opera-2d]{} [@opera2010opera] show excellent agreement, see \[fig:shimeff\]. [Opera-2d]{} calculations are done here assuming linear steel properties: discussion of the effects of saturation is postponed to \[sec:saturation\]. To provide a rule of thumb for how effective Rose shims are, let us consider a required relative field flatness of a few $10^{-3}$ or better. With this assumption one can see in \[fig:shimeff\] that a shim height of 0.03 half gap enables a reduction of the pole width by about 1 half gap. A taller shims can lead to a further reduction of the pole width by another 0.5 half gap. As illustrated in \[fig:optimum\], deviation from the optimum produces a field distribution along the mid-plane which: (1) is closer to the no-shim case for cases with ($a$,$b$) below the optimum curve, and (2) overshoots for cases with ($a$,$b$) above the optimum curve. ![Rose shim ($a=b\approx12\%$) compared to two square shims with non-optimal dimensions.[]{data-label="fig:optimum"}](meaningOfOptimum){width="70.00000%"} Shims on Narrow Poles {#sec:narrowpole} --------------------- The pole geometry used so far, which includes only one edge, is anticipated to be accurate only if the pole is “wide enough”. We show in \[fig:narrow\] that this model remains accurate for pole width down to only about 1 gap height, provided that the contribution from the two edge are superimposed using: $$\label{eq:narrow} \mathcal{H}_y(x)=H_y(x+\frac{w}{2}) + H_y(-x+\frac{w}{2}) - B_0\,,$$ where $w$ is the pole width, $H_y(x)$ is obtained from \[eq:Hy(x)\], and $B_0$ is the nominal field (equal to 1 in the units used here). ![Field distribution along the mid-plane of finite-width poles with Rose shims on both ends ($a=b=0.121$ in unit of half gap). Solid lines are obtained from \[eq:narrow\]; points are obtained from [Opera-2d]{} calculations. []{data-label="fig:narrow"}](narrowPole){width="70.00000%"} Magnetic Saturation {#sec:saturation} ------------------- As the magnetic field in the shim approaches saturation, the surface of the material is no longer equipotential rendering the solution obtained from conformal mapping inaccurate and weakening the effect of the shim. When designing magnets, one must choose the height of the shim such that it does not saturate. To guide this choice we will determine the relation between the shim height and the field level inside the shim. ![ Magnitude of the magnetic field along the surface of two types of shim. Black dotted line: a Rose shim with $b=a=0.121$; Red solid line: a rounded shim with $b=0.121$ and $a=0.274$. The magnitude of the field is given in units of the magnet nominal field.[]{data-label="fig:RoseSurface"}](RoseSurface){width="70.00000%"} Conservation of the magnetic flux imposes continuity of the magnitude of the magnetic field across the material surface. This allows us to use the magnitude of the field along the shim surface as a measure of the field level inside the shim. The field along the surface is obtained from  \[eq:RoseH\] for $t\in\mathbb{R}^+$. In the case of a rectangular shim the magnitude of the field diverges around its two prominent corners (see \[fig:RoseSurface\]) making it hard to discuss field levels inside the steel. To eliminate this singularity we choose to work with a “rounded” version of the Rose shim presented in \[fig:isoRound,tab:isoRound\]. This geometry possesses three corners leading to: $$\label{eq:zisoRound} \frac{{\rm d}z}{{\rm d}t}=\frac{1}{t}C (t-1)^{\frac{\alpha }{2 \pi }} \left((t-1)^{1-\frac{\alpha }{\pi }}+\lambda (t-t_2)^{1-\frac{\alpha }{\pi }}\right)(t-t_2)^{\frac{\alpha }{2 \pi }-\frac{1}{2}}\,,$$ where the term $(t-1)^{1-\frac{\alpha }{\pi }}+\lambda (t-t_2)^{1-\frac{\alpha }{\pi }}$ is used to produce a corner rounded with a parabola (see Ref.[@weber1950electromagnetic]). Like in \[sec:Rose\] we choose the scale of our problem such that both the half-gap height and the magnitude of the field at $t=0$ are equal to 1, leading to: $$\label{eq:isoRoundH} H(t)=\frac{(t-1)^{\frac{\alpha }{2 \pi }} \left(\frac{t}{t_2}-1\right)^{\frac{\alpha +\pi }{2 \pi }} \left(t_2^{\frac{\alpha }{\pi }}+\lambda t_2\right)}{\lambda (t-1)^{\frac{\alpha }{\pi }} (t-t_2)+(t-1) (t-t_2)^{\frac{\alpha }{\pi }}}.$$ Once again expanding \[eq:isoRoundH\] around $t=0$ and cancelling the lowest order dependence in $t$ leads to: $$\lambda=-\frac{t_2^{\frac{\alpha }{\pi }-1} (\alpha -\alpha t_2+2 \pi t_2-\pi )}{-\alpha +\alpha t_2+\pi }\,.$$ Finally, the value of $t_2$ is set such that the shim starts and ends at the same $y$ value by numerically solving: $$\Im\left\{\int_1^{t_2} \frac{{\rm d}z}{{\rm d}t}{\rm d}t\right\}=0\,.$$ The only free parameter left is the value of $\alpha$. The shim width $a$ and height $b$ are obtained numerically from: $$\label{eq:ab2} a=\int_{1}^{t_2} \frac{{\rm d}z}{{\rm d}t}{\rm d}t\,,\quad b=\Im\left\{\int_{1}^{t_{\rm tip}} \frac{{\rm d}z}{{\rm d}t}{\rm d}t\right\}\,,$$ where $t_{\rm tip}$ is the value of $t$ that maximizes $\Im\left\{\int_{1}^{t} \frac{{\rm d}z}{{\rm d}t}{\rm d}t\right\}$ for $t_2<t<1$. ![Proposed rounded shim geometry. The effective width $a$ of the shim is controlled by the value of the angle $\alpha$; the height $b$ depends on $\alpha$ and $\lambda$, see \[eq:zisoRound\]. This plot corresponds to $\alpha=\frac{\pi}{3}$.[]{data-label="fig:isoRound"}](isoRound){width="61.00000%"} $n$ $t_n$ $\alpha_n$ $z$ ----- ------- ---------------------------------- ----------- 0 1 $\pi+\frac{\alpha}{2}$ $i$ 1 $t_2$ $\frac{\pi}{2}+\frac{\alpha}{2}$ $a+i$ 2 0 0 $+\infty$ : Parameters of the proposed rounded shim geometry.[]{data-label="tab:isoRound"} As expected the field along the surface of the rounded shim is continuous, see \[fig:RoseSurface\]. We now know how to calculate the magnitude of the magnetic field at the tip of the shim as a function of the shim height. In \[fig:abac\] we present a plot similar to Fig. 3 of Ref. [@rose1938magnetic]: it gives the optimum shim height as a function of the shim width, given in units of the half-gap height. This essential addition to this figure is the second $y$-axis: it gives the relation between the magnet nominal field and the maximum shim height, assuming a saturation field of 1.5T. ![Abacus: optimum effective shim height $b$ as a function of the shim width $a$, given in units of the half gap height, for rectangular (Rose) and rounded shims. The left-hand $y$-axis gives the relation between the magnet nominal field and the shim height to produce a field of 1.5T on the tip of a rounded shim.[]{data-label="fig:abac"}](abac){width="81.00000%"} Example of application {#sec:bonnie} ====================== The authors were tasked with modifying an existing dipole magnet, not for beam transport, but to serve as a test stand to study penning discharges. The magnet half gap is 238.8mm, and the required nominal field is 0.3T. The existing pole is narrower in the longitudinal ($z$) direction than in the transverse ($x$) direction. To improve field uniformity we considered installing shims on the edges of the pole in the $z$ direction. In \[fig:abac\] we see that with a nominal field of 0.3T we can use a shim height up to about 14.5% of the half gap. We choose to attempt to use a square shim ($a=b$). In \[fig:abac\] the line $a=b$ crosses the Rose shim line at $a=b=12.1\%$, and it crosses the rounded shim line at $a=b=14.5\%$. This defined the narrow range within which to choose our initial guess. We implemented the square shim geometry into [Opera-3d]{}, solved the optimization problem, and found that the optimum square shim dimension is around $a=b=13.3\%$, hence half-way between the two lines in \[fig:abac\]. The corresponding field flatness is presented in \[fig:bonnie\]. As predicted, \[fig:iso\] shows that, although the corners are saturated, the tip of the shim is below 1.5T. ![Effect of a square shim ($a=b$) calculated using [Opera-3D]{} along the $z$-axis with $x=y=0$. The edge of the magnet is at $z=0$, and the magnet center is at $z=400$mm. The optimum square shim size is found to be $a=b=0.133$. Note that with a larger shim ($a=b=0.145$) the field slightly overshoots.[]{data-label="fig:bonnie"}](bonnieSquare){width="60.00000%"} ![\[fig:iso\] Isometric projections of the [Opera-3d]{} model. Shim height and width are both equal to 13.3% of the half-gap height. The colour scale shows the magnitude of the magnetic field on the surface of the steel, dark blue: 0T, green: 0.8T, orange: 1.5T, purple: 2.0T, transparent over 2.0T. The coil is in grey. Thin black lines materialize the finite-element mesh. Non-linear magnetic properties of the steel are modelled using the B(H) curve of a typical C1010 steel. ](ISO){width="60.00000%"} Conclusion ========== Given the nominal field of an iron dominated electro-magnet, we have shown how to determine the optimum dimensions of simple passive shims (see \[fig:abac\]). Passive shims are useful to reduce the size of magnetic and electro-static dipoles. As a rule of thumb we have shown that for a nominal field below 1.2T one can use a shim height of at least 0.03 half gap, which leads to a pole width reduction of at least 1 half gap. In this paper, we restricted ourselves to the study of magnetic dipoles. All results however can be generalized to quadrupoles and any higher-order multipoles by means of an extra conformal transformation [@halbach1968application]. These results can also be applied to electro-static elements, for which the surface field limit ensues, not from a saturation effect, but from the Kilpatrick limit. [^1]: tplanche@triumf.ca [^2]: Now at the University of Toronto

--- author: - '$^1$[^1], $^1$, $^2$[^2], $^{3,4}$, $^2$, $^1$, $^1$, $^1$, $^1$, $^1$, $^5$ and $^1$' title: 'Multiband Superconductivity in KFe$_{2}$As$_{2}$: Evidence for one Isotropic and several Liliputian Energy Gaps' --- Introduction ============ The pairing mechanism in iron-pnictide superconductors is still a subject of intense debate. Similarly to heavy fermions, cuprates and ruthenates, the proximity of these materials to a magnetic instability naturally suggests that spin fluctuations can mediate the formation of Cooper pairs, although other scenarios involving orbital fluctuations are possible. [@Mazin08; @Kontani10; @Chubukov08; @Kuroki08; @Wang09] In this context, the symmetry of the superconducting-state order parameter can have either an $s\pm$ or a $d$-wave symmetry. Unfortunately, these states are almost degenerate and the realization of one of these two states is material-specific, depending on the number and position of Fermi-surface sheets in the Brillouin zone and their mutual interactions. In this context, the interpretation of experimental data is very complicated since the existence (absence) of nodal behavior does not permit the ruling out of $s$-wave ($d$-wave) symmetry. The Ba$_{1-x}$K$_{x}$Fe$_{2}$As$_{2}$ series is a prominent example. Indeed, at the optimal concentration ($x$ $\approx$ 0.4), heat-capacity [@Popovich10] and ARPES [@Ding08; @Evtushinsky09] measurements give strong evidence of an $s$-wave state while in the strongly correlated end-member KFe$_{2}$As$_{2}$, that has only hole pockets (see Fig. \[fig:Fig0\](a)). The situation remains highly controversial. Recently, thermal-conductivity measurements of Reid [*et al.*]{} [@Reid12a] were found to extrapolate at T $\rightarrow$ 0 to a finite residual term $\kappa$(0)/T, independent of sample purity. This was interpreted as a signature of universal heat transport, a property of superconductors with symmetry-imposed line nodes. This hypothetical change from $s$- to $d$-wave symmetry as a function of doping is allowed theoretically via an intermediate $s$ + $id$ state that breaks time-reversal symmetry. [@Lee09; @Stanev10; @Khodas12; @Platt12] On the other hand, laser ARPES [@Okazaki12] have revealed the existence of accidental line nodes on only one of the zone-centered pockets in KFe$_{2}$As$_{2}$, which is only compatible with a nodal $s\pm$ state. However, none of these methods are bulk probes of the superconducting state. In this Article, we report a detailed low-temperature thermodynamic investigation (heat capacity and magnetization) of the superconducting state of KFe$_{2}$As$_{2}$. We show quantitatively that the properties of KFe$_{2}$As$_{2}$, including the upper critical field (H$_{c2}$), are dominated by a relatively large nodeless energy gap of amplitude 1.9 k$_{B}$T$_{c}$ which excludes de facto $d_{x^{2}-y^{2}}$ symmetry (see Fig. \[fig:Fig0\](b)). ![\[fig:Fig0\] (Color online) (a) Schematic Fermi surface of KFe$_{2}$As$_{2}$ inferred from dHvA and ARPES measurements. [@Terashima09; @Yoshida12] (b) Possible symmetry of the superconducting-state order parameter of KFe$_{2}$As$_{2}$ (only one band is shown at the $\Gamma$ point).](Fig0.pdf){width="9cm"} We prove the existence of several additional extremely small gaps ($\Delta_{0}$ $<$ 1.0 k$_{B}$T$_{c}$) and show that they have a profound impact on the low-temperature and low-field behavior, as previously shown experimentally [@Bouquet02; @Sologubenko02; @Pribulova07a; @Seyfarth06; @Seyfarth08] and theoretically [@Tewordt03a; @Tewordt03b; @Barzykin07PRL; @Barzykin07; @Gorkov12] for MgB$_{2}$, CeCoIn$_{5}$ and PrOs$_{4}$Sb$_{12}$. The zero-field heat capacity is analyzed in a realistic self-consistent 4-band BCS model which qualitatively reproduces the recent laser ARPES results of Okazaki [*et al.*]{} [@Okazaki12] We also find that extremely low-temperature measurements, [*i.e.*]{} T $<$ 0.1 K, are required to observe the signature of possible line nodes in KFe$_{2}$As$_{2}$. In accord with recent angle-resolved heat-capacity experiments, [@Kittaka13] our results are compatible with either a $d_{xy}$ or a nodal $s\pm$ state. Experimental details ==================== Single crystals of KFe$_{2}$As$_{2}$ were grown in alumina crucibles using a self-flux method with a molar ratio K:Fe:As=0.3:0.1:0.6. The crucibles were put and sealed into an iron cylinder filled with argon gas. After heating up to 700$\,^{\circ}\mathrm{C}$ and then to 980$\,^{\circ}\mathrm{C}$, the furnace was cooled down slowly at a rate of about 0.5$\,^{\circ}\mathrm{C}$/h. The composition of the samples was checked by energy-dispersive x-ray analysis and four-circle diffractometry. The specific heat was measured with a commercial Quantum Design Physical Property Measurement System (PPMS) for T $>$ 0.4 K and with a home-made calorimeter for T $<$ 0.4 K. For T $>$ 2 K, we used a vibrating sample magnetometer to measure the magnetization. At lower temperature, magnetization measurements were performed using a low-temperature superconducting quantum interference device (SQUID magnetometer) equipped with a miniature dilution refrigerator developed at the Institut Néel-CNRS Grenoble. [@Burger13] Zero-field electron specific heat, C$_{e}$ (T, 0) ================================================= Figure \[fig:Fig1\](a) shows the low-temperature heat capacity of KFe$_{2}$As$_{2}$. We find a large Sommerfeld coefficient $\gamma_{n}$ = 103 mJ mol$^{-1}$ K$^{-2}$ and T$_{c}$ = 3.4 K, in agreement with our previous studies. [@HardyArxiv] Below 0.2K, the high-temperature tail of a Schottky anomaly, probably due to paramagnetic impurities, is observed (see inset Figure \[fig:Fig1\](a)). The electronic contribution C$_{e}$, shown in \[fig:Fig1\](b), is obtained by subtracting, from the measured data, a Debye term (inferred from the 5 T data), and the Schottky contribution. ![\[fig:Fig1\] (Color online) (a) Heat capacity of KFe$_{2}$As$_{2}$. The inset is a close-up of the low-temperature region showing the high-temperature tail of a Schottky anomaly. (b) Zero-field electronic heat capacity C$_{e}$ of KFe$_{2}$As$_{2}$. The green line is the weak-coupling BCS heat capacity for an $s$-wave superconductor ($\Delta_{0}$=1.764 k$_{B}$T$_{c}$).](Fig1.pdf){width="9.0cm"} The overall curve bears a strong similarity with that of MgB$_{2}$. [@Bouquet01] In particular, we observe that: (i) the jump at T$_{c}$, $\Delta$C/$\gamma_{n}$T$_{c}$ $\approx$ 0.54, is substantially smaller than the BCS value ($\Delta$C/C$_{n}$ = 1.43) for a weakly coupled single band $s$-wave superconductor and (ii) there is a steep quasi-linear decrease of C$_{e}$/T with decreasing temperature for T/T$_{c}$ $\leq$ 0.1. A similar linear dependence of the penetration depth was reported by Hashimoto [*et al.*]{} [@Hashimoto10] and was interpreted as evidence of line nodes. In this Article, we argue that this steep feature is instead related to the existence of small energy gaps ($\Delta_{S}$/k$_{B}$T$_{c}$ $\approx$ T/T$_{c}$ $\approx$ 0.2), as inferred from small-angle neutron scattering (SANS) experiments. [@Kawano11] Assuming that all 3 sheets around the $\Gamma$ point exhibit this tiny gap and using the expression of the heat-capacity jump of a two-band $s$-wave superconductor in the weak coupling limit, [@Moskalenko59; @Soda66] $$\label{eq:eq1} \frac{\Delta C}{k_{B}T_{c}}=1.43\cdot\frac{\left(N_{S}\Delta_{S}^{2}+N_{L}\Delta_{L}^{2}\right)^{2}}{\left(N_{S}+N_{L}\right)\left(N_{S}\Delta_{S}^{2}+N_{L}\Delta_{L}^{2}\right)},$$ (where the subscripts S and L refer to the small and large gaps, respectively). We estimate $\Delta_{L}$/k$_{B}$T$_{c}$ $\approx$ 1.8 on the $\epsilon$ band using the individual density of states inferred from dHvA and ARPES measurements [@Terashima09; @Yoshida12] (see Table \[tab:Table1\]). Interestingly, we obtain from this simple estimation a remarkably large gap anisotropy $\Delta_{L}$/$\Delta_{S}$ $\approx$ 9, in comparison with MgB$_{2}$ where it is about 4. [@Bouquet01] C(T) ------------ ------------- ------------------------- ------------------------- ----------------- m$_{i}^{*}$ $\gamma_{i}$ $\gamma_{i}/\gamma_{n}$ N$_{i}$(0)/N(0) (m$_{e}$) (mJ mol$^{-1}$K$^{-2}$) $\alpha$ 6.06 8.8 0.10 0.10 $\beta$ 17.1 24.8 0.28 0.31 $\zeta$ 11.8 17.1 0.19 0.23 $\epsilon$ 6.62 38.4 0.43 0.36 Total - 90.1 1.0 1.0 : Parameters derived from the dHvA and ARPES measurements assuming 2D Fermi-surface sheets, with $\gamma_{i}$=$\frac{\pi N_{A}k_{B}^{2}a^{2}}{3\hbar^{2}}m_{i}^{*}$ (with $a$ = 3.84 Å). [@Terashima09; @Yoshida12] The last column contains the densities of states used in the 4-band BCS model. m$_{e}$ is the bare electron mass.\ []{data-label="tab:Table1"} Thus, KFe$_{2}$As$_{2}$ represents a somewhat extreme case of multiband superconductivity similar to the heavy-fermion compounds CeCoIn$_{5}$ and PrOs$_{4}$Sb$_{12}$. [@Seyfarth06; @Seyfarth08] ![\[fig:Fig2\] (Color online) (a) Raw magnetization curves of KFe$_{2}$As$_{2}$ measured at several temperatures for H $||$ $c$. (b) Reversible magnetization curves. (c) Difference of the superconducting- and normal- states reversible magnetizations. (d) In-plane superfluid density derived from the reversible magnetization curves. The solid line is the superfluid density of an $s$-wave gap with $\Delta_{0}$ = 1.9 k$_{B}$T$_{c}$.](Fig2.pdf){width="9cm"} Mixed-state reversible magnetization, M$_{rev}$ (H) =================================================== Figure \[fig:Fig2\](a) shows magnetization curves of KFe$_{2}$As$_{2}$ for H $||$ $c$ down to 0.6 K. In the normal state (T = 4 K), a sizeable paramagnetic signal is observed with a susceptibility of about 3.3 $\times$ 10$^{-4}$ in agreement with our previous report. [@HardyArxiv] At lower temperatures, the magnetization curves are reversible over a wide field interval ([*e.g.*]{} H$_{c2}$/2.5 $<$ H $<$ H$_{c2}$ = 1.4 T at 0.6 K), indicating a small concentration of pinning centers as confirmed by the observation of a well defined hexagonal vortex lattice by SANS. [@Kawano11] Together with the observation of quantum oscillations and the absence of significant residual density of states in the limit T $\rightarrow$ 0 (see Fig. \[fig:Fig1\](b)) this indicates that our KFe$_{2}$As$_{2}$ single crystals are weakly disordered and are in the clean limit. Note that this is at odds with Co-doped BaFe$_{2}$As$_{2}$ samples in which no vortex lattice could be observed. [@Eskildsen09; @Inosov10] As a result, accurate reversible magnetization curves can be obtained for our sample by averaging the increasing and decreasing branches of the magnetization loop, as illustrated in Fig. \[fig:Fig2\](b). In single-band type II superconductors, M$_{rev}$(H) is entirely defined by H/H$_{c2}$ and the Ginzburg-Landau parameter $\kappa$ = $\lambda$/$\xi$, [@Brandt03] with $$\label{eq:eq2a} M_{rev}=\frac{H-H_{c2}}{(2\kappa^{2}-1)\beta_{A}+1},$$ at high field (Abrikosov regime, with $\beta_{A}$ the Abrikosov coefficient). In the intermediate field range (London regime), [@DeGennesBook; @Kogan88] the reversible magnetization is linear in the logarithm of the applied field with $$\label{eq:eq2b} \mu_{0}M_{rev}=-\frac{\phi_{0}}{8\pi \lambda}\ln{\left(b\frac{H_{c2}}{H}\right)},$$ where $\lambda$ = $\lambda_{ab}$ is the in-plane penetration depth for H $||$ $c$ and $b$ a constant. [@Lang92] Thus, in close analogy to the case of MgB$_{2}$, [@Klein06] Fig. \[fig:Fig2\](c) shows that the linear evolution of M$_{rev}$(H) expected near H$_{c2}$ is not observed and the London dependence dominates up to H$_{c2}$,which therefore allows the determination of $\lambda_{ab}$(T) using Eq.(\[eq:eq2b\]). The derived superfluid density, defined as $\rho_{ab}$(T) = $\left[\lambda_{ab}(\textrm{0.6 K})/\lambda_{ab}(T)\right]^{2}$, is shown in Fig. \[fig:Fig2\](d) together with the calculation for an $s$-wave gap of amplitude $\Delta_{0}$ = 1.9 k$_{B}$T$_{c}$, which accurately reproduces the data. As a consequence, our analysis firmly establishes the existence of a relatively large nodeless gap in KFe$_{2}$As$_{2}$, in agreement with our rough above-mentioned heat-capacity analysis. Contrary to direct penetration-depth measurements, [@Hashimoto10] our estimate of $\rho_{ab}$(T) was inferred from high-field data where the large vortex cores related to the smaller gaps have already overlapped, as observed in MgB$_{2}$, [@Eskildsen02] and discussed hereafter. This explains why this indirect derivation of $\rho_{ab}$(T) is only sensitive to the larger gap as found for MgB$_{2}$. [@Zehetmayer04; @Zehetmayer13]\ Mixed-state specific heat, $\gamma$ (H) ======================================= Evidence for the existence of tiny energy gaps can be found using heat-capacity measurements in the mixed state as previously shown for pure, Al- and C-doped MgB$_{2}$. [@Bouquet02; @Pribulova07a; @Pribulova07b; @Fisher13] Figure \[fig:Fig3\](a) shows the field dependence of the electron heat capacity $\gamma$(H) at 0.12 K ([*i.e.*]{} T/T$_{c}$ = 0.035) for H parallel to $c$. Similar to MgB$_{2}$, we find that $\gamma$(H) is very non-linear with applied magnetic field. In very low fields, $\gamma$(H) increases abruptly and reaches $\gamma$(H)/$\gamma_{n}$ $\approx$ 0.6 at only H/H$_{c2}$ $\approx$ 0.1. In larger fields, $\gamma$(H) closely follows the behavior expected for an individual band (magenta line), [@Ichioka07] indicating that the interband couplings between the sheet with the largest gap and the other bands is rather small. [@Tewordt03b] Using the densities of states derived from dHvA and ARPES (see Table \[tab:Table1\]), we can unambiguously ascribe the largest gap to the $\epsilon$ band and subtract its contribution from $\gamma$(H) to obtain the mixed-state heat capacity of the remaining $\alpha$, $\beta$ and $\zeta$ bands (green curve in Figs. \[fig:Fig3\](a) and \[fig:Fig3\](b)). We find that these bands have almost recovered their normal-state value in a ’crossover’ field H$^{S}_{c2}$ $\approx$ 0.1 $\times$ H$_{c2}$ which is defined here as, $$\label{eq:eq2c} H^{S}_{c2}\approx H_{c2}\left(\frac{\xi_{S}}{\xi_{L}}\right)^{2},$$ and which would correspond to the upper critical field of the small gaps in the absence of interband couplings. [@Tewordt03b] Thus, the disappearance of the small gaps associated with the $\alpha$, $\beta$ and $\zeta$ sheets in an applied field is more rapid than that of the $\epsilon$ band which shows a conventional individual dependence. Following Klein [*et al.*]{}, [@Klein06] we assume that all the excitations are localized in the vortex cores ([*i.e.*]{} we neglect the small-gap Doppler shift [@Bang10]) and that the system can be described by only one field dependent quantity, $\xi_{c}$(H), which is a measure of the vortex-core size. In this context, $\gamma$(H) $\propto$ $\gamma_{n}\cdot \left(\xi_{c}(H)/d\right)^{2}$ (where $d$ $\propto$ 1/$\sqrt{H}$ is the intervortex distance), and we obtain directly $\xi_{c}$(H) as shown in Fig. \[fig:Fig3\](c), with $\xi_{c}$(H=H$_{c2}$)=$\sqrt{\Phi_{0}/2\pi\mu_{0}H_{c2}}$. We find that the vortex-core size smoothly decreases from 50 to 15 nm in high fields, explaining the smooth evolution of the contributions of the $\alpha$, $\beta$ and $\zeta$ bands to $\gamma$(H) near H$^{S}_{c2}$. Thus, the small gaps on these sheets remain finite due to nonzero interband coupling even for H $>>$ H$^{S}_{c2}$ where their vortex cores overlap. In the opposite limit, [*i.e.*]{} where the Doppler shift (Volovik effect) dominates, we note that a similar dependence of $\gamma$(H) for the small gaps (see Fig. \[fig:Fig3\](b)) is expected theoretically for $\Delta_{S}/\Delta_{L}$ $\approx$ 0.1. [@Bang10] ![\[fig:Fig3\] (Color online) (a) Field dependence of the heat capacity of KFe$_{2}$As$_{2}$ (blue symbols) at T = 0.12 K for H $||$ $c$. The magenta line is the theoretical curve of the mixed-state heat capacity of an $s$-wave superconductor [@Ichioka07] normalized by the density of states of the $\epsilon$ band (see Table \[tab:Table1\]). The green curve is the resulting contribution of the 3 small energy gaps $\alpha$, $\beta$ and $\zeta$ obtained by subtracting the heat capacity of the $\epsilon$ band from the data. (b) Close-up of the low-field region. (c) Field dependent vortex core size derived from the mixed-state heat capacity.](Fig3.pdf){width="9cm"} Comparison with thermal-conductivity measurements, $\kappa$ (T, H) ================================================================== In light of our results, we comment here on the interpretation of recent heat-transport experiments in KFe$_{2}$As$_{2}$. In Refs. [@Reid12a] and [@Reid12b], thermal-conductivity measurements $\kappa$(T)/T, performed for T $>$ 0.1 K, were found to extrapolate at T $\rightarrow$ 0 to a finite residual term $\kappa$(0)/T independent of sample purity. This was interpreted as a signature of universal heat transport, a property of superconductors with symmetry-imposed line nodes such as $d$-wave states. Experimentally, these measurements were not strictly realized in zero magnetic field because it was necessary to apply a small field of 0.05 T ([*i.e.*]{} H/H$_{c2}$ $\approx$ 0.03) to suppress superconductivity of the soldered contacts. However, as shown in Fig. \[fig:Fig3\](b), this small field is large enough to produce an enhancement of the density of states, reaching 40% of the normal-state value at 0.12 K, which inexorably leads to a finite value of $\kappa_{0}$/T. In addition, our specific-heat measurements show that a significant increase of $\kappa$(H)/T is also to be expected for H/H$_{c2}$ $<$ 0.1 due to these small gaps. This feature was not observed in Refs. [@Reid12a], [@Reid12b] or in the more recent data of Watanabe [*et al.*]{} [@Watanabe13] while it clearly appears in many other multiband superconductors including MgB$_{2}$, CeCoIn$_{5}$ and PrOs$_{4}$Sb$_{12}$. [@Sologubenko02; @Seyfarth06; @Seyfarth08] Thus, the origin of the finite $\kappa_{0}$/T cannot be attributed to $d$-wave superconductivity in KFe$_{2}$As$_{2}$ in these experimental conditions. We note that the use of superconducting solder was already pointed out to produce spurious results in Ref. [@Seyfarth08]. On the other hand, our observation of a relatively large isotropic gap does not rule out definitively $d$-wave superconductivity in KFe$_{2}$As$_{2}$. Actually, only the $d_{x^{2}-y^{2}}$ order parameter, with nodes located on the diagonals of the Brillouin zone, is excluded while $d_{xy}$ symmetry remains possible if the large gap effectively occurs on the $\epsilon$ sheet. This conclusion is corroborated by recent angle-resolved heat-capacity measurements of Kittaka [*et al*]{} [@Kittaka13]. ![\[fig:Fig4A\] (Color online) Schematics of the 4-band BCS model used to analyze C$_{e}$ (T, 0).](Fig5.pdf){width="9cm"} Four-band BCS analysis of C$_{e}$ (T, 0) ======================================== Recently, laser ARPES measurements, [@Okazaki12] performed at 1.5 K, revealed angle-dependent gaps of the 3 bands around the $\Gamma$ point with accidental line nodes only on the $\zeta$ sheet. These results are compatible with a nodal $s$-state and exclude a possible change of symmetry of the superconducting order parameter as a function of K doping. To check whether these results can be confirmed by bulk measurements, we analyze our zero-field heat capacity, taking into account the observed Fermi surface. In the absence of a sizeable residual density of states for T $\rightarrow$ 0, the modest specific-heat jump clearly shows that KFe$_{2}$As$_{2}$ is close to the weak coupling limit. Therefore, we can model the temperature dependence of C$_{e}$ in a pure 4-band BCS model. Assuming 2D Fermi-surface pockets, we obtain the following system of gap equations: $$\begin{aligned} \label{eq:eq4} &\Delta_{i}(\phi,T)=-\sum_{j=1}^{4} \frac{N_{j}(0)}{2 \pi}\int_{0}^{2 \pi}d\phi'\cdot\nonumber\\ &\int_{0}^{\epsilon_{c}}d\epsilon \frac{V_{ij}(\phi ,\phi')\Delta_{j}(\phi',T)}{\sqrt{\epsilon^{2}+|\Delta_{j}(\phi',T)|^{2}}}\tanh{\frac{\beta}{2}\sqrt{\epsilon^{2}+|\Delta_{j}(\phi',T)|^{2}}},\end{aligned}$$ where N$_{i}$(0) is the density of states of the $i$-th band with $i\in \left\{\alpha,\zeta, \beta,\epsilon\right\}$, $V_{ij}(\phi, \phi')$ are the intraband ($i=j$) and interband ($i\neq j$) pairing potentials, $\beta$=1/k$_{B}$T, and $\phi$ and $\phi'$ the azimuthal angles on the sheets $i$ and $j$, respectively. In the $s$-wave channel, [@Chubukov12; @Maiti12] we write: $$\label{eq:eq3} V_{ij}(\phi,\phi')=V_{ij}^{(0)}+V_{ij}^{(1)}\cdot(\cos{4\phi}+\cos{4\phi'}).$$ Such interactions lead to anisotropic gaps of the form: $$\label{eq:eq5} \Delta_{i}(\phi,T)=\Delta_{i}^{(0)}(T)+\Delta_{i}^{(1)}(T)\cdot\cos(4\phi),$$ which are calculated self-consistently from Eqs. (\[eq:eq4\]) and used to compute the superconducting-state heat capacity. We constrain all the N$_{i}$(0) to match as closely as possible the values inferred from dHvA and ARPES measurements. In this form, the model is parametrized with 20 interaction constants and this number is reduced to 5 by assuming that: $$\begin{aligned} V_{ij}^{(0)}&=&\delta_{ij}\cdot V_{1}+(1-\delta_{ij})\cdot V_{2},\label{eq:eq6a} \\ V_{ij}^{(1)}&=&0, \label{eq:eq6b} \\ V_{\beta \epsilon}(\phi,\phi')&=&V_{\epsilon \epsilon}^{(1)}=V_{\alpha \epsilon}^{(1)}=0,\label{eq:eq6c}\end{aligned}$$ with $i,j\in \left\{\alpha,\zeta, \beta\right\}$. Here, Eqs. (\[eq:eq6a\]) and (\[eq:eq6b\]) impose that the intra- and interband interactions of the zone-centered bands are angle-independent and equal to V$_{1}$ and V$_{2}$, respectively because these bands have a quasi-2D morphology and are centered around the same point. [@Maiti12] On the other hand, inelastic neutron scattering experiments have revealed the persistence of resonant spin excitations in heavily overdoped Ba$_{1-x}$K$_{1-x}$Fe$_{2}$As$_{2}$ (x $\approx$ 0.9) [@Castellan11] and incommensurate spin fluctuations in KFe$_{2}$As$_{2}$ that approximately connect the $\Gamma$ and X bands. [@Lee11] These observations convincingly indicate that the $\Gamma$-X interband interactions remain significant in KFe$_{2}$As$_{2}$, even in the absence of electron pockets.  This is particularly true for the $\alpha$ and $\zeta$ pockets which are strongly involved in nesting in Ba$_{0.6}$K$_{0.4}$Fe$_{2}$As$_{2}$. In KFe$_{2}$As$_{2}$, these bands share a dominant $xy/yz$ orbital character with the $\epsilon$ band while the $\beta$ pocket exhibits mainly $x^{2}-y^{2}$ component. This implies that the $\beta$ sheet plays no decisive role in pairing, as illustrated by the small gap observed on this pocket in both compounds. [@Evtushinsky09; @Okazaki12] Furthermore, the $\zeta$ band shows an additional finite $z^{2}$ component which is absent from the $\alpha$ and $\epsilon$ sheets. Consequently, the $z^{2}$ component in the $\zeta$ pocket has no counterpart for the sign change of the superconducting gap in the $\epsilon$ sheet and this can lead to a sign change of the gap in the parts of the $\zeta$ band where the $z^{2}$ contribution dominates. [@Okazaki12] Thus, the only prominent angle-dependent interaction is V$_{\zeta \epsilon}^{(1)}$. All these assumptions are summarized in Eq. (\[eq:eq6c\]) and Fig.\[fig:Fig4A\]. The heat capacity, as well as the angular and the temperature dependence of the gaps calculated with this model are shown in Fig.\[fig:Fig4\], using the density of states and the remaining non-zero coupling constants given in Tables \[tab:Table1\] and \[tab:Table2\], respectively. These parameters are all given in units of V$_{1}$ because it is not their absolute values that matters, but rather their relative weights. At the $\Gamma$ point, our results are in good agreement with the laser ARPES experiments. V$_{2}$ V$_{\epsilon \epsilon}^{(0)}$ V$_{\alpha \epsilon}^{(0)}$ V$_{\zeta \epsilon}^{(0)}$ V$_{\zeta \epsilon}^{(1)}$ --------- ------------------------------- ----------------------------- ---------------------------- ---------------------------- 0.7 2.5 0.5 0.15 0.6 ARPES SANS C (T) 3.8 0.72 0.57 0.5 0.21 0.22 1.4 - 0.35 - 1.77 1.90 : Fit parameters of the 4-band BCS model. The pairing potentials are given in units of V$_{1}$ (with V$_{1}$=V$_{\alpha \alpha}^{(0)}$=V$_{\zeta \zeta}^{(0)}$=V$_{\beta \beta}^{(0)}$ and V$_{2}$=V$_{\alpha \zeta}^{(0)}$=V$_{\alpha \beta}^{(0)}$=V$_{\zeta \beta}^{(0)}$). Comparison of the energy gaps derived from laser ARPES (at 1.5 K), SANS (at 0.1 T), with those obtained from the 4-band BCS analysis of the zero-field heat capacity. The gaps are given in units of k$_{B}$T$_{C}$. For the $\zeta$ and $\epsilon$ bands, the mean values are given. Assignments for the SANS data is arbitrary.[]{data-label="tab:Table2"} As shown in Fig. \[fig:Fig4\](b), the larger (smaller) of the 3 energy gaps is found on the $\alpha$ ($\beta$) band while the $\zeta$ gap exhibits accidental nodes which arise from the angle-dependent interband interaction with the $\epsilon$ pocket. These results are at odds with all theoretical calculations [@Maiti11a; @Maiti11b; @Suzuki11; @Thomale11] which predict the largest gap on the $\beta$ band. Moreover, at the X point, we recover the large gap $\Delta_{\epsilon}$=1.9 k$_{B}$T$_{c}$ inferred from our magnetization and field-dependent heat-capacity data. This gap was not observed in any other experiments and is due to a significantly larger intraband constant on the $\epsilon$ band (see Table \[tab:Table2\]). We stress that our analysis is not unique and other sets of parameters could fit C$_{e}$(T) equally well. However, they would result necessarily in gap amplitudes close to the values we obtain because we constrain the individual densities of states to match approximately the values inferred from dHvA and ARPES. Moreover, as illustrated in Fig. \[fig:Fig4\](a), each gap has its own role in the temperature dependence of C$_{e}$(T) in the superconducting state. Indeed, the $\epsilon$ gap alone is responsible for the jump at T$_{c}$ and has a vanishing contribution at low temperatures, while the $\beta$ gap is predominantly responsible for the hump observed around T/T$_{c}$ $\approx$ 0.2. However, its decrease below this temperature is too steep to reproduce the experimental data. It is smoothened by the nodal contribution of the $\zeta$ gap. The slight maximum in the contributions from the $\alpha$ and $\zeta$ bands moreover lessen the dip of the high-temperature side of the shoulder due to the $\beta$ band. Although our results do not bring direct evidence of the existence of line nodes, they firmly establish the existence of tiny energy gaps with $\Delta$/k$_{B}$T$_{c}$ $<$ 1.0. Their small amplitude imposes the requirement of cooling the sample below 80 mK to be able to observe the linear nodal behavior, as shown in the inset of Fig. \[fig:Fig4\](a). To our knowledge, no measurements in this temperature range were ever reported. Thus, our results do not exclude $d_{xy}$ symmetry. On the other hand, the agreement with laser ARPES is only qualitative. As shown in Table \[tab:Table2\], Okazaki [*et al.*]{} reported overestimated gap values in comparison to heat-capacity and SANS measurements. Particularly, they find $\Delta_{\alpha}$ = 3.8 k$_{B}$T$_{c}$, which is conspicuously comparable in amplitude to the largest gap observed close to the optimal concentration Ba$_{0.6}$K$_{0.4}$Fe$_{2}$As$_{2}$, while the critical temperature of the latter is 10 times larger. [@Evtushinsky09; @Xu11]\ Conclusions =========== We have shown the existence of a relatively large nodeless energy gap of amplitude 1.9 k$_{B}$T$_{c}$ that excludes the possibility of $d_{x^{2}-y^{2}}$ symmetry for the superconducting-state order parameter in KFe$_{2}$As$_{2}$. Our results do not bring direct evidence for line nodes, they clearly prove the existence of tiny energy gaps ($\Delta$/k$_{B}$T$_{c}$ $<$ 1.0) which strongly govern the low-field and low-temperature heat capacity, much like MgB$_{2}$. Furthermore, the small amplitudes of the gaps indicate that very low-temperature measurements (T $<$ 80 mK) will be required in order to observe the possible signatures of line nodes in this compound; a restriction that also applies to other probes like penetration depth and heat transport. Our results shows qualitative agreement with recent laser ARPES measurements, and strongly suggest the superconducting-state symmetry to be $s$-wave. We thank J.- P. Brison, M. Lang, M. Ichioka, K. Machida, A. Chubukov, T. Shibauchi and S. Kittaka for stimulating and enlightening discussions. This work was supported by the Deutsche Forschungsgemeinschaft through DFG-SPP 1458 “Hochtemperatursupraleitung in Eisenpniktiden”. The work performed in Grenoble was supported by the French ANR Projects (SINUS and CHIRnMAG) and the ERC starting grant NewHeavyFermion. [9]{} I. I. Mazin, D. J. Singh, M. D. Johannes, and M. H. Du: Phys. Rev. Lett. **101** (2008) 057003. H. Kontani and S. Onari: Phys. Rev. Lett. **104** (2012) 157001. A. V. Chubukov, D. V. Efremov, and I. Eremin : Phys. Rev. B **78** (2008) 134512. K. Kuroki, S. Onari, R. Arita, H. Usui, Y. Tanaka, H. Kontani, and H. Aoki: Phys. Rev. Lett. **101** (2008) 087004. F. Wang, H. Zhai, Y. Ran, A. Vishwanath, and D. -H. Lee: Phys. Rev. Lett. **102** (2009) 047005. P. Popovich, A. V. Boris, O. V. Dolgov, A. A. Golubov, D. L. Sun, C. T. Lin, R. K. Kremer, and B. Keimer: Phys. Rev. Lett. **105** (2010) 027003. H. Ding, P. Richard, K. Nakayama, K. Sugawara, T. Arakane, Y. Sekiba, A. Takayama, S. Souma, T. Sato, T. Takahashi, Z. Wang, X. Dai, Z. Fang, G. F. Chen, J. L. Luo and N. L. Wang: Europhys. Lett. **83** (2008) 47001. D. V. Evtushinsky, D. S. Inosov, V. B. Zabolotnyy, A. Koitzsch, M. Knupfer, B. Buechner, M. S. Viazovska, G. L. Sun, V. Hinkov, A. V. Boris, C. T. Lin, B. Keimer, A. Varykhalov, A. A. Kordyuk, and S. V. Borisenko: Phys. Rev. B **79** (2011) 054517. J. -Ph Reid, M. A. Tanatar, A. Juneau-Fecteau, R. T. Gordon, S. René de Cotret, N. Doiron-Leyraud, T. Saito, H. Fukazawa, Y. Kohori, K. Kihou, C. H. Lee, A. Iyo, H. Eisaki, R. Prozorov, and L. Taillefer: Phys. Rev. Lett. **109** (2012) 087001. W. -C. Lee, D.-C. Zhang, and C. Wu: Phys. Rev. Lett. **102** (2009) 217002. V. Stanev and Z. Tesanovic: Phys. Rev. B **81** (2010) 134522. M. Khodas and A. V. Chubukov: Phys. Rev. Lett. **108** (2012) 247003. C. Platt, R. Thomale, C. Honerkamp, S. -C. Zhang, and W. Hanke: Phys. Rev. B **85** (2012) 180502. K. Okazaki, Y. Ota, Y. Kotani, W. Malaeb, Y. Ishida, T. Shimojima, T. Kiss, S. Watanabe, C.-T. Chen, K. Kihou, C. H. Lee, A. Iyo, H. Eisaki, T. Saito, H. Fukazawa, Y. Kohori, K. Hashimoto, T. Shibauchi, Y. Matsuda, H. Ikeda, H. Miyahara, R. Arita, A. Chainani, and S. Shin: Science 337 (2012) 1314. T. Terashima, M. Kimata, H. Satsukawa, A. Harada, K. Hazama, S. Uji, H. Harima, G. -F. Chen, J. -L. Luo, and N. -L. Wang: J. Phys. Soc. Jpn. **79** (2009) 053702. T. Yoshida, S. Ideta, I. Nishi, A. Fujimori, M. Yi, R. G. Moore, S. K. Mo, D. -H. Lu, Z. -X. Shen, Z. Hussain, K. Kihou, P. M. Shirage, H. Kito, C. -H. Lee, A. Iyo, H. Eisaki, and H. Harima:arXiv:1205.6911. F. Bouquet, Y. Wang, I. Sheikin, T. Plackowski, A. Junod, S. Lee, and S. Tajima: Phys. Rev. Lett. **89** (2002) 257001. A. V. Sologubenko, J. Jun, S. M. Kazakov, J. Karpinski, and H. R. Ott: Phys. Rev. B **66** (2002) 014504. Z. Pribulova, T. Klein, J. Marcus, C. Marcenat, F. Levy, M. S. Park, H. G. Lee, B. W. Kang, S. I. Lee, S. Tajima, and S. Lee: Phys. Rev. Lett. **98** (2007) 137001. G. Seyfarth, J.-P. Brison, M.-A. Méasson, D. Braithwaite, G. Lapertot and J. Flouquet: Phys. Rev. Lett. **97** (2006) 236403. G. Seyfarth, J.-P. Brison, G. Knebel, D. Aoki, G. Lapertot, and J. Flouquet: Phys. Rev. Lett. **101** (2008) 046401. L. Tewordt, and D. Fay: Phys. Rev. B **67** (2003) 134524. L. Tewordt, and D. Fay: Phys. Rev. B **68** (2003) 092503. V. Barzykin, and L. P. Gor’kov: Phys. Rev. Lett. **98** (2007) 087004. V. Barzykin, and L. P. Gor’kov: Phys. Rev. B **76** (2007) 014509. L. P. Gor’kov: Phys. Rev. B **86** (2012) 060501. S. Kittaka, Y. Aoki, N. Kase, T. Sakakibara, T. Saito, H. Fukazawa, Y. Kohori, K. Kihou, C. H. Lee, A. Iyo, and H. Eisaki: SCES conference (2013). P. Burger, F. Hardy, D. Aoki, A. E. Böhmer, R. Eder, R. Heid, T. Wolf, P. Schweiss, R. Fromknecht, M. J. Jackson, C. Paulsen, and C. Meingast: Phys. Rev. B **88** (2013) 014517. F. Hardy, A. E. Boehmer, D. Aoki, P. Burger, T. Wolf, P. Schweiss, R. Heid, P. Adelmann, Y. X. Yao, G. Kotliar, J. Schmalian, and C. Meingast: Phys. Rev. Lett. **111** (2013) 027002. F. Bouquet, Y. Wang, R. A. Fisher, D. G. Hinks, J. D. Jorgensen, A. Junod, and N. E. Phillips: Europhys. Lett. **56** (2001) 856. K. Hashimoto, A. Serafin, S. Tonegawa, R. Katsumata, R. Okazaki, T. Saito, H. Fukazawa, Y. Kohori, K. Kihou, C. H. Lee, A. Iyo, H. Eisaki, H. Ikeda, Y. Matsuda, A. Carrington, and T. Shibauchi: Phys. Rev. B **82** (2010) 014526. H. Kawano-Furukawa, C. J. Bowell, J. S. White, R. W. Heslop, A. S. Cameron, E. M. Forgan, K. Kihou, C. H. Lee, A. Iyo, H. Eisaki, T. Saito, H. Fukazawa, Y. Kohori, R. Cubitt, C. D. Dewhurst, J. L. Gavilano, and M. Zolliker: Phys. Rev. B **84** (2011) 024507. V. A. Moskalenko: Sov. Phys. Met. Metallogr. **8** (1959) 25. T. Soda and Y. Wada: Prog. Theor. Phys. **36** (1966) 1111. M. R. Eskildsen, L. Ya. Vinnikov, T. D. Blasius, I. S. Veshchunov, T. M. Artemova, J. M. Densmore, C. D. Dewhurst, N. Ni, A. Kreyssig, S. L. Bud’ko, P. C. Canfield, and A. I. Goldman: Phys. Rev. B **79** (2009) 100501. D. S. Inosov, T. Shapoval, V. Neu, U. Wolff, J. S. White, S. Haindl, J. T. Park, D. L. Sun, C. T. Lin, E. M. Forgan, M. S. Viazovska, J. H. Kim, M. Laver, K. Nenkov, O. Khvostikova, S. Kühnemann, and V. Hinkov: Phys. Rev. B **81** (2010) 014513. E. H. Brandt: Phys. Rev. B **68** (2003) 054506. P. G. de Gennes: in Superconductivity of metals and alloys, ed. D. Pines (W. A. Benjamin, New York 1966) p. 69. V. G. Kogan, M. M. Fang, and Sreeparna Mitra: Phys. Rev. B **38** (1988) 11958. M. Lang, N. Toyota, T. Sasaki, and H. Sato: Phys. Rev. B **46** (1992) 5822 T. Klein, L. Lyard, J. Marcus, Z. Holanova, and C. Marcenat: Phys. Rev. B **73** (2006) 184513. M. R. Eskildsen, M. Kugler, S. Tanaka, J. Jun, S. M. Kazakov, J. Karpinski, and Ø. Fischer: Phys. Rev. Lett. **89** (2002) 187003. M. Zehetmayer, M. Eisterer, J. Jun, S. M. Kazakov, J. Karpinski and H. W. Weber: Phys. Rev. B **70** (2004) 214516. M. Zehetmayer: Supercond. Sci. Technol. **26** (2013) 043001. Z. Pribulova, T. Klein, J. Marcus, C. Marcenat, M. S. Park, H. -S. Lee, H. -G. Lee, and S.-I. Lee: Phys. Rev. B **76** (2007) 180502. R. A. Fisher, J. E. Gordon, F. Hardy, W. E. Mickelson, N. Oeschler, N.E. Phillips, and A. Zettl: Physica C **485** (2013) 168. M. Ichioka, and K. Machida: Phys. Rev. B **76** (2007) 064502. Y. Bang: Phys. Rev. Lett. **104** (2010) 217001. J. -Ph Reid, A. Juneau-Fecteau, R. T. Gordon, S. René de Cotret, N. Doiron-Leyraud, X. G. Luo, H. Shakeripour, J. Chang, M. A. Tanatar, H. Kim, R. Prozorov, T. Saito, H. Fukazawa, Y. Kohori, K. Kihou, C. H. Lee, A. Iyo, H. Eisaki, B. Shen, H.-H. Wen, and L. Taillefer: Supercond. Sci. Technol. **25** (2012) 084013. D. Watanabe, T. Yamashita, Y. Kawamoto, S. Kurata, Y. Mizukami, T. Ohta, S. Kasahara, M. Yamashita, T. Saito, H. Fukazawa, Y. Kohori, S. Ishida, K. Kihou, C. H. Lee, A. Iyo, H. Eisaki, A. B. Vorontsov, T. Shibauchi, and Y. Matsuda: arXiv:1307.3408 \[cond-mat.supr-con\] (2013). A. Chubukov: Annu. Rev. Condens. Matter Phys. **3** (2012) 57. S. Maiti, M. M. Korshunov, and A. V. Chubukov: Phys. Rev. B **85** (2012) 014511. J. -P. Castellan, S. Rosenkranz, E. A. Goremychkin, D. Y. Chung, I. S. Todorov, M. G. Kanatzidis, I. Eremin, J. Knolle, A. V. Chubukov, S. Maiti, M. R. Norman, F. Weber, H. Claus,T. Guidi, R. I. Bewley, and R. Osborn: Phys. Rev. Lett. **107** (2011) 177003. C. H. Lee, K. Kihou, H. Kawano-Furukawa, T. Saito, A. Iyo, H. Eisaki, H. Fukazawa, Y. Kohori, K. Suzuki, H. Usui, K. Kuroki, and K. Yamada: Phys. Rev. Lett. **106** (2011) 067003. S. Maiti, M. M. Korshunov, T. A. Maier, P. J. Hirschfeld, and A. V. Chubukov: Phys. Rev. Lett. **107** (2011) 147002. S. Maiti, M. M. Korshunov, T. A. Maier, P. J. Hirschfeld, and A. V. Chubukov: Phys. Rev. B **84** (2011) 224505. K. Suzuki, H. Usui, and K. Kuroki: Phys. Rev. B **84** (2011) 144514. R. Thomale, C. Platt, W. Hanke, J. Hu, and B. A. Bernevig: Phys. Rev. Lett. **107** (2011) 117001. Y. -M. Xu, Y. -B. Huang, X. -Y. Cui, E. Razzoli, M. Radovic, M. Shi, G. -F. Chen, P. Zheng, N.- L. Wang, C. -L. Zhang, P. -C. Dai, J. -P. Hu, Z. Wang, and H. Ding: Nat. Phys. **7** (2011) 198 [^1]: E-mail: frederic.hardy@kit.edu [^2]: Present address: Department of Low Temperature Physics, Charles University in Prague, 180 00, Praha 8, Czech Republic

--- abstract: 'The jet fragmentation function is measured with direct photon-hadron correlations in $p$$+$$p$ and Au$+$Au collisions at $\sqrt{s_{_{NN}}}=200$ GeV. The $p_{T}$ of the photon is an excellent approximation to the initial $p_{T}$ of the jet and the ratio $z_{T}=p_{T}^{h}/p_{T}^{\gamma}$ is used as a proxy for the jet fragmentation function. A statistical subtraction is used to extract the direct photon-hadron yields in Au$+$Au collisions while a photon isolation cut is applied in $p$$+$$p$. $I_{\rm AA}$, the ratio of jet fragment yield in Au$+$Au to that in $p$$+$$p$, indicates modification of the jet fragmentation function. Suppression, most likely due to energy loss in the medium, is seen at high $z_T$. The fragment yield at low $z_T$ is enhanced at large angles. Such a trend is expected from redistribution of the lost energy into increased production of low-momentum particles.' author: - 'A. Adare' - 'S. Afanasiev' - 'C. Aidala' - 'N.N. Ajitanand' - 'Y. Akiba' - 'R. Akimoto' - 'H. Al-Bataineh' - 'H. Al-Ta’ani' - 'J. Alexander' - 'A. Angerami' - 'K. Aoki' - 'N. Apadula' - 'L. Aphecetche' - 'Y. Aramaki' - 'R. Armendariz' - 'S.H. Aronson' - 'J. Asai' - 'H. Asano' - 'E.C. Aschenauer' - 'E.T. Atomssa' - 'R. Averbeck' - 'T.C. Awes' - 'B. Azmoun' - 'V. Babintsev' - 'M. Bai' - 'G. Baksay' - 'L. Baksay' - 'A. Baldisseri' - 'B. Bannier' - 'K.N. Barish' - 'P.D. Barnes' - 'B. Bassalleck' - 'A.T. Basye' - 'S. Bathe' - 'S. Batsouli' - 'V. Baublis' - 'C. Baumann' - 'S. Baumgart' - 'A. Bazilevsky' - 'S. Belikov' - 'R. Belmont' - 'R. Bennett' - 'A. Berdnikov' - 'Y. Berdnikov' - 'A.A. Bickley' - 'X. Bing' - 'D.S. Blau' - 'J.G. Boissevain' - 'J.S. Bok' - 'H. Borel' - 'K. Boyle' - 'M.L. Brooks' - 'H. Buesching' - 'V. Bumazhnov' - 'G. Bunce' - 'S. Butsyk' - 'C.M. Camacho' - 'S. Campbell' - 'P. Castera' - 'B.S. Chang' - 'W.C. Chang' - 'J.-L. Charvet' - 'C.-H. Chen' - 'S. Chernichenko' - 'C.Y. Chi' - 'J. Chiba' - 'M. Chiu' - 'I.J. Choi' - 'J.B. Choi' - 'S. Choi' - 'R.K. Choudhury' - 'P. Christiansen' - 'T. Chujo' - 'P. Chung' - 'A. Churyn' - 'O. Chvala' - 'V. Cianciolo' - 'Z. Citron' - 'C.R. Cleven' - 'B.A. Cole' - 'M.P. Comets' - 'M. Connors' - 'P. Constantin' - 'M. Csanád' - 'T. Csörgő' - 'T. Dahms' - 'S. Dairaku' - 'I. Danchev' - 'K. Das' - 'A. Datta' - 'M.S. Daugherity' - 'G. David' - 'M.B. Deaton' - 'K. Dehmelt' - 'H. Delagrange' - 'A. Denisov' - 'D. d’Enterria' - 'A. Deshpande' - 'E.J. Desmond' - 'K.V. Dharmawardane' - 'O. Dietzsch' - 'L. Ding' - 'A. Dion' - 'M. Donadelli' - 'O. Drapier' - 'A. Drees' - 'K.A. Drees' - 'A.K. Dubey' - 'J.M. Durham' - 'A. Durum' - 'D. Dutta' - 'V. Dzhordzhadze' - 'L. D’Orazio' - 'S. Edwards' - 'Y.V. Efremenko' - 'J. Egdemir' - 'F. Ellinghaus' - 'W.S. Emam' - 'T. Engelmore' - 'A. Enokizono' - 'H. En’yo' - 'S. Esumi' - 'K.O. Eyser' - 'B. Fadem' - 'D.E. Fields' - 'M. Finger' - 'M. Finger,Jr.' - 'F. Fleuret' - 'S.L. Fokin' - 'Z. Fraenkel' - 'J.E. Frantz' - 'A. Franz' - 'A.D. Frawley' - 'K. Fujiwara' - 'Y. Fukao' - 'T. Fusayasu' - 'S. Gadrat' - 'K. Gainey' - 'C. Gal' - 'A. Garishvili' - 'I. Garishvili' - 'A. Glenn' - 'H. Gong' - 'X. Gong' - 'M. Gonin' - 'J. Gosset' - 'Y. Goto' - 'R. Granier de Cassagnac' - 'N. Grau' - 'S.V. Greene' - 'M. Grosse Perdekamp' - 'T. Gunji' - 'L. Guo' - 'H.-[Å]{}. Gustafsson' - 'T. Hachiya' - 'A. Hadj Henni' - 'C. Haegemann' - 'J.S. Haggerty' - 'K.I. Hahn' - 'H. Hamagaki' - 'J. Hamblen' - 'R. Han' - 'J. Hanks' - 'H. Harada' - 'E.P. Hartouni' - 'K. Haruna' - 'K. Hashimoto' - 'E. Haslum' - 'R. Hayano' - 'X. He' - 'M. Heffner' - 'T.K. Hemmick' - 'T. Hester' - 'H. Hiejima' - 'J.C. Hill' - 'R. Hobbs' - 'M. Hohlmann' - 'R.S. Hollis' - 'W. Holzmann' - 'K. Homma' - 'B. Hong' - 'T. Horaguchi' - 'Y. Hori' - 'D. Hornback' - 'S. Huang' - 'T. Ichihara' - 'R. Ichimiya' - 'J. Ide' - 'H. Iinuma' - 'Y. Ikeda' - 'K. Imai' - 'J. Imrek' - 'M. Inaba' - 'Y. Inoue' - 'A. Iordanova' - 'D. Isenhower' - 'L. Isenhower' - 'M. Ishihara' - 'T. Isobe' - 'M. Issah' - 'A. Isupov' - 'D. Ivanischev' - 'B.V. Jacak' - 'M. Javani' - 'J. Jia' - 'X. Jiang' - 'J. Jin' - 'O. Jinnouchi' - 'B.M. Johnson' - 'K.S. Joo' - 'D. Jouan' - 'D.S. Jumper' - 'F. Kajihara' - 'S. Kametani' - 'N. Kamihara' - 'J. Kamin' - 'M. Kaneta' - 'S. Kaneti' - 'B.H. Kang' - 'J.H. Kang' - 'J.S. Kang' - 'H. Kanou' - 'J. Kapustinsky' - 'K. Karatsu' - 'M. Kasai' - 'D. Kawall' - 'M. Kawashima' - 'A.V. Kazantsev' - 'T. Kempel' - 'A. Khanzadeev' - 'K.M. Kijima' - 'J. Kikuchi' - 'B.I. Kim' - 'C. Kim' - 'D.H. Kim' - 'D.J. Kim' - 'E. Kim' - 'E.-J. Kim' - 'H.J. Kim' - 'K.-B. Kim' - 'S.H. Kim' - 'Y.-J. Kim' - 'Y.J. Kim' - 'Y.K. Kim' - 'E. Kinney' - 'K. Kiriluk' - 'Á. Kiss' - 'E. Kistenev' - 'A. Kiyomichi' - 'J. Klatsky' - 'J. Klay' - 'C. Klein-Boesing' - 'D. Kleinjan' - 'P. Kline' - 'L. Kochenda' - 'V. Kochetkov' - 'Y. Komatsu' - 'B. Komkov' - 'M. Konno' - 'J. Koster' - 'D. Kotchetkov' - 'D. Kotov' - 'A. Kozlov' - 'A. Král' - 'A. Kravitz' - 'F. Krizek' - 'J. Kubart' - 'G.J. Kunde' - 'N. Kurihara' - 'K. Kurita' - 'M. Kurosawa' - 'M.J. Kweon' - 'Y. Kwon' - 'G.S. Kyle' - 'R. Lacey' - 'Y.S. Lai' - 'J.G. Lajoie' - 'D. Layton' - 'A. Lebedev' - 'B. Lee' - 'D.M. Lee' - 'J. Lee' - 'K. Lee' - 'K.B. Lee' - 'K.S. Lee' - 'M.K. Lee' - 'S.H. Lee' - 'S.R. Lee' - 'T. Lee' - 'M.J. Leitch' - 'M.A.L. Leite' - 'M. Leitgab' - 'E. Leitner' - 'B. Lenzi' - 'B. Lewis' - 'X. Li' - 'P. Liebing' - 'S.H. Lim' - 'L.A. Linden Levy' - 'T. Liška' - 'A. Litvinenko' - 'H. Liu' - 'M.X. Liu' - 'B. Love' - 'R. Luechtenborg' - 'D. Lynch' - 'C.F. Maguire' - 'Y.I. Makdisi' - 'M. Makek' - 'A. Malakhov' - 'M.D. Malik' - 'A. Manion' - 'V.I. Manko' - 'E. Mannel' - 'Y. Mao' - 'L. Mašek' - 'H. Masui' - 'S. Masumoto' - 'F. Matathias' - 'M. McCumber' - 'P.L. McGaughey' - 'D. McGlinchey' - 'C. McKinney' - 'N. Means' - 'M. Mendoza' - 'B. Meredith' - 'Y. Miake' - 'T. Mibe' - 'A.C. Mignerey' - 'P. Mikeš' - 'K. Miki' - 'T.E. Miller' - 'A. Milov' - 'S. Mioduszewski' - 'D.K. Mishra' - 'M. Mishra' - 'J.T. Mitchell' - 'M. Mitrovski' - 'Y. Miyachi' - 'S. Miyasaka' - 'A.K. Mohanty' - 'H.J. Moon' - 'Y. Morino' - 'A. Morreale' - 'D.P. Morrison' - 'S. Motschwiller' - 'T.V. Moukhanova' - 'D. Mukhopadhyay' - 'T. Murakami' - 'J. Murata' - 'T. Nagae' - 'S. Nagamiya' - 'Y. Nagata' - 'J.L. Nagle' - 'M. Naglis' - 'M.I. Nagy' - 'I. Nakagawa' - 'Y. Nakamiya' - 'K.R. Nakamura' - 'T. Nakamura' - 'K. Nakano' - 'C. Nattrass' - 'A. Nederlof' - 'J. Newby' - 'M. Nguyen' - 'M. Nihashi' - 'T. Niita' - 'B.E. Norman' - 'R. Nouicer' - 'N. Novitzky' - 'A.S. Nyanin' - 'E. O’Brien' - 'S.X. Oda' - 'C.A. Ogilvie' - 'H. Ohnishi' - 'M. Oka' - 'K. Okada' - 'O.O. Omiwade' - 'Y. Onuki' - 'A. Oskarsson' - 'M. Ouchida' - 'K. Ozawa' - 'R. Pak' - 'D. Pal' - 'A.P.T. Palounek' - 'V. Pantuev' - 'V. Papavassiliou' - 'B.H. Park' - 'I.H. Park' - 'J. Park' - 'S.K. Park' - 'W.J. Park' - 'S.F. Pate' - 'L. Patel' - 'H. Pei' - 'J.-C. Peng' - 'H. Pereira' - 'V. Peresedov' - 'D.Yu. Peressounko' - 'R. Petti' - 'C. Pinkenburg' - 'R.P. Pisani' - 'M. Proissl' - 'M.L. Purschke' - 'A.K. Purwar' - 'H. Qu' - 'J. Rak' - 'A. Rakotozafindrabe' - 'I. Ravinovich' - 'K.F. Read' - 'S. Rembeczki' - 'M. Reuter' - 'K. Reygers' - 'R. Reynolds' - 'V. Riabov' - 'Y. Riabov' - 'E. Richardson' - 'D. Roach' - 'G. Roche' - 'S.D. Rolnick' - 'A. Romana' - 'M. Rosati' - 'C.A. Rosen' - 'S.S.E. Rosendahl' - 'P. Rosnet' - 'P. Rukoyatkin' - 'P. Ružička' - 'V.L. Rykov' - 'B. Sahlmueller' - 'N. Saito' - 'T. Sakaguchi' - 'S. Sakai' - 'K. Sakashita' - 'H. Sakata' - 'V. Samsonov' - 'M. Sano' - 'S. Sano' - 'M. Sarsour' - 'S. Sato' - 'T. Sato' - 'S. Sawada' - 'K. Sedgwick' - 'J. Seele' - 'R. Seidl' - 'A.Yu. Semenov' - 'V. Semenov' - 'A. Sen' - 'R. Seto' - 'D. Sharma' - 'I. Shein' - 'A. Shevel' - 'T.-A. Shibata' - 'K. Shigaki' - 'M. Shimomura' - 'K. Shoji' - 'P. Shukla' - 'A. Sickles' - 'C.L. Silva' - 'D. Silvermyr' - 'C. Silvestre' - 'K.S. Sim' - 'B.K. Singh' - 'C.P. Singh' - 'V. Singh' - 'S. Skutnik' - 'M. Slunečka' - 'A. Soldatov' - 'R.A. Soltz' - 'W.E. Sondheim' - 'S.P. Sorensen' - 'M. Soumya' - 'I.V. Sourikova' - 'N.A. Sparks' - 'F. Staley' - 'P.W. Stankus' - 'E. Stenlund' - 'M. Stepanov' - 'A. Ster' - 'S.P. Stoll' - 'T. Sugitate' - 'C. Suire' - 'A. Sukhanov' - 'J. Sun' - 'J. Sziklai' - 'T. Tabaru' - 'S. Takagi' - 'E.M. Takagui' - 'A. Takahara' - 'A. Taketani' - 'R. Tanabe' - 'Y. Tanaka' - 'S. Taneja' - 'K. Tanida' - 'M.J. Tannenbaum' - 'S. Tarafdar' - 'A. Taranenko' - 'P. Tarján' - 'E. Tennant' - 'H. Themann' - 'T.L. Thomas' - 'T. Todoroki' - 'M. Togawa' - 'A. Toia' - 'J. Tojo' - 'L. Tomášek' - 'M. Tomášek' - 'Y. Tomita' - 'H. Torii' - 'R.S. Towell' - 'V-N. Tram' - 'I. Tserruya' - 'Y. Tsuchimoto' - 'T. Tsuji' - 'C. Vale' - 'H. Valle' - 'H.W. van Hecke' - 'M. Vargyas' - 'E. Vazquez-Zambrano' - 'A. Veicht' - 'J. Velkovska' - 'R. Vértesi' - 'A.A. Vinogradov' - 'M. Virius' - 'A. Vossen' - 'V. Vrba' - 'E. Vznuzdaev' - 'M. Wagner' - 'D. Walker' - 'X.R. Wang' - 'D. Watanabe' - 'K. Watanabe' - 'Y. Watanabe' - 'Y.S. Watanabe' - 'F. Wei' - 'R. Wei' - 'J. Wessels' - 'S.N. White' - 'D. Winter' - 'S. Wolin' - 'J.P. Wood' - 'C.L. Woody' - 'R.M. Wright' - 'M. Wysocki' - 'W. Xie' - 'Y.L. Yamaguchi' - 'K. Yamaura' - 'R. Yang' - 'A. Yanovich' - 'Z. Yasin' - 'J. Ying' - 'S. Yokkaichi' - 'Z. You' - 'G.R. Young' - 'I. Younus' - 'I.E. Yushmanov' - 'W.A. Zajc' - 'O. Zaudtke' - 'A. Zelenski' - 'C. Zhang' - 'S. Zhou' - 'J. Zimányi' - 'L. Zolin' title: 'Medium modification of jet fragmentation in Au$+$Au collisions at $\sqrt{s_{_{NN}}}=200$ GeV measured in direct photon-hadron correlations' --- Experiments at the Relativistic Heavy Ion Collider have observed the formation of a Quark-Gluon Plasma, reported as a fundamentally new state of matter [@whitepaper; @starwp; @phoboswp; @brahmswp]. High momentum quarks and gluons (partons) lose energy as they traverse this matter, resulting in the observed suppression of high transverse momentum (high $p_T$) hadrons in central heavy-ion collisions [@ppg003; @star; @urse; @bdmps]. Direct photons, however, escape the medium unmodified [@raa], since they do not interact via the strong force. This makes them an ideal probe with which to calibrate the energy of an initial hard scattering. At leading order, direct photons are produced via the quantum-chromodynamics analog of Compton scattering, $q+g\rightarrow q+\gamma$. In the limit of negligible initial transverse momentum, the final state quark and photon are emitted back-to-back in azimuth with the photon balancing the transverse momentum of the jet arising from the quark. Determining the initial momentum of the parton is key to measuring the fragmentation function of the quark jet. This initial momentum is provided by the measured energy of the unmodified direct photon, via direct photon-hadron correlations [@wang]. A study of direct photon-hadron correlations in $p$$+$$p$ collisions at $\sqrt{s_{_{NN}}} = 200$ GeV has provided a measurement of the fragmentation function in agreement with measurements from $e^{+}e^{-}$ collisions [@ppg095]. In Au$+$Au collisions contributions from next-to-leading-order processes and medium induced photon production are expected to be small ($\approx$10%) at high $p_{T}$ [@qin]. Parton energy loss in the medium can be observed as a modification to the jet fragmentation function in heavy ion collisions. The fragmentation function is defined as $D(z)=\frac{1}{N_{\rm jet}}\frac{dN(z)}{dz}$, where $z=p^{h}/p^{\rm jet}$; $p^{\rm jet}$ is the initial jet momentum, and $p^h$ is the momentum of a hadronic jet fragment. Experimentally, this is accessible using direct photon-hadron correlations, where $p^{\gamma}_{T} \approx p_{T}^{\rm jet}$. This balance is only approximate due to the transverse momentum, $k_T$, of the colliding partons inside nucleons, which on average introduces a transverse momentum imbalance and acoplanarity to the photon and its opposing jet [@ppg095; @ppg039; @ppg029]. Several energy loss models [@zoww; @qin] predicting direct photon-hadron correlations only track the medium induced parton splitting of the leading parton. Other models follow the lost energy, leading to an increase in low momentum (soft) particle production. In particular, Borghini and Weidemann [@mlla] use the modified leading log approximation (BW-MLLA) and local parton hadron duality to first reproduce the measured fragmentation function in $e^+e^-$ data. Modeling the energy loss in the medium as an increased parton splitting probability, they calculate the suppression of high $p_T$ jet fragments, as well as the redistribution of energy to lower $p_T$ fragments and resulting enhancement at low $z$. The resulting $R_{\rm AA}$ reproduces the PHENIX $\pi^{0}$ measurement for 0–10% central events. The yet-another-jet-energy-loss model (YaJEM) [@renk] traces the energy lost via gluon radiation and redistribution to soft particle production, predicting a suppression of particles at high $z$ and an enhancement at low $z$. This calculation has been done specifically for $\gamma_{\rm dir}$-$h$, making it directly comparable to this data. The predicted low-$z_{T}$ enhancement has not yet been observed within the statistical and systematic limitations of previously published data [@ppg090; @stargjet]. In this letter, we report fragmentation functions measured in Au$+$Au and $p$$+$$p$ collisions determined from the yield of hadrons recoiling opposite to direct photons (i.e. the “away-side”). The extraction of a purely direct-photon sample is complicated by the presence of photons from meson decays (dominantly $\pi^{0} \rightarrow \gamma \gamma$), which must be removed from the inclusive photon-hadron correlations. PHENIX has previously established the extraction of direct photon-hadron correlations via a statistical subtraction procedure in Au$+$Au [@ppg090] collisions and via an isolation cut in $p$$+$$p$ collisions [@ppg095]. This analysis includes 3.9 billion minimum bias Au$+$Au events collected by PHENIX in 2007 and 2.9 billion in 2010, after quality cuts. The $p$$+$$p$ data set comprises 0.5 billion photon-triggered events collected in 2005 and 2006, corresponding to total recorded integrated luminosities of 3.8 (2005) and 10.7 (2006) $\rm{pb}^{-1}$, respectively. Details on the $p$$+$$p$ measurement were previously presented in [@ppg095]. The kinematic reach and improved statistical precision of both data sets allow us to extend previous measurements [@ppg090; @stargjet], reaching a low momentum fraction, $z\approx 0.1$, where interplay between the medium and the deposited energy may be important [@urse]. The Au$+$Au minimum bias events are triggered by particles firing the beam-beam counters (BBCs), which are arrays of Čerenkov counters covering 3.1 $< | \eta | <$ 3.9 and 2$\pi$ in azimuth. These BBCs are also used to determine the collision centrality and the collision vertex position along the beam direction. The 0–40% most central collisions are presented here. Photons and hadrons are measured in two central spectrometers spanning $\pi/2$ in azimuth and $\pm 0.35$ units of pseudorapidity each [@mainNIM]. The photons are measured in one of two electromagnetic calorimeters [@emcNIM] and charged hadrons are measured by reconstructing tracks in the Drift Chambers and Pad Chambers [@trackingNIM]. In Au$+$Au, a statistical subtraction determines the direct (i.e. nondecay) photon-hadron correlations from the measured inclusive photon-hadron correlations. Using the measured associated hadron yield per inclusive photon, $Y_{\rm inc}=1/N_{inc}dN^{h-\gamma_{inc}}/d\Delta\phi$, and per decay photon, $Y_{\rm dec}$, the associated yield per direct photon, $Y_{\rm dir}$, is determined by [@ppg090]: $$\label{eqn:subtraction} Y_{\rm dir} = \frac{R_{\gamma}Y_{\rm inc}-Y_{\rm dec}}{{R_\gamma}-1}.$$ Here $R_\gamma$ is the ratio of inclusive photons to decay photons, reported by PHENIX in [@raa]. Inclusive photon-hadron correlations are determined from the distribution of photon-hadron pairs as a function of their azimuthal angular separation, $\Delta\phi$. The distribution of real pairs is divided by photon-hadron pairs in mixed events to correct for the PHENIX acceptance. The conditional, or per trigger, associated yield is extracted after subtraction of photon-hadron pairs from the bulk underlying event [@ppg039]. Such particles are expected to be correlated to one another only through the bulk anisotropy of the event, which is conventionally characterized by the Fourier coefficients $v_n$, and are removed from the inclusive and decay photon yields using the previously measured $v_{2}$ [@ppg132], and neglecting higher order terms, according to: $$\label{eqn:jetfunction} \frac{1}{N^{t}} \frac{dN^{\rm pair}}{d\Delta\phi} = \frac{1}{N^{t}}\frac{N^{\rm pair}_{real}}{\epsilon^{a}\int{d\Delta\phi}} \Bigl\{\frac{dN^{\rm pair}_{real}/d\Delta\phi}{dN^{\rm pair}_{\rm mix}/d\Delta\phi} - \ b_{0} \left[ 1 + 2 \langle v^{t}_2 v^{a}_2 \rangle \cos(2\Delta\phi) \right] \Bigr\},$$ where the subscripts $t$ and $a$ refer to trigger and associated particle, $\epsilon^{a}$ is the detection efficiency for the associated particle, and $b_{0}$ indicates the level of background pairs. The $\langle v^{t}_2 v^{a}_2 \rangle$ term modulates the background rate as a function of $\Delta\phi$, to account for correlations arising from flow of the bulk. The potential effect of ignoring $v_{3}$ in this extraction was studied by extrapolating the PHENIX hadron $v_{3}$ measurements [@ppg132]. Including a modulation of the background by this $v_{3}$ in addition to the $v_{2}$ results in a change in the away-side yield on the order of a few percent, depending on $\Delta\phi$ and $p_{T}$. The resulting background shape uncertainty is minimal for the highest hadron $p_{T}$ selections used here due to the low level of combinatorial background. The $v_{3}$ effect is included as an additional systematic uncertainty on the background subtraction. An absolute background normalization is used to fix the background level, $b_{0}$, as described in [@ABS]. A [geant]{} simulation of the detector determined the acceptance and efficiency for the measured charged hadrons, $\epsilon^{a}$. The uncertainty on $\epsilon^{a}$ leads to an 8.8% overall and 8.0% normalization uncertainty on the yields for the Au$+$Au and $p$$+$$p$ data, respectively. To measure the decay photon contribution, $\pi^{0}$-$h$ correlations are constructed following the same method as above using $\pi^{0}$s as the trigger. The $\pi^{0}$s are reconstructed from photon pairs whose invariant mass is within the window of 0.12–0.16 GeV/$c^{2}$. The $\pi^{0}$-$h$ correlations are translated into decay photon-hadron correlations according to a Monte Carlo study of the probability that a $\pi^{0}$ with a given $p_{T}$ produces a decay photon within a certain $p_{T}$ bin. This procedure is explained in detail in [@ppg090]. In $p$$+$$p$ collisions, $\gamma_{dir}$-$h$ yields were measured using an isolation cut, and removing decay photons from the inclusive sample on an event-by-event basis [@ppg095]. In the analysis, photons which combine with another photon in the event to produce a mass within the $\pi^{0}$ or $\eta$ mass windows were rejected. Next, an isolation cut was applied, requiring that the transverse electromagnetic energy and charged track momentum within a cone of 0.3 rad around the photon be less than 10% of the photon energy. Finally, a statistical subtraction similar to that in the Au$+$Au analysis eliminated contributions from decay photons which appeared isolated or whose decay partner was lost due to finite detector acceptance or efficiency. In $p$$+$$p$ collisions, the underlying event is subtracted assuming the yield of photon-hadron coincidences is zero at the minimum point in the correlation as a function of $\Delta\phi$ (ZYAM). The lowest three points outside the isolation cut region are averaged; as there is no flow in $p$$+$$p$ collisions, the background is assumed to be flat in $\Delta\phi$. By first eliminating photons from other sources event-by-event, the signal to background ratio is improved and final uncertainties are reduced. The $p$$+$$p$ results using the isolation cut agree with a statistical subtraction analysis in $p$$+$$p$, but have smaller uncertainties. The high multiplicity of the underlying event makes it difficult to perform an isolation cut in Au$+$Au, so a statistical subtraction procedure is used instead. ![(Color online) $\Delta\phi$ distribution for various associated $\xi$ bins for direct photons (circles) for the 0–40% most central Au$+$Au collisions and the $p$$+$$p$ reference (squares) in all panels. Panels (b), (d), and (f) are multiplied by a factor of 2 as indicated. []{data-label="fig:dphifig1"}](jf_combined_final){width="1.0\linewidth"} In order to study the jet fragmentation function, $D(z)$, associated hadron yields are determined as a function of $z_{T}=p_{T}^{h}/p_{T}^{\gamma}$, the ratio of the associated hadron transverse momentum, $p_{T}^{h}$, to the trigger photon transverse momentum, $p_T^{\gamma}$. Here $z_T \approx z$, since direct photon triggers balance the opposing jet. To focus on the low $z_{T}$ region, one can express the fragmentation function as a function of the variable, $\xi=ln(1/z_{T})$. To extend the accessible $z_{T}$ range, hadrons from $0.5<p_{T}<7.0$ GeV/$c$ are used in combination with a single $5<p^{\gamma}_{T}<9$ GeV/$c$ photon bin. Figure \[fig:dphifig1\] shows azimuthal pair angle distributions for the extracted direct $\gamma$-$h$ correlations in 0–40% central Au$+$Au collisions as well as comparison with the direct $\gamma$-$h$ correlations in $p$$+$$p$. Unlike on the away-side, on the trigger side ($|\Delta\phi| < \pi/2 $) the direct $\gamma$-$h$ correlations in Au$+$Au show a negligible yield, indicating that the statistical subtraction method indeed yields direct photons and that the yield of fragmentation photons in Au$+$Au is negligible within uncertainties. ![(Color online) The top panel shows per trigger yield as a function of $\xi$ for $p$$+$$p$ collisions (squares) and 0–40% most central Au$+$Au collisions (circles). The points are shifted for clarity. For reference, the dependence on $z_{T}$ is also indicated. The bottom panel shows $I_{\rm AA}$, the ratio of Au$+$Au to $p$$+$$p$ fragmentation functions. Also shown are predictions from BW-MLLA [@mlla] (dashed line), calculated at $E_{\rm jet}=7$ GeV with $f_{med}=0.8$ selected for 0–10% central Au+Au and from YaJEM-DE [@yajemde; @renkprivate] (dot-dashed curve) for 0–40% centrality and trigger photons from 9–12 GeV/$c$, both for the full away-side ($|\Delta\phi-\pi|<\pi/2$). []{data-label="fig:xidistrib"}](xi_dist_topztaxis){width="1.0\linewidth"} On the away side the associated particle yield is visible, and there is significant variation when comparing the correlations in Au$+$Au to $p$$+$$p$. To further quantify this variation, the yields are integrated over $\Delta\phi$ for $|\pi-\Delta\phi| < \pi/2$, as a function of $\xi$, to obtain the effective fragmentation function. The top panel of Fig. \[fig:xidistrib\] shows the integrated away-side yields in Au$+$Au and $p$$+$$p$ as circles and squares, respectively. The statistical error bars include the point-to-point uncorrelated systematic uncertainty from the background subtraction, while the boxes around the points show the correlated uncertainties. For reference, the dependence on $z_{T}$ is also indicated as the upper scale axis label. To study medium modification of the jet fragmentation function, we take a ratio of the $\xi$ distribution in Au$+$Au to $p$$+$$p$. This ratio, known as $I_{\rm AA}$, is shown in the bottom panel of Fig. \[fig:xidistrib\] and can be written as $I_{\rm AA}=Y^{\rm Au+Au}/Y^{p+p}$. Much of the global scale uncertainty cancels in this ratio, but there is a remaining 6% uncertainty. In the absence of modification, $I_{\rm AA}$ would equal 1. The data instead indicate suppression at low $\xi$ and enhancement at higher $\xi$. Including all systematic uncertainties the $\chi^2/{\rm dof}$ value for the highest 4 points compared to the hypothesis that $I_{\rm AA}$ = 1 is 17.6/4, corresponding to a probability that $I_{\rm AA}$ is 1.0 for $\xi>0.8$ of less than 0.1%. The dashed curve in the bottom panel of Fig. \[fig:xidistrib\] shows $I_{\rm AA}$ calculated at $E_{\rm jet}=7$ GeV using the BW-MLLA model in medium and in vacuum. The vacuum calculation agrees well with the measured $\xi$ distribution in $e^{+}e^{-}$, and the in-medium conditions reproduce the measured $\pi^{0}$ $R_{AA}$ at high-$p_{T}$ for 0–10% central Au+Au events [@mlla]. The dot-dashed curve shows $I_{\rm AA}$ predicted by YaJEM-DE [@yajemde] for trigger photons from 9–12 GeV/$c$ for the same centrality range (0–40%) as the present data [@renkprivate]. Both models, which include all away-side jet fragments, show suppression at low $\xi$ due to parton energy loss in Au$+$Au collisions, and increasing $I_{\rm AA}$ with increasing $\xi$. In both cases, this is due to the lost energy being redistributed into enhanced production of lower momentum particles. ![(Color online) The top panel shows the $I_{\rm AA}$ for the full away-side ($|\Delta\phi-\pi|<\pi/2$) (circles) and for two restricted away-side integration ranges, $|\Delta\phi-\pi|<\pi/3$ (squares) and $|\Delta\phi-\pi|<\pi/6$ (triangles). The points are shifted for clarity. The bottom panel shows the ratio of the $I_{\rm AA}$ for $|\Delta\phi-\pi|<\pi/2$ to $|\Delta\phi-\pi|<\pi/6$.[]{data-label="fig:xiintegrals"}](xi_iaa_integrals){width="1.0\linewidth"} The suppression of $I_{\rm AA}$ at high $z_{T}$ and enhancement at low $z_{T}$ seen in these models agrees with the qualitative trend in the data. However, the models do not reproduce the location in $\xi$ where transition from suppression to enhancement is observed. Understanding the details of this transition can lead to better understanding of how lost energy is being redistributed. One such detail is how $I_{\rm AA}$ depends on the angular distribution of particles about the away-side jet axis. The top panel of Fig. \[fig:xiintegrals\] shows $I_{\rm AA}$ in three integration ranges. Reducing the integration range from $|\Delta\phi-\pi| < \pi/2$ reduces the observed enhancement and shifts the effect to higher $\xi$. If the integration range is restricted to $|\Delta\phi-\pi| < \pi/6$, the enhancement for $\xi>1.0$ becomes negligible, while still showing significant suppression for $\xi<0.8$. To better quantify the angular range of the enhancement, we can look at the ratio of $I_{\rm AA}$’s with different integration ranges, where some of the statistical and systematic uncertainties common to all $I_{\rm AA}$ cancel. The bottom panel of Fig. \[fig:xiintegrals\] shows the ratio of the full away-side integration range to the $|\Delta\phi-\pi| < \pi/6$ case. From this ratio it is clear that there is a significant variation in observed $I_{\rm AA}$ as a function of the integration range. The average ratio for $\xi > 0.8$ is $1.9 \pm 0.3$(stat) $\pm 0.3$(syst), indicating that the enhancement in $I_{\rm AA}$ seen at large $\xi$ is predominately at large angles ($|\Delta\phi-\pi| > \pi/6$). In summary, we have presented evidence for medium modification of jet fragmentation, measured via comparison of direct photon-hadron correlations in $\sqrt{s_{_{NN}}}$ = 200 GeV Au$+$Au and $p$$+$$p$ collisions. The ratio of Au$+$Au to $p$$+$$p$ yields indicates that particles are depleted at low $\xi$ or high momentum fraction, $z_{T}$, due to energy loss of quarks traversing the medium. The ratio exhibits an increasing trend toward high $\xi$, exceeding one at $\xi \ge 1.0$. Restricting the away-side azimuthal integration range reduces the enhancement at high $\xi$ significantly. This suggests that the medium enhances production of soft particles in parton fragmentation, relative to $p$$+$$p$, preferentially at large angles. We thank the staff of the Collider-Accelerator and Physics Departments at Brookhaven National Laboratory and the staff of the other PHENIX participating institutions for their vital contributions. We also thank Thorsten Renk for providing unpublished calculations and for valuable discussions. We acknowledge support from the Office of Nuclear Physics in the Office of Science of the Department of Energy, the National Science Foundation, a sponsored research grant from Renaissance Technologies LLC, Abilene Christian University Research Council, Research Foundation of SUNY, and Dean of the College of Arts and Sciences, Vanderbilt University (U.S.A), Ministry of Education, Culture, Sports, Science, and Technology and the Japan Society for the Promotion of Science (Japan), Conselho Nacional de Desenvolvimento Científico e Tecnol[ó]{}gico and Fundaç[ã]{}o de Amparo [à]{} Pesquisa do Estado de S[ã]{}o Paulo (Brazil), Natural Science Foundation of China (P. R. China), Ministry of Education, Youth and Sports (Czech Republic), Centre National de la Recherche Scientifique, Commissariat [à]{} l’[É]{}nergie Atomique, and Institut National de Physique Nucl[é]{}aire et de Physique des Particules (France), Bundesministerium für Bildung und Forschung, Deutscher Akademischer Austausch Dienst, and Alexander von Humboldt Stiftung (Germany), Hungarian National Science Fund, OTKA (Hungary), Department of Atomic Energy and Department of Science and Technology (India), Israel Science Foundation (Israel), National Research Foundation and WCU program of the Ministry Education Science and Technology (Korea), Ministry of Education and Science, Russian Academy of Sciences, Federal Agency of Atomic Energy (Russia), VR and Wallenberg Foundation (Sweden), the U.S. Civilian Research and Development Foundation for the Independent States of the Former Soviet Union, the US-Hungarian Fulbright Foundation for Educational Exchange, and the US-Israel Binational Science Foundation. [26]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} (), ****, (). (), ****, (). (), ****, (). (), ****, (). (), ****, (). (), ****, (). , **, vol.  (, ). , , , ****, (). (), ****, (). , , , ****, (). (), ****, (). , , , , , ****, (). (), ****, (). (), ****, (). , , , , ****, (). , . , ****, (). (), ****, (). (), ****, (). (), ****, (). (), ****, (). (), ****, (). (), ****, (). , , , ****, (). , ****, (). , .

--- author: - Carlos Gershenson and Francis Heylighen bibliography: - 'carlos.bib' - 'sos.bib' title: 'Protocol Requirements for Self-organizing Artifacts: Towards an Ambient Intelligence' --- A Scenario ========== The diversity and capabilities of devices we use at home, school, or work, are increasing constantly. The functions of different devices often overlap (e.g. a portable computer and a mobile phone have agendas; a radio-clock and a PDA have alarms), but most often we cannot combine their capabilities automatically (e.g. the PDA cannot tell the radio to set its alarm for the early Tuesday’s appointment), and users need to repeat the same tasks for different devices (e.g. setting up an address book in different devices). Moreover, using the functionality of some devices in combination with others would be convenient (e.g. if my computer has an Intelligent User Interface, I would like to use it to ask for coffee, without the need of having speech recognition in the coffee machine: The computer should be able to ask the coffee machine for cappuccino). Could we build devices so that they would *automatically coordinate*, combining their functions, and possibly producing new, “emergent" ones? The technology to achieve this is already at hand. What we lack is a proper design methodology, able to tackle the problems posed by autonomously communicating artifacts in a constantly changing technosphere. In this paper we try to delineate the requirements that such a design paradigm should fulfill. The scenario we imagine considers a nearby future where technological artifacts *self-organize*, in the sense that they are able to communicate and perform desirable tasks with minimal human intervention. This vision is closely related to the concept of “Ambient Intelligence" (AmI)[@ISTAG2001], which envisages a future where people are surrounded by “smart" and “sensitive" devices. AmI would be the result of the integration of three technologies: Ubiquitous Computing [@Weiser1993], Ubiquitous Communication, and Intelligent User Friendly Interfaces. The first one conceives of a seamless integration of computation processes taking place in the variety of artifacts that surround us, being part of “The Grid", the network that would allow anyone anywhere to access the required computing power. The present paper focuses on the aspect of Ubiquitous Communication that attempts to obtain seamless information exchange between devices. Intelligent User Friendly Interfaces should enable an intuitive, effortless interaction between users and devices. With current approaches, this scenario would be possible, since we have the technology, but extremely expensive, since people would need to buy from the same producer all of their devices. We can see a similar casein the area of Home Automation: the technology is available on the market, but it is not possible to buy today ventilation for a house, and in five years integrate the system with a new fire detector. An engineer needs to integrate them manually, so that the ventilation system could be activated if smoke is detected, simply because the ventilation system was not designed to receive such signals. These limitations increase the price and restrict the market of devices for Home Automation, since complete solutions should be bought in order to have full coordination and functionality between devices. People would be more willing to invest in Home Automation if they could have the possibility of acquiring it progressively. Requirements for self-organizing artifacts ========================================== We see self-organization as a paradigm for designing, controlling, and understanding systems [@GershensonHeylighen2003a]. A key characteristic of a self-organizing system is that structure and function of the system “emerge" from interactions between the elements. The purpose should not be explicitly designed, programmed, or controlled. The components should *interact* freely with each other and with the environment, mutually adapting to reach an intrinsically “preferable" or “fit" configuration (attractor), thus defining an emergent purpose for the system [@HeylighenGershenson2003]. By “self-organizing artifacts" we mean a setup where different devices, with different fabrications and functionalities, and moving in and out of different configurations, can communicate and integrate information to produce novel functionalities that the devices by themselves could not achieve. A first requirement for such communication is cross-platform compatibility. This is already achieved for programming with Java, and for documents with XML. Another requirement is wireless communication, which is offered by technologies such as IR, Bluetooth and WiFi. Near Field Communications (NFC) is a newly envisioned standard, proposed by a consortium headed by Sony, Nokia, and Philips, which would allow information to be transmitted between devices that come in close spatial proximity (“touching"). Even with such a standard, the problem remains that the user generally would need to specifically request such communication between devices (e.g. “transfer this file from here to there"). Ideally, the devices would *know* what we want them to do and how to do it. User Interfaces already help us to tell them our wishes. Still, one device cannot tell *another* device what we want, especially if they are produced by different manufacturers. This is a general problem of communication between artifacts: they can recognize standard messages, but they do not “know" what the messages *mean*. To avoid endless debates, we can say that the meaning of a message is determined by its *use* [@Wittgenstein1999]: if a device has received a message, and does “the right thing" (for the user), then it has “understood" the meaning of the message. Thus, the user’s satisfaction is the ultimate measure of the effectiveness of the artifacts’ performance. Another issue is how to deal with changes in technology. We do not want to reconfigure every artifact each time a new device arrives. Moreover, we want the old devices to be able at least to cope with the functionality of new ones. New devices should configure themselves as automatically as possible. Older ones may require user intervention at first (as they cannot know beforehand which functions will be required), but they should be able to cope with new technology being added to the network. The overall system must be *adaptive*, *extensible*, and *open*. An adaptive system can cope with unexpected changes in its environment, as exemplified by the constantly changing technology. Having flexibility built into our systems is desirable: they should at least be able to tolerate events they were not designed for without breaking down, but preferably try to find adapted solutions, or at least ask assistance from the user. For example, home appliances have a limited set of functions. To have them self-organize (e.g. the alarm clock coordinating with the microwave oven, and the oven with the kettle), their functions could be easily programmed to respond to unknown messages. If a new device arrives, and an old one does not know what to do when it receives a message, it can check what the user wants, thus learning how to respond appropriately. The possibility to add more devices to an existing configuration may be called *extensibility*. Suppose that a company develops adaptable and extensible devices that interact seamlessly with each other. This would still leave the problem that customers cannot add devices from other companies, as these would follow their own standards, thus creating compatibility problems. We believe that the solution is to have *open* technologies, in the spirit of GNU. Open means that everyone has free access to their specifications. The advantage is that they can develop much faster, meeting the requirements of more people, because they are developed by a global community that can try out many more approaches than any single company. Still, a company can benefit in promoting an open technology, since this would provide them with free publicity while everyone is using their protocol (e.g. Sun’s Java). Open technology can respond to the needs of the largest variety of people, while allowing problems and errors to be detected and corrected more easily. Another advantage is that it allows people to get updates developed by other users for free. For example, if I program my “old" toaster to integrate with my new mobile phone, it costs me nothing to make the program available on the Internet to anyone else who might need it. Thus, updates, extensions, and specialized applications can flow much more quickly from a global community than from a private company. Still, companies would benefit from this approach, since people would be more willing to buy new devices as integrating them into their existing, open setup, will be easier. Achieving self-organization =========================== We can divide the problem of self-organizing integration into three subproblems: 1) devices should learn to *communicate* with each other, even when they have no a priori shared understanding of what a particular message or function means; 2) devices should learn which other devices they can trust to *cooperate*, avoiding the others; 3) devices should develop an efficient *division of labour* and workflow, so that each performs that part of the overall task that it is most competent at, at the right moment, while delegating the remaining functions to the others. These issues are all part of *collective intelligence* [@Heylighen1999] or *distributed cognition* [@Hutchins1995]: a complex problem cannot be tackled by a single device or agent, but must be solved by them working together, in an efficiently coordinated, yet spatially distributed, system, where information flows from the one agent to the other according to well-adapted rules. Until now, distributed cognition has been studied mostly in existing systems, such as human organizations [@Hutchins1995] or animal “swarms" [@BonabeauEtAl1998], that have evolved over many generations to develop workable rules. Having the rules self-organize from scratch is a much bigger challenge, which has been addressed to some degree in distributed AI and multi-agent simulations of social systems. Inspired by these first explorations, we will propose a number of general mechanisms that could probably tackle the three subproblems. However, extensive simulation will clearly be needed to test and elaborate these mechanisms. Learning to communicate ======================= To communicate effectively, different agents must use the same concepts or categories. To achieve effective coordination, agents must reach a shared understanding of a concept, so that they agree about which situations and actions belong to that category, and which do not. A group of agents negotiating such a consensus may self-organize, so that a globally shared categorisation emerges out of local interactions between agents. Such self-organization has been shown in different simulations of the evolution of language [@Steels1998; @HutchinsHazelhurst1995]. Here, interacting software agents or robots try to develop a shared lexicon, so that they interpret the same expressions, symbols, or “words" in the same way. In these simulations agents interact according to a protocol called a “language *game*". There are many varieties of such games, but the general principle is that two agents “meet" in virtual space, which means that through their sensors they experience the same situation at the same time. Then they try to achieve a consensus on how to designate one of the components of their shared experience by each in turn performing elementary *moves*. In a typical move, the first agent produces an “utterance" referring to a phenomenon that belongs to one of its inbuilt or previously learned categories, and the second one finds the best fitting category for that phenomenon in its knowledge base. The second agent then indicates a phenomenon belonging to that same category. If this phenomenon also belongs to the same category for the first agent, both categorisations are reinforced, otherwise they are reduced in strength. In the next move of the “game", another phenomenon is indicated, which may or may not belong to the category. The corresponding categorisation is strengthened or weakened depending on the degree of agreement. After a number of moves the game is stopped, each agent maintaining the mutually adjusted categories. Each agent in turn is coupled to another agent in the system, to play a new game using different phenomena. After some games a stable and coherent system of categories shared by all agents is likely to emerge through self-organization. A good example of such a set-up can be found in Belpaeme’s [@Belpaeme2001] simulation of the origin of shared colour categories. If for some reason devices are not able to communicate, they should be able to notify the user, and ask for the correct interpretation of the message. This is easy, since devices have a limited functionality. It would be possible to “teach" a device what to do if it receives a particular message, and the device should “learn" the meaning of the message. Research has been done in multi-agent systems where agents negotiate their protocols [@ReedEtAl2001; @DastaniEtAl2001], which could be extended for a setup of self-organizing artifacts. However, agent communication standards, such as FIPA, still do not contemplate adaptation to new meanings. Nevertheless, there is promising research going on in this direction. Learning to cooperate ===================== Integrated devices should not only communicate, but cooperate. Cooperation may seem self-evident in preprogramed systems, where the components are explicitly designed to respond appropriately to requests made by other components. However, this is no longer the case in open, extensible configurations. For example, a person at the airport would like her PDA to collaborate with the devices present at the airport, so that it can automatically warn her when and where she has to go, or tell her which facilities are available in the airport lounge. Yet not all devices at the airport may be ready to help a PDA, e.g. because of security restrictions, because they are proprietary and reserved for paying customers, or because they simply do not care about personal wishes. Moreover, devices may be ready to share certain types of services but not others, e.g. telling users when the flight is scheduled to depart, but not how many passengers will be on it. As another example, devices may not only be uncooperative, but malevolent, in the sense that they try to manipulate other devices in a way detrimental to their user. Such devices may be programmed, e.g. by fraudsters, spies, or terrorists. There exists an extensive literature on the evolution of cooperation between initially “selfish" agents, inspired by the seminal work of Axelrod [@Axelrod1984] that compared different strategies for playing a repeated “Prisoners’ Dilemma" game. However, this game does not seem directly applicable to information exchanging devices. Moreover, the chief result, while sensible, may seem trivial: the most effective strategy to achieve robust cooperation appears to be *tit for tat*, i.e. cooperate with agents that reciprocate the cooperation, stop cooperating with those that do not. More recent, tag-based models (e.g. [@RioloEtAl2001; @HalesEdmonds2003] start from a simpler situation than the Prisoners’ Dilemma, in which one agent “donates" a service to another one, at a small cost to the donor but a larger benefit to the recipient. The main idea is that agents are identified by “tags", and that they cooperate with those agents whose tags are similar to their own. The rationale is that agents with the same type of tag belong to the same group, “family" or “culture", following the same rules, so that they can be trusted to reciprocate. For artifacts, a tag may include such markers as brand, model, and protocols understood. This would show that a device is capable and willing to lend particular services to another one, thus obviating the need for a repeated, “tit-for-tat-like" interaction probing the willingness to reciprocate. Yet extensible environments should allow the addition of very dissimilar devices, made by different companies using different standards and functionalities. Therefore, we propose a different approach, combining some advantages of tags and tit-for-tat strategies. Consider a game with the following moves: an agent makes a request and the other agent either cooperates (donates) or “defects". Agents learn from these interactions in the following manner: if the result is positive (cooperation), the agent will get more “trust" in the other agent’s cooperativeness. Thus, the probability increases that it will make further requests to that agent in the future, or react positively to the other’s requests. Vice-versa, a negative result will lead to more “distrust" and a reduced probability to make or accept requests to/from this agent. Still, to recognise this agent, it has to take its clue from the tag, which is usually not unique to that agent. This means that a later interaction may be initiated with a different agent that carries a similar tag, but that is not necessarily willing to cooperate to the same extent. We may assume that if the first few interactions with agents having similar tags all generate positive (negative) results, the agent will develop a default propensity to react positively (negatively) always to agents characterised by that type of markers. We expect that in this way the initially undirected interactions will produce a differentiation in clusters of similarly marked agents that cooperate with each other (e.g. all devices belonging to the same user or organization), but that are reluctant to interact with members of other groups (e.g. devices belonging to rival organizations). The tags and their association thus develop the function of a mediator [@Heylighen2003] that increases the probability of positive interactions by creating a division between “friends" (in-group) and “strangers" or “foes" (out-group). Note, however, that there is no assumption that an agent only cooperates with agents bearing the same tag as itself: by default it cooperates with anyone having a tag similar to the one of agents that were cooperative in the past. This means that there can be groups with which everyone cooperates (e.g. “public" devices), but also that specific types of “symbiosis" can develop in which one group systematically seeks out members of a different group to cooperate with because of their complementary capabilities. This brings us to the more complex issue of the division of labour. Learning to coordinate ====================== After having ascertained that our devices can communicate and cooperate, we still need to make sure that the functions they perform satisfy the user. This desired functionality can be viewed as a complex of tasks that need to be executed. The tasks are mutually dependent in the sense that a certain task (e.g. locating a file) has to be accomplished before subsequent tasks (e.g. downloading and playing the file) can be initiated. Each agent can either execute a task itself, or delegate it to another agent. Initially, we may assume that all agents that have a certain functionality built in (e.g. playing a sound file) are equally competent at performing that type of task. However, in practice the satisfaction of the user can vary. For example, a recording is likely to be played with a higher sound quality by a music installation than by a PDA or television. By default, devices can use certain preprogramed rules-of-thumb to decide who takes precedence (e.g. newer or more specialized devices are preferred to older, less specialized ones). Yet in an open environment there is no guarantee that such simple heuristics will produce the best result. Again, we may tackle this problem through individual learning coupled with collective self-organization. Assume that the user regularly expresses his/her overall satisfaction with the ambient intelligence environment (e.g. explicitly by clicking on a scale from one to ten, or implicitly by facial or physiological cues that express happiness/unhappiness). This score can be used as a feedback signal to the network of devices, allowing it to reinforce the more successful rules, while weakening the less effective ones. We will assume that the agent who delegated a task will increase its trust in the competence of the agent that performed that task, and thus increase its probability to delegate a similar task to the same agent in the future. Otherwise, it will reduce its trust. As demonstrated by the simulation of Gaines [@Gaines1994], this assumption is sufficient to evolve a self-reinforcing division of labour where tasks are delegated to the most “expert" agents. However, when the tasks are mutually dependent, selecting the right specialist to carry out a task is not sufficient: First the preparatory tasks have to be done by the right agents, in the right order. When the agents do not know a priori what the right order is, they can randomly attempt to execute or delegate a task, and, if this fails, pick out another task. Eventually they will find a task they can execute, either because it requires no preparation, or because a preparatory task has already been accomplished by another agent. Each completed task enables the accomplishment of a series of directly dependent tasks. In this way the overall problem will eventually be solved. In each problem cycle, agents will learn better when to take on which task by themselves, or when to delegate it to a specific other agent. We expect that this learned organisation will eventually stabilise into a system of efficient, coordinated actions, adapted to the task structure. When new devices are added to the system, system and device should mutually adapt, producing a new organization. While no single agent knows how to tackle the entire problem, the knowledge has been “distributed" across the system. The “tags" that identify agents, and the learned associations between a tag and the competence for a particular task, play the role of a mediator [@Heylighen2003], delegating tasks to the right agents and coordinating their interactions so that the problem is tackled as efficiently as possible. Conclusions =========== We cannot keep on adding functions to personal computers. They serve as text editors, game consoles, televisions, home cinemas, radios, agendas, music players, gateway to the Internet, etc. Such general devices will never produce the same quality as specialized appliances. Our PCs are like ducks: they can swim, but not as well as fish; fly, but not as well as hawks; and walk, but not as well as cats. Rather than integrate so many functions in a single device, it seems preferable to entrust them to an ever expanding network of specialized devices that is kept coordinated through an ongoing process of self-organization. We have described a number of general requirements and approaches that may enable our artifacts to learn the most effective way of cooperation. In our overall scenario, we have assumed that standard functions and interaction rules are preprogrammed by a global community to handle the most common, default situations, but that the system is moreover ready to extend its own capabilities, adapting to newly encountered tasks, situations, or devices. This ability to adapt should be already present in the interaction rules. The adaptation may be achieved through the self-organization of the system of agents, using recurrent, “game-like" interactions, in which the agents learn what messages mean and who they can trust to perform which task. Most of this can happen outside of, or in parallel with, their normal “work", using idle processing power to explore many different communication and collaboration configurations. Thus, we can imagine that our future, intelligent devices, like young animals or children, will learn to become more skilful by exploring, “playing games" with each other, and practising uncommon routines, so as to be prepared whenever the need for this kind of coordinated action appears. Acknowledgements ================ We thank Peter McBurney for useful comments. C. G. was supported in part by CONACyT of Mexico.

[**Projective modules over overrings of polynomial rings**]{}\ \ [**Abstract:**]{} Let $A$ be a commutative Noetherian ring of dimension $d$ and let $P$ be a projective $R=A[X_1,\ldots,X_l,Y_1,\ldots,Y_m,\frac {1}{f_1\ldots f_m}]$-module of rank $r\geq$ max $\{2,\dim A+1\}$, where $f_i\in A[Y_i]$. Then $(i)$ ${\mbox{\rm EL}}^1(R{\mbox{$\oplus$}}P)$ acts transitively on ${\mbox{\rm Um}}(R{\mbox{$\oplus$}}P)$. In particular, $P$ is cancellative (\[23\]). $(ii)$ If $A$ is an affine algebra over a field, then $P$ has a unimodular element (\[222\]). $(iii)$ The natural map $\Phi_r : {\mbox{\rm GL}}_r(R)/EL^1_r(R) {\rightarrow}K_1(R)$ is surjective (\[201\]). $(iv)$ Assume $f_i$ is a monic polynomial. Then $\Phi_{r+1}$ is an isomorphism (\[201\]). In the case of Laurent polynomial ring (i.e. $f_i=Y_i$), $(i)$ is due to Lindel [@lindel], $(ii)$ is due to Bhatwadekar, Lindel and Rao [@BLR] and $(iii,\, iv)$ is due to Suslin [@SU]. [**Mathematics Subject Classification (2000):**]{} Primary 13C10, secondary 13B25. [**Key words:**]{} projective module, unimodular element, cancellation problem. Introduction ============ All the rings are assumed to be commutative Noetherian and all the modules are finitely generated. Let $A$ be a ring of dimension $d$ and let $P$ be a projective $A$-module of rank $n$. We say that $P$ is [*cancellative*]{} if $P{\mbox{$\oplus$}}A^m{{\stackrel{\sim}{{\rightarrow}}}}Q{\mbox{$\oplus$}}A^m$ for some projective $A$-module $Q$ implies $P{{\stackrel{\sim}{{\rightarrow}}}}Q$. We say that $P$ has a [*unimodular element*]{} if $P{{\stackrel{\sim}{{\rightarrow}}}}P'{\mbox{$\oplus$}}A$ for some projective $A$-module $P'$. Assume rank $P >\dim A$. Then $(i)$ Bass [@Bas] proved that ${\mbox{\rm EL}}^1 (A\oplus P)$ acts transitively on ${\mbox{\rm Um}}(A\oplus P)$. In particular, $P$ is cancellative and $(ii)$ Serre [@Serre] proved that $P$ has a unimodular element. Later, Plumstead [@P] generalized both the result by proving that if $P$ is a projective $A[T]$-module of rank $> \dim A=\dim A[T]-1$, then $(i)$ $P$ is cancellative and $(ii)$ $P$ has a unimodular element. Let $P$ be a projective $A[X_1,\ldots,X_l]$-module of rank $> \dim A$, then $(i)$ Ravi Rao [@Rao] proved that $P$ is cancellative and $(ii)$ Bhatwadekar and Roy [@BRoy] proved that $P$ has a unimodular element, thus generalizing the Plumstead’s result. Let $P$ be a projective $R=A[X_1,\ldots,X_l,Y_1^{\pm 1},\ldots,Y_m^{\pm 1}]$-module of rank $> \dim A$, then $(i)$ Lindel [@lindel] proved that if rank $P\geq$ max $(2,1+\dim A)$, then ${\mbox{\rm EL}}^1(R\oplus P)$ acts transitively on ${\mbox{\rm Um}}(R\oplus P)$. In particular, $P$ is cancellative and $(ii)$ Bhatwadekar, Lindel and Rao [@BLR] proved that $P$ has a unimodular element. In another direction, Ravi Rao [@R2] generalized Plumstead’s result by proving that if $R=A[T,1/g(T)]$ or $R=A[T,\frac{f_1(T)}{g(T)},\ldots,\frac{f_r(T)}{g(T)}]$, where $g (T)\in A[T]$ is a non-zerodivisor and if $P$ is a projective $R$-module of rank $> \dim A$, then $P$ is cancellative. We will generalize Rao’s result by proving that ${\mbox{\rm EL}}^1(R\oplus P)$ acts transitively on ${\mbox{\rm Um}}(R\oplus P)$ (\[111\]). Let $R[X_1,\ldots,X_l,Y_1,\ldots,Y_m, \frac{1}{f_1\ldots f_m}]$, where $f_i\in A[Y_i]$ and let $P$ be a projective $R$-module of rank $\geq$ max $\{2,\dim A+1\}$ Then we show that $(i)$ ${\mbox{\rm EL}}^1(R\oplus P)$ acts transitively on ${\mbox{\rm Um}}(R\oplus P)$ and $(ii)$ If $A$ is an affine algebra over a field, then $P$ has a unimodular element (\[23\], \[222\]), thus generalizing results of ([@lindel], [@BLR]) where it is proved for $f_i=Y_i$. As an application of the above result, we prove the following result (\[26\]): Let ${\overline}k$ be an algebraically closed field with $1/d!\in {\overline}k$ and let $A$ be an affine ${\overline}k$-algebra of dimension $d$. Let $R=A[T,1/g(T)]$ or $R=A[T,\frac{f_1(T)}{g(T)},\ldots,\frac{f_r(T)}{g(T)}]$, where $g(T)$ is a monic polynomial and $g(T),f_1(T),\ldots,f_r(T)$ is $A[T]$-regular sequence. Then every projective $R$-module of rank $\geq d$ is cancellative. (See[@M2] for motivation) Preliminaries ============= Let $A$ be a ring and let $M$ be an $A$-module. For $m\in M$, we define $O_M(m) = \{ \varphi(m) | \varphi \in {\mbox{\rm Hom}}_R(M,R) \}$. We say that $m$ is [*unimodular*]{} if $O_M(m) = A$. The set of all unimodular elements of $M$ will be denoted by ${\mbox{\rm Um}}(M)$. We denote by ${\mbox{\rm Aut}}_A(M)$, the group of all $A$-automorphism of $M$. For an ideal $J$ of $A$, we denote by ${\mbox{\rm Aut}}_A(M,J)$, the kernel of the natural homomorphism ${\mbox{\rm Aut}}_A(M)\rightarrow {\mbox{\rm Aut}}_A(M/JM)$. We denote by ${\mbox{\rm EL}}^1(A\oplus M,J)$, the subgroup of ${\mbox{\rm Aut}}_A(A\oplus M)$ generated by all the automorphisms $\Delta_{a\varphi}=\left( \begin{smallmatrix} 1 & a\varphi\\ 0 & id_M \end{smallmatrix} \right) $ and $\Gamma_{m}=\left(\begin{smallmatrix} 1&0\\ m&id_M \end{smallmatrix}\right)$ with $a\in J,\varphi \in {\mbox{\rm Hom}}_A(M,A)$ and $m \in M$. We denote by ${\mbox{\rm Um}}^1(A\oplus M, J)$, the set of all $(a,m) \in {\mbox{\rm Um}}(A\oplus M)$ such that $a \in 1 + J$ and by ${\mbox{\rm Um}}(A\oplus M,J)$, the set of all $(a,m) \in {\mbox{\rm Um}}^1(A\oplus M,J)$ with $m \in JM$. We will write ${\mbox{\rm Um}}^1_r(A,J)$ for ${\mbox{\rm Um}}^1(A{\mbox{$\oplus$}}A^{r-1},J)$ and ${\mbox{\rm Um}}_r(A,J)$ for ${\mbox{\rm Um}}(A{\mbox{$\oplus$}}A^{r-1},J)$. We will write ${\mbox{\rm EL}}_{r}^1(A,J)$ for ${\mbox{\rm EL}}^1(A\oplus A^{r-1}, J)$, ${\mbox{\rm EL}}^1_r(A)$ for $ {\mbox{\rm EL}}_{r}^1(A,A)$ and ${\mbox{\rm EL}}^1(A\oplus M)$ for ${\mbox{\rm EL}}^1(A\oplus M,A)$. \[14\] $(i)$ Let $I\subset J$ be ideals of a ring $A$ and let $P$ be a projective $A$-module. Then, it is easy to see that the natural map ${\mbox{\rm EL}}^1(A\oplus P,J)\rightarrow {\mbox{\rm EL}}^1(\frac AI\oplus \frac P{IP},\frac JI)$ is surjective. $(ii)$ Let $E_r(A)$ be the group generated by elementary matrices $E_{i_0j_0}(a)=(a_{ij})$, where $i_0\not= j_0$, $a_{i,j}\in A$, $a_{ii}=1$, $a_{i_0j_0} =a$ and remaining $a_{ij}=0$ for $1\leq i,j\leq r$. Then using ([@W1], Lemma 2.1), it is easy to see that $E_r(A)={\mbox{\rm EL}}^1_r(A)$. The following result is a consequence of a theorem of Eisenbud-Evans as stated in ([@P], p.1420). \[EE\] Let $R$ be a ring and let $P$ be a projective $R$-module of rank $r$. Let $(a,{\alpha})\in (R{\mbox{$\oplus$}}P^*)$. Then there exists ${\beta}\in P^*$ such that ${\mbox{\rm ht\;}}I_a\geq r$, where $I=({\alpha}+a{\beta})(P)$. In particular, if the ideal $({\alpha}(P),a)$ has height $\geq r$, then ${\mbox{\rm ht\;}}I\geq r$. Further, if $({\alpha}(P),a)$ is an ideal of height $\geq r$ and $I$ is a proper ideal of $R$, then ${\mbox{\rm ht\;}}I=r$. The following two results are due to Wiemers ([@W1], Proposition 2.5 and Theorem 3.2). \[4\] Let $A$ be a ring and let $R = A[X_1,\ldots,X_n,Y_1^{\pm 1},\ldots,Y_m^{\pm 1}]$. Let $c$ be the element $1$, $X_n$ or $Y_m - 1$. If $s \in A$ and $r\geqq$ max $\{3, dim A + 2\}$, then ${\mbox{\rm EL}}_r^1(R,sc)$ acts transitively on ${\mbox{\rm Um}}_r^1(R,sc)$. \[5\] Let $A$ be a ring and let $R = A[X_1,\ldots,X_n,Y_1^{\pm 1},\ldots,Y_m^{\pm 1}]$. Let $P$ be a projective $R$-module of rank $r\geqq$ max $\{2, dim A + 1\}$. If $J$ denotes the ideal $R$, $X_nR$ or $(Y_m - 1)R$, then ${\mbox{\rm EL}}^1(R\oplus P, J)$ acts transitively on ${\mbox{\rm Um}}^1(R\oplus P, J)$. The following result is due to Ravi Rao ([@R2], Lemma 2.1). \[rr\] Let $B \subset C$ be rings of dimension $d$ and $x \in B$ such that $B_x = C_x$. Then $(i)$ $ B/(1 + xb) = C/(1 + xb)$ for all $b \in B$. $(ii)$ If $I$ is an ideal of $C$ such that $ht I\geq d$ and $I + xC = C$, then there exists $b \in B$ such that $1 + xb \in I$. $(iii)$ If $c \in C$, then $c = 1 + x + x^2 h$ mod $(1 + xb)$ for some $h \in B$ and for all $b \in B$. Let $A$ be a ring and let $M,N$ be $A$-modules. Suppose $f,g : M{{\stackrel{\sim}{{\rightarrow}}}}N$ be two isomorphisms. We say that “$f$ is [*isotopic*]{} to $g$” if there exists an isomorphism $\phi: M[X]{{\stackrel{\sim}{{\rightarrow}}}}N[X]$ such that $\phi(0) = f$ and $\phi(1) = g$. Note that if $\sigma\in {\mbox{\rm EL}}^1(A{\mbox{$\oplus$}}P)$, then $\sigma$ is isotopic to identity. The following lemma follows from the well known Quillen splitting lemma ([@Q], Lemma 1) and its proof is essentially contained in ([@Q], Theorem 1). \[8\] Let $A$ be a ring and let $P$ be a projective $A$-module. Let $s,t\in A$ be two comaximal elements. Let $\sigma\in {\mbox{\rm Aut}}_{A_{st}}(P_{st})$ which is isotopic to identity. Then $\sigma=\tau_s\theta_t$, where $\tau\in {\mbox{\rm Aut}}_{A_t}(P_t)$ such that $\tau=id$ modulo $sA$ and $\theta\in {\mbox{\rm Aut}}_{A_s}(P_s)$ such that $\theta=id$ modulo $tA$. The following two results are due to Suslin ([@SU], Corrolary 5.7 and Theorem 6.3). \[100\] Let $A$ be any ring and let $f\in A[X]$ be a monic polynomial. Let $\alpha \in {\mbox{\rm GL}}_r(A[X])$ be such that $\alpha_f\in EL^1_r(A[X]_f)$. Then $\alpha\in EL^1_r(A[X])$. \[101\] Let $A$ be a ring and $B=A[X_1,\ldots,X_l]$. Then the canonical map ${\mbox{\rm GL}}_r(B)/EL^1_r(B)\rightarrow K_1(B)$ is an isomorphism for $r\geq$ max $\{3,dimA+2\}$. In particular, if ${\alpha}\in{\mbox{\rm GL}}_r(B)$ is stably elementary, then ${\alpha}$ is elementary. Main Theorem ============ We begin this section with the following result which is easy to prove. We give the proof for the sake of completeness. \[19\] Let $A$ be a ring and let $P$ be a projective $A$-module. Let “bar” denote reduction modulo the nil radical of $A$. For an ideal $J$ of $A$, if ${\mbox{\rm EL}}^1({\overline}A\oplus {\overline}P, {\overline}J)$ acts transitively on ${\mbox{\rm Um}}^1({\overline}A\oplus{\overline}P,{\overline}J)$, then ${\mbox{\rm EL}}^1(A\oplus P,J)$ acts transitively on ${\mbox{\rm Um}}^1(A\oplus P,J)$. Let $(a,p)\in {\mbox{\rm Um}}^1( A\oplus P,J)$. By hypothesis, there exists a $\sigma\in {\mbox{\rm EL}}^1({\overline}A\oplus {\overline}P,{\overline}J)$ such that $\sigma ({\overline}a,{\overline}p) =(1,0)$. Using (\[14\]), let $\varphi \in {\mbox{\rm EL}}^1(A\oplus P,J)$ be a lift of $\sigma$ such that $\varphi(a,p)=(1+b,q)$, where $b\in N= nil(A)$ and $q\in NP$. Note that $b\in N\cap J$. Since $1+b$ is a unit, we get $\Gamma_1=\Gamma_{\frac{-q}{1+b}} \in {\mbox{\rm EL}}^1(A{\mbox{$\oplus$}}P,J)$ such that $\Gamma_1(1+b,q) = (1+b,0)$. It is easy to see that there exists $p_1,\ldots,p_n \in P$ and $\alpha_1,\ldots, \alpha_n \in P^*$ such that $\alpha_1(p_1)+ \ldots +\alpha_n(p_n) = 1$. Write $h=\sum_2^n {\alpha}_i(p_i)$. Note that $(1+b,0)=(1+\sum_1^n b{\alpha}_i(p_i),0)$, $\Gamma_{\frac{p_1}{1+b}}(1+b,0) = (1+b,p_1)$ and $\Delta_{-b\alpha_1} (1+b,p_1) =(1+bh,p_1)$, where $\Delta_{-b\alpha_1}\in {\mbox{\rm EL}}^1(A{\mbox{$\oplus$}}P, J)$. Since $1+bh$ is a unit, $\Gamma_{\frac{-p_1}{1+bh}} (1+bh,p_1)= (1+bh,0)=(1+\sum_2^n b{\alpha}_i(p_i),0) $. Applying further transformations as above, we can take $(1+\sum_2^n b{\alpha}_i(p_i),0)$ to $(1,0)$ by elements of ${\mbox{\rm EL}}^1(A{\mbox{$\oplus$}}P,J)$. $ \hfill \square$ The following lemma is similar to the Quillen’s splitting lemma (\[8\]). We will give the sketch of the proof. \[9\] Let $A$ be a ring and let $u,v$ be two comaximal elements of $A$. For any $s \in A$, every $\alpha \in {\mbox{\rm EL}}_{n}^1(A_{uv},s)$ has a splitting $(\alpha_1)_v \circ(\alpha_2)_u$, where $ \alpha_1\in {\mbox{\rm EL}}_{n}^1(A_u,s)$ and $ \alpha_2\in {\mbox{\rm EL}}_{n}^1(A_v,s)$. If $\alpha \in {\mbox{\rm EL}}_{n}^1(A_{uv},s)$, then $\alpha = \prod_{i = 1}^r \alpha_i$, where $\alpha_i$ is of the form $\left(\begin{smallmatrix} 1 & s\underline{v}\\ 0 & Id_M \\ \end{smallmatrix}\right)$ or $\left(\begin{smallmatrix} 1 & 0\\ \underline{w^t} & Id_M \\ \end{smallmatrix}\right),$ where $M=A_{uv}^{n-1}$, $\underline{v},\underline{w}\in M$. Define $\alpha(X) \in {\mbox{\rm EL}}_{n}^1(A[X]_{uv},s)$ by $\alpha(X) =\prod_{i = 1}^r \alpha_i(X)$, where $\alpha_i(X)$ is of the form $ \left(\begin{smallmatrix} 1 & sX\underline{v}\\ 0 & Id_{M[X]} \\ \end{smallmatrix}\right)$ or $\left(\begin{smallmatrix} 1 & 0\\ X\underline{w^t} & Id_{M[X]} \\ \end{smallmatrix}\right)$ as may by the case above. Since $\alpha(0)=id$ and $\alpha(1)={\alpha}$, $\alpha$ is isotopic to identity. Using proof of (\[8\]) ([@M1], Lemma 2.19), we get that $\alpha(X) = (\psi_1(X))_v \circ(\psi_2(X))_u$, where $\psi_1(X) = \alpha(X)\circ\alpha(\lambda u^kX)^{-1}\in {\mbox{\rm EL}}_n^1(A_u[X],s)$ and $\psi_2(X) = \alpha(\lambda u^kX)\in {\mbox{\rm EL}}_n^1(A_v[X],s)$ with $\lambda \in A$, $k \gg 0$. Write $\psi_1(1)={\alpha}_1 \in {\mbox{\rm EL}}_n^1(A_u,s)$ and $\psi_2(1)={\alpha}_2 \in {\mbox{\rm EL}}^1_n(A_v,s)$, we get that ${\alpha}(1)=\alpha = ({\alpha}_1)_v \circ({\alpha}_2)_u$. $\hfill \square$ Let $A$ be a ring of dimension $d$ and let $l,m,n \in {\mbox{$\mathbb N$}}\cup\{0\}$. We say that a ring $R$ is of the type $A\{d,l,m,n\}$, if $R$ is an $A$-algebra generated by $X_1,\ldots, X_l$, $Y_1,\ldots, Y_m$, $T_1,\ldots, T_n$, $\frac{1}{f_1\ldots f_m}$, $\frac{g_{11}}{h_1},\ldots,\frac{g_{1{t_1}}}{h_1},\ldots,\frac{g_{n1}}{h_n}, \ldots,\frac{g_{n{t_n}}}{h_n}$, where $X_i$’s, $Y_i$’s and $T_i$’s are variables over $A$, $f_i\in A[Y_i]$, $g_{ij}\in A[T_i]$, $h_i\in A[T_i]$ and $h_i$’s are non-zerodivisors. For Laurent polynomial ring (i.e. $f_i=Y_i$), the following result is due to Wiemers (\[4\]). \[20\] Let $A$ be a ring of dimension $d$ and let $R=A[X_1,\ldots,X_l,Y_1,\ldots,Y_m,\frac{1}{f_1\ldots f_m}]$, where $f_i\in A[Y_i]$ (i.e. $R$ is of the type $A\{d,l,m,0\}$). If $s\in A$ and $r\geq $ max $\{3, d+2\}$, then ${\mbox{\rm EL}}_{r}^1(R,s)$ acts transitively on ${\mbox{\rm Um}}_{r}^1(R,s)$. Without loss of generality, we may assume that $A$ is reduced. The case $m =0$ is due to Wiemers (\[4\]). Assume $m\geq 1$ and apply induction on $m$. Let $(a_1,\ldots,a_{r}) \in {\mbox{\rm Um}}_{r}^1(R,s)$. Consider a multiplicative closed subset $S = 1 + f_mA[Y_m]$ of $A[Y_m]$. Then $R_S =B[X_1,\ldots,X_l,Y_1,\ldots,Y_{m-1},\frac{1}{f_1\ldots f_{m-1}}]$, where $B=A[Y_m]_{{f_{m}S}}$ and $dim B = dim A$. Since $R_S$ is of the type $B\{d,l,m-1,0\}$, by induction hypothesis on $m$, there exists $\sigma \in {\mbox{\rm EL}}_{r}^1(R_S,s)$ such that $\sigma (a_1,\ldots,a_{r})= (1,0,\ldots,0)$. We can find $g\in S$ and $\sigma' \in {\mbox{\rm EL}}_{r}^1(R_g,s)$ such that $\sigma' (a_1,\ldots,a_{r})= (1,0,\ldots,0)$. Write $C=A[X_1,\ldots,X_l,Y_1,\ldots,Y_m,\frac{1}{f_1,\ldots,f_{m-1}}]$. Consider the following fiber product diagram $$\xymatrix{ C \ar@{->}[r] \ar@{->}[d] & R=C_{f_m} \ar@{->}[d] \\ C_g \ar@{->}[r] &R_g=C_{gf_m}. }$$ Since $\sigma'\in {\mbox{\rm EL}}_{r}^1(C_{gf_m},s)$, by (\[9\]), $\sigma' =(\sigma_2)_{f_m}\circ (\sigma_1)_{g} $, where $\sigma_2 \in {\mbox{\rm EL}}_{r}^1(C_g,s )$ and $\sigma_1 \in {\mbox{\rm EL}}_{r}^1(R,s)$. Since $(\sigma_1)_{g}(a_1,\ldots,a_{r})=(\sigma_2)_{f_m}^{-1} (1,0,\ldots,0)$, patching $\sigma_1(a_1,\ldots,a_{r}) \in {\mbox{\rm Um}}^1_r(C_{f_m},s)$ and $(\sigma_2)^{-1} (1,0,\ldots,0) \in {\mbox{\rm Um}}^1_r(C_g,s)$, we get a unimodular row $(c_1,\ldots,c_{r}) \in {\mbox{\rm Um}}_{r}^1(C,s)$. Since $C$ is of the type $A\{d,l+1,m-1,0\}$, by induction hypothesis on $m$, there exists $\phi \in {\mbox{\rm EL}}_{r}^1(C,s)$ such that $\phi (c_1,\ldots,c_{r}) = (1,0,\ldots,0)$. Taking projection, we get $\Phi \in {\mbox{\rm EL}}_{r}^1(R,s)$ such that $\Phi\sigma_1(a_1,\ldots,a_{r}) = (1,0,\ldots,0)$. This completes the proof. $ \hfill \square$ \[60\] Let $A$ be a ring of dimension $d$ and let $R=A[X_1,\ldots,X_l,Y_1,\ldots,Y_m,\frac{1}{f_1\ldots f_m}]$, where $f_i\in A[Y_i]$. Let $c$ be $1$ or $X_l$, If $s\in A$ and $r\geq $ max $\{3, d+2\}$, then ${\mbox{\rm EL}}_{r}^1(R,sc)$ acts transitively on ${\mbox{\rm Um}}_{r}^1(R,sc)$. Let $(a_1,\ldots,a_r)\in {\mbox{\rm Um}}_{r}^1(R,sc)$. The case $c=1$ is done by (\[20\]). Assume $c=X_l$. We can assume, after an ${\mbox{\rm EL}}_r^1(R,sX_l)$-transformation, that $a_2,\ldots,a_r \in sX_lR$. Then we can find $(b_1,\ldots,b_r)\in {\mbox{\rm Um}}_r(R,sX_l)$ such that the following equation holds: $$a_1b_1+\ldots +a_rb_r = 1. \hspace*{1in} (i)$$ Now consider the $A$-automorphism $\mu : R\rightarrow R$ defined as follows $ X_i\mapsto X_i$ for $i= 1,...,l-1$,\ $X_l \mapsto X_l(f_1\ldots f_m)^N$ for some large positive integer $N$. Applying $\mu$, we can read the image of equation $(i)$ in the subring $S=A[X_1,\ldots,X_l,Y_1,\ldots,Y_m]$. By (\[4\]), we obtain $\psi\in {\mbox{\rm EL}}_r^1(R,sX_l)$ such that $\psi(\mu(a_1),\ldots,\mu(a_r)) = (1,0,\ldots,0)$. Since $\mu^{-1}(X_l)$ and $X_l$ generate the same ideal in $R$, applying $\mu^{-1}$, the proof follows. $\hfill \square$ \[200\] Let $A$ be a ring of dimension $d$ and let $R=A[X_1,\ldots,X_l,Y_1,\ldots,Y_m,\frac{1}{f_1\ldots f_m}]$, where $f_i\in A[Y_i]$. Then ${\mbox{\rm EL}}^1_r(R)$ acts transitively on ${\mbox{\rm Um}}_r(R)$ for $r\geq max\{3,d+2\}$. The following result is similar to ([@R2], Theorem 5.1). \[201\] Let $A$ be a ring of dimension $d\geq 1$ and let $R=A[X_1,\ldots,X_l,Y_1,\ldots,Y_m,\frac{1}{f_1\ldots f_m}]$, where $f_i\in A[Y_i]$ (i.e. $R$ is of the type $A\{d,l,m,0\}$). Then \(i) the canonical map $\Phi_r: GL_r(R)/{\mbox{\rm EL}}^1_r(R)\rightarrow K_1(R)$ is surjective for $r\geq d+1$. \(ii) Assume $f_i\in A[Y_i]$ is a monic polynomial for all $i$. Then for $r \geq $ max $\{3, d+2\}$, any stably elementary matrix in $GL_r(R)$ is in ${\mbox{\rm EL}}^1_r(R)$. In particular, $\Phi_{d+2}$ is an isomorphism. $(i)$ Let $[M] \in K_1(R)$. We have to show that $[M]=[N]$ in $K_1(R)$ for some $N \in GL_{d+1}(R)$. Without loss of generality, we may assume that $M \in GL_{d+2}(R)$. By (\[20\]), there exists an elementary matrix $\sigma \in {\mbox{\rm EL}}^1_{d+2}(R)$ such that $ M\sigma = \begin{pmatrix} M' & a\\ 0 & 1 \end{pmatrix}$. Applying further $\sigma'\in {\mbox{\rm EL}}^1_{d+2}(R) $, we get $\sigma'M\sigma = \begin{pmatrix} N & 0\\ 0 & 1 \end{pmatrix}$, where $M', N \in GL_{d+1}(R)$. Hence $[M]=[N]$ in $K_1(R)$. This completes the proof of $(i)$. $(ii)$ Let $M \in GL_r(R)$ be a stably elementary matrix. For $m=0$, we are done by (\[101\]). Assume $m\geq 1$. Let $S = 1 + f_mA[Y_m]$. Then $R_S = B[X_1,\ldots,X_l,Y_1,\ldots,Y_{m-1},\frac{1}{{f_1\ldots f_{m-1}}}]$, where $B= A[Y_m]_{{f_{m}S}}$ and $dim B = dim A$. Since $R_S$ is of the type $B\{d,l,m-1,0\}$, by induction hypothesis on $m$, $M \in {\mbox{\rm EL}}^1_r(R_S)$. Hence there exists $g\in S$ such that $M\in {\mbox{\rm EL}}^1_r(R_g)$. Let $\sigma \in {\mbox{\rm EL}}^1_r(R_g)$ be such that $\sigma M = Id$. Write $C=A[X_1,\ldots,X_l,Y_1,\ldots,Y_m,\frac{1}{f_1\ldots f_{m-1}}]$. Consider the following fiber product diagram $$\xymatrix{ C \ar@{->}[r] \ar@{->}[d] & C_{f_m}=R \ar@{->}[d] \\ C_g \ar@{->}[r] &C_{gf_m}=R_g.}$$ By (\[9\]), $\sigma = (\sigma_2)_{f_m}\circ (\sigma_1)_g$, where $\sigma_2 \in {\mbox{\rm EL}}^1_{r}(C_g)$ and $\sigma_1 \in {\mbox{\rm EL}}^1_{r}(C_{f_m})$. Since $(\sigma_1 M)_{g} = (\sigma_2)_{f_m}^{-1}$, patching $\sigma_1M$ and $(\sigma_2)^{-1}$, we get $N\in GL_{r}(C)$ such that $N_{f_m}=\sigma_1 M$. Write $D=A[X_1,\ldots,X_n,Y_1,\ldots,Y_{m-1},\frac{1}{f_1\ldots f_{m-1}}]$. Then $D[Y_m]=C$ and $D[Y_m]_{f_m}=R$. Since $N\in GL_r(D[Y_m])$, $f_m\in D[Y_m]$ is a monic polynomial and $N_{f_m}=\sigma_1 M$ is stably elementary, by (\[100\]), $N$ is stably elementary. Since $C$ is of the type $A\{d,l+1,m-1,0\}$, by induction hypothesis on $m$, $N\in {\mbox{\rm EL}}^1_r(C)$. Since $\sigma_1$ is elementary, we get that $M \in {\mbox{\rm EL}}^1_r(R)$. This completes the proof of $(ii)$. $ \hfill \square$ \[22\] Let $R$ be a ring of the type $A\{d,l,m,n\}$. Let $P$ be a projective $R$-module of rank $r \geq$ max $\{2, 1+d\}$. Then there exists an $s\in A$, $p_1,\ldots ,p_r \in P$ and $\varphi_1,\ldots ,\varphi_r \in {\mbox{\rm Hom}}(P,R)$ such that the following properties holds. \(i) $P_s$ is free. \(ii) $(\varphi_i(p_j) ) =$ diagonal $(s,s,\ldots ,s)$. \(iii) $sP \subset p_1A + \ldots +p_rA$. \(iv) The image of $s$ in $A_{red}$ is a nonzero divisor. \(v) $(0: sA) = (0: s^2A)$. Without loss of generality, we may assume that $A$ is reduced. Let $S$ be the set of all non-zerodivisors in $A$. Since $dim A_S=0$ and projective $R_S$-module $P_S$ has a constant rank, we may assume that $A_S$ is a field. Then it is easy to see that $A_S[T_i,\frac{g_{ij}}{h_i}]=A_S[T_i,\frac{1}{h_i}]$ (assuming gcd $(g_{ij}, h_i)=1$). Therefore $R_S= A_S[X_1,\ldots,X_l,Y_1,\ldots,Y_m,T_1,\ldots ,T_n, \frac{1}{f_1\ldots f_mh_1\ldots h_n}]$ is a localization of a polynomial ring over a field. Hence projective modules over $R_S$ are stably free. Since $P_S$ is stably free of rank $\geq$ max $\{2,1+d\}$, by (\[20\]), $P_S$ is a free $R_S$-module of rank $r$. We can find an $s \in S$ such that $P_s$ is a free $R_s$-module. The remaining properties can be checked by taking a basis $p_1,\ldots ,p_r \in P$ of $P_s$, a basis $\varphi_1,\ldots ,\varphi_r \in {\mbox{\rm Hom}}(P,R)$ of $P_s^*$ and by replacing $s$ by some power of $s$, if needed. This completes the proof. $ \hfill \square$ \[15\] Let $R$ be a ring of the type $A\{d,l,m,n\}$. Let $P$ be a projective $R$-module of rank $r$. Choose $s\in A$, $p_1,\ldots ,p_r \in P$ and $\varphi_1,\ldots ,\varphi_r \in {\mbox{\rm Hom}}(P,R)$ satisfying the properties of (\[22\]). Let $(a,p)\in {\mbox{\rm Um}}(R\oplus P,sA)$ with $p = c_1p_1 +\ldots + c_rp_r$, where $c_i\in sR$ for $i=1$ to $r$. Assume there exists $\phi \in {\mbox{\rm EL}}_{r+1}^1(R,s)$ such that $\phi(a,c_1,\ldots ,c_r) = (1,0,\ldots ,0)$. Then there exists $ \varPhi \in {\mbox{\rm EL}}^1(R\oplus P)$ such that $\varPhi (a,p) = (1,0)$. Since $\phi \in {\mbox{\rm EL}}_{r+1}^1(R,s)$, $\phi =\prod_{j=1}^{n}\phi_j$, where $\phi_j = \Delta_{s\psi_j}$ or $\Gamma_{v^t}$ with $\psi_j = (b_{1j},\ldots ,b_{rj})\in {R^r}^*$ and $v =(f_1,\ldots, f_r)\in {R^{r}} $. Define a map $\Theta : {\mbox{\rm EL}}_{r+1}^1(R,s) \rightarrow {\mbox{\rm EL}}^1(R\oplus P)$ as follows $$\Theta (\Delta_{s\psi_j}) = \begin{pmatrix} 1 & \sum_{i=1}^r {b_{ij}}\varphi_i\\ 0 & id_P \end{pmatrix} \;\;\; and \;\;\;\Theta (\Gamma_{v^t}) =\begin{pmatrix} 1&0\\ \sum_{i=1}^r f_ip_i &id_P \end{pmatrix}.$$ Let $\varPhi = \prod_{j=1}^{n}\Theta(\phi_j) \in {\mbox{\rm EL}}^1(R{\mbox{$\oplus$}}P)$. Then it is easy to see that $\varPhi(a,p) = (1,0)$. This completes the proof. $ \hfill \square$ \[15.1\] From the proof of above lemma, it is clear that if $\phi \in {\mbox{\rm EL}}_{r+1}^1(R,sX_l)$ such that $\phi(a,c_1,\ldots ,c_r) = (1,0,\ldots ,0)$, then $\Phi \in {\mbox{\rm EL}}^1(R\oplus P,X_l)$ such that $\varPhi (a,p) = (1,0)$. For Laurent polynomial ring (i.e. $f_i=Y_i$ and $J=R$), the following result is due to Lindel [@lindel]. \[23\] Let $A$ be a ring of dimension $d$ and let $R=A[X_1,\ldots,X_l,Y_1,\ldots,Y_m,\frac{1}{f_1\ldots f_m}]$, where $f_i\in A[Y_i]$ (i.e. $R$ is of the type $A\{d,l,m,0\}$). Let $P$ be a projective $R$-module of rank $r \geq $ max $\{2, d+1\}$. If $J$ denote the ideal $R$ or $X_lR$, then ${\mbox{\rm EL}}^1(R\oplus P,J)$ acts transitively on ${\mbox{\rm Um}}^1(R\oplus P,J)$. Without loss of generality, we may assume that $A$ is a reduced. We will use induction on $d$. When $d =0$, we may assume that $A$ is a field. Hence projective modules over $R$ are stably free (proof of lemma \[22\]). Using (\[60\]), we are done. Assume $d >0$. By (\[22\]), there exists a non-zerodivisor $s\in A$, $p_1,\ldots ,p_{r} \in P$ and $\phi_1,\ldots ,\phi_{r} \in P^* = {\mbox{\rm Hom}}_R(P,R)$ satisfying the properties of (\[22\]). If $s\in A$ is a unit, then $P$ is a free and the result follows from (\[60\]). Assume $s$ is not a unit. Let $(a,p) \in {\mbox{\rm Um}}^1(R\oplus P,J)$. Let “bar” denotes reduction modulo the ideal $s^2R$. Since $dim{\overline}A< dim A$, by induction hypothesis, there exists $\varphi \in {\mbox{\rm EL}}^1({\overline}R\oplus {\overline}P,J)$ such that $\varphi({\overline}a,{\overline}p) = (1,0)$. Using (\[14\]), let $\Phi \in {\mbox{\rm EL}}^1(R\oplus P,J)$ be a lift of $\varphi$ and $\Phi(a,p)= (b,q)$, where $ b\equiv 1$ mod $s^2JR$ and $q \in s^2JP$. By (\[22\]), there exists $a_1,\ldots ,a_{r} \in sJR$ such that $q = a_1p_1 + \ldots + a_{r}p_{r}$. It follows that $(b,a_1,\ldots ,a_{r}) \in {\mbox{\rm Um}}_{r+1}(R,sJ)$. By (\[60\]), there exists $\phi \in {\mbox{\rm EL}}_{r+1}^1(R,sJ)$ such that $\phi(b,a_1,\ldots ,a_{r})=(1,0,\ldots ,0)$. Applying (\[15.1\]), we get $\Psi \in {\mbox{\rm EL}}^1(R\oplus P,J)$ such that $\Psi(b,q) = (1,0)$. Therefore $\Psi\Phi(a,p)=(1,0)$. This completes the proof. $ \hfill \square$ For Laurent polynomial ring (i.e. $f_i=Y_i$), the following result is due to Bhatwadekar-Lindel-Rao [@BLR]. \[222\] Let $k$ be a field and let $A$ be an affine $k$-algebra of dimension $d$. Let $R=A[X_1,\ldots,X_l,Y_1,\ldots,Y_m,\frac{1}{f_1\ldots f_m}]$, where $f_i\in A[Y_i]$ (i.e. $R$ is of the type $A\{d,l,m,0\}$). Then every projective $R$-module $P$ of rank $\geq d+1$ has a unimodular element. We assume that $A$ is reduced and use induction on $\dim A$. If $\dim A=0$, then every projective module of constant rank is free (\[20\], \[22\]). Assume $\dim A>0$. By (\[22\]), there exists a non-zerodivisor $s\in A$ such that $P_s$ is free $R_s$-module. Let “bar” denote reduction modulo the ideal $sR$. By induction hypothesis, ${\overline}P$ has a unimodular element, say ${\overline}p$. Clearly $(p,s)\in {\mbox{\rm Um}}(P\oplus R)$, where $p\in P$ is a lift of ${\overline}p$. By (\[EE\]), we may assume that ${\mbox{\rm ht\;}}I \geq d+1$, where $I=O_P(p)$. We claim that $I_{(1+sA)} = R_{(1+sA)}$ (i.e. $p\in {\mbox{\rm Um}}(P_{1+sA})$). Since $R$ is a Jacobson ring, $\sqrt I=\cap {\mbox{$\mathfrak m$}}$ is the intersection of all maximal ideals of $R$ containing $I$. Since $I+sR=R$, $s \notin (I\cap A)$. Let ${\mbox{$\mathfrak m$}}$ be any maximal ideal of $R$ which contains $I$. Since $A$ and $R$ are affine $k$-algebras, ${\mbox{$\mathfrak m$}}\cap A$ is a maximal ideal of $A$. Hence ${\mbox{$\mathfrak m$}}\cap A$ contains an element of the form $1+sa$ for some $a\in A$ (as $s\notin {\mbox{$\mathfrak m$}}\cap A$). Hence ${\mbox{$\mathfrak m$}}R_{(1+sA)}=R_{(1+sA)}$ and $I_{(1+sA)}=R_{(1+sA)}$. This proves the claim. Let $S= 1+sA$. Choose $t\in S$ such that $p\in {\mbox{\rm Um}}(P_t)$. Let $p_1\in {\mbox{\rm Um}}(P_s)$ and $p\in {\mbox{\rm Um}}(P_t)$. Since $R_{sS}$ is of the type $A_{sS}\{d-1,l,m,0\}$, by (\[23\]), there exist $\varphi \in {\mbox{\rm EL}}^1(P_{sS})$ such that $\varphi(p_1) = p$. We can choose $t_1=tt_2\in S$ such that $\varphi \in {\mbox{\rm EL}}^1(P_{st_1})$. By (\[8\]), $\varphi = (\varphi_1)_s\circ(\varphi_2)_{t_1}$, where $\varphi_2\in {\mbox{\rm EL}}^1(P_s)$ and $\varphi_1 \in {\mbox{\rm EL}}^1(P_{t_1})$. Consider the following fiber product diagram $$\xymatrix{ P \ar@{->}[r] \ar@{->}[d] &P_s \ar@{->}[d] \\ P_{t_1} \ar@{->}[r] &P_{st_1}. }$$ Since $(\varphi_2)_{t_1}(p_1) = (\varphi_1)_s^{-1}(p)$, patching $\varphi_2(p_1) \in {\mbox{\rm Um}}(P_s)$ and $\varphi_1^{-1}(p) \in {\mbox{\rm Um}}(P_t)$, we get a unimodular element in $P$. This proves the result. $ \hfill \square$ The following result generalizes a result of Ravi Rao [@R2] where it is proved that $P$ is cancellative. \[111\] Let $A$ be a ring of dimension $d$ and let $R=A[X,\frac{f_1}g,\ldots,\frac{f_n}g]$, where $g,f_i\in A[X]$ with $g$ a non-zerodivisor. Let $P$ be a projective $R$-module of rank $r\geq$ max $\{2,d+1\}$. Then ${\mbox{\rm EL}}^1(R{\mbox{$\oplus$}}P)$ acts transitively on ${\mbox{\rm Um}}(R{\mbox{$\oplus$}}P)$. We will assume that $A$ is reduced and apply induction on $\dim A$. If $\dim A=0$, then we may assume that $A$ is a field. Hence $R$ is a PID and $P$ is free. By (\[4\]), we are done. Assume $\dim A=d>0$. By (\[22\]), we can choose a non-zerodivisor $s\in A$, $p_1,\ldots,p_r\in P$ and $\phi_1,\ldots,\phi_r\in P^*$ satisfying the properties of (\[22\]). Let $(a,p)\in {\mbox{\rm Um}}(R{\mbox{$\oplus$}}P)$. Let “bar” denotes reduction modulo $sgR$. Then $\dim {\overline}R <\dim R$ and $r\geq \dim {\overline}R +1$. By Serre’s result [@Serre], ${\overline}P$ has a unimodular element, say ${\overline}q$. Then $(0,{\overline}q)\in {\mbox{\rm Um}}({\overline}R{\mbox{$\oplus$}}{\overline}P)$. By Bass result [@Bas], there exists $\phi\in {\mbox{\rm EL}}^1({\overline}R{\mbox{$\oplus$}}{\overline}P)$ such that $\phi({\overline}a,{\overline}p)=(0,{\overline}q)$. Using (\[14\]), let $\Phi\in {\mbox{\rm EL}}^1(R{\mbox{$\oplus$}}P)$ be a lift of $\phi$ and $\Phi(a,p)=(b,q)$, where $b\in sgR$. By (\[EE\]), we may assume that ${\mbox{\rm ht\;}}O_P(q) \geq d+1$. Write $B=A[X], x=sg,I=O_P(q)$ and $C=R$. Then $\dim B=\dim C$ and $B_{sg}=C_{sg}$. By (\[rr\](ii)), there exists $h\in A[X]$ such that $1+sgh\in O_P(q)$. Hence there exists ${\varphi}\in P^*$ such that ${\varphi}(q)=1+sgh$. By (\[rr\](iii)), there exists $b'\in R$ such that $b-b'(1+sgh)=1+sg+s^2g^2h'$ for some $h'\in A[X]$. Since $\Delta_{-b'{\varphi}}(b,q)=(b-b'{\varphi}(q),q)=(1+sg+s^2g^2h',q)=(b_0,q)$ and $\Gamma_{-q}(b_0,q)=(b_0,q-b_0q)=(b_0,sgq_1)$ for some $q_1\in P$ and $b_0\in A[X]$ with $b_0=1$ mod $sgA[X]$. Write $sgq_1=c_1p_1+\ldots+c_rp_r$ for some $c_i\in R$. Then $(b_0,c_1,\ldots,c_r)\in {\mbox{\rm Um}}^1_{r+1}(R,sg)$. It is easy to see that by adding some multiples of $b_0$ to $c_1,\ldots,c_r$, we may assume that $(b_0,c_1,\ldots,c_r)\in {\mbox{\rm Um}}^1(A[X],sgA[X])$. By (\[4\]), there exists $\Theta\in {\mbox{\rm EL}}^1_{r+1}(A[X],s)$ such that $\Theta(b_0,c_1,\ldots, c_r)=(1,0,\ldots,0)$. Applying (\[15\]), there exists $\Psi\in {\mbox{\rm EL}}^1(R{\mbox{$\oplus$}}P)$ such that $\Psi(b_0,sgq_1)=(1,0)$. This proves the result. $\hfill \square$. Let $R$ be a ring of type $A\{d,l,m,n\}$ and let $P$ be a projective $R$-module of rank $\geq $max $\{2,d+1\}$. $(i)$ Does ${\mbox{\rm EL}}^1(R{\mbox{$\oplus$}}P)$ acts transitively on ${\mbox{\rm Um}}(R{\mbox{$\oplus$}}P)$? In particular, Is $P$ cancellative? $(ii)$ Does $P$ has a unimodular element? Assume $n=0$. Then $(i)$ is (\[23\]) and for affine algebras over a field, $(ii)$ is (\[222\]). When either $P$ is free or ${\overline}k={\overline}{\mbox{$\mathbb F$}}_p$, then the following result is proved in [@M2]. \[26\] Let ${\overline}k$ be an algebraically closed field with $1/d! \in {\overline}k$ and let $A$ be an affine ${\overline}k$-algebra of dimension $d$. Let $f(T) \in A[T]$ be a monic polynomial and assume that either $(i)$ $ R = A[T,\frac{1}{f(T)}]$ or $(ii)$ $R= A[T,\frac{f_1}{f},\ldots ,\frac{f_n}{f}]$, where $f,f_1,\ldots ,f_n$ is $A[T]$-regular sequence.\ Then every projective $R$-module $P$ of rank $d$ is cancellative. By (\[22\]), there exists a non-zerodivisor $s\in A$ satisfying the properties of (\[22\]). Let $(a,p) \in {\mbox{\rm Um}}(R\oplus P)$. Let “bar” denote reduction modulo ideal $s^3A$. Since $dim {\overline}A< dim A$, by (\[23\], \[111\]), there exists a $\phi\in {\mbox{\rm EL}}^1({\overline}R\oplus {\overline}P)$ such that $\phi ({\overline}a,{\overline}p) = (1,0)$. Let $\Phi \in {\mbox{\rm EL}}^1(R\oplus P)$ be a lift of $\phi$. Then $\Phi(a,p) = (b,q)$, where $(b,q) \in {\mbox{\rm Um}}^1(R\oplus P, s^2A)$. Now the proof follows by ([@M2], Theorem 4.4). $ \hfill \square$ The proof of the following result is same as of (\[26\]) using ([@M2], Theorem 5.5). \[270\] Let $k$ be a real closed field and let $A$ be an affine $k$-algebra of dimension $d-2$. Let $f\in A[X,T]$ be a monic polynomial in $T$ which does not belong to any real maximal ideal of $A[X,T]$. Assume that either $(i)$ $R= A[X,T,1/f]$ or $(ii)$ $R = A[X,T,f_1/f,\ldots ,f_n/f]$, where $f,f_1,\ldots ,f_n$ is $A[X,T]$-regular sequence.\ Then every projective $R$-module of rank $d-1$ is cancellative. An analogue of Wiemers result ============================= We begin this section with the following result which can be proved using the same arguments as in ([@W1], Corollary 3.4) and using (\[23\]) \[24\] Let $A$ be a ring of dimension $d$ and $R = A[X_1,\ldots ,X_l,Y_1,\ldots ,Y_m,\frac{1}{f_1\ldots f_m}]$, where $f_i\in A[Y_i]$. Let $P$ be a projective $R$-module of rank $\geq d+1$. Then the natural map ${\mbox{\rm Aut}}_R(P)\rightarrow {\mbox{\rm Aut}}_{{\overline}R}(P/X_lP)$ with ${\overline}R = R/X_lR$ is surjective. Using the automorphism $\mu$ defined in (\[60\]), the following result can be proved using the same arguments as in ([@W1], Proposition 4.1). \[25\] Let $A$ be a ring of dimension $d$, $1/d! \in A$ and $R = A[X_1,\ldots ,X_l,Y_1,\ldots ,Y_m,\frac{1}{f_1\ldots f_m}]$ with $l\geq 1$, $f_i\in A[Y_i]$. Then $GL_{d+1}(R,X_lJR)$ acts transitively on ${\mbox{\rm Um}}_{d+1}(R,X_lJR)$, where $J$ is an ideal of $A$. When $f_i=Y_i$, the following result is due to Wiemers ([@W1], Theorem 4.3). The proof of this result is same as of ([@W1], Theorem 4.3) using (\[24\], \[25\]). \[261\] Let $A$ be a ring of dimension $d$ with $1/d!\in A$ and let $R = A[X_1,\ldots ,X_l,Y_1,\ldots ,Y_m,\frac{1}{f_1\ldots f_m}]$ with $f_i\in A[Y_i]$ for $i =1$ to $m$ . Let $P$ be a projective $R$-module of rank $\geq d$. If $Q$ is another projective $R$-module such that $R\oplus P \cong R\oplus Q$ and ${\overline}P \cong {\overline}Q$, then $P\cong Q$, where “bar” denote reduction modulo the ideal $(X_1,\ldots ,X_l)R$ . Using (\[26\], \[261\]), we get the following result. Let ${\overline}k$ be an algebraically closed field with $1/d! \in {\overline}k$ and let $A$ be an affine ${\overline}k$-algebra of dimension $d$. Let $f(T) \in A[T]$ be a monic polynomial and let $R=A[X_1,\ldots,X_l,T,\frac 1{f(T)}]$. Then every projective $R$-module of rank $\geq d$ is cancellative. H. Bass, [*K-theory and stable algebra*]{}, IHES [**22**]{} (1964), 5-60. S.M. Bhatwadekar, H. Lindel and R.A. Rao, [*The Bass Murthy question: Serre dimension of Laurent polynomial extensions*]{}, Invent. Math. [**81**]{} (1985), 189-203. S.M. Bhatwadekar and A. Roy, [*Some theorems about projective modules over polynomial rings*]{}, J. Algebra [**86**]{} (1984), 150-158. H. Lindel, [*Unimodular elements in projective modules*]{}, J. Algebra [**172**]{} (1995), 301-319. M.K. Keshari, [*Cancellation problem for projective modules over affine algebras*]{}, To appear in Journal of K-Theory. M.K. Keshari, [*Euler class group of a Noetherian ring*]{}, M. Phil Thesis (2001), www.math.iitb.ac.in$\backslash^\sim$keshari. B. Plumstead, [*The conjectures of Eisenbud and Evans*]{}, Amer. J. Math. [**105**]{} (1983), 1417-1433. D. Quillen, [*Projective modules over polynomial rings*]{}, Invent. Math. [**36**]{} (1976), 167-171. Ravi A. Rao, [*A question of H. Bass on the cancellative nature of large projective modules over polynomial rings*]{}, Amer. J. Math. [**110**]{} (1988), 641-657. Ravi A. Rao, [*Stability theorems for overrings of polynomial rings, II*]{}, J. Algebra [**78**]{} (1982), 437-444. J.P. Serre, [*Modules projectifs et espaces fibres a fibre vectorielle*]{}, Sem. Dubreil-Pisot [**23**]{}, 1957/58. A.A. Suslin, [*On the structure of the special linear group over polynomial rings*]{}, Math. USSR Izvestija [**11**]{} (1977), 221-238. A. Wiemers, [*Cancellation properties of projective modules over Laurent polynomial rings*]{}, J. Algebra [**156**]{} (1993), 108-124.

--- abstract: 'We consider the dynamics of one or more self gravitating shells of matter in a centrally symmetric gravitational field in the Painlevé family of gauges. We give the reduced hamiltonian for two intersecting shells, both massless and massive. Such a formulation is applied to the computation of the semiclassical action of two intersecting shells. The relation of the imaginary part of the space-part of the action to the computation of the Bogoliubov coefficients is revisited.' address: 'Department of Physics, University of Pisa, Italy and INFN Sezione di Pisa, Italy' author: - Pietro Menotti title: Reduced Hamiltonian for intersecting shells and Hawking radiation --- Introduction {#introSec} ============ In this paper we shall study the dynamics of one or more self-gravitating spherical shells of matter subject to a centrally symmetric gravitational field and the application of the ensuing formalism to the semiclassical treatment of Hawking radiation. Such field of research was started by the papers by Kraus and Wilczek [@KW1; @KW2]. The main results we shall report here are the extension of the treatment to one massive shell of matter and also the extension to two or more massive shells of matter which in the time development can also intersect [@FM]. The main appeal of the approach of [@KW1; @KW2] is that energy conservation is taken exactly into account and thus the back reaction effects can be computed. We shall keep the formalism for more that one shell as close as possible to the original formalism of [@KW1] which can be summarized in words that we shall adopt a Painlevé gauge. The extraction of the reduced hamiltonian is often classified as a very complicated procedure. We shall show that by introducing a “generating function” the derivation can be drastically simplified and also by the same token it can be extended to deal not only with massive shells but also with a finite number of massive shells. The treatment of massless shells is just a particular case. In Sec.\[reducedSec\] we shall give the derivation of the reduced action in the case of one shell while proving the independence of the canonical momentum within the Painlevé class of gauges. In Sec.\[eqmotionSec\] we shall discuss the equations of motion. In Sec.\[analyticSec\] we work out the analytic properties of the conjugate momentum $p_c$ which appears in the reduced action and in Sec.\[twoshellSec\] we shall briefly describe the extension of the treatment to more than one shell. An application of the results is given in Sec.\[exchangeSec\] giving a derivation within such a formalism of the Dray-’t Hooft- Redmount exchange relations. An important integrability relation is reported in Sec.\[integrabilitySec\] which allows to compute the imaginary part of the space integral of the canonical momenta for two shells. In Sec.\[modeSec\] we revisit the role of the imaginary part of the canonical momentum in determining the Bogoliubov coefficients and thus the most important features of Hawking radiation. In Sec.\[conclusionSec\] we give some concluding remarks. The reduced action {#reducedSec} ================== As usual we write the metric for a spherically symmetric configuration in the ADM form $$ds^2=-N^2 dt^2+L^2(dr+N^r dt)^2+R^2d\Omega^2.$$ were following [@KW1; @KW2; @FLW; @LWF] we shall choose the functions $N, N^r, L, R$ as continuous functions of the coordinates. We shall work on a finite region of space time $(t_i, t_f)\times( r_0, r_m)$ . On the two initial and final surfaces we give the intrinsic metric by specifying $R(r,t_i)$ and $L(r,t_i)$ and similarly $R(r,t_f)$ and $L(r,t_f)$. The complete action in hamiltonian form, boundary terms included is [@KW1; @KW2; @hawkinghunter; @FMP] $$\begin{aligned} \label{completeaction} S=S_{shell}+\int_{t_i}^{t_f} dt \int_{r_0}^{r_m} dr (\pi_L \dot L + \pi_R \dot R - N{\cal H}_t-N^r {\cal H}_r) + \nonumber \\ \int_{t_i}^{t_f} dt \left.(-N^r \pi_L L+ \frac{NRR'}{L})\right|^{r_m}_{r_0}\end{aligned}$$ where $$\label{shellaction} S_{shell}=\int_{t_i}^{t_f} dt ~\hat p~\dot{\hat r}.$$ $\hat r$ is the shell position and $\hat p$ its canonical conjugate momentum. ${\cal H}_r$ and ${\cal H}_t$ are the constraints $$\label{constraintr} {\cal H}_r = \pi_R R'- \pi_L'L -\hat p~\delta(r-\hat r),$$ $$\label{constraintt} {\cal H}_t = \frac{R R''}{L}+\frac{{R'}^2}{2 L}+\frac{L \pi_L^2}{2R^2}-\frac{R R' L'}{L^2}- \frac{\pi_L\pi_R}{R}-\frac{L}{2}+ \sqrt{{\hat p}^2 L^{-2}+m^2}~\delta(r-\hat r).$$ Action (\[completeaction\]) is immediately generalized to a finite number of shells. The shell action as given by eqs.(\[shellaction\],\[constraintr\],\[constraintt\]) refers to a dust shell even though generalizations to more complicated equations of state have been considered [@FMP; @goncalves; @nunez]. The boundary terms are those given in the paper by Hawking and Hunter [@hawkinghunter] and will play a very important role in the following. The function $F$ [@FM] $$\label{Fgeneral} F=R L \sqrt{\left(\frac{R'}{L}\right)^2 -1+ \frac{2{\cal M}}{R}} + RR'\log\left(\frac{R'}{L}- \sqrt{\left(\frac{R'}{L}\right)^2 -1+ \frac{2{\cal M}}{R}}\right)$$ has the remarkable property of generating the conjugate momenta as solutions of the constraints as follows $$\label{piLfromF} \pi_L =\frac{\delta F}{\delta L}=\frac{\partial F}{\partial L}$$ $$\label{piRfromF} \pi_R = \frac{\delta F}{\delta R}=\frac{\partial F}{\partial R}-\frac{\partial }{\partial r}\frac{\partial F}{\partial R'}$$ where ${\cal M}$ is a mass which is constant in $r$ except at the shell position $\hat r$ and that we shall denote by $H$ for $r$ above all the shell positions and by $M$ for $r$ below all the shell positions. We shall adopt a Painlevé gauge defined by $L=1$ everywhere. With regard to the remaining freedom in the choice of the gauge we shall choose $R=r$ except for a deformation region near the shell positions. Such deformation is unavoidable because the constraints impose a discontinuity in the derivative $R'(r)$ at $r=\hat r$ as we shall see in the following. In the Painlevé gauges $F$ becomes $$F = R W(R,R',{\cal M}) + RR'({\cal L}(R,R',{\cal M}) - {\cal B}(R,{\cal M}))$$ where $$\label{Wdefinition} W(R,R',{\cal M})= \sqrt{R'^2-1+\frac{2{\cal M}}{R}};~~ {\cal L}(R,R',{\cal M}) = \log(R'-W(R,R',{\cal M}))$$ and $${\cal B}(R,{\cal M}) = \sqrt{\frac{2{\cal M}}{R}}+ \log\left(1-\sqrt{\frac{2{\cal M}}{R}}\right)$$ where we exploited the freedom of adding to (\[Fgeneral\]) a total derivative, thus gaining for $F$ the useful property of vanishing wherever $R'=1$. For $L=1$ eqs.(\[piLfromF\],\[piRfromF\]) become $$\pi_L=R\sqrt{R'^2-1+\frac {2{\cal M}}{R}}\equiv R W;~~~~ \pi_R= \frac{[R R''+R'^2-1+{\cal M}/R]}{W}.$$ With regard to $R(r,t)$ one can choose several gauges, within the Painlevé family. ![\[inner\]Inner gauge](outer.eps){width="14pc"} ![\[inner\]Inner gauge](inner.eps){width="14pc"} ![\[generic\]Generic gauge](generic.eps){width="14pc"} Typical are the “outer gauge” characterized by $R(r,t)=r$ for $r\geq \hat r(t)$ and shown in Fig.\[outer\] and the inner gauge characterized by $R(r,t)=r$ for $r\leq \hat r(t)$ shown in Fig.\[inner\], but there are also more general choices as shown in Fig.\[generic\]. We stress that such deformation $g$ is not related to the thickness of the shell which is always zero. Using the solutions (\[piLfromF\],\[piRfromF\]) of the constraints one obtains in the outer gauge the following reduced [@FM] action where only the shell position $\hat r$ appears as degree of freedom $$\label{outerreducedaction} \int_{t_i}^{t_f} \left(p_{c}~ \dot{\hat r} -\dot M(t)\int_{r_0}^{\hat r(t)}\frac{\partial F}{\partial M} dr+ \left.(-N^r \pi_L + NRR')\right|^{r_m}_{r_0}\right)dt.$$ The $p_c$ is easily computed $$\label{pcgeneral} p_c = \hat r (\Delta {\cal L}-\Delta {\cal B})$$ where $\Delta {\cal L}={\cal L}(\hat r+\varepsilon)-{\cal L}(\hat r-\varepsilon)$ and similarly for $\Delta{\cal B}$. In deriving eq.(\[pcgeneral\]) we used the consequences of the constraints (\[constraintr\],\[constraintt\]) $$\label{disceq} \Delta R' =-\frac{V}{R};~~{\rm where~~} V=\sqrt{\hat p^2+m^2};~~~~\Delta\pi_L=-\hat p.$$ In the outer gauge we find $$\label{pc} p_c= \sqrt{2M\,\hat r}-\sqrt{2H \,\hat r}-\hat r\log\left(\frac{\hat r+\sqrt{{\hat p}^2+m^2}-\hat p- \sqrt{2H \,\hat r}}{\hat r-\sqrt{2M \,\hat r}}\right)$$ with $\hat p$ given implicitly by $$\label{fundamentalH} H-M= V +\frac{m^2}{2\hat r}-\hat p\sqrt{\frac{2H}{\hat r}};~~~~~ V=\sqrt{{\hat p}^2+m^2}.$$ This is the result obtained by Friedman, Louko, Winters-Hilt [@FLW] through a limit procedure in which the support of the deformation function goes to zero but actually it is independent of the deformation. For working out the dynamics of one shell the limit procedure is all right, but in dealing with two or more shell it is important to have a deformation on a finite range because otherwise a limit procedure would give a result dependent on the order in which the two limits, deformation and $\hat r_1\rightarrow \hat r_2$, are taken. Similarly one can compute using the general formula the reduced canonical momentum of the system in the inner gauge $$p_c^i= \sqrt{2M\,\hat r}-\sqrt{2H \,\hat r}-\hat r\log\left(\frac{\hat r-\sqrt{2H \,\hat r}} {\hat r- V + \hat p-\sqrt{2M\hat r}}\right)$$ and $\hat p$, again determined by the discontinuity equation (\[disceq\]), is given now by the implicit equation $$\label{fundamentalM} H-M=V-\frac{m^2}{2\hat r}-\hat p \sqrt{\frac{2M}{\hat r}};~~~~V=\sqrt{\hat p^2+m^2}.$$ The two reduced canonical momenta $p_c,~p_c^i$ appear to be completely different but they can be proven to be the same function of $\hat r$. One can consider also the more general gauges depicted in Fig.3 and it can be proven [@menotti] that $p_c$ is always the same. However in inner gauge a term $\dot H(t)$ appears in the reduced action while in the more general gauges both a term $\dot M$ and a term $\dot H$ appear in the action. The boundary term given in eq.(\[completeaction\]) is equivalent to $$- H N(r_m) + M N(r_0).$$ Equations of motion {#eqmotionSec} =================== In deriving the equation of motion from action (\[outerreducedaction\]) one can consider $M(t)=M$, the interior mass, as a datum of the problem and vary $H$ to obtain $$\dot{\hat r} \frac{\partial p_c}{\partial H} - N(r_m)=0$$ which using the expression of $p_c$ and the relation between $N$ and $N^r$ imposed by the gravitational equations [@FM; @FLW] can be written as $$\label{1sheqmotion} \dot{\hat r} = \frac{\hat p}{V}N(\hat r)- N^r(\hat r) = \left(\frac{\hat p}{V}-\sqrt{\frac{2H}{\hat r}}\right)N(\hat r).$$ Alternatively one can consider $H$, the total energy as a datum of the problem and vary $M(t)$. The calculation is far more complicated due to the presence of $\dot M$ in the action (\[outerreducedaction\]), but using the equations for the gravitational field one reaches [@FM] the same equation of motion (\[1sheqmotion\]). In addition a consequence of the gravitational equations of motion is the constancy in time of $M(t)$ and $H(t)$. Analytic properties of $p_c$ {#analyticSec} ============================ We saw that in the outer gauge $p_c$ is given by (\[pc\],\[fundamentalH\]). The solution of eq.(\[fundamentalH\]) for $\hat p$ is $$\frac{\hat p}{\hat r} =\frac{A \sqrt{\frac{2H}{\hat r}}~ \pm\sqrt{A^2 -(1-\frac{2H}{\hat r})\frac{m^2}{\hat r^2}}} {1-\frac{2H}{\hat r}}$$ where $$A =\frac{H-M}{\hat r}-\frac{m^2}{2 \hat r^2}.$$ If we want $\hat p$ to describe an outgoing shell we must choose the plus sign in front of the square root. Moreover the shell reaches $r=+\infty$ if and only if $H-M>m$ as expected. The logarithm in $p_c$, eq.(\[pc\]), has branch points at zero and infinity and thus we must investigate for which values or $\hat r$ such values are reached. At $\hat r=2H$, $\hat p$ has a simple pole with positive residue. Thus the numerator in the argument of the logarithm in $p_c$ goes to zero and below $2H$ it becomes $$\label{contnumerator} \hat r - V - \hat p -\sqrt{2H\hat r}$$ where here $V$ is the absolute value of the square root. Expression (\[contnumerator\]) is negative irrespective of the sign of $\hat p$ and stays so for $\hat r<2H$ because $\hat p$ is no longer singular. As a consequence $p_c$ below $2H$ acquires the imaginary part $i\pi \hat r$. Below $\hat r = 2M$ the denominator of the argument of the logarithm in eq.(\[pc\]) becomes negative so that the argument of the logarithm reverts to positive values. Thus the so called classically forbidden region where $p_c$ becomes complex is $2M<\hat r <2H$ independent of $m$ and of the deformation $g$ and the integral of the imaginary part of $p_c$ for any deformation $g$ and for any mass $m$ of the shell is $$\label{integratedimpart} {\rm Im}~\int p_c dr = \pi \int_{2M}^{2H} r dr = 2\pi(H^2-M^2)= \frac{\Delta S}{2}$$ which is the original result derived in [@KW1; @PW] in the zero mass case. Parikh and Wilczek [@PW] gave to $$\exp (-2 {\rm Im}\int p_c dr )$$ the interpretation of the tunneling probability for the emission of a quantum of energy $\omega$. Criticism and alternative proposals for the emission probability like $$\exp (- {\rm Im} \oint p_cdr )$$ $$\exp \left(-2 {\rm Im}(\int p_c dr + {\rm temporal~contribution} )\right)$$ followed [@chowdhury; @zerbini; @akhmedov; @akhmedova; @zerbini1; @zerbini2; @zerbini3; @pizzi; @belinski; @padmana; @banerjee; @kerner]. Here instead we shall discuss the role of eq.(\[pc\]) in the framework of mode analysis. This will be done in Sec.(\[modeSec\]). Two shell reduced action {#twoshellSec} ======================== ![\[2sh\] Two shell dynamics](twoshell.eps) Now we have three characteristic masses, $M$, $H$ and an intermediate mass $M_0$, see Fig.4, which can change only if the two shells cross. Working as before with $M={\rm const}$ considered as a datum of the problem we reach the reduced action [@FM] $$\begin{aligned} \label{twoshellreducedaction} && \dot{\hat r}_1p_{c1} + \dot{\hat r}_2p_{c2}+ \dot H (R(\hat r_1)-\hat r_1)\frac{\partial T}{\partial H}{\cal D} + \dot M_0 (R(\hat r_1)-\hat r_1)\frac{\partial T}{\partial M_0} {\cal D}+ \nonumber\\ && +\frac{d}{dt}\int_{r_0}^{\hat r_2} F dr-\dot M_0\int^{\hat r_2}_{\hat r_1}\frac{\partial F}{\partial M_0}dr +(-N^r\pi_L+N R R')|^{r_m}_{r_0} \end{aligned}$$ where $$\begin{aligned} \label{pc1} T=\log \frac{V_2}{R(\hat r_2)};~~~~ {\cal D} = R (\Delta{\cal L}- \Delta{\cal B})|_{\hat r_1};\\ p_{c1}= R'(\hat r_1+\varepsilon){\cal D};~~~~ p_{c2}= p_{c2}^0+\frac{d}{d\hat r_2}(R(\hat r_1)-\hat r_1){\cal D}\end{aligned}$$ and $p^0_{c2}$ is given by eq.(\[pc\]) with $M$ replaced by $M_0$. The novelty is that now even in the outer gauge the time derivative of $H$ intervenes in addition to $\dot M_0$ and $p_{c1}$ and $p_{c2}$ depend both on $\hat r_1$ and $\hat r_2$. We can vary $\hat r_1$, $\hat r_2$, $H$ and $M_0$ independently obtaining the correct equations of motion [@FM]. As expected one finds that the exterior shell moves irrespective of the dynamics which develops at lower values of $r$ until a crossing occurs. The exchange relations {#exchangeSec} ====================== In case of crossing of the two shells from the equations of Sec.(\[twoshellSec\]) we can obtain relations between $M, M_0,H, \hat r_e$ and $M'_0$ being $\hat r_e$ the the shell position at the crossing and $M'_0$ the intermediate mass after the crossing. During the crossing the masses of the shells can change, provided they satisfy a relation analogous to the energy-momentum conservation is special relativity $$\begin{aligned} & & \hat p_1+\hat p_2=\hat p'_1+\hat p'_2\\ & & V_1+V_2=V'_1+V'_2;~~~~V_n = \sqrt{\hat p_n^2+m_n^2};~~~~V'_n = \sqrt{\hat {p'}_n^2+{m'}_n^2}. \nonumber\end{aligned}$$ This is an outcome of the constraints. It is not possible to predict the final masses of the two shells as it depends on the details of the interaction which has to be specified. A relatively simple case is when the masses are unchanged during the crossing (transparent crossing) [@FM] and the simplest case is the crossing of two massless shells which remain massless, thus re-obtaining the well known Dray-’t Hooft-Redmount relations [@DtH; @redmount] which we report here below $$H \hat r_e + M \hat r_e-2 H M = M_0 \hat r_e- 2 M_0 M'_0 + M'_0 \hat r_e$$ being $\hat r_e$ the crossing radius. ![\[DHR\]The exchange diagram](DHR.eps) Integrability of the form $p_{c1}d\hat r_1+p_{c2}d\hat r_2$ and imaginary part of the space- component of the action {#integrabilitySec} ==================================================================================================================== The space-part of the on-shell action for two massive shells of matter is given from eq.(\[twoshellreducedaction\]) by $$\int_{t_i}^{t_f} (p_{c1}~\dot{\hat r}_1+ p_{c2}~\dot{\hat r}_2)~dt$$ as on the equations of motion $\dot H(t)=\dot M(t)=0$. It is possible to prove a theorem analogous to the one found in the books of Whittaker and Arnold [@whittaker; @arnold] i.e. that in presence of a constant of motion, in addition to the hamiltonian, the form $p_{c1} d\hat r_1+ p_{c2} d\hat r_2$ is closed even though the proof is somewhat different [@FM]. The intermediate mass $M_0$ plays the role of the additional constant of motion. The above result allows to deform the integration path as to bring $\hat r_1$ immediately below or above $\hat r_2$. This is allowed by the absence of discontinuities in our scheme. But, in words, two coaleshed shell have the same properties of a single shell with energy $H-M$. The final result is that $$\label{intpc1pc2} {\rm Im}\int_{t_i}^{t_f} dt (p_{c1}~\dot{\hat r}_1+ p_{c2}~\dot{\hat r}_{c2})= {\rm Im} \int_{r_{1i}, r_{2i}}^{r_{1f}, r_{2f}} (p_{c1}~d\hat r_1+p_{c2}~d\hat r_2) = 2\pi(H^2-M^2).$$ The result (\[intpc1pc2\]) holds also for in the case when at the crossing the two shells can change their mass. For details we refer to [@FM]. Mode analysis {#modeSec} ============= The original way to extract information on the spectrum of the radiation is mode analysis [@hawking; @KW1; @KVK]. In the massless case $$\int^{\hat r} p_c d\hat r' = f(\hat r, M)-f(\hat r, H)$$ can be computed exactly and its expansion to first order in $\omega=H-M$ is $$\int^{\hat r} p_c d\hat r'= 4 M \omega\log(\hat r - 2M)+~{\rm regular~terms}.$$ Given the semiclassical mode $$\phi(r,t)=e^{i\int^{\hat r} p_c d\hat r' - i\omega t}$$ we can perform the analysis of the mode regular at the horizon in terms of the above mode. This analysis can be performed either by scalar product [@BD] i.e. space integration where one has to keep into account that the background metric is Painlevé or by time Fourier analysis. In this way one obtains the well known formulas for the Bogoliubov coefficients. The regular outgoing modes near the horizon are given by $$\psi(\hat r,t)=e^{ik(\hat r-2M)e^{-\frac{t}{4M}}}.$$ Computing the scalar product, and taking into account that the background metric is the Painlevé metric, we have $$\label{scalarproduct} -i\int(\psi^*\partial_\rho\phi-\phi\partial_\rho\psi^*)g^{\rho 0}\varepsilon_{0r\theta\phi}\sqrt{-g} ~d\hat r d\theta d\phi$$ with $$g^{rt}=N^r=\sqrt{\frac{2M}{\hat r}}~~~~{\rm and}~~~~\sqrt{-g}=1$$ and the integration region is outside the horizon. Keeping only the most singular terms at the horizon eq.(\[scalarproduct\]) reduces, with $\tau=\exp(-t/4M)$, to $$\int_0^\infty e^{-ikx\tau} e^{4iM\omega \log x -i \omega t} ~\frac{dx}{x}.$$ We can compute such scalar products at $t=0$ obtaining the standard Hawking integral $$\label{alphakomega} \int_0^{\infty}e^{-ikx} e^{4iM\omega \log x} ~\frac{dx}{x}= e^{2\pi\omega M}(k)^{-4i\omega M}\Gamma(4i\omega M)= {\rm const}~\alpha^*_{k\omega}$$ which gives the dominant contribution for large $k$. The coefficient $\beta_{k\omega}$ is obtained by changing in (\[alphakomega\]) $\omega$ into $-\omega$. Alternatively we can extract the Bogoliubov coefficients by performing a time Fourier transform i.e. $$\int_{-\infty}^{+\infty}e^{-ikx\tau} e^{4iM\omega\log x -i\omega t}dt= \int_0^{+\infty} e^{-ikx\tau} e^{4iM\omega \log(x\tau)} ~\frac{d\tau}{\tau}$$ which is, as it should be, a result independent of $r$ (i.e. $x$) and reproduces eq.(\[alphakomega\]). The derivation is valid to first order in $\omega$ which is the realm of validity of the external field approximation. For finite $\omega$ it is problematic to perform a space integration on modes, because we have not a well defined background metric. Thus Kraus and Wilczek [@KW1] followed the time Fourier analysis method. In order to do so one has to construct the non-perturbative modes regular at the horizon. This is not a completely trivial task. The regular modes are given by $$e^{iS}=e^{i k \hat r(0) + i \int_{\hat r(0)}^r p_c d\hat r - i(H-M) t}$$ where $S$ is the on shell action computed with the following boundary conditions: 1) At time $t$ the shell position is $\hat r$, outside the horizon. 2) At time $t=0$ the conjugate momentum is $k$, i.e. $S(0,\hat r)=k\hat r$. Due to these boundary conditions $\hat r(0)$, $H$ and as a consequence $p_c$ depend on $k,t,\hat r$ even if along each trajectory $H$ is always a constant of motion. Using the saddle point approximation [@KW1; @KVK] one obtains that the absolute value of the Bogoliubov coefficient $\alpha_{k\omega}$ is given by $$\label{kvkintegral} \left|e^{i\int_{r(0)}^r p_c(r',H, M) dr'}\right|= e^{-{\rm Im}\int_{r(0)}^r p_c(r',H, M) dr'}$$ computed for $H=M+\omega$. What is important here is that only the space part of the action appears, as the time part $(H-M)t$ cancels with $\omega t$ at the saddle point. $\hat r(0)$ is given by the condition $$k = \hat r(0) \log\frac{\sqrt{\hat r(0)}-\sqrt{2M}} {\sqrt{\hat r(0)}-\sqrt{2H}}$$ which for $H=M+\omega>M$ is solved by $$2M<2H<\hat r(0)$$ and thus there is no imaginary contribution to the integral from the gap $2M,2H$ as we discussed in Sect.(\[analyticSec\]). The Bogoliubov coefficient $\beta_{k\omega}$ is obtained by changing in (\[kvkintegral\]) $\omega$ in $-\omega$ (always $\omega>0$). Then we have $$\hat r(0)<2H<2M$$ The value of the saddle point $t$, now complex, is given by $$t = 4H\log\frac{\sqrt{\hat r}-\sqrt{2H}}{\sqrt{\hat r(0)}-\sqrt{2H}}$$ and the r.h.s. term of eq.(\[kvkintegral\]) becomes $$\label{correctedbeta} e^{-4\pi M \omega(1 -\omega/2 M)}$$ where the last passage is due to eq.(\[integratedimpart\]) for which we gave a general proof within the family of Painlevé gauges even in the massive case. An explicit derivation of the result (\[correctedbeta\]) has been given in [@menotti] by working out the late time expansion of the time development of the action. The main point is that in the mode analysis what comes in is not the total action of a model particle crossing the horizon, but only the “space part” of it. The analysis and the results depend on the interpretation of the expression $$\exp{(iS)}$$ as modes of the field dressed by the gravitational interaction with non vanishing quanta of energy. The result eq.(\[intpc1pc2\]) proven for the emission of two shells [@FM] was interpreted [@parikh] as the absence of correlations among quanta in the emitted radiation. It would be of interest to give a similar “mode interpretation” of the result (\[intpc1pc2\]) derived for the emission of two shells which during the time evolution can also interact [@FM]. To this end one should compute the two shell modes which are regular at the horizon, and perform a time Fourier analysis of them. This has not yet been accomplished. Concluding remarks {#conclusionSec} ================== In this paper we gave a general treatment of the dynamics of one or more self- gravitating spherical shell of matter in a spherical gravitational field. We extended the treatment of [@KW1] to more than one shell [@FM] proving on the way the universality of the reduced conjugate momentum within the family of Painlevé gauges [@menotti]. This allows to give a general derivation of the exchange relations and also, exploiting an integrability result, to extend the result on the imaginary part of the space part of the action to more than one shell. Instead of following the tunneling picture here we have revisited the treatment which follows by interpreting the exponential of the action, properly subtracted, as the dressed modes of the Hawking radiation. The main point here is that according such a mode interpretation only the space part of the action and in particular its imaginary part comes in determining the Bogoliubov coefficient and thus in determining the Hawking spectrum corrected for the back reaction effects. References {#references .unnumbered} ========== [9]{} P. Kraus, F. Wilczek, [*Nucl. Phys.*]{} [**B 433**]{} (1995) 403, e-Print: arXiv:gr-qc/9408003 P. Kraus, F. Wilczek, [*Nucl. Phys.*]{} [**B 437**]{} (1995) 231, e-Print: arXiv:hep-th/9411219 F. Fiamberti, P. Menotti, [*Nucl. Phys.*]{} [**B 794**]{} (2008) 512, e-Print: arXiv:0708.2868 \[hep-th\] J. L. Friedman , J. Louko S. Winters-Hilt, [*Phys. Rev.*]{} [**D 56**]{} (1997) 7674, e-Print: arXiv:gr-qc/9706051 J. Louko, B. F. Whiting, J. L. Friedman, [*Phys. Rev.*]{} [**D 57**]{} (1998) 2279, e-Print: arXiv:gr-qc/9708012 S. W. Hawking, C. J. Hunter, [*Class. Quantum Grav.*]{} [**13**]{} (1996) 2735, e-Print: arXiv:gr-qc/9603050 W. Fischler, D. Morgan, J. Polchinski, [*Phys. Rev.*]{} [**D 41**]{} (1990) 2638; [*Phys. Rev.*]{} [**D 42**]{} (1990) 4042 S. Goncalves, [*Phys. Rev.*]{} [**D 66**]{} (2002) 084021, e-Print: arXiv:gr-qc/0212124 D. Nunez, H. P. Oliveira, J. Salim, [*Class. Quantum Grav.*]{} [**10**]{} (1993) 1117, e-Print: arXiv:gr-qc/9302003 P. Menotti e-Print: arXiv:0911.4358 \[hep-th\] M. Parikh, F. Wilczek, [*Phys. Rev. Lett.*]{} [**85**]{} (2000) 5042, e-Print: arXiv:hep-th/9907001 B. Chowdhury, [*Pramana*]{} [**70**]{} (2008) 593; [*Pramana*]{} [**70**]{} (2008) 3, e-Print: arXix:hep-th/0605197; P. Mitra, [*Phys. Lett.*]{} [**B 648**]{} (2007) 240, e-Print: arXiv:hep-th/0611265 M. Angheben, M. Nadalini, L. Vanzo, S. Zerbini, JHEP 0505:014 (2005), e-Print: arXiv:hep-th/0503081; A.J.M. Medved, E.C. Vagenas, [*Mod. Phys. Lett.*]{} [**A 20**]{} (2005) 2449, e-Print: arXiv:gr-qc/0504113 E. T. Akhmedov, V. Akhmedova, D. Singleton, [*Phys. Lett.*]{} [**B 642**]{} (2006) 124, e-Print: arXiv:hep-th/0608098 V. Akhmedova, T. Pilling, A. de Gill, D. Singleton, [*Phys. Lett.*]{} [**B 666**]{} (2008) 269, e-Print: arXiv:0804.2289 \[hep-th\] S. A. Hayward , R. Di Criscienzo, L. Vanzo, M. Nadalini, S. Zerbini, [*Class. Quant. Grav.*]{} [**26**]{} (2009) 062001, e-Print: arXiv:0806.0014 \[gr-qc\] R. Di Criscienzo, S. A. Hayward, M. Nadalini, L. Vanzo, S. Zerbini, e-Print: arXiv:0906.1725 \[gr-qc\]; e-Print: arXiv:0911.4417 S.A. Hayward, R. Di Criscienzo, M. Nadalini, L. Vanzo, S. Zerbini,e-Print: arXiv:0909.2956 \[gr-qc\]; M. Pizzi, e-Print: arXiv:0904.4572 \[gr-qc\]; e-Print: arXiv:0907.2020 \[gr-qc\]; e-Print: arXiv:0909.3800 \[gr-qc\] V.A. Belinski, e-Print: arXiv:0910.3934 \[gr-qc\] T. Dray, G. ’t Hooft, [*Comm. Math. Phys.*]{} [**99**]{} (1985) 613 I. H. Redmount, [*Prog. Theor. Phys.*]{} [**73**]{} (1985) 1401 E. T. Whittaker, [*A treatise on the analytical mechanics of particles and rigid bodies*]{}, Chapt. X (Cambridge University Press, Cambridge U.K., 1947) V. I. Arnol’d, [*Mathematical Methods of Classical Mechanics*]{}, Chapt. IX (Springer, Berlin, 1989) N.D. Birrel and P.C.W. Davies, [*Quantum field theory in curved space*]{} Cambridge Monographs on Mathematical Physics, (Cambridge University Press, Cambridge U.K. 1982); R. Wald, [*Quantum field theory in curved spacetime and black- hole thermodynamics*]{}, (The University of Chicago Press, Chicago USA 1994) S. Hawking, [*Comm. Math. Phys.*]{} [**43**]{} (1975) 199 E. Keski-Vakkuri, P. Kraus, [*Nucl. Phys.*]{} [**B 491**]{} (1997) 249, e-Print: arXiv:hep-th/9610045 M. Parikh, [*Rio de Janeiro 2003, Recent developments in theoretical and experimental general relativity, gravitation, and relativistic field theories*]{}, B 1585, e-Print: arXiv:hep-th/0402166 K. Srinivasan, T. Padmanabhan, [*Phys. Rev.*]{} [**D 60**]{} (1999) 024007, e-Print: arXiv:gr-qc/9812028 R. Banerjee, B. R. Majhi, [*Phys. Lett.*]{} [**B 675**]{} (2009) 243, e-Print:arXiv:0903.0250 \[hep-th\] R. Kerner and R.B. Mann, [*Phys. Rev.*]{} [**D 73**]{} (2006) 104010 e-Print: arXiv:gr-qc/0603019

--- abstract: 'We study the interaction of a Bose-Einstein condensate, which is confined in an optical lattice, with a largely detuned light field propagating through the condensate. If the condensate is in its ground state it acts as a periodic dielectric and gives rise to photonic band gaps at optical frequencies. The band structure of the combined system of condensed lattice-atoms and photons is studied by using the concept of polaritons. If elementary excitations of the condensate are present, they will produce defect states inside the photonic band gaps. The frequency of localized defect states is calculated using the Koster-Slater model.' address: ' School of Mathematics, Physics, Computing and Electronics, Macquarie University, Sydney, NSW 2109, Australia ' author: - 'Karl-Peter Marzlin and Weiping Zhang' title: 'Photonic band gaps and defect states induced by excitations of Bose-Einstein condensates in optical lattices' --- 15.5cm 0.2cm $ $\ Introduction ============ The achievement of Bose-Einstein condensation in magnetic traps [@experimente] has induced a great interest in the properties of quantum atomic gases and their manipulation by atom optic techniques. Although the latter are usually used for laser cooling in the formation process of a Bose-Einstein condensate (BEC), confinement in an optical dipole-trap has been demonstrated only recently [@ketterle98]. The all-optical confinement of a BEC provides a great potential for the manipulation and application of BEC. In particular, it opens the new opportunity to study atomic BECs in optical lattices. In recent years experimentalists have made great efforts to create a BEC in optical lattices. Although there are presently still some technical problems to achieve this goal the rich physics of uncondensed ultracold atoms in optical lattices [@deutsch96] and quasi-crystals [@grynberg97] has attracted great interest for both experimentalists and theorists. Recently, several theoretical papers dealing with condensates in optical potentials have been published [@molmer98; @zoller98] In this paper we focus on a new aspect of this subject: light propagation through a coherent condensate confined in an optical lattice. Since the ground state of the condensate in a lattice potential is periodic, it will act as a periodic dielectric for laser light propagating through it. Thus it will give rise to photonic band gaps at optical frequencies. The phenomenon of photonic band gaps is a natural consequence of the periodicity of the condensate. In fact, it also should occur for uncondensed ultracold atoms in optical lattices. However, in the case of a condensed atomic lattice what is interesting is that, because of the macroscopic occupation of the ground state, a proper description of photonic band gaps is given in terms of polaritons (an entangled coherent system composed of superpositions of photons and excited atoms). Furthermore, elementary excitations may be present in the lattice BEC. In general these excitations are no longer periodic and will cause distortions of the perfect periodic structure of the condensed atomic lattice. An excitation-induced defect in the atomic lattice in turn causes the occurence of defect states inside photonic band gaps of light propagating through the BEC. In this sense, elementary excitations have a close analogy to lattice defects in solid state physics which cause defect states in the electronic band structure. The paper is organized as follows. In Sec. II we will derive the equations of motion. In Sec. III we consider the case where only the lattice laser beams are present and seek for a periodic solution to the coupled equations of motion describing the ground-state BEC and the lattice laser beams. This solution shows that the optical potential experiences no back-reaction from the condensate if the latter has settled down into its ground state. In this sense, the lattice laser beams just effectively act as a constant periodic potential for the BEC. In Sec. IV we consider the propagation of a weak probe laser beam through the ground-state BEC and derive the form of the lowest photonic band gap for this beam by using polariton modes. To examine the behaviour of a probe laser beam propagating through a weakly non-periodic BEC a theory of defect states for photonic band gaps is developed in Sec. V which is applied in Sec. VI to the Koster-Slater model for a localized elementary excitation. Sec. VII concludes the paper. Equations of motion =================== The system under consideration consists of interacting two-level atoms which are coupled to the electromagnetic field. This coupling is described by using the electric-dipole and rotating-wave approximation so that the corresponding second quantized Hamiltonian is given by $$H = H_A + H_{\mbox{{\scriptsize NL}}} + H_{\mbox{{\scriptsize E.M.}}} + H_{\mbox{{\scriptsize int}}} \; , \label{hamil}$$ where $$H_A := \int d^3 x \sum_{i=e,g} \Psi_i^\dagger \{ \frac{\vec{p}^2}{2M} +V(\vec{x}) +E_i \} \Psi_i$$ describes the atomic center-of-mass motion. $V(\vec{x})$ denotes an external potential. $M$ represents the atomic mass, and $E_i$, $i=e,g$ are the internal energy levels for ground-state and excited atoms, respectively. The corresponding field operators $\Psi_g$ and $\Psi_e$ fulfill the commutation relations $[\Psi_i(\vec{ x})^\dagger , \Psi_j(\vec{y})] = \delta_{ij} \delta(\vec{x}-\vec{y})$. $H_{\mbox{{\scriptsize NL}}}$ is the nonlinear part of the atomic Hamiltonian which describes two-body collisions. For a dilute Bose gas it can be approximated by $$H_A := {1\over 2} \int d^3 x \sum_{i,j=e,g} g_{ij} \Psi_i^\dagger \Psi_j^\dagger \Psi_j \Psi_i \; ,$$ where $g_{ij} := 4\pi \hbar^2 a_{\mbox{{\scriptsize sc}}}^{(ij)}/M$ are coupling constants and $ a_{\mbox{{\scriptsize sc}}}^{(ij)}$ denote the scattering lengths for scattering between atoms in the internal state $i$ and $j$. For the description of the electromagnetic field we use the representation in terms of positive end negative frequency parts of the vector potential, $\vec{A}(\vec{ x}) = \vec{A}^{(+)} (\vec{ x})+ \vec{A}^{(-)}(\vec{x})$. This representation will turn out to be convenient for the adiabatic elimination of excited atoms. The Hamiltonian for the free electromagnetic field then takes the simple form $$H_{\mbox{{\scriptsize E.M.}}} = 2 \varepsilon_0 \int d^3 x \sum_{a,b=1}^3 A_a^{(-)} (\hat{\omega}^2)_{ab} A_b^{(+)} \; .$$ The positive and negative frequency parts are related by $(\vec{A}^{(+)})^\dagger =\vec{A}^{(-)}$ and fulfill the commutation relation $[A_a^{(+)}(\vec{x}), A_b^{(-)}(\vec{y}) ] = (\hbar/2 \varepsilon_0) \hat{\omega}^{-1} \delta_{ab}^T(\vec{ x}-\vec{ y})$, where $\delta_{ab}^T(\vec{ x}-\vec{ y})$ is the transverse delta function. The [*frequency operator*]{} $\hat{\omega}$ is a pseudo-differential operator whose action is defined in momentum space by $$[\hat{\omega} \vec{A}](\vec{x}) = (2\pi)^{-3/2} \int d^3 k e^{i \vec{k}\cdot \vec{x}} c |\vec{k}| \vec{A}(\vec{k})\; .$$ The physical interpretation of the frequency operator is simple. It just multiplies a photon mode with its frequency $\omega(\vec{k}) = c |\vec{k}|$. A more compact representation of $\hat{\omega}$ in position space is given by $\hat{\omega} = c\sqrt{-\hat{\Delta}} = c|-i\nabla|$, where $\hat{\Delta}$ denotes the Laplace operator. Using the positive and negative frequency part of the vector potential the electric dipole coupling between the atoms and the electromagnetic field in rotating-wave approximation can be written as $$H_{\mbox{{\scriptsize int}}} = i \int d^3 x \{ \Psi_g^\dagger \Psi_e (\vec{d}^* \cdot \hat{\omega} \vec{A}^{(-)} ) - \Psi_e^\dagger \Psi_g (\vec{d}\cdot \hat{\omega} \vec{A}^{(+)}) \} \; .$$ The Heisenberg equations of motion derived from the Hamiltonian (\[hamil\]) are given by $$\begin{aligned} i \hbar \dot{\Psi}_e &=& \left \{ \frac{\vec{p}^2}{2M} + V + E_e + \sum_{j=e,g} g_{ej} \Psi_j^\dagger \Psi_j \right \} \Psi_e -i \Psi_g (\vec{d}\cdot \hat{\omega} \vec{A}^{(+)}) \label{edgl} \\ i\hbar \dot{\Psi}_g &=& \left \{ \frac{\vec{p}^2}{2M} + V + E_g + \sum_{j=e,g} g_{gj} \Psi_j^\dagger \Psi_j \right \} \Psi_g +i \Psi_e (\vec{d}^* \cdot \hat{\omega} \vec{A}^{(-)}) \label{gdgl} \\ i \dot{A}^{(+)}_a(\vec{x}) &=& \hat{\omega} A_a^{(+)}(\vec{x}) + \frac{i}{2 \varepsilon_0} \int d^3 y \Psi_g^\dagger (\vec{y}) \Psi_e(\vec{y})\sum_{b=1}^3 d^*_b \delta_{ab}^T (\vec{x}-\vec{y}) \label{adgl}\end{aligned}$$ Lattice laser beams and BEC in ground state: decoupling of the fields ===================================================================== To analyse the interaction between the atoms and the lattice laser beams in absence of an external potential ($V(\vec{x})=0$) we restrict us to the particular case where the atomic field is composed of condensed atoms, i.e., a Bose-Einstein condensate. This allows us to make further substantial simplifications. As is well-known a condensate can be described by assuming that all atoms are in the same quantum state $\psi_g$. This amounts in replacing the field operator $\Psi_g$ in Eq. (\[gdgl\]) by the c-number field $\psi_g$ which then fulfills a nonlinear Schrödinger equation. In addition, we assume that the photon fluctuations of the lattice laser beams are small and therefore not important for our case. This allows us to replace the operator $\vec{A}^{(+)}$ by a corresponding classical field vector $\vec{A}^{(+)}_L$ of the lattice laser beams. We consider the regime of coherent interaction where the electromagnetic field is detuned far away from the atomic resonance frequency $\omega_{\mbox{{\scriptsize res}}} := (E_e-E_g )/\hbar$. Specifically, we assume that the detuning $\Delta_L := \omega_L - \omega_{\mbox{{\scriptsize res}}}$ (with $\omega_L := c|\vec{k}_L|$) of the lattice laser beams is negative (red detuning) and its absolute value much larger than any other characeristic frequency of our system so that we can adiabatically eliminate the excited atoms [@zhang94; @lenz94; @lewenstein94; @marzlin98b]. This amounts in replacing the field operator for excited atoms by $$\Psi_e \approx \frac{-i}{\hbar \Delta_L} \psi_g (\vec{d}\cdot \hat{\omega} \vec{A}^{(+)}_L) \; . \label{adiab}$$ Inserting Eq. (\[adiab\]) into Eqs. (\[gdgl\]) and (\[adgl\]) one easily finds (to first order in $1/\Delta_L $) $$\begin{aligned} i\hbar \dot{\psi}_g &=& \left \{ \frac{\vec{p}^2}{2M} + V + E_g + g_{gg} \psi_g^\dagger \psi_g + \frac{1}{\hbar \Delta_L} (\vec{d}\cdot \hat{\omega} \vec{A}^{(-)}_L) (\vec{d}\cdot \hat{\omega} \vec{A}^{(+)}_L) \right \} \psi_g \label{gdgl2} \\ i (\dot{A}^{(+)}_L(\vec{x}))_a &=& \hat{\omega} (A_L^{(+)}(\vec{x}))_a + \frac{1}{2 \varepsilon_0 \hbar \Delta_L} \int d^3 y |\psi_g (\vec{y})|^2 (\vec{d}\cdot \hat{\omega} \vec{A}_L^{(+)}(\vec{y})) \sum_{b=1}^3 d^*_b \delta_{ab}^T (\vec{x}-\vec{y}) \label{adgl2}\end{aligned}$$ Eqs.(\[gdgl2\]) and (\[adgl2\]) describe the coherent coupling of a ground-state atomic field to the lattice laser beams. The physics implicit in these equations is straightforward. The laser beams induce an optical potential for ground state atoms which is proportional to the light intensity ($\propto (\hat{\omega}\vec{A}_L)^2$). The atoms in turn act on the electromagnetic field like a dielectric, where the index of refraction is determined by the density $|\psi_g|^2$ of ground-state atoms. We are interested in how the condensate affects the lattice laser beams and the corresponding back-reaction in the optical potential. For simplicity, we take the laser beam to be parallel to the x-axis. Further, we assume that the BEC has settled to its ground-state which, because of the periodicity of the optical potential provided by the lattice laser beams, is periodic. It is then convenient to decompose the fields into a discrete Fourier series $$\begin{aligned} \psi_g(x) &=& \sum_{l\in {\bf Z}} \psi_l \exp [ i l \vec{k}_L\cdot \vec{x}] \nonumber \\ \Omega^{(L)} (x) &:=& {1\over \hbar} \vec{d}\cdot \hat{\omega} \vec{A}^{(+)}_L(x) = \sum_{l\in {\bf Z}} \Omega_l^{(L)} \exp [ i l \vec{k}_L\cdot \vec{x}] \; . \nonumber \end{aligned}$$ We remark that $\vec{k}_L$ does [*not*]{} denote the wavevector of the laser beams. It is defined by its relation to the spatial period $x_L$ of the optical lattice by $\vec{k}_L = \vec{e}_x 2\pi/x_L$. This period $x_L$ differs in general slightly from the wavelength of the laser beams outside the atomic medium [@fourwave]. Transforming Eqs. (\[gdgl2\]) and (\[adgl2\]) to momentum space we arrive at the following one-dimensional set of equations, $$\begin{aligned} i \hbar \dot{\psi}_l &=& \frac{\hbar^2 k_L^2}{2M} l^2 \psi_l + \frac{g_{gg}}{(2\pi)^2} \sum_{m,n \in {\bf Z}} \psi^*_{m+n-l} \psi_m \psi_n + \frac{\hbar}{(2\pi)^3 \hbar \Delta_L} \sum_{m,n \in {\bf Z}} \psi_{l+m-n} \Omega^{(L)*}_m \Omega^{(L)}_n \label{momg}\\ i\hbar \dot{\Omega}^{(L)}_l &=& \hbar \omega_L |l| \Omega^{(L)}_l + \frac{\omega_L |l|\vec{d}_\perp^2 }{2 (2\pi)^3 \varepsilon_0 \Delta_L } \sum_{m,n \in {\bf Z}} \Omega^{(L)}_m \psi^*_{m+n-l} \psi_n \; , \label{moma}\end{aligned}$$ where we have defined the transversal dipole moment of the atoms $\vec{d}_\perp := \vec{d} - \vec{k}_L (\vec{d}\cdot \vec{k}_L) /\vec{k}_L^2$. We can now exploit the fact that the optical frequency $\omega_L$ is typically (very) much larger than any other frequency scale involved in the system. This allows us to perform a rotating-wave approximation by inserting $\tilde{\Omega}^{(L)}_l := \exp \{i \omega_L t |l|\} \Omega^{(L)}_l$ into Eqs. (\[momg\]) and (\[moma\]) and neglecting all terms which rotate at multiples of the frequency $\omega_L$. This procedure results in the simplified equations $$\begin{aligned} i \hbar \dot{\psi}_l &=& \left \{ \frac{\hbar^2 k_L^2}{2M} l^2 + \frac{\hbar}{(2\pi)^3 \Delta_L} \sum_{m \in {\bf Z}} | \Omega^{(L)}_m|^2 \right \}\psi_l + \frac{g_{gg}}{(2\pi)^2} \sum_{m,n \in {\bf Z}} \psi^*_{m+n-l} \psi_m \psi_n \nonumber \\ & & + \frac{\hbar}{(2\pi)^3 \Delta_L} \sum_{m\in {\bf Z}} \psi_{l+2m} \Omega^{(L)*}_m \Omega^{(L)}_{-m} \label{mom2g}\\ i \hbar \dot{\Omega}^{(L)}_l &=& \hbar \omega_L |l|\left \{ 1 + \frac{\vec{d}_\perp^2}{2 (2\pi)^3 \varepsilon_0 \Delta_L } \sum_{n \in {\bf Z}} | \psi_n|^2 \right \} \Omega^{(L)}_l + \frac{\vec{d}_\perp^2}{2 (2\pi)^3 \varepsilon_0 \Delta_L } \omega_L |l| \Omega^{(L)}_{-l} \sum_{n \in {\bf Z}} \psi^*_{n-2l} \psi_n \; . \label{mom2a}\end{aligned}$$ These equations have some properties which allow to decouple the system for some physically interesting cases. The most important one is that the atoms do only couple counterpropagating modes, i.e., $\Omega^{(L)}_l$ and $\Omega^{(L)}_{-l}$. This is a direct consequence of energy conservation since a transition to any other mode would require an amount of energy in the order of $\hbar \omega_L$ which cannot be provided by the interaction with the condensate. In addition, it is not difficult to see that both the mean density $\bar{\rho} := \sum_n |\psi_n|^2$ and the mean light intensity per mode, $\bar{I}_l := |\Omega^{(L)}_l|^2 + |\Omega^{(L)}_{-l}|^2$, are conserved quantities, reflecting the conservation of the total number of atoms and the number of photons with energy $\hbar \omega_L |l|$, respectively. Therefore, the first sums on the right-hand-side of Eqs. (\[mom2g\]) and (\[mom2a\]) just produce a constant shift of the energy levels. We now consider a solution of the system of equations (\[mom2g\]) and (\[mom2a\]) which corresponds to a standing-wave lattice laser beam interacting with a BEC in its ground state in the coherent regime. For a standing-wave lattice, we can make the ansatz $\Omega^{(L)}_l = \Omega^{(L)}_{-l}$. In addition, in the ground state of the BEC the time dependence of all coefficients $\psi_l$ is given by $\psi_l(t) = \exp[-i \mu t/\hbar] \psi_l(0)$, where $\mu$ denotes the chemical potential and $\psi_l(0)$ can be chosen to be real. It is then not difficult to show that $\Omega^{(L)}_l = \Omega^{(L)}_{-l}$ holds for all times (by differentiating both sides and comparing the results). In addition, the expression $\Omega^{(L)}_l \Omega^{(L)*}_{-l}$, which describes the optical potential in Eq. (\[mom2g\]), is time-independent, too. An immediate consequence of this fact is that in the coherent regime the optical potential created by the lattice laser beams for the BEC in Eq. (\[mom2g\]) is not altered by the condensate itself. Thus, the problem is decoupled and the condensate behaves as if it were moving in a given external periodic potential. In this sense the influence of the lattice laser beams on the BEC can be replaced by an external potential $V(\vec{x})$ with periodicity $\pi/k_L$. Polariton band theory for light interacting with a condensate ============================================================= In the preceding section we have shown how the combined system of a BEC and lattice laser beams behaves in its ground state. We are now interested in a different situation where a running probe laser beam propagates through the “lattice condensate” (condensate plus lattice laser beams). The behaviour of the probe laser can intuitively understood by considering the BEC as a kind of dielectric for the probe laser beam. Since the BEC is periodic the probe laser beam will effectively propagate through a periodic dielectric. We thus expect it to show the phenomenon of photonic band gaps [@yablonovitch91]. To describe the interaction of the probe laser beam with the condensate we will assume that the ground-state BEC changes little so that $\psi_g$ enters as a given external field into the equations of motion for excited atoms (\[edgl\]) and for the probe laser beam (\[adgl\]). Solutions of these coupled equation describe polariton modes, i.e., entangled superpositions of matter and light fields [@shlyapnikov91; @politzer91]. Thus, it is really “polaritonic” band gaps rather than photonic band gaps that we are studying. However, for sufficiently large detuning of the lattice laser beams the entanglement is very small so that the result indeed can be considered as photonic band gaps. To find a suitable expression for the ground-state wavefunction $\psi_g$ we use the results of the preceding section, i.e., consider the case where the BEC is confined by a periodic potential of the form $$V(\vec{x}) = -V_0 \cos (2 k_L x)$$ (we choose the factor of $2k_L$ since the potential created by an optical lattice of wavevector $k_L$ would create such a potential [@marzlin98b]). In addition, we consider a very weak probe beam and neglect the four-wave mixing effect due to the interference between the probe laser and the lattice lasers [@fourwave]. As a result, the ground state of the BEC can effectively be described by the Gross-Pitaevskii equation $$\mu \psi_g = \left \{ \frac{\vec{p}^2}{2M} + V(\vec{x}) \right \} \psi_g + g_{gg} |\psi_g|^2 \psi_g\; , \label{gpe}$$ where $\mu$ is the chemical potential. In the experimentally realized dilute Bose condensates the interaction energy of the two-body collisions between atoms is usually large, so that one can perform the Thomas-Fermi approximation by neglecting the kinetic energy. This transforms Eq. (\[gpe\]) into a simple algebraic equation whose solution is of the form $ |\psi_g(x)|^2 = \rho_0 + \rho_1 \cos (2 k_L x)$. This solution is valid for all $x$ if the optical potential is not too strong, so that $\rho_1$ is smaller than $\rho_0$. If $\rho_1$ is not too close to $\rho_0$ we furthermore can simplify the wavefunction to $$\psi_g(x) \approx \sqrt{\rho_0} + \frac{\rho_1}{2 \sqrt{\rho_0}} \cos (2 k_L x) \label{rho}$$ (this time-independent expression is valid in a frame rotating at frequency $E_g/\hbar$). We remark that this expression produces qualitatively correct results for the lowest photonic band even if the kinetic energy is not negligible or $\rho_1\approx \rho_0$ because the corresponding correction essentially introduce higher coefficients in the Fourier series of Eq. (\[rho\]). Since these higher coefficients do only couple higher bands, they do not affect the results for the lowest band. Since $\psi_g$ does not depend on $y$ and $z$ it is advantageous to rescale the wavefunctions as $\psi \rightarrow L_\perp \psi$, where $L_\perp$ is the typical extension of the BEC in the $y$ and $z$ direction. This guarantees that the one-dimensional integral $\int dx |\psi|^2$ is dimensionless and can be interpreted as a particle number. In the actual calculations this rescaling leads to the appearance of various factors of $L_\perp$. $L_\perp$ will not enter the final results, though. Introducing the (classical) field $\Omega^{(P)}(x) := \vec{d}\cdot \hat{\omega} \vec{A}^{(+)}_P(x)$ for the probe laser’s Rabi frequency Eqs. (\[edgl\]) and (\[adgl\]) can be reduced to the polariton equations of motions [@politzer91] $$\begin{aligned} i \hbar \dot{\psi}_e &=& \left \{ \frac{\vec{p}^2}{2M} + \hbar \omega_{\mbox{{\scriptsize res}}}+ {g_{eg}\over L_\perp^2} |\psi_g|^2 \right \} \psi_e -i\hbar \psi_g \Omega^{(P)} \label{edgl3} \\ i \hbar \dot{\Omega}^{(P)} &=& \hbar \hat{\omega} \Omega^{(P)} + \frac{i |\vec{d}_\perp|^2}{2 \varepsilon_0 L_\perp^2} \hat{\omega} \{ \psi_g^* \psi_e\} \; , \label{adgl3}\end{aligned}$$ where $\psi_g$ (divided by $L_\perp$) is given by Eq. (\[rho\]). Since the density of excited atoms should be very small we have neglected two-body collisions between excited atoms ($g_{ee}=0$ in Eq. (\[edgl\])). Because $\psi_g$ only depends on $x$, and since we consider the case that $\psi_e$ and $\Omega^{(P)}$ do not depend on $y$ and $z$ either, the transverse delta function of Eq. (\[adgl\]) can be reduced to an ordinary delta function (to prove this one can transform Eq. (\[adgl\]) to momentum space). For later use it will be convenient to consider the solutions $\langle x | \phi \rangle := ( \psi_e(x), \Omega^{(P)}(x)\; )$ of Eqs. (\[edgl3\]) and (\[adgl3\]) as elements of a polariton Hilbert space with conserved scalar product $$\langle \phi^\prime | \phi \rangle := \int dx \left \{ \psi_e^{\prime *} \psi_e + \frac{2\varepsilon_0 \hbar L_\perp^2}{ |\vec{d}_\perp|^2} \Omega^{(P)\prime *} \hat{\omega}^{-1} \Omega^{(P)} \right \} \;. \label{sprod}$$ Physically the quantity $\langle \phi | \phi \rangle$ is related to the number of excitations (number of excited atoms plus number of photons) in our system. It is a conserved quantity because of the rotating-wave approximation made in Sec. II. Eqs. (\[edgl3\]) and (\[adgl3\]) can be rewritten in the form $i\hbar \partial_t |\phi \rangle = H_{\mbox{{\scriptsize pol}}}|\phi \rangle $ with the polariton Hamiltonian $$\begin{aligned} H_{\mbox{{\scriptsize pol}}} & := & {1\over 2} ({\bf 1} + \sigma_3) \left \{ \frac{\vec{p}^2}{2M} + \hbar \omega_{\mbox{{\scriptsize res}}} + {g_{eg}\over L_\perp^2} |\psi_g|^2 \right \} + {1\over 2} ({\bf 1} - \sigma_3) \hbar \hat{\omega} -i\hbar \psi_g \sigma_+ + \frac{i |\vec{d}_\perp|^2}{2 \varepsilon_0 L_\perp^2} \hat{\omega} \psi_g^* \sigma_- \; ,\end{aligned}$$ where $\sigma_i$ are the Pauli matrices. We remark that $ H_{\mbox{{\scriptsize pol}}}$ is Hermitean with respect to the scalar product (\[sprod\]), i.e., $\langle \phi^\prime | H_{\mbox{{\scriptsize pol}}} \phi \rangle = \langle H_{\mbox{{\scriptsize pol}}} \phi^\prime | \phi$. To derive the polariton band-structure we have to find the eigenvalues of the operator $ H_{\mbox{{\scriptsize pol}}}$. Since $\psi_g$ is periodic $H_{\mbox{{\scriptsize pol}}}$ commutes with the operator of discrete translations of amount $\pi/k_L$ and thus has a common set of eigenvectors with this operator. The eigenvectors $|\phi_{n,q} \rangle$ therefore can be characterized by two quantum numbers $n\in \{ 0, 1, 2, \cdots \} $ and $q \in [-k_L, k_L ] $ which denote the band index and the quasi-momentum, respectively. The eigenvalues of the discrete translation operator are given by $\exp [i q \pi/k_L]$ and belong to eigenvectors which are simply given by momentum eigenstates with momentum $ \hbar k_m := \hbar (q + 2 m k_L) $ for integer $m$. The eigenvalues $\hbar \omega_{n,q}$ of the Hamiltonian can be found by expanding Eqs. (\[edgl3\]) and (\[adgl3\]) in this basis and searching for stationary solutions. The corresponding equations, $$\begin{aligned} \hbar \omega \psi_e(k_m) &=& \left \{ \frac{\hbar^2 k_m^2}{2M} + \hbar \omega_{\mbox{{\scriptsize res}}} \right \} \psi_e(k_m) + {g_{eg}\over L_\perp^2} \left \{ \rho_0 \psi_e(k_m) + {\rho_1 \over 2} [ \psi_e(k_{m+1}) + \psi_e(k_{m-1})]\right \} \nonumber \\ & & -i\hbar L_\perp \sqrt{\rho_0} \Omega^{(P)}(k_m) -i\hbar L_\perp \frac{\rho_1}{4 \sqrt{ \rho_0}} [ \Omega^{(P)}(k_{m+1}) + \Omega^{(P)}(k_{m-1}) ] \\ \hbar \omega \Omega^{(P)}(k_m) &=& \hbar c |k_m| \Omega^{(P)}(k_m) + i \frac{|\vec{d}_\perp|^2 c |k_m|}{2 \varepsilon_0 L_\perp} \left \{ \sqrt{\rho_0} \psi_e(k_m) + \frac{\rho_1}{4 \sqrt{\rho_0}} [ \psi_e(k_{m+1}) + \psi_e(k_{m-1})] \right \} \; ,\end{aligned}$$ can easily be solved numerically. Fig. 1c) shows the resulting band structure near the upper band edge of the lowest frequency band for a condensate of density $\rho_0 = 1.1 \rho_1 = 10^{14}$ cm$^{-3}$. In order to describe the limit of a photonic instead of a polariton band-structure we have assumed a very large detuning of $\Delta_L = 100$ GHz of the lattice. However, the results given below do not change very much if a smaller detuning is assumed. We furthermore have set $|\vec{d}_\perp| \approx e a_0$, with $e$ being the electron’s charge and $a_0$ denoting Bohr’s radius. The wavevector of the lattice was taken to be $k_L = 10^{7}$ m$^{-1}$. An excellent analytical approximation for the band structure can be made by assuming that for $q\in (0,k_L)$ only the modes $\Omega^{(P)}(q)$, $\Omega^{(P)}(q-2k_L)$, $\psi_e(q)$, and $\psi_e(q-2k_L)$ are important. The problem is then reduced to finding the eigenvalues of a $4\times 4$ matrix. For $q=k_L$ the eigenvalues have a simple form and allow to derive the following expression for the band gap $\Delta \omega$ separating the two lowest energy bands, $$\Delta \omega = s_+ - s_- \; ,$$ where we have defined the frequencies $\nu_i := |\vec{d}_\perp|^2 \rho_i /(2 \hbar \varepsilon_0)$ and introduced the abbreviations $s_{\pm} := \sqrt{(\Delta_L / 2 )^2 + \omega_{\mbox{{\scriptsize res}}} (\sqrt{\nu_0}\pm\sqrt{\nu_1}/4)^2}$. For a large detuning $|\Delta_L|$, i.e., in the limit of a photonic band gap, this expression simplifies to $\Delta \omega = \omega_0 \nu_1/|\Delta_L|$. For the numerical values given above the band gap takes the value $\Delta \omega \approx$ 40 GHz. For $q\neq k_L$ the band structure is given by a rather complicated expression. We therefore have further simplified the analytical result by fitting it to a square root [@fitting]. The lowest polariton band then takes the form $$\omega_{0,q} \approx \omega_{0,\mbox{{\scriptsize max}}} + \bar{\nu} -\sqrt{c^2(|q|-k_L)^2 + \bar{\nu}^2} \; , \label{band}$$ where $$\omega_{0,\mbox{{\scriptsize max}}} = \omega_{\mbox{{\scriptsize res}}} - \frac{|\Delta_L |}{2} - s_+$$ denotes the upper edge of the lowest frequency band, and $$\bar{\nu} := \frac{\Delta \omega}{ (\omega_{0,\mbox{{\scriptsize max}}} -\omega_{\mbox{{\scriptsize res}}})^2} s_+(s_+ + s_-)$$ determines the curvature of the band. In the limit of a large detuning this simplifies to $\bar{\nu} \approx \Delta \omega /2$. Theory of localized defects =========================== In the previous section we have studied polariton band gaps of light generated by the lattice condensate in its ground state. However, in general the BEC might be in a state corresponding to a (coherent) elementary excitation which usually are not periodic. Thus, we expect defects in the lattice condensate. As is well-known from solid state theory a defect or an impurity in an otherwise periodic potential can lead to defect states, i.e., states whose energy eigenvalue lies inside the gap between two energy bands (see, e.g., Ref. [@callaway91]). In the system under consideration this phenomenon could be exploited to acquire knowledge about non-periodic elementary excitations of the Bose condensate by observing light propagation through the BEC. In addition, defect theory can also be applied to study the back-reaction of an excited BEC on the optical potential confining it. The existence of defect states for photonic band gaps has been examined in the microwave regime for ordinary dielectric materials [@yablonovitch91; @defectpaper]. The method of calculation that we adopt is closely related to the Green’s function approach of Ref. [@callaway91]. Specifically we consider the situation that the condensate’s wavefunction is given by $\psi_G(x) = \psi_g(x) + \delta \psi_g(x)$, where $\psi_g$ (divided by $L_\perp$) is given by Eq. (\[rho\]) and $\delta \psi_g$ describes a coherent elementary excitation of the condensate which we assume to be localized in the x-direction. This allows us to estimate the resulting energy eigenstates by adapting the Koster-Slater model [@koster54] to the case of polariton band gaps. Our aim is to find solutions of the equation $$\hbar \omega_{\mbox{{\scriptsize defect}}} |\phi \rangle = (H_{\mbox{{\scriptsize pol}}} + H_{\mbox{{\scriptsize defect}}}) |\phi \rangle\; , \label{defectdgl}$$ where $$H_{\mbox{{\scriptsize defect}}} := -i\hbar \delta \psi_g \sigma_+ + \frac{i |\vec{d}_\perp|^2}{2 \varepsilon_0 L_\perp^2} \hat{\omega} \delta \psi_g^* \sigma_-$$ describes the influence of the elementary excitation $\delta \psi_g$ and $\omega_{\mbox{{\scriptsize defect}}}$ is the defect eigenfrequency. To find the solutions of Eq. (\[defectdgl\]) we expand $|\phi \rangle $ in terms of Wannier functions, $$|W_{n,\nu} \rangle := {1\over \sqrt{2k_L}} \int_{-k_L}^{k_L} dq e^{-\pi i \nu q/k_L} |\phi_{n,q} \rangle \; ,$$ where $\nu$ is an integer number. The functions $W_{n,\nu}(x)$ are localized around the lattice point $\pi \nu/k_L$ which makes them a convenient tool to study localized defects. Inserting $|\phi \rangle = \sum_{n,\nu} \phi_{n,\nu} |W_{n,\nu} \rangle $ into Eq. (\[defectdgl\]) one easily deduces the equation $$\omega_{\mbox{{\scriptsize defect}}} \phi_{n,\nu} = \sum_{\mu} \omega_{n,\nu-\mu} \phi_{n,\mu} + \sum_{m,\mu} \langle W_{n,\nu} | H_{\mbox{{\scriptsize defect}}}/\hbar |W_{m,\mu} \rangle \phi_{m,\mu}\; , \label{w1}$$ with $$\omega_{n,\nu-\mu} := {1\over 2 k_L} \int_{-k_L}^{k_L} dq e^{\pi i(\nu-\mu)/k_L} \omega_{n,q} \; .$$ Since the Wannier basis is countable Eq. (\[w1\]) can be interpreted as a matrix equation. In particular, one now can exploit the fact that both the Wannier functions and $H_{\mbox{{\scriptsize defect}}}$ are localized. This implies that, at least approximately, only a finite number, say an $N\times N$ submatrix, of the matrix elements $ \langle W_{n,\nu} |H_{\mbox{{\scriptsize defect}}} |W_{m,\mu} \rangle$ is nonvanishing. Introducing the Green’s function $G := ( \omega_{\mbox{{\scriptsize defect}}} {\bf 1} - H_{\mbox{{\scriptsize pol}}}/\hbar )^{-1}$ whose matrix elements are given by $$\langle W_{n,\nu} | G | W_{m,\mu} \rangle = \delta_{n,m} {1\over 2k_L} \int_{-k_L}^{k_L} dq \frac{e^{\pi i (\nu-\mu)/k_L}}{ \omega_{\mbox{{\scriptsize defect}}} -\omega_{n,q}} =: \delta_{n,m} G_{n, \nu-\mu} \label{gmatelem}$$ the eigenvalue problem can be reduced to $$\phi_{n,\nu} = \sum_{m,\mu,\lambda} G_{n, \nu-\mu} \langle W_{n,\mu} | H_{\mbox{{\scriptsize defect}}}/\hbar |W_{m,\lambda} \rangle \phi_{m,\lambda} \; . \label{evbed}$$ As is well-known in solid state theory [@callaway91] it is sufficient to consider only the eigenvalue problem of the $N\times N$ subspace where the matrix elements of $ H_{\mbox{{\scriptsize defect}}}$ are non-zero to derive the frequencies of the defect states. Koster-Slater model for photonic band gaps ========================================== Having derived the matrix eigenvalue equation (\[evbed\]) for the determination of the defect frequency it is straightforward to apply the Koster-Slater model [@koster54] to the problem at hand. In this model the assumption is made that both the Wannier functions and the perturbation $U_{\mbox{{\scriptsize np}}}$ are localized in such a way that only one matrix element of the perturbation is nonzero, $$\langle W_{n,\nu} | H_{\mbox{{\scriptsize defect}}} |W_{m,\mu} \rangle = U_0 \delta_{n,0} \delta_{m,0} \delta_{\nu,0} \delta_{\mu,0} \; . \label{ksm}$$ This model is not valid if $ \delta \psi_g$ is too strongly localized (i.e., on a scale much smaller than the lattice spacing $\pi/k_L$) [@stoneham75], but should produce qualitative estimates of defect frequencies for moderately localized perturbations. Inserting Eq. (\[ksm\]) into the eigenvalue equation (\[evbed\]) and using Eq. (\[gmatelem\]) to evaluate the only relevant matrix element of $G$, $\langle W_{0,0}| G | W_{0,0} \rangle $, we find that the defect frequency $\omega_{\mbox{{\scriptsize defect}}}$ has to fulfill the condition $$1 = {U_0\over \hbar} \frac{1}{2k_L} \int_{-k_L}^{k_L} \frac{dq}{\omega_{\mbox{{\scriptsize defect}}} -\omega_{0,q}} \; . \label{ksbed}$$ The integral can be calculated exactly for the photonic band of Eq. (\[band\]), but since the resulting expression is somewhat complicated it is more instructive to use the following approximation which is valid if the defect frequency $\omega_{\mbox{{\scriptsize defect}}}$ is close to the upper edge $\omega_{0,\mbox{{\scriptsize max}}}$ of the lowest frequency band, $$\frac{1}{2k_L} \int_{-k_L}^{k_L} \frac{dq}{ \omega_{\mbox{{\scriptsize defect}}} -\omega_{0,q}} \approx \frac{\pi}{\omega_L} \sqrt{\frac{\bar{\nu}}{2( \omega_{\mbox{{\scriptsize defect}}}- \omega_{0,\mbox{{\scriptsize max}}} ) }} \; .$$ Inserting this into Eq. (\[ksbed\]) we find for the frequency of the defect state the expression $$\omega_{\mbox{{\scriptsize defect}}} - \omega_{0,\mbox{{\scriptsize max}}} \approx \frac{\pi^2}{2} \frac{\bar{\nu}}{\omega_L^2} \frac{U_0^2}{\hbar^2} \; . \label{deffreq}$$ It is of interest to know how large this frequency difference is for realistic systems. To achieve this we first have to estimate the value of $U_0 = \langle W_{0,0} | H_{\mbox{{\scriptsize defect}}} | W_{0,0} \rangle = i\hbar \int dx \{ W_\Omega^* W_e \delta \psi_g^* - W_e^* W_\Omega \delta \psi_g \}$ with $\langle x|W_{0,0} \rangle = (W_e(x), W_\Omega(x))$. A rough estimate for this integral can be made by setting both $\delta \psi_g$ and the Wannier function to be constant over one wavelength $\lambda_L$ and to be zero outside this range. The normalization condition $ \langle W_{0,0}|W_{0,0} \rangle =1$ then leads approximately to $1= \lambda_L \{ |W_e|^2 + |W_\Omega|^2/(\omega_L |d_\perp|^2 /2\hbar \varepsilon_0 L_\perp^2) \}$. Using this condition $U_0$ takes its maximal value for $W_e = 1/(\sqrt{2}L_\perp)$, $$U_0 \approx 2 \hbar \delta \psi_g \sqrt{\omega_L \frac{ |d_\perp|^2}{ 2 \hbar \varepsilon_0 L_\perp^2}} \; .$$ Assuming a defect amplitude of $\delta \psi_g = \epsilon \sqrt{\rho_0} L_\perp$ over the extent of $W_{0,0}(x)$, where $\epsilon$ is small compared to one, we can derive an estimate for the defect frequency (\[deffreq\]) of $$\omega_{\mbox{{\scriptsize defect}}} - \omega_{0,\mbox{{\scriptsize max}}} \approx 2 \pi^2 \epsilon^2 \frac{\bar{\nu}}{\omega_L} \nu_0 \; .$$ Using the same numbers as in Sec. IV (that is, $\rho_i = 10^{14}$ cm$^{-3}$ and $\Delta_L =10^{11}$ s$^{-1}$) as well as $\epsilon =0.3$ this frequency difference can be shown to be of the order of 100 Hz. Though this number is too small to be measurable it should be pointed out that it applies only to a defect corresponding to an elementary excitation over one wavelength. A different type of defect can produce a much different result. For example, if we do not consider a weak elementary excitation but a strong localized excitation we can estimate its effect but assuming a larger value for $\epsilon$. A threefold increase of the local density ($\epsilon = 3$) would lead to a defect frequency $\omega_{\mbox{{\scriptsize defect}}} - \omega_{0,\mbox{{\scriptsize max}}}$ of about 10 KHz, for instance. Conclusions =========== In this paper we have analysed the interaction of a lattice (or “crystalized”) Bose-Einstein condensate with largely detuned laser beams. We have derived a periodic solution of the coupled equations of motion, corresponding to a free BEC and a standing lattice laser beam. We found that, if the condensate is in its ground state, these equations decouple and the effect of the lattice laser beam on the condensate is not affected by the condensate itself (no back-reaction). In this situation it is thus equivalent to consider a condensate in some external periodic potential with the same periodicity. Building on this result we then assumed that the condensate is confined by an external periodic potential. Since the condensate’s ground state is then periodic, too, it forms a kind of periodic dielectric. A probe laser beam propagating through this dielectric will then experience the formation of photonic band gaps. We have analysed this situation using the concept of polaritons, i.e., entangled superpositions of excited atoms and photons. If the condensate is not in its ground state but in a state corresponding to a localized elementary excitation the periodicity of the system is perturbed. This leads to the formation of defect states inside a polariton band gap. [**Acknowledgement**]{}: We thank Michael Steel for helpful discussions. This work has been supported by the Australian Research Council. [99]{} M. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, Science [**269**]{}, 198 (1995); C.C. Bradley, C.A. Sackett, J.J. Tollet and R. Hulet, Phys. Rev. Lett. [**75**]{}, 1687 (1995); M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.M. Kurn, D.S. Durfee, C.G. Townsend and W. Ketterle, Phys. Rev. Lett. [**77**]{}, 416 (1996); D.S. Jin, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, Phys. Rev. Lett. [**77**]{}, 420 (1996). D.M. Stamper-Kurn, M.R. Andrews, A.P. Chikkatur, S. Inouye, H.-J. Miesner, J. Stenger, and W. Ketterle, Phys. Rev. Lett. [**80**]{}, 2027 (1998). P.S. Jessen and I.H. Deutsch, [*Optical lattices*]{}, in [*Advances in atomic, molecular, and optical Physics*]{} [**37**]{}, 95 (1996), edited by B. Bederson and H. Walther, Cambridge 1996. L. Guidoni, C. Triché, P. Verkerk, and G. Grynberg, Phys. Rev. Lett. [**79**]{}, 3363 (1997). K. Berg-S[ø]{}rensen and K. M[ø]{}lmer, to appear in Phys. Rev. A. D. Jaksch, C. Bruder, J.I. Cirac, C.W. Gardiner, and P. Zoller, preprint cond-mat/9805037. W. Zhang and D.F. Walls, Phys. Rev. A [**49**]{}, 3799 (1994). G. Lenz, P. Meystre, and E.M. Wright, Phys. Rev. A [**50**]{}, 1681 (1994). M. Lewenstein, L. You, J. Cooper, and K. Burnett, Phys. Rev. A [**50**]{}, 2207 (1994). K.-P. Marzlin und W. Zhang, Phys. Rev. A. [**57**]{}, p. 4761 (1998). I.H. Deutsch, R.J.C. Spreeuw, S.L. Rolston, and W.D. Phillips, Phys. Rev. A [**52**]{}, 1394 (1995). E. Yablonovitch, T.J. Gmitter, R.D. Meade, A.M. Rappe, K.D. Brommer, and J.D. Joannopoulos, Phys. Rev. Lett. [**67**]{}, 3380 (1991). B.V. Svistunov and G.V. Shlyapnikov, Sov. Phys. JETP [**71**]{}, 71 (1990). H.D. Politzer, Phys. Rev. A [**43**]{}, 6444 (1991). More specifically, we assumed that the lowest frequency band $\omega_{0,q}$ has the form $\omega_{0,q} \approx f(q) := A - \sqrt{c^2(|q|-k_L)^2 +B^2}$. The parameters $A$ and $B$ are determined by comparing $f$ and $d^2 f/dq^2$ at $q=k_L$ with $\omega_{0,q}$ and $d^2 \omega_{0,q}/dq^2$ found in the 4-level approximation. The latter can be found by taking the derivative of the characteristic polynomial of the $4\times 4$ matrix with respect to $q$ at $q=k_L$. J. Callaway, [*Quantum theory of the solid state*]{}, 2nd edition, Academic Press, Boston 1991. R.D. Meade, K.D. Brommer, A.M. Rappe, and J.D. Joannopoulos, Phys. Rev. B [**44**]{}, 13772 (1991); K.M. Leung, J. Opt. Soc. Am. B [**10**]{}, 303 (1993); D.R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S.L. McCall, and P.M. Platzman, J. Opt. Soc. Am. B [**10**]{}, 314 (1993); H.G. Algul, M. Khazhinsky, A.R. McGurn, and J. Kapenga, J. Phys.: Condens. Matter [**7**]{}, 447 (1995); N.-H. Liu, Phys. Rev. B [**55**]{}, 4097 (1997). G.F. Koster and J.C. Slater, Phys. Rev. [**96**]{}, 1208 (1954). A.M. Stoneham, [*Theory of defects in solids*]{}, Clarendon Press, Oxford 1975. $ $ $ $

--- abstract: 'SIDH is a post-quantum key exchange algorithm based on the presumed difficulty of computing isogenies between supersingular elliptic curves. However, the exact hardness assumption SIDH relies on is not the pure isogeny problem; attackers are also provided with the action of the secret isogeny restricted to a subgroup of the curve. Petit [@Petit2017] leverages this information to break variants of SIDH in polynomial time, thus demonstrating that exploiting torsion-point information can lead to an attack in some cases. The contribution of this paper is twofold: First, we revisit and improve the techniques of [@Petit2017] to span a broader range of parameters. Second, we construct SIDH variants designed to be weak against the resulting attacks; this includes weak choices of starting curve under moderately imbalanced parameters as well as weak choices of base field under balanced parameters. We stress that our results do not reveal any weakness in the NIST submission SIKE [@nistpqc]. However, they do get closer than previous attacks in several ways and may have an impact on the security of SIDH-based group key exchange [@DBLP:journals/iacr/AzarderakhshJJS19] and certain instantiations of B-SIDH [@DBLP:journals/iacr/Costello19].' author: - - Péter Kutas - Chloe Martindale - Lorenz Panny - | \ Christophe Petit - 'Katherine E. Stange' bibliography: - 'bib.bib' title: | Weak instances of SIDH variants\ under improved torsion-point attacks --- ### Acknowledgements. {#acknowledgements. .unnumbered} We thank Victoria de Quehen for her invaluable input, especially for sharing her ideas from concurrent work and for her careful reading and advice during editing. Thanks to Daniel J. Bernstein for his help with Section \[sec:3.3\], and to John Voight for answering a question concerning Section \[sec:numweak\]. We would also like to thank the anonymous reviewers for their useful feedback.

--- abstract: 'Let $M$ be a manifold with pinched negative sectional curvature. We show that when $M$ is geometrically finite and the geodesic flow on $T^1 M$ is topologically mixing then the set of mixing invariant measures is dense in the set $\M^1(T^1M)$ of invariant probability measures. This implies that the set of weak-mixing measures which are invariant by the geodesic flow is a dense $G_{\delta}$ subset of $\M^1(T^1 M)$. We also show how to extend these results to manifolds with cusps or with constant negative curvature.' author: - Kamel Belarif title: 'Genericity of weak-mixing measures on geometrically finite manifolds' --- Introduction ============ Let $\widetilde{M}$ be a complete, simply connected manifold with pinched sectional curvature (*i.e* there exists $b > a > 0$ such that $-b^2 \leq \kappa \leq -a^2$) and $\Gamma$ a non elementary group of isometries of $\widetilde{M}$. We will denote by $M$ the quotient manifold $\widetilde{M} / \Gamma$ and $\phi_t$ the geodesic flow on the unit tangent bundle $T^1 M$. The interesting behavior of this flow occurs on its non-wandering set $\Omega$.\ Let us recall a few definitions related to the mixing property of the geodesic flow. 1. $\phi_t$ is topologically mixing if for all open sets $\U, \jV \subset \Omega$ there exists $T>0$ such that $\phi_t(\U) \cap \jV \neq \emptyset$ for all $|t| >T$, 2. Given a finite measure $\mu$ the geodesic flow is mixing with respect to $\mu$ if for all $f \in L^2(T^1 M,\mu)$, $$\lim\limits_{t \to \infty} \int_{T^1M} f \circ \phi_t \cdot f d\mu = \left( \int_{T^1 M} f d\mu \right)^2,$$ 3. it is weakly-mixing if for all continuous function $f$ with compact support we have $$\lim\limits_{T \to \infty} \frac{1}{T} \int_0^T \left| { \int_{T^1 M} f \circ \phi_t (x)f(x) d\mu(x) - \left( \int_{T^1 M} f d\mu \right) ^2 } \right| dt = 0.$$ Finally, let us recall what is the weak topology: a sequence $\mu_n$ of probability measures converges to a probability measure $\mu$ if for all bounded continuous functions $f$, $$\int_{T^1 M} f d\mu_n \xrightarrow{n \to \infty} \int_{T^1 M} f d\mu.$$\ In this article, we will show that the weak-mixing property is generic in the set $\M^1(T^1 M)$ of probability measures invariant by the geodesic flow and supported on $\Omega$. We endow $\M^1(T^1 M)$ with the weak topology.\ In [@Si72], K. Sigmund studies this question for Anosov flows defined on compact manifolds and shows that the set of ergodic probability measures is a dense $G_{\delta}$ set (*i.e* a countable intersection of dense sets) in $\M^1 (T^1 M)$. On non-compact manifolds, the question has been studied by Y. Coudène and B. Schapira in [@MR2735038] and [@MR3322793]. It appears that ergodicity and zero entropy are typical properties for the geodesic flow on negatively curved manifolds.\ Since the set of mixing measures with respect to the geodesic flow is contained in a meager set (see [@MR3322793]) it is natural to consider the set of weak-mixing measures from the generic point of view.\ Let us recall that a manifold $M$ is geometrically finite if it is negatively curved, complete and has finitely many ends, all of which are cusps of finite volume and funnels.\ Here is our main result. \[thm:princ\] Let $M$ be a geometrically finite manifold with pinched curvature and $\phi_t$ the geodesic flow defined on its unit tangent bundle $T^1 M$.\ If $\phi_t$ is topologically mixing on $\Omega$, then the set of mixing probability measures invariant by the geodesic flow is dense in $\M^1 (T^1 M)$ for the weak topology. \[cor:gef\] Let $M$ be a geometrically finite manifold with pinched negative curvature and $\phi_t$ the geodesic flow defined on its unit tangent bundle $T^1M$.\ If $\phi_t$ is topologically mixing on the non-wandering set of $T^1M$, then the set of invariant weak-mixing probability measures with full support on $\Omega$ is a dense $G_{\delta}$ subset of $\M^1 (T^1 M)$. To prove theorem \[thm:princ\], we will use the fact that Dirac measures supported on periodic orbits are dense in $\M^1(T^1 M)$. This comes from [@MR2735038] where the result is shown for any metric space $X$ admitting a local product structure and satisfying the closing lemma.\ The rest of the proof relies on the approximation of a single Dirac measure supported on a periodic orbit $\O(p)$ using a sequence of Gibbs measures associated to $(\Gamma,F_n)$ where $F_n:T^1 M \to \R$ is a Hölder-continuous potential.\ The notion of Gibbs measures which is related to the construction of\ Patterson-Sullivan densities on the boundary at infinity of $\widetilde{M}$ will be recalled in $\S. 2$ .\ In $\S. 4$ we will prove a criterion connecting the divergence of some subgroups of $\Gamma$ with the finiteness of the Gibbs measures which comes from [@MR1776078] for the potential $F=0$.\ After this, we will construct in $\S. 5$ a sequence of bounded potentials satisfying the desired property. The main step of this paragraph builds on a result of [@C04] which claims that there exists a bounded potential such that the Gibbs measure is finite.\ Finally, we will prove in $\S. 6$ the convergence of the Gibbs measures using the variational principle which is recalled in $\S. 2$.\ Now, assume that Theorem \[thm:princ\] is true. The proof of corollary \[cor:gef\] is a consequence of [@MR3322793] where it is shown that the set of weak-mixing measures with full support is a $G_{\delta}$ subset of the set of invariant Borel probability measures supported on $\Omega$.\ In the previous theorem, we restricted ourselves to the case of geometrically finite manifolds but we make the following conjecture: the result is still true for non geometrically finite manifolds as soon as the geodesic flow is topologically mixing on its non wandering set.\ The conjecture is supported by the following two results. \[cor:geinf\] Let $M$ be a connected, complete pinched negatively curved manifold with a cusp then the set of probability measures fully supported on $\Omega$ that are weakly mixing with respect to the geodesic flow is a dense $G_{\delta}$ set of $\M^1 (T^1 M)$. \[cor:surf\] Let $S$ be a pinched negatively curved surface or a manifold with constant negative curvature then the set of probability measures fully supported on $\Omega$ that are weakly mixing with respect to the geodesic flow is a dense $G_{\delta}$ set of $\M^1 (T^1 M)$. Let $M$ be a manifold such that $dim(M)= 2$, $\kappa_M = -1$ or $M$ possesses a cusp and denote by $\phi_t$ the geodesic flow on $T^1M$. We will show in $\S.7$ that we can find a geometrically finite manifold $\hat{M}$ on which theorem \[thm:princ\] applies and for which $\phi_t$ is a factor of the geodesic flow $\hat{\phi}_t$ on $T^1 \hat{M}$.\ One way to confirm the conjecture is to find a positive answer to the following question:\ Let $M$ be a connected, complete manifold with pinched negative curvature. We will suppose that $M$ is not geometrically finite and has no cusp. Does there exist a potential $F: T^1 M \to \R$ such that the Gibbs measure associated with $(\Gamma, F)$ is finite? Preliminaries ============= Geometry on $T^1 \widetilde{M}$ ------------------------------- We first recall a few notations and results related to the geometry of negatively curved manifolds.\ Let $\partial_{\infty} \widetilde{M}$ be the boundary at infinity of $\widetilde{M}$, we define the limit set of $\Gamma$ by $$\Lambda \Gamma = \overline{ \Gamma x} \cap \partial_{\infty} \widetilde{M},$$ where $x$ is any point of $\widetilde{M}$.\ An element $\xi_p \in \Lambda \Gamma$ is **parabolic** if there exists a parabolic isometry $\gamma \in \Gamma$ satisfying $\gamma \xi = \xi$.\ A parabolic point $\xi \in \Lambda \Gamma$ is **bounded** if $\Lambda \Gamma / \Gamma_{\xi_p}$ is compact where $\Gamma_{\xi_p}$ is the maximal subgroup of $\Gamma$ fixing $\xi_p$. In this case, let $\jH_{\xi}$ be a horoball centered at $\xi$ then, $$\jC_{\xi} = \jH_{\xi} / \Gamma_{\xi_p}$$ is called the cusp associated with $\xi$.\ We say that $\Lambda_c \Gamma \subset \Lambda \Gamma$ is the conical limit set if for all $\xi \in \Lambda_c \Gamma$ for some $\widetilde{x} \in \widetilde{M}$ there exists $\epsilon >0$ and a sequence $(\gamma_n)_{n \in \N}$ such that $(\gamma_n (\widetilde{x}))_{n \in \N}$ converges to $\xi$ and stays at distance at most $\epsilon$ from the geodesic ray $(\widetilde{x} \xi)$.\ We define the parabolic limit set as follows. $$\Lambda_p \Gamma = \{ \eta \in \Lambda \Gamma : \exists \gamma \in \Gamma \text{ parabolic }, \gamma \cdot \eta = \eta \}$$ Let us choose an origin $\widetilde{x}_0$ in $\widetilde{M}$ once and for all. We define the Dirichlet domain of $\Gamma$, centered on $\widetilde{x}_0$ as follows. $$\jD = \bigcap\limits_{\gamma \in \Gamma, \gamma \neq Id} \{ \widetilde{x} \in \widetilde{M} : d(\widetilde{x},\widetilde{x}_0) \leq d(\widetilde{x},\gamma \widetilde{x}_0) \}.$$ It is a convex domain having the following properties. - $\bigcup\limits_{\gamma \in \Gamma} \gamma \jD = \widetilde{M},$ - for all $\gamma \in \Gamma \backslash \{Id\}, \mathring{\jD} \cap \gamma \mathring{\jD} = \emptyset.$ We define the diagonal of $\Lambda \Gamma \times \Lambda \Gamma$ the set of points $(x,y) \in \Lambda \Gamma \times \Lambda \Gamma$ such that $x=y$. We denote by $\Delta$ this set.\ For all $\xi, \eta \in \partial_{\infty} \widetilde{M}$, we denote by $(\xi \eta)$ the geodesic joining $\xi$ to $\eta$. We define the lift of the non wandering set on $T^1 \widetilde{M}$ by $$\widetilde{\Omega} = \{\widetilde{x} \in T^1 \widetilde{M}: \exists (\xi,\eta)\in \Lambda \Gamma \times \Lambda \Gamma \backslash \Delta, \text{ } \widetilde{x} \in (\xi\eta) \}.$$ Let $\pi:T^1\widetilde{M} \to \widetilde{M}$ be the natural projection of the unit tangent bundle to the associated manifold. We denote by $\jC\Lambda\Gamma$ the smallest convex set in $\widetilde{M}$ containing $\pi(\widetilde{\Omega})$. $M$ is geometrically finite if one of the following equivalent conditions is satisfied 1. $ \Lambda \Gamma = \Lambda_c \Gamma \cup \Lambda_p \Gamma $ = $\Lambda_c \Gamma \cup \{ \text{bounded parabolic fixed points} \}$, 2. For some $\epsilon >0$, the $\epsilon-$neighborhood of $\jC \Lambda\Gamma / \Gamma$ has finite volume, 3. $M$ has finitely many ends, all of which are funnels and cusps of finite volume. We define a map $$C_F: \partial_{\infty} \widetilde{M} \times \widetilde{M}^2 \to \R$$ called the Gibbs cocycle of $(\Gamma,F)$ by $$C_{F,\xi}(x,y)= C_F (\xi,x,y) = \lim\limits_{t \to \infty} \int_y^{\xi(t)}\widetilde{F} - \int_x^{\xi(t)} \widetilde{F}$$ where $t \mapsto \xi(t)$ is any geodesic ending at $\xi$.\ Here is a technical lemma of [@PPS] giving estimates for the Gibbs cocycle. \[prop:major\]([@PPS]) For every $r_0>0$, there exists $c_1,c_2,c_3,c_4>0$ with $c_2,c_4\leq 1$ such that the following assertions hold. \(1) For all $x,y\in \widetilde{M}$ and $\xi\in\partial_\infty \widetilde{M}$, $$|\;C_{F,\,\xi}(x,y)\;|\leq \;c_1\,e^{d(x,\,y)} \;+\; d(x,y)\max_{\pi^{-1}(B(x,\,d(x,\,y)))}|\widetilde{F}|\;,$$ and if furthermore $d(x,y)\leq r_0$, then $$|\;C_{F,\,\xi}(x,y)\;|\leq c_1\,d(x,y)^{c_2}+ d(x,y)\max_{\pi^{-1}(B(x,\,d(x,\,y)))}|\widetilde{F}|\;.$$ \(2) For every $r\in\mathopen{[}0,r_0\mathclose{]}$, for all $x,y'$ in $\widetilde{M}$, for every $\xi$ in the shadow $\O_xB(y',r)$ of the ball $B(y',r)$ seen from $x$, we have $$\Big|\;C_{F,\,\xi}(x,y')+\int_x^{y'} \widetilde{F}\;\Big|\leq c_3 \;r^{c_4}+2r\max_{\pi^{-1}(B(y',\,r))}|\widetilde{F}|\;.$$ Thermodynamic formalism for negatively curved manifolds ------------------------------------------------------- We start by recalling a few facts on Gibbs measures on negatively curved manifolds. The results of this paragraph come from [@PPS]. Let $\widetilde{F}: T^1 \widetilde{M} \to \R$ be a $\Gamma-$invariant Hölder function. We will say that the induced function $F$ on $T^1 M = T^1 \widetilde{M} / \Gamma$ is a potential.\ The Poincaré series associated with $(\Gamma,F)$ is defined by $$P_{x,\Gamma,F} (s) = \sum\limits_{\gamma \in \Gamma} e^{\int_x^{\gamma x}\widetilde{F}-s}.$$ Its critical exponent is given by $$\delta_{\Gamma,F} = \limsup\limits_{n \to \infty} \frac{1}{n} \log (\sum\limits_{\gamma \in \Gamma, n-1 \leq d(x,\gamma x)\leq n} e^{\int_x^{\gamma x} \widetilde{F}}).$$ We say that $(\Gamma,F)$ is of divergence type if\ $P_{x,\Gamma,F}(\delta_{\Gamma,F})$ diverges.\ When $F=0$ on $T^1M$, we will denote by $\delta_{\Gamma}$ the critical exponent associated to $(\Gamma,F)$. \[prop:exp\] Let $F$ be the potential on $T^1 M = (T^1 \widetilde{M}) / \Gamma$ induced by the $\Gamma-$invariant potential $\widetilde{F}:T^1 \widetilde{M} \to \R$. 1. The Poincaré series associated with $(\Gamma,F)$ converges if $s >\delta_{\Gamma,F}$ and diverges if $s<\delta_{\Gamma,F}$, 2. We have the upper bound $$\delta_{\Gamma,F} \leq \delta_{\Gamma} + \sup\limits_{\pi^{-1}(\jC \Lambda \Gamma)} \widetilde{F},$$ 3. For every $c>0$, we have $$\delta_{\Gamma,F} = \limsup\limits_{n \to \infty} \frac{1}{n} \log (\sum\limits_{\gamma \in \Gamma, n-c \leq d(x,\gamma x)\leq n} e^{\int_x^{\gamma x} \widetilde{F}}).$$ We define a set of measures on $\partial_{\infty} \widetilde{M}$ as the limit points when $s \to \delta_{\Gamma,F}^+$ of $$\frac{1}{P_{x,\Gamma,F}(s)} \sum\limits_{\gamma \in \Gamma} e^{\int_x^{\gamma x} \widetilde{F}-s} h(d(x,\gamma x)) D_{\gamma x} = \mu^F_{x,s}.$$ where $h: \R_+ \to \R_+^*$ is a well chosen non-decreasing map and $D_{\gamma x}$ is the Dirac measure supported on $\gamma x$. If $\delta_{\Gamma,F}< \infty$ then 1. $\{ \mu^F_{x,s} \}$ has at least one limit point when $s \to \delta_{\Gamma,F}^+$ with support $\Lambda \Gamma$, 2. If $\mu^F_x$ is a limit point then it is a Patterson density *i.e*\ $\forall \gamma \in \Gamma, x,y \in \widetilde{M}, \xi \in \partial_{\infty} \widetilde{M}$\ $$\gamma_* \mu^F_x = \mu^F_{\gamma x},$$ $$d\mu^F_x (\xi) = e^{-C_{F-\delta_{\Gamma,F}, \xi}(x,y)} d\mu^F_y (\xi).$$ Using the Hopf parametrization on $T^1 \widetilde{M}$, each unit tangent vector $v$ can be written as $v= (v_+,v_-,t) \in \partial_{\infty} \widetilde{M} \times \partial_{\infty} \widetilde{M} \times \R$. We define a measure on $T^1 \widetilde{M}$ by $$d \widetilde{m}_{\widetilde{F}} (v) = \frac{d \widetilde{\mu}_x^{\widetilde{F} \circ \iota} (v_-) d \widetilde{\mu}_x^{\widetilde{F}} (v) dt}{D_{F,x} (v_+,v_-) }$$ where $$D_{F,x} (v_+,v_-) = e^{- \frac{1}{2} (C_{F,v_-} (x, \pi (v)) + (C_{F\circ \iota,v_+} (x,\pi(v)))}$$ is the potential gap and $$\iota \left\{ \begin{aligned} T^1 \widetilde{M} &\to &T^1 \widetilde{M}\\ v &\mapsto& -v \end{aligned} \right.$$ is the antipodal map.\ This measure is called the Gibbs measure associated to $(\Gamma,F)$.\ It is a measure independent of $x$, invariant under the action of $\Gamma$ and invariant by the geodesic flow. Hence it defines a measure $m^F$ on $T^1M$ invariant by the geodesic flow.\ Let $m \in \M^1 (T^1 M)$ be a measure with finite entropy $h_{m}(\phi_t)$. We define the **metric pressure** of a potential $F$ with respect to the measure $m$ as the quantity $$P_{\Gamma,F}(m) = h_{m}(\phi_t) +\int_{T^1 M} F dm .$$ We say that the supremum $$P(\Gamma,F) = \sup\limits_{m \in \M(T^1M)} P_{\Gamma,F}(m)$$ is the **topological pressure** of the potential $F$. An element realizing this upper bound is called an **equilibrium state** for $(\Gamma,F)$. [@otal2004; @PPS]\[thm:varp\] Let $\widetilde{M}$ be a complete, simply connected Riemannian manifold with pinched negative curvature, $\Gamma$ a non-elementary discrete group of isometries of $\widetilde{M}$ and $\widetilde{F}: T^1 \widetilde{M} \to \R$ a Hölder-continuous $\Gamma$-invariant map with $\delta_{\Gamma,F} < \infty$. 1. We have $$P(\Gamma,F) = \delta_{\Gamma,F}.$$ 2. If there exists a finite Gibbs measure $m_F$ for $(\Gamma,F)$ such that the negative part of $F$ is $m_F -$integrable, then $m^F = \frac{m_F}{\norm{m_F}}$ is the unique equilibrium state for $(\Gamma,F)$. Otherwise, there exists no equilibrium state for $(\Gamma,F)$. Mixing property for the geodesic flow ===================================== The question of the topological mixing of the geodesic flow on a negatively curved manifold is still open in full generality. This question is closely related to mixing with respect to a Gibbs measure (see [@PPS]).\ We define the length of an element $\gamma \in \Gamma$ by $\ell (\gamma) = \inf\limits_{z \in M} d(z, \gamma z)$. If $\delta_{\Gamma,F} < \infty$ and $m^F$ is finite then the following propositions are equivalent. 1. The geodesic flow is topologically mixing on $\Omega $, 2. The geodesic flow is mixing with respect to $m^F$, 3. $L(\Gamma) = \{ \ell(\gamma); \gamma \in \Gamma \}$ is not contained in a discrete subgroup of $\R$. Here are some cases where the geodesic flow is known to be topologically mixing [@MR1617430],[@MR1703039],[@MR1779902]. \[lem:mix\] Let $\Gamma$ be a non elementary group of isometries of a Hadamard manifold $\widetilde{M}$ with pinched negative curvature. If $M = \widetilde{M} / \Gamma$ satisfies one of the following properties then the restriction of the geodesic flow to its non-wandering set is topologically mixing. 1. The curvature of $M$ is constant, 2. $\dim M = 2$, 3. There exists a parabolic isometry in $\Gamma$, 4. $\Omega = T^1 M$. To conclude this section, let us recall the Hopf-Tsuji-Sullivan criterion for the ergodicity of the geodesic flow with respect to the Gibbs measure (see [@PPS] for a proof) \[thm:HTS\] The following assertions are equivalent: 1. $(\Gamma,F)$ is of divergence type, 2. $\forall x\in \widetilde{M}, \mu_x^{F} (\partial_{\infty} \widetilde{M} \backslash \Lambda_c \Gamma) = 0$, 3. The dynamical system $(T^1M, (\phi_t)_{t \in \R},m_F)$ is ergodic. As a consequence of this theorem, one can show that if $\delta_{\Gamma,F} < \infty$, the Patterson density $(\mu_x^{F})_{x \in \widetilde{M}}$ associated with $(\Gamma,F)$ is non-atomic (see [@PPS] Proposition 5.13). A finiteness criterion ====================== First, let us give a criterion for the finiteness of the Gibbs measure. This result comes from [@MR1776078] for a potential $F=0$. For the general case where $F$ is an Hölder potential, the proof is given in [@PPS]. Suppose that $\Gamma$ is a geometrically finite group with $(\Gamma,F)$ of divergence type and $\delta_{\Gamma,F}< \infty$. The Gibbs measure $m_{F}$ is finite if and only if for every parabolic fixed point $\xi_p$ $\sum\limits_{\gamma \in \Gamma_{\xi_p}} d(x,\gamma x) e^{\int_x^{\gamma x} (\widetilde{F}-\delta_{\Gamma,F})} $ converges. $(\Gamma ,F)$ satisfies the spectral gap property if for all parabolic points $ \xi_p \in \partial_{\infty} \widetilde{M}$, $$\delta_{\Gamma_{\xi_p},F} < \delta_{\Gamma,F}.$$ Proposition 2 of [@MR1776078] gives a criterion for this property for the zero potential. The following proposition is more general and applies to all Hölderian potentials. Let $\widetilde{M}$ be a Hadamard manifold with pinched negative curvature and $\Gamma$ a geometrically finite discrete group acting on it. Suppose there exists a bounded Hölderian potential $\widetilde{F}: T^1 \widetilde{M} \to \R$.\ If for all parabolic fixed point $\xi_p$ the couple $(\Gamma_{\xi_p},F)$ is of divergence type, then $(\Gamma,F)$ satisfies the spectral gap property. Moreover, the Gibbs measure $m^F$ associated to $(\Gamma,F)$ is finite. To prove the first claim, we follow the ideas of [@MR1776078] when $F=0$.\ Since the action of $\Gamma_{\xi_p}$ on $\partial_{\infty} \widetilde{M}$ has a fundamental domain $\G$ in $\partial_{\infty} \widetilde{M}$, we have $$\mu_x^F (\partial_{\infty} \widetilde{M}) = \sum\limits_{g \in \Gamma_{\xi_p}} \mu_x^F (g \G) + \mu_x^F (\Lambda \Gamma_{\xi_p}).$$ Moreover, since there exists $K\in \R$ such that $$\left\{ \begin{aligned} \mu_x^F (g \G) = \int_{\G} e^{-C_{F-\delta_{\Gamma,F}, \xi}(x,gx)} d\mu_{gx} (\xi)\\ |C_{F,\xi} (x,gx) + \int_x^{gx} \widetilde{F}| \leq K, \end{aligned} \right.$$ we have $$\mu_x^F (g \G) \geq (e^{\int_x^{gx} \widetilde{F} - \delta_{\Gamma,F}}) ( e^{-K} \mu_x^F(\G) )$$ and $$\infty > \mu_x^F (\partial_{\infty} \widetilde{X}) \geq \sum\limits_{g \in \Gamma_{\xi_p}} \mu_x^F (g \G) \geq C_0 \cdot P_{x,\Gamma_{\xi_p},F}(\delta_{\Gamma,F}).$$ So, $\delta_{\Gamma,F} > \delta_{\Gamma_{\xi_p},F}.$\ For the second claim, since $(\Gamma,F)$ satisfies the spectral gap property for $M = \widetilde{M} / \Gamma$ and $$\forall \epsilon >0, \exists C_{\epsilon}>0, d(x,\gamma x) \geq C_{\epsilon} \Rightarrow e^{\epsilon d(x,\gamma x)} > d(x,\gamma x),$$ we have $$\sum\limits_{\gamma \in \Gamma_{\xi_p}} d(x,\gamma x) e^{\int_x^{\gamma x} \widetilde{F} - \delta_{\Gamma, F}} \leq \sum\limits_{\gamma \in \Gamma_{\xi_p}} e^{\int_x^{\gamma x} \widetilde{F} - (\delta_{\Gamma, F} - \epsilon)}$$ Choosing $\epsilon$ small enough such that $\frac{\delta_{\Gamma,F} - \delta_{\Gamma_{\xi_p},F}}{2} > \epsilon$, the series converges and the Gibbs measure is finite. Construction of the potentials ============================== We now construct a $\Gamma-$invariant potential $\widetilde{H}: T^1 \widetilde{M} \to \R$ such that the associated Gibbs measure is finite and which critical exponent associated to $(\Gamma_{\xi_p},H)$ is of divergence type.\ Since $M$ is geometrically finite, the set $Par_{\Gamma}$ of parabolic points $\xi_p \in \partial_{\infty} \widetilde{M}$ intersecting the boundary of the Dirichlet domain is finite.\ We define for those parabolic points a family of disjoint horoballs $\{\jH_{\xi_p} (u_{0,\xi_p}) \}_{\xi_p}$ on $\widetilde{M}$ passing through a well chosen point $\widetilde{u}_{0,\xi_p}$ of the cusp. For any $\widetilde{u}$ in $\jH_{\xi_p} (\widetilde{u}_{0,\xi_p})$, we define a height function $\rho: \widetilde{M} \to \R$ by the Buseman cocycle at $\widetilde{u}_{0,\xi_p}$: $$\rho(\widetilde{u}) = \beta_{\xi_p}(\widetilde{u},\widetilde{u}_{0,\xi_p}).$$ This cocycle coincides with the Gibbs cocycle for the potential $F= -1$.\ The curve levels of this function are the horocycles based on $\xi_p$ passing through $\widetilde{u}$.\ Let $t_n$ be a decreasing sequence of positive numbers converging to $0$.\ One can construct a sequence $Y_n$ of positive numbers such that $$\left\{ \begin{aligned} Y_{n+1} \geq Y_n + t_n - t_{n+1}, \\ \sum \limits_{p \in \rho^{-1}(]Y_n , Y_{n+1}])} e^{d(x_0,px_0)(t_n - \delta_{\Gamma_{\xi_p}})} \geq 1. \end{aligned} \right.$$ for all $\xi_p \in Par_{\Gamma}$, we define a $\Gamma$-invariant map on $\jD \cap \jH_{\xi_p}(u_{0,\xi_p})$ as follows:\ for every $\widetilde{u}$ in $\jH_{\xi_p} (u_{0,\xi_p})$, $$\widetilde{H}_1(\widetilde{u})= \left\{ \begin{aligned} t_n + Y_n -\rho (\widetilde{u}) \text{on } \rho^{-1}(]Y_n,Y_n+t_n-t_{n+1}]) , \\ t_{n+1} \text{on } \rho^{-1}(]Y_n+t_n-t_{n+1},Y_{n+1}]). \end{aligned} \right.$$ We extend $\widetilde{H}_1$ to $\widetilde{M}$ as follows.\ First, we extend it to $\jD$ by a constant function such that $$\widetilde{H}_1: \jD \to \R$$ is Hölder-continuous. Next, we extend it to $\widetilde{M}$ as follows. For all $\gamma \in \Gamma$, if $\widetilde{x} \in \gamma \jD$ then $$\widetilde{H}_1 (\widetilde{x}) = \widetilde{H}_1 (\gamma^{-1} \widetilde{x}).$$ Let $\widetilde{H}$ be the $\Gamma-$invariant potential obtained by pulling back ${H}_1$ on $T^1 \widetilde{M}$. We denote by $H: T^1 M \to \R$ the map induced by $\widetilde{H}$ on $T^1 M$. (Coudène [@C04]) $\widetilde{H}$ is a $\Gamma-$invariant Hölderian potential such that for all parabolic fixed point $\xi_p$, the critical exponent associated with $(\Gamma_{\xi_p},H)$ is of divergence type. We have therefore constructed a $\Gamma$-invariant Hölderian potential on $T^1 M$ such that the associated Gibbs measure is finite and is supported on $\Omega$.\ Remark that the divergence of the parabolic subgroups only depends on the value of the potential in the cusp. In the previous construction, we made the assumption that the potential was constant on $$T^1 M_0 = T^1 (\jC \Lambda \Gamma \bigcup\limits_{\xi_p \in Par_{\Gamma}} \jH_{\xi_p}) / \Gamma .$$ However, taking any bounded potential such that the resulting function on $T^1 M_0$ is Hölder-continuous, the associated Gibbs measure will still be finite.\ We now choose $p\in T^1 M = T^1\widetilde{M} / \Gamma$ such that $\phi_{t} (p)$ is periodic. We will denote by $\O(p)$ the closed curve $\phi_{\R}(p)$ and assume that $\O(p) \subset T^1 M_0$.\ For all $n \in \N$, we define a Lipschitz-continuous potential by $$F_n (x) = \max \{ c_n - c_n d (\O(p),x) ; H(x) \}.$$ For all $n \in \N$, the critical exponent $\delta_{\Gamma,F_n}$ is finite. Moreover, we have $$c_n \leq \delta_{\Gamma,F_n} \leq \delta_{\Gamma} + c_n.$$ Since $c_k = \sup\limits_{x \in T^1 M} F_k(x)$, the upper bound $\delta_{\Gamma,F_k} \leq \delta_{\Gamma} + c_k$ is evident by the second claim of Proposition \[prop:exp\].\ Let $p$ be the periodic point of $T^1M$ defined above and $h\in \Gamma$ the generator of the isometry group fixing the periodic orbit $\phi_{\R}(p)$. Let $H= <h>$ and denote by $\ell$ the length of $h$. By the very definition of critical exponents, we have $$\begin{aligned} \delta_{\Gamma,F_k} &= & \limsup\limits_{n \to \infty} \frac{1}{n} \log \sum\limits_{\gamma \in \Gamma,\;n- \ell< d(x,\,\gamma x)\leq n} e^{\int_x^{\gamma x}\widetilde{F}_k} \\ &\geq & \limsup\limits_{n \to \infty} \frac{1}{n} \log \sum\limits_{\gamma \in H ,\;n-\ell< d(x,\,\gamma x)\leq n} e^{\int_x^{\gamma x}\widetilde{F}_k}.\end{aligned}$$ Since the critical exponent does not depend on the choice of a base point, one can choose $x = \widetilde{p}$ where $\widetilde{p}$ is a lift of $p$ on $T^1 \widetilde{M}$. Therefore, the fact that the value of the potential on $\O(p)$ is constant, equal to $c_k$ implies that $$\int_{\widetilde{p}}^{\gamma {\widetilde{p}}}\widetilde{F}_k = d(x, \gamma x ) c_k \geq (n- \ell) c_k$$ which gives $$\begin{aligned} \delta_{\Gamma,F_k} &\geq \limsup\limits_{n \to \infty} \frac{1}{n} \log \sum\limits_{\gamma \in H ,\;n-\ell< d(x,\,\gamma x)\leq n} e^{(n-\ell) c_k} \geq \limsup\limits_{n \to \infty} \frac{1}{n} \log \left( A_x e^{(n- \ell)c_k} \right),\end{aligned}$$ where $A_x = \sharp \{\gamma \in H : n- \ell < d(x,\gamma x) \leq n \}$.\ Since the group $H$ is generated by a hyperbolic isometry $h$, for all $\gamma \in H$, there exists $i \in \N$ such that $\gamma = h^i$ and $\ell (\gamma)= \ell (h^i) = i \ell(h).$ Therefore the quantity $A_x$ does not depend on $n$ and $$\delta_{\Gamma,F_k} \geq c_k.$$ which concludes the proof. Therefore, the Gibbs measure $m_{F_n}$ associated with $(\Gamma,F_n)$ of dimension $\delta_{\Gamma,F_n}$ exists for all $n\in \N$.\ Proof of the main theorem ========================= Let $D_{\O(p)}$ be the Dirac measure supported on $\O(p)$. We prove Theorem \[thm:princ\] which states the following.\ *Let $M$ be a geometrically finite, negatively curved manifold and $\phi_t$ its geodesic flow. If $\phi_t$ is topologically mixing on $\Omega$, then the set of probability measures that are mixing with respect to the geodesic flow is dense in $\M^1 (T^1 M)$ for the weak topology.*\ Here is our strategy: since the geodesic flow on a manifold with pinched negative curvature satisfies the closing lemma (see [@MR1441541]) and admits a local product structure, we use the following result. (Coudène-Schapira [@MR2735038]) Let $M$ be a complete, connected Riemannian manifold with pinched negative curvature and $\phi_t$ its geodesic flow. Then the set of normalized Dirac measures on periodic orbits is dense in the set of all invariant measures $\M^1 (T^1 M)$. It is therefore clear that the following proposition implies Theorem \[thm:princ\]. \[prop:conv\] For all $p \in T^1 M$ such that $\O(p)= \phi_{\R} (p)$ is periodic, there exists a sequence $\{m_k \}_{k \in \N}$ of measures satisfying the following properties 1. $m_k$ is a probability measure which is mixing with respect to the geodesic flow, 2. $m_k \rightharpoonup D_{\O(p)}$. Let $\htop(\phi_t)$ be the topological entropy of the geodesic flow on $T^1 M$ and $h_{\mu}(\phi_t)$ the measure theoretic entropy of the geodesic flow with respect to $\mu$. D. Sullivan proved in [@Su] that if the Bowen-Margulis measure $m_{BM}$ (*i.e* the Gibbs measure associated with the potential $F=0$) is finite then $$\delta_{\Gamma} = h_{m_{BM}}(\phi_t) .$$ Using a result of C.J.Bishop and P.W.Jones [@BJ] connecting the critical exponent $\delta_{\Gamma}$ with the Hausdorff dimension of the conical limit set of $\Gamma$, J.P.Otal and M.Peigné proved that for all $\phi_t-$invariant probability measures $m \in \M^1(T^1\widetilde{M})$ which are not the Bowen-Margulis measure, we have the strict inequality, $$h_{\mu}(\phi_t) < \delta_{\Gamma} .$$ We refer to F. Ledrappier [@ENSML] for a survey of these results. [@otal2004] Let $\widetilde{M}$ be a simply connected, complete Riemannian manifold with pinched negative curvature and $\Gamma$ be a non-elementary discrete group of isometries of $\widetilde{M}$, then $$\htop (\phi_t)= \delta_{\Gamma}.$$ Moreover, there exists a probability measure $\mu$ maximizing the entropy if and only if the Bowen-Margulis measure is finite and $\mu= m_{BM}$. We begin the proof of the main theorem. First, we state a general result which holds true for any metric space $X$ satisfying a variational principle. In the next claim, we suppose that $F: X \to \R$ is a measurable function such that there exist an invariant compact set $K \subset X$ and a neighborhood $V$ of $K$ satisfying the following assumptions: $ \left\{ \begin{aligned} \forall x \in K, F(x) = c = \sup\limits_{x \in X} F(x)\\ \sup\limits_{x\in X \backslash V} F (x)= c' < c. \end{aligned} \right.$ We say that a probability measure $\mu$ with finite entropy is an equilibrium state for a potential $F:X \to \R$ if it achieves the supremum of $$m \mapsto h_m (\phi_t) + \int_X F dm$$ over all invariant probability measures with finite entropy. \[lem:majvar\] Let $X$ be a metric space, $\phi_t$ a flow defined on $X$ and $F:X \to \R$, $K$, $V$ defined as above.\ Suppose there exists an equilibrium state $\mu$ for $F$, then $$\mu(X \backslash V) \leq \frac{h_{\mu}(\phi_t)}{c-c'}.$$ Let $m_K$ be a probability measure supported on $K$. Since $\mu$ realises the supremum of $$m \in \M^1(X) \mapsto h_m (\phi_t) + \int_X F dm$$ we have $$h_{\mu} (\phi_t) + \int_X F d\mu \geq h_{m_K} (\phi_t) + \int_X F dm_K$$ which implies $$\label{eqn:1} h_{\mu} (\phi_t) + \int_X F d\mu \geq \int_X F dm_K$$ since $h_{m_K} (\phi_t) \geq 0$. Moreover, since the potential $F$ is constant on $K$ we have $$\int_X F dm_K = c$$ which can be written as $$\label{eqn:2} \int_X F dm_K = c (\mu(V) + \mu(X \backslash V)).$$ Combining equations \[eqn:1\] and \[eqn:2\], we obtain $$\begin{array}{rcl} h_{\mu} (\phi_t) &\geq& \int_X F dm_K - \int_X F d\mu \\ &\geq & c (\mu(V) + \mu(X \backslash V)) - c\mu(V) - c' \mu(X \backslash V) ) \end{array}$$ which finally gives $$\frac{h_{\mu}(\phi_t)}{c-c'} \geq \mu(X \backslash V).$$ Recall that a sequence $\{\mu_n\}_{n \in \N}$ of probability measures on a Polish space $X$ is tight if for all $\epsilon >0$ there exists a compact set $K \subset X$ such that $$\forall n \in \N, \text{ } \mu_n(X \backslash K)< \epsilon .$$ We give a criterion for the convergence of a sequence of probability measures to the Dirac measure supported on the periodic orbit $\O(p)$. We denote by $V_{\epsilon}$ the subset of $T^1 M$ defined by $$V_{\epsilon}= \{ x \in T^1M : d(x, \O(p)) \leq \epsilon \}.$$ \[lem:cv\] The following assertions are equivalent: 1. The sequence of probability measures $\{m^{F_n} \}_{n \in \N}$ converges to the Dirac measure supported on $\O(p)$, 2. for all $\epsilon >0$, $$\lim\limits_{n\to \infty} m^{F_n}(T^1 M \backslash V_{\epsilon}) =0 .$$ It is clear that $(1) \Rightarrow (2)$. Let us show that $(2) \Rightarrow (1)$.\ We first notice that $\{m^{F_n} \}_{n \in \N}$ is tight. Indeed, let $V$ be a compact subset of $T^1 M$ containing $\O(p)$. Since condition $(2)$ is satisfied, for all $\epsilon >0$, there exists $N_0 >0$ such that for all $n \geq N_0,$ $$m^{F_n} (T^1 M \backslash V ) < \epsilon.$$ For all $n \in \{1,..,N_0-1 \}$, we can also find a compact set $K_n$ such that $$m^{F_n} (T^1 M \backslash K_n) < \epsilon.$$ Define the compact set $K = (\bigcup\limits_{n = 1}^{N_0 -1} K_n) \cup V$.\ For all $n \in \N$, $K$ satisfies $$m^{F_n}(T^1 M \backslash K) \leq \epsilon.$$ Therefore, the sequence $\{m^{F_n} \}_{n \in \N}$ is tight.\ Since condition $(2)$ is satisfied and using the fact that the unique invariant probability measure supported on $\O(p)$ is $D_{\O(p)}$, each converging subsequence of $\{m^{F_n}\}$ converges to $D_{\O(p)}$.\ Therefore, by Prokhorov’s Theorem [@Pro], each sub sequence of $\{m^{F_n}\}$ possesses a further subsequence converging weakly to $D_{\O(p)}$ so the sequence $m^{F_n}$ converges weakly to $D_{\O(p)}. $ We are now able to prove our main result of convergence. The sequence $\{m^{F_n}\}_{n \in \N}$ of probability measures converges to the Dirac measure supported on $\O(p)$. Let $D_{\O(p)}$ be the Dirac measure supported on the periodic orbit $\O(p)$. Recall that $c_n = \sup\limits_{x \in T^1 M} F_n(x)$.\ By lemma \[lem:cv\], we have to show that $$\lim\limits_{n\to \infty} m^{F_n}(T^1 M \backslash V_{\epsilon}) =0 .$$ Using the variational principle described in Theorem \[thm:varp\], we have $$\delta_{\Gamma,F_n} = \sup\limits_{\mu \in \M^1(T^1M)} P_{\Gamma,F_n}(\mu) = P_{\Gamma,F_n}(m^{F_n})$$ and $$\delta_{\Gamma,F_n} \geq P_{\Gamma,F_n}(D_{\O(p)}) = h_{D_{\O(p)}}(\phi_t) +\int_{T^1 M} F_n dD_{\O(p)} .$$ Since $\O(p)$ is invariant by the action of the geodesic flow and $$h_{m_{F_n}} (\phi_t) \leq \delta_{\Gamma} < \infty,$$ one can use the claim of lemma \[lem:majvar\] to obtain the following inequality $$m^{F_n}(T^1 M \backslash V_{\epsilon}) \leq \frac{\delta_{\Gamma}}{c_n -c'_n}$$ where $$c'_n = \sup\limits_{x \in T^1M \backslash V_{\epsilon}} F_n(x) .$$ By the definition of the potential $F_n$, we know that $$c'_n \leq \max\{c_0,c_n(1-\epsilon) \}.$$ Therefore, for all $\epsilon >0$ and $n$ large enough, $$m^{F_n}(T^1 M \backslash V_{\epsilon}) \leq \frac{\delta_{\Gamma}}{c_n \epsilon}$$ implies that $$\lim\limits_{n\to \infty} m^{F_n}(T^1 M \backslash V_{\epsilon}) =0 .$$ Which concludes the proof. Finally, we are able to prove corollary \[cor:gef\] which states that the set of weak-mixing measures is a dense $G_{\delta}$ subset of $\M^1(T^1 M)$.\ Our proof relies on the following theorem. (Coudène-Schapira [@MR3322793]) Let $(\varphi^t)_{t\in \R}$ be a continuous flow on a Polish space. The set of weak-mixing measures on $X$ is a $G_{\delta}$ subset of the set of Borel invariant probability measures on $X$. (of corollary \[cor:gef\])\ Since mixing measures are weak-mixing, Theorem \[thm:princ\] implies that weak-mixing measures are dense in the set of probability measures on $\Omega$. The previous theorem insures us that it is a $G_{\delta}$ set. Since negatively curved manifolds with pinched curvature satisfy the closing lemma, it is shown in [@MR2735038] that the set of invariant measures with full support on $\Omega$ is a dense $G_{\delta}$ subset of the set of invariant probability measures on $\Omega$. The intersection of those two dense $G_{\delta} $ sets is a dense $G_{\delta} $ set by the Baire Category Theorem. Non-geometrically finite manifolds with cusps ============================================= We now prove corollaries \[cor:geinf\] and \[cor:surf\]. First of all, remark that since the manifold possesses a cusp then from lemma \[lem:mix\], the geodesic flow is topologically mixing on $T^1M= T^1\widetilde{M} / \Gamma$.\ Let $\gamma \in \Gamma$ and $D(\gamma)$ the subset of $\widetilde{M}$ bounded by $$\{ \widetilde{x} \in \widetilde{M} : d(\widetilde{x},\widetilde{x}_0) = d(\widetilde{x},\gamma \widetilde{x}_0) \}$$ and containing $\gamma \widetilde{x_0}$. We define the subset $C_{\gamma}$ on $\partial_{\infty} \widetilde{M}$ as follows. $$C_{\gamma} = \partial_{\infty} \widetilde{M} \cap \overline{D(\gamma)}.$$ The proof of corollary \[cor:geinf\] is deduced from the following result. \[lem:V\_0\] Let $M$ be a connected, complete pinched negatively curved manifold with a cusp and $\phi_t$ the geodesic flow defined on $T^1 M$. There exists a geometrically finite manifold $\hat{M}$ such that its geodesic flow $\hat{\phi}_t$ is topologically mixing and a covering map $ \rho: T^1 \hat{M} \to T^1M $ such that the diagram $$\begin{CD} T^1 \hat{M} @>{\hat{\phi}_t}>> T^1 \hat{M} \\ @V{\rho}VV @VV{\rho}V \\ T^1 M @>{\phi_t}>> T^1 M \end{CD}$$ commutes. Let $\xi_p \in \partial_{\infty}\widetilde{M}$ be a bounded parabolic point fixed by a parabolic isometry $\gamma_p \in \Gamma$ and $h \in \Gamma$ be a hyperbolic transformation.\ Let $N>0$ be defined such that the sets $C_{\gamma_p^N} , C_{h^N}, C_{\gamma_p^{-N}} , C_{h^{-N}}$ have disjoint interiors. We define $\Gamma_0 =$ $<\gamma_p^N , h^N>$, a subgroup of $\Gamma$. The ping-pong lemma shows that $\Gamma_0$ is a discrete group which acts freely discontinuously on $\widetilde{M}$. So, the quotient $\widetilde{M} / \Gamma_0$ is a geometrically finite manifold. (of corollary \[cor:geinf\]) Let $\hat{M}$ be a geometrically finite manifold constructed as in lemma \[lem:V\_0\] and $\rho$ its associated covering map. We can use the proof of Theorem \[thm:princ\] on $\hat{M}$ and construct a sequence $\hat{m}_k$ of invariant mixing measures for $\hat{\phi}_t$ such that $\hat{m}_k \rightharpoonup D_{\O(p)}$.\ Since the geodesic flow $\phi_t$ is a factor of $\hat{\phi}_t$, we can define $\nu_k$ to be the push-forward by $\rho$ of $\hat{m}_k$, then it is an invariant mixing measure on $T^1M$ and for all bounded continuous function $g$ on $T^1 M$, $$\begin{array}{rcl} \lim\limits_{k \to \infty} \int_{T^1M} g d\nu_k &=& \lim\limits_{k \to \infty} \int_{T^1\hat{M}} g\circ \rho dm_k \\ &=& \int_{T^1 \hat{M}} g\circ \rho dD_{\O(p)} \\ &=& \int_{T^1 M} g d(\rho_* D_{\O(p)}). \end{array}$$ We end up by the proof of corollary \[cor:surf\]. In the case of a surface $S$ (or a constant negatively curved manifold), we don’t need to ask for the existence of a bounded parabolic point. Since the geodesic flow is always topologically mixing in restriction to its non wandering set, we can choose two hyperbolic isometries $h_1, h_2$ in $\Gamma$ such that the subgroup $\Gamma_0 = <h_1, h_2>$ is convex-cocompact.\ The same proof as lemma \[lem:V\_0\] shows us that the geodesic flow $\phi_t$ on $S$ is a factor of the geodesic flow $\hat {\phi_t}$ on the convex-cocompact manifold $T^1 S_0 = T^1\widetilde{S} / \Gamma_0 $. Therefore, the previous proof gives us the density result. [PPS15]{} Coud[è]{}ne, Yves and Schapira, Barbara. “Generic measures for geodesic flows on nonpositively curved manifolds” [*J. Éc. polytech. Math.*]{} (2014), 387-408. Coud[è]{}ne, Yves and Schapira, Barbara. “Generic measures for hyperbolic flows on non-compact spaces” [*Israel J. Math.*]{} [**179**]{} (2010), 157-172. K. Sigmund. “On the space of invariant measures for hyperbolic flows” [*Proceedings of the American Math. Soc.*]{} [**36**]{} (1972). K. Sigmund. “Séries de [P]{}oincaré des groupes géométriquement finis” [*Israel J. Math.*]{} [**118**]{} (2000), 109-124. Paulin, F. and Pollicott, M. and Schapira, B. “Equilibrium states in negative curvature” [*Soc. Math. France* ]{} [**Astérique $n^0373$**]{} (2015), . Sullivan, Dennis. “The density at infinity of a discrete group of hyperbolic motions” [*Publications Mathématiques de l’Institut des Hautes Études Scientifiques*]{} [**50**]{} ,171–202 . Dal’bo, Fran[ç]{}oise, Peigné, Marc. “Some negatively curved manifolds with cusps, mixing and counting” [*Journal für die Reine und Angewandte Mathematik*]{} [**497**]{} (1998), 141-169. Dal’bo, Fran[ç]{}oise. “Remarques sur le spectre des longueurs d’une surface et comptages” [*Bol. Soc. Brasil. Mat. (N.S.)*]{} [**30**]{} (1999), 199-221. Dal’bo, Fran[ç]{}oise. “Topologie du feuilletage fortement stable” [*Ann. Inst. Fourier (Grenoble)*]{} [**50**]{} (2000), 981-993. Coudene, Yves. “Gibbs Measures on Negatively Curved Manifolds” [*Journal of Dynamical and Control Systems*]{} [**9**]{} (2003), 89–101 . Otal, Jean-Pierre and Peigné, Marc. “Principe variationnel et groupes Kleiniens” [*Duke Mathematical Journal* ]{} [**125**]{} (2004), 15–44. Bishop, Christopher J. and Jones, Peter W.. “Hausdorff dimension and Kleinian groups” [*Acta Mathematica*]{} [**179**]{} , 1–39. Ledrappier, François. “Entropie et principe variationnel pour le flot géodésique en courbure négative pincée” In [*Géométrie ergodique*]{} in [**Monogr. Enseign. Math., volume $n^0 43$**]{} (2013), 117–144. Prokhorov, Yu. V. “Convergence of Random Processes and Limit Theorems in Probability Theory” [*Theory of Probability & Its Applications*]{} [**1**]{} (1956), 157–214.

--- abstract: 'In this paper, the existence of positive solutions for a nonlinear fourth-order two-point boundary value problem with integral condition is investigated. By using Krasnoselskii’s fixed point theorem on cones, sufficient conditions for the existence of at least one positive solutions are obtained.' address: - | Slimane Benaicha\ Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO)\ Department of Mathematics, University of Oran 1 Ahmed Benbella, 31000 Oran, Algeria - | Faouzi Haddouchi\ Department of Physics, University of Sciences and Technology of Oran-MB, El Mnaouar, BP 1505, 31000 Oran, Algeria\ Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO)\ Department of Mathematics, University of Oran 1 Ahmed Benbella, 31000 Oran, Algeria author: - 'Slimane Benaicha, Faouzi Haddouchi' title: 'Positive solutions of a nonlinear fourth-order integral boundary value problem' --- \[section\] \[theorem\][Lemma]{} \[theorem\][Defnition]{} \[theorem\][Proposition]{} \[theorem\][Corollary]{} \[theorem\][Remark]{} \[section\] Introduction ============ In this work, we study the existence of positive solutions of a nonlinear two-point boundary value problem (BVP) for the following fourth-order differential equation: $$\label{eq-1.1} {u^{\prime \prime \prime \prime }}(t)+f(u(t))=0,\ t\in(0,1),$$ $$\label{eq-1.2} u^{\prime}(0)=u^{\prime}(1)=u^{\prime \prime}(0)=0, \ u(0)=\int_{0}^{1}a(s)u(s)ds,$$ where - $f\in C([0,\infty),[0,\infty))$; - $a\in C([0,1],[0,\infty))$ and $0<\int_{0}^{1}a(s)ds<1$. Fourth-order ordinary differential equations are models for bending or deformation of elastic beams, and therefore have important applications in engineering and physical sciences. Recently, the two-point and multi-point boundary value problems for fourth-order nonlinear differential equations have received much attention from many authors. Many authors have studied the beam equation under various boundary conditions and by different approaches. In 2009, Graef et al.[@Graef3] considered the fourth order three-point boundary value problem $$\label{eq-1.3} {u^{\prime \prime \prime \prime }}(t)=g(t)f(u(t)), \ t\in(0,1),$$ together with the boundary conditions $$\label{eq-1.4} u(0)=u^{\prime}(0)=u^{\prime \prime}(\beta)=u^{\prime \prime}(1)=0.$$ In 2006, Anderson and Avery [@Ander], studied the following fourth order right focal four-point boundary value problem $$\label{eq-1.5} {u^{\prime \prime \prime \prime }}(t)+f(u(t))=0, \ 0<t<1,$$ $$\label{eq-1.6} u(0)=u^{\prime}(q)=u^{\prime \prime}(r)=u^{\prime \prime \prime}(1)=0.$$ In 2011, Xiading Han et al.[@Han], considered the nonlocal fourth-order boundary value problem with variable parameter $$\label{eq-1.7} {u^{\prime \prime \prime \prime }}(t)+B(t)u^{\prime \prime}(t)=\lambda f(t,u(t), u^{\prime \prime}(t)), \ 0<t<1,$$ $$\label{eq-1.8} u(0)=u(1)=\int_{0}^{1}p(s)u(s)ds, \ u^{\prime \prime}(0)=u^{\prime \prime}(1)=\int_{0}^{1}q(s)u^{\prime \prime}(s)ds.$$ In 2013, Yan Sun and Cun Zhu [@Sun], considered the singular fourth-order three-point boundary value problem $$\label{eq-1.9} {u^{\prime \prime \prime \prime }}(t)+f(t,u(t))=0, \ 0<t<1,$$ $$\label{eq-1.10} u(0)=u^{\prime}(0)=u^{\prime \prime}(0)=0 , \ u^{\prime \prime}(1)-\alpha u^{\prime \prime}(\eta)=\lambda.$$ In 2014, Xiaorui Liu and Dexiang Ma [@Liu], considered the third-order two-point boundary value problem $$\label{eq-1.11} {u^{\prime \prime \prime }}(t)+f(u(t))=0, \ 0<t<1,$$ $$\label{eq-1.12} u^{\prime}(0)=u^{\prime}(1)=0 , \ u(0)=\int_{0}^{1}k(s)u(s)ds,$$ and in 2015, Wenguo shen [@Shen], studied the fourth-order second-point nonhomogeneous singular boundary value problem $$\label{eq-1.9} {u^{\prime \prime \prime \prime }}(t)+a(t)f(u(t))=0, \ 0<t<1,$$ $$\label{eq-1.10} u(0)=\alpha, \ u(1)=\beta, \ u^{\prime}(0)=\lambda, \ u^{\prime}(1)=-\mu.$$ For some other results on boundary value problem, we refer the reader to the papers [@Alves; @Graef1; @Graef2; @Hend; @Kos; @Ma1; @Ma2; @Ping; @Webb; @Yao; @Yang; @Zhang]. To the authors’ knowledge, there are few papers that have considered the existence of solutions for fourth-order two-point boundary value problem with integral condition. Motivated by the works mentioned above, the aim of this paper is to establish some sufficient conditions for the existence of at least one positive solutions of the BVP and . We shall first construct the Green’s function for the associated linear boundary value problem and then determine the properties of the Green’s function for associated linear boundary value problem. Finally, existence results for at least one positive solution for the above problem are established when $f$ is superlinear or sublinear. As applications, some interesting examples are presented to illustrate the main results. Preliminaries ============= We shall consider the Banach space $C([0,1])$ equipped with the sup norm $$\|u\|=\\sup_{t\in[0, 1]}|u(t)|$$ \[def 2.1\] Let $E$ be a real Banach space. A nonempty, closed, convex set $ K\subset E$ is a cone if it satisfies the following two conditions: \(i) $x\in K$, $\lambda \geq 0$ imply $\lambda x\in K$; \(ii) $x\in K$, $-x\in K$ imply $x=0$. \[def 2.2\] An operator $T:E\rightarrow E$  is completely continuous if it is continuous and maps bounded sets into relatively compact sets. To prove some of our results, we will use the following fixed point theorem, which is due to Krasnoselskii’s [@Krasn]. \[thm 2.3\][@Krasn]. Let $E$ be a Banach space, and let $K\subset E$, be a cone. Assume that $% \Omega_{1}$ and $\Omega_{2}$ are open subsets of $E$ with $0\in \Omega _{1}$, $\Omega _{1}\subset \Omega_{2}$ and let $$A:K\cap (\overline{ \Omega_{2}}\backslash \Omega_{1})\rightarrow K$$ be a completely continuous operator such that \(i) $\ \left\Vert Au\right\Vert \leq \left\Vert u\right\Vert ,$ $u\in K\cap \partial \Omega _{1}$, and $\left\Vert Au\right\Vert \geq \left\Vert u\right\Vert ,$ $u\in K\cap \partial \Omega_{2}$; or \(ii) $\left\Vert Au\right\Vert \geq \left\Vert u\right\Vert ,$ $u\in K\cap \partial \Omega_{1}$, and  $\left\Vert Au\right\Vert \leq \left\Vert u\right\Vert ,$ $u\in K\cap \partial \Omega_{2}.$ Then $A$ has a fixed point in $K\cap (\overline{\Omega _{2}}$ $\backslash $ $ \Omega_{1})$. Consider the two-point boundary value problem $$\label{eq-2.1} {u^{\prime \prime \prime \prime }}(t)+y(t)=0,\ t\in(0,1),$$ $$\label{eq-2.2} u^{\prime}(0)=u^{\prime}(1)=u^{\prime \prime}(0)=0, \ u(0)=\int_{0}^{1}a(s)u(s)ds.$$ \[lem 2.4\] The problem - has a unique solution $$u(t)=\int_{0}^{1}\Big(G(t, s)+\frac{1}{1-\alpha}\int_{0}^{1}a(\tau)G(\tau, s)d\tau \Big)y(s)ds,$$ where $G(t, s):[0, 1]\times[0, 1]\rightarrow \mathbb{R}$ is the Green’s function defined by $$\label{eq-2.3} G(t, s)=\frac{1}{6}\begin{cases} t^{3}(1-s)^{2}-(t-s)^{3},&0\leq s\leq t \leq1; \\ t^{3}(1-s)^{2}, &0\leq t \leq s\leq 1, \end{cases}$$ and $$\alpha=\int_{0}^{1}a(t)dt.$$ Integrating over the interval $[0, t]$ for $t\in[0, 1]$, we obtain $$\begin{aligned} &u^{\prime \prime \prime}(t)=-\int_{0}^{t}y(s)ds+C_{1},\\ &u^{\prime \prime}(t)=-\int_{0}^{t}(t-s)y(s)ds+C_{1}t+C_{2},\\ &u^{\prime}(t)=-\frac{1}{2}\int_{0}^{t}(t-s)^{2}y(s)ds+\frac{1}{2}C_{1}t^{2}+C_{2}t+C_{3},\\\end{aligned}$$ $$\begin{aligned} \label{eq-2.4} &u(t)=-\frac{1}{6}\int_{0}^{t}(t-s)^{3}y(s)ds+\frac{1}{6}C_{1}t^{3}+\frac{1}{2}C_{2}t^{2}+C_{3}t+C_{4}.\end{aligned}$$ From the boundary conditions we get $$C_{2}=C_{3}=0, \ \ C_{1}=\int_{0}^{1}(1-s)^{2}y(s)ds,$$ and $$\begin{aligned} C_{4}&=&u(0)\\ &=&\int_{0}^{1}a(\tau)\Big(-\frac{1}{6}\int_{0}^{\tau}(\tau-s)^{3}y(s)ds+ \frac{1}{6}\tau^{3}\int_{0}^{1}(1-s)^{2}y(s)ds+C_{4}\Big)d\tau \\ &=&-\frac{1}{6}\int_{0}^{1}a(\tau)\Big(\int_{0}^{\tau}(\tau-s)^{3}y(s)ds\Big)d\tau+ \frac{1}{6}\int_{0}^{1}a(\tau)\tau^{3}\Big(\int_{0}^{1}(1-s)^{2}y(s)ds\Big)d\tau\\ &&+ C_{4}\int_{0}^{1}a(\tau)d\tau,\end{aligned}$$ which implies $$C_{4}=\frac{1}{6(1-\alpha)}\Big(\int_{0}^{1}a(\tau)\tau^{3}\Big(\int_{0}^{1}(1-s)^{2}y(s)ds\Big)d\tau- \int_{0}^{1}a(\tau)\Big(\int_{0}^{\tau}(\tau-s)^{3}y(s)ds\Big)d\tau\Big).$$ Replacing these expressions in , we get $$\begin{aligned} u(t)&=&-\frac{1}{6}\int_{0}^{t}(t-s)^{3}y(s)ds+\frac{1}{6}t^{3}\int_{0}^{1}(1-s)^{2}y(s)ds\\ &&+\frac{1}{6(1-\alpha)}\Big[\int_{0}^{1}a(\tau)\tau^{3}\Big(\int_{0}^{1}(1-s)^{2}y(s)ds\Big)d\tau\\ &&-\int_{0}^{1}a(\tau)\Big(\int_{0}^{\tau}(\tau-s)^{3}y(s)ds\Big)d\tau\Big]\\ &=&\frac{1}{6}\int_{0}^{t}\Big[t^{3}(1-s)^{2}-(t-s)^{3}\Big]y(s)ds+\frac{1}{6}\int_{t}^{1}t^{3}(1-s)^{2}y(s)ds\\ &&+\frac{1}{6(1-\alpha)}\Big[\int_{0}^{1}a(\tau)\Big(\int_{0}^{\tau}\Big[\tau^{3}(1-s)^{2}-(\tau-s)^{3}\Big]y(s)ds\\ &&+\int_{\tau}^{1}\tau^{3}(1-s)^{2}y(s)ds\Big)d\tau\Big]\\ &=&\frac{1}{6}\int_{0}^{t}\Big[t^{3}(1-s)^{2}-(t-s)^{3}\Big]y(s)ds+\frac{1}{6}\int_{t}^{1}t^{3}(1-s)^{2}y(s)ds\\ &&+\frac{1}{1-\alpha}\int_{0}^{1}\Big(\frac{1}{6}\int_{0}^{\tau}a(\tau)\Big[\tau^{3}(1-s)^{2}-(\tau-s)^{3}\Big]y(s)ds\\ &&+\frac{1}{6}\int_{\tau}^{1}a(\tau)\tau^{3}(1-s)^{2}y(s)ds\Big)d\tau\\ &=&\int_{0}^{1}G(t, s)y(s)ds+\frac{1}{1-\alpha}\int_{0}^{1}\Big(\int_{0}^{1}a(\tau)G(\tau, s)y(s)ds\Big)d\tau\\ &=&\int_{0}^{1}\Big(G(t, s)+\frac{1}{1-\alpha}\int_{0}^{1}a(\tau)G(\tau, s)d\tau\Big)y(s)ds.\end{aligned}$$ \[lem 2.5\] Let $\theta\in]0, \frac{1}{2}[$ be fixed. Then - $G(t, s)\geq0$,  for all $t, s\in[0, 1];$ - $\frac{1}{6}\theta^{3}s(1-s)^{2}\leq G(t, s)\leq \frac{1}{6}s(1-s)^{2}$,  for all $(t, s)\in[\theta, 1-\theta]\times[0, 1]$. [(i)]{} We will show that $G(t, s)\geq0$,  for all $(t, s)\in[0, 1]\times[0, 1]$. Since it is obvious for $t\leq s$, we only need to prove the case $s\leq t$. Now we suppose that $s\leq t$. Then $$\begin{gathered} \begin{aligned} G(t, s)&=\frac{1}{6}\Big[t^{3}(1-s)^{2}-(t-s)^{3}\Big]=\frac{1}{6}\Big[t(t-ts)^{2}-(t-s)^{3}\Big]\\ &\geq\frac{1}{6}\Big[t(t-s)^{2}-(t-s)^{3}\Big]\\ &\geq\frac{1}{6}(t-s)^{2}\Big[t-(t-s)\Big]\\ &=\frac{1}{6}s(t-s)^{2}\geq 0. \end{aligned}\label{eq-2.5}\end{gathered}$$ [(ii)]{} If $s\leq t$, from we have $$\begin{gathered} \begin{aligned} G(t, s)&=\frac{1}{6}\Big[t^{3}(1-s)^{2}-(t-s)^{3}\Big]\geq\frac{1}{6}\Big[t^{3}(1-s)^{3}-(t-s)^{3}\Big]\\ &=\frac{1}{6}\Big[(t-ts)^{3}-(t-s)^{3}\Big]\\ &=\frac{1}{6}s(1-t)\Big[t^{2}(1-s)^{2}+t(1-s)(t-s)+(t-s)^{2}\Big]\\ &\geq\frac{1}{6}t^{2}(1-t)s(1-s)^{2}. \end{aligned}\label{eq-2.6}\end{gathered}$$ On the other hand $$\begin{gathered} \begin{aligned} G(t,s)-\frac{1}{6}s(1-s)^{2}&=\frac{1}{6}t^{3}(1-s)^{2}-\frac{1}{6}(t-s)^{3}- \frac{1}{6}s(1-s)^{2}\\ &=\frac{1}{6}s\Big(-2t^{3}+t^{3}s+3t^{2}-3ts-1+2s\Big)\\ &=\frac{1}{6}s(t-1)^{2}\Big[(s-2)t+2s-1\Big]\\ &\leq\frac{1}{6}s(t-1)^{2}\Big[(s-2)t+2t-1\Big]\\ &=\frac{1}{6}s(t-1)^{2}(st-1)\leq0.\\ \end{aligned}\label{eq-2.7}\end{gathered}$$ If $t\leq s$, from , we have $$\label{eq-2.8} G(t, s)=\frac{1}{6}t^{3}(1-s)^{2}\geq \frac{1}{6}t^{3}s(1-s)^{2},$$ and, $$\label{eq-2.9} G(t, s)=\frac{1}{6}t^{3}(1-s)^{2}\leq\frac{1}{6}s^{3}(1-s)^{2}\leq \frac{1}{6}s(1-s)^{2}.$$ Let $$\rho(t)=\frac{1}{6}\min\{t^{3}, t^{2}(1-t)\}=\frac{1}{6}\begin{cases} t^{3},& t\leq \frac{1}{2}; \\ t^{2}(1-t), & t \geq \frac{1}{2}. \end{cases}$$ From , , and we have $$\rho(t)s(1-s)^{2}\leq G(t, s)\leq \frac{1}{6}s(1-s)^{2}, \ \text{for all} \ (t, s)\in[0, 1]\times[0, 1].$$ For $\theta\in]0, \frac{1}{2}[$, we have $$\frac{\theta^{3}}{6}s(1-s)^{2}\leq G(t, s)\leq \frac{1}{6}s(1-s)^{2}, \ \text{for all} \ (t, s)\in[\theta, 1-\theta]\times[0, 1].$$ \[lem 2.6\] Let $y(t)\in C([0, 1], [0, \infty))$ and $\theta\in]0, \frac{1}{2}[$. The unique solution of - is nonnegative and satisfies $$\min_{t\in[\theta,1-\theta]}u(t)\geq \theta^{3}(1-\alpha+\beta) \|u\|,$$ where $\beta=\int_{\theta}^{1-\theta}a(t)dt$,  $\alpha=\int_{0}^{1}a(t)dt$. From Lemma \[lem 2.4\] and Lemma \[lem 2.5\], $u(t)$ is nonnegative. For $t\in[0, 1]$, from Lemma \[lem 2.4\] and Lemma \[lem 2.5\], we have that $$\begin{aligned} u(t)&=&\int_{0}^{1}\Big(G(t, s)+\frac{1}{1-\alpha}\int_{0}^{1}a(\tau)G(\tau, s)d\tau \Big)y(s)ds\\ &\leq&\frac{1}{6}\int_{0}^{1}s(1-s)^{2}\Big(1+\frac{\alpha}{1-\alpha}\Big)y(s)ds\\ &=&\frac{1}{6(1-\alpha)}\int_{0}^{1}s(1-s)^{2}y(s)ds.\end{aligned}$$ Then $$\label{eq-2.10} \|u\|\leq\frac{1}{6(1-\alpha)}\int_{0}^{1}s(1-s)^{2}y(s)ds,$$ and for $t\in[\theta,1-\theta]$, we have $$\begin{gathered} \begin{aligned} u(t)&=\int_{0}^{1}\Big(G(t, s)+\frac{1}{1-\alpha}\int_{0}^{1}a(\tau)G(\tau, s)d\tau\Big)y(s)ds\\ &\geq\frac{\theta^{3}}{6}\int_{0}^{1}\Big[s(1-s)^{2}+\frac{1}{1-\alpha}\int_{\theta}^{1-\theta}s(1-s)^{2}a(\tau)d\tau\Big]y(s)ds\\ &=\frac{\theta^{3}}{6}\int_{0}^{1}s(1-s)^{2}\Big(1+\frac{\beta}{1-\alpha}\Big)y(s)ds\\ &=\frac{\theta^{3}}{6}\cdot\frac{1-\alpha+\beta}{1-\alpha}\int_{0}^{1}s(1-s)^{2}y(s)ds. \end{aligned}\label{eq-2.11}\end{gathered}$$ From , , we obtain $$\min_{t\in[\theta,1-\theta]}u(t)\geq \theta^{3}(1-\alpha+\beta) \|u\|.$$ Let $\theta\in]0, \frac{1}{2}[$. We define the cone $$K=\left\{u\in C([0, 1], \mathbb{R}), u\geq0: \min_{t\in[\theta,1-\theta]}u(t)\geq \theta^{3}(1-\alpha+\beta) \|u\|\right\},$$ and the operator $A:K\rightarrow C[0, 1]$ by $$\label{eq-2.12} Au(t)=\int_{0}^{1}\Big(G(t, s)+\frac{1}{1-\alpha}\int_{0}^{1}a(\tau)G(\tau, s)d\tau\Big)f(u(s))ds.$$ By Lemma \[lem 2.4\], problem , has a positive solution $u(t)$ if and only if $u$ is a fixed point of $A$. \[lem 2.7\] The operator $A$ defined in is completely continuous and satisfies $AK\subset K$. From Lemma \[lem 2.6\], we obtain $AK\subset K$. $A$ is completely continuous by an application of Arzela-Ascoli theorem. In what follows, we will use the following notations\ $$f_{0}=\lim_{u\to 0+}\frac{f(u)}{u}, \ \ f_{\infty}=\lim_{u\to\infty}\frac{f(u)}{u}.$$ We note that the case $f_{0}=0$ and $f_{\infty}=\infty$ corresponds to the superlinear case and $f_{0}=\infty$ and $f_{\infty}=0$ corresponds to the sublinear case. Existence of positive solutions =============================== In this section, we will state and prove our main results. \[theo 3.1\] Assume that $f_{0}=0$ and $f_{\infty}=\infty$. Then BVP and has at least one positive solution. Since $f_{0}=0$, there exists $\rho_{1}>0$ such that $f(u)\leq \epsilon u$, for $0<u\leq \rho_{1}$, where $\epsilon>0$ satisfies $$\frac{\epsilon}{6(1-\alpha)}\leq1.$$ Thus, if we let $$\Omega_{1}=\left\{u\in C[0,1]: \|u\|<\rho_{1}\right\},$$ then, for $u\in K\cap \partial\Omega_{1}$, we have $$\begin{gathered} \begin{aligned} Au(t)&=\int_{0}^{1}\Big(G(t, s)+\frac{1}{1-\alpha}\int_{0}^{1}a(\tau)G(\tau, s)d\tau\Big)f(u(s))ds\\ &\leq\frac{1}{6}\int_{0}^{1}\Big(s(1-s)^{2}+\frac{1}{1-\alpha}\int_{0}^{1}s(1-s)^{2}a(\tau)d\tau\Big)\epsilon u(s)ds\\ &\leq\frac{1}{6}\cdot\frac{\epsilon}{1-\alpha}\|u\|\int_{0}^{1}s(1-s)^{2}ds\\ &\leq\frac{1}{6}\cdot\frac{\epsilon}{1-\alpha}\|u\|\\ &\leq\|u\|. \end{aligned}\label{eq-3.1}\end{gathered}$$ Therefore $$\|Au\|\leq\|u\|, \ \ u\in K\cap\partial\Omega_{1}.$$ Further, since $f_{\infty}=\infty$, there exists $\widehat{\rho}_{2}>0$ such that $f(u)\geq \delta u$, for $u> \widehat{\rho}_{2}$, where $ \delta>0$ is chosen so that $$\delta\frac{\theta^{6}}{36}\cdot\frac{(1-\alpha+\beta)^{2}}{1-\alpha}(1-2\theta)(\frac{1}{2}+\theta-\theta^{2})\geq1.$$ Let $\rho_{2}=\max\Big\{2\rho_{1}, \frac{\widehat{\rho}_{2}}{\theta^{3}(1-\alpha+\beta)}\Big\}$ and $\Omega_{2}=\left\{u\in C[0, 1]: \|u\|<\rho_{2}\right\}$. Then $u\in K\cap \partial\Omega_{2}$ implies that $$\min_{t\in[\theta,1-\theta]}u(t)\geq \theta^{3}(1-\alpha+\beta) \|u\|=\theta^{3}(1-\alpha+\beta)\rho_{2}\geq\widehat{\rho}_{2},$$ so, by and for $t\in[\theta, 1-\theta]$, we obtain $$\begin{gathered} \begin{aligned} Au(t)&=\int_{0}^{1}\Big(G(t, s)+\frac{1}{1-\alpha}\int_{0}^{1}a(\tau)G(\tau, s)d\tau\Big)f(u(s))ds\\ &\geq\int_{\theta}^{1-\theta}\Big[\frac{\theta^{3}}{6}s(1-s)^{2}+ \frac{1}{1-\alpha}\int_{\theta}^{1-\theta}\frac{\theta^{3}}{6}s(1-s)^{2}a(\tau)d\tau\Big]\delta u(s)ds\\ &=\frac{\theta^{3}}{6}\delta\int_{\theta}^{1-\theta}s(1-s)^{2}\Big(1+\frac{\beta}{1-\alpha}\Big)u(s)ds\\ &=\frac{\theta^{3}\delta}{6}\cdot\frac{(1-\alpha+\beta)}{1-\alpha}\int_{\theta}^{1-\theta}s(1-s)^{2}u(s)ds\\ &\geq\frac{\theta^{3}\delta}{6}\cdot\frac{(1-\alpha+\beta)}{1-\alpha}\min_{t\in[\theta,1-\theta]}u(t)\int_{\theta}^{1-\theta}s(1-s)^{2}ds\\ &\geq\delta\frac{\theta^{6}}{36}\cdot\frac{(1-\alpha+\beta)^{2}}{1-\alpha}(1-2\theta)(\frac{1}{2}+\theta-\theta^{2})\|u\|\\ &\geq\|u\|. \end{aligned}\label{eq-3.2}\end{gathered}$$ Hence, $\|Au\|\geq\|u\|, \ u\in K\cap\partial\Omega_{2}$. By Theorem \[thm 2.3\], the operator $A$ has a fixed point in $K\cap (\overline{\Omega _{2}}$ $\backslash $ $ \Omega_{1})$ such that $\rho_{1}\leq \|u\|\leq \rho_{2}$. \[theo 3.2\] Assume that $f_{0}=\infty$ and $f_{\infty}=0$. Then BVP and has at least one positive solution. Since $f_{0}=\infty$, there exists $\rho_{1}>0$ such that $f(u)\geq \lambda u$, for $0<u\leq \rho_{1}$, where $ \lambda>0$ is chosen so that $$\lambda\frac{\theta^{6}}{36}\cdot\frac{(1-\alpha+\beta)^{2}}{1-\alpha}(1-2\theta)(\frac{1}{2}+\theta-\theta^{2})\geq1.$$ Thus, for $u\in K\cap \partial\Omega_{1}$ with $$\Omega_{1}=\left\{u\in C[0, 1]: \|u\|<\rho_{1}\right\},$$ we have from $$\begin{aligned} Au(t)&=&\int_{0}^{1}\Big(G(t, s)+\frac{1}{1-\alpha}\int_{0}^{1}a(\tau)G(\tau, s)d\tau\Big)f(u(s))ds\\ &\geq&\int_{\theta}^{1-\theta}\Big[\frac{\theta^{3}}{6}s(1-s)^{2}+ \frac{1}{1-\alpha}\int_{\theta}^{1-\theta}\frac{\theta^{3}}{6}s(1-s)^{2}a(\tau)d\tau\Big]\lambda u(s)ds\\ &\geq&\lambda\frac{\theta^{6}}{36}\cdot\frac{(1-\alpha+\beta)^{2}}{1-\alpha}(1-2\theta)(\frac{1}{2}+\theta-\theta^{2})\|u\|\\ &\geq&\|u\|.\end{aligned}$$ Then, $ Au(t)\geq \|u\|$ for $ t\in[\theta, 1-\theta]$ , which implies that $$\|Au\|\geq\|u\|, \ \ u\in K\cap\partial\Omega_{1}.$$ Next we construct the set $\Omega_{2}$. We discuss two possible cases: Case 1. If $f$ is bounded. Then, there exists $L>0$ such that $f(u)\leq L$. Set $\Omega_{2}=\left\{u\in C[0, 1]: \|u\|<\rho_{2}\right\}$, where $\rho_{2}=\max\Big\{2\rho_{1}, \frac{L}{6(1-\alpha)}\Big\}$. If $u\in K\cap\partial\Omega_{2}$, similar to the estimates of , we obtain $$\begin{aligned} Au(t)&\leq&\frac{1}{6}\cdot\frac{L}{1-\alpha}\int_{0}^{1}s(1-s)^{2}ds\\ &\leq&\frac{1}{6}\cdot\frac{L}{1-\alpha}\leq \rho_{2}=\|u\|,\end{aligned}$$ and consequently, $\|Au\|\leq\|u\|, \ \ u\in K\cap\partial\Omega_{2}$. Case 2. Suppose that $f$ is unbounded, since $f_{\infty}=0$, there exists $\widehat{\rho}_{2}>0$ ($\widehat{\rho}_{2}>\rho_{1}$) such that $f(u)\leq \eta u$ for $u>\widehat{\rho}_{2}$, where $\eta>0$ satisfies $$\frac{\eta}{6(1-\alpha)}\leq 1.$$ On the other hand, from condition [(H1)]{}, there is $\sigma>0$ such that $f(u)\leq \eta\sigma$,  with $0\leq u\leq \widehat{\rho}_{2}$. Now, set $\Omega_{2}=\left\{u\in C[0, 1]: \|u\|<\rho_{2}\right\}$, where $\rho_{2}=\max\{\sigma,\widehat{\rho}_{2}\}$. If $u\in K\cap\partial\Omega_{2}$, then we have $f(u)\leq\eta\rho_{2}$. Similar to , we obtain $$\begin{aligned} Au(t)&\leq&\frac{1}{6}\cdot\frac{\eta \rho_{2}}{1-\alpha}\int_{0}^{1}s(1-s)^{2}ds\\ &\leq&\frac{1}{6}\cdot\frac{\eta \rho_{2}}{1-\alpha}\leq \rho_{2}=\|u\|,\end{aligned}$$ so, $\|Au\|\leq\|u\|$, for $u\in K\cap\partial\Omega_{2}$. Therefore by Theorem \[thm 2.3\], $A$ has at least one fixed point, which is a positive solution of and . Examples ======== Consider the boundary value problem $$\label{eq-4.1} {u^{\prime \prime \prime \prime }}(t)+u^{2}(e^{-u}+1)=0, \ \ 0<t<1,$$ $$\label{eq-4.2} u^{\prime}(0)=u^{\prime}(1)=u^{\prime \prime}(0)=0, \ u(0)=\int_{0}^{1}s^{2}u(s)ds,$$ where $f(u)=u^{2}(e^{-u}+1)\in C([0,\infty),[0,\infty))$ and $a(t)=t^{2}\geq0$, $\int_{0}^{1}a(s)ds=\int_{0}^{1}s^{2}ds=\frac{1}{3}$. We have $$\begin{aligned} \lim_{u\to 0+}\frac{f(u)}{u}&=\lim_{u\to 0+}\frac{u^{2}(e^{-u}+1)}{u}=0,\\ \lim_{u\to+\infty}\frac{f(u)}{u}&=\lim_{u\to+\infty}\frac{u^{2}(e^{-u}+1)}{u}=+\infty.\end{aligned}$$ From Theorem \[theo 3.1\], the problem and has at least one positive solution. Consider the boundary value problem $$\label{eq-4.3} {u^{\prime \prime \prime \prime }}(t)+\sqrt{1+u}+\sin{u}=0, \ \ 0<t<1,$$ $$\label{eq-4.4} u^{\prime}(0)=u^{\prime}(1)=u^{\prime \prime}(0)=0, \ u(0)=\int_{0}^{1}su(s)ds,$$ where $f(u)=\sqrt{1+u}+\sin{u}\in C([0,\infty),[0,\infty))$ and $a(t)=t\geq0$, $\int_{0}^{1}a(s)ds=\int_{0}^{1}sds=\frac{1}{2}$. We have $$\begin{aligned} \lim_{u\to 0+}\frac{f(u)}{u}&=\lim_{u\to 0+}\frac{\sqrt{1+u}+\sin{u}}{u}=+\infty,\\ \lim_{u\to+\infty}\frac{f(u)}{u}&=\lim_{u\to+\infty}\frac{\sqrt{1+u}+\sin{u}}{u}=0.\end{aligned}$$ Therefore, by Theorem \[theo 3.2\], the problem and has at least one positive solution. E. Alves, T. F. Ma, M. L. Pelicer, *Monotone positive solutions for a fourth order equation with nonlinear boundary conditions*, Nonlinear Anal., **71** (2009), 3834–3841. D. R. Anderson, R. I. Avery, *A fourth-order four-point right focal boundary value problem*, Rocky Mountain J. Math., **36** (2006), 367–-380. J. R. Graef, C. Qian, B. Yang, *A three point boundary value problem for nonlinear fourth order differential equations*, J. Math. Anal. Appl., **287** (2003), 217–233. J. R. Graef, B. Yang, *Existence and nonexistence of positive solutions of fourth order nonlinear boundary value problems*, Appl. Anal., **74** (2000), 201–214. J. R. Graef, J. Henderson, B. Yang, *Positive solutions to a fourth-order three point boundary value problem*, Discrete Contin. Dyn. Syst., Supplement (2009), 269–275. X. Han, H. Gao, J. Xu, *Existence of positive solutions for nonlocal fourth-order boundary value problem with variable parameter*, Fixed Point Theory Appl., **2011** (2011), Art. ID 604046, 11 pages. J. Henderson, D. Ma, *Uniqueness of solutions for fourth-order nonlocal boundary value problems*, Bound. Value Probl., **2006** (2006), Art. ID 23875, 12 pages. N. Kosmatov, *Countably many solutions of a fourth-order boundary value problem*, Electron. J. Qual. Theory Differ. Equ., **12** (2004), 1–15. M. A. Krasnoselskii, *Positive Solutions of Operator Equations*, P. Noordhoff, Groningen, The Netherlands, 1964. X. Liu, D. Ma, *The existence of positive solution for a third-order two-point boundary value problem with integral boundary conditions*, Scientific Journal of Mathematics Research. February 2014, V.4, Issue 1, 1–7. R. Ma, *Multiple positive solutions for a semipositone fourth-order boundary value problem*, Hiroshima Math. J., **33** (2003), 217–227. R. Ma, W. Haiyan, *On the existence of positive solutions of fourth-order ordinary differential equations*, Appl. Anal., **59** (1995), 225–231. W. Shen, *Positive solutions for fourth-order second-point nonhomogeneous singular boundary value problems*, Adv. Fixed Point Theory., 1(**5**) (2015), 88–100. Y. Sun, C. Zhu, *Existence of positive solutions for singular fourth-order three-point boundary value problems*, Adv. Difference Equ., 51(**2013**) (2013), 13 pages. J. R. Webb, G. Infante, D. Franco, *Positive solutions of nonlinear fourth-order boundary value problems with local and nonlocal boundary conditions*, Proc. Roy. Soc. Edinburgh Sect. A. **138** (2008), 427–446. B. Yang, *Positive solutions for a fourth-order boundary value problem*, Electron. J. Qual. Theory Differ. Equ., **3** (2005), 1–17. Q. Yao, *Local existence of multiple positive solutions to a singular cantilever beam equation*, J. Math. Anal. Appl., **363** (2010), 138–154. S. Yong-Ping, *Existence and multiplicity of positive solutions for an elastic beam equation*, Appl. Math. J. Chinese. Univ., 26(**3**) (2011), 253–264. Q. Zhang, S. Chen, J. Lü *Upper and lower solution method for fourth-order four-point boundary value problems*, J. Comput. Appl. Math., **196** (2006), 387–393.

--- abstract: 'We consider the problem of option pricing and hedging when stock returns are correlated in time. Within a quadratic–risk minimisation scheme, we obtain a general formula, valid for weakly correlated non–Gaussian processes. We show that for Gaussian price increments, the correlations are irrelevant, and the Black–Scholes formula holds with the volatility of the price increments on the scale of the re–hedging. For non–Gaussian processes, further non trivial corrections to the ‘smile’ are brought about by the correlations, even when the hedge is the Black–Scholes $\Delta $–hedge. We introduce a compact notation which eases the computations and could be of use to deal with more complicated models.' author: - 'Lorenzo Cornalba$^{1}$, Jean–Philippe Bouchaud$^{2,3}$ and Marc Potters$^{3} $' date: | $^{1}$* *Laboratoire de Physique Théorique de l’École Normale Supérieure\ 24 rue Lhomond, 75231 Paris Cedex 5, France\ $^{2}$ Service de Physique de l’État Condensé, Centre d’études de Saclay\ Orme des Merisiers, 91191 Gif–sur–Yvette Cedex, France\ $^{3}$ Science & Finance\ The Research division of Capital Fund Management\ 109–111 rue Victor-Hugo, 92532 Levallois Cedex, France\ title: '**OPTION PRICING AND HEDGING WITH TEMPORAL CORRELATIONS**' --- Introduction ============ The assumptions underlying the Black–Scholes model [@Hull; @Wilmott] – that the dynamics of financial assets can be modelled by a continuous time Gaussian process – are very far from reality. It is well known, in particular, that the empirical distribution of returns exhibits ‘fat–tails’. The presence of these extreme events not only induces a ‘smile’ in the implied volatility of the options, but also destroys the essence of the Black–Scholes pricing procedure: the possibility of constructing a riskless hedge which replicates perfectly the option pay–off [@BS2; @Sch; @us]. A more subtle further effect, completely discarded by the standard Black–Scholes model, is the presence of small, but significant, temporal *anticorrelations* in the series of stock returns. Correspondingly, the variance of the $N$–day distribution of returns is smaller than $N$ times the variance of the daily distribution (see *e.g.* [@Lo]). On other financial assets, such as stock indices, one observes the opposite effect of weak positive correlations. The order of magnitude of these correlations or anticorrelations is $10\%$ from one day to the next. These correlations lead, in principle, to the possibility of statistical arbitrage. However, one should also take into account transaction costs which significantly reduce the possibility of using these correlation in practice [@us]. In other words, the presence of transaction costs allow the existence of non zero correlations, which can be significant and therefore must be dealt with consistently in the context of option pricing. What is the influence of these correlations on the price of an option and on the corresponding hedging strategy? The aim of this paper is to provide a general answer to this question within a quadratic risk minimization sheme, in the case where these correlations are small [^1] (which is, in most cases of interest, amply justified), with only very few assumptions on the actual distribution of returns. We calculate the first order corrections both to the price and optimal hedge. In the Gaussian case, we find that the effect of correlations can be compensated by a change in the hedging strategy and therefore options should be priced using the standard uncorrelated Black–Scholes model [^2]. The correct volatility to be used is the one measured on the time scale of the rehedging, and not the one corresponding to the terminal distribution – *i.e.* corresponding to the maturity of the option. In other words, the ‘risk–neutral’ measure that one should use to price derivatives is simply an uncorrelated Gaussian measure with the same elementary variance. These conclusions are not valid if one considers a process for which the underlying distribution is non–Gaussian. One then finds contributions to the option prices which depend in general on the strength of the correlations, and on the higher moments of the distribution (in particular on the kurtosis). The corrections to the option price also depend on the past history of the price–increments of the underlying. The detailed analysis of a specific example shows, in particular, that anticorrelations change the effective kurtosis to be used when computing ‘smile’ corrections to the options prices. In general, the correct kurtosis to be used is neither the local nor the global one, but some effective kurtosis which depends on the strength of the correlations. \[GS\]General setting ===================== Our starting point is the global wealth balance approach discussed in [@BS2; @us]. We set, for simplicity, the interest rate to zero, and consider a discrete time model with time interval $\tau $, within which no rehedging takes place. The global wealth balance at time $T=N\tau $ for the writer of the option reads $$W=\mathcal{C}+H-\Theta ,$$ where $\mathcal{C}$ is the option premium, $\Theta $ is the pay–off of the option, and $H$ is the result of the hedging strategy $$H=\sum_{n=0}^{N-1}\phi _{n}\Delta _{n+1}. \label{eq1}$$ In equation (\[eq100\]\[eq1\]), $\Delta _{n+1}=x_{n+1}-x_{n}$ denotes the change of the price $x$ of the underlying asset between time $n$ and $n+1 $, and $\phi _{n}$ is the amount of underlying in the portfolio at time $n$ [^3]. In incomplete markets, the price of an option is not unique since a risk–premium should be added. However, a convenient framework is the risk minimisation scheme proposed in [@BS2; @Sch; @us], that fixes both the price and strategy $\mathcal{C}$, $\phi _{n}$ so as to minimize the total risk $\mathcal{R}$ associated to option writing, which we simply define as $$\mathcal{R}^{2}=\left\langle W^{2}\right\rangle ,$$ where $\langle ...\rangle $ is an average over the objective probability at time $0$. Minimizing with respect to $\mathcal{C}$ fixes the price of the call option to be $$\mathcal{C}=\left\langle \Theta -H\right\rangle . \label{eq200}$$ If, on the other hand, we minimize $\mathcal{R}^{2}$ with respect to $\phi _{n}$, we obtain an equation implicitly determining the optimal strategy $\phi _{n}^{\star }$, equation that reads, for $0\leq n<N$ $$\langle \Delta _{n+1}\left( \Theta -H\right) \rangle _{n}=\mathcal{C}\langle \Delta _{n+1}\rangle _{n} \label{general}$$ The notation $\langle ...\rangle _{n}$ in the above equation means that the average (again over the objective probability) is performed at time $n$ – *i.e.* with all the information available at this instant of time. This notation turns out to be very powerful, and could be used to handle very general situations. We now analyze the above general and compact equations (\[eq200\]) and (\[general\]) in several simple but important limiting cases. \[zeroD\]The case of zero conditional drift =========================================== Let us define the conditional drift $\mu _{n}$ and the conditional variance $D_{n}$ at time $n$ as $$\begin{aligned} \mu _{n} &=&\left\langle \Delta _{n+1}\right\rangle _{n} \\ D_{n} &=&\left\langle (\Delta _{n+1}-\mu _{n})^{2}\right\rangle _{n}.\end{aligned}$$ Note that these quantities are the expected drift and variance for the next time interval, and depend in general on the past (realized) history of price changes. We start by assuming that the drift $\mu _{n}$ is identically zero (this is to say, the price process is a ‘martingale’). Then, since $\phi _{n}$ is known at time $n$, one has that $$\left\langle \phi _{n}\Delta _{n+1}\right\rangle =\left\langle \phi _{n}\langle \Delta _{n+1}\rangle _{n}\right\rangle =0,$$ and therefore that $\left\langle H\right\rangle =0$. This implies that the option price is nothing but the expectation value of the pay–off $$\mathcal{C}=\left\langle \Theta \right\rangle .$$ To determine the optimal hedge, we note that, in general, one has $$\begin{aligned} \left\langle \Delta _{n+1}H\right\rangle _{n} &=&\sum_{m=0}^{N-1}\left\langle \phi _{m}\Delta _{m+1}\Delta _{n+1}\right\rangle _{n} \label{corr} \\ &=&\phi _{n}\left\langle \Delta _{n+1}^{2}\right\rangle _{n}+\sum_{m=0}^{n-1}\phi _{m}\Delta _{m+1}\mu _{n}+\sum_{m=n+1}^{N-1}\left\langle \phi _{m}\mu _{m}\Delta _{n+1}\right\rangle _{n}. \notag\end{aligned}$$ In the case of vanishing drift $\mu _{n}=0$ we therefore have that $$\left\langle \Delta _{n+1}H\right\rangle _{n}=\phi _{n}D_{n}$$ so that the optimal hedging strategy can be computed, from equation (\[general\]), as [@BS2; @us] $$\phi _{n}^{\star }=\frac{1}{D_{n}}\left\langle \Delta _{n+1}\Theta \right\rangle _{n}.$$ \[pert\]Perturbative expansion for small drift ============================================== Let us now turn to the case where the conditional bias $\mu _{n}$ is non–zero but small. Using equation (\[corr\]), it is easy to see that the basic equations (\[eq200\]) and (\[general\]), which determine the optimal price and the hedging strategy, can be solved perturbatively in $\mu _{n}$. For our purposes, we will only need the first correction in the conditional drift, which we now derive. First note that equation (\[eq200\]) can be rewritten as $$\mathcal{C}=\left\langle \Theta \right\rangle -\sum_{n=0}^{N-1}\left\langle \phi _{n}^{\ast }\mu _{n}\right\rangle .$$ In particular, to first order in $\mu _{n}$, one finds that $$\mathcal{C}=\left\langle \Theta \right\rangle -\sum_{n=0}^{N-1}\left\langle \frac{\Delta _{n+1}\mu _{n}}{D_{n}}\Theta \right\rangle . \label{eq1000}$$ One can also compute the first correction to the hedging strategy, by combining equation (\[corr\]) with the zero–th order results obtained in the last section. One obtains then $$\phi _{0}^{\star }=\frac{1}{D_{0}}\left\langle \Delta _{1}\Theta \right\rangle -\frac{1}{D_{0}}\sum_{n=1}^{N-1}\left\langle \frac{\Delta _{n+1}\mu _{n}}{D_{n}}\Delta _{1}\Theta \right\rangle -\frac{1}{D_{0}}\mu _{0}\left\langle \Theta \right\rangle . \label{eq1001}$$ In the above equation, we have only shown the optimal hedge for the first time period $\phi _{0}^{\star }$. The explicit expression for the optimal hedge for the subsequent time periods is also of a similar form, but it is quite intricate and not very illuminating. More importantly, it is not necessary to determine the correct hedging strategy. This important point will be discussed, in a slightly more general context, in section \[other\]. \[model\]A general model for weakly correlated processes ======================================================== Let us introduce a rather general model for correlated price increments. We first define an auxiliary set of *uncorrelated i.i.d. random variables* $\{\eta _{n}\}$, distributed according to an arbitrary probability distribution[^4] $P(\eta )$. The joint distribution is then given by $$P(\{\eta _{n}\})=\prod_{n}P(\eta _{n}).$$ We will assume that $P(\eta )$ is symmetrical[^5] – *i.e.* that $$\int \eta ^{2q+1}P(\eta )d\eta =0, \label{eq600}$$ and we will denote the second moment of $P$ by $$D=\int \eta ^{2}P(\eta )d\eta .$$ We now construct the set of correlated price increments $\{\Delta _{n}\}$ by writing $$\Delta _{n}=\sum_{m}M_{nm}\eta _{m}+\mu , \label{eq400}$$ with $$M_{nm}=\delta _{n,m}+\frac{1}{2D}c_{n-m}.$$ The coefficients $c_{n}$ are assumed to satisfy $$c_{0}=0\qquad \ \ \ \ \ \ \ \ \ \ \ \ \ c_{-k}=c_{k}.$$ In the sequel, we assume that both the $c_{k}$’s and $\mu $ are small, and will work only to first order in both $\mu $ and$\ c$. Let us show the significance of equation (\[eq400\]), by first noting that, since $\left\langle \eta _{n}\right\rangle =0$, we have that $$\left\langle \Delta _{n}\right\rangle =\mu .$$ Therefore $\mu $ is nothing but the average *unconditional* drift of the price process. Moreover, to first order in $c$, the following holds $$\left\langle (\Delta _{n}-\mu )(\Delta _{m}-\mu )\right\rangle =\sum_{i,j}M_{ni}M_{mj}\left\langle \eta _{i}\eta _{j}\right\rangle =D\delta _{n,m}+c_{n-m}.$$ The coefficients $D$ and $c$ denote then, respectively, the *unconditional* variance and correlation of the price increments. Finally, one has that, independently of $c_{n-m}$, and for all $p$, $$\left\langle (\Delta _{n}-\mu )^{p}\right\rangle =\left\langle \eta ^{p}\right\rangle .$$ We note that the variance of the price differences over a time scale $n$, given by $$\sigma ^{2}(n)=\left\langle \left( \sum_{k=m+1}^{m+n}(\Delta _{k}-\mu )\right) ^{2}\right\rangle \ ,\qquad (n\geq 1)$$ is related to the correlation coefficients $c_{k}$ as follows $$\begin{aligned} \sigma ^{2}(n) &=&nD+\sum_{k=1}^{n-1}2(n-k)c_{k} \notag \\ c_{n} &=&\frac{1}{2}\left( \sigma ^{2}(n+1)-2\sigma ^{2}(n)+\sigma ^{2}(n-1)\right) . \label{eq33}\end{aligned}$$ In the case where $c_{n}=0$, one finds $\sigma ^{2}(n)=nD$, whereas, in the case of very short range anticorrelations ($c_{n}=-c\delta _{n,1}$), one has $\sigma ^{2}(n)=n(D-2c)+2c$. In this case, the global volatility $\sigma ^{2}(n)/n\rightarrow D-2c$ is lower than the short–range volatility $\sigma ^{2}\left( 1\right) =D$. The probability distribution of the price increments can be written, to first order in $\mu ,c$, as $$\begin{aligned} &&\prod_{n}P\left(\Delta _{n}-\mu -\frac{1}{2D}\sum_{m}c_{n-m}\Delta _{m}\right) \label{dist1} \\ &=&\left[ 1-\mu \sum_{n}\frac{\partial \ln P}{\partial \Delta _{n}}-\frac{1}{2D}\sum_{n,m}\frac{\partial \ln P}{\partial \Delta _{n}}c_{n-m}\Delta _{m}\right] \prod_{n}P(\Delta _{n}). \notag\end{aligned}$$ It is now easy to write the marginal distribution at time $n=0$, given all previous returns $\Delta _{n}$ with $n\leq 0$ $$\left[ 1-\mu \sum_{n>0}\frac{\partial \ln P}{\partial \Delta _{n}}-\frac{1}{2D}\sum_{\max (n,m)>0}\frac{\partial \ln P}{\partial \Delta _{n}}c_{n-m}\Delta _{m}\right] \prod_{n>0}P(\Delta _{n}). \label{condi}$$ The equation above follows immediately from the general expression for the probability distribution, up to a possible normalization factor. To show that (\[condi\]) is actually correctly normalized, we have only to note the obvious fact that $$\int d\Delta \,P(\Delta )\frac{\partial \ln P}{\partial \Delta }=0.$$ Moreover, using that $$\begin{aligned} \int d\Delta \,P(\Delta )\Delta \frac{\partial \ln P}{\partial \Delta } &=&-1 \\ \int d\Delta \,P(\Delta )\Delta ^{2}\frac{\partial \ln P}{\partial \Delta } &=&0\end{aligned}$$ it is then easy to show that the *conditional* drift and variance $\mu _{n}$ and $D_{n}$ are given, to first order in $\mu$ and $c$, by $$\begin{aligned} \mu _{n-1} &=&\mu +\sum_{m<n}c_{n-m}\left( \frac{1}{2D}\Delta _{m}-\frac{1}{2}\frac{\partial \ln P}{\partial \Delta _{m}}\right) \\ D_{n} &=&D.\end{aligned}$$ \[pricing\]Derivative pricing with small correlations ===================================================== A general formula ----------------- Let us now apply the general perturbative formula (\[eq1000\]) for the price of the contract to the specific model of correlations described in the previous section . One finds $$\begin{aligned} \mathcal{C} &=&\left\langle \Theta \right\rangle -\frac{1}{D}\sum_{n=1}^{N}\left\langle \mu _{n-1}\Delta _{n}\Theta \right\rangle \\ &=&\left\langle \Theta \right\rangle -\frac{\mu }{D}\sum_{n=1}^{N}\left\langle \Delta _{n}\Theta \right\rangle -\frac{1}{2D}\sum_{n=1}^{N}\sum_{m<n}c_{n-m}\left\langle \left( \frac{1}{D}\Delta _{m}-\frac{\partial \ln P}{\partial \Delta _{m}}\right) \Delta _{n}\Theta \right\rangle .\end{aligned}$$ Now, the above averaging $\left\langle \cdots \right\rangle $ is performed over the ‘correlated’ historical price increment distribution, equation (\[condi\]). It is convenient to re–express all of the following formulæ in terms of an auxiliary probability distribution $\prod_{n>0}P\left( \Delta _{n}\right) $, which describes an unbiased and uncorrelated process, and is obtained by setting $c_{k}=\mu =0$. We will denote the corresponding averages by $\langle ...\rangle ^{0}$, where the superscript indicates that the conditional drift is now set to zero (in the case of a risk–free Gaussian model, this auxiliary probability distribution is called the equivalent martingale measure in the financial mathematics literature). Equation (\[condi\]) can then be reinterpreted as relating the two averaging procedures $\left\langle \cdots \right\rangle $ and $\left\langle \cdots \right\rangle ^{0}$ as follows $$\left\langle \cdots \right\rangle =\left\langle \cdots \right\rangle^0 -\mu \sum_{n>0}\left\langle \cdots \frac{\partial \ln P}{\partial \Delta _{n}}\right\rangle ^{0}-\frac{1}{2D}\sum_{\max (n,m)>0}\left\langle \cdots \frac{\partial \ln P}{\partial \Delta _{n}}c_{n-m}\Delta _{m}\right\rangle ^{0}.$$ After a some algebra we then find that $$\mathcal{C}=\left\langle \Theta \right\rangle ^{0}+\mu \sum_{n=1}^{N}\left\langle F\left( \Delta _{n}\right) \Theta \right\rangle ^{0}+\frac{1}{2D}\sum_{n=1}^{N}\sum_{m<n}c_{n-m}\left\langle F(\Delta _{n})\Delta _{m}\Theta \right\rangle ^{0}, \label{price}$$ where $$F(\Delta )=-\frac{\partial \ln P}{\partial \Delta }-\frac{1}{D}\Delta .$$ The above equation is the central result of this paper, that we now comment in various limits. Let us first consider the purely Gaussian case, where $\ln P(\Delta )=-\Delta ^{2}/2D-1/2\ln (2\pi D)$. In this situation, it is easy to check that $F(\Delta )=0$. Therefore, all corrections brought about by the drift and correlations strictly vanish, and the price can be calculated by assuming that the process is correlation–free. As shown in the Appendix, this is indeed expected in a continuous time framework. In other words, the price of the option is given by the Black–Scholes price, with a volatility given by the small scale volatility $D$ (that corresponds to the hedging time scale), and [*not*]{} with the volatility corresponding to the terminal distribution, $\sigma ^{2}(N)/N$. As we have already shown, when price increments are anticorrelated, $D$ is larger than $\sigma ^{2}(N)/N$, and the price of the contract is higher than the objective average pay–off. This is due to the fact that the hedging strategy is on average losing money because of the anticorrelations: a rise of the price of the underlying leads to an increase of the hedge, which is followed (on average) by a drop of the price. For an uncorrelated non–Gaussian process, the effect of a non–zero drift $\mu $ does not vanish, a situation already discussed in details in [@us]. The same is true with correlations: the correction no longer disappears and one cannot price the option by considering a process with the same statistics of price increments but no correlations. Note that the correction term involves all $\Delta _{m}$ with $m<0$. This means, as expected a priori, that, in the presence of correlations, the price of the option does not only depend on the current price of the underlying, but also on all past price increments. The case of short range correlations ------------------------------------ Let us illustrate the general formula in the case of very short range anticorrelations $c_{n}=-c\delta _{n,1}$. We will furthermore consider the simple case of path–*independent* pay–offs $\Theta $, which are functions only on the terminal price $x_{N}$ of the underlying. The calculation of the price will therefore involve the following quantities $$I_{1}=\langle F\left( \Delta _{n}\right) |x_{N}\rangle ^{0}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ I_{2}=\langle \Delta _{n}F\left( \Delta _{m}\right) |x_{N}\rangle ^{0},$$ where the notation $\langle ...|x_{N}\rangle ^{0}$ means that we are conditioning on a given value of the terminal price $x_{N}$. The computation of the two quantities $I_{1},I_{2}$ is easily done in momentum space, with the aid the following formulae $$\begin{aligned} \int d\Delta \;\Delta P(\Delta) e^{iz\Delta } &=&iDz-\frac{i}{6}D^{2}\left( 3+\kappa \right) z^{3}+o\left( z^{5}\right) \\ \int d\Delta \;F\left( \Delta \right)P(\Delta) \,e^{iz\Delta } &=&\frac{i}{6}\kappa D\,z^{3}+o\left( z^{5}\right) ,\end{aligned}$$ where $\kappa$ is the kurtosis of the distribution of elementary increments. The result reads, for large $N$, $$I_{1}=-\frac{\kappa }{N\sqrt{DN}}p_{1}\;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ I_{2}=-\frac{4\kappa }{N^{2}}p_{2},$$ where $p_{1}$ and $p_{2}$ are, respectively, the skewness and kurtosis polynomials $$p_{1}=\frac{1}{6}\left( \widetilde{x}^{3}-3\widetilde{x}\right) \;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p_{2}=\frac{1}{24}\left( \widetilde{x}^{4}-6\widetilde{x}^{2}+3\right)$$ and where we have consider the natural scaling $\widetilde{x}^{2}=\left( x_{N}-x_{0}\right) ^{2}/DN$ : $\widetilde{x}$ is the moneyness of the option counted in standard deviations. The general equation (\[price\]) then takes the form $\mathcal{C}=\langle \widetilde{\Theta}\rangle ^{0}$, with a modified pay–off function $$\widetilde{\Theta}(x_{N})=\Theta (x_{N})\left[ 1-\frac{\kappa }{\sqrt{DN}}\left( \mu -\frac{c\Delta _{0}}{2DN}\right) p_{1}+\frac{2c\kappa }{DN}p_{2}\right] ,$$ where the explicit dependence on the last past increment $\Delta _{0}$ appears. Note that the correction terms vanish for at the money options ($x_{N}=x_{0}$). Let us concentrate on the last term in the above equation. It represents an *increase* in the global kurtosis $\kappa /N$ of the ‘risk neutral’ distribution that must be used to price the option: $$\frac{\kappa }{N}\rightarrow \frac{\kappa }{N}\left( 1+\frac{2c}{D}\right) . \label{res20}$$ This is means that the ‘implied’ kurtosis is increased, in percent terms, as much as the global variance is decreased from the local one by correlation effects. Therefore the volatility ‘smile’, which is proportional to the kurtosis, is enhanced by the presence of anti–correlations, and reduced by the presence of correlations. Note that the above model neglects all volatility fluctuations, which lead to a terminal kurtosis decaying much more slowly than $\kappa/N$ (see [@us]). The above calculations could in principle be extended to deal with this effect, which is crucial for financial applications. An alternative model of correlations ------------------------------------ Let us comment more on the above result. To this end, let us briefly discuss an other possible model of correlated price increments. One could consider, instead of the probability distribution (\[dist1\]), the following distribution for price increments (we set for simplicity $\mu =0$) $$\left[ 1+\frac{1}{2D^{2}}\sum_{n,m}c_{n-m}\Delta _{n}\Delta _{m}\right] \prod_{n}P(\Delta _{n}). \label{dist2}$$ The above measure induces statistics similar to (\[dist1\]), and in particular one still has that $\left\langle \Delta _{n}\Delta _{m}\right\rangle =D\delta _{n,m}+c_{n-m}$, and that $\left\langle \Delta _{n}^{p}\right\rangle =\left\langle \eta ^{p}\right\rangle $. On the other hand, it is not hard to show, following an analysis similar to the one in section \[general\], that, to first order in the correlations, the price of an option is *unchanged,* regardless of the basic day–to–day probability distribution $P\left( \eta \right) $. How can one decide which model more accurately describes a specific time–series of price increments? To answer this question, we first note that (\[dist1\]) and (\[dist2\]) coincide when $P\left( \eta \right) $ is Gaussian. It must then be that the deviations involve the higher moments of $P$. Let us denote with $\lambda _{i}$ the $i$–th cumulant of $P$ (in particular $\lambda _{2}=D$ and $\lambda _{4}=D^{2}\kappa $, with $\kappa $ the kurtosis), and with $\lambda _{i}\left( n\right) $ the $i$–th cumulant of the return $\Delta _{1}+\cdots +\Delta _{n}$ of the stock in $n$ time periods. If the price increments where independent, then $\lambda _{i}\left( n\right) =n\lambda _{i}$. In the presence of correlations, this equation is not valid any more, but, if the correlations are short–range, we can define global cumulants as $$\widetilde{\lambda }_{i}=\frac{1}{n}\lambda _{i}\left( n\right) .\;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( n\rightarrow \infty \right)$$ To be definite, let us consider the case already discussed of short–range anticorrelations $c_{1}=-c$. We have already shown that $\lambda _{2}\left( n\right) =n\left( D-2c\right) +2c$ and therefore that $$\widetilde{\lambda }_{2}=D\left( 1-\frac{2c}{D}\right) .$$ It is then not hard to show that, if one considers the model (\[dist1\]), then the global forth moment is given by $$\widetilde{\lambda }_{4}=D^{2}\kappa \left( 1-\frac{4c}{D}\right) .$$ On the other hand, if one uses (\[dist2\]) one finds that $$\widetilde{\lambda }_{4}=D^{2}\kappa \left( 1-\frac{8c}{D}\right) .$$ In both cases, the global historical kurtosis is reduced by anti–correlation effects, but the amount varies in the two models. Recalling that $2c/D\simeq 0.10$, we see that the two models differ by $10\%$ in the measurement of the global kurtosis $\widetilde{\lambda }_{4}/D^{2}\kappa $. The case of Delta-hedging ------------------------- Le us conclude this section by discussing other possible hedging schemes. Instead of choosing the optimal hedge that minimizes the variance, one can follow the (sub–optimal) Black–Scholes $\Delta $–hedge. As emphasized in [@us], this leads in general to a negligible risk increase, and has the advantage of removing the effect of the drift $\mu $ on the price, and therefore reduces the risk associated to change of long time trends, encoded in $\mu $. One might therefore expect that $\Delta $–hedging also allows one to get rid of the correlations. However, the following calculation shows that this is not the case. Up to first order in $c$, it is sufficient to consider that the $\Delta $–hedge is given by $$\phi _{BS,n}=\frac{\partial }{\partial x_{n}}\langle \Theta \rangle _{n}^{0}=-\langle \frac{\partial \ln P}{\partial \Delta }(\Delta _{n+1})\Theta \rangle _{n}^{0}.$$ The price of the contract is then given by $$\begin{aligned} \mathcal{C} &=&\left\langle \Theta \right\rangle -\sum_{n=0}^{N-1}\left\langle \mu _{n}\phi _{BS,n}\right\rangle \\ &=&\left\langle \Theta \right\rangle +\sum_{n=0}^{N-1}\left\langle \mu _{n}\frac{\partial \ln P}{\partial \Delta _{n+1}}\Theta \right\rangle \\ &=&\langle \Theta \rangle ^{0}+\frac{1}{2}\sum_{n=1}^{N}\sum_{m<n}c_{m-n}\langle F(\Delta _{n})\frac{\partial \ln P}{\partial \Delta _{m}}\Theta \rangle ^{0}.\end{aligned}$$ Note that, in this case, the drift $\mu $ indeed disappears to first order even if $F(\Delta )$ is not zero; the correlations, on the other hand, are still present within a $\Delta $–hedging scheme. \[other\]Other optimization schemes =================================== Let us now consider a slightly different problem, where the price of the option is fixed by the market to a certain value $\mathcal{C}_{M}$. The option trader knows that price increments have a non–zero conditional drift and wishes to buy and hedge the option in order to optimize a certain risk adjusted return. The wealth balance associated to selling the option now reads $$W=\mathcal{C}_{M}+H-\Theta ,$$ so that the expected return is equal to $$M=\mathcal{C}_{M}+\langle H-\Theta \rangle =\mathcal{C}_{M}-\mathcal{C}$$ and the expected risk is $$\mathcal{R}^{2}=\left\langle W^{2}\right\rangle -M^{2}.$$ We will assume that the trader wants to maximize a certain function $G$ of $M $ and $\mathcal{R}$, for example the Sharpe ratio $G_{1}=M/\mathcal{R}$, or a certain risk corrected return $G_{2}=\lambda M-\mathcal{R}$. Consider a small variation $\delta \phi _{n}$ of the strategy $\phi _{n}$. The corresponding changes in $M$ and $\mathcal{R}$ read $$\frac{\delta M}{\delta \phi _{n}}=\mu _{n}\qquad \mathcal{R}\frac{\delta \mathcal{R}}{\delta \phi _{n}}=\langle \Delta _{n+1}(H-\Theta )\rangle _{n}+\mathcal{C}\,\mu _{n}.$$ We may now extremize $G$ with respect to the hedging strategy and obtain the following equation $$\frac{\delta M}{\delta \phi _{n}}\frac{\partial G}{\partial M}+\frac{\delta \mathcal{R}}{\delta \phi _{n}}\frac{\partial G}{\partial \mathcal{R}}=0,$$ or $$\langle \Delta _{n+1}(\Theta -H)\rangle _{n}=\mu _{n}\mathcal{C}+\mu _{n}\mathcal{G\;\ \ \ \ \ \ \ \ G}=\mathcal{R}\frac{\partial _{M}G}{\partial _{\mathcal{R}}G}. \label{eq2000}$$ In the two examples considered before, one has that $$\mathcal{G}_{1}=-\frac{\mathcal{R}^{2}}{M}\qquad \ \ \ \ \ \mathcal{G}_{2}=-\lambda \mathcal{R}.$$ Equation (\[eq2000\]) is a generalization of equation (\[general\]), and is again well suited for a perturbation in $\mu $. We will work as always to first order in $\mu $, and we will call $\widetilde{\phi}$ the solution to (\[eq2000\]) with the function$\mathcal{\ G}$ set to zero – that is, the solution to the problem solved in section \[pert\]). It is then easy to see, using equation (\[corr\]), that the optimal strategy is given by $$\phi _{n}^{\ast }=\widetilde{\phi}_{n}-\frac{\mu _{n}}{D_{n}}\left. \mathcal{G}\right| _{\widetilde{\phi}}. \label{eq3000}$$ We conclude this section by showing that the more general formalism developed above actually has some interesting consequences for the original problem of option pricing and hedging which was considered in the rest of the paper. Let us in particular consider, as we have done in the previous sections, a specific contract sold at time $0$ at the optimal price $\mathcal{C}_{0}$ and with optimized hedging strategy $\phi _{n,0}^{\star }\,$(we will explicitly insert the time–index for the date of *sale* of the contract in the last part of this section). More generally, we may sell the same contract at some other time $P$ at a price $\mathcal{C}_{P}$ with a hedging strategy $\phi _{n,P}^{\star }$ (with $n\geq P$). It is a natural question to ask what is the relation of the various hedging strategies $\phi _{n,P}^{\star }$ for *fixed* time $n$, as we vary the sale time of the contract $P$. In particular, one may ask if the optimal hedging strategy $\phi _{n,P}^{\star }$ at time $n$ should be equal to the optimal hedge $\phi _{n,n}^{\star }$ which one is to adopt if one is selling the contract at time $n$. This question can be easily answered in the framework of the present section, by noting that, at time $n$, the option traders wealth balance is given by the original price of the contract $\mathcal{C}_{P}$ plus the returns of the hedging strategy up to time $n$$$\mathcal{W}_{n}=\mathcal{C}_{P}+\phi _{P,P}^{\star }\Delta _{P+1}+\cdots +\phi _{n-1,P}^{\star }\Delta _{n}.$$ From the point of view of the trader, the above quantity acts as an effective market price $\mathcal{C}_{M}$, and the optimization problem after time $n$ is of the type described at the beginning of this section. Recalling that the trader is trying to minimize $\left\langle W^{2}\right\rangle $, we deduce that the function $G$ is given by $M^{2}+\mathcal{R}^{2}$, and therefore that $\mathcal{G}=M=\mathcal{C}_{M}-\mathcal{C}=\mathcal{W}_{n}-\mathcal{C}_{n}$. We can then use equation (\[eq3000\]) to deduce that $$\phi _{n,P}^{\star }=\phi _{n,n}^{\star }-\frac{\mu _{n}}{D_{n}}\left( \mathcal{W}_{n}-\mathcal{C}_{n}\right).$$ The above equation has a clear interpretation. In a risk–free model, the quantity $\left( \mathcal{W}_{n}-\mathcal{C}_{n}\right) $ vanishes, since we can, by assumption, perfectly reproduce the price–process of an option by hedging correctly. Therefore the correct hedge is independent of the time of sale of the contract. On the other hand, in a model with non–zero risk, one needs to correct the hedging strategy whenever the past hedge has not completely compensated the price change of the option. In particular, if the hedging strategy has made too much money and $\mathcal{W}_{n}>\mathcal{C}_{n}$has to decrease the amount of stock in the portfolio whenever the conditional drift $\mu _{n}$ is positive, and increase it otherwise. A similar result was obtained in [@us], section 3.4; it was however noted there that this history dependent strategy, although reducing the variance, actually increases the probabilty of large losses. \[conc\]Summary and conclusion ============================== We have considered the problem of option pricing and hedging when stock returns are correlated in time. Using a variance minimization framework, we have obtained a general formula, valid for weakly correlated non–Gaussian processes. We have shown that in the limit of Gaussian price increments, the correlations are irrelevant, in the sense that the Black–Scholes formula holds with volatility that of the price increments on the time–scale of the re–hedging. For non–Gaussian processes, however, further non trivial corrections to the ‘smile’ effect are brought about by the correlations, even when the hedge is the Black–Scholes $\Delta $–hedge. The kurtosis to be used in option–pricing is neither the local one, nor the global one, but some effective kurtosis which depends on the strength of the correlations. The above formalism can be extended to treat the case where the distribution of price increments depends explicitly on time, as would be needed to treat the case of a time dependent volatility. Finally, we have given formula for the optimal hedge when the expected return associated to options is non zero. Appendix: Correlations within the Black–Scholes framework {#appendix-correlations-within-the-blackscholes-framework .unnumbered} ========================================================= The presence of correlations in the continuous time limit can be modelled as a history dependent drift term in the stochastic differential equation for the price (or log-price process). More precisely, the dynamical evolution of the price reads: $$dX(t) = \mu(t) dt + \sigma_0 dW(t)$$ where the drift $\mu(t)$ depends on the whole past history and $dW$ is a Brownian noise of unit variance per unit time. A simple model of correlations is to assume that $\mu(t)$ is constructed from the past values of $dW(t')$ as an exponential moving average: $$\mu(t)= \epsilon \Gamma \int_{-\infty}^t dW(t') e^{-\Gamma(t-t')}.$$ When $\epsilon > 0$, this corresponds to correlations in price increments extending over a time scale $\Gamma^{-1}$. Conversely, $\epsilon < 0$ corresponds to anticorrelations. It is easy to see that in this model, the time dependent square volatility $\sigma^2(t)$ is given by: $$\sigma^2(t)=\frac{1}{t} \int_0^{t} dt' \left(\sigma_0 + \epsilon (1-e^{-\Gamma{t-t'}}) \right)^2,$$ that interpolates between the short term volatility $\sigma_0$ and the long term volatility $|\sigma_0+\epsilon|$. In the Black–Scholes formalism, however, the drift is totally absent from the price and hedge of an option. This is intimately related to the fact that the Ito formula does not involve the drift $\mu(t)$ [@Baxter]. Hence, it is immediate that in this case, correlations are irrelevant and only the short term volatility $\sigma_0$ is needed to price the option. This is what we find within our general formalism in the case of Gaussian increments. Note that the difference between the short term volatility and long term volatility is particularly striking in the case of an Ornstein-Uhlenbeck (mean reverting) process for which $\mu(t)=-K X(t)$. In this case, the long term volatility tends to zero: $X(t)$ has bounded fluctuations, even when $t \to \infty$. Correspondingly, the average pay-off over the objective probability distribution is finite for $t \to \infty$, whereas the option price, given by the standard Black–Scholes formula with a volatility $\sigma_0$, tends to infinity. The difference between the two comes from the hedging strategy that loses, because of the anticorrelations, an infinite amount of money in the limit $t \to \infty$. [9]{} see *e.g.* : J.C. Hull *Futures, Options and Other Derivative Securities*, Prentice Hall (1997). P. Wilmott, *Derivatives, The theory and practice of financial engineering*, John Wiley (1998). J. P. Bouchaud & D. Sornette, *Journal de Physique I*, **4**, 863–881 (1994). M. Schweizer, *The Mathematics of Operations Research*, **20**, 1–32 (1995). J. P. Bouchaud & M. Potters, *Theory of Financial Risks*, Cambridge University Press (2000). see *e.g.* : J.Y. Campbell, A.W. Lo and A.C. MacKinlay *The Econometrics of Financial Markets*, Princeton (1997). see *e.g.* : M. Baxter, A. Rennie, *Financial Calculus*, Cambridge University Press (1996). [^1]: The general correlated Gaussian case was considered in [@BS2], and leads to rather complex formulae for which no simple interpretation was given. [^2]: This can be easily seen within a Black–Scholes framework: see Appendix. [^3]: In the following, we assume for simplicity that the interest rate is zero. It is easy to generalize the formulae obtained here when this rate is non-zero: see [@us] for details. [^4]: The following results can be generalized to the case where the distribution $P(\eta )$ explicitly depends on $n$. [^5]: In the sequel, we will actually only need that equation (\[eq600\]) holds for $q=0$,$1$.

--- abstract: '[ Short-range ordering (SRO) tendency for disordered alloys is considered as competition between chemical ordering and geometric (mainly, difference in atomic radius for constituents) effects. Especially for multicomponent (including the so-called high entropy alloys (HEAs) near equiatomic composition), it has been considered as difficult to systematically characterize the SRO tendency only by geometric effects, due mainly to the fact that (i) chemical effects typically plays significant role, (ii) near equiatomic composition, we cannnot classify which elements belong to solute or solvent, and (iii) underlying lattice for pure elements can typically differ from each other. Despite these facts, we here show that SRO tendency for seven fcc-based alloys including subsystems of Ni-based HEAs, can be well characterized by geometric effects, where corresponding atomic radius is defined based on atomic configuration with special fluctuation, measured from ideally random structure. The present findings strongly indicate the significant role of geometry in underlying lattice on SRO for multicomponent alloys. ]{}' author: - Koretaka Yuge - Shouno Ohta title: Microscopic Geometry Rules Ordering Tendency for Multicomponent Disordered Alloys --- Introduction ============ For binary disordered alloys, there have been experimental and/or theoretical attempts to universally characterize short-range order (SRO) tendency based on the difference (or ratio) of constituent atomic radius. These are commonly based on the intuition that ordering tendency (i.e., unlike-atom pairs are preffered) is enhanced when constituent atomic radius exhibit significant differences due to reducing strain energy. This intuition fails, i.e., SRO tendency for binary alloys cannot be generally characterized only by the difference between constituent atomic radius, since chemical effects, coming from many-body interactions in alloys, play central role on SRO.[@sro-j] Following theoretical investigations quantitatively point out the significant chemical effect on SRO,[@chem1; @chem2] which naturally makes the trends that the relationship between SRO for binary alloys and geometric effects (including differences in atomic radius) have not been focused on so far. Despite these facts, we recently reveal[@em-sro] that although trends in SRO for fcc-based binary alloys cannot be characterized by conventional Goldschmidt or DFT-based atomic radius even qualitatively, those are well-characterized by effective atomic radius of specially-selected microscopic structure derived only from information of underlying geometry (i.e., lattice). The results strongly indicate that ratio of constituent atomic radius in short-range ordered structure covariance deviation, w.r.t. configurational density of states (CDOS) for *non-interacting* system, from linear average for constituent pure elements can characterize the SRO tendency. Although recent DFT-based theoretical investigations amply address SRO tendencies not only for binary, but also for multicomponent alloys at or near equiatomic composition such as CoCrNi and CoFeNi for ternary, and Mo- and Ni-based quaternary and quinary alloys due to the recently attractive attention to the so-called high entropy alloys (HEA) that can exhibit super-high strength compared to conventional alloys, very little has been focused on SRO in multicomponent (three or more components) systems in terms of geometric contributions, due mainly to the fact that (i) near equiatomic composition, we cannot clearly classify whether the selected element is solute or solvent, (ii) underlying lattice for pure constituent can differ from that for HEA and (iii) it has been believed that chemical effects other than geometric effects play central role on SRO for binary systems. Here we extend our previous approach on binary system to systematically characterizing ordering tendency for seven equiatomic ternary alloys, by explicitly employing non-conventional coordinates (i.e., not conventional in generalized Ising systems) along chosen pair probability, where quantitative formulation for relationship between conventional and non-conventional coordinates for multicomponent discrete systems under *constant composition* has recently been clarified by our study. We find that SRO tendencies in terms of neighboring pair probability can be universally, well characterized by ratio of atomic radius between of specially-fluctuated ordered structure and of ideally random structure, which is discussed in detail below. Methodology =========== In order to quantitatively represent any atomic configuration for multicomponent systems, we here employ generalized Ising model[@ce] (GIM) that provides complete orthonormal basis functions enables to exactly describe physical quantities as a function of atomic configuration for classical discrete systems under constant composition. In the present A-B-C ternary systems, we define spin variables $\sigma_i$ to specify atomic occupation of A, B or C at lattice point $i$ as $\sigma_i=+1$ for A, $\sigma_i=0$ for B and $\sigma_i=-1$ for C element. Under this definition, basis functions at a single lattice point are given by $$\begin{aligned} \label{eq:basis} \phi_0 = 1,\quad \phi_1 =\sqrt{\frac{3}{2}}\sigma_i,\quad \phi_2 = \sqrt{2}\left(\frac{3}{2}\sigma_1^2 - 1\right),\end{aligned}$$ and higher-order multisite correlations $\Phi$ can be obtained by taking average of corresponding products of the above basis functions over all lattice points. Once we construct complete set of basis, we recently find that for any number of components, canonical average of multisite correlation function can be given by $$\begin{aligned} \Braket{\Phi_r}\left(T\right) \simeq \Braket{\Phi_r}_1 - \sqrt{\frac{\pi}{2}} \Braket{\Phi_r}_2\frac{U_r - U_0 }{k_{\textrm{B}}T},\end{aligned}$$ where $\Braket{\quad}_1$ and $\Braket{\quad}_2$ respectively denotes taking arithmetic average and standard deviation over all possible atomic configurations for *non-interacting* system. $U_r$ and $U_0$ represents potential energy of the so-called “projection state (PS)”[@em2] and special quasirandom structure (SQS, that mimic perfect random state).[@sqs] Since we have shown that structures of PS and SQS, and $\Braket{\quad}_1$ and $\Braket{\quad}_2$ can be known from system *before* applying many-body interactions, we can *a priori* know their information without any thermodynamic information. This directly means that once we obtain the structure of PS and SQS, we can systematically characterize temperature dependence of ordering tendency for multicomponent alloys. However, for multicomponent system, basis functions in Eq.  does not provide intuitive interpretation of which like- and/or unlike-atom pairs are preferred and/or disfavored, which is contrary to the case of binary system. For multicomponent systems, practical problemes are (i) pair probabilities are obtained by linear combination of the basis functions, leading to accumulating predictive error and (ii) a set of pair probability does not form orthonormal basis. To overcome these problems, we recently modify Eq.  that can be applied to canonical average of any linear combination of orthonormal basis functions. For pair probability of $Y_{IJ}$, this is given by $$\begin{aligned} \label{eq:y} \Braket{Y_{IJ}}\left(T\right) \simeq \Braket{Y_{IJ}}_1 - \sqrt{\frac{\pi}{2}} \frac{\Braket{Y_{IJ}}_2}{k_{\textrm{B}}T}\sum_M \Braket{U|\Phi_M} \left( \Braket{\Phi_M}_{Y_{IJ}}^{\left(+\right)} - \Braket{\Phi_M}_1 \right),\end{aligned}$$ where $\Braket{\quad}_{Y_{IJ}}^{\left(+\right)}$ denotes taking linear average over all possible atomic configuration satisfying $Y_{IJ}\ge \Braket{Y_{IJ}}_1$. Eq.  directly means that we can construct PS to characterize temperature dependence of selected pair probability $Y_{IJ}$, where its structure is given by $\left\{ \Braket{\Phi_1}_{Y_{IJ}}^{\left(+\right)},\cdots, \Braket{\Phi_f}_{Y_{IJ}}^{\left(+\right)} \right\}$ for $f$-dimensional configuration space considered, and structure of SQS is given by $\left\{\Braket{\Phi_1}_1,\cdots,\Braket{\Phi_f}_1\right\}$. Note that defiition of the above pair probabily for unlike-atom pair includes permutation of constituent pair probability $y_{IJ}$ and $y_{JI}$, namely, $$\begin{aligned} Y_{IJ} = y_{IJ} + y_{JI},\end{aligned}$$ where constituent pair probabilities satisfy the following summation for composition of $I$, $c_I$: $$\begin{aligned} \sum_{J} y_{IJ} = c_I.\end{aligned}$$ We here consider ordering tendency in terms of 1st nearest-neighbor (1NN) pair probability for seven fcc-based equiatomic ternary alloys of CrCoNi, CrFeNi, CrNiMn, CoNiMn, FeNiMn, AgAuCu and AgPdRh: The reasons for choosing these alloys are (i) Ni-based ternarly alloys are all subsystems of HEAs whose short-range order is considered as key role to characterize its extreme mechanical properties, and (ii) compared to binary alloys, very little has been known for short-range order even qualitatively, where short-range ordering tendency for Ag-based two alloys, CrCoNi and CrFeNi are qualitatively available for previous experimental and/or theoretical studies. In the present study, PSs along possible six pairs (A-A, A-B, B-B, A-C, B-C and C-C) and SQS are constructed for supercell of 480-atom ($4\times5\times6$ expantion of convensional unit cell) based on Monte Carlo simulation to minimize Euclidean distance between multisite correlation functions for practically-constructed configuration and that for ideal values,[@em1] where we consider up to 6NN pair, and all triplets and quartets consisting of up to 4NN pairs that can well characterize ordering tendency for fcc-based binary alloys. We emphasize again that structure of PSs and SQS are constructed only for geometric information of underlying lattice, which are all common for the seven ternary alloys. The constructed PSs and SQS are applied to density functional theory (DFT) calculation to obtain total energy, which is perfomed by the VASP[@vasp] code using the projector-augmented wave method,[@paw] with the exchange-correlation functional treated within the generalized-gradient approximation of Perdew-Burke- Ernzerhof (GGA-PBE).[@pbe] The plane wave cutoff of 360 eV is used. Structural optimization is performed until the residual forces less than 0.005 eV/Å. Results and Discussions ======================= ![Color plot of sign and magnitude of $\alpha_3$, $\alpha_2$ and $\alpha_1$. For $\alpha_1$, dark blue and dark red color triangles denotes pairts with the top three highest absolute value of $\alpha_1$. []{data-label="fig:a"}](SRO-Ternary-ColorMap.eps){width="0.93\linewidth"} For multicomponent system, information about whether a chosen unlike-atom pair is preferred or disfavored w.r.t. random state is not sufficient to characterize whether the system undergoes ordered structure or phase separation at low temperature, which is in contrast to binary systems. This is simply because for multicomponent systems, there are multiple (for ternary, three) different kinds of unlike-atom pair. Therefore, we here introduce two type of quantity to measure the ordering (or separating) tendency as follows: $$\begin{aligned} \alpha_3 &=& -\frac{1}{2} \sum_{I\neq J} U_{IJ} + \sum_J U_{JJ}, \nonumber \\ \alpha_2^{\left(JK\right)} &=& -U_{JK} + \left(U_{JJ} + U_{KK}\right), \nonumber \\ \alpha_1^{\left(JK\right)} &=& -U_{JK}, \nonumber \\ &&\left(I, J, K = \textrm{A, B, C} \right),\end{aligned}$$ where $U_{JK}$ denotes potential energy of PS for $JK$ pair, given in Eq. . Note that here and hereinafter, for simplicity PS energy and its correlation functins are always measured from SQS energy and correlation functions. From the definition, we can see that (i) $\alpha_3$ denotes the measure of whole preference (disfavor) for three unlike-atom pairs w.r.t. other three like-atom pairs when $\alpha_3$ exhibit positive (negative) sign, (ii) $\alpha_2$ denotes the measure of preference (disfavor) for a selected unlike-atom pair w.r.t. corresponding two like-atom pairs when $\alpha_2$ exhibits positive (negative) sign, (iii) $\alpha_1$ denotes the measure of preference (disfavor) for a selected unlike-atom or like-atom pair w.r.t. random states with its positive (negative) sign, and (iv) for ideally random states, $\alpha_3=\alpha_2=\alpha_1=0$. Therefore, $\alpha_3$ and $\alpha_2$ respectively represent the measure of overall preference (disfavor) of unlike-atom pair(s) for ternary system and three subsystems, i.e., constituent binary systems. Figure \[fig:a\] shows the color plot of sign and magnitude of $\alpha_3$, $\alpha_2$ and $\alpha_1$ for the present seven ternary alloys. We can clearly see that ordering (or separating) tendency for a chosen pair in terms of $\alpha_1$ can strongly depend on combination of the rest element: For instance, Ni-Ni like-atom pair is strongly preferred for FeMnNi and CoMnNi, while it is disfavored for CrFeNi alloy. The fact that ordering tendency as ternary (i.e., $\alpha_3 > 0$) does not always holds for its subsystems, e.g., for CrCoNi, $\alpha_3 > 0$ while for Co-Ni subsystem, $\alpha_2 < 0$. Moreover, we can clearly see for Fe-Ni subsystem in CrFeNi and Co-Mn subsystem in CoMnNi alloys the counterbalance breaking of like- and unlike-atom pair that is not broken for binary system: Fe-Ni subsystem exhibit the same negative sign for Fe-Fe, Fe-Ni and Ni-Ni pair, and Co-Mn exhibit the same positive sign for Co-Co, Mn-Mn and Co-Mn pairs. ![Ternary ordering tendency in terms of averaged ratio of atomic radius under three different definitions.[]{data-label="fig:a3"}](R-vs-SRO-total.eps){width="0.97\linewidth"} ![Binary-subsystem ordering tendency in terms of ratio of atomic radius under three different definitions.[]{data-label="fig:a2"}](R-vs-SRO-decompose.eps){width="0.97\linewidth"} In order to characterize the above ordering tendencies in terms of underlying geometry, we also introduce three kinds of atomic radius ratio as follows: $$\begin{aligned} r_g^{\left(JK\right)} &=& \left[ \frac{R_{g}^{\left(J\right)}}{R_{g}^{\left(K\right)}} \right]_1 \nonumber \\ r_{PS/PS}^{\left(JK\right)} &=& \left[ \frac{R_{PS}^{\left(J\right)}}{R_{PS}^{\left(K\right)}} \right]_1 \nonumber \\ r_{PS/RD}^{\left(JK\right)} &=& \frac{1}{2}\left( \left[ \frac{R_{PS}^{\left(J\right)}}{R_{RD}} \right]_1 + \left[ \frac{R_{PS}^{\left(K\right)}}{R_{RD}} \right]_1 \right), \end{aligned}$$ where $\left[\quad\right]_1$ denotes that internal numerator and enumerator can be reversed so that resultant value of fraction should always be greater or equal to 1, and $R_g^{\left(K\right)}$, $R_{PS}^{\left(K\right)}$ and $R_{RD}$ respectively denotes Goldschmidt atomic radius[@gold] for element $K$, effective atomic radius of $K$ obtained from projection state, and effective atomic radius for random states (i.e., from SQS structure). Note that since PS contains multiple elements other than considered $K$, we define $R_{PS}^{\left(K\right)}$ from the volume of the following conditions: $$\begin{aligned} \label{eq:vjk} V_{JK} = V_{SQS} + C\cdot \left(V_{PS}^{\left(JK\right)} - V_{SQS}\right), \end{aligned}$$ where constant $C$ is determined so that $$\begin{aligned} \label{eq:ykk} \left |Y_{KK}\left( C\cdot\left\{ \Braket{\Phi_1}_{Y_{IJ}}^{\left(+\right)},\cdots, \Braket{\Phi_f}_{Y_{IJ}}^{\left(+\right)} \right\} \right) -1 \right| = \textrm{min}.\end{aligned}$$ With these prerations, $R_{PS}^{\left(K\right)}$ can be obtained through $$\begin{aligned} \label{eq:rps} R_{PS}^{\left(K\right)} \propto \left\{ \left(\sum_{I\neq J} V_{IJ}/2\right) - V_{LM}\left(L\neq K, M\neq K\right) \right\}^{1/3}\end{aligned}$$ for system with $\alpha_3 > 0$, and $$\begin{aligned} \label{eq:rps2} R_{PS}^{\left(K\right)} \propto \left\{ V_{KK}/2 \right\}^{1/3}\end{aligned}$$ for system with $\alpha_3 < 0$. The reason why we do not employ sign of $\alpha_2$ or $\alpha_1$ to determine effective atomic radius is that (i) there can be multiple values for atomic radius for a chosen element of a given system, and (ii) algebraic equations for atomic radius can be linear dependent (e.g., CrMnNi system: there is four preference pairs) with each other when we focus on the preference of ordering tendency of subsystems or cannot be solved (e.g., CrFeNi system: there is two preference pairs). Eqs.  and  means that (i) since projection state in Eq.  corresponds a specially fluctuated structure (i.e., covariance fluctuation) from random state along considered pair (similarly to binary system), we define effective atomic radius from information about such fluctuated structure, (ii) it has uncertainity to multiply scalar constant $C$ for projection state since only the direction of fluctuation is essential to determine pair probability (invariant to the choice of $C$), while resultant effective volume (or atomic radius) depends on $C$, and (iii) therefore, we determine the magnitude of fluctuation $C$ so that the fluctuated structure has pair probability corresponding to the value of correlation function for considered pure element (Eq. ). Eqs.  and  means that effective atomic radius are determined from information about volume of fluctuated structure with prefered direction (i.e., for $\alpha > 0$, fluctuation along unlike-atom pair, and $\alpha < 0$, along like-atom pair). Figure \[fig:a3\] shows the resultant ternary atomic ordering tendency, $\alpha_3$, in terms of the above three different kinds of definition of atomic radius ratio. $\Braket{\quad}_{\textrm{ABC}}$ means taking linear average over possible set of pairs, e.g., for Goldschmidt radius, $$\begin{aligned} \Braket{r_g}_{\textrm{ABC}} = \frac{1}{3}\left( r_g^{\left(\textrm{AB}\right)} + r_g^{\left(\textrm{AC}\right)} + r_g^{\left(\textrm{BC}\right)} \right).\end{aligned}$$ We can clearly see that Goldshmidt atomic radius ratio does not exhibit effective correlation with ordering tendency: systems with larger positive value of $\alpha_3$ does not have larger value of $\Braket{r_g}_{\textrm{ABC}}$. The correlation is most strong when we take $\Braket{r_{PS/RD}}_{\textrm{ABC}}$ compared with $\Braket{r_{PS/PS}}_{\textrm{ABC}}$, and when we decompose these tendency into subsystems (see Fig. \[fig:a2\]), the same correlation relationships holds true. These strongly indicate that for multicomponent disordered alloys, geometric effects on ordering tendency in terms of strain due to difference in atomic radius should be naturally considered as difference in atomic radius between effective constituent radius in covariance-fluctuated strucre and that for random states. Conclusions =========== Based on specially fluctuated structure and ideally random structure, we here show that introduced definition of atomic radius can reasonablly characterize SRO tendency for multicomponent alloys not only for whole (ternary) system as well as their constituent subsystems, while conventional definition of atomic radius fails to explain such SRO tendency. The present findings strongly indicate the significant role of geometry in underlying lattice on SRO for multicomponent alloys, where quantitative formulation of SRO parameter in terms of the present geometric factors should be performed in our future study. Acknowledgement =============== This work was supported by Grant-in-Aids for Scientific Research on Innovative Areas on High Entropy Alloys through the grant number JP18H05453 and a Grant-in-Aid for Scientific Research (16K06704) from the MEXT of Japan, Research Grant from Hitachi Metals$\cdot$Materials Science Foundation, and Advanced Low Carbon Technology Research and Development Program of the Japan Science and Technology Agency (JST). [9]{} J. Cryst. Soc. Jpn. **12**, 186 (1970). R. V. Chepulskii, J. Phys.: Condens. Matter **10**, 1505 (1998). R. V. Chepulskii and V. N. Bugaev, J. Phys.: Condens. Matter **10**, 7309 (1998). K. Yuge, J. Phys. Soc. Jpn. **87**, 044804 (2018). K. Yuge, J. Phys. Soc. Jpn. **85**, 024802 (2016). J.M. Sanchez, F. Ducastelle, and D. Gratias, Physica A **128**, 334 (1984). S.-H. Wei, L. G. Ferreira, J. E. Bernard, and A. Zunger, Phys. Rev. B **42**, 9622 (1990). G. Kresse and J. Hafner, Phys. Rev. B [**47**]{}, R558 (1993). G. Kresse and D. Joubert, Phys. Rev. B [**59**]{}, 1758 (1999). J.P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. [**77**]{}, 3865 (1996). V. M. Goldschmidt, Z. Phys. Chem. **133**, 397 (1928).

--- abstract: 'This paper describes Task 2 of the DCASE 2018 Challenge, titled “General-purpose audio tagging of Freesound content with AudioSet labels”. This task was hosted on the Kaggle platform as “Freesound General-Purpose Audio Tagging Challenge”. The goal of the task is to build an audio tagging system that can recognize the category of an audio clip from a subset of 41 diverse categories drawn from the AudioSet Ontology. We present the task, the dataset prepared for the competition, and a baseline system.' address: | $^1$Music Technology Group, Universitat Pompeu Fabra, Barcelona {name.surname}@upf.edu\ $^2$ Google, Inc., New York, NY, USA {plakal,dpwe}@google.com\ bibliography: - 'refs.bib' title: | General-purpose tagging of Freesound audio with AudioSet labels:\ Task Description, Dataset, and Baseline --- Audio tagging, audio dataset, data collection Introduction {#sec:intro} ============ Task Setup {#sec:challenge} ========== Dataset {#sec:dataset} ======= Baseline System {#sec:baseline} =============== Conclusion {#sec:conclusion} ========== ACKNOWLEDGMENT {#sec:ack} ============== We thank Addison Howard and Walter Reade of Kaggle for their invaluable assistance with the task design and Kaggle platform, and everyone who contributed to FSDKaggle2018 with annotations. Eduardo Fonseca is also grateful for the GPU donated by NVidia.

--- abstract: 'In this paper, we study the boundedness of pseudodifferential operators with symbols in the Hörmander class $S^0_{\rho,\rho}$ on $\alpha$-modulation spaces $M_{p,q}^{s,\alpha}$, and consider the relation between $\alpha$ and $\rho$. In particular, we show that pseudodifferential operators with symbols in $S^0_{\alpha,\alpha}$ are bounded on all $\alpha$-modulation spaces $M^{s,\alpha}_{p,q}$, for arbitrary $s\in{\mathbb{R}}$ and for the whole range of exponents $0 < p,q \leq \infty$.' address: 'Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan' author: - Tomoya Kato - Naohito Tomita title: 'Pseudodifferential operators with symbols in the Hörmander class $S^0_{\alpha,\alpha}$ on $\alpha$-modulation spaces' --- Introduction {#sec1} ============ In Gröbner’s Ph.D. thesis [@grobner; @1992], $\alpha$-modulation spaces $M_{p,q}^{s,\alpha}$ were introduced as intermediate spaces between modulation spaces $M_{p,q}^s$ and (inhomogeneous) Besov spaces $B^s_{p,q}$. The parameter $\alpha \in [0,1)$ determines how the frequency space is decomposed. Modulation spaces which are constructed by the uniform frequency decomposition correspond to the case $\alpha = 0$ and Besov spaces which are constructed by the dyadic decomposition can be regarded as the limiting case $\alpha \to 1$. See the next section for the precise definition of $\alpha$-modulation spaces. Let $b \in {\mathbb{R}}$, $0 \le \delta \le \rho \le 1$, $\delta<1$, $1<p,q<\infty$ and $s \in {\mathbb{R}}$. It is known that all operators of class ${\mathrm{Op}}(S^b_{\rho,\delta})$ are bounded on $L^p({\mathbb{R}}^n)$ if and only if $b \le -|1/p-1/2|(1-\rho)n$ (see [@stein; @1993 Chapter VII, Section 5.12]), and the same condition assures the $B^s_{p,q}$-boundedness, namely the boundedness of operators of class ${\mathrm{Op}}(S^b_{\rho,\delta})$, $b \leq -|1/p-1/2|(1-\rho)n$, on $B^s_{p,q}({\mathbb{R}}^n)$ holds (see, e.g., Bourdaud [@bourdaud; @1982], Gibbons [@gibbons; @1978] and Sugimoto [@sugimoto; @1988]). It should be remarked that the boundedness of operators of class ${\mathrm{Op}}(S^0_{1,1})$ on $B^s_{p,q}$ also holds for $s>0$ (see the references above). On the other hand, as a difference between boundedness on Besov and modulation spaces, it is known that all operators of class ${\mathrm{Op}}(S^0_{0,0})$ are bounded on $M^s_{p,q}$ (see, e.g., Gröchenig and Heil [@grochenig; @heil; @1999], Tachizawa [@tachizawa; @1994] and Toft [@toft; @2004]). Moreover, Sugimoto and Tomita [@sugimoto; @tomita; @PAMS; @2008 Theorem 2.1] proved that the boundedness of operators of class ${\mathrm{Op}}(S^0_{1,\delta})$, $0<\delta<1$, on $M^0_{p,q}$, $q \neq 2$, does not hold in general, and also Let $1 < q < \infty$, $b \in {\mathbb{R}}$, $0 \leq \delta \leq \rho \leq 1$ and $ \delta < 1$. Then, all pseudodifferential operators with symbols in $S_{\rho, \delta}^b$ are bounded on $M_{2,q}^{0}({\mathbb{R}}^n)$ if and only if $b \leq - | 1/q - 1/2 | \delta n$. In this paper, we discuss the $M_{p,q}^{s,\alpha}$-boundedness of pseudodifferential operators with symbols in the so-called exotic class $S^0_{\rho, \rho}$, and try to clarify the relation between $\alpha$ and $\rho$. For the boundedness of pseudodifferential operators, we will use the following terminology with a slight abuse. Let $s,t \in {\mathbb{R}}$. If there exist a constant $C_\sigma$ such that the estimate $$\left\| \sigma(X,D) f \right\|_{M_{p,q}^{t,\alpha}} \leq C_\sigma \left\| f \right\|_{M_{p,q}^{s,\alpha} } {\quad\textrm{for}\quad}f \in {\mathcal{S}}({\mathbb{R}}^n)$$ holds, then we simply say that the pseudodifferential operator $\sigma(X,D)$ is bounded from $M_{p,q}^{s,\alpha} $ to $M_{p,q}^{t,\alpha}$. In particular, if $s=t$, then we say that $\sigma(X,D)$ is bounded on $M_{p,q}^{s,\alpha}$. If $p,q < \infty$, the boundedness mentioned above can be extended to the formal one by density. Borup [@borup; @2003] proved that all pseudodifferential operators of class ${\mathrm{Op}}(S_{\alpha, \alpha}^0)$ are bounded from $M_{p,q}^{s,\alpha}$ to $M_{p,q}^{s-(1-\alpha),\alpha}$ for the space dimension $n=1$. Borup and Nielsen [@borup; @nielsen; @AM; @2006; @borup; @nielsen; @2008] also obtained the boundedness of operators of class ${\mathrm{Op}}(S_{\alpha, 0}^0)$ on $M_{p,q}^{s,\alpha}$. Our purpose is to improve the result of [@borup; @2003] by removing the loss of the smoothness $1-\alpha$ and that of [@borup; @nielsen; @AM; @2006; @borup; @nielsen; @2008] by replacing $\delta=0$ with $\delta=\alpha$ in the full range $0<p,q \le \infty$. Our main result is the following: \[main theorem\] Let $0 \leq \alpha < 1$, $ 0 < p , q \leq \infty$ and $s \in {\mathbb{R}}$. Then, all pseudodifferential operators with symbols in $S_{\alpha, \alpha}^0$ are bounded on $M_{p,q}^{s,\alpha} ({\mathbb{R}}^n)$. 0 that is, there exists a positive integer $N$ such that the estimate $$\label{main theorem estimate} \left\| \sigma(X,D) f \right\|_{M_{p,q}^{s,\alpha}} \lesssim \left\|\sigma ; S^0_{\alpha,\alpha}\right\|_N \left\| f \right\|_{M_{p,q}^{s,\alpha} }$$ holds for all $\sigma \in S_{\alpha, \alpha}^0$ and all $f \in {\mathcal{S}}({\mathbb{R}}^n)$. More precisely, we will prove that there exists a positive integer $N$ such that the estimate $$\label{main theorem estimate} \left\| \sigma(X,D) f \right\|_{M_{p,q}^{s,\alpha}} \lesssim \left\|\sigma ; S^0_{\alpha,\alpha}\right\|_N \left\| f \right\|_{M_{p,q}^{s,\alpha} }$$ holds for all $\sigma \in S_{\alpha, \alpha}^0$ and all $f \in {\mathcal{S}}({\mathbb{R}}^n)$. Recalling the relation $S^0_{\rho, \delta_1} \subset S^{0}_{\rho, \delta_2}$ for $\delta_1 \leq \delta _2$, we see that the class $S_{\alpha, \alpha}^0 $ in Theorem \[main theorem\] is wider than $S_{\alpha, 0}^0$ in [@borup; @nielsen; @AM; @2006]. Furthermore, we recall the well-known result by Calderón and Vaillancourt [@calderon; @vaillancourt; @1972], where it was stated that for any $0 \leq \alpha < 1$, ${\mathrm{Op}}(S^0_{\alpha, \alpha}) \subset \mathcal{L} (L^2)$. Since we have $M^{0,\alpha}_{2,2} = L^2$ for any $0 \leq \alpha < 1 $, we see that Theorem \[main theorem\] contains their result. Also, more generally, we have the following. \[main cor\] Let $0 \leq \alpha < 1$, $ 0 < p , q \leq \infty$, $s,b \in {\mathbb{R}}$ and $0 \leq \delta \leq \alpha \leq \rho \leq 1$. Then, all pseudodifferential operators with symbols in $S_{\rho, \delta}^b$ are bounded from $M_{p,q}^{s,\alpha} ({\mathbb{R}}^n)$ to $M_{p,q}^{s-b,\alpha} ({\mathbb{R}}^n)$. 0 that is, there exists a positive integer $N$ such that the estimate $$\left\| \sigma(X,D) f \right\|_{M_{p,q}^{s-b,\alpha}} \lesssim \left\|\sigma ; S^b_{\rho,\delta}\right\|_N \left\| f \right\|_{M_{p,q}^{s,\alpha} }$$ holds for all $\sigma \in S_{\rho, \delta}^b$ and all $f \in {\mathcal{S}}({\mathbb{R}}^n)$. As a direct consequence of Theorem \[main theorem\], Theorem A and inclusion relations between modulation and $\alpha$-modulation spaces, we immediately have the following statement. \[sharp cor\] Let $1 < q < \infty$, $q\neq 2$, $s \in {\mathbb{R}}$, $0 \leq \delta \leq \rho \leq 1$, $0 \leq \delta, \alpha < 1$ and $\alpha \leq \rho$. Then, all pseudodifferential operators with symbols in $S_{\rho, \delta}^0$ are bounded on $M_{2,q}^{s,\alpha} ({\mathbb{R}}^n)$ if and only if $\delta \leq \alpha$. To conclude the overview of our results, we comment on the optimality of the symbol class in Theorem \[main theorem\]. Corollary \[sharp cor\] implies that ${\mathrm{Op}}(S_{\alpha + \varepsilon, \alpha + \varepsilon}^0) \not \subset \mathcal{L}(M_{2,q}^{s,\alpha})$, $q \neq 2$, for any $0 < \varepsilon < 1 - \alpha$. On the other hand, ${\mathrm{Op}}(S_{\alpha - \varepsilon, \alpha - \varepsilon}^0) \not \subset \mathcal{L}(M_{p,q}^{s,\alpha})$, $0 < p < 1$, for any $0 < \varepsilon < \alpha$ (see Remark \[alpha -\]). Therefore, the class $S_{\alpha, \alpha}^0$ in Theorem \[main theorem\] seems to be optimal to obtain the $M_{p,q}^{s, \alpha}$-boundedness. The plan of this paper is as follows. In Section \[sec2\], we will state basic notations which will be used throughout this paper, and then introduce the definition and some basic properties of $\alpha$-modulation spaces. After stating and proving some lemmas needed to show the main theorem in Section \[sec3\], we will actually prove it in Section \[sec4\]. We end this section by mentioning a remark on arguments to give a proof of the boundedness. If we prove estimate for all Schwartz functions $\sigma$ on $ {\mathbb{R}}^{2n} $, then the same estimate holds for all $\sigma \in S_{\alpha, \alpha}^0$ by a limiting argument (see, e.g., the beginning of the proof of [@stein; @1993 Chapter VII, Section 2.5, Theorem 2]). Hence, in the following statements, we will prove Theorem \[main theorem\] for symbols $\sigma$ belonging to ${\mathcal{S}}({\mathbb{R}}^{2n})$. Preliminaries {#sec2} ============= Basic notations {#sec2.1} --------------- In this section, we collect notations which will be used throughout this paper. We denote by ${\mathbb{R}}$, ${\mathbb{Z}}$ and ${\mathbb{Z}}_+$ the sets of reals, integers and non-negative integers, respectively. The notation $a \lesssim b$ means $a \leq C b$ with a constant $C > 0$ which may be different in each occasion, and $a \sim b $ means $a \lesssim b$ and $b \lesssim a$. We will write $\langle x \rangle = (1 + | x |)$ for $x \in {\mathbb{R}}^n$ and $[s] = \max\{ n \in {\mathbb{Z}}: n \leq s \}$ for $s \in {\mathbb{R}}$. We will also use the notation $A = A(\alpha) = \frac{ \alpha }{ 1 - \alpha }$ (especially, in Sections \[sec3\] and \[sec4\]). We denote the Schwartz space of rapidly decreasing smooth functions on ${\mathbb{R}}^n$ by ${\mathcal{S}}= {\mathcal{S}}({\mathbb{R}}^n)$ and its dual, the space of tempered distributions, by ${\mathcal{S}}^\prime = {\mathcal{S}}^\prime({\mathbb{R}}^n)$. The Fourier transform of $ f \in {\mathcal{S}}({\mathbb{R}}^n) $ is given by $${\mathcal{F}}f (\xi) = \widehat {f} (\xi) = \int_{{\mathbb{R}}^n} e^{-i \xi \cdot x } f(x) d x,$$ and the inverse Fourier transform of $ f \in {\mathcal{S}}({\mathbb{R}}^n) $ is given by $${\mathcal{F}}^{-1} f (x) = \check {f} (x) = (2\pi)^{-n} \int_{{\mathbb{R}}^n} e^{i x \cdot \xi } f( \xi ) d\xi.$$ We next recall the symbol class $S_{\rho, \delta}^b = S_{\rho, \delta}^b ( {\mathbb{R}}^n \times {\mathbb{R}}^n ) $ for $b \in {\mathbb{R}}$ and $0 \leq \delta \leq \rho \leq 1$, which consists of all functions $\sigma \in C^\infty ({\mathbb{R}}^n \times {\mathbb{R}}^n)$ satisfying $$| \partial_x^\beta \partial_\xi^\gamma \sigma (x , \xi ) | \leq C_{\beta, \gamma} \langle \xi \rangle^{b + \delta | \beta | - \rho |\gamma|}$$ for all multi-indices $\beta, \gamma \in {\mathbb{Z}}_+^n$, and set $$\left\|\sigma ; S^b_{\rho,\delta}\right\|_{N} =\max_{|\beta|+|\gamma| \leq N} \left(\sup_{(x, \xi) \in {\mathbb{R}}^n \times {\mathbb{R}}^n} \langle \xi \rangle^{-(b + \delta | \beta | - \rho |\gamma|)} | \partial_x^\beta \partial_\xi^\gamma \sigma (x , \xi ) |\right)$$ for $N \in {\mathbb{Z}}_+$. Note that $S^{b_1}_{\rho_1, \delta_1} \subset S^{b_2}_{\rho_2, \delta_2}$ holds if $b_1 \leq b_2$, $\rho_1 \geq \rho_2$ and $\delta_1 \leq \delta _2$. For $\sigma \in S_{\rho, \delta}^b$, the pseudodifferential operator $\sigma(X,D)$ is defined by $$\sigma (X,D) f (x) =(2\pi)^{-n} \int_{{\mathbb{R}}^n} e^{i x \cdot \xi} \sigma (x,\xi) \widehat f (\xi) d\xi$$ for $ f \in {\mathcal{S}}({\mathbb{R}}^n) $. We denote by ${\mathrm{Op}}(S^b_{\rho,\delta})$ the class of all pseudodifferential operators with symbols in $S^b_{\rho,\delta}$. For the case $0\leq \delta <1$ and $\delta \leq \rho \leq 1$, we know the statement of the composition rule about the class $S^b_{\rho,\delta}$ called the symbolic calculus. That is, if $\sigma \in S^b_{\rho,\delta}$ and $\tau \in S^c_{\rho,\delta}$, then there exists a symbol $\theta \in S^{b+c}_{\rho,\delta}$ satisfying that $\theta (X,D) = \sigma (X,D) \circ \tau (X,D)$. Moreover, $\theta$ can be chosen so that $$\left\| \theta ; S^{b+c}_{\rho,\delta}\right\|_{N} \lesssim \left\| \sigma ; S^b_{\rho,\delta}\right\|_{M} \cdot \left\| \tau ; S^c_{\rho,\delta}\right\|_{M},$$ where $M$ depends on $N$, $b$, $c$, $\delta$ and $\rho$. See Stein [@stein; @1993 Chapter VII, Section 5.8]. The estimate for the symbols just above can be found in Kumano-go [@kumanogo; @1981 Lemma 2.4]. For $m \in L^\infty ({\mathbb{R}}^n)$, we write the associated Fourier multiplier operator as $$m(D) f = {\mathcal{F}}^{-1} \left[m \cdot {\mathcal{F}}f \right]$$ for $ f \in {\mathcal{S}}({\mathbb{R}}^n) $, and especially the Bessel potential as $(I - \Delta )^{s/2} f = {\mathcal{F}}^{-1} \left[(1 + | \cdot |^2)^{s/2} \cdot {\mathcal{F}}f \right]$ for $ f \in {\mathcal{S}}({\mathbb{R}}^n) $ and $s \in {\mathbb{R}}$. In the following, we recall the definitions and properties of function spaces which we will use. The Lebesgue space $L^p = L^p({\mathbb{R}}^n) $ is equipped with the (quasi)-norm $$\| f \|_{L^p} = \left( \int_{{\mathbb{R}}^n} \big| f(x) \big|^p dx \right)^{1/p}$$ for $0 < p < \infty$. If $p = \infty$, $\| f \|_{\infty} = \textrm{ess}\sup_{x\in{\mathbb{R}}^n} |f(x)|$. Moreover, for a compact subset $\Omega \subset {\mathbb{R}}^n$, $L^p_\Omega = L^p_\Omega({\mathbb{R}}^n) = \{ f \in L^p ({\mathbb{R}}^n) \cap {\mathcal{S}}^\prime ({\mathbb{R}}^n) : {\mathrm{supp}\hspace{0.5mm}}( {\mathcal{F}}f ) \subset \Omega \}$. For $0 < q \leq \infty$, we denote by $\ell^q$ the set of all complex number sequences $\{ a_k \}_{ k \in {\mathbb{Z}}^n }$ such that $$\| \{ a_k \}_{ k\in{\mathbb{Z}}^n } \|_{ \ell^q } = \left( \sum_{ k \in {\mathbb{Z}}^n } | a_k |^q \right)^{ 1 / q } < \infty,$$ if $q < \infty$, and $\| \{ a_k \}_{ k\in{\mathbb{Z}}^n } \|_{ \ell^\infty } = \sup_{k \in {\mathbb{Z}}^n} |a_k| < \infty$ if $q = \infty$. For the sake of simplicity, we will write $ \| a_k \|_{ \ell^q } $ instead of the more correct notation $ \| \{ a_k \}_{ k\in{\mathbb{Z}}^n } \|_{ \ell^q } $. For a function space $X$, we denote by $\mathcal{L}(X)$ the space of all bounded linear operators on $X$. We end this subsection with stating the following lemmas from [@triebel; @1983]. \[[[@triebel; @1983 Section 1.5.3]]{}\] \[convolution 0&lt;p&lt;1\] Let $0 < p \leq 1$. Then we have $$\| f \ast g \|_{L^p} \leq C R^{n(1/p-1)} \| f \|_{L^p} \| g \|_{L^p}$$ for all $f,g \in L^p_\Omega$, where $\Omega = \{ x \in {\mathbb{R}}^n : | x - x_0 | \leq R \}$ and the constant $C > 0$ is independent of $x_0$ and $R$. \[maximal inequality\] Let $0 < p \leq \infty$. If $0 < r < p$, then we have $$\left\| \sup_{y \in {\mathbb{R}}^n } \frac{ | f (x-y) | }{ 1 + | R y |^{n/r} } \right\|_{L^p({\mathbb{R}}^n_x ) } \leq C \| f \|_{L^p}$$ for all $f \in L^p_\Omega$, where $\Omega = \{ x \in {\mathbb{R}}^n : | x - x_0 | \leq R \}$ and the constant $C>0$ is independent of $x_0$ and $R$. $\alpha$-modulation spaces {#sec2.2} -------------------------- We give the definition of $\alpha$-modulation spaces and their basic properties. Let $C > 1 $ be a constant which depends on the space dimension and $\alpha \in [0,1)$. Suppose that a sequence of Schwartz functions $\{ \eta_k^\alpha \}_{k\in{\mathbb{Z}}^n}$ satisfies that - ${\mathrm{supp}\hspace{0.5mm}}\eta_k^\alpha \subset \left\{ \xi \in {\mathbb{R}}^n : \left| \xi - \langle k \rangle^{\alpha/(1-\alpha)}k \right| \leq C \langle k \rangle^{\alpha/(1-\alpha)} \right\}$; - $\left| \partial^\beta_\xi \eta_k^\alpha (\xi) \right| \leq C'_\beta \langle k \rangle^{-|\beta| \alpha/(1-\alpha)}$ for every multi-index $\beta \in {\mathbb{Z}}^n_+$; - $\displaystyle{\sum_{k\in{\mathbb{Z}}^n}} \eta_k^\alpha (\xi) = 1$ for any $\xi \in {\mathbb{R}}^n$. Then, for $0< p, q \leq \infty$, $s\in{\mathbb{R}}$, and $\alpha\in[0,1)$, we denote the $\alpha$-modulation space $M_{p,q}^{s,\alpha} $ by $$M_{p,q}^{s,\alpha} ({\mathbb{R}}^n) = \left\{ f\in{{\mathcal{S}}}^\prime({\mathbb{R}}^n) : \left\| f \right\|_{M^{s, \alpha}_{p,q}} = \left\| \langle k \rangle^{s / (1-\alpha)} \left\| \eta_k^\alpha(D) f \right\|_{L^p} \right\|_{\ell^q({\mathbb{Z}}^n_k)} < +\infty \right\}.$$ See Borup and Nielsen [@borup; @nielsen; @JMAA; @2006; @borup; @nielsen; @AM; @2006] for the abstract definition including the end point case $\alpha = 1$. We remark that $M_{p,q}^{s,\alpha}$ is a quasi-Banach space (a Banach space if $1 \leq p,q \leq \infty$) and ${\mathcal{S}}\subset M_{p,q}^{s,\alpha} \subset {\mathcal{S}}^\prime$. In particular, ${\mathcal{S}}$ is dense in $M_{p,q}^{s,\alpha}$ for $0 < p,q < \infty$ (see Borup and Nielsen [@borup; @nielsen; @MN; @2006]). Moreover, if we choose different decompositions satisfying the conditions above, they determine equivalent (quasi)-norms of $\alpha$-modulation spaces, so that the definition of $\alpha$-modulation spaces is independent of the choice of the sequence $\{ \eta_k^\alpha \}_{k\in{\mathbb{Z}}^n}$. Next, we recall some basic properties of $\alpha$-modulation spaces. \[lift op\] Let $0 < p,q \leq \infty$, $s, t \in{\mathbb{R}}$ and $0\leq \alpha < 1$. Then the mapping $ (I-\Delta)^{t/2} : M_{p,q}^{s,\alpha} \hookrightarrow M_{p,q}^{s-t,\alpha}$ is isomorphic. The proof of Proposition \[lift op\] is similar to that for Besov spaces in [@triebel; @1983 Section 2.3.8]. One can find the explicit proof in [@kato; @2017 Appendix A]. \[[[@han; @wang; @2014 Proposition 6.1]]{}\] \[equivalent norm 0\] Let $0 < p,q \leq \infty$, $s \in {\mathbb{R}}$ and $0\leq\alpha<1$. Let a smooth radial bump function $\varrho$ satisfy that $\varrho (\xi) = 1$ on $|\xi| < 1$, and $\varrho (\xi) = 0$ on $|\xi| \geq 2$. Then, we have $$\| f \|_{M_{p,q}^{s,\alpha}} \sim \left\| \langle k \rangle^{s/(1-\alpha)} \left\| \varrho_k^\alpha(D) f \right\|_{L^p} \right\|_{\ell^q ({\mathbb{Z}}^n_k)}$$ for all $f \in M_{p,q}^{s,\alpha}$, where $$\varrho_k^\alpha(\xi) = \varrho \left( \frac{\xi - \langle k \rangle^{ \alpha/(1-\alpha)} k}{C\langle k \rangle^{ \alpha/(1-\alpha)}} \right).$$ Here, the constant $C > 1$ is the same as in the definition of the sequence $\{ \eta_k^\alpha \}_{k\in{\mathbb{Z}}^n}$. \[maximal inequality for alpha modulation\] Let $0 < p \leq \infty$ and $0 \leq \alpha < 1 $. If $0 < r < p$, then we have $$\label{max ineq alpha} \left\| \sup_{y \in {\mathbb{R}}^n} \frac{ \left| \left[ \eta_k^\alpha(D) f \right] (x-y) \right| } { 1 + \left( \langle k \rangle^{ \alpha/(1-\alpha)} | y | \right)^{n/r} } \right\|_{L^p({\mathbb{R}}^n_x)} \lesssim \left\| \eta_k^\alpha(D) f \right\|_{L^p}$$ for all $f \in {\mathcal{S}}^\prime ({\mathbb{R}}^n) $ and all $k \in {\mathbb{Z}}^n$. It follows from the definition of the decomposition $\{ \eta_k^\alpha \}_{k\in{\mathbb{Z}}^n}$ that $${\mathrm{supp}\hspace{0.5mm}}{\mathcal{F}}\left[ \eta_k^\alpha(D) f \right] \subset \left\{ \xi \in {\mathbb{R}}^n : \left| \xi - \langle k \rangle^{ \alpha/(1-\alpha) } k \right| \leq C \langle k \rangle^{ \alpha/(1-\alpha) } \right\},$$ so that Lemma \[maximal inequality for alpha modulation\] follows from Proposition \[maximal inequality\]. Taking the $\ell^q({\mathbb{Z}}^n_k)$ (quasi)-norm of both sides of , we have for $0 \leq \alpha < 1 $, $0 < p,q \leq \infty$ and $0 < r < p$ $$\left\| \left\| \sup_{y \in {\mathbb{R}}^n} \frac{ \left| \left[ \eta_k^\alpha(D) f \right] (x-y) \right| }{ 1 + \left( \langle k \rangle^{ \alpha/(1-\alpha) } | y | \right)^{n/r} } \right\|_{L^p({\mathbb{R}}^n_x)} \right\|_{\ell^q({\mathbb{Z}}^n_k)} \lesssim \left\| f \right\|_{ M_{p,q}^{0,\alpha} }.$$ We end this subsection by stating the definition of modulation spaces, which arise as special $\alpha$-modulation spaces for the choice $\alpha=0$. Another definition and basic properties of modulation spaces can be found in [@feichtinger; @1983; @grochenig; @2001; @kobayashi; @2006; @kobayashi; @2007; @wang; @hudzik; @2007]. Let a sequence of Schwartz functions $\{ \varphi_k \}_{k\in{\mathbb{Z}}^n}$ satisfy that $${\mathrm{supp}\hspace{0.5mm}}\varphi \subset \left\{ \xi \in {\mathbb{R}}^n : | \xi | \leq \sqrt{ n } \right\} {\quad\mathrm{and}\quad}\sum_{k\in{\mathbb{Z}}^n} \varphi_k (\xi) = 1 \textrm{ for any } \xi \in {\mathbb{R}}^n,$$ where $\varphi_k = \varphi(\cdot - k ) $. Then, for $0 < p,q \leq \infty$ and $s \in {\mathbb{R}}$, we denote the modulation space $M_{ p , q }^s$ by $$M_{p,q}^s ({\mathbb{R}}^n) = \left\{ f\in{{\mathcal{S}}}^\prime({\mathbb{R}}^n) : \left\| f \right\|_{M^{s}_{p,q}} = \Big\| \langle k \rangle^{s} \left\| \varphi_k(D) f \right\|_{L^p} \Big\|_{\ell^q({\mathbb{Z}}^n_k)} < +\infty \right\}.$$ We finally note that $M_{ p , q }^{s,0} = M_{ p , q }^s$. Lemmas {#sec3} ====== As stated in the beginning of Section \[sec2\], we will use the notation $A = \frac{ \alpha }{ 1 - \alpha }$ in the remainder of the paper. In this section, we prepare some lemmas to use in the proof of Theorem \[main theorem\]. As mentioned in the end of Section \[sec1\], we may assume $\sigma \in {\mathcal{S}}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$ in the following statements. We remark that for the partition of unity $\{ \varphi_\ell \}_{\ell \in {\mathbb{Z}}^n}$ used to construct modulation spaces, it holds that $$\sum_{\ell \in {\mathbb{Z}}^n} \varphi_\ell \left( \xi \right) =1 \textrm{ for any } \xi \in {\mathbb{R}}^n \quad\Longrightarrow\quad \sum_{\ell \in {\mathbb{Z}}^n} \varphi_\ell \left( \frac{\xi}{\langle m \rangle^A } \right) = 1 \textrm{ for any } \xi \in {\mathbb{R}}^n \textrm{ and } m \in {\mathbb{Z}}^n.$$ Then, we can decompose the symbols $\sigma$ as $$\begin{aligned} \sigma(x,\xi) = \sum_{ m \in {\mathbb{Z}}^n} \sigma(x,\xi) \cdot \eta_m^\alpha (\xi) = \sum_{ \ell,m \in {\mathbb{Z}}^n} \left( \varphi_\ell \left( \frac{D_x}{\langle m \rangle^A } \right) \sigma \right) (x,\xi) \cdot \eta_m^\alpha (\xi) ,\end{aligned}$$ where $\{ \eta_m^\alpha \}_{m \in {\mathbb{Z}}^n}$ is the partition of unity used for defining the $\alpha$-modulation spaces. Put $$\label{def of sigma} \sigma_{\ell,m}(x,\xi) = \left( \varphi_\ell \left( \frac{D_x}{\langle m \rangle^A } \right) \sigma \right) (x,\xi) \cdot \eta_m^\alpha (\xi).$$ Then, we have $\sigma_{\ell,m} \in {\mathcal{S}}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$ and also by Proposition \[equivalent norm 0\] $$\begin{aligned} \label{teinei} \begin{split} \left\| \sigma(X,D) f \right\|_{M_{p,q}^{s,\alpha}} &\sim \left\| \langle k \rangle^{\frac{s}{1-\alpha}} \left\| \varrho_k^\alpha(D) \left[ \sigma(X,D) f \right] \right \|_{L^p} \right\|_{\ell^q ({\mathbb{Z}}^n_k)} \\ &= \left\| \langle k \rangle^{\frac{ s }{1-\alpha}} \left\| \sum_{\ell,m \in {\mathbb{Z}}^n} \varrho_k^\alpha(D) \left[ \sigma_{\ell,m} (X,D) f \right] \right\|_{L^p} \right\|_{\ell^q ({\mathbb{Z}}^n_k)}. \end{split}\end{aligned}$$ In the following, we investigate some properties of $\varrho_k^\alpha(D) \left[ \sigma_{\ell,m} (X,D) f \right]$. We first determine the relations among $k,\ell,m \in {\mathbb{Z}}^n$ by considering the support of $ {\mathcal{F}}[\sigma_{\ell,m} (X,D) f ] $. \[region\] It holds that $${\mathrm{supp}\hspace{0.5mm}}{\mathcal{F}}[\sigma_{\ell,m} (X,D) f ] \subset \left\{ \zeta \in {\mathbb{R}}^n : \left| \zeta - \langle m \rangle^A (\ell + m ) \right| \leq (C+ \sqrt{ n } )\langle m \rangle^A \right\}$$ for all $f \in {\mathcal{S}}({\mathbb{R}}^n)$ and all $\ell,m \in {\mathbb{Z}}^n$. Here, $C$ is the constant in the definition of the functions $\{ \eta_k^\alpha \}_{k\in{\mathbb{Z}}^n}$. Furthermore, it holds that $${\mathrm{supp}\hspace{0.5mm}}\varrho_k^\alpha \cap {\mathrm{supp}\hspace{0.5mm}}{\mathcal{F}}\left[ \sigma_{\ell,m} (X,D) f \right] \neq \varnothing \quad\Longrightarrow \quad | k - m | \lesssim \langle \ell \rangle$$ for all $f \in {\mathcal{S}}({\mathbb{R}}^n)$ and all $k,\ell,m \in {\mathbb{Z}}^n$. \[cor region\] Lemma \[region\] implies that $\varrho_k^\alpha(D) \left[ \sigma_{\ell,m} (X,D) f \right] $ always vanishes unless $ | k - m | \lesssim \langle \ell \rangle $ is satisfied. Before beginning with the proof of Lemma \[region\], we prepare one lemma. \[k and m\] It holds that $$(\langle k \rangle ^A + \langle m \rangle^A ) | k - m | \lesssim \left| \langle k \rangle^A k - \langle m \rangle^A m \right|$$ for all $k,m \in {\mathbb{Z}}^n$. By the symmetry of $k, m \in {\mathbb{Z}}^n$ in the desired inequality, we may assume that $|k| \ge |m|$, and divide the argument into three cases. [Case 1:]{} $|k| \geq 2|m|$. Since $$(\langle k \rangle ^A + \langle m \rangle^A ) | k - m | \lesssim \langle k \rangle ^A | k |$$ and $$\left| \langle k \rangle^A k - \langle m \rangle^A m \right| \geq \langle k \rangle^A |k| - \left( 1 + \frac{ | k | }{2} \right)^A \cdot \frac{ | k | }{2} \gtrsim \langle k \rangle^A | k |,$$ we have $ (\langle k \rangle ^A + \langle m \rangle^A ) | k - m | \lesssim \left| \langle k \rangle^A k - \langle m \rangle^A m \right| $. [Case 2:]{} $|k|=|m|$. Obviously, $(\langle k \rangle ^A + \langle m \rangle^A ) | k - m | = 2 | \langle k \rangle ^A k - \langle m \rangle^A m | $. [Case 3:]{} $|m| < |k| < 2|m|$ ($\Rightarrow 1 < \langle k \rangle / \langle m \rangle < 2$). Note that $$\label{k m equiv} | k - m | \leq \left| \frac{ \langle k \rangle^A }{ \langle m \rangle^A } k - m \right|$$ holds in this case. In fact, if $A=0$ ($\Leftrightarrow \alpha=0$), then holds obviously true. Assume that $0 < A < \infty$ ($\Leftrightarrow 0 < \alpha < 1$). Since $\langle k \rangle^A / \langle m \rangle^A > 1 $, we have the following equivalences by squaring both sides of the just above estimate and by rewriting the euclidean norm $|x|^2 = x \cdot x$ in terms of the standard inner product on ${\mathbb{R}}^n$: $$\begin{aligned} && | k - m | \leq \left| \frac{ \langle k \rangle^A }{ \langle m \rangle^A }k - m \right| \\ &\Longleftrightarrow& 2 \left( \frac{ \langle k \rangle^A }{ \langle m \rangle ^A } -1 \right) k \cdot m \leq \left( \frac{ \langle k \rangle^{2A} }{ \langle m \rangle ^{2A} } -1 \right ) | k |^2 \\ &\Longleftrightarrow& 2 k \cdot m \leq \left( \frac{ \langle k \rangle^{A} }{ \langle m \rangle ^{A} } + 1 \right) | k |^2.\end{aligned}$$ The last statement is justified from the facts $k \cdot m < | k |^2$ and $2 < \frac{ \langle k \rangle^{A} }{ \langle m \rangle ^{A} } + 1 $. Therefore, it follows that $$\begin{aligned} (\langle k \rangle^A + \langle m \rangle^A ) | k - m | & \sim \langle m \rangle^A | k - m | \leq \langle m \rangle^A\left| \frac{ \langle k \rangle^A }{ \langle m \rangle^A } k - m \right| = | \langle k \rangle ^A k - \langle m \rangle^A m | .\end{aligned}$$ Gathering all the cases, we obtain the desired estimate. Now, we start the proof of Lemma \[region\]. We first consider the former part, that is, the support of ${\mathcal{F}}\left[ \sigma_{\ell,m} (X,D) f \right] $. By the Fubini–Tonelli theorem and the definition of $\sigma_{\ell,m}$ in , we have $$\begin{aligned} {\mathcal{F}}[\sigma_{\ell,m} (X,D) f ] (\zeta) & = (2\pi)^{-n} \int_{{\mathbb{R}}^n_\xi} \widehat f (\xi) \int_{{\mathbb{R}}^n_x} e^{- i x \cdot ( \zeta - \xi ) } \sigma_{\ell,m} (x,\xi) dx d\xi \\ & = (2\pi)^{-n} \int_{{\mathbb{R}}^n_\xi} \eta_m^\alpha(\xi) \cdot \widehat f (\xi) \int_{{\mathbb{R}}^n_x} e^{ - i x \cdot ( \zeta - \xi ) } \left( \varphi_\ell \left( \frac{D_x}{\langle m \rangle^A } \right) \sigma \right) (x,\xi) dx d\xi \\ &= (2\pi)^{-n} \int_{{\mathbb{R}}^n_\xi} \eta_m^\alpha(\xi) \cdot \varphi \left( \frac{ \zeta - \xi }{\langle m \rangle^A } - \ell \right) \cdot \big( {\mathcal{F}}_x \sigma \big) ( \zeta - \xi , \xi) \cdot \widehat f (\xi) d\xi,\end{aligned}$$ where ${\mathcal{F}}_x \sigma$ is the partial Fourier transform of $\sigma(x,\xi)$ with respect to the $x$-variable. Hence, the facts $$\begin{aligned} {\mathrm{supp}\hspace{0.5mm}}\eta_m^\alpha &\subset \left\{ \xi \in {\mathbb{R}}^n : \left| \xi - \langle m \rangle^A m \right| \leq C \langle m \rangle^A \right\} ; \\ {\mathrm{supp}\hspace{0.5mm}}\varphi \left( \frac{ \cdot }{\langle m \rangle^A } - \ell \right) & \subset \left\{ \xi \in {\mathbb{R}}^n : \left| \xi - \langle m \rangle^A \ell \right| \leq \sqrt{ n } \, \langle m \rangle^A \right\} \end{aligned}$$ yield that $$\label{support of 2nd} {\mathrm{supp}\hspace{0.5mm}}{\mathcal{F}}[\sigma_{\ell,m} (X,D) f ] \subset \left\{ \zeta \in {\mathbb{R}}^n : \left| \zeta - \langle m \rangle^A (\ell + m ) \right| \leq (C+ \sqrt{ n } )\langle m \rangle^A \right\}.$$ This is the former part of this lemma. Next, we consider the latter part. Assume that $ {\mathrm{supp}\hspace{0.5mm}}\varrho_k^\alpha \cap {\mathrm{supp}\hspace{0.5mm}}{\mathcal{F}}\left[ \sigma_{\ell,m} (X,D) f \right] \neq \varnothing, $ and recall from Proposition \[equivalent norm 0\] that $$\label{support of 1st} {\mathrm{supp}\hspace{0.5mm}}\varrho_k^\alpha \subset \left\{ \zeta \in {\mathbb{R}}^n : \left| \zeta - \langle k \rangle^A k \right| \leq 2C\langle k \rangle^A \right\}.$$ Then, combining with , we obtain $$\left| \langle k \rangle^A k - \langle m \rangle^A (\ell + m ) \right| \lesssim \langle m \rangle^A + \langle k \rangle^A ,$$ which implies that $$\left| \langle k \rangle^A k - \langle m \rangle^A m \right| \lesssim (\langle m \rangle^A + \langle k \rangle^A ) \langle \ell \rangle .$$ Hence, we conclude from Lemma \[k and m\] $$| k - m | \lesssim \frac{1}{\langle k \rangle ^A + \langle m \rangle^A } \cdot \left| \langle k \rangle^A k - \langle m \rangle^A m \right| \lesssim \langle \ell \rangle,$$ which completes the proof. We next prove that $\sigma_{\ell,m} (X,D) f$ has high decay rate with respect to $| \ell |$. For technical purposes to prove our main theorem, we slightly change the formulation of $\sigma_{\ell,m} (X,D) f $ as follows. Choose a function $\kappa \in {\mathcal{S}}({\mathbb{R}}^n)$ satisfying that $ \kappa (\xi) = 1$ on $|\xi| \leq 1$ and $\kappa (\xi) = 0$ on $|\xi| \geq 2$, and set $$\kappa_m^\alpha(\xi) = \kappa \left( \frac{\xi - \langle m \rangle^A m }{ C\langle m \rangle^A } \right)$$ with the constant $C>1$ as in the definition of $\alpha$-modulation spaces (see Section \[sec2.2\]). Then, $\kappa_m^\alpha = 1$ on the support of $\eta_m^\alpha$ and thus $$\begin{aligned} \label{change to tilde sigma} \begin{split} [ \sigma_{\ell,m} (X,D) f ] (x) &= (2\pi)^{-n} \int_{{\mathbb{R}}^n_\xi} e^{i x \cdot \xi} \left( \varphi_\ell \left( \frac{D_x}{\langle m \rangle^A } \right) \sigma \right) (x,\xi) \cdot \eta_m^\alpha (\xi) \cdot \widehat f (\xi) d\xi \\ &= (2\pi)^{-n} \int_{{\mathbb{R}}^n_\xi} e^{i x \cdot \xi} \left( \varphi_\ell \left( \frac{D_x}{\langle m \rangle^A } \right) \sigma \right) (x,\xi) \cdot \eta_m^\alpha (\xi) \kappa_m^\alpha (\xi) \cdot \widehat f (\xi) d\xi \\ &= [ \widetilde \sigma_{\ell,m} (X,D) \eta_m^\alpha(D) f ](x) , \end{split}\end{aligned}$$ where $ \widetilde \sigma_{\ell,m} (x,\xi) = \left( \varphi_\ell \left( \frac{D_x}{\langle m \rangle^A } \right) \sigma \right) (x,\xi) \cdot \kappa_m^\alpha (\xi)$. For the symbol $\widetilde \sigma_{\ell,m}$, we have the following lemma. \[L\^p to L\^p\] Let $0 < p \leq \infty$. For an arbitrary integer $N \in {\mathbb{Z}}_+$, there exists a constant $N^\prime \in {\mathbb{Z}}_+$ such that $$\left\| \widetilde \sigma_{\ell,m} (X,D) \eta_m^\alpha(D) f \right\|_{L^p} \lesssim \langle \ell \rangle^{-N} \| \sigma ; S^{0}_{\alpha, \alpha} \|_{N^\prime} \cdot \left\| \eta_m^\alpha(D) f \right\|_{L^p}$$ holds for all $\sigma \in S_{\alpha, \alpha}^0 ({\mathbb{R}}^n \times {\mathbb{R}}^n)$, all $f \in {\mathcal{S}}({\mathbb{R}}^n)$ and all $\ell,m \in {\mathbb{Z}}^n$. Before starting the proof, we prepare one lemma. In the first step of the proof of Lemma \[L\^p to L\^p\], we will use the following estimate: $$\left| [ \widetilde \sigma_{\ell,m} (X,D) \eta_m^\alpha(D) f ](x) \right| \leq (2\pi)^{-n} \int_{{\mathbb{R}}^n_y} \left| \eta_m^\alpha(D) f (y) \right| \cdot \left| \int_{{\mathbb{R}}^n_\xi} e^{i ( x - y ) \cdot \xi} \widetilde \sigma_{\ell,m} (x,\xi) d\xi \right| dy ,$$ which is justified by the Fubini–Tonelli theorem and the fact $\widetilde \sigma_{\ell,m} \in {\mathcal{S}}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$. In view of this estimate, the following lemma will be helpful. \[decay of l and x-y\] For arbitrary integers $M,N \in {\mathbb{Z}}_+$, we have $$\left| \int_{{\mathbb{R}}^n_\xi} e^{i y \cdot \xi} \widetilde \sigma_{\ell,m}(x,\xi) d\xi \right| \lesssim \langle \ell \rangle^{-N} \, \| \sigma ; S^{0}_{\alpha, \alpha} \|_{M+N} \cdot \frac{ \langle m \rangle^{An} }{\left(1+\langle m \rangle^A |y| \right)^{M} }$$ for all $\sigma \in S_{\alpha, \alpha}^0 ({\mathbb{R}}^n \times {\mathbb{R}}^n)$, all $x,y \in {\mathbb{R}}^n$ and all $\ell, m \in {\mathbb{Z}}^n$. In order to obtain the decay with respect to $|y|$, we will use integration by parts with respect to the $\xi$-variable, so that we first consider the derivatives of $\widetilde \sigma_{\ell,m}$. For any multi-index $\gamma \in {\mathbb{Z}}_+^n$ with $| \gamma | = M$ and $N \in {\mathbb{Z}}_+$, we have $$\left| \partial_\xi^\gamma \left( \widetilde \sigma_{\ell,m} (x,\xi) \right) \right| \lesssim \langle \ell \rangle ^{-N} \langle m \rangle^{-A M } \, \| \sigma ; S^{0}_{\alpha, \alpha} \|_{M+N} \cdot \chi_{\{ \xi\in{\mathbb{R}}^n: | \xi - \langle m \rangle^A m | \leq 2C \langle m \rangle^A \} }(\xi),$$ where $\chi_\Omega$ is the characteristic function on the set $\Omega$. In fact, the Leibniz rule yields that $$\label{derivative of sigma} \left| \partial_\xi^\gamma \left( \widetilde \sigma_{\ell,m} (x,\xi) \right) \right| \lesssim \sum_{\beta \leq \gamma} \left| \left( \varphi_\ell \left( \frac{D_x}{\langle m \rangle^A } \right) \left(\partial_\xi^\beta \sigma \right)\right) (x,\xi) \right| \cdot \left| \left( \partial_{\xi}^{\gamma - \beta}\kappa \right) \left(\frac{\xi - \langle m \rangle^A m}{ C \langle m \rangle^A } \right) \right| \cdot \langle m \rangle^{ - A | \gamma - \beta | }.$$ Let us fix $\beta \leq \gamma$ for the moment. Next, note that $\langle \xi \rangle \sim \langle m \rangle^{ \frac{ 1 }{ 1 - \alpha } }$ if $\xi \in {\mathrm{supp}\hspace{0.5mm}}\left( \partial_{\xi}^{\gamma - \beta}\kappa \right) \left(\frac{\cdot - \langle m \rangle^A m}{ C \langle m \rangle^A } \right)$ in . Then, using the Fubini–Tonelli theorem, we have $$\begin{aligned} \left| \left( \varphi_\ell \left( \frac{D_x}{\langle m \rangle^A } \right) \left(\partial_\xi^\beta \sigma \right)\right) (x,\xi) \right| &= (2\pi)^{-n} \left| \int_{{\mathbb{R}}^n_z } \left(\partial_\xi^\beta \sigma \right) (z , \xi ) \int_{{\mathbb{R}}^n_{\zeta}} e^{i ( x - z ) \cdot \zeta } \varphi \left( \frac{\zeta}{\langle m \rangle^A } -\ell \right) d\zeta dz \right| \\ &= (2\pi)^{-n} \langle m \rangle^{An} \left| \int_{{\mathbb{R}}^n_z } \left(\partial_\xi^\beta \sigma \right) (z , \xi ) \int_{{\mathbb{R}}^n_{\zeta}} e^{i \langle m \rangle^A ( x - z ) \cdot ( \zeta + \ell ) } \varphi \left( \zeta \right) d\zeta dz \right| \\ &= \langle m \rangle^{An} \left| \int_{{\mathbb{R}}^n_z } e^{ - i \langle m \rangle^A \ell \cdot z } \left(\partial_\xi^\beta \sigma \right) (z , \xi ) \cdot \check \varphi \left( \langle m \rangle^A ( x - z ) \right) dz \right| , \end{aligned}$$ where, in the second identity, we used the changes of variables: $\zeta' = \zeta / \langle m \rangle^A - \ell $. If $\ell \neq 0$, we apply an $N$-fold integration by parts with respect to the $z$-variable to obtain $$\begin{aligned} & \left| \int_{{\mathbb{R}}^n_z } e^{ - i \langle m \rangle^A \ell \cdot z } \left(\partial_\xi^\beta \sigma \right) (z , \xi ) \cdot \check \varphi \left( \langle m \rangle^A ( x - z ) \right) dz \right| \\ &\lesssim \left( \langle m \rangle^A | \ell | \right)^{-N} \sum_{ \substack{ \widetilde \beta \leq \widetilde \gamma \\ | \widetilde \gamma | = N } } \langle m \rangle^{ A | \widetilde \gamma - \widetilde \beta | } \int_{{\mathbb{R}}^n_z } \left| \left( \partial_{z}^{\widetilde \beta} \partial_\xi^\beta \sigma \right) (z , \xi ) \right| \cdot \left| \big(\partial_z^{ \widetilde \gamma - \widetilde \beta} \check \varphi \big) \big( \langle m \rangle^A ( x - z ) \big) \right| dz \\ &\lesssim \left( \langle m \rangle^A | \ell | \right)^{-N} \sum_{ \substack{ \widetilde \beta \leq \widetilde \gamma \\ |\widetilde \gamma | = N } } \langle m \rangle^{ A | \widetilde \gamma - \widetilde \beta | } \cdot \langle m \rangle^{A|\widetilde \beta| - A | \beta | } \, \| \sigma ; S^{0}_{\alpha, \alpha} \|_{M+N} \cdot \langle m \rangle^{-An} \\ &\sim \langle \ell \rangle^{-N} \langle m \rangle^{-A|\beta|-An} \, \| \sigma ; S^{0}_{\alpha, \alpha} \|_{M+N}.\end{aligned}$$ Here, we used $ \langle \xi \rangle^\alpha \sim \langle m \rangle^{ A }$ in the second inequality, and the facts $| \widetilde \beta | + | \widetilde\gamma - \widetilde\beta | = | \widetilde\gamma| =N$ and $ | \ell | \sim \langle \ell \rangle$ for $| \ell | \geq 1$ in the last equivalence. On the other hand, if $\ell = 0$, then it similarly follows that $$\left| \int_{{\mathbb{R}}^n_z } e^{ - i \langle m \rangle^A \ell \cdot z } \left( \partial_\xi^\beta \sigma \right) (z , \xi ) \cdot \check \varphi \left( \langle m \rangle^A ( x - z ) \right) dz \right| \lesssim \langle m \rangle^{-A|\beta|-An} \, \| \sigma ; S^{0}_{\alpha, \alpha} \|_{M}.$$ Hence, we obtain $$\label{derivative phi} \left| \left( \varphi_\ell \left( \frac{D_x}{\langle m \rangle^A } \right) \left(\partial_\xi^\beta \sigma \right) \right) (x,\xi) \right| \lesssim \langle \ell \rangle^{-N} \langle m \rangle^{-A|\beta|} \, \| \sigma ; S^{0}_{\alpha, \alpha} \|_{M+N}$$ for all $\ell \in{\mathbb{Z}}^n$. Substituting into , we have $$\begin{aligned} \left| \partial_\xi^\gamma \left( \widetilde \sigma_{\ell,m} (x,\xi) \right) \right| &\lesssim \sum_{\beta \leq \gamma} \langle \ell \rangle^{-N} \langle m \rangle^{-A|\beta|} \, \| \sigma ; S^{0}_{\alpha, \alpha} \|_{M+N} \cdot \left| \left( \partial_{\xi}^{\gamma - \beta}\kappa \right) \left(\frac{\xi - \langle m \rangle^A m}{ C \langle m \rangle^A } \right) \right| \cdot \langle m \rangle^{ - A | \gamma - \beta | } \\ &\lesssim \langle \ell \rangle^{-N} \langle m \rangle^{-A M} \, \| \sigma ; S^{0}_{\alpha, \alpha} \|_{M+N} \cdot \chi_{\{ \xi\in{\mathbb{R}}^n: | \xi - \langle m \rangle^A m | \leq 2C \langle m \rangle^A \} } (\xi),\end{aligned}$$ where we used the identity $|\beta|+ |\gamma - \beta | = | \gamma | = M$ to obtain the last inequality. This concludes the result in this step. Next, we actually investigate the decay of the given integral with respect to $|y|$ and obtain the desired estimate for arbitrary $M , N \in {\mathbb{Z}}_+$. Obviously, using the conclusion in Step 1 with $\gamma=0$, we have $$\begin{aligned} \label{y=0} \begin{split} \left| \int_{{\mathbb{R}}^n_\xi} e^{i y \cdot \xi} \widetilde \sigma_{\ell,m}(x,\xi) d\xi \right| &\lesssim \langle \ell \rangle^{-N} \, \| \sigma ; S^{0}_{\alpha, \alpha} \|_{N} \cdot \int_{{\mathbb{R}}^n_\xi} \chi_{\{ \xi\in{\mathbb{R}}^n: | \xi - \langle m \rangle^A m | \leq 2C \langle m \rangle^A \} } (\xi) d\xi \\ & \lesssim \langle \ell \rangle^{-N} \langle m \rangle^{An} \, \| \sigma ; S^{0}_{\alpha, \alpha} \|_{N} . \end{split}\end{aligned}$$ On the other hand, by an $M$-fold integration by parts with respect to the $\xi$-variable, we get $$\begin{aligned} \label{y neq 0} \begin{split} \left| \int_{{\mathbb{R}}^n_\xi} e^{i y \cdot \xi} \widetilde \sigma_{\ell,m}(x,\xi) d\xi \right| &\leq | y |^{-M} \sum_{|\gamma|=M} \int_{{\mathbb{R}}^n_\xi} \left| \partial_\xi^\gamma \left( \widetilde \sigma_{\ell,m} (x,\xi) \right) \right| d\xi \\ &\lesssim | y |^{-M} \langle \ell \rangle^{-N} \langle m \rangle^{-AM} \, \| \sigma ; S^{0}_{\alpha, \alpha} \|_{M+N} \int_{{\mathbb{R}}^n_\xi} \chi_{\{ \xi\in{\mathbb{R}}^n: | \xi - \langle m \rangle^A m | \leq 2C \langle m \rangle^A \} } (\xi) d\xi \\ &\sim \langle \ell \rangle^{-N} \langle m \rangle^{An} \, \| \sigma ; S^{0}_{\alpha, \alpha} \|_{M+N} \left(\langle m \rangle^{A} | y |\right)^{-M} \end{split}\end{aligned}$$ for $y \neq 0$. Combining the conclusions and , we obtain $$\left| \int_{{\mathbb{R}}^n_\xi} e^{i y \cdot \xi} \widetilde \sigma_{\ell,m}(x,\xi) d\xi \right| \lesssim \langle \ell \rangle ^{-N} \, \| \sigma ; S^{0}_{\alpha, \alpha} \|_{M+N} \cdot \frac{ \langle m \rangle^{An} } { \left( 1 + \langle m \rangle^A | y | \right)^{M} }$$ for all $x, y \in {\mathbb{R}}^n$ and all $\ell, m \in {\mathbb{Z}}^n$. We are now in a position to prove Lemma \[L\^p to L\^p\]. We choose $M = ( n + 1) + [n/r] + 1$ for $0 < r < p \leq \infty$ in Lemma \[decay of l and x-y\]. Then we have by the Fubini–Tonelli theorem and Lemma \[decay of l and x-y\] $$\begin{aligned} \left| [ \widetilde \sigma_{\ell,m} (X,D) \eta_m^\alpha(D) f ](x) \right| &\leq (2\pi)^{-n} \int_{{\mathbb{R}}^n_y} \left| \eta_m^\alpha(D) f (y) \right| \cdot \left| \int_{{\mathbb{R}}^n_\xi} e^{i ( x - y ) \cdot \xi} \widetilde \sigma_{\ell,m} (x,\xi) d\xi \right| dy \\ &\lesssim \langle \ell \rangle ^{-N} \, \| \sigma ; S^{0}_{\alpha, \alpha} \|_{N'} \int_{{\mathbb{R}}^n_y} \left| \eta_m^\alpha(D) f (y) \right| \cdot \frac{ \langle m \rangle^{An} }{ \left( 1 + \langle m \rangle^A | x-y | \right)^{( n + 1) + [n/r] +1} } dy \\ &\lesssim \langle \ell \rangle ^{-N} \, \| \sigma ; S^{0}_{\alpha, \alpha} \|_{N'} \cdot \sup_{y\in{\mathbb{R}}^n} \frac{ \left| \eta_m^\alpha(D) f (x-y) \right| }{ 1 + \left( \langle m \rangle^A | y | \right)^{ n/r} } \cdot \int_{{\mathbb{R}}^n_y} \frac{ \langle m \rangle^{An} }{ \left( 1 + \langle m \rangle^A | y | \right)^{ n + 1 } } dy \\ &\sim \langle \ell \rangle ^{-N} \, \| \sigma ; S^{0}_{\alpha, \alpha} \|_{N'} \cdot \sup_{y\in{\mathbb{R}}^n} \frac{ \left| \eta_m^\alpha(D) f (x-y) \right| }{ 1 + \left( \langle m \rangle^A | y | \right)^{ n/r} } ,\end{aligned}$$ where $N'= n + [n/r] + 2 + N \in {\mathbb{Z}}_+$. Then, taking the $L^p$ (quasi)-norm of both sides and applying Lemma \[maximal inequality for alpha modulation\], we obtain $$\left\| \widetilde \sigma_{\ell,m} (X,D) \eta_m^\alpha(D) f \right\|_{L^p } \lesssim \langle \ell \rangle ^{-N} \| \sigma ; S^{0}_{\alpha, \alpha} \|_{N'} \cdot \left\| \eta_m^\alpha(D) f \right\|_{L^p } ,$$ which is the desired result. Proof of the main theorems {#sec4} ========================== In this section, we prove Theorem \[main theorem\] and Corollaries \[main cor\] and \[sharp cor\] stated in Section \[sec1\]. We first give a proof of Theorem \[main theorem\]. To this end, we prepare two facts. The first one is as follows. \[convolution 0&lt;p&lt;1 alpha modulation\] Let $ 0 < p \leq \infty$. Then we have $$\begin{aligned} \left\| \varrho_k^\alpha(D) \left[ \widetilde \sigma_{\ell,m} (X,D) \eta_m^\alpha(D) f \right] \right \|_{L^p} \lesssim \langle \ell \rangle^{ A n ( \frac{ 1 }{ \min(1,p) } - 1 ) } \left\| \widetilde \sigma_{\ell,m} (X,D) \eta_m^\alpha(D) f \right \|_{L^p}\end{aligned}$$ for all $f \in {\mathcal{S}}({\mathbb{R}}^n)$ and all $k,\ell,m \in {\mathbb{Z}}^n$. The case $1\leq p \leq \infty$ follows from the Young inequality. Assume that $0 < p < 1$. We recall from Proposition \[equivalent norm 0\] and Lemma \[region\] that $$\begin{aligned} {\mathrm{supp}\hspace{0.5mm}}\varrho_k^\alpha &\subset \left\{ \zeta \in {\mathbb{R}}^n : \left| \zeta - \langle k \rangle^A k \right| \leq 2C\langle k \rangle^A \right\} ; \\ {\mathrm{supp}\hspace{0.5mm}}{\mathcal{F}}[\sigma_{\ell,m} (X,D) f ] &\subset \left\{ \zeta \in {\mathbb{R}}^n : \left| \zeta - \langle m \rangle^A (\ell + m ) \right| \leq (C+ \sqrt{ n } )\langle m \rangle^A \right\},\end{aligned}$$ and from that $$[ \sigma_{\ell,m} (X,D) f ] (x) = [ \widetilde \sigma_{\ell,m} (X,D) \eta_m^\alpha(D) f ](x).$$ Combining these, we see that $\varrho_k^\alpha(D) \left[ \widetilde \sigma_{\ell,m} (X,D) \eta_m^\alpha(D) f \right] $ always vanishes unless $$\left| \langle m \rangle^A (\ell + m ) - \langle k \rangle^A k \right| \leq 2C\langle k \rangle^A + (C+ \sqrt{ n } )\langle m \rangle^A.$$ Hence, in those cases when $\varrho_k^\alpha(D) \left[ \widetilde \sigma_{\ell,m} (X,D) \eta_m^\alpha(D) f \right] $ does not vanish identically, we obtain $$\begin{aligned} {\mathrm{supp}\hspace{0.5mm}}{\mathcal{F}}[\sigma_{\ell,m} (X,D) f ] &\subset \left\{ \zeta \in {\mathbb{R}}^n : \left| \zeta - \langle k \rangle^A k \right| \leq 2C\langle k \rangle^A + 2(C+ \sqrt{ n } )\langle m \rangle^A \right\} ;\\ {\mathrm{supp}\hspace{0.5mm}}\varrho_k^\alpha &\subset \left\{ \zeta \in {\mathbb{R}}^n : \left| \zeta - \langle k \rangle^A k \right| \leq 2C\langle k \rangle^A + 2(C+ \sqrt{ n } )\langle m \rangle^A \right\}.\end{aligned}$$ Moreover, recalling Lemma \[region\] (or Remark \[cor region\]), we have $| k - m | \lesssim \langle \ell \rangle$, which implies $\langle m \rangle^A \lesssim \langle k \rangle^A + \langle \ell \rangle^A$. Hence, we have by Proposition \[convolution 0&lt;p&lt;1\] $$\begin{aligned} \left\| \varrho_k^\alpha(D) \left[ \widetilde \sigma_{\ell,m} (X,D) \eta_m^\alpha(D) f \right] \right \|_{L^p} &\lesssim \left( \langle k \rangle^A + \langle m \rangle^A \right)^{ n ( \frac{ 1 }{ p } - 1 ) } \left\| {\mathcal{F}}^{-1} [\varrho_k^\alpha] \right\|_{L^p} \cdot \left\| \widetilde \sigma_{\ell,m} (X,D) \eta_m^\alpha(D) f \right \|_{L^p} \\ &\lesssim \left( \langle k \rangle^A + \langle \ell \rangle^A \right)^{ n ( \frac{ 1 }{ p } - 1 ) } \langle k \rangle^{ - A n ( \frac{1}{p} - 1 ) } \cdot \left\| \widetilde \sigma_{\ell,m} (X,D) \eta_m^\alpha(D) f \right \|_{L^p} \\ &\lesssim \langle \ell \rangle^{ A n ( \frac{ 1 }{ p } - 1 ) } \left\| \widetilde \sigma_{\ell,m} (X,D) \eta_m^\alpha(D) f \right \|_{L^p},\end{aligned}$$ which completes the proof. As the second preparation for the proof of Theorem \[main theorem\], we note that $$\begin{aligned} \label{triangle} \begin{split} \left\| \sum_{\ell\in{\mathbb{Z}}^n} f_{\ell} (x) \right\|_{L^p({\mathbb{R}}^n_x)} &\leq \left( \sum_{\ell\in{\mathbb{Z}}^n} \left\| f_{\ell} (x) \right\|_{L^p({\mathbb{R}}^n_x)}^{\min(1,p)} \right)^{\frac{1}{\min(1,p)}}; \\ \left\| \sum_{\ell\in{\mathbb{Z}}^n} a_{k,\ell} \right\|_{\ell^q({\mathbb{Z}}^n_k)} &\leq \left( \sum_{\ell\in{\mathbb{Z}}^n} \left\| a_{k,\ell} \right\|_{\ell^q({\mathbb{Z}}^n_k)}^{\min(1,q)} \right)^{\frac{1}{\min(1,q)}} \end{split}\end{aligned}$$ hold for $0 < p,q \leq \infty$. For $p \geq 1$ or $q \geq 1$, these are just the triangle inequality. For $0 < p < 1$ or $ 0 < q < 1$, these estimates follow from the fact that $| \sum a_k |^p \leq \sum | a_k |^p$, i.e., the embedding $\ell^p \hookrightarrow \ell^1$. Now, we begin with the proof of Theorem \[main theorem\]. Due to Proposition \[lift op\] and the symbolic calculus (see Section \[sec2.1\]), it suffices to prove Theorem \[main theorem\] only for $s=0$. In fact, if $\sigma \in S^0_{\alpha, \alpha}$, then there is a symbol $\tau \in S^0_{\alpha, \alpha}$ such that $J^s \sigma (X,D) J^{-s} = \tau (X,D)$ and the estimate $ \| \tau ; S^0_{\alpha, \alpha} \|_{L} \leq C \| \sigma ; S^0_{\alpha, \alpha} \|_{L'} $ holds, where the constants $C$ and $L'$ depend on $L,s,n$. Here, we set $J = (I-\Delta)^{1/2}$. Hence, assuming that Theorem \[main theorem\] (or more precisely ) with $s=0$ holds, we have by Proposition \[lift op\] for some $L,L'$ $$\begin{aligned} \left\| \sigma (X,D) f \right\|_{ M_{ p , q }^{ s , \alpha } } \sim \left\| J^s \sigma (X,D) J^{-s} J^s f \right\|_{ M_{ p , q }^{ 0 , \alpha } } &= \left\| \tau (X,D) \left[ J^s f \right] \right\|_{ M_{ p , q }^{ 0 , \alpha } } \\ &\lesssim \| \tau ; S^0_{\alpha, \alpha} \|_{L} \left\| J^s f \right\|_{ M_{ p , q }^{ 0 , \alpha } } \lesssim \| \sigma ; S^0_{\alpha, \alpha} \|_{L'} \left\| f \right\|_{ M_{ p , q }^{ s , \alpha } }.\end{aligned}$$ We actually prove Theorem \[main theorem\] for $s =0$. We first estimate the $L^p$ (quasi)-norm of $\varrho_k^\alpha(D) [ \sigma(X,D) f ] $. Set $p^\ast = \min (1,p)$. Then, we have $$\begin{aligned} \left\| \varrho_k^\alpha(D) \left[ \sigma(X,D) f \right] \right\|_{L^p} &= \left\| \sum_{ \ell\in {\mathbb{Z}}^n } \sum_{ \substack{ m \in {\mathbb{Z}}^n \\ | k - m | \lesssim \langle \ell \rangle } } \varrho_k^\alpha(D) \left[ \widetilde \sigma_{\ell,m} (X,D) \eta_m^\alpha(D) f \right] \right \|_{L^p} \\ &\leq \left( \sum_{ \ell\in {\mathbb{Z}}^n } \sum_{ \substack{ m \in {\mathbb{Z}}^n \\ | k - m | \lesssim \langle \ell \rangle } } \left\| \varrho_k^\alpha(D) \left[ \widetilde \sigma_{\ell,m} (X,D) \eta_m^\alpha(D) f \right] \right \|_{L^p}^{p^\ast} \right)^{1/p^\ast} \\ &\lesssim \left( \sum_{ \ell\in {\mathbb{Z}}^n } \sum_{ \substack{ \widetilde m \in {\mathbb{Z}}^n \\ | \widetilde m | \lesssim \langle \ell \rangle } } \langle \ell \rangle^{ \left\{ A n ( \frac{ 1 }{ p^\ast } - 1 ) - N \right\} \cdot p^\ast} \left\|\sigma ; S^0_{\alpha,\alpha}\right\|_{N'}^{p^\ast} \left\| \eta_{k-\widetilde m}^\alpha(D) f \right \|_{L^p}^{p^\ast} \right)^{1/p^\ast}\end{aligned}$$ for all $k\in{\mathbb{Z}}^n$, where $N' = N'_{n,p,N} \in {\mathbb{Z}}_+$ is the constant given in Lemma \[L\^p to L\^p\]. Also, we applied , and Lemma \[region\] to obtain the first line. In the second line, we used the first estimate in . In the last line, we invoked Lemmas \[L\^p to L\^p\] and \[convolution 0&lt;p&lt;1 alpha modulation\]. Next, recalling Proposition \[equivalent norm 0\] and taking the $\ell^q ({\mathbb{Z}}^n_k)$ (quasi)-norm of the above estimate, we obtain $$\begin{aligned} \left\| \sigma(X,D) f \right\|_{M^{0,\alpha}_{p,q}} &\lesssim \left\|\sigma ; S^0_{\alpha,\alpha}\right\|_{N'} \left\| \sum_{ \ell\in {\mathbb{Z}}^n } \sum_{ \substack{ \widetilde m \in {\mathbb{Z}}^n \\ | \widetilde m | \lesssim \langle \ell \rangle } } \langle \ell \rangle^{ \left\{ A n ( \frac{ 1 }{ p^\ast } - 1 ) - N \right\}\cdot p^\ast} \left\| \eta_{k-\widetilde m}^\alpha(D) f \right \|_{L^p}^{p^\ast} \right\|_{\ell^{q/p^\ast} ({\mathbb{Z}}^n_k)}^{1/p^\ast} \\ &\leq \left\|\sigma ; S^0_{\alpha,\alpha}\right\|_{N'} \left( \sum_{ \ell\in {\mathbb{Z}}^n } \sum_{ \substack{ \widetilde m \in {\mathbb{Z}}^n \\ | \widetilde m | \lesssim \langle \ell \rangle } } \langle \ell \rangle^{ \left\{ A n ( \frac{ 1 }{ p^\ast } - 1 ) - N \right\} \cdot \min(1,p,q)} \Big\| \left\| \eta_{k-\widetilde m}^\alpha(D) f \right \|_{L^p} \Big\|_{\ell^{q}({\mathbb{Z}}^n_k)}^{\min (1,p,q) } \right)^{\frac{1}{\min(1, p, q)}} \\ &\lesssim \left\|\sigma ; S^0_{\alpha,\alpha}\right\|_{N'} \left\| f \right\|_{M_{p,q}^{0,\alpha} } \left(\sum_{ \ell\in {\mathbb{Z}}^n } \langle \ell \rangle^{ \left\{ A n ( \frac{ 1 }{ p^\ast } - 1 ) - N \right\} \cdot \min(1,p,q) + n } \right)^{\frac{1}{\min(1, p, q)}},\end{aligned}$$ where in the second inequality we used the second inequality in and the identity $$p^\ast \cdot \left( \frac{q}{p^\ast} \right)^\ast = p^\ast \cdot \min \left(1, \frac{q}{p^\ast} \right) = \min (p^\ast, q) = \min (1,p,q).$$ Therefore, choosing $N \in {\mathbb{Z}}_+$ such that $ \left\{ A n ( \frac{ 1 }{ p^\ast } - 1 ) - N \right\} \cdot \min(1,p,q) + n < -n$, then we have $$\left\| \sigma(X,D) f \right\|_{M^{0,\alpha}_{p,q}} \lesssim \left\|\sigma ; S^0_{\alpha,\alpha}\right\|_{N'} \| f \|_{M_{p,q}^{0,\alpha} },$$ which completes the proof of the main theorem. Next, we prove Corollary \[main cor\]. This is immediately given from Theorem \[main theorem\] and the symbolic calculus. We observe that if $\sigma \in S_{\rho, \delta}^b$, then $\sigma \in S_{\alpha, \alpha}^b$, since $0 \leq \delta \leq \alpha \leq \rho$. Here, recall that $0\leq \alpha <1$. Then, the symbolic calculus shows that there is a symbol $\tau \in S_{\alpha, \alpha}^0$ such that $J^{-b} \sigma(X,D) = \tau(X,D)$, where $J = (I-\Delta)^{1/2}$. Thus, Theorem \[main theorem\] shows that $\| \tau(X,D) f \|_{M_{p,q}^{s,\alpha}} \lesssim \| f \|_{M_{p,q}^{s,\alpha}}$ for all $f \in {\mathcal{S}}({\mathbb{R}}^n)$. In combination with Proposition \[lift op\], this implies that $$\| \sigma(X,D) f \|_{M_{p,q}^{s-b,\alpha}} \sim \| J^{-b} \sigma(X,D) f \|_{M_{p,q}^{s,\alpha}} = \| \tau(X,D) f \|_{M_{p,q}^{s,\alpha}} \lesssim \| f \|_{M_{p,q}^{s,\alpha}}.$$ This completes the proof. We next prove Corollary \[sharp cor\]. In order to achieve this, we recall the following inclusion relations between modulation spaces and $\alpha$-modulation spaces given by [@han; @wang; @2014 Theorem 4.1] and [@toft; @wahlberg; @2012 Section 1]. \[embedding 0-1\] Let $0 < q \leq \infty$ and $0 \leq \alpha < 1$. [(1)]{} $M_{2,q}^{s_1,\alpha} ({\mathbb{R}}^n) \subset M_{2,q}^{0} ({\mathbb{R}}^n)$ holds for $s_1 = n \alpha \cdot \max(0, 1/q-1/2)$; [(2)]{} $M_{2,q}^{0} ({\mathbb{R}}^n) \subset M_{2,q}^{s_2,\alpha} ({\mathbb{R}}^n)$ holds for $s_2 = n \alpha \cdot \min(0, 1/q-1/2)$. Now, let us start the proof. The “IF” part immediately follows from the relation $S^{0}_{\rho, \delta} \subset S^{0}_{\alpha, \alpha}$ for $\delta \leq \alpha \leq \rho$ and Theorem \[main theorem\], so that we only consider the “ONLY IF” part. We assume that all $\sigma(X,D) \in {\mathrm{Op}}(S_{\rho, \delta}^0)$ are bounded on $M_{2,q}^{s,\alpha} ({\mathbb{R}}^n)$. Then all $\widetilde{\sigma}(X,D) \in {\mathrm{Op}}(S_{\rho, \delta}^{-s_1+s_2})$ are also bounded on $M^0_{2,q} ({\mathbb{R}}^n)$, where $s_1,s_2$ are as in Lemma \[embedding 0-1\]. Indeed, the symbolic calculus shows that $J^{- s + s_1} \widetilde \sigma (X,D) J^{s - s_2} \in {\mathrm{Op}}(S_{\rho, \delta}^0)$ for $\widetilde{\sigma} \in S_{\rho, \delta}^{-s_1+s_2}$, where $J = (I-\Delta)^{1/2}$. Therefore, we have by Proposition \[lift op\] and Lemma \[embedding 0-1\] $$\begin{aligned} \left\| \widetilde \sigma (X,D) f \right\|_{M_{2,q}^{0}} &\lesssim \left\| \widetilde \sigma (X,D) f \right\|_{M_{2,q}^{s_1,\alpha}} \lesssim \left\| J^{- s + s_1} \widetilde \sigma (X,D) f \right\|_{M_{2,q}^{s, \alpha}} \\ &= \left\| J^{- s + s_1} \widetilde \sigma (X,D) J^{s - s_2} J^{-s + s_2} f \right\|_{M_{2,q}^{s, \alpha}} \lesssim \left\| J^{ - s + s_2} f \right\|_{M_{2,q}^{s, \alpha}} \sim \left\| f \right\|_{M_{2,q}^{s_2, \alpha}} \lesssim \left\| f \right\|_{M_{2,q}^{0}} .\end{aligned}$$ This yields that ${\mathrm{Op}}(S_{\rho, \delta}^{-s_1+s_2}) \subset \mathcal{L} \big(M_{2,q}^{0} ({\mathbb{R}}^n) \big)$, and thus Theorem A gives $- s_1 + s_2 \leq - | 1/q - 1/2 | \delta n$. Here, since $-s_1+s_2 = -n\alpha|1/q-1/2|$, this is equivalent to $0 \leq - | 1/q - 1/2 | ( \delta - \alpha ) n $. Hence, because of $q \neq 2$, we obtain $\delta \leq \alpha$, which concludes the proof of “ONLY IF” part in Corollary \[sharp cor\]. \[alpha -\] In this remark, we find a counterexample to the inclusion $ {\mathrm{Op}}(S_{\alpha - \varepsilon, \alpha - \varepsilon}^0) \subset \mathcal{L}(M_{p,q}^{s,\alpha}) $ for $0 < \varepsilon < \alpha$ and $0 < p < 1$. We write $A_\varepsilon = \frac{\alpha - \varepsilon}{ 1 - \alpha}$ for $0 < \varepsilon < \alpha$. Choose $\psi , \widetilde \psi \in {\mathcal{S}}( {\mathbb{R}}^n ) $ satisfying that ${\mathrm{supp}\hspace{0.5mm}}\psi \subset \{ \xi \in {\mathbb{R}}^n : | \xi | \leq c \}$, $\widetilde \psi (\xi) = 1$ on $\{ \xi \in {\mathbb{R}}^n : | \xi | \leq c \}$ and ${\mathrm{supp}\hspace{0.5mm}}\widetilde \psi \subset \{ \xi \in {\mathbb{R}}^n : | \xi | \leq 2c \}$. Here, the constant $c=c_\alpha > 0$ is so small that the sets $B_k$, $k\in{\mathbb{Z}}^n$, are pairwise disjoint, where $B_k = \{ \xi \in {\mathbb{R}}^n: | \xi-\langle k \rangle^A k | \leq 2 c \langle k \rangle^A \}$. We will see in Remark \[small c\] that such a constant $c>0$ exists. Set $$\sigma(\xi) = \sum_{m \in {\mathbb{Z}}^n} \psi \left( \frac{ \xi-\langle m \rangle^A m}{ \langle m \rangle^{A_\varepsilon} } \right) {\quad\mathrm{and}\quad}\widehat{ f_{\ell} } (\xi) = \widetilde \psi \left( \frac{ \xi - \langle \ell \rangle^A \ell}{ \langle \ell \rangle^{A} } \right)$$ for all $\ell \in {\mathbb{Z}}^n$. Here, $\psi ( \frac{ \cdot - \langle m \rangle^A m}{ \langle m \rangle^{A_\varepsilon} } ) \subset B_m$, since $A > A_\varepsilon $. Then, it follows that $$\label{psi2} {\mathrm{supp}\hspace{0.5mm}}\psi \left( \frac{ \cdot - \langle m \rangle^A m}{ \langle m \rangle^{A_\varepsilon} } \right) \cap {\mathrm{supp}\hspace{0.5mm}}\widetilde\psi \left( \frac{ \cdot - \langle \ell \rangle^A \ell }{ \langle \ell \rangle^{A} } \right) \subset B_m \cap B_\ell = \varnothing {\quad\textrm{if}\quad}m \neq \ell$$ and $$\label{psi3} \widetilde \psi \left( \frac{ \xi - \langle m \rangle^A m}{ \langle m \rangle^{A} } \right) = 1 \textrm{ on } {\mathrm{supp}\hspace{0.5mm}}\psi \left( \frac{ \cdot - \langle m \rangle^A m}{ \langle m \rangle^{A_\varepsilon} } \right) .$$ In addition, note that at most one term in the sum defining $\sigma$ is non-zero for each $\xi$, since $$\label{psi4} {\mathrm{supp}\hspace{0.5mm}}\psi \left( \frac{ \cdot - \langle m \rangle^A m}{ \langle m \rangle^{A_\varepsilon} } \right) \cap {\mathrm{supp}\hspace{0.5mm}}\psi \left( \frac{ \cdot - \langle m^\prime \rangle^A m^\prime }{ \langle m^\prime \rangle^{A_\varepsilon} } \right) \subset B_m \cap B_{m'} = \varnothing {\quad\textrm{if}\quad}m \neq m^\prime.$$ Then, since $\langle \xi \rangle \sim \langle m \rangle^{ \frac{ 1 }{ 1 - \alpha } }$ if $\xi \in {\mathrm{supp}\hspace{0.5mm}}\psi \left( \frac{ \cdot - \langle m \rangle^A m}{ \langle m \rangle^{A_\varepsilon} } \right)$ and $\sigma$ is the $x$-independent symbol, we see that $\sigma \in S_{\alpha - \varepsilon, \delta}^0$ for any $0 \leq \delta \leq 1$. In particular, $\sigma \in S_{\alpha - \varepsilon, \alpha - \varepsilon}^0$. Now, by using these functions $\sigma$ and $f_\ell$, we will prove that ${\mathrm{Op}}(S_{\alpha - \varepsilon, \alpha - \varepsilon}^0) \subset \mathcal{L}(M_{p,q}^{s,\alpha})$ does not hold for $0 < p < 1$. We first estimate the $\alpha$-modulation space quasi-norm of $f_\ell$. By Proposition \[equivalent norm 0\], we have $$\begin{aligned} \| f_\ell \|_{M_{p,q}^{s,\alpha}} &\sim \left\| \langle k \rangle^{s/(1-\alpha)} \left\| \varrho_k^\alpha(D) f_\ell \right\|_{L^p} \right\|_{\ell^q ({\mathbb{Z}}^n_k)} \\ &= \Bigg\| \langle k \rangle^{\frac{s}{1-\alpha}} \left\| \left ({\mathcal{F}}^{-1} \left[ \varrho_k^\alpha \right] \right) \ast \left({\mathcal{F}}^{-1} \left[ \widetilde \psi \left( \frac{ \cdot - \langle \ell \rangle^A \ell}{ \langle \ell \rangle^{A} } \right)\right] \right) \right\|_{L^p} \Bigg\|_{\ell^q ({\mathbb{Z}}^n_k:| k - \ell | \lesssim 1)}\end{aligned}$$ for all $\ell \in {\mathbb{Z}}^n$. Here, the summation in $\ell^q({\mathbb{Z}}^n_k)$ is restricted to $| k - \ell | \lesssim 1$ (otherwise, $\varrho_k^\alpha(D) f_\ell$ vanishes). This restriction is due to the relation: $$\label{fellrhok} |\langle k \rangle^A k - \langle \ell \rangle^A \ell | \lesssim \langle k \rangle^A+\langle \ell \rangle^A,$$ which is obtained from the information of ${\mathrm{supp}\hspace{0.5mm}}\varrho_k^\alpha$ and ${\mathrm{supp}\hspace{0.5mm}}\widehat{f_\ell}$: $$|\xi-\langle k \rangle^A k| \lesssim \langle k \rangle^A {\quad\mathrm{and}\quad}|\xi-\langle \ell \rangle^A \ell| \lesssim \langle \ell \rangle^A,$$ and Lemma \[k and m\]. Then, since this restriction leads $\langle \ell \rangle \sim \langle k \rangle$, it follows that $${\mathrm{supp}\hspace{0.5mm}}\varrho_k^\alpha \subset \left\{ \xi \in {\mathbb{R}}^n : | \xi - \langle k \rangle^A k | \lesssim \langle k \rangle^A \right\} \subset \left\{ \xi \in {\mathbb{R}}^n : | \xi - \langle \ell \rangle^A \ell | \lesssim \langle \ell \rangle^A \right\}.$$ This is because we have by $$\begin{aligned} | \xi - \langle k \rangle^A k | \lesssim \langle k \rangle^A &\Longrightarrow | \xi - \langle \ell \rangle^A \ell | \lesssim \langle k \rangle^A +| \langle k \rangle^A k - \langle \ell \rangle^A \ell | \\&\Longrightarrow | \xi - \langle \ell \rangle^A \ell | \lesssim \langle \ell \rangle^A.\end{aligned}$$ Hence, Proposition \[convolution 0&lt;p&lt;1\] gives that $$\| f_\ell \|_{M_{p,q}^{s,\alpha}} \lesssim \Bigg\| \langle \ell \rangle^{\frac{s}{1-\alpha} + An(\frac{1}{p} - 1)} \left\| {\mathcal{F}}^{-1} \left[ \varrho_k^\alpha \right]\right\|_{L^p} \cdot \left\| {\mathcal{F}}^{-1} \left[ \widetilde \psi \left( \frac{ \cdot - \langle \ell \rangle^A \ell}{ \langle \ell \rangle^{A} } \right)\right] \right\|_{L^p} \Bigg\|_{\ell^q ({\mathbb{Z}}^n_k:| k - \ell | \lesssim 1)}.$$ Performing the changes of the variables for both $L^p$ (quasi)-norms and using the fact that $\langle \ell \rangle \sim \langle k \rangle$, we conclude that $$\begin{aligned} \| f_\ell \|_{M_{p,q}^{s,\alpha}} \lesssim \langle \ell \rangle^{\frac{s}{(1-\alpha)}} \cdot \langle \ell \rangle^{An(1 - \frac{1}{p})} \left\| 1 \right\|_{\ell^q ({\mathbb{Z}}^n_k:| k - \ell | \lesssim 1)} \sim \langle \ell \rangle^{\frac{s}{1-\alpha}} \cdot \langle \ell \rangle^{An(1 - \frac{1}{p})} \end{aligned}$$ for all $\ell \in {\mathbb{Z}}^n$. We next consider the $\alpha$-modulation space quasi-norm of $\sigma (X,D) f_\ell$. Using , and Proposition \[equivalent norm 0\], we have $$\begin{aligned} \| \sigma(X,D) f_\ell \|_{M_{p,q}^{s,\alpha}} &\sim \left\| \langle k \rangle^{\frac{s}{1-\alpha}} \left\| {\mathcal{F}}^{-1} \left[ \varrho_k^\alpha \cdot \left( \sum_{m \in {\mathbb{Z}}^n} \psi \left( \frac{ \cdot - \langle m \rangle^A m}{ \langle m \rangle^{A_\varepsilon} } \right) \right ) \cdot \widetilde \psi \left( \frac{ \cdot - \langle \ell \rangle^A \ell}{ \langle \ell \rangle^{A} } \right) \right] \right\|_{L^p} \right\|_{\ell^q ({\mathbb{Z}}^n_k)} \\ &\geq \langle \ell \rangle^{\frac{s}{1-\alpha}} \left\| {\mathcal{F}}^{-1} \left[ \varrho_\ell^\alpha \cdot \psi \left( \frac{ \cdot - \langle \ell \rangle^A \ell}{ \langle \ell \rangle^{A_\varepsilon} } \right) \right] \right\|_{L^p} .\end{aligned}$$ If we recall the definition of the function $\varrho$ in Proposition \[equivalent norm 0\], we see that $\varrho_\ell^\alpha (\xi) = 1 $ on the support of $\psi \left( \frac{ \cdot - \langle \ell \rangle^A \ell}{ \langle \ell \rangle^{A_\varepsilon} } \right)$ (possibly after shrinking the constant $c=c_\alpha$ further). Hence, we obtain $$\begin{aligned} \| \sigma(X,D) f_\ell \|_{M_{p,q}^{s,\alpha}} \gtrsim \langle \ell \rangle^{\frac{s}{1-\alpha}} \left\| {\mathcal{F}}^{-1} \left[ \psi \left( \frac{ \cdot - \langle \ell \rangle^A \ell}{ \langle \ell \rangle^{A_\varepsilon} } \right) \right] \right\|_{L^p} \sim \langle \ell \rangle^{\frac{s}{1-\alpha}} \cdot \langle \ell \rangle^{A_\varepsilon n ( 1 - \frac{1 }{p} ) }.\end{aligned}$$ We are now in position to prove the conclusion of this remark. We assume toward a contradiction that $\sigma (X,D)$ is bounded on $M_{p,q}^{s,\alpha}$. Then, we have $$\langle \ell \rangle^{\frac{s}{1-\alpha}} \cdot\langle \ell \rangle^{A_\varepsilon n (1-\frac{1}{p})} \lesssim \| \sigma(X,D) f_\ell \|_{M_{p,q}^{s,\alpha}} \lesssim \| f_\ell \|_{M_{p,q}^{s,\alpha}} \lesssim \langle \ell \rangle^{\frac{s}{1-\alpha}} \cdot\langle \ell \rangle^{An (1-\frac{1}{p})}$$ for all $\ell \in {\mathbb{Z}}^n$. However, since $A_\varepsilon < A$ and $0 < p < 1$, this is a contradiction. Therefore, $\sigma$ belongs to $S_{\alpha - \varepsilon, \alpha - \varepsilon}^0$, but $\sigma(X,D)$ is not bounded on $M_{p,q}^{s,\alpha}$. \[small c\] We determine the detail quantity of the small constant $c=c_\alpha > 0$ which was used to ensure that the sets $B_{k}$ in Remark \[alpha -\] are pairwise disjoint. Assume that $B_m \cap B_\ell \neq \varnothing$, which implies $$| \langle \ell \rangle^A \ell - \langle m \rangle^A m | \leq 2c (\langle \ell \rangle ^A + \langle m \rangle^A ) .$$ We here recall from Lemma \[k and m\] that $$(\langle \ell \rangle ^A + \langle m \rangle^A ) | \ell - m | \leq K \left| \langle \ell \rangle^A \ell - \langle m \rangle^A m \right|$$ holds for all $m,\ell \in {\mathbb{Z}}^n$, where the constant $K = K_\alpha > 0$ (careful reading gives that this constant is $\max (6, 1 + 2^A)$ at most). Then, we have $$| m - \ell | \leq K \left| \langle \ell \rangle^A \ell - \langle m \rangle^A m \right| / (\langle \ell \rangle ^A + \langle m \rangle^A ) \leq 2cK.$$ Choosing the constant $c$ satisfying $2cK < 1$, we have $|m-\ell|<1$, i.e., $m=\ell$. Acknowledgments {#acknowledgments .unnumbered} =============== The authors sincerely express deep gratitude to the anonymous referees for their careful reading and giving fruitful suggestions and comments. The first author is supported by Grant-in-Aid for JSPS Research Fellow (No. 17J00359). The second author is partially supported by Grant-in-aid for Scientific Research from JSPS (No. 16K05201). [n]{} Borup, L.: Pseudodifferential operators on $\alpha$-modulation spaces. J. Funct. Spaces Appl. [**2**]{}, 107–123 (2004). Borup, L., Nielsen, M.: Banach frames for multivariate $\alpha$-modulation spaces. J. Math. Anal. Appl. [**321**]{}, 880–895 (2006). Borup, L., Nielsen, M.: Boundedness for pseudodifferential operators on multivariate $\alpha$-modulation spaces. Ark. Mat. [**44**]{}, 241-259 (2006). Borup, L., Nielsen, M.: Nonlinear approximation in $\alpha$-modulation spaces. Math. Nachr. [**279**]{}, 101–120 (2006). Borup, L., Nielsen, M.: On anisotropic Triebel-Lizorkin type spaces, with applications to the study of pseudo-differential operators. J. Funct. Spaces Appl. [**6**]{} (2008), no. 2, 107–154. Bourdaud, G.: $L^p$ estimates for certain nonregular pseudodifferential operators. Comm. Partial Differential Equations [**7**]{}, 1023–1033 (1982). Calderón A.-P., Vaillancourt, R.: A class of bounded pseudo-differential operators. Proc. Nat. Acad. Sci. U.S.A. [**69**]{}, 1185–1187 (1972). Feichtinger, H.G.: Modulation spaces on locally compact Abelian groups. Technical Report, University of Vienna (1983). Gibbons, G.: Opérateurs pseudo-différentiels et espaces de Besov. C. R. Acad. Sci. Paris Sér. A-B [**286**]{}, A895–A897 (1978). Gröbner, P.: Banachräume glatter Funktionen und Zerlegungsmethoden. Ph.D. thesis, University of Vienna (1992). Gröchenig K.: [*Foundations of time-frequency analysis*]{}. Birkhäuser Boston (2001). Gröchenig, K., Heil, C.: Modulation spaces and pseudo differential operators. Integral Equations Operator Theory [**34**]{}, 439–457 (1999). Han, J., Wang, B.: $\alpha$-modulation spaces (I) scaling, embedding and algebraic properties. J. Math. Soc. Japan [**66**]{}, 1315–1373 (2014). Kato, T.: The inclusion relations between $\alpha$-modulation spaces and $L^p$-Sobolev spaces or local Hardy spaces. J. Funct. Anal. [**272**]{}, 1340–1405 (2017). Kobayashi, M.: Modulation spaces $M^{p,q}$ for $0 < p,q \leq \infty$. J. Funct. Spaces Appl. [**4**]{}, 329–341 (2006). Kobayashi, M.: Dual of modulation spaces. J. Funct. Spaces Appl. [**5**]{}, 1–8 (2007). Kumano-go, H.: [*Pseudo-differential operators*]{}. MIT Press, Cambridge (1981). Stein, E.M.: [*Harmonic Analysis*]{}. Princeton Univ. Press (1993). Sugimoto, M.: Pseudo-differential operators on Besov spaces. Tsukuba J. Math. [**12**]{}, 43–63 (1988). Sugimoto, M., Tomita, N.: A counterexample for boundedness of pseudo-differential operators on modulation spaces. Proc. Amer. Math. Soc. [**136**]{}, 1681–1690 (2008). Sugimoto, M., Tomita, N.: Boundedness properties of pseudo-differential and Calderón-Zygmund operators on modulation spaces. J. Fourier Anal. Appl. [**14**]{}, 124–143 (2008). Tachizawa, K.: The boundedness of pseudodifferential operators on modulation spaces. Math. Nachr. [**168**]{}, 263–277 (1994). Toft, J.: Continuity properties for modulation spaces, with applications to pseudo-differential calculus—I. J. Funct. Anal. [**207**]{}, 399–429 (2004). Toft, J., Wahlberg, P.: Embeddings of $\alpha$-modulation spaces. Pliska Stud. Math. Bulgar. [**21**]{}, 25–46 (2012). Triebel, H.: [*Theory of Function Spaces*]{}. Birkhäuser Verlag (1983). Wang, B., Hudzik, H.: The global Cauchy problem for the NLS and NLKG with small rough data. J. Differential Equations [**232**]{}, 36–73 (2007).

--- abstract: 'In this paper we continue the project of generalizing tilting theory to the category of contravariant functors $\mathrm{Mod}(\mathcal{C})$, from a skeletally small preadditive category $\mathcal{C}$ to the category of abelian groups, initiated in \[17\]. In \[18\] we introduced the notion of a a generalized tilting category $\mathcal{T}$, and extended Happel’s theorem to $\mathrm{Mod}(\mathcal{C})$. We proved that there is an equivalence of triangulated categories $D^{b}(\mathrm{Mod}(C))\cong D^{b}(\mathrm{Mod}(\mathcal{T}))$. In the case of dualizing varieties, we proved a version of Happel’s theorem for the categories of finitely presented functors. We also proved in this paper, that there exists a relation between covariantly finite coresolving categories, and generalized tilting categories. Extending theorems for artin algebras proved in \[4\], \[5\]. In this article we consider the category of maps, and relate tilting categories in the category of functors, with relative tilting in the category of maps. Of special interest is the category $\mathrm{mod}(\mathrm{mod}\Lambda)$ with $\Lambda$ an artin algebra.' address: - 'Instituto de Matemáticas UNAM, Unidad Morelia, Mexico' - 'Instituto de Matemáticas UNAM, Unidad Morelia, Mexico' author: - 'R. Martínez-Villa' - 'M. Ortiz-Morales' date: 'January 21, 2010' title: | Tilting theory and functor categories III.\ The Maps Category. --- [^1] [^2] [^3] Introduction and basic results ============================== This is the last article in a series of three in which, having in mind applications to the category of functors from subcategories of modules over a finite dimensional algebra to the category of abelian groups, we generalize tilting theory, from rings to functor categories. In the first paper \[17\] we generalized classical tilting to the category of contravariant functors from a preadditive skeletally small category $\mathcal{C}$, to the category of abelian groups and generalized Bongartz’s proof \[10\] of Brenner-Butler’s theorem \[11\]. We then applied the theory so far developed, to the study of locally finite infinite quivers with no relations, and computed the Auslander-Reiten components of infinite Dynkin diagrams. Finally, we applied our results to calculate the Auslander-Reiten components of the category of Koszul functors (see \[19\], \[20\], \[21\]) on a regular component of a finite dimensional algebra over a field. These results generalize the theorems on the preprojective algebra obtained in \[15\]. Following \[12\], in \[18\] we generalized the proof of Happel’s theorem given by Cline, Parshall and Scott: given a generalized tilting subcategory $\mathcal{T}$ of $\mathrm{Mod}(\mathcal{C})$, the derived categories of bounded complexes $D^{b}(\mathrm{Mod}(\mathcal{C}))$ and $D^{b}(\mathrm{Mod}(\mathcal{T}))$ are equivalent, and we discussed a partial converse \[14\]. We also saw that for a dualizing variety $\mathcal{C}$ and a tilting subcategory $\mathcal{T}\subset \mathrm{mod}(\mathcal{C})$ with pseudokerneles, the categories of finitely presented functors $\mathrm{mod}(\mathcal{C})$ and $\mathrm{mod}(\mathcal{T})$ have equivalent derived bounded categories, $D^{b}(\mathrm{mod}(\mathcal{C}))\cong D^{b}(\mathrm{mod}(\mathcal{T}))$. Following closely the results for artin algebras obtained in \[3\], \[4\], \[5\], by Auslander, Buchweits and Reiten, we end the paper proving that for a Krull-Schmidt dualizing variety $\mathcal{C}$, there are analogous relations between covariantly finite subcategories and generalized tilting subcategories of $\mathrm{mod}(\mathcal{C})$. This paper is dedicated to study tilting subcategories of $\mathrm{mod}(\mathcal{C})$. In order to have a better understanding of these categories, we use the relation between the categories $\mathrm{mod}(\mathcal{C})$ and the category of maps, $\mathrm{maps}(\mathcal{C})$, given by Auslander in \[1\]. Of special interest is the case when $\mathcal{C}$ is the category of finitely generated left $\Lambda $-modules over an artin algebra $\Lambda $, since in this case the category $\mathrm{maps}(\mathcal{C})$ is equivalent to the category of finitely generated $\Gamma $modules, $\mathrm{mod}(\Gamma ), $ over the artin algebra of triangular matrices $\Gamma =\left( \begin{array}{cc} \Lambda & 0 \\ \Lambda & \Lambda\end{array}\right) $. In this situation, tilting subcategories on $\mathrm{mod}(\mathrm{mod}(\Lambda) \mathcal{)}$ will correspond to relative tilting subcategories of $\mathrm{mod}(\Gamma )$, which in principle, are easier to compute. The paper consists of three sections: In the first section we establish the notation and recall some basic concepts. In the second one, for a variety of annuli with pseudokerneles $\mathcal{C}$, we prove that generalized tilting subcategories of $\mathrm{mod}(\mathcal{C})$ are in correspondence with relative tilting subcategories of $\mathrm{maps}(\mathcal{C})$ \[9\]. In the third section, we explore the connections between $\mathrm{mod}\ \Gamma $, with $\Gamma =\left( \begin{array}{cc} \Lambda & 0 \\ \Lambda & \Lambda\end{array}\right) $ and the category $\mathrm{mod}(\mathrm{mod}(\Lambda) )$. We compare the Auslander-Reiten sequences in $\mathrm{\mathrm{mod}}(\Gamma )$ with Auslander-Reiten sequences in $\mathrm{mod}(\mathrm{mod}(\Lambda) )$. We end the paper proving that, some important subcategories of $\mathrm{mod}(\mathcal{C})$ related with tilting, like: contravariantly, covariantly, functorially finite \[see 18\], correspond to subcategories of $\mathrm{maps}(\mathcal{C})$ with similar properties. Functor Categories ------------------ In this subsection we will denote by $\mathcal{C}$ an arbitrary skeletally small pre additive category, and $\mathrm{Mod}(\mathcal{C})$ will be the category of contravariant functors from $\mathcal{C}$ to the category of abelian groups. The subcategory of $\mathrm{Mod}(\mathcal{C})$ consisting of all finitely generated projective objects, $\mathfrak{p}(\mathcal{C})$, is a skeletally small additive category in which idempotents split, the functor $P:\mathcal{C}\rightarrow \mathfrak{p}(\mathcal{C})$, $P(C)=\mathcal{C}(-,C)$, is fully faithful and induces by restriction $\mathrm{res}:\mathrm{Mod}(\mathfrak{p}(\mathcal{C}))\rightarrow \mathrm{Mod}(\mathcal{C})$, an equivalence of categories. For this reason, we may assume that our categories are skeletally small, additive categories, such that idempotents split. Such categories were called **annuli varieties** in \[2\], for short, varieties. To fix the notation, we recall known results on functors and categories that we use through the paper, referring for the proofs to the papers by Auslander and Reiten \[1\], \[4\], \[5\]. Given a category $\mathcal{C}$ we will write for short, $\mathcal{C}(-,?)$ instead of $\mathrm{Hom}_{\mathcal{C}}(-,?)$ and when it is clear from the context we use just $(-,?).$ Given a variety $\mathcal{C}$, we say $\mathcal{C}$ has **pseudokernels**; if given a map $f : C_1\rightarrow C_0$, there exists a map $g : C_2 \rightarrow C_1$ such that the sequence of representable functors $\mathcal{C}(-, C_2 )\xrightarrow{(-,g)}\mathcal{C}( -,C_1 )\xrightarrow{(-,f)} \mathcal{C}(-, C_0 )$ is exact. A functor $M$ is **finitely presented**; if there exists an exact sequence $$\mathcal{C}( -,C_1 )\rightarrow \mathcal{C}(-, C_0 )\rightarrow M\rightarrow 0$$ We denote by $\mathrm{mod}(\mathcal{C})$ the full subcategory of $\mathrm{Mod}(\mathcal{C})$ consisting of finitely presented functors. It was proved in \[1\] $\mathrm{mod}(C)$ is abelian, if and only if, $\mathcal{C}$ has pseudokernels. Krull-Schmidt Categories ------------------------- We start giving some definitions from \[6\]. Let $R$ be a commutative artin ring. An $R$-variety $\mathcal{C}$, is a variety such that $\mathcal{C}(C_{1},C_{2})$ is an $R$-module, and composition is $R$-bilinear. Under these conditions $\mathrm{\mathrm{Mod}}(\mathcal{C})$ is an $R$-variety, which we identify with the category of contravariant functors $(\mathcal{C}^{op},\mathrm{Mod}(R))$. An $R$-variety $\mathcal{C}$ is $\mathrm{Hom}$-**finite**, if for each pair of objects $C_{1},C_{2}$ in $\mathcal{C},$ the $R$-module $\mathcal{C}(C_{1},C_{2})$ is finitely generated. We denote by $(\mathcal{C}^{op},\mathrm{mod}(R))$, the full subcategory of $(\mathcal{C}^{op},\mathrm{\mathrm{Mod}}(R))$ consisting of the $\mathcal{C}$-modules such that; for every $C$ in $\mathcal{C}$ the $R$-module $M(C)$ is finitely generated. The category $(\mathcal{C}^{op},\mathrm{mod}(R))$ is abelian and the inclusion $(\mathcal{C}^{op},\mathrm{mod}(R))\rightarrow (\mathcal{C}^{op},\mathrm{\mathrm{Mod}}(R))$ is exact. The category $\mathrm{mod}(C)$ is a full subcategory of $(\mathcal{C}^{op},\mathrm{mod}(R))$. The functors $D:(\mathcal{C}^{op},\mathrm{mod}(R))\rightarrow (\mathcal{C},\mathrm{mod}(R))$, and $D:(\mathcal{C},\mathrm{mod}(R))\rightarrow (\mathcal{C}^{op},\mathrm{mod}(R))$, are defined as follows: for any $C$ in $\mathcal{C}$, $D(M)(C)=\mathrm{Hom}_{R}(M(C),I(R/r)) $, with $r$ the Jacobson radical of $R$, and $I(R/r)$ is the injective envelope of $R/r$. The functor $D$ defines a duality between $(\mathcal{C},\mathrm{mod}(R))$ and $(\mathcal{C}^{op},\mathrm{mod}(R))$. If $\mathcal{C}$ is an $\mathrm{Hom}$-finite $R$-category and $M$ is in $\mathrm{mod}(\mathcal{C})$, then $M(C)$ is a finitely generated $R$-module and it is therefore in $\mathrm{mod}(R)$. An $\mathrm{Hom}$-finite $R$-variety $\mathcal{C}$ is **dualizing**, if the functor $$D:(\mathcal{C}^{op},\mathrm{mod}(R))\rightarrow (\mathcal{C},\mathrm{mod}(R))$$ induces a duality between the categories $\mathrm{mod}(\mathcal{C})$ and $\mathrm{mod}(\mathcal{C}^{op}).$ It is clear from the definition that for dualizing categories $\mathcal{C}$ the category $\mathrm{mod}(\mathcal{C})$ has enough injectives. To finish, we recall the following definition: An additive category $\mathcal{C}$ is **Krull-Schmidt**, if every object in $\mathcal{C}$ decomposes in a finite sum of objects whose endomorphism ring is local. In \[18 Theo. 2\] we see that for a dualizing Krull-Schmidt variety the finitely presented functors have projective covers. Let $\mathcal{C}$ a dualizing Krull-Schmidt $R$-variety. Then $\mathrm{mod}(\mathcal{C})$ is a dualizing Krull-Schmidt variety. Contravariantly finite categories --------------------------------- \[4\] Let $\mathscr X$ be a subcategory of $\mathrm{mod}(\mathcal{C})$, which is closed under summands and isomorphisms. A morphism $f:X\rightarrow M $ in $\mathrm{mod}(\mathcal{C})$, with $X$ in $\mathscr X$, is a *right* $\mathscr X$-*approximation* of $M$, if $(-,X)_{\mathscr X}\xrightarrow{(-,h)_\mathscr X}(-,M)_{\mathscr X}\rightarrow 0$ is an exact sequence, where $(-,?)_{\mathscr X}$ denotes the restriction of $(-,?)$ to the category $\mathscr X$. Dually, a morphism $g:M\rightarrow X$, with $X$ in $\mathscr X$, is a *left* $\mathscr X$-*approximation* of $M$, if $(X,-)_{\mathscr X}\xrightarrow{(g,-)_\mathscr X}(M,-)_{\mathscr X}\rightarrow 0$ is exact. A subcategory $\mathscr X$ of $\mathrm{mod}(\mathcal{C})$ is called *contravariantly* (*covariantly*) finite in $\mathrm{mod}(\mathcal{C})$, if every object $M$ in $\mathrm{mod}(\mathcal{C})$ has a right (left) $\mathscr X$-approximation; and *functorially finite*, if it is both contravariantly and covariantly finite. A subcategory $\mathscr X$ of $\mathrm{mod}(\mathcal{C})$ is *resolving* (*coresolving*), if it satisfies the following three conditions: (a) it is closed under extensions, (b) it is closed under kernels of epimorphisms (cokernels of monomorphisms), and (c) it contains the projective (injective) objects. Relative Homological Algebra and Frobenius Categories ----------------------------------------------------- In this subsection we recall some results on relative homological algebra introduced by Auslander and Solberg in \[9\],\[see also 14, 23\]. Let $\mathcal{C}$ be an additive category which is embedded as a full subcategory of an abelian category $\mathcal{A}$, and suppose that $\mathcal{C}$ is closed under extensions in $\mathcal{A}$. Let $\mathcal{S}$ be a collection of exact sequences in $\mathcal{A}$ $$0\rightarrow X\xrightarrow{f}Y\xrightarrow{g}Z\rightarrow 0$$$f$ is called an *admissible monomorphism*, and $g$ is called an *admissible epimorphism*. A pair $(\mathcal{C},\mathcal{S})$ is called an *exact category* provided that: (a) Any split exact sequence whose terms are in $\mathcal{C}$ is in $\mathcal{S}$. (b) The composition of admissible monomorphisms (resp., epimorphisms) is an admissible monomorphism (resp., epimorphism). (c) It is closed under pullbacks (pushouts) of admissible epimorphisms (admissible monomorphisms). Let $(\mathcal{C},\mathcal{S})$ be an exact subcategory of an abelian category $\mathcal{A}$. Since the collection $\mathcal{S}$ is closed under pushouts, pullbacks and Baer sums, it gives rise to a subfunctor $F$ of the additive bifunctor $\mathrm{Ext}_{\mathcal{C}}^{1}(-,-):\mathcal{C}\times \mathcal{C}^{op}\rightarrow \mathbf{Ab}$ \[9\]. Given such a functor $F$, we say that an exact sequence $\eta :0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ in $\mathcal{C}$ is $F$-exact, if $\eta $ is in $F(C,A)$, we will write some times $\mathrm{Ext}_{F}^{1}(-,?)$ instead of $F(-,?)$. An object $P$ in $\mathcal{C}$ is $F$-projective, if for each $F$-exact sequence $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$, the sequence $0\rightarrow (P,N)\rightarrow (P,E)\rightarrow (P,M)\rightarrow 0$ is exact. Analogously we have the definition of an $F$-injective object. If for any object $C$ in $\mathcal{C}$ there is an $F$-exact sequence $0\rightarrow A\rightarrow P\rightarrow C\rightarrow 0$, with $P$ an $F$-projective, then we say $(\mathcal{C},\mathcal{S})$ has enough $F$- projectives. Dually, if for any object $C$ in $\mathcal{C}$ there is an $F$-exact sequence $0\rightarrow C\rightarrow I\rightarrow A\rightarrow 0$, with $I$ an $F$-injective, then $(\mathcal{C},\mathcal{S})$ has enough $F-$ injectives. An exact category $(\mathcal{C},\mathcal{S})$ is called *Frobenius*, if the category $(\mathcal{C},\mathcal{S})$ has enough $F$-projectives and enough $F$-injectives and they coincide. Let $F$ be a subfunctor of $\mathrm{Ext}^1_{\mathcal{C}}(-,-)$. Suppose $F$ has enough projectives. Then for any $C$ in $\mathcal{C}$ there is an exact sequence in $\mathcal{C}$ of the form $$\cdots P_n\xrightarrow{d_n}P_{n-1}\xrightarrow{d_{n-1}}\cdots\rightarrow P_1\xrightarrow{d_1} P_0\xrightarrow{d_0} C\rightarrow 0$$ where $P_i$ is $F$-projective for $i\ge 0$ and $0\rightarrow \mathrm{Im}d_{i+1}\rightarrow P_i\rightarrow \mathrm{Im}d_{i}\rightarrow 0$ is $F$-exact for all $i\ge 0$. Such sequence is called an $F$-*exact projective resolution*. Analogously we have the definition of an $F$-*exact injective resolution*. When $(\mathcal{C},\mathcal{S})$ has enough $F$-injectives $($enough $F$- projectives), using $F$-exact injective resolutions (respectively, $F$-exact projective resolutions), we can prove that for any object $C$ in $\mathcal{C} $, ($A$ in $\mathcal{C}$ ), there exists a right derived functor of $\mathrm{Hom}_{\mathcal{C}}(C,-)$ ( $\mathrm{Hom}_{\mathcal{C}}(-,A)$ ). We denote by $\mathrm{Ext}_{F}^{i}(C,-)$ the right derived functors of $\mathrm{Hom}_{\mathcal{C}}(C,-)$ and by $\mathrm{Ext}_{F}^{i}(-,A)$ the right derived functors of $\mathrm{Hom}_{\mathcal{C}}(-,A)$. The maps category, $\mathrm{maps}(\mathcal{C})$ =============================================== In this section $\mathcal{C}$ is an annuli variety with pseudokerneles. We will study tilting subcategories of $\mathrm{mod}(\mathcal{C})$ via the equivalence of categories between the maps category, module the homotopy relation, and the category of functors, $\mathrm{mod}(\mathcal{C})$, given by Auslander in \[1\]. We will provide $\mathrm{maps}(\mathcal{C})$ with a structure of exact category such that, tilting subcategories of $\mathrm{mod}(\mathcal{C})$ will correspond to relative tilting subcategories of $\mathrm{maps}(\mathcal{C})$. We begin the section recalling concepts and results from \[1\], \[14\] and \[23\]. The objects in $\mathrm{maps}(\mathcal{C})$ are morphisms $(f_{1},A_{1},A_{0}):A_{1}\xrightarrow{f_1}A_{0}$, and the maps are pairs $(h_{1},h_{0}):(f_{1},A_{1},A_{0})\rightarrow (g_{1},B_{1},B_{0})$, such that the following square commutes $$\begin{diagram}\dgARROWLENGTH=1em \node{A_1}\arrow{e,t}{f_1}\arrow{s,l}{h_1} \node{A_0}\arrow{s,l}{h_0}\\ \node{B_1}\arrow{e,t}{g_1} \node{B_0} \end{diagram}$$We say that two maps $(h_{1},h_{0})$, $(h_{1}^{\prime },h_{0}^{\prime }):(f_{1},A_{1},A_{0})\rightarrow (g_{1},B_{1},B_{0})$ are homotopic, if there exist a morphisms $s:A_{0}\rightarrow B_{1}$ such that $h_{0}-h_{0}^{\prime }=g_{1}s$. Denote by $\underline{\mathrm{maps}}(\mathcal{C})$ the category of maps modulo the homotopy relation. It was proved in \[1\] that the categories $\underline{\mathrm{maps}}(\mathcal{C})$ and $\mathrm{mod}(\mathcal{C})$ are equivalent. The equivalence is given by a functor $\underline{\varPhi}:\underline{\mathrm{maps}}(\mathcal{C})\rightarrow \mathrm{mod}(\mathcal{C})$ induced by the functor $\varPhi:\mathrm{map}(\mathcal{C})\rightarrow \mathrm{mod}(\mathcal{C})$ given by $$\varPhi(A_{1}\xrightarrow{f_1}A_{0})=\mathrm{Coker}((-,A_{1})\xrightarrow{(-,f_1)}(-,A_{0}))\text{.}$$ The category $\mathrm{maps}(\mathcal{C})$ is not in general an exact category, we will use instead the exact category $P^{0}(\mathcal{A})$ of projective resolutions, which module the homotopy relation, is equivalent to $\underline{\mathrm{maps}}(\mathcal{C})$. Since we are assuming $\mathcal{C}$ has pseudokerneles, the category $\mathcal{A}=\mathrm{mod}(\mathcal{C})$ is abelian. We can consider the categories of complexes $C(\mathcal{A})$, and its subcategory $C^{-}(\mathcal{A})$, of bounded above complexes, both are abelian. Moreover, if we consider the class of exact sequences $\mathcal{S}$: $0\rightarrow {L{.}}\xrightarrow{j}M{.}\xrightarrow{\pi}N{.}\rightarrow 0$, such that, for every $k$, the exact sequences $0\rightarrow L_{k}\xrightarrow{j_k}M_{k}\xrightarrow{\pi_k}N_{k}\rightarrow 0$ split, then $(\mathcal{S},C(\mathcal{A}))$, $(\mathcal{S},C^{-}(\mathcal{A}))$ are exact categories with enough projectives, in fact they are both Frobenius. In the first case the projective are summands of complexes of the form: $$\cdots B_{k+2}\coprod B_{k+1}\xrightarrow{\left[\begin{matrix} 0&1\\ 0 &0 \end{matrix}\right]}B_{k+1}\coprod B_{k}\xrightarrow{\left[\begin{matrix} 0&1\\ 0 &0 \end{matrix}\right]}B_{k}\coprod B_{k-1}\cdots$$ In the second case of the form: $$\cdots B_{k+3}\coprod B_{k+2}\xrightarrow{\left[\begin{matrix} 0&1\\ 0&0 \end{matrix}\right]}B_{k+2}\coprod B_{k+1}\xrightarrow{\left[\begin{matrix} 0&1\\ 0 &0 \end{matrix}\right]}B_{k+1}\coprod B_{k}\xrightarrow{\left[\begin{matrix} 0&1 \end{matrix}\right]}B_{k}\rightarrow 0$$ If we denote by $\underline{C^{-}}(\mathcal{A})$ the stable category, it is well known \[23\], \[14\], that the homotopy category ${K}^{-}(\mathcal{A})$ and $\underline{C^{-}}(\mathcal{A})$ are equivalent. Now, denote by $P^{0}(\mathcal{A})$ the full subcategory of $C^{-}(\mathcal{A})$ consisting of projective resolutions, this is, complexes of projectives $P.$: $$\cdots P_{k}\rightarrow P_{k-1}\rightarrow \cdots \rightarrow P_{1}\rightarrow P_{0}\rightarrow 0$$such that $H^{i}(P.)=0$ for $i\neq 0$. Then we have the following: The category $P^{0}(\mathcal{A})$ is closed under extensions and kernels of epimorphisms. If $0\rightarrow P.\rightarrow E.\rightarrow Q.\rightarrow 0$ is an exact sequence in $P^{0}(\mathcal{A})$, then $0\rightarrow P_{j}\rightarrow E_{j}\rightarrow Q_{j}\rightarrow 0$ is a splitting exact sequence in $\mathcal{A}$ with $P_{j}$, $Q_{j}$ projectives, hence $E_{j}$ is also projective. By the long homology sequence we have the exact sequence: $\cdots \rightarrow H^{i}(P.)\rightarrow H^{i}(E.)\rightarrow H^{i}(Q.)\rightarrow H^{i-1}(P.)\rightarrow \cdots $, with $H^{i}(P.)=H^{i}(Q.)=0$, for $i\neq 0$. This implies $E.\in P^{0}(\mathcal{A}) $. Now, let $0\rightarrow T.\rightarrow Q.\rightarrow P.\rightarrow 0$ be an exact sequence with $Q.$, $P.$ in $P^{0}(\mathcal{A})$. This implies that for each $k$, $0\rightarrow T_{k}\rightarrow Q_{k}\rightarrow P_{k}\rightarrow 0$ is an exact and splittable sequence, hence each $T_{k}$ is projective and, by the long homology sequence, we have the following exact sequence $$\cdots \rightarrow H^{1}(T.)\rightarrow H^{1}(Q.)\rightarrow H^{1}(P.)\rightarrow H^{0}(T.)\rightarrow H^{0}(Q.)\rightarrow H^{0}(P.)\rightarrow 0$$with $H^{i+1}(P.)=H^{i}(Q.)=0$ for $i\geq 1$. This implies $H^{i}(T.)=0$, for $i\neq 0$. If $\mathcal{S}_{P^{0}(\mathcal{A})}$ denotes the collection of exact sequences with objects in $P^{0}(\mathcal{A})$, then $(P^{0}(\mathcal{A}),\mathcal{S}_{P^{0}(\mathcal{A})})$ is an exact subcategory of $(C^{-}(\mathcal{A}),\mathcal{S})$. The category $P^{0}(\mathcal{A})$ has enough projectives, they are the complexes of the form: $$\label{relativeproj} \cdots \rightarrow P_{3}\coprod P_{2}\xrightarrow{\left[\begin{matrix} 0&1\\ 0&0 \end{matrix}\right]}P_{2}\coprod P_{1}\xrightarrow{\left[\begin{matrix} 0&1\\ 0&0 \end{matrix}\right]}P_{1}\coprod P_{0}\rightarrow 0$$ Denote by $R^{0}(\mathcal{A})$ the category $P^{0}(\mathcal{A})$ module the homotopy relation. This is: $R^{0}(\mathcal{A})$ is a full subcategory of $\underline{\mathcal{C}^{-}}(\mathcal{A})=K^{-}(\mathcal{A})$. It is easy to check that $R^{0}(\mathcal{A})$ is the category with objects in $P^{0}(\mathcal{A})$ and maps the maps of complexes, module the maps that factor through a complex of the form: $$\cdots \rightarrow P_{3}\coprod P_{2}\xrightarrow{\left[\begin{matrix} 0&1\\ 0&0 \end{matrix}\right]}P_{2}\coprod P_{1}\xrightarrow{\left[\begin{matrix} 0&1\\ 0&0 \end{matrix}\right]}P_{1}\coprod P_{0}\xrightarrow{\left[\begin{matrix} 0&1 \end{matrix}\right]}P_{0}\rightarrow 0$$ We have the following: There is a functor $\varPsi:P^{0}(\mathcal{A})\rightarrow \mathrm{maps}(\mathcal{C})$ which induces an equivalence of categories $\underline{\varPsi}:R^{0}(\mathcal{A})\rightarrow \underline{\mathrm{maps}}(\mathcal{C})$ given by: $$\varPsi(P.)=\varPsi(\cdots \rightarrow (-,A_{2})\xrightarrow{(-,f_2)}(-,A_{1})\xrightarrow{(-,f_1)}(-,A_{0})\rightarrow 0)=A_{1}\xrightarrow{f_1}A_{0}$$ Since $\mathcal{C}$ has pseudokerneles, any map $A_{1}\xrightarrow{f_1}A_{0}$ induces an exact sequence $$(-,A_{n})\xrightarrow{(-,f_n)}(-,A_{n-1})\rightarrow \cdots \rightarrow (-,A_{2})\xrightarrow{(-,f_2)}(-,A_{1})\xrightarrow{(-,f_1)}(-,A_{0})$$and $\varPsi$ is clearly dense. Let $(-,\varphi ):P.\rightarrow Q.$ be a map of complexes in $P^{0}(\mathcal{A})$: $$\begin{diagram}\dgARROWLENGTH=1em \node{\cdots}\arrow{e} \node{(-,A_2)}\arrow{e,t}{(-,f_2)}\arrow{s,l}{(-,\varphi_2)} \node{(-,A_1)}\arrow{e,t}{(-,f_1)}\arrow{s,l}{(-,\varphi_1)} \node{(-,A_0)}\arrow{e}\arrow{s,l}{(-,\varphi_0)} \node{0}\\ \node{\cdots}\arrow{e} \node{(-,B_2)}\arrow{e,t}{(-,g_2)} \node{(-,B_1)}\arrow{e,t}{(-,g_1)} \node{(-,B_0)}\arrow{e} \node{0} \end{diagram} \label{maps1}$$ If $\varPsi(P.\xrightarrow{(-,\varphi)}Q.)$ is homotopic to zero, then we have a map $s_{0}:A_{0}\rightarrow A_{1}$ such that $g_{0}s_{0}=\varphi _{0}$: $$\begin{diagram}\dgARROWLENGTH=1em \node{A_1}\arrow{e,t}{f_1}\arrow{s,l}{\varphi_1} \node{A_0}\arrow{s,l}{\varphi_0}\arrow{sw,t}{s_0}\\ \node{B_1}\arrow{e,t}{g_1} \node{B_0} \end{diagram}$$and $s_{0}$ lifts to a homotopy $s:P.\rightarrow Q.$. Conversely, any homotopy $s:P.\rightarrow Q.$ induces an homotopy in $\mathrm{maps}(\mathcal{C})$. Then $\varPsi$ is faithful. If $\varPsi(P.)=(f_{1},A_{1},A_{0})$, $\varPsi(Q.)=(g_{1},B_{1},B_{0})$ and $(h_{0},h_{1}):\varPsi(P.)\rightarrow \varPsi(Q.)$ is a map in $\mathrm{maps}(\mathcal{C})$, then $(h_{0},h_{1})$ lifts to a map $(-,h)={(-,h_{i})}:P.\rightarrow Q.$, and $\varPsi$ is full. There is an equivalence of categories $\Theta :R^{0}(\mathcal{A})\rightarrow \mathrm{mod}(\mathcal{C})$ given by $\underline{\varTheta}=\underline{\varPhi}\underline{\varPsi}$, with $\varTheta=\varPhi\varPsi$. \[omegaP\] Let $P.$ be an object in $P^{0}(\mathcal{A})$, denote by $\mathrm{rpdim}P.$ the relative projective dimension of $P.$ , and by $\mathrm{pdim}\varTheta(P.)$ the projective dimension of $\varTheta(P.)$. Then we have $\mathrm{rpdim}P.=\mathrm{pdim}\varTheta(P.)$. Moreover, if $\Omega ^{i}(P.)$ is the relative syzygy of $P.$, then for all $i\geq 0$, we have $\Omega ^{i}(\varTheta(P.))=\varTheta(\Omega ^{i}(P.))$. Let $P.$ be the complex resolution $$\text{0}\rightarrow \text{(-,}A_{n})\xrightarrow{(-,f_n)}(\text{-,}A_{n-1})\rightarrow \cdots \rightarrow (\text{-,}A_{2})\xrightarrow{(-,f_2)}(\text{-,}A_{1})\xrightarrow{(-,f_1)}\text{(-,}A_{0}\text{)}\rightarrow \text{0}$$and $M=\mathrm{Coker}(-,f_{1})$, then $\mathrm{pdim}M\leq n$. Now, consider the following commutative diagram $$\begin{diagram}\dgARROWLENGTH=.9em \node{} \node{} \node{} \node{0}\arrow{e,t}{}\arrow{s,=,-} \node{A_n}\arrow{s,l}{\big( ^{1}_{0}\big)}\\ \node{} \node{} \node{} \node{A_n}\arrow{e,t}{-\big( ^{1}_{0}\big)}\arrow{s,..,-} \node{A_n\coprod A_{n-1}}\arrow{s,..,-}\\ \node{} \node{} \node{A_n}\arrow{s,r}{\big( ^{1}_{f_n}\big)} \arrow{e,..,-} \node{A_3}\arrow{s,r}{\big( ^{1}_{f_3}\big)}\arrow{e,t}{f_3} \node{A_2}\arrow{s,r}{\big( ^{1}_{f_2}\big)}\\ \node{} \node{A_n}\arrow{s,=}\arrow{e,t}{-\big( ^{1}_{0}\big)} \node{A_n\coprod A_{n-1}}\arrow{s,r}{(-f_n\; 1)}\arrow{e,..,-} \node{A_3\coprod A_2}\arrow{s,r}{(-f_3\; 1)}\arrow{e,t}{\big( _{0\; 0}^{0\; 1} \big)} \node{A_2\coprod A_1}\arrow{s,r}{(f_2\; -1)}\\ \node{} \node{A_n}\arrow{s,r}{\big( ^{1}_{f_n}\big)} \arrow{e,t}{f_n} \node{A_{n-1}}\arrow{s,r}{\big( ^{1}_{f_{n-1}}\big)}\arrow{e,..,-} \node{A_2}\arrow{s,r}{\big( ^{1}_{f_2}\big)}\arrow{e,t}{f_2} \node{A_1}\arrow{s,r}{\big( ^{1}_{f_1}\big)}\\ \node{A_n}\arrow{s,=}\arrow{e,t}{-\big( ^{1}_{0}\big)} \node{A_n\coprod A_{n-1}}\arrow{s,r}{(-f_n\; 1)}\arrow{e,t}{\big( _{0\; 0}^{0\; 1} \big)} \node{A_{n-1}\coprod A_{n-2}}\arrow{s,r}{(f_{n-1}\; -1)}\arrow{e,..,-} \node{A_2\coprod A_1}\arrow{e,t}{\big( _{0\; 0}^{0\; 1} \big)}\arrow{s,r}{(f_2\; -1)} \node{A_1\coprod A_{0}}\arrow{s,r}{(-f_1\; 1)}\\ \node{A_n}\arrow{e,t}{f_n} \node{A_{n-1}}\arrow{e,t}{f_{n-1}} \node{A_{n-2}}\arrow{e,..,-} \node{A_1}\arrow{e,t}{f_1} \node{A_0} \end{diagram}$$ Set $\mathbf{Q}_{n}=0\rightarrow (-,A_{n})\rightarrow 0$, and for $n-1\geq i\geq 1$ consider the following complex $\mathbf{Q}_{i}$: $$\begin{aligned} 0\rightarrow (-,A_{n})\rightarrow (-,A_{n})\coprod (-,A_{n-1})\rightarrow\cdots\rightarrow\\ \rightarrow (-,A_{i+2})\coprod (-,A_{i+1})\rightarrow (-,A_{i+1})\coprod (-,A_{i})\rightarrow 0\end{aligned}$$ Then we have a relative projective resolution $$0\rightarrow \mathbf{Q}_{n}\rightarrow \mathbf{Q}_{n-1}\rightarrow \cdots \rightarrow \mathbf{Q}_{1}\rightarrow \mathbf{Q}_{0}\rightarrow P.\rightarrow 0$$with relative syzygy the complex: $$\Omega ^{i}(P.):0\rightarrow (-,A_{n})\rightarrow (-,A_{n-1})\rightarrow (-,A_{n-2})\cdots (-,A_{i+2})\rightarrow (-,A_{i+1})\rightarrow 0$$for $n-1\geq i\geq 0$. Therefore: we have an exact sequence $$0\rightarrow \varTheta(\Omega (P.))\rightarrow \varTheta(\mathbf{Q}_{0})\rightarrow \varTheta(P.)\rightarrow 0$$ in $\mathrm{mod}(\mathcal{C})$. Since $\varTheta(\mathbf{Q}_{i})=(-,A_{i})$ and $\varTheta(P.)=M$, we have $\Omega (\varTheta(P.))=\varTheta(\Omega (P.))$, and we can prove by induction that $\Omega ^{i}(\varTheta(P.))=\varTheta(\Omega ^{i}P.)$, for all $i\geq 0$. It follows $\mathrm{rpdim}P.\geq \mathrm{pdim}\varTheta(P.).$ Conversely, applying $\varTheta$ to a relative projective resolution $$0\rightarrow \mathbf{Q}_{n}\rightarrow \mathbf{Q}_{n-1}\rightarrow \cdots \rightarrow \mathbf{Q}_{1}\rightarrow \mathbf{Q}_{0}\rightarrow P.\rightarrow 0,$$ we obtain a projective resolution of $\varTheta(P.)$ $$0\rightarrow \varTheta(\mathbf{Q}_{n})\rightarrow \varTheta(\mathbf{Q}_{n-1})\rightarrow \cdots \rightarrow \varTheta(\mathbf{Q}_{1})\rightarrow \varTheta(\mathbf{Q}_{0})\rightarrow \varTheta (P.)\rightarrow 0.$$ It follows $\mathrm{rpdim}P.\leq \mathrm{pdim}\varTheta(P.).$ As a corollary we have: Let $\mathcal{C}$ a dualizing Krull-Schmidt variety. If $P.$ and $Q$ are are complexes in $P^{0}(\mathrm{mod}(\mathcal{C}))$ without summands of the form (\[relativeproj\]), then there is an isomorphism $$\mathrm{Ext}_{C^{-}(\mathrm{mod}(\mathcal{C}))}^{k}(P.,Q.)=\mathrm{Ext}_{\mathrm{mod}(\mathcal{C})}^{k}(\varTheta(P.),\varTheta(Q.))$$ By Proposition \[omegaP\], we see that $\varTheta(\Omega ^{i}P.)=\Omega ^{i}(\varTheta(P.))$, $i\geq 0$. It is enough to prove the corollary for $k=1 $. Assume that (\*) $0\rightarrow Q.\xrightarrow {(-,j_i)}E.\xrightarrow{(-,p_i)}P.\rightarrow 0$ is a exact sequence in $\mathrm{Ext}_{C^{-}(\mathrm{mod}(\mathcal{C})}^{k}(P.,Q.)$, with $Q.=\cdots \rightarrow (-,B_{2})\xrightarrow{(-,g_2)}(-,B_{1})\xrightarrow{(-,g_1)}(-,B_{0})\rightarrow 0$, $P.=\cdots \rightarrow (-,A_{2})\xrightarrow{(-,f_2)}(-,A_{1})\xrightarrow{(-,f_1)}(-,A_{0})\rightarrow 0$, $E.=\cdots \rightarrow (-,E_{2})\xrightarrow{(-,h_2)}(-,E_{1})\xrightarrow{(-,h_1)}(-,E_{0})\rightarrow 0$. Since the exact sequence $0\rightarrow (-,B_{i})\xrightarrow {(-,j_i)}(-,E_{i})\xrightarrow{(-,p_i)}(-,A_{i})\rightarrow 0$ splits $E_{i}=A_{i}\coprod B_{i}$, $i\geq 0$. Then we have an exact sequence in $\mathrm{mod}(\mathcal{C})$ $$0\rightarrow \varTheta(Q.)\xrightarrow{\rho}\varTheta(E.)\xrightarrow{\sigma}\varTheta(P.)\rightarrow 0 \label{splits}$$If (\[splits\]) splits, then there exist a map $\delta :\varTheta(E.)\rightarrow \varTheta(Q.)$ such that $\delta \rho =1_{\varTheta(Q.)}$, We have a lifting of $\delta $, ${(-,l_{i})}_{i\in \mathbb{Z}}:E.\rightarrow Q.$ such that the following diagram is commutative $$\begin{diagram} \node{\cdots}\arrow{e} \node{(-,B_1)}\arrow{e,t}{(-,g_1)}\arrow{s,l}{(-,l_1j_i)} \node{(-,B_0)}\arrow{e,t}{\pi}\arrow{s,l}{(-,l_0j_0)} \node{\varTheta(Q.)}\arrow{e}\arrow{s,-,=} \node{0}\\ \node{\cdots}\arrow{e} \node{(-,B_1)}\arrow{e,t}{(-,g_1)} \node{(-,B_0)}\arrow{e,t}{\pi} \node{\varTheta(Q.)}\arrow{e} \node{0} \end{diagram}$$The complex $Q.$ has not summand of the form (\[relativeproj\]), hence, $Q.$ is a minimal projective resolution of $\varTheta(Q.)$. Since $\pi :(-,B_{0})\rightarrow \varTheta(Q.)$ is a projective cover, the map $(-,l_{0}j_{0}):(-,B_{0})\rightarrow (-,B_{0})$ is an isomorphism, and it follows by induction that all maps $(-,l_{i}j_{i})$ are isomorphisms, which implies that the map $\{(-,j_{i})\}_{i\in\mathbb{Z}}:Q.\rightarrow E.$ is a splitting homomorphism of complexes. Given an exact sequence (\*\*) $0\rightarrow G\rightarrow H\rightarrow F\rightarrow 0$, in $\mathrm{mod}(\mathcal{C})$, we take minimal projective resolutions $P.$ and $Q.$ of $F$ and $G,$ respectively, by the Horseshoe’s Lemma, we have a projective resolution $E.$ for $H$, with $E_{i}=Q_{i}\oplus P_{i}$, and $0\rightarrow \varTheta(Q.)\rightarrow \varTheta(E.)\rightarrow \varTheta(P.)\rightarrow 0$ is a exact sequence in $\mathrm{mod}(\mathcal{C})$ isomorphic to (\*\*). Relative Tilting in $\mathrm{maps}(\mathcal{C})$ ------------------------------------------------ Let $\mathcal{C}$ a dualizing Krull-Schmidt variety. In order to define an exact structure on $\mathrm{maps}(\mathcal{C})$ we proceed as follows: we identify first $\mathcal{C}$ with the category $\mathfrak{p(}\mathcal{C})$ of projective objects of $\mathcal{A}=\mathrm{mod}(\mathcal{C})$, in this way $\mathrm{maps}(\mathcal{C})$ is equivalent to $\mathrm{maps}(\mathfrak{p(}\mathcal{C}))$ which is embedded in the abelian category $\mathcal{B}=$ $\mathrm{maps}(\mathcal{A}).$ We can define an exact structure ($\mathrm{maps}(\mathcal{C}),\mathcal{S})$ giving a subfunctor $F$of $\mathrm{Ext}_{\mathcal{B}}^{1}(-,?)$. Let $\varPsi:P^{0}(\mathcal{A})\rightarrow \mathrm{maps}(\mathcal{C})$ be the functor given above and $\alpha :$ $\mathrm{maps}(\mathcal{C})$ $\rightarrow $ $\mathrm{maps}(\mathfrak{p(}\mathcal{C}))$ the natural equivalence. Since $\varPsi$ is dense any object in $\mathrm{maps}(\mathcal{C})$ is of the form $\varPsi(P.)$ and we define $\mathrm{Ext}_{F}^{1}$($\alpha \varPsi(P.)$ , $\alpha \varPsi(Q.))$ as $\alpha \varPsi(\mathrm{Ext}_{C^{-}(\mathcal{A})}^{1}(P.,Q.))$. We obtain the exact structure on $\mathrm{maps}(\mathcal{C})$ using the identification $\alpha .$ Once we have the exact structure on $\mathrm{maps}(\mathcal{C})$ the definition of a relative tilting subcategory $\mathcal{T}_{\mathcal{C}}$ of $\mathrm{maps}(\mathcal{C})$ is very natural, it will be equivalent to the following: A relative tilting category in the category of maps, $\mathrm{maps}(\mathcal{C})$, is a subcategory $\mathcal{T}_{\mathcal{C}}$ such that : - Given $T:T_{1}\rightarrow T_{0}$ in $\mathcal{T}_{\mathcal{C}}$ , and $P.\in P^{0}(\mathcal{C})$ such that $\varPsi(P.)=T$, there exist an integer $n$ such that $\mathrm{rpdim}P.\leq n$. - Given $T:T_{1}\rightarrow T_{0}$, $T^{\prime }:T_{1}^{\prime }\rightarrow T_{0}^{\prime }$ in $\mathcal{T}_{\mathcal{C}}$ and $\varPsi(P.)=T$, $\varPsi(Q.)=T^{\prime }$, $P.,Q.\in P^{0}(\mathrm{mod}(\mathcal{C}))$. Then $\mathrm{Ext}_{C^{-}(\mathrm{mod}(\mathcal{C}))}^{k}(P.,Q.)=0$ for all $k\geq 1$. - Given an object $C$ in $\mathcal{C}$, denote by $(-,C)_{\circ }$ the complex $0\rightarrow (-,C)\rightarrow 0$ concentrated in degree zero. Then there exists an exact sequence $$0\rightarrow (-,C)_{\circ }\rightarrow P_{0}\rightarrow P_{1}\rightarrow \cdots \rightarrow P_{n}\rightarrow 0$$with $P_{i}\in P^{0}(\mathrm{mod}(\mathcal{C}))$ and $\varPsi(P_{i})\in \mathcal{T}_{\mathcal{C}}$. By definition, the following is clear \[maps2\] Let $\varPhi:\mathrm{maps}(\mathcal{C})\rightarrow \mathrm{mod}(\mathcal{C})$ be functor above, $\mathcal{T}_{\mathcal{C}}$ is a relative tilting subcategory of $\mathrm{maps}(\mathcal{C})$ if and only if $\varPhi(\mathcal{T}_{\mathcal{C}})$ is a tilting subcategory of $\mathrm{mod}(\mathcal{C})$ The Algebra of Triangular Matrices ================================== Let $\Lambda$ be an artin algebra. We want to explore the connections between $\mathrm{mod}\ \Gamma $, with $\Gamma =\left( \begin{array}{cc} \Lambda & 0 \\ \Lambda & \Lambda\end{array}\right) $ and the category $\mathrm{mod}(\mathrm{mod}\Lambda )$. In particular we want to compare the Auslander-Reiten quivers and subcategories which are tilting, contravariantly, covariantly and functorially finite. We identify $\mathrm{mod}\;\Gamma $ with the category of $\Lambda $-maps, $\mathrm{maps}(\Lambda )$ \[see 7 Prop. 2.2\]. We refer to the book by Fossum, Griffits and Reiten \[13\] or to \[16\] for properties of modules over triangular matrix rings. Almost Split Sequences ---------------------- In this subsection we want to study the relation between the almost split sequences in $\mathrm{mod}\ \Gamma $ and almost split sequences in $\mathrm{mod}(\mathrm{mod}\Lambda ).$We will see that except for a few special objects in $\mathrm{mod}\ \Gamma $, the almost split sequences will belong to the class $\mathcal{S}$ of the exact structure, so in particular will be relative almost split sequences. For any indecomposable non projective $\Gamma $-module $M=(M_{1},M_{2},f)$ we can compute $DtrM$ as follows: To construct a minimal projective resolution of $M$ (\[13\], \[16\]), let $P_{1}\xrightarrow{p_1}P_{0}\rightarrow M_{1}\rightarrow 0$ be a minimal projective presentation. Taking the cokernel, we have an exact sequence $M_{1}\xrightarrow{f}M_{2}\xrightarrow{f_2}M_{3}\rightarrow 0$, and a commutative diagram $$\begin{diagram}\dgARROWLENGTH=.5em \node{0}\arrow{e} \node{P_1}\arrow{e,t}{} \arrow{s,l}{p_1} \node{P_1\oplus Q_1}\arrow{s,l}{}\arrow{e} \node{Q_1}\arrow{e}\arrow{s,l}{q_1} \node{0}\\ \node{0}\arrow{e} \node{P_0}\arrow{e,t}{} \arrow{s,l}{p_0} \node{P_0\oplus Q_0}\arrow{s,l}{}\arrow{e} \node{Q_0}\arrow{e}\arrow{s,l}{q_0} \node{0}\\ \node{} \node{M_1}\arrow{e,t}{f}\arrow{s} \node{M_2}\arrow{e,t}{f_2}\arrow{s} \node{M_3}\arrow{e}\arrow{s} \node{0}\\ \node{} \node{0} \node{0} \node{0} \end{diagram}$$with $Q_{0}$ the projective cover of $M_{3}.$The presentation can be written as: $\bigskip $$$\left( \begin{array}{c} P_{1} \\ P_{1}\oplus Q_{1}\end{array}\right) \rightarrow \left( \begin{array}{c} P_{0} \\ P_{0}\oplus Q_{0}\end{array}\right) \rightarrow \left( \begin{array}{c} M_{1} \\ M_{2}\end{array}\right) \rightarrow 0$$ and $trM$ will look as as follows: $$\left( \begin{array}{c} P_{0}^{\ast }\oplus Q_{0}^{\ast } \\ Q_{0}^{\ast }\end{array}\right) \rightarrow \left( \begin{array}{c} P_{1}^{\ast }\oplus Q_{1}^{\ast } \\ Q_{1}^{\ast }\end{array}\right) \rightarrow tr\left( \begin{array}{c} M_{1} \\ M_{2}\end{array}\right) \rightarrow 0$$ which corresponds to the commutative exact diagram: $$\begin{diagram}\dgARROWLENGTH=.5em \node{0}\arrow{e} \node{Q_0^{\ast}}\arrow{e,t}{} \arrow{s,l}{q_1^{\ast}} \node{Q_0^{\ast}\oplus P_0^{\ast}}\arrow{s,l}{}\arrow{e} \node{P_0^{\ast}}\arrow{e}\arrow{s,l}{p_1^{\ast}} \node{0}\\ \node{0}\arrow{e} \node{Q_1^{\ast}}\arrow{e,t}{} \arrow{s,l}{} \node{Q_1^{\ast}\oplus P_1^{\ast}}\arrow{s,l}{}\arrow{e} \node{P_1^{\ast}}\arrow{e}\arrow{s} \node{0}\\ \node{} \node{tr M_3\oplus Q^{\ast}}\arrow{e,t}{}\arrow{s} \node{tr M_2\oplus P^{\ast}}\arrow{e,t}{}\arrow{s} \node{tr M_1}\arrow{e}\arrow{s} \node{0}\\ \node{} \node{0} \node{0} \node{0} \end{diagram}$$ with $Q^{\ast }$, $P^{\ast }$, projectives coming from the fact that the presentations of $M_{2}$ and $M_{3}$ in the first diagram are not necessary minimal. Then $\tau M$ is obtained as $\tau (M_{1},M_{2},f)=\tau M_{2}\oplus D(P^{\ast })\rightarrow \tau M_{3}\oplus D(Q^{\ast })$, with kernel $0\rightarrow \tau M\rightarrow \tau M_{2}\oplus D(P^{\ast })\rightarrow \tau M_{3}\oplus D(Q^{\ast })$. We consider now the special cases of indecomposable $\Gamma $-modules of the form: $M\xrightarrow{1_M}M$, $(M,0,0)$, $(0,M,0)$, with $M$ a non projective indecomposable $\Lambda $-module. Let $0\rightarrow \tau M\xrightarrow{j}E\xrightarrow{\pi}M\rightarrow 0$ be an almost split sequence of $\Lambda $-modules. - Then the exact sequences of $\Gamma $-modules: - $0\rightarrow (\tau M,0,0)\xrightarrow{(j\ 0)}(E,M,\pi )\xrightarrow{(\pi\ 0)}(M,M,1_{M})\rightarrow 0$, - $0\rightarrow (\tau M,\tau M,1_{\tau M})\xrightarrow {(1_{\tau M}\ j)}(\tau M,E,j)\xrightarrow {(0\ \pi)}(0,M,0)\rightarrow 0$, are almost split. - Given a minimal projective resolution $P_{1}\xrightarrow{p_1}P_{0}\xrightarrow {p_0}M\rightarrow 0$, we obtain a commutative diagram: $$\begin{diagram}\dgARROWLENGTH=1.5em \node{0}\arrow{e} \node{\tau M}\arrow{e,t}{j}\arrow{s,=,-} \node{E}\arrow{e,t}{\pi}\arrow{s,l}{\overline{t}} \node{M}\arrow{e}\arrow{s,l}{t} \node{0}\\ \node{0}\arrow{e} \node{\tau M}\arrow{e,t}{u} \node{D(P_1^{\ast})}\arrow{e,t}{D(p_1^{\ast})} \node{D(P_0^{\ast})}\arrow{e,!} \node{} \end{diagram} \label{almostspecial}$$ Then the exact sequence $$0\rightarrow (N_{1},N_{2},g)\xrightarrow {(j_2\ j_1)}(E_{1},E_{2},h)\xrightarrow{(\pi_1\ \pi_2)}(M_{1},M_{2},f)\rightarrow 0\text{,}$$ with $(N_{1},N_{2},g)=(D(P_{1}^{\ast }),D(P_{0}^{\ast }),D(p_{1}^{\ast }))$,    $(M_{1},M_{2},f)=(M,0,0)$, $(E_{1},E_{2},h)=(D(P_{1}^{\ast })\oplus M,D(P_{0}^{\ast }),(D(p_{1}^{\ast })\ t))$ and   $(j_2\ j_1)=(\left( ^1_0\right)\ 1)$, $(\pi_1\ \pi_2)=(0\ -1)$, is an almost split sequence. \(a) (1) Since $\pi :E\rightarrow M$ does not splits, the map $(\pi ,1_{M}):(E,M,\pi )\rightarrow (M,M,1_{M})$ does not split. Let $(q_{1},q_{2}):(X_{1},X_{2},f)\rightarrow (M,M,1_{M})$ be a map that is not a splittable epimorphism. Then $q_{2}f=q_{1}1_{M}=q_{1}$. We claim $q_{1}$ is not a splittable epimorphism. Indeed, if $q_{1}$ is a splittable epimorphism, then there exists a morphism $s:M\rightarrow X_{1}$, such that $q_{1}s=1_{M}$ and we have the following commutative diagram: $$\begin{diagram}\dgARROWLENGTH=1em \node{M}\arrow{s,l}{s}\arrow{e,t}{1_M} \node{M}\arrow{s,l}{fs}\\ \node{X_1}\arrow{s,l}{q_1}\arrow{e,t}{f} \node{X_2}\arrow{s,l}{q_2}\\ \node{M}\arrow{e,t}{1_M} \node{M} \end{diagram}$$with $q_{2}fs=q_{1}s=1_{M}$, and $(q_{1},q_{2}):(X_{1},X_{2},f)\rightarrow (M,M,1_{M})$ is a splittable epimorphism, a contradiction. Since $\pi :E\rightarrow M$ is a right almost split morphism, there exists a map $h:X_{1}\rightarrow E$ such that $\pi h=q_{1}$, and $q_{2}f=q_{1}=\pi h$ . We have the following commutative diagram: $$\begin{diagram}\dgARROWLENGTH=1em \node{X_1}\arrow{s,l}{h}\arrow{e,t}{f} \node{X_2}\arrow{s,l}{q_2}\\ \node{E}\arrow{s,l}{\pi}\arrow{e,t}{\pi} \node{M}\arrow{s,l}{1_M}\\ \node{M}\arrow{e,t}{1_M} \node{M} \end{diagram}$$with $(\pi \ 1_{M})(h\ q_{2})=(q_{1}\ q_{2}).$ We get a lifting $(h\ q_{2}):(X_{1},X_{2},f)\rightarrow (E,M,\pi )$ of $(q_{1},q_{2})$. We have proved $\tau (M,M,1_{M})=(\tau M,0,0)$. \(2) It is clear, $\tau (0,M,0)=(\tau M,\tau M,1_{\tau M})$ and the exact sequence does not split. Now, let $(0,\rho ):(0,M,0)\rightarrow (0,M,0)$ be a non isomorphism. Then there exists a map $h:M\rightarrow E$ with $\pi h=\rho $. We have $(0\ \pi )(0\ h)=(0\ \rho )$. \(b) We have the following commutative diagram: $$\begin{diagram}\dgARROWLENGTH=1em \node{P_1}\arrow{s,r}{p_1}\arrow{e,t}{\left( ^{1_{P_1}}_0\right)} \node{P_1\oplus P_0}\arrow{s,r}{(p_1\; 1_{P_0})}\arrow{e,t}{(p_1\; 1_{P_0})} \node{P_0}\\ \node{P_0}\arrow{s,l}{p_0}\arrow{e,t}{1_{P_0}} \node{P_0}\arrow{s}\\ \node{M}\arrow{e,t}{}\arrow{s} \node{0}\\ \node{0} \node{} \end{diagram}$$ which implies the existence of the following commutative diagram: $$\begin{diagram}\dgARROWLENGTH=1.5em \node{0}\arrow{s,r}{}\arrow{e,t}{} \node{P_0^{\ast}}\arrow{s,r}{\left( ^{1_{P_0^{\ast}}}_{p_1^{\ast}}\right)}\arrow{e,t}{1_{P_0^{\ast}}} \node{P_0^{\ast}}\arrow{s,r}{p_1^{\ast}}\\ \node{P_0^{\ast}}\arrow{s,r}{1_{P_0^{\ast}} }\arrow{e,t}{\left( ^{1_{P_0^{\ast}}}_{p_1^{\ast}}\right)} \node{P_0^{\ast}\oplus P_1^{\ast}}\arrow{e,t}{(0\;1_{P_1^{\ast}})}\arrow{s,r}{(p_1^{\ast}\;-1_{P_1^{\ast}})} \node{P_1^{\ast}}\arrow{s}\\ \node{P_0^{\ast}}\arrow{e,t}{p_1^{\ast}}\arrow{s} \node{P_1^{\ast}}\arrow{e,t}{}\arrow{s} \node{tr M}\arrow{s}\\ \node{0} \node{0} \node{0} \end{diagram}$$and $Dtr(M,0,0)=(D(P_{1}^{\ast })\xrightarrow{D(p_1^{\ast})}D(P_{0}^{\ast })) $ Since $j:\tau M\rightarrow E$ is a left almost split map, it extends to the map $\tau M\rightarrow D(P_{1}^{\ast })$. We have the commutative diagram (\[almostspecial\]). Hence, $E$ is the pullback of the maps $t:M\rightarrow D(P_0^{\ast})$, $D(P_1^{\ast})\rightarrow D(P_0^{\ast})$. We have an exact sequence: $$0\rightarrow E\xrightarrow{\left(^{\overline{t}}_{-\pi}\right)}D(P_1^{\ast})\oplus M\xrightarrow {(D(p_1^{\ast})\;t)} D(P_0^{\ast})$$ from which we built an exact commutative diagram: $$\begin{diagram}\dgARROWLENGTH=1.5em \node{\tau M}\arrow{s,r}{j}\arrow{e,t}{u} \node{D(P_1^{\ast})}\arrow{s,l}{\left( ^{1_{D(P_1^{\ast})}}_0\right)}\arrow{e,t}{D(p_1^{\ast})} \node{D(P_0^{\ast})}\arrow{s,r}{1_{D(P_0^{\ast})}}\\ \node{E}\arrow{s,l}{\pi}\arrow{e,t}{\left(^{\overline{t}}_{-\pi}\right)} \node{D(P_1^{\ast})\oplus M}\arrow{s,l}{(0\; -1_{M})}\arrow{e,t}{(D(p_1^{\ast})\; t)} \node{D(P_0^{\ast})}\arrow{s}\\ \node{M}\arrow{e,t}{1_{M}}\arrow{s} \node{M}\arrow{e,t}{}\arrow{s} \node{0}\\ \node{0} \node{0} \node{} \end{diagram}$$ We claim that the exact sequence $$0\rightarrow (D(P_{1}^{\ast }),D(P_{0}^{\ast }),D(p_{1}^{\ast }))\rightarrow (D(P_{1}^{\ast })\oplus M,D(P_{0}^{\ast }),(D(p_{1}^{\ast })\;t))\rightarrow (M,0,0)\rightarrow 0$$is an almost split sequence. We need to prove first that it does not split. Suppose there exists a map $\left( _{v}^{\mu }\right) :(M,0,0)\rightarrow (D(P_{1}^{\ast })\oplus M,D(P_{0}^{\ast }),(D(p_{1}^{\ast })\;t))$ such that $((0\;-1_{M})\;0)(\left( _{v}^{\mu }\right) \;0)=(1_{M}\;0)$, then $(D(p_{1}^{\ast })\;t)\left( _{v}^{\mu }\right) =0$ and $v=-1_{M}$. It follows that there exists $s:M\rightarrow E$ such that $\left( _{-\pi }^{\overline{t}}\right) s=\left( _{v}^{\mu }\right) $, therefore $-\pi s=v=-1$, which implies $0\rightarrow \tau M\rightarrow E\xrightarrow{\pi}M\rightarrow 0$ splits. It will be enough to prove that any automorphism $(\sigma ,0):(M,0,0)\rightarrow (M,0,0)$, which is not an isomorphism, lifts to $D(P_{1}^{\ast })\oplus M\rightarrow D(P_{0}^{\ast })$. But there exist a map $s:M\rightarrow E$ with $\pi s=\sigma $. We have a map $$(\left( _{-\pi s}^{\overline{t}s}\right) \ 0):(M,0,0)\rightarrow (D(P_{1}^{\ast })\oplus M,D(P_{0}^{\ast }),(D(p_{1}^{\ast })\;t))$$such that $((0\;-1_{M})\;0)(\left( _{-\pi s}^{\overline{t}s}\right) \ 0)=(\sigma \;0)$. Dually, we consider almost split sequences of the form $$0\rightarrow (N_{1},N_{2},g)\xrightarrow {(j_2\ j_1)}(E_{1},E_{2},h)\xrightarrow{(\pi_1\ \pi_2)}(M_{1},M_{2},f)\rightarrow 0\text{,}$$such that $(N_{1},N_{2},g)$ is one of the following cases $(N,N,1_{N}),(N,0,0),(0,N,0)$ , with $N$ a non injective indecomposable $\Lambda $-module to have the following: Let $0\rightarrow N\xrightarrow{j}E\xrightarrow {\pi}\tau^{-1}N\rightarrow 0$ an almost split sequence of $\Lambda$-modules. - Then the exact sequences of $\Gamma$-modules - $0\rightarrow (N,N,1_N)\xrightarrow{(1_N\ j)}(\tau M,E,j)\xrightarrow{(0\ \pi)}(0,M,0)\rightarrow 0$ - $0\rightarrow (N,0,0)\xrightarrow {(j\ 0)} (E,\tau^{-1}N,\pi)\xrightarrow {(\pi \ 1)} (\tau^{-1}N,\tau^{-1}N,1_{\tau^{-1}N})\rightarrow 0$ are almost split. - Given a minimal injective resolution $0\rightarrow N\xrightarrow{q_0}I_{0}\xrightarrow{q_1}I_{1}$ , we obtain a commutative diagram $$\begin{diagram}\dgARROWLENGTH=1.5em \node{}\arrow{e,!} \node{D(I_0)^{\ast}}\arrow{e,t}{D(q_1)^{\ast}}\arrow{s,l}{v} \node{D(I_1)^{\ast}}\arrow{e,t}{}\arrow{s,l}{\overline{v}} \node{\tau^{-1}N}\arrow{e}\arrow{s,=}{} \node{0}\\ \node{0}\arrow{e} \node{N}\arrow{e,t}{j} \node{E}\arrow{e,t}{\pi} \node{\tau^{-1}N}\arrow{e} \node{0} \end{diagram}$$Then the exact sequence $$0\rightarrow (N_{1},N_{2},g)\xrightarrow {(j_2\ j_1)}(E_{1},E_{2},h)\xrightarrow{(\pi_1\ \pi_2)}(M_{1},M_{2},f)\rightarrow 0\text{,}$$ with $(N_{1},N_{2},g)=(0,N,0)$, $(E_{1},E_{2},h)=(D(I_{0})^{\ast },D(I_{1})^{\ast }\oplus N,\left( _{v}^{D(q_{1})^{\ast }}\right) )$, $(M_{1},M_{2},f)=(D(I_{0})^{\ast },D(I_{0})^{\ast },D(q_{1})^{\ast })$ and $(j_2\ j_1)=(0\ \left( ^0_1\right))$, $(\pi_1\ \pi_2)=(1\ (1\ 0))$, is an almost split sequence. We will prove next that almost split sequences of objects which do not belong to the special cases consider before, are exact sequences in the relative structure $\mathcal{S}.$ Let $$0\rightarrow (N_{1},N_{2},g)\xrightarrow{(j_1\ j_2)}(E_{1},E_{2},h)\xrightarrow{(p_1\ p_2)}(M_{1},M_{2},f)\rightarrow 0$$be an almost split sequence of $\Gamma $-modules and assume that both $g,f,$ are neither splittable epimorphisms, nor splittable monomorphisms. Consider the following commutative commutative exact diagram: $$\begin{diagram}\dgARROWLENGTH=1em \node{} \node{0}\arrow{s} \node{0}\arrow{s} \node{0}\arrow{s} \node{} \node{}\\ \node{0}\arrow{e,t}{} \node{N_0}\arrow{s,l}{j_0}\arrow{e,t}{} \node{N_1}\arrow{s,l}{j_1}\arrow{e,t}{} \node{N_2}\arrow{s,l}{j_2}\arrow{e,t}{} \node{N_3}\arrow{s,l}{j_3}\arrow{e,t}{} \node{0}\\ \node{0}\arrow{e,t}{} \node{E_0}\arrow{s,l}{p_0}\arrow{e,t}{} \node{E_1}\arrow{s,l}{p_1}\arrow{e,t}{} \node{E_2}\arrow{s,l}{p_2}\arrow{e,t}{} \node{E_3}\arrow{s,l}{p_3}\arrow{e,t}{} \node{0}\\ \node{0}\arrow{e,t}{} \node{M_0}\arrow{e,t}{} \node{M_1}\arrow{s}\arrow{e,t}{} \node{M_2}\arrow{s}\arrow{e,t}{} \node{M_3}\arrow{s}\arrow{e,t}{} \node{0}\\ \node{} \node{} \node{0} \node{0} \node{0} \node{} \end{diagram} \label{diagramalmost}$$ Then, the map $p_0:E_0\rightarrow M_0$ is an epimorphism, $j_3:N_3\rightarrow E_3$ is a monomorphism, and the exact sequences $$0\rightarrow N_i\xrightarrow{j_i} E_i\xrightarrow{p_i} M_i\rightarrow 0,\ 0\le i\le 3$$ split. The map $(1_{M_{1}}\ f):(M_{1},M_{1},1_{M_{1}})\rightarrow (M_{1},M_{2},f)$ is not a splittable epimorphism. Therefore it factors through $E_{1}\xrightarrow{h}E_{2}$. Hence; there exists a map $(t_{1}\ t_{2}):(M_{1},M_{2},f)\rightarrow (E_{1},E_{2},h)$, with $(p_{1}\ p_{2})(t_{1}\ t_{2})=(1_{M_{1}}\ f)$. Similarly, the map $(0\ 1_{M_{2}}):(0,M_{2},0)\rightarrow (M_{1},M_{2},f)$ is not a splittable epimorphism. Hence; there exists a map $(t_{1}\ t_{2}):(M_{1},M_{2},f)\rightarrow (E_{1},E_{2},h)$ with $(p_{1}\ p_{2})(t_{1}\ t_{2})=(0\ 1_{M_{2}})$. We have proved that for $i=1,2,$ the exact sequences $0\rightarrow N_{i}\xrightarrow{j_i}E_{i}\xrightarrow{p_i}M_{i}\rightarrow 0$ split. The diagram (\[diagramalmost\]) induces the following commutative diagram $$\begin{diagram}\dgARROWLENGTH=1em \node{} \node{0}\arrow{s} \node{0}\arrow{s} \node{0}\arrow{s} \node{} \node{}\\ \node{0}\arrow{e,t}{} \node{(-,N_0)}\arrow{s,l}{(-,j_0)}\arrow{e,t}{} \node{(-,N_1)}\arrow{s,l}{(-,j_1)}\arrow{e,t}{(-,g)} \node{(-,N_2)}\arrow{s,l}{(-,j_2)}\arrow{e,t}{\tau} \node{G}\arrow{s,l}{\rho}\arrow{e,t}{} \node{0}\\ \node{0}\arrow{e,t}{} \node{(-,E_0)}\arrow{s,l}{(-,p_0)}\arrow{e,t}{} \node{(-,E_1)}\arrow{s,l}{(-,p_1)}\arrow{e,t}{(-,h)} \node{(-,E_2)}\arrow{s,l}{(-,p_2)}\arrow{e,t}{\pi} \node{H}\arrow{s,l}{\sigma}\arrow{e,t}{} \node{0}\\ \node{0}\arrow{e,t}{} \node{(-,M_0)}\arrow{e,t}{} \node{(-,M_1)}\arrow{s}\arrow{e,t}{(-,f)} \node{(-,M_2)}\arrow{s}\arrow{e,t}{\eta} \node{F}\arrow{s}\arrow{e,t}{} \node{0}\\ \node{} \node{} \node{0} \node{0} \node{0} \node{} \end{diagram}$$By the Snake’s Lemma, we have a connecting map $\delta $, $$\cdots \rightarrow (-,E_{0})\xrightarrow{(-,p_0)}(-,M_{0})\xrightarrow{\delta}G\xrightarrow{\rho}H\rightarrow \cdots$$We want to prove $\rho $ is a monomorphism. Let $\rho :G\xrightarrow{\rho_1}\mathrm{Im}\rho \xrightarrow{\rho_2}H$ be a factorization through its image. Since $\mathrm{mod}(\mathrm{mod}\Lambda )$ is an abelian category, $\mathrm{Im}\rho $ is a finitely presented functor, with presentation $$(-,X_{1})\xrightarrow{(-,t)}(-,X_{2})\rightarrow \mathrm{Im}\rho \rightarrow 0.$$ Lifting the maps $\rho _{1},\rho _{2}$ we obtain a commutative diagram with exact rows: $$\begin{diagram}\dgARROWLENGTH=1em \node{(-,N_1)}\arrow{e,t}{(-,g)}\arrow{s,l}{(-,u_1)} \node{(-,N_2)}\arrow{e,t}{}\arrow{s,l}{(-,u_2)} \node{G}\arrow{e,t}{}\arrow{s,l}{\rho_1} \node{0}\\ \node{(-,X_1)}\arrow{e,t}{(-,t)}\arrow{s,l}{(-,v_1)} \node{(-,X_2)}\arrow{e,t}{}\arrow{s,l}{(-,v_2)} \node{\mathrm{Im}\rho}\arrow{e,t}{}\arrow{s,l}{\rho_1} \node{0}\\ \node{(-,E_1)}\arrow{e,t}{(-,h)} \node{(-,E_2)}\arrow{e,t}{} \node{H}\arrow{e,t}{} \node{0} \end{diagram}$$whose composition is another lifting of $\rho $. Then the two liftings are homotopic and there exist maps $(-,s_{1}):(-,N_{2})\rightarrow (-,E_{1})$, $(-,s_{2}):(-,N_{1})\rightarrow (-,E_{0})$ such that $(-,j_{2})=(-,h)(-,s_{1})+(-,v_{1}u_{1})$, $(-,j_{2})=(-,h_{1})(-,s_{2})+(-,s_{1})(-,g)+(-,v_{2}u_{2})$. This is $j_{2}=hs_{1}+v_{1}u_{1}$, $j_{1}=h_{1}s_{2}+s_{1}g+v_{2}u_{2}$. Consider the following commutative diagram $$\begin{diagram}\dgARROWLENGTH=2.5em \node{N_1}\arrow{s,l}{\left(\begin{matrix} u_1\\ s_1g\\ s_2 \end{matrix}\right) }\arrow{e,t}{g} \node{N_2}\arrow{s,r}{\left(\begin{matrix} u_2\\ s_1 \end{matrix}\right) }\\ \node{X_1\oplus E_1\oplus E_0}\arrow{e,t}{\left(\begin{matrix} t&0&0\\ 0&1_{E_1}&0 \end{matrix}\right) }\arrow{s,l}{(v_1\ 1_{E_1}\ h_1)} \node{X_2\oplus E_1}\arrow{s,r}{(v_2\ h)}\\ \node{E_1}\arrow{e,t}{h} \node{E_2} \end{diagram}$$ But $$(-,X_{1}\oplus E_{1}\oplus E_{0})\xrightarrow{\left(-,\left(\begin{matrix} t&0&0\\ 0&1_{E_1}&0 \end{matrix}\right)\right)}(-,X_{2}\oplus E_{1})\rightarrow \mathrm{Im}\rho \rightarrow 0 \label{diagramalmost1}$$is exact. Changing $(-,X_{1})\xrightarrow{(-,t)}X_{2}$ by ([diagramalmost1]{}), we can assume $v_{1}u_{1}=l_{i}$, $i=1,2$; but being $(l_{1},l_{2}):(N_{1},N_{2},g)\rightarrow (E_{1},E_{2},h)$ an irreducible map, this implies either $(u_{1},u_{2}):(N_{1},N_{2},g)\rightarrow (X_{1},X_{2},t)$ is a splittable monomorphism or $(v_{1},v_{2}):(X_{1},X_{2},t)\rightarrow (E_{1},E_{2},h)$ is a splittable epimorphism. In the second case we have a map $(s_{1},s_{2}):(E_{1},E_{2},h)\rightarrow (X_{1},X_{2},t),$ with $(v_{1}\ v_{2})(s_{1}\ s_{2})=(1_{E_{1}}\ 1_{E_{2}})$. Then there exists a map $\sigma :H\rightarrow \mathrm{Im}\rho $, such that $\rho _{2}\sigma =1_{H}$. It follows $\rho _{2}$ is an isomorphism. Hence; $F=0$ and $f:M_{1}\rightarrow M_{2}$ is a splittable epimorphism. A contradiction. Now, if $(u_{1}\ u_{2})$ is a splittable monomorphism, then there exists a map $(q_{1}\ q_{2}):(X_{1},X_{2},t)\rightarrow (N_{1},N_{2},g)$, with $(q_{1}\ q_{2})(u_{1}\ u_{2})=(1_{N_{1}}\ 1_{N_{2}})$. Then, there exists $\sigma :\mathrm{Im}\rho \rightarrow G$ such that $\sigma \rho =1_{G}$, and $\rho _{1}$ is an isomorphism, in particular $\rho $ is a monomorphism.It follows $(-,E_{0})\xrightarrow{(-,p_0)}(-,M_{0})$ is an epimorphism. Therefore: $0\rightarrow N_{0}\xrightarrow{j_0}E_{0}\xrightarrow{p_0}M_{0}\rightarrow 0$ is an exact sequence that splits. A contradiction. Dualizing the diagram, we obtain, the exact sequence $0\rightarrow D(M_{3})\rightarrow D(E_{3})\rightarrow D(N_{3})\rightarrow 0$ splits. Therefore the exact sequence $0\rightarrow N_{3}\rightarrow E_{3}\rightarrow M_{3}\rightarrow 0$ splits. We can see now that the functor $\Phi $ preserves almost split sequences. Let $$0\rightarrow (N_{1},N_{2},g)\xrightarrow{(j_1\ j_2)}(E_{1},E_{2},h)\xrightarrow{(p_1\ p_2)}(M_{1},M_{2},f)\rightarrow 0$$ be an almost split sequence, such that $g,f,$ are neither splittable epimorphisms nor splittable monomorphisms. Then the exact sequence $$0\rightarrow G\xrightarrow{\rho}H\xrightarrow{\theta}F\rightarrow 0$$ obtained from the commutative diagram: $$\begin{diagram}\dgARROWLENGTH=.7em \node{0}\arrow{s} \node{0}\arrow{s} \node{0}\arrow{s} \node{}\\ \node{(-,N_1)}\arrow{s,l}{(-,j_1)}\arrow{e,t}{(-,g)} \node{(-,N_2)}\arrow{s,l}{(-,j_2)}\arrow{e,t}{} \node{G}\arrow{s,l}{\rho}\arrow{e,t}{} \node{0}\\ \node{(-,E_1)}\arrow{s,l}{(-,p_1)}\arrow{e,t}{(-,h)} \node{(-,E_2)}\arrow{s,l}{(-,p_2)}\arrow{e,t}{} \node{H}\arrow{s,l}{\theta}\arrow{e,t}{} \node{0}\\ \node{(-,M_1)}\arrow{s}\arrow{e,t}{(-,f)} \node{(-,M_2)}\arrow{s}\arrow{e,t}{} \node{F}\arrow{s}\arrow{e,t}{} \node{0}\\ \node{0} \node{0} \node{0} \node{} \end{diagram}$$is an almost split sequence \(1) The sequence $0\rightarrow G\xrightarrow{\rho}H\xrightarrow{\theta}E\rightarrow 0$ does not split. Assume it does split and let $u:F\rightarrow H$, with $\theta u=1_{F}$ be the splitting. There is a lifting of $u$ making the following diagram, with exact raws, commute: $$\begin{diagram}\dgARROWLENGTH=.7em \node{(-,M_0)} \arrow{e,t}{(-,f_1)}\arrow{s,l}{(-,s_0)} \node{(-,M_1)} \arrow{e,t}{(-,f)}\arrow{s,l}{(-,s_1)} \node{(-,M_2)}\arrow{e}\arrow{s,l}{(-,s_2)} \node{F}\arrow{e}\arrow{s} \node{0}\\ \node{(-,E_0)} \arrow{e,t}{(-,h_1)}\arrow{s,l}{(-,p_0)} \node{(-,E_1)} \arrow{e,t}{(-,h)}\arrow{s,l}{(-,p_1)} \node{(-,E_2)}\arrow{e}\arrow{s,l}{(-,p_2)} \node{H}\arrow{e}\arrow{s} \node{0}\\ \node{(-,M_0)} \arrow{e,t}{(-,f_1)} \node{(-,M_1)} \arrow{e,t}{(-,f)} \node{(-,M_2)}\arrow{e} \node{F}\arrow{e} \node{0} \end{diagram}$$The composition is a lifting of the identity, and as before, it is homotopic to the identity. By Yoneda’s lemma, there exist maps, $w_{2}:M_{2}\rightarrow M_{1}$, $w_{1}:M_{1}\rightarrow M_{0}$, such that $fw_{2}+p_{2}s_{2}=1_{M_{2}}$, $w_{2}f+f_{1}w_{1}+p_{1}s_{1}=1_{M_{1}}$. Since $f\in \mathrm{rad}(M_{1},M_{2})$, $f_{1}\in \mathrm{rad}(M_{0},M_{1})$, this implies $w_{2}f,f_{1}w_{1}\in \mathrm{rad}\mathrm{End}(M_{1})$ and $fw_{2}\in \mathrm{rad}\mathrm{End}(M_{2})$. It follows $p_{2}s_{2}=1_{M_{2}}-fw_{2}$ and $p_{1}s_{1}=1_{M_{1}}-(w_{2}f+f_{1}w_{1})$ are invertible. Therefore: $(p_{1}\ p_{2}):(E_{1},E_{2},h)\rightarrow (M_{1},M_{2},f)$ is a splittable epimorphism. A contradiction. \(2) Let $\eta :L\rightarrow F$ be a non splittable epimorphism, and $(-,X_{1})\xrightarrow{(-,t)}(-,X_{2})\rightarrow L\rightarrow 0$ a projective presentation of $L$. The map $\eta $ lifts to a map $(-,\eta _{i}):(-,X_{i})\rightarrow (-,M_{i})$, $i=1,2$. Then the map $(\eta _{1}\ \eta _{2}):(X_{1},X_{2},t)\rightarrow (M_{1},M_{2},f)$ is not a splittable epimorphism., and there exists a map $(v_{1}\ v_{2}):(X_{1},X_{2},t)\rightarrow (E_{1},E_{2},h)$, with $(p_{1}\ p_{2})(v_{1}\ v_{2})=(\eta _{1}\ \eta _{2})$. The map $(v_{1}\ v_{2})$ induces a map $\sigma :L\rightarrow H$ with $\theta \sigma =\eta $. In a similar way we prove $0\rightarrow G\rightarrow H$ is left almost split. Assume now (\*) $0\rightarrow G\rightarrow H\rightarrow F\rightarrow 0$ is an almost split sequence in $\mathrm{mod}(\mathrm{mod}\Lambda )$. Let $(-,M_{1})\xrightarrow{(-,f)}(-,M_{2})\rightarrow F\rightarrow 0$ be a minimal projective projective presentation of $F$. The map $M_{1}\xrightarrow{f}M_{2} $ is an indecomposable object in $\mathrm{mod}\left( \begin{matrix} \Lambda & 0 \\ \Lambda & \Lambda\end{matrix}\right) $ and is not projective. Then we have an almost split sequence in $\mathrm{maps}(\Lambda )$: $$0\rightarrow N=(N_{1},N_{2},g)\xrightarrow{(j_1\ j_2)}E=(E_{1},E_{2},h)\xrightarrow{(p_1\ p_2)}M=(M_{1},M_{2},f)\rightarrow 0$$where $f,g$ are both neither splittable monomorphisms and nor splittable epimorphisms. Applying the functor $\varPhi$ to the above sequence we obtain an almost split sequence (\*\*) $0\rightarrow \varPhi(N)\rightarrow \varPhi(E)\rightarrow \varPhi(M)\rightarrow 0$ with $\varPhi(M)=F$. By the uniqueness of the almost split sequence $\varPhi(N)=G$, $\varPhi(E)=H$ and (\*) is isomorphic to (\*\*). ### An example Let $\Lambda =\left( \begin{matrix} K & 0 \\ K & K\end{matrix}\right) $ be the algebra isomorphic to to the quiver algebra $KQ$, where $Q$ is: $1\rightarrow 2$ and $\Gamma =\left( \begin{array}{cc} \Lambda & 0 \\ \Lambda & \Lambda \end{array}\right) $. The algebra $\Lambda $ has a simple projective $S_{2}$, a simple injective $S_{1}$ and a projective injective $P_{1}$ . The projective $\Gamma $-modules correspond to the maps: $0\rightarrow S_{2}$, $0\rightarrow P_{1}$, $P_{1}\xrightarrow{1_{P_1}}P_{1}$ and $S_{2}\xrightarrow{1_{S_2}}S_{2}$. We compute the almost split sequences in $\Lambda ))$ to obtain exact sequences: $$0\rightarrow N\xrightarrow{j}E\xrightarrow{\pi}M\rightarrow 0$$with: \(a) $N=(0,S_{2},0)$, $E=(S_{2},S_{2}\oplus P_{1},\left( _{0}^{1_{S_{2}}}\right) )$, $M=(S_{2},P_{1},f)$, $j=(0\ $,$\left( _{f}^{1_{S_{2}}}\right) )$, $\pi =(1_{S_{2}}$,$(f\ $,$-1_{P_{1}}))$ \(b) $N=(S_{2},P_{1},f)$, $E=(P_{1}\oplus S_{2},P_{1}\oplus S_{1},\left( _{0}^{1_{P_{1}}}\ _{0}^{0}\right) )$, $M=(P_{1},S_{1},g)$, $j=(\left( _{1_{S_{2}}}^{f}\right) $,$\left( _{g}^{1_{P_{1}}}\right) )$, $\pi =((-1_{P_{1}},\ f),\ (-g\ ,1_{S_{1}})$. \(c) $N=(P_{1},S_{1},g)$, $E=(S_{1}\oplus P_{1},S_{1},(1_{S_{1}}\ 0))$, $M=(S_{1},0,0)$, $j=(\left( _{1_{P_{1}}}^{g}\right) \ 1_{S_{1}})$, $\pi =((-1_{S_{1}}\ g)\ 0)$. The Auslander-Reiten quiver in $\mathrm{maps}(\Lambda)$ is: $$\begin{diagram}\dgARROWLENGTH=.05em \dgTEXTARROWLENGTH=.5em \dgHORIZPAD=.6em \dgVERTPAD=.6ex \node{} \node{(S_2,S_2,1)}\arrow{se} \node{} \node{(P_1,P_1,1)}\arrow{se} \node{} \node{(S_1,S_1,1)}\arrow{se} \node{}\\ \node{(0,S_2,0)}\arrow{ne}\arrow{se} \node{} \node{(S_2,P_1,f)}\arrow{ne}\arrow{e}\arrow{se} \node{(S_2,0,0)}\arrow{e} \node{(P_1,S_1,g)}\arrow{ne}\arrow{se} \node{} \node{(S_1,0,0)}\\ \node{} \node{(0,P_1,0)}\arrow{ne} \node{} \node{(0,S_1,0)}\arrow{ne} \node{} \node{(P_1,0,0)}\arrow{ne} \end{diagram}$$ Applying the functor $\varPhi$ we obtain the following Auslander-Reiten quiver of $\mathrm{mod}(\mathrm{mod}\Lambda ):$ $$0\rightarrow (-,S_{2})\rightarrow (-,P_{1})\rightarrow \mathrm{rad}(-,S_{1})\rightarrow (-,S_{1})\rightarrow S_{S_{1}}\rightarrow 0,$$which is isomorphic to the Auslander-Reiten quiver of $1\xrightarrow{\alpha}2\xrightarrow{\beta}3$, with $\beta \alpha =0$, and this is the Auslander algebra of $\Lambda $. Tilting in $\mathrm{mod}(\mathrm{mod}\varLambda)$ ------------------------------------------------- Let $\Lambda $ be an artin algebra. Since $\mathrm{maps}(\mathrm{mod}\Lambda )$ is equivalent to the category $\mathrm{mod}\ \Gamma $, with $\Gamma =\left( \begin{matrix} \Lambda & 0 \\ \Lambda & \Lambda \end{matrix}\right) $, it is abelian, dualizing Krull-Schmidt, and it has kernels. Hence it has pseudokernels, and we can apply the theory so far developed. In this case the exact structure is easy to describe. The collection of exact sequences $\mathcal{S}$ consists of the exact sequences in the category $\mathrm{maps}(\mathrm{mod}\Lambda )$ $$0\rightarrow (N_{1},N_{2},g)\rightarrow (E_{1},E_{2},h)\rightarrow (M_{1},M_{2},f)\rightarrow 0$$such that in the following exact commutative diagram $$\begin{diagram}\dgARROWLENGTH=.5em \node{}\node{0}\arrow{s,l}{} \node{0}\arrow{s,l}{} \node{0}\arrow{s,l}{}\\ \node{0}\arrow{e,t}{} \node{N_0}\arrow{e,t}{g_0}\arrow{s,l}{} \node{N_1}\arrow{e,t}{g}\arrow{s,l}{} \node{N_2}\arrow{s,l}{}\\ \node{0}\arrow{e,t}{} \node{E_0}\arrow{e,t}{h_0}\arrow{s,l}{} \node{E_1}\arrow{e,t}{h}\arrow{s,l}{} \node{E_2}\arrow{s,l}{}\\ \node{0}\arrow{e,t}{} \node{M_0}\arrow{e,t}{f_0}\arrow{s,l}{} \node{M_1}\arrow{e,t}{f}\arrow{s,l}{} \node{M_2}\arrow{s,l}{}\\ \node{}\node{0} \node{0} \node{0} \end{diagram} \label{triangle3}$$the columns split . Here $(N_{0},g_{0})$, $(E_{0},h_{0})$, $(M_{0},f_{0})$ are the kerneles of the maps $g$, $h$ and $f$, respectively. The collection   $\mathcal{S}$  gives rise  to a subfunctor $F$   of the additive bifunctor $\mathrm{Ext}_{\Gamma }^{1}(-,-):(\mathrm{mod}\Gamma )\times (\mathrm{mod}\Gamma )^{op}\rightarrow \mathbf{Ab}$. The category $\mathrm{maps}(\mathrm{mod}\Lambda )$ has enough $F$-projectives and enough $F$-injectives, the $F$-projectives are the maps of the form $M\xrightarrow{1_M}M$ and $0\rightarrow M$, and the $F$-injectives are of the maps of the form $M\xrightarrow{1_M}M$ and $M\rightarrow 0$. According to Theorem \[maps2\] we have the following: Classical tilting subcategories in $\mathrm{mod}(\mathrm{mod}\Lambda )$ correspond under $\Psi $ with relative tilting subcategories $\mathcal{T}_{\mathrm{mod}\Lambda }$ of $\mathrm{maps}(\mathrm{mod}\Lambda )$, such that the following statements hold: - The maps $f:T_{0}\xrightarrow{f}T_{1}$ of objects in $\mathcal{T}_{\mathrm{mod}\Lambda }$ are monomorphisms. - Given $T:f:T_{0}\xrightarrow{f}T_{1}$ and $T^{\prime }:g:T_{0}^{\prime }\xrightarrow{g}T_{1}^{\prime }$ in $\mathcal{T}_{\mathcal{C}}$. Then $\mathrm{Ext}_{F}^{1}(T,T^{\prime })=0$. - For each object $C$ in $\mathcal{C}$, there exist a exact sequence in $\mathrm{maps}(\mathrm{mod}\Lambda )$: $$\begin{diagram}\dgARROWLENGTH=1em \node{0}\arrow{s} \node{0}\arrow{s}\\ \node{0}\arrow{s,l}{}\arrow{e,t}{} \node{C}\arrow{s,l}{}\\ \node{T_0}\arrow{s,l}{}\arrow{e,t}{f} \node{T_1}\arrow{s,l}{}\\ \node{T_0}\arrow{e,t}{g}\arrow{s} \node{T_1^{\prime}}\arrow{s}\\ \node{0} \node{0} \end{diagram}$$ such that the second column splits and $T:f:T_{0}\rightarrow T_{1}$, $T^{\prime }:g:T_{0}\rightarrow T_{1}^{\prime }$ are in $\mathcal{T}_{\mathcal{C}}$. Since $\mathrm{gdim}(\mathrm{mod}\Lambda )\leq 2$, the relative global dimension of $\mathrm{maps}(\mathrm{mod}\Lambda )$ is $\leq 2$. For generalized tilting subcategories of $\mathrm{maps}(\mathrm{mod}\Lambda ) $ there is the following analogous to the previous theorem: Generalized tilting subcategories of $\mathrm{mod}(\mathrm{mod}\Lambda )$ correspond under $\Psi $ with relative tilting subcategories $\mathcal{T}_{\mathrm{mod}\Lambda }$ of $\mathrm{maps}(\mathrm{mod}\Lambda )$ such that the following statements hold: - Given $T:T_{1}\rightarrow T_{0}$, $T^{\prime }:T_{1}^{\prime }\rightarrow T_{0}^{\prime }$ in $\mathcal{T}_{\mathrm{mod}\Lambda }$. Then $\mathrm{Ext}_{F}^{k}(T,T^{\prime })=0$, for $0<k\leq 2.$ - For each   object $C$   in $\mathcal{C}$,   there exists a relative exact sequence in $\mathrm{maps}(\mathrm{mod}\Lambda )$: $$0\rightarrow (0,C,0)\rightarrow T^{0}\rightarrow T^{1}\rightarrow T^{2}\rightarrow 0$$with $T^{i}\in \mathcal{T}_{\mathrm{mod}\Lambda }$. Contravariantly Finite Categories in $\mathrm{mod}(\mathrm{mod} \varLambda)$ --------------------------------------------------------------- In this subsection we will see that some properties like: contravariently, covariantly, functorially finite subcategories of $\mathrm{maps}(\mathrm{mod}\Lambda ),$ are preserved by the functor $\varPhi$. The following theorem was proved in \[18\]. \[See also 4 Theo. 5.5\]. Let $\mathcal{C}$ be a dualizing Krull-Schmidt variety. The assignments $\mathcal{T}\mapsto \mathcal{T}^{\bot }$ and $\mathscr Y\mapsto \mathscr Y\cap ^{\bot }\mathscr Y$ induce a bijection between equivalence classes of tilting subcategories $\mathcal{T}$ of $\mathrm{mod}(\mathcal{C})$, with $\mathrm{pdim}\mathcal{T}\leq n$, such that $\mathcal{T}$ is a generator of $\mathcal{T}^{\bot }$ and classes of subcategories $\mathscr Y$ of $\mathrm{mod}(\mathcal{C})$ which are covariantly finite, coresolving, and whose orthogonal complement $^{\bot }\mathscr Y$ has projective dimension $\leq n$. Of course, the dual of the above theorem is true. Hence it is clear the importance of studying; covariantly, contravariantly and functorially finite subcategories in $\mathrm{mod}(\mathcal{C})$. We are specially interested in the case $\mathcal{C}$ is the category of finitely generated left modules over an artin algebra $\Lambda .$ In this situation we can study them via the functor $\Psi $ relating them with the corresponding subcategories of $\mathrm{maps}(\mathrm{mod}\Lambda )$, which in principle are easier to study, since $\mathrm{maps}(\mathrm{mod}\Lambda )$ and the category of finitely generated left $\Gamma $- modules, with $\Gamma $ the triangular matrix ring, are equivalent. Such is the content of our next theorem. Let $\mathscr C\subset \mathrm{maps}(\mathrm{mod}\Lambda )$ be a subcategory. Then the following statements hold: - If $\mathscr C$ is contravariantly finite in $\mathrm{maps}(\mathrm{mod}\Lambda )$, then $\varPhi(\mathscr C)$ is a contravariantly finite subcategory of $\mathrm{mod}(\mathrm{mod}\Lambda )$. - If $\mathscr C$ is covariantly finite in $\mathrm{maps}(\mathrm{mod}\Lambda )$, then $\varPhi(\mathscr C)$ is a covariantly finite subcategory of $\mathrm{mod}(\mathrm{mod}\Lambda )$. - If $\mathscr C$ is functorially finite in $\mathrm{maps}(\mathrm{mod}\Lambda )$, then $\varPhi(\mathscr C)$ is a functorially finite subcategory of $\mathrm{mod}(\mathrm{mod}\Lambda )$. \(a) Assume $\mathscr C\subset \mathrm{maps}(\mathrm{mod}\Lambda )$ is contravariantly finite. Let $F$ be a functor in $\mathrm{mod}(\mathrm{mod}\Lambda ))$ and $(\;,M_{1})\xrightarrow{(\;,f)}(\;,M_{2})\rightarrow F\rightarrow 0$ a minimal projective presentation of $F$. Then, there exist a map $Z:Z_{1}\xrightarrow{h}Z_{2}$ and a map $$\begin{diagram}\dgARROWLENGTH=1em\label{contramaps1} \node{Z_1}\arrow{e,t}{h}\arrow{s,l}{q_1} \node{Z_2}\arrow{s,l}{q_2}\\ \node{M_1}\arrow{e,t}{f} \node{M_2} \end{diagram}$$ in $\mathrm{maps}(\mathrm{mod}\Lambda )$ such that $Z=(Z_{1},Z_{2},h)$ is a right $\mathscr C$-approximation of $M=(M_{1},M_{2},f)$. The diagram ([contramaps1]{}) induce the following commutative exact diagram: $$\begin{diagram}\dgARROWLENGTH=1em \node{(\;,Z_1)}\arrow{e,t}{(\;,h)}\arrow{s,l}{(\;,q_1)} \node{(\;,Z_2)}\arrow{e,t}{}\arrow{s,l}{(\;,q_2)} \node{\varPhi(Z)}\arrow{e}\arrow{s,l}{\rho} \node{0}\\ \node{(\;,M_1)}\arrow{e,t}{(\;,f)} \node{(\;,M_2)}\arrow{e,t}{} \node{F}\arrow{e} \node{0} \end{diagram}$$ We claim that $\rho $ is a right $\varPhi(\mathscr{C})$-approximation of $F$. Let $H\in \varPhi(\mathscr C)$, $\eta :H\rightarrow F$ a map and $(\;,X_{1})\xrightarrow{(\;,r)}(\;,X_{2})\rightarrow H\rightarrow 0$ a minimal projective presentation of $H$. We have a lifting of $\eta :$ $$\begin{diagram}\dgARROWLENGTH=1em \node{(\;,X_1)}\arrow{e,t}{(\;,r)}\arrow{s,l}{(\;,s_1)} \node{(\;,X_2)}\arrow{e,t}{}\arrow{s,l}{(\;,s_2)} \node{H}\arrow{e}\arrow{s,l}{\eta} \node{0}\\ \node{(\;,M_1)}\arrow{e,t}{(\;,f)} \node{(\;,M_2)}\arrow{e,t}{} \node{F}\arrow{e} \node{0} \end{diagram}$$ By Yoneda’s Lemma, there is the following commutative square: $$\begin{diagram}\dgARROWLENGTH=1em \node{X_1}\arrow{e,t}{r}\arrow{s,l}{s_1} \node{X_2}\arrow{s,l}{s_2}\\ \node{M_1}\arrow{e,t}{f} \node{M_2} \end{diagram}$$with $X=(X_{1},X_{2},r)\in \mathscr C$. Since $Z=(Z_{1},Z_{2},h)$ is a right $\mathscr C$-approximation of $M=(M_{1},M_{2},f)$, there exists a morphism $(t_{1},t_{2}):(X_{1},X_{2},r)\rightarrow (Z_{1},Z_{2},h)$, such that the following diagram: $$\begin{diagram}\dgARROWLENGTH=1em \node{X_1}\arrow{e,t}{r}\arrow{s,l}{t_1} \node{X_2}\arrow{s,l}{t_2}\\ \node{Z_1}\arrow{e,t}{h}\arrow{s,l}{q_1} \node{Z_2}\arrow{s,l}{q_2}\\ \node{M_1}\arrow{e,t}{f} \node{M_2} \end{diagram}$$is commutative, with $q_{i}t_{i}=s_{i}$ for $i=1,2$. Which implies the existence of a map $\theta :H\rightarrow \Psi (H)=G$, such that the diagram $$\begin{diagram}\dgARROWLENGTH=1em \node{(\;,X_1)}\arrow{e,t}{(\;,r)}\arrow{s,l}{(\;,t_1)} \node{(\;,X_2)}\arrow{e}\arrow{s,l}{(\;,t_2)} \node{H}\arrow{e}\arrow{s,l}{\theta} \node{0}\\ \node{(\;,Z_1)}\arrow{e,t}{(\;,h)}\arrow{s,l}{(\;,q_1)} \node{(\;,Z_2)}\arrow{e}\arrow{s,l}{(\;,q_2)} \node{G}\arrow{e}\arrow{s,l}{\rho} \node{0}\\ \node{(\;,M_1)}\arrow{e,t}{(\;,f)} \node{(\;,M_2)}\arrow{e} \node{F}\arrow{e} \node{0} \end{diagram}$$with exact raws, is commutative, this is: $\rho \theta =\eta $. The proof of (b) is dual to (a), and (c) follows from (a) and (b). Let $\mathscr{C}\subset \mathrm{maps}(\mathrm{mod}\Lambda )$ be a category which contains the objects of the form $(M,0,0)$, $(M,M,1_{M})$, and assume $\varPhi(\mathscr{C})$ is contravariantly finite. Then $\mathscr{C}$ is contravariantly finite. Let $M_{1}\xrightarrow{f}M_{2}$ a map in $\mathrm{maps}(\mathrm{mod}\Lambda ) $, then we have an exact sequence $(\;,M_{1})\xrightarrow{(\;,f)}(\;,M_{2})\rightarrow F\rightarrow 0$. There exist $G\in \varPhi(\mathcal{C})$ such that $\rho :G\rightarrow F$ is a right $\varPhi(\mathscr{ C})$-approximation. Let $$(\;,Z_{1})\xrightarrow{(\;,h)}(\;,Z_{2})\rightarrow G\rightarrow 0$$be a minimal projective presentation of $G$. Then, there exists a map $(r_{1},r_{2}):(Z_{1},Z_{2},h)\rightarrow (M_{1},M_{2},f)$ such that$$\begin{diagram}\dgARROWLENGTH=1em \node{(\;,Z_1)}\arrow{e,t}{(\;,h)}\arrow{s,l}{(\;,r_1)} \node{(\;,Z_2)}\arrow{e}\arrow{s,l}{(\;,r_2)} \node{G}\arrow{e}\arrow{s,l}{\rho} \node{0}\\ \node{(\;,M_1)}\arrow{e,t}{(\;,f)} \node{(\;,M_2)}\arrow{e} \node{F}\arrow{e} \node{0} \end{diagram}$$is a lifting of $\rho $. Let $(X_{1},X_{2},g)$ be an object in $\mathscr{C}$ and a map $(v_1,v_2):(X_{1},X_{2},g)\rightarrow (M_1,M_2,f)$, which  induces  the  following  commutative exact diagram in  $\mathrm{mod}(\mathrm{mod}\Lambda):$ $$\begin{diagram}\dgARROWLENGTH=1em \node{(\;,X_1)}\arrow{e,t}{(\;,g)}\arrow{s,l}{(\;,v_1)} \node{(\;,X_2)}\arrow{s,l}{(\;,v_2)}\arrow{e} \node{H}\arrow{e}\arrow{s,l}{\eta} \node{0} \\ \node{(\;,M_1)}\arrow{e,t}{(\;,f)} \node{(\;,M_2)}\arrow{e} \node{F}\arrow{e} \node{0} \end{diagram}$$ Since $\rho :G\rightarrow F$ is a right $\varPhi(\mathscr{C})$-approximation, there exists a morphism $\theta :H\rightarrow G$ such taht $\rho \theta =\eta $. Therefore: $\theta $ induces a morphism $$\begin{diagram}\dgARROWLENGTH=1em \node{X_1}\arrow{e,t}{g}\arrow{s,l}{t_1} \node{X_2}\arrow{s,l}{t_2}\\ \node{Z_1}\arrow{e,t}{h} \node{Z_2} \end{diagram}$$ in $\mathrm{maps}(\mathcal{C})$ such that $\varPhi(t_{1},t_{2})=\theta $. We have two liftings of $\rho :$ $$\begin{diagram}\dgARROWLENGTH=1.5em \node{0}\arrow{e} \node{(\;,X_0)}\arrow{e,t}{g_0}\arrow{s,l}{(\;,v_0)}\arrow{s,r}{(\;,r_0t_0)} \node{(\;,X_1)}\arrow{e,t}{(\;,g)}\arrow{s,l}{(\;,v_1)}\arrow{s,r}{(\;,r_1t_1)} \node{(\;,X_2)}\arrow{s,l}{(\;,v_2)}\arrow{e}\arrow{s,r}{(\;,r_2t_2)} \node{H}\arrow{e}\arrow{s,l}{\rho} \node{0} \\ \node{0}\arrow{e} \node{(\;,M_0)}\arrow{e,t}{f_0} \node{(\;,M_1)}\arrow{e,t}{(\;,f)} \node{(\;,M_2)}\arrow{e} \node{F}\arrow{e} \node{0} \end{diagram}$$Then they are homotopic, and there exist maps $(\;,\lambda _{1}):(\;,X_{1})\rightarrow (\;,M_{0})$ and $(\;,\lambda _{2}):(\;,X_{2})\rightarrow (\;,M_{1})$, such that $$\begin{aligned} v_{1} &=&r_{1}t_{1}+f_{0}\lambda _{1}+\lambda _{2}g, \\ v_{2} &=&r_{2}t_{2}+f\lambda _{2}.\end{aligned}$$ We have the following commutative diagram: $$\begin{diagram}\dgARROWLENGTH=1em \node{X_1}\arrow{e,t}{g}\arrow{s,l}{m_1} \node{X_2}\arrow{s,l}{m_2}\\ \node{Z_1\coprod M_0\coprod M_1}\arrow{e,t}{w}\arrow{s,l}{n_1} \node{Z_2\coprod M_1}\arrow{s,l}{n_2}\\ \node{M_1}\arrow{e,t}{f} \node{M_2} \end{diagram}$$with morphisms $n_{1}=(r_{1}\;f_{0}\ 1_{M_{1}}),\;n_{2}=(r_{2}\;f_{2})$ and $$m_{1}=\left( \begin{array}{c} t_{1} \\ \lambda _{1} \\ \lambda _{2}g\end{array}\right) ,m_{2}=\left( \begin{array}{c} t_{2} \\ \lambda _{2}\end{array}\right) ,w=\left( \begin{array}{ccc} h & 0 & 0 \\ 0 & 0 & 1_{M_{1}}\end{array}\right) .$$But $w:Z_{1}\coprod M_{0}\coprod M_{1}\rightarrow Z_{2}\coprod M_{1}$ is in $\mathscr C,$ and $$\begin{diagram}\dgARROWLENGTH=1em \node{Z_1\coprod M_0\coprod M_1}\arrow{e,t}{w}\arrow{s,l}{n_1} \node{Z_2\coprod M_1}\arrow{s,l}{n_2}\\ \node{M_1}\arrow{e,t}{f} \node{M_2} \end{diagram}$$is a right $\mathscr{C}$-approximation of $M_{1}\xrightarrow{f}M_{2}$. We can define the functor $\varPhi^{op}:\mathrm{maps}(\mathrm{mod}\Lambda )\rightarrow \mathrm{mod(}(\mathrm{mod}\Lambda \mathcal{)}^{op})$ as: $$\varPhi^{op}(A_{1}\xrightarrow{f}A_{0})=\mathrm{Coker}((A_{0},-)\xrightarrow{(f,-)}(A_{1},-))$$We have the following dual of the above theorem, whose proof we leave to the reader: Let $\mathscr{C}\subset \mathrm{maps}(\mathrm{mod}\Lambda )$ be a subcategory that contains the objects of the form $(0,M,0)$ and $(M,M,1_{M})$. If $\varPhi^{op}(\mathscr C)$ is contravariantly finite, then $\mathscr{C}$ is covariantly finite. The subcategory $\mathscr{C}$ of $\mathrm{maps}(\mathrm{mod}\Lambda )$ consisting  of   all   maps $(M_{1},M_{2},f)$, such that $f$ is an epimorphism, will be called the category of epimaps, $\mathrm{epimaps}(\mathrm{mod}\Lambda )$. Dually, the subcategory $\mathscr{C}$ of $\mathrm{maps}(\mathrm{mod}\Lambda )$ consisting of all maps $(M_{1},M_{2},f)$, such that  $f$ is a monomorphism, will   be called   the   category of monomaps, $\mathrm{monomaps}(\mathrm{mod}\Lambda ).$ We have the following examples of functorially finite subcategories of the category $\mathrm{maps}(\mathrm{mod}\Lambda )$: The categories $\mathrm{epimaps}(\mathrm{mod}\Lambda )$ and $\mathrm{monomaps}(\mathrm{mod}\Lambda )$ are functorially finite in $\mathrm{maps}(\mathrm{mod}\Lambda ).$ Let $M_{1}\xrightarrow{f}M_{2}$ be an object in $\mathrm{maps}(\mathrm{mod}\Lambda )$. Then we have the following right approximation $$\begin{diagram}\dgARROWLENGTH=1em \node{M_1}\arrow{e,t}{f'}\arrow{s,l,=}{} \node{\mathrm{Im}(f)}\arrow{e,t}{} \arrow{s,l}{j} \node{0}\\ \node{M_1}\arrow{e,t}{f} \node{M_2} \end{diagram}$$ Consider an epimorphism $X_{2}\xrightarrow{g}X_{2}\rightarrow 0$ and a map in $\mathrm{maps}(\mathrm{mod}\Lambda )$ $(t_{1},t_{2}):(X_{1},X_{2},g)\rightarrow (M_{1},M_{2},f)$. Then we have the following commutative diagram: $$\begin{diagram}\dgARROWLENGTH=1em \node{X_1}\arrow{e,t}{g}\arrow{s,l}{t_1} \node{X_2}\arrow{e,t}{} \arrow{s,l}{t_2} \node{0}\arrow{s} \node{}\\ \node{M_1}\arrow{e,t}{f} \node{M_2}\arrow{e,t}{\pi} \node{\mathrm{Coker}(f)}\arrow{e} \node{0} \end{diagram}$$Since $\pi t_{2}=0$, the map $t_{2}:X_{2}\rightarrow M_{2}$ factors through $j:\mathrm{Im}(f)\rightarrow M_{2}$, this is: there is a map $u:X_{2}\rightarrow \mathrm{Im}(f)$ such that $ju=t_{2}$, and we have the commutative diagram: $$\begin{diagram}\dgARROWLENGTH=1em \node{X_1}\arrow{e,t}{g}\arrow{s,l}{t_1} \node{X_2}\arrow{e,t}{} \arrow{s,l}{u} \node{0}\\ \node{M_1}\arrow{e,t}{f'}\arrow{s,l,=}{} \node{\mathrm{Im}(f)}\arrow{s,l}{j}\\ \node{M_1}\arrow{e,t}{f} \node{M_2} \end{diagram}$$ Now, let $P\xrightarrow{p}M_{2}\rightarrow 0$ be the projective cover of $M_{2}$. We get a commutative exact diagram: $$\begin{diagram}\label{epimap} \dgARROWLENGTH=1em \node{M_1}\arrow{e,t}{f}\arrow{s,l}{(^1_0)} \node{M_2} \arrow{s,l}{1_{M_2}}\\ \node{M_1\oplus P}\arrow{e,t}{[f\; p]} \node{M_2}\arrow{e} \node{0} \end{diagram}$$ Let $X_{1}\xrightarrow{g}X_{2}\rightarrow 0$ be an epimorphism and $$\begin{diagram}\dgARROWLENGTH=1em \node{M_1}\arrow{e,t}{f}\arrow{s,l}{s_1} \node{M_2} \arrow{s,l}{s_2}\\ \node{X_1}\arrow{e,t}{g} \node{X_2}\arrow{e} \node{0} \end{diagram}$$ a map of maps. Since $P$ is projective, we have the following commutative square $$\begin{diagram}\dgARROWLENGTH=1em \node{P}\arrow{e,t}{p}\arrow{s,l}{\mu} \node{M_2} \arrow{s,l}{s_2}\\ \node{X_1}\arrow{e,t}{g} \node{X_2}\arrow{e} \node{0} \end{diagram}$$Then gluing the two squares: $$\begin{diagram}\dgARROWLENGTH=1em \node{M_1}\arrow{e,t}{f}\arrow{s,l}{(^1_0)} \node{M_2} \arrow{s,l,=}{}\\ \node{M_1\oplus P}\arrow{e,t}{[f\;p]}\arrow{s,l}{[s_1\; \mu]} \node{M_2}\arrow{s,l}{s_2}\arrow{e,t}{} \node{0}\\ \node{X_1}\arrow{e,t}{g} \node{X_2}\arrow{e} \node{0} \end{diagram}$$ we obtain the map $(s_{1},s_{2})$. Then (\[epimap\]) is a left approximation. The second part is dual. - The category $\mathrm{mod(}\Lambda )^{O}$ of functors vanishing on projectives is functorially finite. - The category $\hat{\mathrm{mod(}\Lambda )}$ of functors with $\mathrm{pd}\leq 1$ is functorially finite. The proof of this follows immediately from $$\mathrm{mod(}\Lambda )^{O}=\varPhi(\mathrm{epimaps}(\mathrm{mod}\Lambda ),\hat{\mathrm{mod(}\Lambda )}=\varPhi(\mathrm{monomaps}(\mathrm{mod}\Lambda ).$$ In view of the previous theorem it is of special interest to characterize the functors in $\mathrm{\mathrm{mod}}(\mathrm{mod}\Lambda )$ of projective dimension less or equal to one. The radical $t_{H}(F)$ of a finetly presented functor $F$, is defined as $t_{H}(F)=\underset{L\in \Theta }{\Sigma }L$, where $\Theta $ is the collection of subfunctors of $F$ of finite length and with composition factors the simple objects of the form $S_{M}$, with $M$ a non projective indecomposable module. Let $F$ be a finitely presented functor. Then $F$ is torsion free if and only if $t_{H}(F)=0.$ We end the paper with the following result, whose proof is essentially in \[19, Lemma 5.4.\] Let $F$ be a functor in $\mathrm{\mathrm{mod}}(\mathrm{mod}\Lambda )$. Then $F$ has projective dimension less or equal to one, if and only if, $F$ is torsion free. [**ACKNOWLEDGEMENTS.**]{} The second author thanks CONACYT for giving him financial support during his graduate studies. This paper is in final form and no version of it will be submitted for publication elsewhere. [MVO1]{} 505 - 574. (1974), 177 - 268. (1989), 5 - 37. (1991), 111 - 151. (1973), 8 - 71. (1974), 306 - 366. (1995). (1993), 3033 - 3097. (1993), 2995 - 3031. (1981), 26 - 38. (1980), 103 - 169. (1986), 397 - 409. (1975), 193 - 215. (1988). (1996), 487 - 504. (2007), 441 - 467. (2010), 1369 - 1407. (2009), 355 - 375. [^1]: [^2]: The second author thanks CONACYT for giving him financial support during his graduate studies [^3]: This paper is in final form and no version of it will be submitted for publication elsewhere.

--- author: - | T. Kasemets[^1] and T. Sjöstrand[^2]\ \ *Theoretical High Energy Physics*\ *Department of Astronomy and Theoretical Physics,*\ *Lund University,*\ *Sölvegatan 14A,*\ *SE 223 62 Lund, Sweden*\ title: 'A Comparison of new MC-adapted Parton Densities' --- Introduction ============ Precision measurements of cross sections should be compared with precision theoretical calculations. The state of the art is next-to-leading order (NLO) calculations of matrix elements (MEs), which then should be combined with NLO parton distribution functions (PDFs) to obtain the theoretical predictions for various processes at hadron colliders. Such an approach works well for sufficiently inclusive quantities, say the total cross section for $W^{\pm}$ production. At the other extreme, say for the production of jets associated with the $W^{\pm}$, experimental jet finding will be based on the clustering of a complex hadronic final state. Currently such states can only be modeled by the use of event generators, where many components are only formulated to leading order (LO), such as multiparton interactions (MPIs), initial-state radiation (ISR) and final-state radiation (FSR). In these descriptions it is thus not apropriate to use NLO PDFs. To be more specific, the combination of LO MEs with NLO PDFs is accurate to LO, just as LO MEs with LO PDFs would be. So from a formal point of view the use of NLO or LO PDFs is equivalent. There is a key point, however: LO PDFs have a simple probabilistic interpretation and are always positive, as are LO MEs. At NLO positivity is no longer required, neither for PDFs nor for MEs. The convolution of the two hopefully should still be positive, but even that is strictly not guaranteed over all of phase space. Characteristic for NLO PDFs is especially that the gluon distribution at small $x$ and $Q^2$ tends to come out negative, or at least very small, in order to describe the evolution of $F_2$ with $Q^2$. This means that the LO ME $+$ NLO PDF combination breaks down in this region, which is where most of the underlying-event activity originates (from MPIs, ISR and FSR). Thus there is a need to continue the development and use of LO PDFs. In recent years this field has received renewed attention [@mrstmod]-[@cteqmc]. In particular, attempts have been made to find LO PDFs that, when combined with LO MEs, attach better to the complete NLO behavior for a selection of cross sections. This is accomplished primarily by relaxing the momentum sum rule, in order to allow the PDFs to have a large small-$x$ gluon distribution without compromising the large-$x$ quark distributions. Studying these new PDFs can shed more light on which features of the PDFs are ideal for leading-order MC generators and how further improvements can be made. To this end we have incorporated ten new PDFs into <span style="font-variant:small-caps;">Pythia8</span> [@Pythia8] and compare the new MC-adapted PDFs with regular LO and NLO PDFs. We also study how the differences between them affect results of simulations, and compare the results with data collected by the CDF experiment. The observables studied are somewhat different from the ones used in the construction of the PDFs and thus give a complementary picture. The structure of the article is as follows. Section \[sec:PDFPYTHIA\] describes the inclusion of the PDFs in <span style="font-variant:small-caps;">Pythia8</span>. The different PDFs are compared in section \[sec:Compare\]. The results from simulations of minimum-bias events and hard QCD events are studied in sections \[sec:minbias\] and \[sec:jet\], respectively. Finally the article concludes with a summary and outlook in section \[sec:sum\]. PDFs in PYTHIA8 {#sec:PDFPYTHIA} =============== <span style="font-variant:small-caps;">Pythia8</span> has so far been distributed with the option to choose between two PDFs, GRV94L [@grv] and CTEQ5L [@cteq5l], which are both fairly old. Many new and improved PDFs have been released and made available to <span style="font-variant:small-caps;">Pythia8</span> simulations only through LHAPDF [@lhapdf]. The LHAPDF package has grown quite large and in the process also somewhat slow, and the code is written in Fortran while the community is changing to C++. It is desirable to include some PDFs directly into <span style="font-variant:small-caps;">Pythia8</span> since it can speed up simulations, make <span style="font-variant:small-caps;">Pythia8</span> more complete and facilitate the switching between different frequently used PDFs. We therefore incorporate ten new PDFs from the MRST [@mrstmod]-[@mrstmod2], MSTW 2008 [@mstw2008], CTEQ6 [@cteq6] and CTEQ MC [@cteqmc] distributions into <span style="font-variant:small-caps;">Pythia8</span>, listed in Tab. \[pdfrange\]. Two of them are NLO, which are not intended for LO MC use, but included for comparison. Inclusion of the PDFs was done in close contact with the MSTW and CTEQ collaborations. Including additional PDFs proved to be less straightforward than might first be expected, see further below. A major reason for this is the need to, in MC simulations, go outside the range of the PDF grids; specifically to smaller $x$ and $Q^2$ values. MSTW provides routines not only for interpolation but also for extrapolation outside this grid, while the CTEQ collaboration has recommended a freeze of the PDFs at the value just inside the grid. The range of the grids for the different PDFs are shown in Tab. \[pdfrange\], together with their $\alpha_S$ values and running. The code supplied by the authors had to be modified to fit natively into <span style="font-variant:small-caps;">Pythia8</span> and we carried out extensive tests. When possible the tests included comparisons with the corresponding PDFs in the LHAPDF package. The <span style="font-variant:small-caps;">Pythia8</span>-included PDFs run about a factor two faster than they do going the way via the LHAPDF package. PDF $x$ range $Q^2$ range \[GeV$^2$\] $\alpha_S$ $\alpha_S(M_Z)$ -------------- --------------- ------------------------- ------------ ----------------- GRV94L $10^{-5} - 1$ $0.40 - 10^6$ LO 0.128 CTEQ5L $10^{-6} - 1$ $1.00 - 10^8$ LO 0.127 MRST LO\* $10^{-6} - 1$ $1.00 - 10^9$ NLO 0.12032 MRST LO\*\* $10^{-6} - 1$ $1.00 - 10^9$ NLO 0.11517 MSTW LO $10^{-6} - 1$ $1.00 - 10^9$ LO 0.13939 MSTW NLO $10^{-6} - 1$ $1.00 - 10^9$ NLO 0.12018 CTEQ6L $10^{-6} - 1$ $1.69 - 10^8$ NLO 0.1180 CTEQ6L1 $10^{-6} - 1$ $1.69 - 10^8$ LO 0.1298 CTEQ66 (NLO) $10^{-8} - 1$ $1.69 - 10^{10}$ NLO 0.1180 CT09MC1 $10^{-8} - 1$ $1.69 - 10^{10}$ LO 0.1300 CT09MC2 $10^{-8} - 1$ $1.69 - 10^{10}$ NLO 0.1180 CT09MCS $10^{-8} - 1$ $1.69 - 10^{10}$ NLO 0.1180 MRST/MSTW --------- The PDFs supplied to us from MSTW were in some respects improved compared to the versions available in LHAPDF. Our implementation for the MRST LO\* and LO\*\* PDFs makex use of the new MSTW grid ($64\times48$) ranging down to $x=10^{-6}$ while the LHAPDF versions use the original grid with fewer ($49\times37$) grid points and shorter $x$ range ($x_{min}=10^{-5}$). The new grid results in a less steep gluon distribution towards small $x$, and at $x=10^{-8}$ the difference reaches a factor of two. The values of $\alpha_S$ are slightly different in the new LO\* and LO\*\* grid files than in the corresponding LHAPDF ones. The LHAPDF versions use $\Lambda_{QCD}$ for four active flavors, and the change to $\Lambda_{QCD}$ for five active flavors yields a slightly different value. Also worth noticing is that LO\* and LO\*\* both use the unorthodox value of the $Z$ boson mass, $M_Z = 91.71$ GeV, unlike the MSTW 2008 distribution which uses $M_Z=91.19$ GeV [@lhapdf]. For MSTW 2008 LO, LO\* and LO\*\* the interpolation gave negative gluon values at some large-$x$ intervals and for a wide range of $Q^2$ scales. The gluon distribution is very small in this region and therefore the negative values do not affect the results of the simulations. Furthermore, LHAPDF can give negative values for the up quark for $0.9 < x < 1.0$ at large $Q^2$, which is worse since the up quark dominates for large $x$ values. The MSTW NLO distribution gave very large negative values for the anomalous dimension $\frac{d\log(xf)}{d\log(Q^2)}$ at small $Q^2$ and $x$ values around $10^{-5}$, which resulted in a huge $\bar{s}$ distribution when extrapolated to low $Q^2$. This could also be a problem for the gluon which could get large negative anomalous dimensions in the $x$ region where the distribution is negative. To avoid this the anomalous dimension is manually forced to be larger than $-2.5$. Although this does fix the issue at hand, it is also an example of the dangers of using NLO PDFs in LO MC simulations, and an indication that one has to be very careful with such use. CTEQ 6/MC --------- The CTEQ distributions work well inside the grid but outside or near the edges some problems occurred. The $tv = \log(\log(Q))$ values in the grid file were discovered not to exactly correspond to the $Q$ values in the same file, and hence some points inside the $Q$ grid would end up outside the $tv$ grid. This caused some strange errors, for example the $b$ quark distribution, after being zero below the threshold, suddenly became huge at $Q^2$ values just inside the grid. Therefore we choose to read in only the $Q$ grid points and then calculate $tv$. Outside the grid there can be differences between the CTEQ6 PDFs in <span style="font-variant:small-caps;">Pythia8</span> and the corresponding ones in LHAPDF. This is because LHAPDF provides the option to use extrapolation routines where <span style="font-variant:small-caps;">Pythia8</span>, by recommendation from the CTEQ authors, freezes the values. Comparison of PDFs {#sec:Compare} ================== There are strong similarities between the PDFs but also large differences, especially in the small $x$ region. Broadly the LO PDFs are similar, except for MSTW LO which is much larger than the others for small $x$ values, the MC-adapted PDFs are similar and the NLO PDFs are similar. Comparing the two groups, the CTEQ distributions have smaller distributions at small $x$, both for the PDFs that freeze and for the ones with the grid ranging down to $10^{-8}$, except for the NLO distributions. The two NLO PDFs have a smaller gluon distribution at small $x$, and MSTW NLO is negative at $Q^2=4$ GeV$^2$. The gluon distribution is dominating in the region of small $x$ while the valence quarks, and then especially the up quarks, dominate for large $x$. For a first comparison we therefore choose to focus mainly on these two distributions. The PDFs are different from one another in several aspects. Four of them do not obey the momentum sum rule. LO\* carries 1.12 times the proton momentum, LO\*\* a $Q^2$-dependent number between 1.17 and 1.14, MC1 1.10 and MC2 1.15. The special behavior of LO\*\* is related to the use of $\alpha_S(p_{\perp}^2)$ instead of $\alpha_S(Q^2)$ in the evolution. The MC-adapted distributions from CTEQ have pseudodata from full NLO calculations included in their fit. MCS is the only MC-adapted PDF that does not break the momentum sum rule, but instead has more freedom in the parameterization: renormalization and factorization scales are allowed to vary in the global fit. Minimum-bias events are sensitive to low $Q$; a $Q^2$ around $4$ GeV$^2$ is a typical scale for such simulations. The up distributions for large $x$ at $Q^2=4$ GeV$^2$ in Fig. \[fig:xfQ24\] are all similar, with slight differences for the two NLO PDFs and CT09 MC1 and MC2. The gluon distributions, on the other hand, show large differences in the small-$x$ region: note the difference in horizontal scale. Especially MSTW LO has a much steeper rise and becomes much larger than the others. All the MRST/MSTW distributions give larger values at small $x$ than the CTEQ ones. The MC-adapted PDFs from the respective collaboration follow each other, except for MCS which is more similar to CTEQ6L and CTEQ6L1. The two NLO PDFs stand apart from the rest, and MSTW NLO is negative in a large region. One can also note that CTEQ5L, CTEQ6L and CTEQ6L1 all freeze at $x=10^{-6}$. Fig. \[fig:xfQ21e3\] shows the distributions at $Q^2=10^3$. Both the up and the gluon distributions show the same relative patterns as at $Q^2=4$. However, the differences between the PDFs are smaller, and especially the difference between the two groups are no longer as prominent, except where some PDFs freeze. At this $Q^2$ the CTEQ MC-adapted PDFs, in Fig. \[fig:gLogxQ21e3-2\], are all similar. \ \ Minbias Events {#sec:minbias} ============== Minbias events are interesting in their own right, but also because they constitute a background when studying hard interactions. They tend to have low average transverse energy, low particle multiplicity and consist largely of soft inelastic interactions. Because of the low $p_{\perp}$ the interacting partons only need a small portion of the momenta of the incoming hadrons, and hence minbias events probe parton distributions in the small-$x$ region dominated by the gluon distribution. We examine the rapidity and multiplicity distributions from simulations with different PDFs, both at Tevatron and LHC energies. With the aid of Rivet [@Rivet] we also compare $p_{\perp}$ and $\sum E_{\perp}$ particle spectra, as well as average $p_{\perp}$ evolution with multiplicity, with data collected by the CDF experiment at Tevatron Run 2 [@Aaltonen2]. Multiplicity and Tuning ----------------------- A generator using the different PDFs needs to be retuned separately for each of them before we can make any reasonable comparisons. Specifically, the larger momentum carried by the partons in LO\*, LO\*\*, MC1 and MC2 allows a larger activity and hence a larger multiplicity than with the ordinary leading-order PDFs. Furthermore the NLO PDFs give less activity, owing to the small gluon at low $x$ and $Q^2$. We have chosen to tune <span style="font-variant:small-caps;">Pythia8</span> so that all PDFs have the same average charged-particle multiplicity as CTEQ5L. This PDF is taken as reference because it is the default PDF in <span style="font-variant:small-caps;">Pythia8</span> and is most commonly used in <span style="font-variant:small-caps;">Pythia8</span> simulations. We are not making a complete tune and only intend to get a first impression of relative differences, under comparable conditions. The tuning is accomplished by tweaking the $p_{\perp 0}^{{\mbox{\ssmall{Ref}}}}$ parameter in <span style="font-variant:small-caps;">Pythia8</span> $$p_{\perp 0}=p_{\perp 0}^{{\mbox{\ssmall{Ref}}}}\left( \frac{E_{{\mbox{\ssmall{CM}}}}}{E_{{\mbox{\ssmall{CM}}}}^{{\mbox{\ssmall{Ref}}}}}\right)^p$$ where $E_{{\mbox{\ssmall{CM}}}}^{{\mbox{\ssmall{Ref}}}} = 1800$ GeV and $p = 0.24$. $p_{\perp 0}$ is used for the regularization of the divergence of the QCD cross section as $p_{\perp}\rightarrow 0$. A smaller $p_{\perp0}$ cause the regularization to take effect at a lower $p_{\perp}$, increasing the charged particle multiplicity, $n_{ch}$. The tuning was done for the $\alpha_S$ value and leading-order running which is default in <span style="font-variant:small-caps;">Pythia8</span>. Results are shown in Tab. \[charmult1\]. The largest multiplicity before retuning is obtained with LO\*\*, followed by MC2 and LO\* which are also the PDFs with most momentum. Further tuning could well bring some of the results closer together, but it is well known that $p_{\perp 0}^{{\mbox{\ssmall{Ref}}}}$ is the single most crucial parameter for obtaining overall agreement with minimum-bias data at a given energy. It is also a parameter without any constraints from theory. Other key min-bias parameters, related to the impact-parameter picture, the colour-reconnection mechanism or the beam-remnant handling, do not have as direct a coupling to the choice of PDF. The parton showers, by contrast, are better controlled by basic principles. Even if there are different shower schemes being proposed and used, within a given scheme there is no degree of freedom intended to counteract the PDF (and associated $\alpha_S$) choice, in the way that $p_{\perp 0}^{{\mbox{\ssmall{Ref}}}}$ is. The exception would be $K$ factors for cross sections, which we will comment on later. Hadronization, finally, is fixed by the LEP data, and should not be touched. PDF $\langle n_{ch} \rangle$ $p_{\perp 0}^{{\mbox{\ssmall{Ref}}}}$ ------------- -------------------------- --------------------------------------- CTEQ5L 54.48 2.25 MRST LO\* 59.74 2.50 MRST LO\*\* 63.52 2.63 MSTW LO 49.10 2.06 MSTW NLO 48.02 1.56 CTEQ6L 54.92 2.25 CTEQ6L1 51.71 2.13 CTEQ66 42.85 1.75 CT09MC1 53.92 2.25 CT09MC2 60.37 2.50 CT09MCS 54.87 2.25 : Average charged particle multiplicity for the different PDFs with the default value of $p_{\perp 0}^{{\mbox{\ssmall{Ref}}}} = 2.25$ and also the $p_{\perp 0}^{{\mbox{\ssmall{Ref}}}}$ required to tune the charge multiplicity to be equal to the value for CTEQ5L[]{data-label="charmult1"} Generator-level --------------- In this section all simulations are for $p\bar{p}$ collisions at 1960 GeV, if not explicitly stated otherwise. So as not to crowd the plots, not all sets are shown all the time. The selection of PDFs that are shown each figure represents both the extremes and the middle way. The rapidity distributions of the outgoing particles at the parton level, when only the $2 \rightarrow 2$ sub-process is considered, are presented in Fig. \[fig:Rapid-strip\]. MSTW LO gives a broader distribution than the rest of the PDFs as an effect of the large gluon distribution at small $x$. The distributions with LO\*\* and MC2 closely resemble each other, as do the distributions with LO\* and MC1. This may be explained by the similarities in the violation of the momentum sum rule. The NLO PDFs give lower values in the central rapidity region. The remaining leading-order PDFs give results which are all similar to the MC-adapted ones. Turning on the rest of the <span style="font-variant:small-caps;">Pythia8</span> machinery, the distribution of charged particles after hadronization, shown in Fig. \[fig:Rapid\], changes in shape. There are now more particles at larger rapidities as a result of the fragmenting color field strings stretched out to the beam remnants, and most of the differences between the PDFs get blurred. Some differences still remain. MSTW LO still gives smaller multiplicity at central rapidities, and as a remnant of the wider distribution lack the inward dents that all other PDFs give at rapidities around $\pm5$. The peaks for LO\*\* and MC2 are somewhat sharper than for LO\* and MC1, but the trace of the lower value with NLO PDFs at central rapidity has vanished. The multiplicity distributions are similar for most PDFs. The two NLO distributions stand out as two extremes in different directions, MSTW NLO here gives the highest peak and the shortest tail as shown in Fig. \[fig:Multdist\]. All MC-adapted PDFs, except MCS, give a peak slightly shifted to larger multiplicities but are different in height, where the two from MRST give a larger peak value. The three normal leading-order distributions are all similar and we only show the CTEQ6L. Increasing the energy to the level of a fully operational LHC enhances the differences seen at Tevatron energy, especially for MSTW LO and the two NLO PDFs. The multiplicity of these three evolve with energy in a different way than for the other PDFs. The rapidity distribution, shown in Fig. \[fig:Rapid-LHC\], naturally extends to larger rapidities and shows a higher central activity. MSTW LO here gives a much broader distribution. This is because as the energy increases even lower values of $x$ come into play, so that the effect of the gluon distribution in this region has larger impact on the results. The two NLO PDFs result in flatter peaks than the MC-adapted PDFs and are similar in shape to MSTW LO. This can be explained by the smaller gluon distribution at small $x$. The rest of the PDFs evolve in a fashion similar to the MC-adapted PDFs shown in the figure, but with some more variation. MC1 and LO\* results in a little bit larger distributions at central rapidities than MC2 and LO\*\*. The multiplicity distributions in Fig. \[fig:Mult-LHC\] also show the effect of the low-$x$ gluon distribution. MSTW LO has a much higher total multiplicity. The two NLO PDFs converge at this energy but they have smaller multiplicity than MSTW LO because of their small gluon distribution at small $x$. In general the same distributions that stand out with their rapidities also do so with their charge particle multiplicity distributions. Comparison with CDF Run 2 data ------------------------------ The analyses in Rivet ensure that the comparisons to data have the same cuts and corrections as the original experiment. Therefore only the central pseudorapidity region is used and cuts in transverse momentum are implemented [@Aaltonen2]. $p_{\perp}$ spectra of charged particles in Fig. \[fig:SigmapT-Riv\] show the same relative shape for all PDFs, which gives too large values at the low-$p_{\perp}$ end, then decreases compared to data and gives too small differential cross sections at the high end. The slope shows some differences depending on the choice of PDF. MC-adapted PDFs and the CTEQ6L give results that are the closest to data, while MSTW LO and NLO are further away than the rest. The $\sum E_{\perp}$ spectrum of particles, neutral particles included, shows larger dependence on the PDFs, but the trend of disagreement of the PDFs with the data is the same as that noted for $p_{\perp}$. Since we have not done a complete tune, differences do indicate the importance of the PDFs. Further tuning could well bring curves closer to each other, but the $\sum E_{\perp}$ distribution is less dependent on details of the MC and therefore easier for PDF developers to consider in tunes. The MC-adapted PDFs reproduce data well while the LO and NLO PDFs give results which are further away. The exception is CTEQ6L which also gives results close to data, while MSTW LO goes down to less than half the cross section of data at the larger energy end. MSTW NLO results peak at higher energies than the rest. All PDFs give a too large value at the peak, but then decrease too fast and differ the most from data at the high energy end. Although different PDFs result in differences in many observables, they sometimes give more similar results. For example the evolution of the average transverse momentum with charge multiplicity reproduce the data fairly well, independent of the choice of PDF. They all give slightly too low $\langle p_{\perp}\rangle$ at low multiplicity and then increase relative to data so that they get closer as the multiplicity increases, see Fig. \[fig:pTCMult-Riv\]. The only PDF that gives a slightly different evolution is MSTW LO. Inclusive Jet Cross Section {#sec:jet} =========================== We compared the inclusive jet cross sections from our MC simulations with data collected by the CDF experiment at Tevatron Run 2 [@Aaltonen], over five pseudorapidity intervals ranging up to $\eta \leq 2.1$. In the experimental analysis the jets are identified with the midpoint cone algorithm and also compared to results with the $k_T$ algorithm [@Salam]. We also examined the rapidity, multiplicity and transverse momentum distributions for the individual hadrons. Two rapidity intervals and the transverse momentum distribution have been selected to show the cross section and illustrate our findings. The inclusive jet cross section drops rapidly with increasing $p_{\perp}$ and spans over several orders of magnitude. Since this makes differences between experiments and simulations difficult to distinguish, we only show the MC/data ratio in the following figures. The results with MRST LO\*\*, MSTW LO, CTEQ6L1, CTEQ66, CT09MC1 and CT09MC2 are shown in Fig. \[fig:jetCross2\]. Generally the LO PDFs give similar results, as do the MC-adapted PDFs and the NLO PDFs are similar as well. MRST LO\*\* starts with a much too large cross section and the ratio quickly decreases when $p_{\perp}$ rises. This behavior is the strongest for LO\*\* at low pseudorapidity; at larger $\eta$ the ratio gets smaller and flatter. All MC-adapted PDFs, except MCS, show this type of behavior. MC2 and MC1 give results with very similar shapes but the MC2 cross section is larger, and MRST LO\*\* and LO\* are related much in the same fashion. MSTW LO and CTEQ6L give cross sections which have similar behavior, and the ratio is much less dependent on $p_{\perp}$ than with the MC-adapted PDFs. CTEQ6L1 gives a ratio which starts to decrease with $p_{\perp}$ at larger rapidities. CT09MCS gives a too low cross section, is once again different from the other MC-adapted PDFs, and gives results which behave in a way more similar to those of the normal leading-order distributions. In the central rapidity regions the NLO PDFs actually give the cross sections closest to data with a ratio close to $1$, but their ratios decrease towards $0.5$ at larger $\eta$. \ \ Summary and Outlook {#sec:sum} =================== Including the very latest parton distribution functions into <span style="font-variant:small-caps;">Pythia8</span> both caused some technical troubles and gave some surprises, especially while venturing outside the grid of the PDFs. At several occasions we were reminded that it can be risky to use NLO PDFs in LO MC generators. In addition we found that there is a need for improved numerical stability at large $x$, in order to keep the leading-order PDFs from going negative. There is also a need for better understanding of parton distribution functions at small $x$, where the PDFs are now very different from each other. Excluding the NLO PDFs, the MRST/MSTW distributions are much larger than the ones from CTEQ, and MSTW LO goes sky high compared to all other PDFs. At larger $Q^2$ the differences are smaller between the two collaborations, except for the distributions that freeze their values at the end of the grid, and for MSTW LO which is still much larger than the rest. The quarks, and in particular the up distributions, at large $x$ show smaller differences, but MSTW LO is once again larger at small $x$. The large differences in the PDFs get blurred when looking at simulation results, but do nonetheless sometimes cause large variations. The MC-adapted PDFs frequently give results closer to data than the LO and NLO PDFs in their respective group. The CTEQ PDFs have distributions and give results that are more homogeneous, while the MRST/MSTW PDFs give a broader spectrum. The MC-adapted PDFs from MRST give results that show resemblance to results with the CTEQ distributions. MSTW LO have a much larger gluon distribution at small $x$, which affect results of, especially, minbias simulations. The rapidity distribution is wide at the parton level ($2\rightarrow 2$ subprocess) and at hadron level lack the inward dents at larger rapidities, which all other PDFs give. The $p_{\perp}$ and $\sum E_{\perp}$ spectrum with this PDF have large deviations from experimental data. The inclusive jet cross section is too low and the ratio to data is fairly constant with both $p_{\perp}$ and $\eta$. The multiplicity increases with energy at a more rapid rate than for all other PDFs. CTEQ6L and CTEQ6L1 have a larger gluon distribution than the CTEQ MC-adapted PDFs at small $x$ for low $Q^2$, down to the freezing point, but much smaller than MSTW LO. CTEQ6L cause a rapidity distribution with a narrow peak at parton level and a slightly larger peak value at hadron level. The two distributions give similar results, but 6L is usually closer to data and hence also closer to results with the MC-adapted PDFs from CTEQ. The two NLO PDFs, MSTW NLO and CTEQ66 are similar and also give similar results to each other for almost all observables in this study. They have the small gluon distributions characteristic of NLO PDFs, and give rapidity distributions with a low peak value at central rapidities. They are generally further away from the data in the minbias simulations, and give a slower multiplicity evolution with energy than the LO PDFs. Their inclusive jet cross section is close to data at small pseudorapidities but too low at large. The MC-adapted PDFs which break the momentum sum rule, i.e. MRST LO\*/\*\* and CT09 MC1/2, generally give similar results to each other. They give high and narrow peaks in the rapidity distributions and are generally closer to data than the ordinary LO and NLO PDFs from their respective collaboration. The main exception is the inclusive jet cross section at low pseudorapidities and $p_{\perp}$, where they give too large values. Here they also give a relative decrease with $p_{\perp}$, a rather surprising trend which is most prominent for LO\* and LO\*\*, especially at low pseudorapidities. In the simulations of hard QCD events rapidity distributions with the MC-adapted PDFs were narrower, and their multiplicity distributions were shifted to lower multiplicity. Interesting to note is that the CT09MCS seems to have some of the features of the other MC-adapted PDFs, but in some contexts gives results more similar to ordinary leading-order PDFs. For the leading-order PDFs a constant $K$-factor could improve the fit to the inclusive jet data, but for the MC-adapted PDFs the difference in shape makes it more complicated. Differences in the PDFs have a larger impact when the CM-energy of the collisions increases, and this can cause large uncertainties in simulations at LHC energies. Many of the differences found can be explained by the differences in the gluon distribution at small $x$, where we see that the middle way represented by the MC-adapted PDFs give results closer to data for many observables. It is reasonable to suspect that the results with the LO PDFs from CTEQ would resemble those of MSTW LO if extrapolated towards small $x$. At this point there is no final answer as to which PDF gives the best results. In order to answer this question a much broader spectrum of observables is needed and complete tunes for the different PDFs. This study does however highlight some of the relative differences between the PDFs when they are used under comparable conditions. During the last years there has been a renewed interest in LO tunes with focus on the applicability in MC generators. The MC-adapted PDFs resulted in some remarkable differences compared with leading-order PDF. They give results that are closer to data for many observables, although not for all. Thus there is space for further improvements. With these new PDFs a broader spectrum of tools is gained in <span style="font-variant:small-caps;">Pythia8</span> and in examining the origin of differences and similarities between simulations and experiments. Investigations are likely to continue in relation to LHC physics. [99]{} A. Sherstnev and R. S. Thorne, (2008), arXiv:0807.2132v1 \[hep-ph\]. A. Sherstnev and R. S. Thorne, *Eur. Phys. J.* **C55**, 553 (2008), arXiv:0711.2473 \[hep-ph\]. H. L. Lai, J. Huston, S. Mrenna, P. Nadolsky, D. Stump, W. K. Tung and C. P. Yuan, (2009), arXiv:0910.4183 \[hep-ph\]. T. Sjöstrand, S. Mrenna and P. Skands, *Comput. Phys. Commun* **178** (2008) 852-867, arXiv:0170.3820 \[hep-ph\]. M. Glueck, E. Reya and A. Vogt, *Z.Phys.* **C67** (1995) 433. H. L. Lai et al. \[CTEQ Collaboration\], *Eur. Phys. J.* **C12** (2000) 375 LHAPDF, http://hepforge.cedar.ac.uk/lhapdf/. A. D. Martin, W. J. Stirling, R. S. Thorne and G. Watt, *Eur. Phys. J.* **C63** (2009) 189, arXiv:0901.0002 \[hep-ph\]. J. Pumplin, D. R. Stump, J. Huston, H. L. Lai, P. Nadolsky and W. K. Tung, *JHEP* **0207** (2002) 012, arXiv:hep-ph/0201195. M. Bähr et.al., *Eur. Phys. J* **C58** (2009) 639, arXiv:0803.0883 \[hep-ph\] A. Buckley et.al., *Eur. Phys. J* **C65** (2010) 331, arXiv:0907.2973 \[hep-ph\]. T. Aaltonen et al. \[CDF Collaboration\], *Phys. Rev.* **D79** (2009) 112005, arxiv:0904.1098v2 \[hep-ex\]. T. Aaltonen et al. \[CDF Collaboration\], *Phys. Rev.* **D78** (2008) 052006, arXiv:0807.2204 \[hep-ex\]. G. P. Salam, (2009), arXiv:0906.1833v1 \[hep-ph\]. [^1]: tomas@kasemets.se [^2]: torbjorn@thep.lu.se

--- abstract: 'The multipath-rich wireless environment associated with typical wireless usage scenarios is characterized by a fading channel response that is time-varying, location-sensitive, and uniquely shared by a given transmitter-receiver pair. The complexity associated with a richly scattering environment implies that the short-term fading process is inherently hard to predict and best modeled stochastically, with rapid decorrelation properties in space, time and frequency. In this paper, we demonstrate how the channel state between a wireless transmitter and receiver can be used as the basis for building practical secret key generation protocols between two entities. We begin by presenting a scheme based on level crossings of the fading process, which is well-suited for the Rayleigh and Rician fading models associated with a richly scattering environment. Our level crossing algorithm is simple, and incorporates a self-authenticating mechanism to prevent adversarial manipulation of message exchanges during the protocol. Since the level crossing algorithm is best suited for fading processes that exhibit symmetry in their underlying distribution, we present a second and more powerful approach that is suited for more general channel state distributions. This second approach is motivated by observations from quantizing jointly Gaussian processes, but exploits empirical measurements to set quantization boundaries and a heuristic log likelihood ratio estimate to achieve an improved secret key generation rate. We validate both proposed protocols through experimentations using a customized 802.11a platform, and show for the typical WiFi channel that reliable secret key establishment can be accomplished at rates on the order of 10 bits/second.' author: - 'Chunxuan Ye$^\dag$, Suhas Mathur$^\ddag$, Alex Reznik$^\dag$, Yogendra Shah$^\dag$, Wade Trappe$^\ddag$and Narayan Mandayam$^\ddag$ [^1] [^2] [^3][^4] [^5][^6] [^7]' title: 'Information-theoretically Secret Key Generation for Fading Wireless Channels' --- Introduction ============ The problem of secret key generation from correlated information was first studied by Maurer [@Mau93], and Ahlswede and Csiszár [@AhlCsi93]. In a basic secret key generation problem, called the [*basic source model*]{}, two legitimate terminals (Alice and Bob)[^8] observe a common random source that is inaccessible to an eavesdropper. Modeling the observations as memoryless, we can define the model as follows: Alice and Bob respectively observe $n$ independent and identically distributed (i.i.d.) repetitions of the dependent random variables $X$ and $Y$, denoted by $X^n=(X_1,\cdots, X_n)$ and $Y^n=(Y_1, \cdots , Y_n)$. In any given time instance, the observation pair $(X_i, Y_i)$ is highly statistically dependent. Based on their dependent observations, Alice and Bob generate a common secret key by communicating over a public error-free channel, with the communication denoted collectively by ${\bf V}$. A random variable $K$ with finite range ${\cal K}$ represents an [*$\varepsilon$-secret key*]{} for Alice and Bob, achievable with communication ${\bf V}$, if there exist two functions $f_A$, $f_B$ such that $K_A=f_A(X^n, {\bf V})$, $K_B=f_B(Y^n, {\bf V})$, and for any $\varepsilon>0$, $$\Pr(K=K_A=K_B)\geq 1-\varepsilon,\label{e2.1}$$ $$I(K; {\bf V})\leq \varepsilon, \label{e2.2}$$ $$H(K) \geq \log |{\cal K}|-\varepsilon. \label{e2.3}$$ Here, condition (\[e2.1\]) ensures that Alice and Bob generate the same secret key with high probability; condition (\[e2.2\]) ensures such secret key is effectively concealed from the eavesdropper observing the public communication ${\bf V}$; and condition (\[e2.3\]) ensures such a secret key is nearly uniformly distributed. An achievable secret key rate $R$ is defined [@Mau93], [@AhlCsi93] to be a value such that for every $\varepsilon>0$ and sufficiently large $n$, an $\varepsilon$-secret key $K$ is achievable with suitable communication such that $\frac{1}{n}H(K)\geq R-\varepsilon$. The supremum of all achievable secret key rates is the [*secret key capacity*]{} denoted by $C_{SK}$. For the model presented above, this is given by [@Mau93], [@AhlCsi93], [@Mau94], [@MauWol00] $$C_{SK} = I(X;Y). \label{e2.3a}$$ This result holds for both discrete and continuous random variables $X$ and $Y$, as long as $I(X; Y)$ is finite (cf. [@YeRez06], [@Nit08]). The model defined above assumes the eavesdropper (i.e. Eve) may observe the transmissions on the public channel, but is unable to tamper with them and has no access to any other useful side information. The case of an eavesdropper with access to side information has received significant attention (see, e.g., [@Mau93], [@AhlCsi93], [@RSW_IT03], [@MRW_IT03]); unfortunately the capacity problem remains open in this case. The case of an eavesdropper with the ability to tamper with the transmissions on the public channel has been addressed in a comprehensive analysis by Maurer and Wolf [@Mau97], [@MauWol03a], [@MauWol03b], [@MauWol03c]. A practical implementation of secret-key agreement schemes follows a basic 3-phase protocol defined by Maurer *et.al.*. The first phase, [*advantage distillation*]{} [@Mau93], [@CacMau97a], is aimed at providing two terminals an advantage over the eavesdropper when the eavesdropper has access to side information. We do not consider this scenario (as we shall see shortly, it is not necessary for secrecy generation from wireless channels) and, therefore, do not address [*advantage distillation*]{}. The second phase, [*information reconciliation*]{} [@BenBra86], [@BenBes92], [@BraSal94], is aimed at generating an identical random sequence between the two terminals by exploiting the public channel. For a better secret key rate, the entropy of this random sequence should be maximized, while the amount of information transmitted on the public channel should be minimized. This suggests an innate connection between the information reconciliation phase of the secrecy agreement protocol and Slepian-Wolf data compression. This connection was formalized by [@CsiNar04] in the general setting of multi-terminal secrecy generation. The connection between secrecy generation and data compression is of significant practical, as well as theoretical interest. Considering the duality between Slepian-Wolf data compression and channel coding (e.g., [@GarZha01], [@LivXio02], [@PraRam03], [@ColLee06], [@CheHe08], etc), the relationship between secrecy generation and data compression allows capacity-achieving channel codes, like Turbo codes or LDPC codes, to be used for the information reconciliation phase. Moreover, the capacity-achieving capabilities of such codes in the channel coding sense carry over to the secrecy generation problem. A comprehensive treatment of the application and optimality of such codes to the secrecy generation problem can be found in [@BloTha06], [@BloBar08]. The last phase of Maurer’s protocol, [*privacy amplification*]{} [@BenBra88], [@BenBra95], extracts a secret key from the identical random sequence agreed to by two terminals in the information reconciliation phase. This can be implemented by linear mapping and universal hashing [@CarWeg79], [@WegCar81], [@BenBra95], [@MauWol03c], or by an extractor [@RazRei99], [@MauWol03c], [@DodKat06], [@DodOst08], [@CraDod08]. The combination of the information reconciliation phase and the privacy amplification phase has been considered in [@CacMau97a], [@YeNar05a]. Perhaps the first practical application of the basic source model is quantum cryptography (cf. e.g., [@BenBra92],[@NieChu00]), where non-orthogonal states of a quantum system provide two terminals correlated observations of randomness which are at least partially secret from a potential eavesdropper. Quantum key distribution schemes based on continuous random variables have been discussed in [@GroVan03], [@VanCar04], [@BloTha06], [@LodBlo07]. Less realized is the fact that wireless fading channel provides another source [@HerHas95], [@YeRez06], [@BloBar08] of secrecy which can be used to generate information-theoretically secure keys. Because the source model for secrecy establishment essentially requires *a priori* existence of a “dirty secret” which is then just cleaned up, such sources of secrecy are hard to find. To our knowledge no such sources other than quantum entanglement and wireless channel reciprocity have been identified to date. Further, we note that although there have been several implementations of quantum cryptographic key establishment, little work has been done to provide a system validation of this process for wireless channels. This paper examines both theoretical and practical aspects of key establishment using wireless channels and represents one of the first validation efforts to this effect. An alternative approach to secrecy generation from wireless channels is based on the wiretap channel models, see e.g.[@BloBar08]. However, this approach suffers from a need to make certain assumptions as part of the security model that are hard to satisfy in practice and has not, to date, led to a practical implementation. A (narrowband) wireless channel is well modeled as a flat fading channel. The fading coefficient changes in time, but the change is rather slow (on the order of 1 msec to 1 sec, depending on terminal velocities and other factors). For simplicity, let us consider frequency flat fading. Roughly speaking, for a fixed time and location, the transmitted signal $t$ and the received signal $r$ are related via $r=Ft+Z$, where $F$ is the channel fading coefficient and $Z$ is the additive independent noise. If the transmitted signal $t$ is known at the receiver beforehand, (e.g., it is a training sequence) then the receiver is able to obtain a noisy estimate of the fading coefficient $F$. Furthermore, if both terminals send the training sequence at approximately the same time (more precisely, well within one channel coherence time of each other), then they can obtain channel estimates that are highly correlated due to channel reciprocity. This suggests the following model: let the random variables $X$ and $Y$ be defined by $ X=F+Z_A $, $ Y=F+Z_B $, where $F$, $Z_A$, $Z_B$ are three independent random variables. In data communications application, it is common to model the channel as Rayleigh or Rician, in which case, $F$, $Z_A$ and $Z_B$ are Gaussian. Let these be distributed as ${\cal N}(0,P)$, ${\cal N}(0,N_A)$ and ${\cal N}(0,N_B)$ respectively. A simple calculation shows that the secret key capacity [@YeRez06] of this jointly Gaussian model is $$C_{SK}= \log_2\left(1+\frac{P}{N_A+N_B+\frac{N_AN_B}{P}}\right)\ bits/sample. \label{e2.6}$$ If we let $N_A= N_B =N$ in this setting, then we get a natural definition of SNR as ${\mbox{\sffamily\upshape{\scriptsize{SNR}}}}= \frac{P}{N}$, and the above secret key capacity reduces to $\log_2\left(1+\frac{SNR}{2+1/SNR}\right)$ bits/sample. As noted, the above calculation is relevant for the traditional Rayleigh or Rician fading model, and serves as an upper bound on the secret key establishment rate, but does not provide insight into how one can practically extract such secret bits from the underlying fading process. In this paper, we examine two different approaches for secrecy extraction from the channel state between a transmitter and receiver in a richly scattering wireless environment. Our first approach, which is based on level-crossings, is a simple algorithm that is well-suited for environments that can be characterized as Rayleigh or Rician. However, we recognize that such a method might not apply to other, general fading cases. One way to address this problem is to consider more complex fading distribution models, such as those appropriate for ultrawideband channels. This has been addressed in a previous work by Wilson [*et. al*]{} [@WilTse07] (see also [@AonHig05], [@ImaKob06], [@AziKia07]). However, we take a different approach in this paper. Inspired by our prior work on Gaussian-based approaches, we propose a [*universal*]{} reconciliation approaches for wireless channels. This second, and more powerful method, only assumes that the channel impulse responses (CIRs) measured at both terminals are highly correlated, and their measurement noise is very low. Whereas the first of our two approaches was simple, and able to achieve a limited secret key establishment rate, our second approach is more complex, but is able to take better advantage of the secrecy capabilities offered by CIR measurements, which tend to have high SNR (due to a high processing gain associated with such measurements in modern communication systems). In both of these cases, our goal is to come up with a practical approach to secrecy generation from wireless channel measurements. In particular, because the statistics of the real channel sources we utilize are not known (and that is the major challenge we believe addressed by our work), it is impossible to make any quantitative statements about optimality of our approaches. Nevertheless, we do want to make sure that our solution is based on solid theoretical foundation. To do so, we include discussion of the motivating algorithms and their performance in idealized models when necessary. Several previous attempts to use wireless channels for encrypting communications have been proposed. Notably, [@KooHas00] exploited reciprocity of a wireless channel for secure data transformation; [@HasSta96] discussed a secrecy extraction scheme based on the phase information of received signals; the application of the reciprocity of a wireless channel for terminal authentication purpose was studied in [@PatKas07], [@XiaGre07], [@XiaGre08], etc. Unlike these and other approaches, our approach for direct secrecy generation allows the key generation component to become a “black box” within a larger communication system. Its output (a secret bit stream) can then be used within the communication system for various purposes. This is important, as the key generation rate is likely to be quite low, and thus direct encryption of data will either severely limit throughput (to less than 1 kbps in indoor channels) or result in extremely weak secrecy. The adversary model assumed in this paper focuses mainly on passive attacks. We do not consider authentication attacks, such as the man-in-the-middle attack, since these require an explicit authentication mechanism between Alice and Bob and cannot be addressed by key-extraction alone. The starting point for algorithms presented in this paper is the successive probing of the wireless channel by the terminals that wish to extract a secret key. Implicitly, we assume that the adversary is not engaging in an active attack against the probing process, though we note that physical layer authentication techniques, such as presented in [@XiaGre08] might be applicable in such an adversarial setting. The infeasibility of passive eavesdropping attacks on the key generation procedures is based on the rapid spatial decorrelation of the wireless channel. We demonstrate this using empirically computed mutual information from the channel-probing stage, between the signals received at Bob and Eve and comparing it with the mutual information between the signals received at Alice and Bob. Beyond the basic eavesdropping attack, we do consider a particular type of active attack in our level-crossing algorithm in Section II, where the adversary attempts to disrupt the key extraction protocol by replacing or altering the protocol messages. In this case, we provide a method to deal with this type of active attack by cleverly using the shared fading process between Alice and Bob. One of the goals of our work is to demonstrate that secrecy generation can be accomplished in real-time over real channels (and not simulation models) and in real communication systems. To that end, results based on implementations on actual wireless platforms (a modified commercial 802.11 a/g implementation platform) and using over-the-air protocols are presented. To accomplish this, we had to work with several severe limitations of the *experimental system* at our disposal. Consequently certain parameters (e.g. code block length) had to be selected to be somewhat below what they should be for a well-designed system. This, however, does not reflect on the feasibility of proper implementation in a system with these features designed in. For example, nothing would prevent a design with the code block length sufficiently long to guarantee desired performance. On the contrary, we believe the demonstration of a practical implementation to be one of the major contributions of our work. The rest of this paper is organized as follows. Section II discusses the simpler of our algorithms based on level crossings. Section III presents a more complex and more powerful approach to extracting secret bits from the channel response, as well as some new results on secrecy generation for Gaussian sources which motivate our solution. We conclude the paper with some final remarks in Section IV. Level Crossing Secret Key Generation System =========================================== In this section we describe a simple and lightweight algorithm in [@MatTra08] for extracting secret bits from the wireless channel that does not explicitly involve the use of coding techniques. While this comes at the expense of a lower secret key rate, it reduces the complexity of the system and it still provides a sufficiently good rate in typical indoor environments. The algorithm uses excursions in the fading channel for generating bits and the timing of excursions for reconciliation. Further, the system does not require i.i.d. inputs and, therefore, does not require knowledge of the channel coherence time a priori. We refer to this secret key generation system as the [*level crossing system*]{}. We evaluate the performance of the level crossing system and test it using customized 802.11 hardware. System and Algorithm Description -------------------------------- Let $F(t)$ be a stochastic process corresponding to a time-varying parameter $F$ that describes the wireless channel shared by, and unique to Alice and Bob. Alice and Bob transmit a known signal (a probe) to one another in quick succession in order to derive correlated estimates of the parameter $F$, using the received signal by exploiting reciprocity of the wireless link. Let $X$ and $Y$ denote the (noisy) estimates of the parameter $F$ obtained by Alice and Bob respectively. Alice and Bob generate a sequence of $n$ correlated estimates $\hat{X}^n = (\hat{X}_1, \hat{X}_2, \ldots , \hat{X}_n)$ and $\hat{Y}^n = (\hat{Y}_1, \hat{Y}_2, \ldots , \hat{Y}_n)$, respectively, by probing the channel repeatedly in a time division duplex (TDD) manner. Note however, that $\hat{X}_i$ (and $\hat{Y}_i$) are no longer i.i.d. for $i=1, \ldots n$ since the channel may be strongly correlated between successive channel estimates. Alice and Bob first low-pass filter their sequence of channel estimates, $\hat{X}^n$ and $\hat{Y}^n$ respectively, by subtracting a windowed moving average. This removes the dependence of the channel estimates on large-scale shadow fading changes and leaves only the small scale fading variations (see Figure \[IDs2\]). The resulting sequences, ${X}^n$ and ${Y}^n$ have approximately zero mean and contain excursions in positive and negative directions with respect to the mean. The subtraction of the windowed mean ensures that the level-crossing algorithm below does not output long strings of ones or zeros and that the bias towards one type of bit is removed. The filtered sequences are then used by Alice and Bob to build a 1-bit quantizer $\psi^u(\cdot)$ quantizer based on the scalars $q^u_+$ and $q^u_-$ that serve as threshold levels for the quantizer: $$\begin{aligned} \label{q_def_u} q_{+}^u &=& mean(U^n) + \alpha\cdot\sigma(U^n) \\ \label{q_def_u2} q_{-}^u &=& mean(U^n) - \alpha\cdot\sigma(U^n),\end{aligned}$$ where the sequence $U^n = {X}^n$ for Alice and $U^n = {Y}^n$ for Bob. $\sigma(\cdot)$ is the standard deviation and the factor $\alpha$ can be selected to control the quantizer thresholds. The sequences ${X}^n$ and ${Y}^n$ are then fed into the following locally-computed quantizer at Alice and Bob respectively: $$\psi^u(x) = \left\{ \begin{array}{ll} 1 & \textrm{if $x>q^u_{+}$}\\ 0 & \textrm{if $x<q^u_{-}$} \\ e & \textrm{Otherwise} \end{array} \right.\nonumber$$ where $e$ represents an undefined state. The superscript $u$ stands for *user* and may refer to either Alice, in which case the quantizer function is $\psi^A(\cdot)$, or to Bob, for which the quantizer is $\psi^B(\cdot)$. This quantizer forms the basis for quantizing positive and negative excursions. Values between $q_-^u$ and $q_+^u$ are not assigned a bit. It is assumed that the number $n$ of channel observations is sufficiently large before using the level crossing system, and that the $i^{th}$ element $X_i$ and $Y_i$ correspond to successive probes sent by Bob and Alice respectively, for each $i = 1, \ldots, n$. The level crossing algorithm consists of the following steps: 1. Alice parses the vector $X^n$ containing her filtered channel estimates to find instances where $m$ or more successive estimates lie in an excursion above $q_{+}$ or below $q_{-}$. Here, $m$ is a parameter used to denote the minimum number of channel estimates in an excursion. 2. Alice selects a random subset of the excursions found in step 1 and, for each selected excursion, she sends Bob the index of the channel estimate lying in the center of the excursion, as a list $L$. Therefore, if $X_i > q_{+}$ or $< q_{-}$ for some $i=i_{start}, \ldots, i_{end}$, then she sends Bob the index $i_{center}=\lfloor\frac{i_{start} + i_{end}}{2} \rfloor$. 3. To make sure the $L$-message received is from Alice, Bob computes the fraction of indices in $L$ where $Y^n$ lies in an excursion spanning $(m-1)$ or more estimates. If this fraction is less than $\frac{1}{2} + \epsilon$, for some fixed parameter $0 < \epsilon < \frac{1}{2}$, Bob concludes that the message was not sent by Alice, implying an adversary has injected a fake $L$-message. 4. If the check above passes, Bob replies to Alice with a message $\tilde{L}$ containing those indices in $L$ at which $Y^n$ lies in an excursion. Bob computes $K_B = \psi^B(Y_i; i\in \tilde{L})$ to obtain $N$ bits. The first $N_{au}$ bits are used as an authentication key to compute a message authentication code (MAC) of $\tilde{L}$. The remaining $N-N_{au}$ bits are kept as the extracted secret key. The overall message sent by Bob is $\left\{\tilde{L}, MAC\left(K_{au},\tilde{L}\right)\right\}$. Practical implementations, for example, one could use CBC-MAC as the implementation for MAC, and use a key $K_{au}$ of length $N_{au} = 128$ bits. 5. Upon receiving this message from Bob, Alice uses $\tilde{L}$ to form the sequence of bits $K_A = \psi^A(X_i; i\in \tilde{L})$. She uses the first $N_{au}$ bits of $K_A$ as the authentication key $K_{au} = K_A(1, \ldots, N_{au})$, and, using $K_{au}$, she verifies the MAC to confirm that the package was indeed sent by Bob. Since Eve does not know the bits in $K_{au}$ generated by Bob, she cannot modify the $\tilde{L}$-message without failing the MAC verification at Alice. Figure \[LC\_sys\] shows the system-level operation of the level crossing algorithm. We show later that provided the levels $q_+, q_-$ and the parameter $m$ are properly chosen, the bits generated by the two users are identical with very high probability. In this case, both Alice and Bob are able to compute identical key bits and identical authentication key bits $K_{au}$, thereby allowing Alice to verify that the protocol message $\tilde{L}$ did indeed come from Bob. Since Eve’s observations from the channel probing do not provide her with any useful information about $X^n$ and $Y^n$, the messages $L$ and $\tilde{L}$ do not provide her any useful information either. This is because they contain time indices only, whereas the generated bits depend upon the values of the channel estimates at those indices. Security Discussion for the Level-crossing Algorithm ---------------------------------------------------- The secrecy of our key establishment method is based on the assumption that Alice and Bob have confidence that there is no eavesdropper Eve located near either Alice or Bob. Or equivalently, any eavesdropper is located a sufficient distance away from both Alice and Bob. In particular, the fading process associated with a wireless channel in a richly scattering environment decorrelates rapidly with distance and, for two receivers located at a distance of roughly the carrier wavelength from each other, the fading processes they each witness with respect to a transmitter will be nearly independent of each other[@Jak74]. For a Rayleigh fading channel model, if $h_{ba}$ and $h_{be}$ are the jointly Gaussian channels observed by Alice and Eve due to a probe transmitted by Bob, then the correlation between $h_{ba}$ and $h_{be}$ can be expressed as a function of the distance $d$ between Alice and Eve, and is given by $J_0({2\pi d/\lambda})$, where $J_0(x)$ is the zeroth-order Bessel function of the first kind, $d$ is the distance between Alice and Eve, and $\lambda$ is the carrier wavelength. Hence, because of the decay of $J_0(x)$ versus the argument $x$, if we are given any $\epsilon>0$, it is possible to find the minimum distance $d$ that Eve must be from both Alice and Bob such that the mutual information $I(h_{ba}; h_{be}) \leq \epsilon$. Further, we note that the statistical uniformity of the bit sequences that are extracted by Alice and Bob using our level-crossing algorithm is based on the statistical uniformity of positive and negative excursions in the distribution of the common stochastic channel between them. This inherently requires that the channel state representation for the fading process be symmetrically distributed about the distribution’s mean. Many well-accepted fading models satisfy this property. Notably, Rayleigh and Rician fading channels[@AndreaGS], which result from the multiple paths in a rich scattering environment adding up at the receiver with random phases, fall into this category. Consequently, we believe that the reliance of level-crossing algorithm on the underlying distribution symmetry, suggests that the level-crossing algorithm is best suited for Rayleigh or Rician fading environments. The independence of successive extracted bits follows from the fact that the excursions used for each bit are naturally separated by a coherence time interval or more, allowing the channel to decorrelate in time. Finally, we note that our approach does not preclude a final privacy amplification step, though application of such a post-processing step is straightforward and might be desirable in order to ensure that no information is gleaned by an eavesdropper. Performance Evaluation and Experimental Validation -------------------------------------------------- The central quantities of interest in our protocol are the rate of generation of secret bits and the probability of error. The controls available to us are the parameters: $q_+^u, q_-^u, m$ and the rate at which Alice and Bob probe the channel between themselves, $f_s$. We assume the channel is not under our control and the rate at which the channel varies can be represented by the maximum Doppler frequency, $f_d$. The typical Doppler frequency for indoor wireless environments at the carrier frequency of $2.4$ GHz is $f_d = \frac{v}{\lambda} \sim \frac{2.4 \times 10^9}{{3 \times 10^8}} = 8 \hspace{0.1cm}$ Hz, assuming a velocity $v$ of $1$ m/s. We thus expect typical Doppler frequencies in indoor environments in the $2.4$ GHz range to be roughly $10$ Hz. For automobile scenarios, we can expect a Doppler of $\sim 200$ Hz in the $2.4$ GHz range. We assume, for the sake of discussion, that the parameter of interest, $F$ is a Gaussian random variable and the underlying stochastic process $F(t)$ is a stationary Gaussian process. A Gaussian distribution for $F$ may be obtained, for example, by taking $F$ to be the magnitude of the in-phase component of a Rayleigh fading process between Alice and Bob [@Rap01]. We note that the assumption of a Gaussian distribution on $F$ is for ease of discussion and performance analysis, and our algorithm is valid in the general case where the distribution is symmetric about the mean. The probability of error, $p_e$ is critical to our protocol. In order to achieve a robust key-mismatch probability $p_k$, the bit-error probability $p_e$ must be much lower than $p_k$. A bit-error probability of $p_e = 10^{-7} \sim 10^{-8}$ is desirable for keys of length $N=128$ bits. The probability of bit-error, $p_e$ is the probability that a single bit generated by Alice and Bob is different at the two users. Consider the probability that the $i^{th}$ bit generated by Bob is “$K_B^i = 0$” at some index given that Alice has chosen this index, but she has generated the bit “$K_A^i = 1$”. As per our Gaussian assumption on the parameter $F$ and estimates $X$ and $Y$, this probability can be expanded as $$\begin{aligned} \label{dejavu2} &&\Pr(K_B^i = 0 | K_A^i = 1) = \frac{\Pr(K_B^i=0, K_A^i = 1)}{\Pr(K_A^i = 1)} = \\ \nonumber && \frac{ \underbrace{\int_{q_+^X}^{\infty}\int_{-\infty}^{q_{-}^Y} \ldots \int_{q_+^X}^{\infty}}_{(2m-1)~terms} \frac{(2 \pi)^{{(1-2m)/2}}}{|K_{2m-1}|^{1/2}}\exp{\left\{-\frac{1}{2}x^TK_{2m-1}^{-1}x \right\}} d^{(2m-1)}x}{ \underbrace{\int_{q_+^X}^{\infty}\ldots \int_{q_+^X}^{\infty}}_{(m)~terms} \frac{(2 \pi)^{-m/2}}{ |K_{m}|^{1/2}}\exp{\left\{-\frac{1}{2}x^TK_{m}^{-1}x \right\}} d^{(m)}x},\end{aligned}$$ where $K_m$ is the covariance matrix of $m$ successive Gaussian channel estimates of Alice and $K_{2m-1}$ is the covariance matrix of the Gaussian vector $(X_1, Y_1, X_2, \ldots, Y_{m-1}, X_{m})$ formed by combining the $m$ channel estimates of Alice and the $m-1$ estimates of Bob in chronological order. The numerator in (\[dejavu2\]) is the probability that of $2m-1$ successive channel estimates ($m$ belonging to Alice, and $m-1$ for Bob), all $m$ of Alice’s estimates lie in an excursion above $q_{+}$ while all $m-1$ of Bob’s estimates lie in an excursion below $q_{-}$. The denominator is simply the probability that all of Alice’s $m$ estimates lie in an excursion above $q_+$. We compute these probabilities for various values of $m$ and present the results of the probability of error computations in Figure \[pe2\]. The results confirm that a larger value of $m$ will result in a lower probability of error, as a larger $m$ makes it less likely that Alice’s and Bob’s estimates lie in opposite types of excursions. Note that if either user’s estimates do not lie in an excursion at a given index, a bit error is avoided because that index is discarded by both users. How many secret bits/second (bps) can we expect to derive from a fading channel using level crossings? An approximate analysis can be done using the level-crossing rate for a Rayleigh fading process, given by $LCR = \sqrt{2\pi}f_d \rho e^{- \rho^2}$ [@Rap01], where $f_d$ is the maximum Doppler frequency and $\rho$ is the threshold level, normalized to the root mean square signal level. Setting $\rho =1$, gives $LCR \sim f_d$. This tells us that we cannot expect to obtain more secret bits per second than the order of $f_d$. In Figure \[rate1\] (a) and (b), we plot the rate in s-bits/sec as a function of the channel probing rate for a Rayleigh fading channel with maximum Doppler frequencies of $f_d = 10$ Hz and $f_d = 100$ Hz respectively. As expected, the number of s-bits the channel yields increases with the probing rate, but saturates at a value on the order of $f_d$. In order for successive bits to be statistically independent, they must be separated in time by more than one coherence time interval. While the precise relationship between coherence time and Doppler frequency is only empirical, they are inversely related and it is generally agreed that the coherence time is smaller in magnitude (Coherence time $T_c$, is sometimes expressed in terms of $f_d$ as $T_c \thickapprox \sqrt{\frac{9}{16 \pi f_d^2}}$) than $1/f_d$. Therefore, on average, if successive bits are separated by a time interval of $1/f_d$, then they should be statistically independent. More precisely, the number of secret bps is the number of secret bits per observation times the probing rate. Therefore $$\begin{aligned} R_k & \hspace{-0.25cm}= &\hspace{-0.2cm} H(bins) \times \Pr(K_A^i=K_B^i) \times \frac{f_s}{m} \\ \nonumber &\hspace{-0.25cm} = &\hspace{-0.2cm} 2 \frac{f_s}{m} \times \Pr(K_A^i = 1, K_B^i=1) \nonumber \\ &\hspace{-0.25cm}= &\hspace{-0.3cm} 2\frac{f_s}{m}. \hspace{-0.15cm}\underbrace{\int_{q_+^X}^{\infty}\hspace{-0.23cm} \ldots \int_{q_+^X}^{\infty}}_{(2m-1)~terms} \hspace{-0.1cm}\frac{(2 \pi)^{\frac{1-2m}{2}}}{|K_{2m-1}|^{1/2}}e^{\left\{-\frac{1}{2}x^TK_{2m-1}^{-1}x \right\}} d^{2m-1}x ,\nonumber\end{aligned}$$ where $H(bins)$ is the entropy of the random variable that determines which bin ($>q_+$ or $<q_-$) of the quantizer the observation lies in, which in our case equals $1$ assuming that the two bins are equally likely[^9]. The probing rate $f_s$ is normalized by a factor of $m$ because a single ‘observation’ in our algorithm is a sequence of $m$ channel estimates. Figure \[rate1\] confirms the intuition that the secret bit rate must fall with increasing $m$, since the longer duration excursions required by a larger value of $m$ are less frequent. In Figure \[rate6\](a), we investigate how the secret-bit rate $R_k$ varies with the maximum Doppler frequency $f_d$, i.e., the channel time-variation. We found that for a fixed channel probing rate (in this case, $f_s = 4000$ probes/sec), increasing $f_d$ results in a greater rate but only up to a point, after which the secret-bit rate begins to fall. Thus, ‘running faster’ does not necessarily help unless we can increase the probing rate $f_s$ proportionally. Figure \[rate6\](b) shows the expected decrease in secret-bit rate as the quantizer levels the value of $\alpha$ is varied to move $q_+^u$ and $q_-^u$ further apart. Here, $\alpha$ denotes the number of standard deviations from the mean at which the quantizer levels are placed. We examined the performance of the secrecy generation system through experiments. The experiments involved three terminals, Alice, Bob and Eve, each equipped with an 802.11a development board. In the experiments, Alice was configured to be an access point (AP), and Bob was configured to be a station (STA). Bob sends Probe Request messages to Alice, who replies with Probe Response messages as quickly as possible. Both terminals used the long preamble segment [@80211a] of their received Probe Request or Probe Response messages to compute 64-point CIRs. The tallest peak in each CIR (the dominant multipath) was used as the channel parameter of interest, i.e., the $X$ and $Y$ sample inputs to the secret key generation system. To access such peak data, FPGA-based customized logic was added to the 802.11 development platform. Eve was configured to capture the Probe Response messages sent from Alice in the experiments. Two experiments were conducted. In the first experiment, Alice and Eve were placed in a laboratory. In a second experiment, Alice and Eve remained in the same positions while Bob circled the cubicle area of the office. Figure \[ABECIR\](a) shows an example of Alice’s, Bob’s, and Eve’s 64-point CIRs obtained through a single common pair of Probe Request and Probe Response messages. It is seen from the figure that Alice’s and Bob’s CIRs look similar, while they both look different from Eve’s CIR. We show the traces for Alice and Bob resulting from 200 consecutive CIRs in Figure \[ABECIR\](b). The similarity of Alice’s and Bob’s samples, as well as their difference from Eve’s samples, are evident from the figure. While our experiments ran for $\sim22$ minutes, in the interest of space and clarity we show only $700$ CIRs collected over a duration of $\sim77$ seconds. Each user locally computes $q_{+}$ and $q_{-}$ as in (\[q\_def\_u\]), (\[q\_def\_u2\]). We chose $\alpha = \frac{1}{8}$ for our experiments. Figure \[IDs2\] shows the traces collected by Alice and Bob after removal of slow shadow fading components using a simple local windowed mean. This is to prevent long strings of $1$s and $0$s, and to prevent the predictable component of the average signal power from affecting our key generation process. Using the small scale fading traces, our algorithm generates $N=125$ bits in $110$ seconds ($m=4$), yielding a key rate of about $1.13$ bps. Figure \[IDs2\] shows the bits that Eve would generate if she carried through with the key-generation procedure. The results from our second experiment with a moving Bob are very similar to the ones shown for the first experiment, producing $1.17$ bps. with $m=4$ and $\alpha = \frac{1}{8}$. Note that while figures \[rate1\] and \[rate6\] depict the secret bit rate that can be achieved for the specified values of Doppler frequency, our experimental setup does not allow us to measureably control the precise Doppler frequency and the secret bits rates we report from our experiments correspond only the indoor channel described. In order to verify the assumption that Eve does not gain any useful information by passive observation of the probes transmitted by Alice and Bob, we empirically computed the mutual information using the method in [@WanKul06] between the signals received at the legitimate users and compare this with that between the signals received by Eve and a legitimate user. The results of this computation, summarized in Table \[TableMI\], serve as an upper bound to confirm that Eve does not gather any significant information about the signals received at Alice and Bob. Although this information leakage is minimal relative to the mutual information shared between Alice and Bob, it might nonetheless be prudent to employ privacy amplification as a post-processing to have a stronger assurance that Eve has learned no information about the key established between Alice and Bob. Finally, we note that with suitable values of the parameters chosen for the level crossing algorithm, the bits extracted by Alice and Bob are statistically random and have high-entropy per bit. This has been tested for and previously reported in [@MatTra08] using a suite of statistical randomness tests provided by NIST [@NIST]. Quantization-Based Secret Key Generation for Wireless Channels ============================================================== We now present a more powerful and general approach than the level-crossing approach discussed in Section II for obtaining secret keys from the underlying fading phenomena associated with a with a richly scattering wireless environment. Whereas the level-crossing algorithm was best suited for extracting keys from channel states whose distributions are inherently symmetric, our second approach is applicable to more general channel state distributions. Further, this second approach approach is capable of generating significantly more than a single bit per independent channel realization, especially when the channel estimation SNRs are high. To accomplish this we propose a new approach for the quantization of sources whose statistics are not known, but are believed to be similar in the sense of having “high SNR" - a notion we shall define more precisely below. Our quantization approach is motivated by considering a simpler setting of a Gaussian source model and addressing certain deficiencies which can be observed in that model. This problem has been addressed by [@YeRez06] using a simple “BICM-like“ approach [@BloTha06] to the problem. A more general treatment which introduces multi-level coding can be found in [@BloTha06] and also [@BloBar08], however for our purposes, the simple ”BICM-like" approach of [@YeRez06] and [@BloTha06] is sufficient. To motivate our approach to “universal” quantization we need to take this solution and improve on it - the process which we describe next. Over-quantized Gaussian Key Generation System --------------------------------------------- We begin our discussion of the over-quantized Gaussian Key Generation System by reviewing the simple approach to the problem described in [@YeRez06]. A block diagram of a basic secret key generation system is shown in Figure \[fig1\]. Alice’s secrecy processing consists of four blocks: Quantizer, Source Coder, Channel Coder and the Privacy Amplification (PA) process. The Quantizer quantizes Alice’s Gaussian samples $X^n$. The Source Coder converts the quantized samples to a bit string ${\bf X}_b$. The Channel Coder computes the syndrome ${\bf S}$ of the bit string ${\bf X}_b$. A rate $1/2$ LDPC code is used in [@YeRez06]. This syndrome is sent to Bob for his decoding of ${\bf X}_b$. As discussed in Section I, the transmission of the syndrome is assumed to take place through an error-free public channel; in practice this can be accomplished through the wireless channel with the use of standard reliability techniques (e.g., CRC error control and ARQ). Finally, privacy amplification (if needed) is implemented in the PA block. Figure \[KeyratesGaussian\](a) present the results obtained by using various algorithm options discussed in [@YeRez06]. We observe from this figure that at high SNR ($>15 $dB), the secret key rates resulting from Gray coding are within 1.1 bits of the secret key capacity (\[e2.6\]). However, the gap between the achieved secret key rates and the secret key capacity is larger at low SNR. In this sub-section, we demonstrate how the basic system can be improved such that the gap at low SNR is reduced. We restrict ourselves to Gray coding, as this is clearly the better source coding approach. We start with the observation that the quantization performed by Alice involves some information loss. To compensate for this, Alice could quantize her samples at a higher level than the one apparently required for the basic secret key generation purpose. Suppose that quantization to $v$ bits is required by the baseline secrecy generation scheme. Alice then quantizes to $v+m$ bits using Gray coding as a source coder. We refer to the $v$ most significant bits as the *regularly quantized bits* and the $m$ least significant bits as the *over-quantized bits*. The over-quantized bits ${\bf B}$ are sent directly to Bob through the error-free public channel. The Channel Decoder (at Bob) uses the syndrome ${\bf S}$ of the regularly quantized bits ${\bf X}_b$, the over-quantized bits ${\bf B}$ and Bob’s Gaussian samples $Y^n$ to decode ${\bf X}_b$. Again, it applies the modified belief-propagation algorithm (cf. [@LivXio02]), which requires the per-bit LLR. The LLR calculation is based on both $Y^n$ and ${\bf B}$. Suppose one of Alice’s Gaussian samples $X$ is quantized and Gray coded to bits $(X_{b,1}, \cdots, X_{b, v+m})$. With Bob’s corresponding Gaussian sample $Y$ and Alice’s over-quantized bits $(X_{b,v+1}, \cdots, X_{b, v+m}) = (a_{v+1}, \cdots, a_{v+m})$, the probability of $X_{b,i}$, $1\leq i\leq v$, being 0 is derived below: $$\begin{aligned} &&\Pr(X_{b,i}=0|Y=y,X_{b,v+1}=a_{v+1}, \cdots, X_{b,v+m} = a_{v+m})\nonumber\\ & = & \frac{\Pr(X_{b,i}= 0, X_{b, v+1}= a_{v+1},\cdots, X_{b, v_m}=a_{v+m}| Y=y)} {\Pr(X_{b, v+1}= a_{v+1},\cdots, X_{b, v_m}=a_{v+m}| Y=y)}\nonumber\\ &=& \frac{\sum_{j=1}^{2^{v+m}}\Pr(\bar{q}_{j-1}\leq X< \bar{q}_j|Y=y){\bf 1}_{G_{v+m}^{i}(j-1)=0} \cdot {\bf 1}_{G_{v+m}^{v+1}(j-1)=a_{v+1}} \cdots {\bf 1}_{G_{v+m}^{v+m}(j-1)=a_{v+m}}} {\sum_{j=1}^{2^{v+m}}\Pr(\bar{q}_{j-1}\leq X< \bar{q}_j|Y=y)\cdot {\bf 1}_{G_{v+m}^{v+1}(j-1)=a_{v+1}} \cdots {\bf 1}_{G_{v+m}^{v+m}(j-1)=a_{v+m}}}, \label{e3.8}\end{aligned}$$ where ${\bf 1}$ is an indicator function and the function $G_{k}^{i}(j)$, $1\leq i\leq k$, $0\leq j\leq 2^k-1$, denotes the $i^{th}$ bit of the $k$-bit Gray codeword representing the integer $j$. The quantization boundaries $\bar{q}_0 < \cdots <\bar{q}_{2^{v+m}}$ depend on the quantization scheme used. For instance, the quantization boundaries of the [*equiprobable quantizer*]{} satisfy $$\int_{\bar{q}_{j-1}}^{\bar{q}_j} \frac{1}{{\sqrt {2\pi N} }}e^{-\frac{x^2}{2N}}dx = \frac{1}{2^{v+m}}, \ \ j= 1, \cdots , 2^{v+m}.$$ Now, $$\begin{aligned} \Pr(\bar{q}_{j-1}\leq X< \bar{q}_j|Y=y) &=& \Pr(\bar{q}_{j-1}\leq \frac{P}{P+N}Y+Z_0<\bar{q}_j|Y=y)\nonumber\\ &=& \Pr(\bar{q}_{j-1}-\frac{P}{P+N}y\leq Z_0<\bar{q}_j-\frac{P}{P+N}y) \nonumber\\ &=& Q\left(\frac{\bar{q}_{j-1}-\frac{P}{P+N}y}{{\sqrt \frac{2PN+N^2}{P+N}}}\right)- Q\left(\frac{\bar{q}_{j}-\frac{P}{P+N}y}{{\sqrt \frac{2PN+N^2}{P+N}}}\right)\nonumber\\ &=& g(j-1, y)-g(j,y)\nonumber,\end{aligned}$$ where the function $g(k,y)$, $0\leq k\leq 2^{v+m}$, is defined as $$g(k,y)=Q\left(\frac{\bar{q}_k-\frac{P}{P+N}y}{\sqrt{(2PN+N^2)/(P+N)}}\right), $$ and $Q$ is the usual Gaussian tail function [@Proakis00]. Hence, the probability of (\[e3.8\]) is given by $$\frac {\sum_{j=1}^{2^{v+m}}\left[g(j-1,y)-g(j,y)\right]\cdot {\bf 1}_{G_{v+m}^{i}(j-1)=0} \cdot {\bf 1}_{G_{v+m}^{v+1}(j-1)=a_{v+1}} \cdots {\bf 1}_{G_{v+m}^{v+m}(j-1)=a_{v+m}}} {\sum_{j=1}^{2^{v+m}}\left[g(j-1,y)-g(j,y)\right]\cdot {\bf 1}_{G_{v+m}^{v+1}(j-1)=a_{v+1}} \cdots {\bf 1}_{G_{v+m}^{v+m}(j-1)=a_{v+m}}}. \label{LLR}$$ It should be noted that when equiprobable quantization is used, the over-quantized bits ${\bf B}$ and the regularly quantized bits ${\bf X}_b$ are independent as shown below. Suppose a sample $X$ is equiprobably quantized and source coded to $t$ bits $(X_{b,1}, \cdots , X_{b,t})$. For an arbitrary bit sequence $(a_1, \cdots, a_{t})$ and a set ${\cal S}\subseteq {\cal T}= \{1,\cdots , t\}$, we have $$\begin{aligned} &&\Pr \left(\{X_{b,i} = a_i: i\in {\cal S}\}|\{X_{b,i} = a_i: i\in {\cal T}\setminus{\cal S}\} \right) = \frac{\Pr\left(\{X_{b,i} = a_i: i\in {\cal T}\}\right)}{\Pr\left(\{X_{b,i} = a_i: i\in {\cal T}\setminus{\cal S}\}\right)}\nonumber\\ &=&\frac{2^{-t}}{2^{-(t-|{\cal S}|)}}=2^{-|{\cal S}|}=\Pr\left(\{X_{b,i} = a_i: i\in {\cal S}\}\right), \nonumber\end{aligned}$$ which implies the amount of secrecy information remaining in ${\bf X}_b$ after the public transmission is at least $|{\bf X}_b|- |{\bf S}|$ bits.[^10] Note that this conclusion does not hold for other quantization approaches (e.g., MMSE quantization) and, therefore, equiprobable quantization should be used if over-quantization is applied. On the other hand, it is implied by (\[LLR\]) that the over-quantized bits ${\bf B}$ and the regularly quantized bits ${\bf X}_b$ are dependent given Bob’s samples $Y^n$. Hence, $I({\bf X}_b; {\bf B}|Y^n)>0$. It follows from the Slepian-Wolf theorem (cf. [@CovTho91]) that with the availability of the over-quantized bits ${\bf B}$, the number of syndrome bits $|{\bf S}|$ required by Bob to successfully decode ${\bf X}_b$ is approximately $H({\bf X}_b|Y^n, {\bf B})$, which is less than $H({\bf X}_b|Y^n)$, the number of syndrome bits transmitted in the basic system. In other words, the secret key rate achieved by the over-quantized system is approximated by $\frac{1}{n}I({\bf X}_b; Y^n, {\bf B})$, which is larger than $\frac{1}{n}I({\bf X}_b; Y^n)$, the secret key rate achieved by the basic system. To obtain an upper limit on the performance improvement that over-quantization may provide us, we can imagine sending the entire (real-valued) quantization error as a side information. There are a number of issues with this approach. Clearly, distortion-free transmission of real-valued quantities is not practically feasible. However, as we are looking for a bound, we can ignore this. More importantly, the transmission of raw quantization errors may reveal information about ${\bf X}_b$. For example, to equiprobably quantize a zero mean, unit variance Gaussian random variable with 1 bit per sample, the quantization intervals are $(-\infty, 0]$ and $(0,\infty)$, with respective representative value -0.6745 and 0.6745. Suppose a sample $X$ is of value 2, then its quantization error is $2-0.6745=1.3255$. This implies that $X$ must be in the interval $(0, \infty)$, since otherwise, the quantization error does not exceed 0.6745. Thereby, it is necessary to process the raw quantization errors such that the processed quantization errors do not contain any information about ${\bf X}_b$. For this purpose, it is desirable to transform quantization errors to uniform distribution. To do so, we first process an input sample $X$ with the cumulative distribution function (CDF) of its distribution and then quantize. The transformed quantization error is then given by $E=\phi\left(X\right)-\phi\left(q(X)\right)$, where $\phi(x)$ is the CDF for $X$ and $q(X)$ is the representative value of the interval to which $X$ belongs. The quantization errors $E^n=(E_1,\cdots,E_n)$, which are then uniformly distributed on $\left[-2^{-(v+1)}, 2^{-(v+1)}\right]$, are sent to Bob through the error-free public channel. The rest of the process (encoding/decoding and PA) proceeds as before. However, the LLR computation must be modified to use probability density functions, rather than probabilities: $$\ln \frac{\Pr(X_{b,i}=0|Y=y, E=e)}{\Pr(X_{b,i}=1|Y=y, E=e)} = \sum_{j=1}^{2^v} (-1)^{{\bf 1}_{G_{v}^{i}(j-1)=0}}\cdot h(e,j,y),\label{e3.12}$$ where the function $G_{k}^{i}(j)$ is defined in (\[e3.8\]) and the function $h(e,j,y)$ is defined as $$h(e,j,y)=\frac{P+N}{2(2PN+N^2)} \left(\phi^{-1} \left(e+\frac{j-0.5}{2^v}\right) -\frac{P}{P+N} y \right)^2,$$ for $-2^{-(v+1)}\leq e\leq 2^{-(v+1)}$, $1\leq j\leq 2^v$, with the function $\phi$ being the CDF for $X$. The derivation of (\[e3.12\]) is similar to that of (\[LLR\]), which is omitted here. Figure \[KeyratesGaussian\](b) shows simulation results for 2-bit over-quantization and the upper bound. We note, as expected, that the overall gap to capacity has been reduced to about 1.1 dB at the low-SNR. A Universal Secret Key Generation System ---------------------------------------- In the previous sub-section we discussed secret key generation for a jointly Gaussian model. The random variables $X$ and $Y$ in the model are jointly Gaussian distributed and the distribution parameter SNR is known at both terminals. However, in many practical conditions, the correlated random variables at the two terminals may not be subject to a jointly Gaussian distribution, and the distribution parameters are usually unknown or estimated inaccurately. We address this problem by describing a method for LLR generation and subsequent secrecy generation that makes very few assumptions on the underlying distribution. As we shall see this method is largely based on the over-quantization idea we introduced above. ### System Description Compared to the basic system (Figure \[fig1\]) developed for the Gaussian model, the universal system includes two additional Data Converter blocks (one at Alice; the other at Bob), and modified Quantizer and Channel Decoder blocks. The inputs to Alice’s Data Converter blocks are $X^n$ and the outputs of Alice’s Data Converter block are sent to the modified Quantizer block. The inputs to Bob’s Data Converter blocks are $Y^n$ and the outputs of Bob’s Data Converter block are sent to the modified Channel Decoder block. The purpose of the Data Converter is to convert the input samples $X^n$, $Y^n$ to uniformly distributed samples $U^n$, $V^n$, where $U_i, V_i\in [0,1)$. The conversion is based on the empirical distribution of input samples. Given the $i^{th}$ sample $X_i$ of input samples $X^n$, denote by $K_n(X_i)$ the number of samples in $X^n$ which are strictly less than $X_i$ plus the number of samples in $X^n$ which are equal to $X_i$ but their indices are less than $i$. The output of the Data conversion block corresponding to $X_i$ is given by $U_i=\frac{K_n(X_i)}{n}$. To justify the use of this approach, we show that $U^n$ asymptotically tends to an i.i.d. sequence, each uniformly distributed between 0 and 1. Thus, while for any finite block length the sequence $U^n$ is not comprised of independent variables, it is assymtotically i.i.d. uniform. Consider an i.i.d. sequence $X^n=(X_1, \cdots ,X_n)$ . Denote by $\phi$ the actual CDF of $X_i$. Let $W_i=\phi(X_i)$, $i=1,\cdots, n$. Then $W_1, \cdots, W_n$ is an i.i.d. sequence, each uniformly distributed between 0 and 1. Hence, it suffices to show that the sequence $U^n$ converges to the sequence $W^n$. Convergence of the empirical distribution to the true distribution is a well-established fact in probability known as the Glivenko-Cantelli Theorem [@Shorak86]. However, we need a stronger statement which gives the rate of such convergence. This is known as the Dvoretzky-Kiefer-Wolfowitz Theorem [@DKW56] and is stated in the following lemma. [@DKW56] Let $X_1,\cdots, X_n$ be real-valued, i.i.d. random variables with distribution function $F$. Let $F_n$ denote the associate empirical distribution function defined by $$F_n(x) = \frac{1}{n} \sum_{i=1}^n 1_{(-\infty, x]}(X_i),\ \ x\in {\cal R}.$$ For any $\varepsilon>0$, $$\Pr\left(\sup_{x\in {\cal R}}|F_n(x)-F(x)|>\varepsilon \right)\leq 2e^{-2n\varepsilon^2}. \label{DKW}$$ $\hfill \Box$ \[LemmaDKW\] We will also need the notion of a $L^p$ convergence of random sequences [@Chung01]. The $L^p$-norm of a sequence $X^n$, $p\geq 1$, is defined by $||X^n||_p = \left(\sum_i^n |X_i|^p\right)^{\frac{1}{p}}$. A sequence $X^n$ is said to converge in $L^p$ to $Y^n$, $0\leq p\leq \infty$, if $\lim_{n\rightarrow \infty} {\cal E}\left[||X^n-Y^n||_p\right]= 0$. We then have the following lemma [@Chung01 Theorem 4.1.4]. If a sequence $X^n$ converges to another sequence $Y^n$ in $L^p$, $0\leq p\leq \infty$, then $X^n$ converges to $Y^n$ in probability.$\hfill \Box$ \[lemmaLP\] We can now show the desired statement. The sequence $U^n$ converges to the sequence $W^n$ in probability. [**Proof**]{}: According to Lemma \[lemmaLP\], we only need to show $ \lim_{n\rightarrow \infty} {\cal E}[||U^n-W^n||_4]= 0$. Here, $$\begin{aligned} {\cal E}[||U^n-W^n||_4] & = & {\cal E}\left[\left(\sum_{i=1}^n |U_i-W_i|^4\right)^{\frac{1}{4}}\right] \leq \left({\cal E}\left[\sum_i^n |U_i-W_i|^4\right]\right)^{\frac{1}{4}}\nonumber\\ &=& \left(\sum_i^n {\cal E}\left[|U_i-W_i|^4\right]\right)^{\frac{1}{4}}, \label{allupperbound}\end{aligned}$$ For any $i = 1,\cdots , n$, we have $$\begin{aligned} {\cal E}\left[|U_i-W_i|^4\right] &=& \int_0^1 \Pr\left(|U_i-W_i|^4> u\right) du = \int_0^1 \Pr\left(|U_i-W_i|> u^{\frac{1}{4}}\right) du \nonumber\\ &\leq & \int_0^1 2e^{-2nu^{\frac{1}{2}}}du, \label{callDKW}\end{aligned}$$ where (\[callDKW\]) follows from (\[DKW\]). By letting $t={\sqrt u}$ and integrating by parts, we show $${\cal E}\left[|U_i-W_i|^4\right] \leq 4 \int_0^1 te^{-2nt}dt =\frac{1}{n^2}-\frac{e^{-2n}}{n}(2+\frac{1}{n}) \leq \frac{1}{n^2}. \label{indupperbound}$$ Combining (\[allupperbound\]) and (\[indupperbound\]), we obtain $${\cal E}\left[||U^n-W^n||_4\right] \leq \left(\sum_{i=1}^n \frac{1}{n^2}\right)^{\frac{1}{4}} = n^{-\frac{1}{4}},$$ which tends to 0 as $n\rightarrow \infty$. This completes the proof of the theorem. $\hfill \Box$ The conversion from $X^n$ (or $Y^n$) to $U^n$ (or $V^n$) can be accomplished using a procedure that requires no computation and relies only on a sorting algorithm. It has the important side benefit that the output is inherently fixed-point, which is critical in the implementation of most modern communication systems. Let $A$ be the number of bits to be used for each output sample $U_i$. This implies that $U_i$ is of value $\frac{j}{2^A}$, $0\leq j\leq 2^A-1$. Denote by $C(j)$, $0\leq j\leq 2^A$, the number of output samples of value $\frac{j-1}{2^A}$. The values of $C(j)$ are determined by the following pseudo-code: where $\lfloor x \rfloor$ is the largest integer less than $x$. For an input sample $X_i$ with $$\sum_{j=0}^{k}C(j)\leq K_n(X_i)<\sum_{j=0}^{k+1}C(j),$$ the corresponding output $U_i$ is given by $\frac{k}{2^A}$. To efficiently implement this process, we follow a three step process: i) sort the input samples $X^n$ in ascending order; ii) convert sorted samples to values $\frac{j}{2^A}$, $0\leq j\leq 2^A-1$; iii) associate each input sample with its converted value. Suppose input samples $X^n$ are sorted to $\widetilde{X}^n$, where $\widetilde{X}_1\leq \cdots \leq \widetilde{X}_n$. The index mapping between $X^n$ and $\widetilde{X}^n$ is also recorded for the use in the association step. The values of $\widetilde{X}^n$ are converted to $\widetilde{U}^n$ using the algorithm defined via the pseudo-code below. The algorithm distributes $n$ items among $A$ bins in a “uniform" way even when $A$ does not divide $n$. The process is based on the rate-matching algorithms used in modern cellular systems, e.g. [@3GPP], and is also similar to line-drawing algorithms in computer graphics. The last step rearranges $\widetilde{U}^n$ to outputs $U^n$ such that the $i^{th}$ output sample $U_i$ is associated with the $i^{th}$ input sample $X_i$. Although the above procedures use $2^A$ as the total number of possible values to be assigned, in general, any integer $M$ may be substituted for $2^A$, in which case the unit interval $[0,1)$ is partitioned into $M$ equal sub-intervals, with the data distributed among them as uniformly as possible. To equiprobably quantize uniformly distributed samples $U^n$ with $v$ bits per sample, the Quantizer determines the quantization boundaries as $$q_i=\frac{i}{2^v}, \ \ \ 0\leq i\leq 2^v.$$ For a simple decoding process, the quantization error $E$ is defined as the difference between $U$ and the lower bound of the interval to which $U$ belongs. Hence, the quantization error $E$ is uniformly distributed between 0 and $\frac{1}{2^v}$. The transmission of such quantization errors $E^n=(E_1,\cdots , E_n)$ over the public channel does not reveal any information about ${\bf X}_b$. For the case of fixed point inputs $U^n$, if the number of bits per sample $v$ in the Quantizer block used for generating ${\bf X}_b$ is less than the number $A$ of bits used for $U$, then the Quantizer block obtains the quantized value and the quantization error for $U$ simply from the first $v$ bits and the last $A-v$ bits out of the $A$ bits for $U$, respectively. Bob’s Data Converter performs the same operations as Alice’s. The Channel Decoder calculates the per-bit LLR based on the outputs of Bob’s Data Converter block $V^n$ and the received quantization errors $E^n$. Unlike the jointly Gaussian model, the joint distribution of $X$ and $Y$ in this case is unknown and the accurate LLR is generally incomputable. We provide an extremely simple but effective way of computing the LLR. Heuristically, the LLR is related to the distances from $V$ to the possible $U$ values that cause $X_{b,i}=1$ and that cause $X_{b,i}=0$. Suppose a uniform sample $U$ is quantized and Gray coded to bits $(X_{b,1},\cdots ,X_{b,v})$ and the quantization error of $U$ is $E$. The heuristic LLR $L_i$ for $X_{b,i}$, $1\leq i\leq v$, is derived through the following pseudo-code: Consider an example of $E=0.2$ and $v=1$. This quantization error indicates the two possible values of $U$ are 0.2 and 0.7, which corresponds to $X_{b,1}=0$ and $X_{b,1}=1$, respectively. If $V=0.3$, which is closer to the possible $U$ value 0.2, then it is more likely that $X_{b,1}$ is equal to ‘0’ and the LLR for $X_{b,1}$ should be positive. It follows from the pseudo-code above that $L_1= 0.3$. If $V=0.5$, which is closer to the possible $U$ value 0.7, then it is more likely that $X_{b,1}$ is equal to ‘1’ and the LLR for $X_{b,1}$ should be negative. It follows from the codes above that $L_1= -0.1$. As the $L_i$ obtained in the codes above is generally within the range of $[-1,1]$, the likelihood probability of each bit is restricted to the range of $[0.27, 0.73]$. Hence, it is desirable to re-scale $L_i$ to the operational range of the modified belief-propagation algorithm by multiplying with a constant. ### Simulation and Experimental Validation We examine the performance of the proposed approach in a simulation environment with the jointly Gaussian channel model and with real channels. In order to examine the performance of the universal system, we apply it to the jointly Gaussian model, though noting that the parameters $P$, $N$ of the jointly Gaussian model are not utilized in the universal system. The secret key rates achieved by the universal system are shown in Figure \[fig8\]. For comparison, the secret key capacity and the upper bound for the secret key rates achieved by the over-quantized system are also plotted in the same figure. It is seen from the figure that the universal system performs well at low SNR, but deviates at high SNR. The deviation may be due to the trade-off made between the regularly quantized bits and the over-quantized bits. A different trade-off can push the deviation point higher at the expense of more communication (of over-quantized bits) and higher LDPC decoding complexity. We experimentally validated the feasibility of the above universal approach using 802.11 setup described earlier. In the two experiments stated in Section II, Bob sent Probe Request messages at an average rate of 110 ms.[^11] Typically, Bob received the corresponding Probe Response message from Alice within 7 ms after a Probe Request message was sent. It is reported in Table I that in the first experiment, the mutual information between Alice and Bob’s samples is about 3.294 bits/sample, while the mutual information between Bob and Eve’s samples is about 0.047 bit/sample. In the second experiment, the mutual information between Alice and Bob’s samples is about 1.218 bits/sample, while the mutual information between Bob and Eve’s samples is 0 within the accuracy of the measurement. This suggests that the respective secret key capacities[^12] of the first and the second experimental environments are about 30 ($\approx$ (3.294-0.047) bits/sample $\div$ 0.11 second/sample) bps and 11 bps, provided that the channel coherence time is around 110 ms. Next, we check the secret key rates achieved by the universal system. For the purpose of generating keys in a short time duration, we apply a LDPC code with a shorter block length in the universal system. The code is a (3,6) regular LDPC code of codeword length 400 bits. The quantization parameter $v$ is chosen as 3 for the first experiment and 2 for the second experiment. This implies that for each run of the system, a block of 134 ($\approx 400/3$) first experimental samples or 200 second experimental samples is sent to the universal system. Our experimental results show that in both cases, Bob is able to successfully decode Alice’s bit sequence ${\bf X}_b$ of 400 bits. With the reduction of 200 bits, revealed as syndrome bits over the public channel, both terminals remain with 200 secret bits. In order to remove the correlation between the 200 secret bits and Eve’s samples in the first experiment, which shows non-zero mutual information, we may need to squash out an additional 7 ($\approx 0.047 * 134$) bits from the 200 secret bits, resulting in 193 secret bits. Considering the period of collecting these 134 or 200 samples, we conclude that the secret key rate achieved by the universal system is about 13 bps for the first experiment and 9 bps for the second experiment. Conclusions =========== The wireless medium creates the unique opportunity to exploit location-specific and time-varying information present in the channel response to generate information-theoretically secret bits, which may be used as cryptographic keys in other security services. This ability follows from the property that in a multipath scattering environment, the channel impulse response decorrelates in space over a distance that is of the order of the wavelength, and that it also decorrelates in time, providing a resource for fresh randomness. In this paper, we have studied secret key extraction, under the assumption of a Rayleigh or Rician fading channel, and under a more general setting where we do not make any assumption on the channel distribution. We have developed two techniques for producing identical secret bits at either end of a wireless communication link and have evaluated each technique using channel measurements made using a modified 802.11 system. The first technique is based on the observation of correlated excursions in the measurements at the two users while the second technique employs error-correction codes. The former method trades off the performance of the latter with a lower complexity and does not require knowledge of the channel coherence time. Since the time-varying nature of the channel acts as the source of randomness, it limits the number of random bits that can be extracted from the channel for the purpose of a cryptographic key. The second method applies to more general distributions for the shared channel information between a transmitter and receiver, and is able to achieve improved secret key rates at the tradeoff of increased complexity. Our evaluations indicate that typical indoor wireless channels allow us to extract secret bits at a practically useable rate, with minimal information about these secret bits being learned by an eavesdropper. Lastly, we note that as a final step, the legitimate participants in the protocol may wish to employ privacy amplification to provide added assurance that the eavesdropper cannot infer the bits being generated. [p[6 cm]{}p[6cm]{}]{} \ & \ \[rate6\]  Traces of Alice and Bob after subtracting average signal power. Using $m=5$, $N=59$ bits were generated in $110$ seconds ($R_k = 0.54$ s-bits/sec) while $m=4$ gives $N=125$ bits ($R_k = 1.13$ s-bits/sec.) with no errors in each case. (b) A magnified portion of (a)](fig11_home_july1_2.eps){width="5in" height="3.2in"} [p[7 cm]{}p[7.3cm]{}]{} \ &  Secret key rates achieved by the basic system. (b) Secret key rates achieved by the improved system.](Errorforwarding2.eps "fig:"){width="3.2in" height="2.8in"}\ ![Secret key rates achieved by the universal system.[]{data-label="fig8"}](Universal2.eps){width="4in" height="3.2in"} [^1]: Manuscript first submitted to the IEEE Transactions on Information Forensics and Security on 23 February, 2009. [^2]: $^\dag$ InterDigital Communications, LLC, King of Prussia, PA 19406, USA. [^3]: E-mail:`{chunxuan.ye, alex.reznik, yogendra.shah}@interdigital.com` [^4]: $^\ddag$ WINLAB, Rutgers University, 671 Route 1 South, North Brunswick, NJ 08902, USA. [^5]: E-mail: `{suhas, trappe, narayan}@winlab.rutgers.edu` [^6]: S. Mathur, W. Trappe and N. Mandayam are supported in part by by NSF grant CNS-0626439 and DARPA grant W31P4Q-07-1-002 [^7]: Portions of this work have been previous presented at the *IEEE International Symposium on Information Theory*, Seattle, WA, July 2006 and *ACM Conference on Mobile Computing and Networking*, San Francisco, CA, Sept. 2008. [^8]: Unless otherwise specified, all the terminals in this paper refer to legitimate terminals, and hence the term “legitimate” will be omitted henceforth. [^9]: The levels $q_+$ and $q_-$ are chosen so as to maintain equal probabilities for the two bins. [^10]: Relying on hash functions for privacy amplification requires the use of Rényi entropy. However, we can use [@BenBra95 Theorem 3] to equivocate Rényi and Shannon entropies. [^11]: Here, we assume the channel coherence time is less than or equal to 110 ms. Hence, two consecutive CIRs at either terminal are assumed to be mutually independent. [^12]: We abuse the notion of capacity a bit as this “capacity” assumes i.i.d. channel samples.

--- abstract: 'We consider the problem of two-player zero-sum game. In this setting, there are two agents working against each other. Both the agents observe the same state and the objective of the agents is to compute a strategy profile that maximizes their rewards. However, the reward of the second agent is negative of reward obtained by the first agent. Therefore, the objective of the second agent is to minimize the total reward obtained by the first agent. This problem is formulated as a min-max Markov game in the literature. The solution of this game, which is the max-min reward (of first player), starting from a given state is called the equilibrium value of the state. In this work, we compute the solution of the two-player zero-sum game utilizing the technique of successive relaxation. Successive relaxation has been successfully applied in the literature to compute a faster value iteration algorithm in the context of Markov Decision Processes. We extend the concept of successive relaxation to the two-player zero-sum games. We prove that, under a special structure, this technique computes the optimal solution faster than the techniques in the literature. We then derive a generalized minimax Q-learning algorithm that computes the optimal policy when the model information is not known. Finally, we prove the convergence of the proposed generalized minimax Q-learning algorithm.' author: - 'Raghuram Bharadwaj Diddigi$^{*}$ Chandramouli Kamanchi$^{*}$ Shalabh Bhatnagar[^1][^2] [^3][^4]' bibliography: - 'references.bib' title: '**Solution of Two-Player Zero-Sum Game by Successive Relaxation** ' --- **Keywords -  **Zero-Sum games, Stochastic Approximation, minimax Q-learning. [^1]: $^{*}$ Equal Contribution. [^2]: This work was supported by Robert Bosch Centre for Cyber-Physical Systems, Indian Institute of Science and a grant from the Department of Science and Technology, India. [^3]: C. Kamanchi and R. B. Diddigi are with the Department of Computer Science and Automation, Indian Institute of Science, Bengaluru 560012, India (e-mail: chandramouli@iisc.ac.in; raghub@iisc.ac.in). [^4]: S. Bhatnagar is with the Department of Computer Science and Automation, Indian Institute of Science, Bengaluru 560012, India, and also with the Department of Robert Bosch Centre for Cyber-Physical Systems, Indian Institute of Science, Bengaluru 560012, India (e-mail: shalabh@iisc.ac.in).

--- author: - Lingxiao Huang - Yifei Jin - 'Jian Li[^1]' - 'Haitao Wang[^2]' bibliography: - 'Citation.bib' title: Improved Algorithms For Structured Sparse Recovery --- [^1]: lijian83@mail.tsinghua.edu.cn [^2]: haitao.wang@usu.edu

--- abstract: 'Throughout this paper we investigate the complex structure of the conifold $C(T^{1,1})$ basically making use of the interplay between symplectic and complex approaches of the Kähler toric manifolds. The description of the Calabi-Yau manifold $C(T^{1,1})$ using toric data allows us to write explicitly the complex coordinates and apply standard methods for extracting special Killing forms on the base manifold. As an outcome, we obtain the complete set of special Killing forms on the five-dimensional Sasaki-Einstein space $T^{1,1}$.' author: - 'Vladimir Slesar[^1]' - 'Mihai Visinescu[^2]' - 'Gabriel Eduard Vîlcu[^3][^4]' title: 'Toric data and Killing forms on homogeneous Sasaki-Einstein manifold $T^{1,1}$' --- Introduction ============ Symmetries are widely used as a useful tool in modeling physical systems. The ordinary symmetries are associated with isometries, that are spacetime diffeomorphisms that leave the metric invariant. A one-parameter continuous isometry is connected with a Killing vector field. An extension of the Killing vector fields is represented by conformal Killing vector fields [@K-Y] with flows preserving a given class of metrics. However, it has been proved that the investigation of symmetries in the whole phase space of a system is exceedingly useful. Such transformations of the whole phase space for which the dynamics of the system is left invariant are often referred as [*hidden symmetries*]{}. The hidden symmetries of curved manifolds are represented by Killing tensors and Killing-Yano tensors. Thanks to such symmetries many complicated physical problems become tractable taking into account that the equation of motion are separable and integrable. Analogously, conformal Killing-Yano tensors are associated with conserved quantities along null geodesics and integrability of massless field equations. The purpose of this paper is to present a method to construct Killing forms on toric Sasaki-Einstein manifolds. We exemplify the procedure in the case of the five-dimensional homogeneous Sasaki-Einstein manifold $T^{1,1}$. Until recently the only explicitly known non-trivial Sasaki-Einstein metric in dimension five was $T^{1,1}$ [@C-O]. The five-dimensional manifolds $T^{p,q}$ which are the coset spaces $(SU(2)\times SU(2))/U(1)$ have been considered by Romans [@LJR] in the context of Kaluza-Klein supergravity. Romans found that for $p=q=1$ the compactification preserves $8$ supersymmetries, while for other $p$ and $q$ all supersymmetries are broken. In light of the $AdS/CFT$ correspondence the $AdS_5 \times T^{1,1}$ model of [@KW] is the first example of a supersymmetric holographic theory based on a compact manifold which is not locally $S^5$. The approach we take in order to achieve our goal basically relies on the interplay between symplectic and complex coordinates on toric manifolds [@M-S-Y; @Abr]. In our particular case, this description of the conifold is slightly different from [@M-S; @H-K-O], when the geometric features of the conifold are mainly exhibited. In turn, our approach enable us to use the correspondence between special Killing forms and parallel forms on the metric cone which was introduced by Semmelmann [@Semm]. For more details concerning this method in the case of toric manifolds we refer to [@Vis; @S-V-V]. The paper is organized in the following manner: In the second Section we introduces the main concepts and technical tools we use in the rest of the paper. We also give the necessary preliminaries regarding the special Killing forms, toric Sasaki-Einstein manifolds and the way their Calabi-Yau cones spaces can be regarded as complex manifolds. In Section 3 we apply these results constructing complex coordinates in the particular case of the conifold $C(T^{1,1})$. In Section 4 we extract the complete set of special Killing forms on the Sasaki-Einstein space $T^{1,1}$. Finally, our conclusions are presented within the last Section. Preliminaries ============= Special Killing forms --------------------- A natural generalization of conformal Killing vector fields is given by the conformal Killing forms which are sometimes referred as twistor forms or conformal Killing-Yano tensors. A conformal Killing-Yano tensor of rank $p$ on a $n$-dimensional Riemannian manifold $(M,g)$ is a $p$-form $\psi$ which satisfies $$\label{CKY} \nabla_X\psi=\frac{1}{p+1}X \lrcorner d\psi-\frac{1}{n-p+1}X^*\wedge d^*\psi \,,$$ for any vector field $X$ on $M$. Here we used the standard conventions: $\nabla$ is the Levi-Civita connection with respect to the metric $g$, $X^*$ is the $1$-form dual to the vector field $X$, $\lrcorner$ is the operator dual to the wedge product and $d^*$ is the adjoint of the exterior derivative $d$. In component notation, the conformal Killing-Yano tensor equation is given by $$\nabla_{(i_1} \psi_{i_2)i_3 \cdots i_{p+1}} = \frac{1}{n-p+1} \left( g_{i_1 i_2} \nabla_j \psi^{j}_{\phantom{j} i_3 \cdots i_{p+1}} - (p-1)g_{[i_3(i_1} \nabla_j \psi^{j}_{\phantom{j} i_2)i_4 \cdots i_{p+1}]}\right)\,.$$ We used round brackets to denote symmetrization over the indices within. For $p=1$ we recover the usual definition of a Killing vector: $$\nabla_{(j}\psi_{i)} = 0 \,.$$ If $\psi$ is co-closed in (\[CKY\]), then we obtain the definition of a Killing-Yano tensor [@K-Y] which, in component notation, satisfies the equation: $$\nabla_{(j}\psi_{i_1)i_2 \dots i_p} = 0 \,.$$ A particular class of Killing forms is represented by the special Killing forms: A Killing form $\psi$ is said to be a special Killing form if it satisfies for some constant $c$ the additional equation $$\nabla_X(d\psi) = c X^* \wedge \psi \,,$$ for any vector field $X$ on $M$. It is worth mentioning the fact that the most known Killing forms are actually special. There is also a symmetric generalization of the Killing vectors: A symmetric tensor $K_{i_1 \cdots i_r}$ of rank $r>1$ satisfying the generalized Killing equation $$\nabla_{(j}K_{i_1 \cdots i_r)} =0\,,$$ is called a Stäckel-Killing tensor. The analogue of the conserved quantities associated with Killing vectors is given by the following proposition: For any geodesic $\gamma$ with tangent vector $\dot{\gamma}^i$ $$Q_K =K_{i_1 \cdots i_r} \dot{\gamma}^{i_1} \cdots \dot{\gamma}^{i_r}\,,$$ is constant along $\gamma$. Let us note that there is an important connection between these two generalizations of the Killing vectors. To wit, given two Killing-Yano tensors $\psi^{i_1, \dots, i_k}$ and $\sigma^{i_1, \dots, i_k}$ there is a Stäckel-Killing tensor of rank $2$: $$K^{(\psi,\sigma)}_{ij} = \psi_{i i_2 \dots i_k} \sigma_{j}^{\phantom{j}i_2 \dots i_k}+ \sigma_{i i_2 \dots i_k} \psi_{j}^{\phantom{j}i_2 \dots i_k} \,.$$ This fact offers a method to generate higher order integrals of motion by identifying the complete set of Killing forms. Sasaki-Einstein manifolds ------------------------- An *almost contact structure* on a smooth manifold $M$ is a triple $(\varphi,B,\eta)$, where $\varphi$ is a field of endomorphisms of the tangent spaces, $B$ is a vector field and $\eta$ is a 1-form on $M$ satisfying (see [@Sas]) $$\label{gg1} \varphi^2=-I+\eta\otimes B,\ \ \ \eta(B)=1 \,.$$ We remark that many authors also include in the above definition the conditions that $\varphi B=0$ and $\eta\circ\varphi=0$, although these are deducible from (\[gg1\]) (see [@Bl]). A Riemannian metric $g$ on $M$ is said to be *compatible* with the almost contact structure $(\varphi,B,\eta)$ if and only if the relation $$g(\varphi X, \varphi Y)=g(X,Y)-\eta(X)\eta(Y)\,,$$ holds for all pairs of vector fields $X,Y$ on $M$. In this case $(\varphi,B,\eta,g)$ is called an *almost contact metric structure*. Moreover, if the Levi-Civita connection $\nabla$ of the metric $g$ satisfies $$(\nabla_X\varphi) Y=g(X,Y)B-\eta(Y)X\,,$$ for all vector fields $X,Y$ on $M$, then $(\varphi,B,\eta,g)$ is said to be a *Sasakian structure* [@Bl]. It is also important to note that Sasakian geometry is in fact the odd-dimensional counterpart of Kähler geometry, since a Sasakian structure may be reinterpreted and characterized in terms of the metric cone as follows. The metric cone of a Riemannian manifold $(M,g)$ is the Riemannian manifold $C(M)=(0,\infty)\times M$ with the metric given by $$\bar{g}=dr^2+r^2g\,,$$ where $r$ is a coordinate on $(0,\infty)$. Then $M$ is a Sasaki manifold if and only if its metric cone $C(M)$ is Kähler [@B-G-1999]. We note that the one form $\eta$ extends to a one form on $C(M)$ by $\eta(X)=\frac12\bar{g}(B,X)$ and $M$ is identified with the subset $r=1$ of $C(M)$. In particular, the cone $C(M)$ is equipped with an integrable complex structure $J$ defined by $$Jr\partial_r=B,\ \ JY=\varphi Y-\eta(Y)r\partial_r,\ Y\in TM \,,$$ and a Kähler 2-form $\omega$ given by $$\omega = \frac12 d (r^2 \eta)=\frac12 dd^cr^2 \,,$$ where $d^c=\frac{i}{2}(\bar{\partial}-\partial)$, both $J$ and $\omega$ being parallel with respect to the Levi-Civita connection $\bar{\nabla}$ of $\bar{g}$. Moreover, $M$ has odd dimension $2n+1$, where $n+1$ is the complex dimension of the Kähler cone. Conversely, given any algebraic Kähler orbifold, we can naturally associate to it a quasi-regular Sasakian manifold [@B-G-1999]. An *Einstein manifold* is a Riemannian manifold $(M,g)$ satisfying the Einstein condition $$\label{Einstein} Ric_{g} = \lambda g \,,$$ for some real constant $\lambda$, where $Ric_{g}$ denotes the Ricci tensor of $g$. Einstein manifolds with $\lambda=0$ are called *Ricci-flat manifolds*. A *Sasaki-Einstein manifold* is a Riemannian manifold $(M,g)$ that is both Sasaki and Einstein. We note that in the case of Sasaki-Einstein manifolds one always has with the Einstein constant $\lambda=2n$. We also remark that Gauss equation relating the curvature of submanifolds to the second fundamental form shows that a Sasaki manifold $M$ is Einstein if and only if the metric cone $C(M)$ is Kähler Ricci-flat. In particular the Kähler cone of an Sasaki-Einstein manifold has trivial canonical bundle [@B-G-2010; @Sp]. We note that one of the most familiar example of homogeneous Sasaki-Einstein five-manifold is the space $T^{1,1}=S^2\times S^3$ endowed with the following metric [@M-S; @C-O] $$\begin{split} ds^2(T^{1,1}) = & \frac16 (d \theta^2_1 + \sin^2 \theta_1 d \phi^2_1 + d \theta^2_2 + \sin^2 \theta_2 d \phi^2_2) +\\ & \frac19 (d \psi + \cos \theta_1 d \phi_1 + \cos \theta_2 d \phi_2)^2 \,. \end{split}$$ A *toric Sasaki manifold* $M$ is a Sasaki manifold whose Kähler cone $C(M)$ is a toric Kähler manifold [@Gu]. In particular, a five-dimensional toric Sasaki-Einstein manifold is a Sasaki-Einstein manifold with three $U(1)$ isometries: $G=\mathbb{T}^3$. In this case one can construct canonical coordinates based on the symplectic geometry of the cone $C(M)$ and specify the Sasaki-Einstein structure in terms of toric data together with a single function $G$, a symplectic potential, on the three-dimensional image of the momentum map [@O-Y]. It is known that one of the simplest example of a toric non-orbifold singularity is the conifold $C(T^{1,1})$, i.e. the Calabi-Yau cone over $T^{1,1}$. On the other hand, according to Semmelman [@Semm], there is a correspondence between special Killing forms defined on the Sasaki-Einstein manifold $M$ and the parallel forms defined on the metric cone $C(M)$. More exactly, a $p-$dimensional differential form $\Psi $ is a special Killing form on $M$ if and only if the corresponding form $$\Psi _{cone}:=r^pdr\wedge \Psi +\frac{r^{p+1}}{p+1}d\Psi \,, \label{eq Semmelmann}$$ is parallel on $C(M)$. In particular, on a five-dimensional Sasaki manifold $M$ with the Reeb vector field $B$ and $1-$form $\eta := B^\ast$, there are the following two special Killing forms: $$\label{sKPsi} \Psi_1 = \eta \wedge d\eta,\quad \Psi_2=\eta \wedge (d\eta)^2 \,.$$ Besides these Killing forms, there are two closed conformal Killing forms, also called $\ast$-Killing forms, given by $$\label{sKPhi} \Phi_1 = d\eta,\quad \Phi_2=(d\eta)^2 \,.$$ Moreover, in the case of the Calabi-Yau cone $C(M)$ it follows that we have two additional Killing forms on $M$ connected with the additional parallel forms of the cone given by the holomorphic complex volume form $\Omega$ of $C(M)$ and its conjugate [@Semm]. Symplectic and complex coordinates on toric manifolds ----------------------------------------------------- Let us consider a toric Sasaki-Einstein manifold. In the spirit of [@Abr] we use in our further considerations the symplectic (action-angle) coordinates $(y ^i,\Phi ^i)$; here the angular coordinates $\Phi ^i$ will generate the toric action. The $y^i$ coordinates are obtained using the momentum map $\mu =\frac 12r^2\eta $, with the correspondence $$y^i=\mu (\partial /\partial \Phi ^i)\,. \label{xi}$$ The Kähler form $\omega $ can be written in the simple manner [@Abr; @M-S-Y] $$\omega =dy^i\wedge d\Phi ^i\,.$$ In turn, the corresponding Kähler metric on the cone $C(M)$ is constructed using the *symplectic potential* $G$, which is a strictly convex function $G=G(y)$ of homogeneous degree $-1$ in $y$ [@M-S-Y; @Abr]. We get $$ds^2=G_{ij}dy^idy^j+G^{ij}d\Phi ^id\Phi ^j\,,$$ where the metric coefficients are computed $$G_{ij}=\frac{\partial ^2G}{\partial y^i\partial y^j}\,,$$ with $\left( G^{ij}\right) =\left( G_{ij}\right) ^{-1}$. The complex structure $J$ can be described using the above symplectic coordinates metric coefficients, namely $$J=\left( \begin{array}{cc} 0 & -G^{ij} \\ G_{ij} & 0 \end{array} \right) \,.$$ Now we return to the construction of the symplectic potential $G$. A classical result of Delzant associates to any Delzant polytope $P\in \mathbb{R}^n$ a close connected symplectic manifold $M$, together with a Hamiltonian $\mathbb{T}^n$ action on the manifold, showing that the polytope turns out to be the image of the momentum map, $P=\mu (M)$. We remaind that a Delzant polytope is a convex polytope such that there are $n$ edges meeting at each vertex, each edge meeting at the vertex is of form $1+tu_i$, where $u_i\in \mathbb{Z}^n$, and $\{u_i\}$ can be chosen to form a basis in $\mathbb{Z}^n$. A Delzant polytope can be described by the inequalities $$l_A(y):=\left\langle y,v_A\right\rangle \ge 0\text{, for }1\le A\le d\,,$$ where $\{v_A\}$ are inward pointing normal vectors to the facets of the polytope, $d$ is the number of facets [@Abr; @Gu]. In the case of the Calabi-Yau cone we take $C(M)$ to be [*Gorenstein*]{} which is a necessary condition to admit a Ricci-flat Kähler metric and $M$ to admit a Sasaki-Einstein metric. For affine toric varieties it is well-known that $C(M)$ being Gorenstein is equivalent to the existence of a basis for the torus $\mathbb{T}^n$ for which $$\label{Gorenstein} v_i=(+1,w_i)\,,$$ for each $a,\cdots ,d$ and $w_a \in\mathbb{Z}^{n-1}$ [@M-S; @M-S-Y]. If $B$ is the Reeb vector field, let us point out the following relation which links this geometric object to the metric coefficients $$B_i=2G_{ij}y^j.$$ Now let us also define the affine function $l_B:=\left\langle B,\cdot \right\rangle $, and $l_\infty :=\left\langle \sum_Av_A,\cdot \right\rangle $. Then, the symplectic potential $G$ can be written in terms of the toric data [@Abr; @Gu] $$G=G^{can}+G^B+h, \label{G}$$ where $$\begin{aligned} G^{can} &=&\frac 12\sum_Al_A(y )\log l_A(y), \label{G_can_G_B} \\ G^B &=&\frac 12l_B(y)\log l_B(y)-\frac 12l_\infty (y)\log l_\infty (y)\,, \nonumber\end{aligned}$$ and $h$ is a homogeneous function of degree $1$ in variables $y$. In the general case, as $G$ needs to satisfy the Monge-Ampère equation, the function $h$ is added. For a complete determination of the symplectic potential $G$ it is necessary to compute the Reeb vector $B$ and the function $h$. There are two known different algebraic procedures to extract the components of the Reeb vector from the toric data. According to the AdS/CFT correspondence the volume of the Sasaki-Einstein space corresponds to the central charge of the dual conformal field theory. The first procedure is based on *the maximization of the central charge* ($a$-maximization) [@Int-Wec] used in connection with the computation of the Weyl anomaly in 4-dimensional field theory. The second one is known as *volume minimization* (or $Z$-minimization) [@M-S-Y]. The symplectic potential allow us to pass to the coordinate patch $(x^i,\Phi^i)$ obtained from complex coordinates $z^i:=x^i+\mathrm{i}\Phi ^i$, with $\mathrm{i}:=\sqrt{-1}$; this is possible via the Legendre transform which relates the symplectic potential $G$ and the Kähler potential $F$ $$F(x)=\left( y^i\frac{\partial G}{\partial y^i}-G\right) : \left(y=\partial F/\partial x\right) \,.$$ Consequently $F$ and $G$ are Legendre dual to each other $$F(x)+G(y)=\sum_j\frac{\partial F}{\partial x^j}\frac{\partial G}{\partial y^i}\,\,,$$ and $$x^i=\frac{\partial G}{\partial y^i},\mbox{\,\,}y^i=\frac{\partial F}{\partial x^i}\,.$$ The metric structure is now written in the following manner $$ds^2=F_{ij}dx^idx^j+F_{ij}d\Phi ^id\Phi ^j\,,$$ where the metric coefficients are again obtained using the Hessian of the Kähler potential $F$, i.e. $$F_{ij}=\frac{\partial ^2F}{\partial x^i\partial x^j}\,.$$ Note also that $\left( F_{ij}\right) =\left( G^{ij}\right) $ [@Abr]. With respect to the coordinates $(x^i,\Phi ^i)$, the Kähler form is $$\omega =\left( \begin{array}{cc} 0 & F_{ij} \\ -F_{ij} & 0 \end{array} \right) \,.$$ We use in the following the fact that the Calabi-Yau metric cone is Ricci flat. From the classical formula $$\rho =-\mathrm{i}\partial \bar \partial \log \det (F_{ij})\,,$$ we get $$\det (G_{ij})=\exp{\left(2\gamma _i\frac{\partial G}{\partial y^i}-c\right)}\,, \label{det_G}$$ with constants $\gamma _i$, and $c$. Using -, we are able to express the coordinates $x^i$ and the metric coefficients $G_{ij}$ $$\begin{split} x^i &=\frac{\partial G}{\partial y^i}=\frac 12\sum_Av_A^i\log l_A(y) +\frac 12B^i(1+\log l_B(y)) \\ &-\frac 12\sum_Av_A^i\log l_\infty (y)+\lambda _i \,,\\ G_{ij} &=\frac 12\sum_A\frac{v_A^iv_A^j}{l_A(y)} +\frac 12\frac{B_iB_j}{l_B(y)} -\frac 12\frac{\sum_Av_A^i\sum_Av_A^j}{l_\infty (y)}\,. \end{split}$$ Now, as the metric has to be smooth, from (\[det\_G\]) it turns out that [@O-Y] $$\gamma =(-1,0,..,0) \,,$$ and $$\det (F_{ij})=\exp{\left(2x^1+c\right)}.$$ If $\mathrm{Vol}$ is the volume form on the metric cone, then the holomorphic volume form $\Omega$ satisfies $$\mathrm{Vol}=\frac{i^{n+1}}{2^{n+1}} (-1)^{n(n+1)/2}\Omega \wedge \bar{\Omega} =\frac{1}{(n+1)!} \omega ^{n+1}\,.$$ Then, eventually ignoring the multiplicative constant, in complex coordinates $\Omega $ can be written as [@M-S-Y] $$\begin{aligned} \Omega &=&\exp({i\alpha })\det (F_{ij})^{1/2}dz^1\wedge ..\wedge dz^n \\ &=&\exp({x^1+i\alpha})dz^1\wedge ..\wedge dz^n\,.\end{aligned}$$ As $\Omega $ is parallel, it is also closed. Then we can fix the phase $\alpha $ to be $\Phi ^1$, and we obtain the following simple formula for the holomorphic volume form, which motivates the interest for complex (and consequently, symplectic) coordinates $$\Omega =\exp (z^1)dz^1\wedge ..\wedge dz^n. \label{Omega}$$ Employing the above relation, in the next sections we show that it is possible to extract the special Killing forms on manifolds of Sasaki-Einstein type. Symplectic and complex coordinates on conifold $C(T^{1,1})$ =========================================================== Throughout this section we introduce complex coordinates on $C(T^{1,1})$ using the classical procedure exposed above. We start out by considering the global defined contact 1-form $$\eta =\frac 13(d\psi +\cos \theta _1d\phi _1+\cos \theta _2d\phi _2)\,. \label{eta}$$ This form allows us to construct on $C(T^{1,1})$ the symplectic form (see e.g. [@M-S]) $$\begin{split} \omega & =-\frac{r^2}6(\sin \theta _1d\theta _1\wedge d\phi_1 +\sin \theta_2d\theta _2\wedge d\phi _2) \\ & +\frac 13rdr\wedge (d\psi +\cos \theta _1d\phi _1+\cos \theta _2d\phi_2)\,. \end{split}$$ Furthermore, if we employ the basis [@M-S] for an effectively acting $\mathbb{T}^3$ action $$\begin{split} e_1& =\frac \partial {\partial \phi _1}+\frac 12\frac \partial {\partial \nu}\,, \\ e_2& =\frac \partial {\partial \phi _2}+\frac 12\frac \partial {\partial \nu}\,, \\ e_3& =\frac \partial {\partial \nu }\,, \end{split}$$ where $2\nu =\psi $, then, considering action coordinates associated with this basis, we get the momentum map using (see also [@M-S]) $$\mu =\left( \frac 16r^2(\cos \theta _1+1),\frac 16r^2(\cos \theta_2+1),\frac 13r^2\right) \,. \label{mom}$$ The Reeb vector field $B$ has the form $$\label{Reeb} B = 3 \frac{\partial}{\partial \psi} = \frac{3}{2} \frac{\partial}{\partial \nu}\,,$$ and is easy to see that $\eta(B)=1$. Now let us consider the “inward pointing” primitive normal vectors to the cone $$v_1^{\prime}=[-1,0,1]\,,\,v_2^{\prime}=[0,-1,1]\,,\,v_3^{\prime}=[1,0,0]\, \,,\,v_4^{\prime}=[0,1,0]\,. \label{v'}$$ We apply a $SL(3;\mathbb{Z})$ transformation $T$ (see also [@M-S]) $$T=\left( \begin{array}{ccc} 1 & 1 & 2 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array} \right) \,,$$ to bring the vectors $v_i=Tv_i^{\prime }$ in the form [@O-Y] $$v_1=[1,1,1]\,,\,v_2=[1,0,1]\,,\,v_3=[1,0,0]\,,\,v_4=[1,1,0]\,. \label{v}$$ According to the above transformation we obtain the new basis $$\label{eprime} \begin{split} e_1^{\prime }& =\frac \partial {\partial \phi _1}+\frac 12\frac \partial{\partial \nu }\,, \\ e_2^{\prime }& =\frac \partial {\partial \phi _2}-\frac \partial {\partial\phi _1}\,, \\ e_3^{\prime }& =-\frac \partial {\partial \phi _1}-\frac \partial {\partial\phi _2}\,, \end{split}$$ where $e_i^{\prime }:=e_iT^{-1}$. We consider the new angle coordinates $$\begin{split} \Phi ^1& :=2\nu =\psi \,, \\ \Phi ^2& :=-\frac 12\phi _1+\frac 12\phi _2+\nu =-\frac 12\phi _1+\frac12\phi _2 +\frac 12\psi,\\ \Phi ^3& :=-\frac 12\phi _1-\frac 12\phi _2+\nu =-\frac 12\phi _1-\frac12\phi _2 +\frac 12\psi\,. \end{split}$$ and it is easy to check that $$e_i^{\prime }=\frac \partial {\partial \Phi ^i}\,,$$ In this new basis, applying , the momentum map becomes: $$\mu ^{\prime }= y =\left( \frac 16r^2(\cos \theta _1+1), \frac16r^2(\cos \theta _2-\cos \theta _1), -\frac 16r^2(\cos \theta _1+\cos \theta_2)\right)\,.$$ This way we end up with the symplectic action-angle coordinates $(y^i$, $\Phi ^i)$, for $1\le i\le 3$. Now, in order to introduce the complex coordinates on conifold we need the symplectic potential $G$. In the particular case of the conifold $T^{1,1}$, the sum $G^{can}+G^B$ is already a solution of the Monge-Ampère equation. Consequently the function $h$ is not needed anymore, and the equations - simplify as (see e.g. [@O-Y]) $$\label{G_conifold} G=G^{can}+G^B\,,$$ where $$G^{can}=\sum_{A=1}^4\frac 12\left\langle v_A,y \right\rangle \log\left\langle v_A,y \right\rangle \,,$$ and $$G^B=\frac 12\left\langle B,y \right\rangle \log \left\langle B,y \right\rangle -\frac 12\left\langle B^{can},y \right\rangle \log \left\langle B^{can},y \right\rangle \,.$$ In the above relation the vectors $v_A$ are just and $$B^{can}=\sum_1^4v_A=(4,2,2)\,.$$ Concerning the Reeb vector field on $T^{1,1}$ written in the new basis $\{e_i^{\prime}\}$ , it has the components $$B=(3,3/2,3/2)\,,$$ consistent with the determination from toric data using $Z$-minimization [@Int-Wec] or $a$-maximization [@M-S-Y]. We construct complex coordinates using and the Legendre transform $$\begin{split} x^i=\frac{\partial G}{\partial y^i}=& \frac 12\sum_1^4 v_{A}^{i}\log <v_A,y >+\frac 12B^i\log <B, >+\frac 12B^i \\ & -\frac 12(B^{can})^i\log <B^{can},y >\,. \end{split}$$ But it is easy to see that $$\begin{split} \sum_1^4 v_{A}^{1}\log <v_A,y >=& 8\log r+2\log \sin \theta _1+2\log \sin \theta _2 \\ & -4\log 2-4\log 3, \\ \sum_1^4 v_{A}^{2}\log <v_A,y >=& 4\log r+\log (1-\cos \theta _1)+\log (1+\cos \theta _2) \\ & -2\log 2-2\log 3, \\ \sum_1^4 v_{A}^{3}\log <v_A,y >=& 4\log r+\log (1-\cos \theta _1)+\log (1-\cos \theta _2) \\ & -2\log 2-2\log 3, \\ \log <B,y >=& 2\log r-\log 2, \\ \log <B^{can},y >=& 2\log r+\log 2-\log 3\,. \end{split} \label{g1}$$ Using now (\[g1\]) and basic trigonometric formulas, we derive $$\begin{split} x^1& =3\log r+\log \sin \theta _1+\log \sin \theta _2+\frac 32-\frac{11} 2\log 2\,, \\ x^2& =\frac 32\log r+\log \sin \frac{\theta _1}2+\log\cos\frac{\theta _2}2 +\frac 34-\frac{11}4\log 2\,, \\ x^3& =\frac 32\log r + \log \sin \frac{\theta _1}2+\log\sin\frac{\theta_2}2 \frac 34-\frac{11}4\log 2\,. \end{split}$$ In the sequel, for the sake of simplicity we will ignore the additive constants. Therefore, in accordance with [@M-S-Y] we can introduce on conifold $C(T^{1,1})$ the following patch of complex coordinates $$\label{Z} \begin{split} z^1 &=3\log r+\log \sin \theta _1+\log \sin \theta _2+\mathrm{i}\psi , \\ z^2 &=\frac 32\log r+\log \sin \frac{\theta _1}2+\log \cos \frac{\theta _2}2\, \\ &+\frac{\mathrm{i}}2(\psi +\phi _1+\phi _2), \\ z^3 &=\frac 32\log r+\log \sin \frac{\theta _1}2+\log \sin \frac{\theta _2}2\, \\ &+\frac{\mathrm{i}}2(\psi -\phi _1-\phi _2)\,. \end{split}$$ Now, regarding , we see that the above coordinates are precisely the necessary ingredient in order to extract the special Killing forms on our homogeneous Sasaki-Einstein manifold. Special Killing forms on $T^{1,1}$ ================================== Applying the above general results, in this section we obtain the complete set of special Killing forms on the manifold $T^{1,1}$. First of all, we have to calculate the holomorphic volume form $\Omega $. Starting out with , we obtain $$\begin{split} & \exp( {z^1})=r^3\sin \theta _1\sin \theta _2\exp \mathrm{i}{\psi }\,, \\ & dz^1=\frac 3rdr+T_1, \\ & dz^2=\frac 3{2r}dr+T_2\,, \\ & dz^3=\frac 3{2r}dr+T_3\,. \end{split}$$ where $$\label{T} \begin{split} T_1&:=\cot \theta _1d\theta _1+\cot \theta _2d\theta _2+\mathrm{i}d\psi ,\\ T_2 &:=\frac 12\cot \frac{\theta_1}{2} d\theta _1-\frac 12\tan \frac{\theta _2}2d\theta_2 +\frac{\mathrm{i}}2(d\psi -d\phi _1+d\phi _2), \\ T_3 &:=\frac 12\cot \frac{\theta_1}{2} d\theta _1+\frac 12\cot \frac{\theta _2}2d\theta_2 +\frac{\mathrm{i}}2(d\psi -d\phi _1-d\phi _2)\,. \end{split}$$ Now we calculate the holomorphic volume form (see e.g. [@M-S-Y]) $$\begin{split} \Omega &=\exp (z^1)dz^1\wedge dz^2\wedge dz^3 \\ \ &=\exp (z^1)(\frac 3rdr+T_1)\wedge (\frac 3{2r}dr+T_2)\wedge (\frac 3{2r}dr+T_3) \,. \nonumber \end{split}$$ In our particular framework the equation becomes $$\Omega =r^2dr\wedge \Psi +\frac{r^3}3d\Psi \,.$$ In order to extract $\Psi$ we have to keep the trace of the differential form $dr$ in the above equation. We clearly get $$\Psi =3\sin \theta _1\sin \theta _2 \exp({\mathrm{i}\psi })(\frac 14T_2\wedge T_3-\frac 12T_1\wedge T_3+\frac 12T_1\wedge T_2)\,. \label{TT}$$ We calculate the above wedge products using . For the first wedge product in , after calculations we get $$\begin{split} T_2\wedge T_3 &=\frac 12\cot \frac{\theta _1}2\frac 1{\sin \theta_2}d\theta _1\wedge d\theta _2- \frac{\mathrm{i}}2\cot \frac{\theta _1}2d\theta _1\wedge d\phi _2 \label{T2T3} \\ &\ -\frac{\mathrm{i}}2\frac 1{\sin \theta _2}d\theta _2\wedge d\psi +\frac {\mathrm{i}}{2}\frac 1{\sin \theta _2}d\theta _2\wedge d\phi _1-\frac{\mathrm{i}}2\cot \theta _2d\theta _2\wedge d\phi _2 \\ &\ -\frac 12d\phi _1\wedge d\phi _2+\frac 12d\psi \wedge d\phi _2\,. \end{split}$$ For the second product we obtain $$\begin{split} T_1\wedge T_3 &=\frac 12(\cot \theta _1\cot \frac{\theta _2}2 -\cot \theta_2\cot \frac{\theta _1}2)d\theta _1\wedge d\theta_2 -\frac{\mathrm{i}}2\frac1{\sin \theta _1}d\theta _1\wedge d\psi \label{T1T3} \\ &-\frac{\mathrm{i}}2\cot \theta _1d\theta _1\wedge d\phi_1 -\frac{\mathrm{i}}2\cot \theta _1d\theta _1\wedge d\phi _2 -\frac{\mathrm{i}}2\frac 1{\sin\theta _2}d\theta _2\wedge d\psi \\ &-\frac{\mathrm{i}}2\cot \theta _2d\theta _2\wedge d\phi _1 -\frac{\mathrm{i}}2\cot \theta _2d\theta _2\wedge d\phi _2+\frac 12d\psi \wedge d\phi_1 +\frac 12d\psi \wedge d\phi _2\,. \end{split}$$ Finally, for the last product we get $$\begin{split} \label{T1T2} T_1\wedge T_2 &=-\frac 12(\cot \theta _1\tan \frac{\theta _2}2 -\cot \theta_2\cot \frac{\theta _1}2)d\theta _1\wedge d\theta _2 -\frac{\mathrm{i}}2\frac1{\sin \theta _1}d\theta _1\wedge d\psi \\ &\ \ -\frac{\mathrm{i}}2\cot \theta _1d\theta _1\wedge d\phi_1 +\frac{\mathrm{i}}2\cot \theta _1d\theta _1\wedge d\phi_2 +\frac{\mathrm{i}}2\frac1{\sin \theta _2}d\theta _2\wedge d\psi \\ &\ -\frac{\mathrm{i}}2\cot \theta _2d\theta _2\wedge d\phi_1 +\frac{\mathrm{i}}2\cot \theta _2d\theta _2\wedge d\phi _2+\frac 12d\psi \wedge d\phi_1 -\frac 12d\psi \wedge d\phi _2\,. \end{split}$$ Now we plug - in and we end up with the simple formula bellow for the special complex Killing form (in what follows we ignore the multiplicative constants) $$\begin{split} \Psi & =\exp({\mathrm{i}\psi}) \left[ 2 d\theta_1\wedge d\theta_2 -2\mathrm{i}\sin \theta_2 d\theta_1\wedge d\theta_2 \right. \\ & \left. ~~~~+2\mathrm{i}\sin \theta_1 d\theta_2\wedge d\phi_1 -2\sin \theta_1 \sin \theta_2 d\phi _1\wedge d\phi_2 \right] \,. \end{split}$$ From here, we can easily derive the real special Killing forms computing the real and imaginary part of $\Psi $: $$\begin{split} \Re \Psi & =\cos \psi \,d\theta _1\wedge d\theta _2+\sin \theta _2\sin \psi \,d\theta _1\wedge d\phi _2 \\ & ~~-\sin \theta _1\sin \psi \,d\theta _2\wedge d\phi _1-\sin \theta _1\sin \theta _2\cos \psi \,d\phi _1\wedge d\phi _2\,, \end{split}$$ $$\begin{split} \Im \Psi & =\sin \psi \,d\theta _1\wedge d\theta _2-\sin \theta _2\cos \psi \,d\theta _1\wedge d\phi _2 \\ & ~~+\sin \theta _1\cos \psi \,d\theta _2\wedge d\phi _1-\sin \theta _1\sin \theta _2\sin \psi \,d\phi _1\wedge d\phi _2\,. \end{split}$$ Finally, we calculate the Killing forms $\Phi _1$ and $\Phi _2$ , $\Psi _1$ and $\Psi _2$ using the contact 1-form $\eta $ . We obtain $$\Phi _1=d\eta =-\frac 13(\sin \theta _1d\theta _1\wedge d\phi _1+ \sin \theta_2d\theta _2\wedge d\phi _2)\,, \label{1}$$ $$\Phi _2=(d\eta )^2=-\frac 29\sin \theta _1\sin \theta _2d\theta _1\wedge d\theta _2\wedge d\phi _1\wedge d\phi _2 \,, \label{2}$$ and respectively $$\begin{split} \Psi _1 =&\eta \wedge d\eta =\frac 19(\sin \theta _1d\psi \wedge d\theta _1\wedge d\phi _1+\sin \theta _2d\psi \wedge d\theta _2\wedge d\phi _2 \\ & -\cos \theta _1\sin \theta _2d\theta _2\wedge d\phi _1\wedge d\phi _2+\cos \theta _2\sin \theta _1d\theta _1\wedge d\phi _1\wedge d\phi _2)\,,\\ \Psi _2=&\eta \wedge (d\eta )^2=-\frac 2{27}\sin \theta _1\sin \theta _2d\psi \wedge d\theta _1\wedge d\theta _2\wedge d\phi _1\wedge d\phi _2 \,. \end{split}$$ Conclusions =========== The significance of the present work relies on unquestionable relevance of conformal Killing-Yano tensors in both mathematics and physics. It is known that it is a very difficult problem to find solutions of the conformal Killing-Yano equations on arbitrary Riemannian manifolds, but fortunately, in the case of spaces endowed with remarkable geometrical structures the explicit construction of the Killing forms is permitted. In this paper we have obtained the complete set of special Killing forms on the five-dimensional Sasaki-Einstein space $T^{1,1}$ with an approach based on the interplay between symplectic and complex coordinates on Kähler toric manifolds. On the other hand, concerning the potential of the present paper, as a lot of non-trivial examples of toric Sasaki-Einstein manifolds occurs in the recent literature (see, e.g., [@CFO; @MS; @VC]), it is both natural and useful to extend the present work to other spaces of interest. In fact, using toric geometry many examples of Sasaki-Einstein manifolds can be constructed, and these spaces are a good testing ground for the predictions of the AdS/CFT correspondence [@MSY]. Acknowledgments {#acknowledgments .unnumbered} =============== MV was supported by CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0137. The work of GEV was supported by CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0118. [99]{} M. Abreu, *Kähler geometry of toric manifolds in symplectic coordinates*, Fields Institute Commun. **35** (2003), 1–24, AMS, Providence, RI; arXiv:math.DG/0004122. D.E. Blair, *Contact manifolds in Riemannian Geometry.* Lectures Notes in Math. **509**, Springer-Verlag, 1976. C. Boyer and K. Galicki, *3-Sasakian manifolds*, Surveys in Differential Geometry: Essays on Einstein Manifolds, Surv. Differ. Geom., Vol. VI, 123–184, Int. Press, Boston, MA, 1999; arXiv:hep-th/9810250. C. Boyer and K. Galicki, *Sasakian geometry, holonomy, and supersymmetry*, Handbook of pseudo-Riemannian geometry and supersymmetry, 39–83, IRMA Lect. Math. Theor. Phys. **16**, Eur. Math. Soc., Zürich, 2010; arXiv:math/0703231. P. Candelas and X. de la Ossa, *Comments on conifolds*, Nucl. Phys. **B342** (1990) 246–268. K. Cho, A. Futaki and H. Ono, *Uniqueness and examples of compact toric Sasaki-Einstein metrics*, Comm. Math. Phys. **277** (2008) 439–458; arXiv:math/0701122. V. Guillemin, *Kaehler structures on toric varieties*, J. Diff. Geom. **40** (1994) 285–309. C.P. Herzog, I.R. Klebanov and P. Ouyang, *Remarks on the warped deformed conifold*; arXiv:hep-th/0108101. K. Intriligator and B. Wecht, *The exact superconformal R-symmetry maximizes a*, Nucl. Phys. **B667** (2003) 183–200; arXiv:hep-th/0304128. I. R. Klebanov and E. Witten, *Superconformal field theory on threebranes at a Calabi-Yau singularity*, Nucl. Phys.**B536** (1998) 199–218; arXiv:hep-th/9807080. D. Martelli and J. Sparks, *Toric geometry, Sasaki-Einstein manifolds and a new infinite class of AdS/CFT duals*, Commun. Math. Phys. **262** (2006) 51–89; arXiv:hep-th/0411238. D. Martelli and J. Sparks, *Notes on toric Sasaki-Einstein seven-manifolds and AdS4/CFT3*, J. High Energy Phys. **11** (2008), 016, 26; arXiv:hep-th/08080904. D. Martelli, J. Sparks and S.-T. Yau, *The geometric dual of a-maximisation for toric Sasaki-Einstein manifolds*, Commun. Math. Phys. **268** (2006) 39–65; arXiv:hep-th/0503183. D. Martelli, J. Sparks and S.-T. Yau, *Sasaki-Einstein manifolds and volume minimisation*, Comm. Math. Phys. **280** (2008) 611–673; arXiv:hep-th/0603021. T. Oota and Y. Yasui, *Toric Sasaki-Einstein manifolds and Heun equations*, Nucl. Phys. **B742** (2006) 275–294; arXiv:hep-th/0512124. L. J. Romans, *New compactifications of chiral $N=2, d=10$ supergravity*, Phys. Lett. **B153** (1985) 392–396. S. Sasaki, *On differentiable manifolds with certain structures which are closely related to almost contact structure I*, Tohoku Math. J. **12** (1960) 459–476. U. Semmelmann, *Conformal Killing forms on Riemannian manifolds*, Math. Z. **245** (2003) 503–527; arXiv:math.DG/0206117. C. van Coevering, *Some examples of toric Sasaki-Einstein manifolds*, Riemannian topology and geometric structures on manifolds, 185–232, Progr. Math. **271** (2009), Birkhäuser Boston, Boston, MA; arXiv:math/0703501. V. Slesar, M. Visinescu and G.E. Vîlcu *Special Killing forms on Sasaki-Einstein manifolds*, [*Phys. Scripta*]{} **89** (2014) 125205; arXiv: 1403.1015. J. Sparks, *Sasaki-Einstein Manifolds*, Surv. Diff. Geom. **16** (2011) 265–324; arXiv:1004.2461. M. Visinescu, *Killing forms on the five dimensional Einstein-Sasaki $Y(p,q)$ spaces*, Mod. Phys. Lett. A **27** (2012) 1250217; arXiv:1207.2581. K. Yano, *Some remarks on tensor fields and curvature*, Ann. Math. **55** (1952) 328–347. [^1]: vlslesar@central.ucv.ro [^2]: mvisin@theory.nipne.ro [^3]: gvilcu@upg-ploiesti.ro [^4]: gvilcu@gta.math.unibuc.ro

--- abstract: 'We investigate the problem of determining the parameters that describe a quantum channel. It is assumed that the users of the channel have at best only partial knowledge of it and make use of a finite amount of resources to estimate it. We discuss simple protocols for the estimation of the parameters of several classes of channels that are studied in the current literature. We define two different quantitative measures of the quality of the estimation schemes, one based on the standard deviation, the other one on the fidelity. The possibility of protocols that employ entangled particles is also considered. It turns out that the use of entangled particles as a new kind of nonclassical resource enhances the estimation quality of some classes of quantum channel. Further, the investigated methods allow us to extend them to higher dimensional quantum systems.' address: 'Abteilung für Quantenphysik, Universität Ulm, D-89069 Ulm, Germany' author: - 'Markus A. Cirone, Aldo Delgado, Dietmar G. Fischer, Matthias Freyberger, Holger Mack, and Michael Mussinger' title: Estimation of quantum channels with finite resources --- \#1[|\#1]{} \#1[\#1|]{} \#1\#2[\#1|\#2]{} \#1\#2\#3[\#1|\#2|\#3]{} Introduction ============ Quantum information processing has attracted a lot of interest in recent years, following Deutsch’s investigations [@deu] concerning the potentiality of a quantum computer, i.e., a computer where information is stored and processed in quantum systems. Their application as quantum information carriers gives rise to outstanding possibilities, like secret communication (quantum cryptography) and the implementation of quantum networks and quantum algorithms that are more efficient than classical ones [@bell]. Many investigations concern the transmission of quantum information from one party (usually called Alice) to another (Bob) through a communication channel. In the most basic configuration the information is encoded in qubits. If the qubits are perfectly protected from environmental influence, Bob receives them in the same state prepared by Alice. In the more realistic case, however, the qubits have a nontrivial dynamics during the transmission because of their interaction with the environment [@bell]. Therefore, Bob receives a set of distorted qubits because of the disturbing action of the channel. Up to now investigations have focused mainly on two subjects: Determination of the channel capacity [@capa] and reconstruction schemes for the original quantum state under the assumption that the action of the quantum channel is known [@reco]. Here we focus our attention on the problem that precedes, both from a logical and a practical point of view, all those schemes: The problem of determining the properties of the quantum channel. This problem has not been investigated so far, with the exception of very recent articles [@fuj2; @us]. The reliable transfer of quantum information requires a well known intermediate device. The knowledge of the behaviour of a channel is also essential to construct quantum codes [@code; @knil]. In particular, we consider the case when Alice and Bob use a finite amount $N$ of qubits, as this is the realistic case. We assume that Alice and Bob have, if ever, only a partial knowledge of the properties of the quantum channel and they want to estimate the parameters that characterize it. The article is organized as follows. In section \[GeneralDescript\] we shall give the basic idea of quantum channel estimation and introduce the notation as well as the tools to quantify the quality of channel estimation protocols. We shall then continue with the problem of parametrizing quantum channels appropriately in section \[Parametrization\]. Then we are in a position to envisage the estimation protocol for the case of one parameter channels in section \[OneParameter\]. In particular, we shall investigate the optimal estimation protocols for the depolarizing channel, the phase damping channel and the amplitude damping channel. We shall also give the estimation scheme for an arbitrary qubit channel. In section \[QubitPauli\] we explore the use of entanglement as a powerful nonclassical resource in the context of quantum channel estimation. section \[QuditPauli\] deals with higher dimensional quantum channels before we conclude in section \[Conclude\]. A General Description of Channel Estimation {#GeneralDescript} =========================================== The determination of all properties of a quantum channel is of considerable importance for any quantum communication protocol. In practice such a quantum channel can be a transmission line, the storage for a quantum system, or an uncontrolled time evolution of the underlying quantum system. The behaviour of such channels is generally not known from the beginning, so we have to find methods to gain this knowledge. This is in an exact way only possible if one has infinite resources, which means an infinite amount of well prepared quantum systems. The influence of the channel on each member of such an ensemble can then be studied, i.e., the corresponding statistics allows us to characterize the channel. In a pratical application, however, such a condition will never be fulfilled. Instead we have to come along with low numbers of available quantum systems. We therefore cannot determine the action of a quantum channel perfectly, but only up to some accuracy. We therefore speak of channel estimation rather than channel determination, which would be the case for infinite resources. A quantum channel describes the evolution affecting the state of a quantum system. It can describe effects like decoherence or interaction with the environment as well as controlled or uncontrolled time evolution occuring during storage or transmission. In mathematical terms a quantum channel is a completely positive linear map $\cal C$ (CP-map) [@chan; @stine], which transforms a density operator $\rho$ to another density operator $$\rho\,'={\cal C}\rho.$$ Each quantum channel $\cal C$ can be parametrized by a vector $\vec\lambda$ with $L$ components. For a specific channel we shall therefore write ${\cal C}_{\vec\lambda}$ throughout the paper. Depending on the initial knowledge about the channel, the number of parameters differs. The goal of channel estimation is to specify the parameter vector $\vec\lambda$. The protocol Alice and Bob have to follow in order to estimate the properties of a quantum channel is depicted in figure \[figurescheme\]. Alice and Bob agree on a set of $N$ quantum states $\rho_i, i=1,2,\ldots ,N$, which are prepared by Alice and then sent through the quantum channel ${\cal C}_{\vec\lambda}$. Therefore, Bob receives the $N$ states $\rho\,'_i={\cal C}_{\vec\lambda}\,\rho_i$. He can now perform measurements on them. From the results he has to deduce an estimated vector $\vec\lambda^{\rm est}$ which should be as close as possible to the underlying parameter vector $\vec\lambda$ of the quantum channel. ![Basic scheme of channel estimation. Alice sends $N$ quantum state $\rho_i$ to Bob. The channel maps these states onto the states $\rho_i'={\cal C}_{\vec\lambda}\rho_i$, on which Bob can perform arbitrary measurements. Note that Bob’s measurements are designed with the knowledge of the original quantum states $\rho_i$. His final aim will be to present an estimated vector $\vec\lambda^{\rm est}$ being as close as possible to the underlying parameter vector $\vec\lambda$.[]{data-label="figurescheme"}](protocol.eps){width="8cm"} How can we quantify Bob’s estimation? To answer this we introduce two errors or cost functions which describe how good the channel is estimated. The first obvious cost function is the [*statistical error*]{} c\_s(N,)\_[=1]{}\^[L]{} (\_-\_\^[est]{}(N))\^2 \[ciesse\] in the estimation of the parameter vector $\vec\lambda$. Note that the elements of the estimated parameter vector $\vec\lambda^{\rm est}$ strongly depend on the available resources, i.e. the number $N$ of systems prepared by Alice. We also emphasize that $c_s$ describes the error for one single run of an estimation protocol. However, we are not interested in the single run error (\[ciesse\]) but in the average error of a given protocol. Therefore, we sum over all possible measurement outcomes $\cal J$ to get the [*mean statistical error*]{} $$\overline{c}_s(N,\vec\lambda)\equiv\langle c_s(N,\vec\lambda)\rangle_{\cal J}$$ while keeping the number $N$ of resources fixed. Though this looks as a good benchmark to quantify the quality of an estimation protocol it has a major drawback. The cost function $c_s$ strongly depends on the parametrization of the quantum channel. While this is not so important if one compares different protocols using the same parametrization it anyhow would be much better if we could give a cost function which is independent of any specifications. We define such a cost function with the help of the average overlap ([C]{}\_1,[C]{}\_2)\_ \[channelfidelity\] between two quantum channels ${\cal C}_1$ and ${\cal C}_2$, where we average the fidelity [@fid] F(\_1,\_2)\^2 \[mixedstatefidelity\] between two mixed states $\rho_1 = {\cal C}_1\,\rho$ and $\rho_2= {\cal C}_2\,\rho$ over all possible pure quantum states $\rho = \ket\psi\bra\psi$ [^1]. Since the fidelity ranges from zero to one, the [*fidelity error*]{} is given by c\_f(N,)1-[F]{}([C]{}\_,[C]{}\_[\^[est]{}(N)]{}) \[costfct\_f\] which now is zero for identical quantum channels. Again we average over all possible measurement outcomes to get the [*mean fidelity error*]{} \_f(N,)c\_f(N,)\_[J]{} \[costfct\_fav\] which quantifies the whole protocol and not a specific single run. In the first part of this paper we are only dealing with qubits described by the density operator 12(1+s) \[blochvector\] with Bloch vector $\vec s={\rm Tr}(\rho \, \vec\sigma)$ and the Pauli matrices $\vec\sigma\equiv(\sigma_x,\sigma_y,\sigma_z)$. The action of a channel is then completely described by the function s’=[C]{}(s), \[MapBloch\] which maps the Bloch vector $\vec s$ to a new Bloch vector $\vec s\, '$. In particular, the mixed state fidelity equation (\[mixedstatefidelity\]), for two qubits with Bloch vectors $\vec s_1$ and $\vec s_2$, see equation (\[blochvector\]), then simplifies to [@fid] F(s\_1,s\_2)=12\[fidss1\] which leads to an average channel overlap, equation (\[channelfidelity\]), ([C]{}\_1,[C]{}\_2) F([C]{}\_1(n),[C]{}\_2(n)) \[fidss2\] for the two qubit channels ${\cal C}_1$ and ${\cal C}_2$. As emphasized above we only average over pure input states with unit Bloch vector $$\vec n=(\cos\Phi\sin\Theta,\sin\Phi\sin\Theta,\cos\Theta) \label{enne}$$ and Bloch sphere element $d\Omega=\sin\Theta\,d\Theta\,d\Phi$. By inserting equations (\[fidss1\]) and (\[fidss2\]) into equations (\[costfct\_f\]) and (\[costfct\_fav\]) we get a cost function for the comparison of two qubit channels which is independent of the chosen parametrization. Parametrization of a quantum channel {#Parametrization} ==================================== As we have already mentioned in section \[GeneralDescript\], the qubits that Alice sends to Bob are fully characterized by their Bloch vector. Therefore, the disturbing action of the channel modifies the initial Bloch vector $\vec{s}$ according to equation (\[MapBloch\]). It has been shown that for completely positive operators the action of the quantum channel ${\cal C}$ is given by an affine transformation [@fuji] $$\vec{s}'={\cal M} \; \vec{s}+\vec{v} \label{eq12}$$ where ${\cal M}$ denotes a $3\times 3$ invertible matrix and $\vec{v}$ is a vector. The transformation is thus described by 12 parameters, 9 for the matrix ${\cal M}$ and 3 for the vector $\vec{v}$. These 12 parameters have to fullfill some constraints to guarantee the complete positivity of $\cal C$ [@fuji]. The definition of the parameters is somewhat arbitrary. We will not use the parameters as defined in [@fuji], but adopt a different parametrization $$\left( \begin{array}{c} s'_{1} \\ s'_{2} \\ s'_{3} \end{array} \right) = \left( \begin{array}{ccc} 2\lambda_{7}-\lambda_{1}-\lambda_{4}, \; \; 2\lambda_{10}-\lambda_{1}-\lambda_{4}, \; \; \lambda_{4}-\lambda_{1} \\ 2\lambda_{8}-\lambda_{2}-\lambda_{5}, \; \; 2\lambda_{11}-\lambda_{2}-\lambda_{5}, \; \; \lambda_{5}-\lambda_{2} \\ 2\lambda_{9}-\lambda_{3}-\lambda_{6}, \; \; 2\lambda_{12}-\lambda_{3}-\lambda_{6}, \; \; \lambda_{6}-\lambda_{3} \\ \end{array} \right) \left( \begin{array}{c} s_{1} \\ s_{2} \\ s_{3} \end{array} \right) + \left( \begin{array}{c} \lambda_{1}+\lambda_{4}-1 \\ \lambda_{2}+\lambda_{5}-1 \\ \lambda_{3}+\lambda_{6}-1 \end{array} \right) \label{ssprime}$$ in terms of the parameter vector $\vec{\lambda}= (\lambda_{1},\ldots ,\lambda_{12})^T$. In general, the choice of the best parametrization of quantum channels depends on the relevant features of the channel and also on which observables can be measured. The reason for the choice of the parametrization, equation (\[ssprime\]), will become clear at the end of the section \[OneParameter\], in which a protocol for the general channel is described. Estimation of channel parameters {#OneParameter} ================================ Although the characterization of the general quantum channel requires the determination of 12 parameters, only a smaller number of parameters must be determined in practice for a given class of quantum channels. Indeed, the knowledge of the properties of the physical devices used for quantum communication gives information on some parameters and allows to reduce the number of parameters to be estimated. We shall now examine in detail some known channels described by only one parameter. The depolarizing channel ------------------------ The first channel we consider is the depolarizing channel. The relation (\[eq12\]) between the Bloch vectors $\vec{s}$ and $\vec{s}\,'$ of Alice’s and Bob’s qubit, respectively, reduces to the simple form $$\vec{s}_{\lambda}\,'=(1-2\lambda)\vec{s},$$ where $0 \le \lambda \le 1/2 $ is the only parameter that describes this quantum channel. This channel is a good model when quantum information is encoded in the photon polarization that can change along the transmission fiber via random rotations of the polarization direction. If Alice prepares the qubit in the pure state $\rho= | \psi \rangle \langle \psi |$, the qubit Bob receives is described by the state $$\rho'=(1-\lambda) | \psi \rangle \langle \psi | +\lambda | \bar{\psi} \rangle \langle \bar{\psi} |, \label{dep}$$ where $\mid \!\! \bar{\psi} \rangle$ denotes the state orthogonal to $\mid \!\! \psi \rangle$. We note that the depolarizing channel has no preferred basis. Therefore, its action is isotropic in the direction of the input state. This means that the action of the channel is described by equation (\[dep\]) even after changing the basis of states. Bob must estimate $\lambda$, which ranges between 0 (noiseless channel) and 1/2 (total depolarization). The estimation protocol is the following: Alice prepares $N$ qubits in the pure state $$\mid \uparrow_{\vec{n}} \rangle =e^{-i\Phi/2}\cos \frac{\Theta}{2}\mid \uparrow_z \rangle +e^{i\Phi/2}\sin\frac{\Theta}{2}\mid \downarrow_z \rangle$$ which denotes spin up in the direction $\vec{n}$, equation (\[enne\]), in terms of eigenvectors $\mid \uparrow_z \rangle$ and $\mid \downarrow_z \rangle$ of $\sigma_z$. The state $\mid \uparrow_{\vec{n}} \rangle$ is sent to Bob through the channel. Bob knows the direction $\vec{n}$ and measures the spin of the qubit he receives along $\vec{n}$. The outcome probabilities are $$P(\uparrow_{\vec{n}})=1-\lambda , \;\;\; P(\downarrow_{\vec{n}})=\lambda.$$ Since Alice sends a finite number of qubits, Bob can only determine frequencies of measurements instead of probabilities. After Bob has measured $i \le N$ qubits with spin down and the remaining $N-i$ with spin up, his estimate of $\lambda$ is $\lambda^{\rm{est}}=i/N$. Note that in a single run of the probabilistic estimation method we can get results $\lambda^{\rm{est}}>1/2$, which is outside the range of the depolarizing channel. However, for the calculation of the average errors we do not have to take that into account. The cost function $c_{s}$, equation (\[ciesse\]), is thus $$c_{s}(N,\lambda)=\left( \lambda-\frac{i}{N} \right)^2.$$ The average error is easily obtained when one considers that Bob’s frequencies of measurements occur according to a binomial probability distribution, since each of the $i$ measurements of spin down occurs with probability $\lambda$ and each of the spin up measurements occurs with probability $1-\lambda$. Therefore, the mean statistical error reads $$\begin{aligned} \bar{c}_{s}(N,\lambda) = \sum_{i=0}^{N} { N \choose i} \lambda^{i}(1-\lambda)^{N-i} \left( \lambda-\lambda^{\rm{est}}\right)^2 \nonumber \\ = \sum_{i=0}^{N} { N \choose i} \lambda^{i}(1-\lambda)^{N-i} \left( \lambda-\frac{i}{N} \right)^2= \frac{\lambda(1-\lambda)}{N}. \label{eq16}\end{aligned}$$ This function, shown in figure \[DepolarizingFigure\] a), scales with the available finite resources $N$. It vanishes when the channel faithfully preserves the polarization, $\lambda=0$. The largest average error occurs for $\lambda=1/2$, when the two actions of preserving and changing the polarization have the same probability to occur. ![a) The average statistical mean error $\bar{c}_s (N, \lambda)$ equation (\[eq16\]), and b) the fidelity mean error $\bar{c}_f (N,\lambda)$, equation (\[EqDepolarizingCf\]), for a depolarizing channel with parameter $\lambda$, are shown for different values of the number of qubits N.[]{data-label="DepolarizingFigure"}](gfig2.eps){width="12cm"} The cost function $c_{f}$ is obtained from equations (\[costfct\_f\]) and (\[fidss2\]) $$c_{f}(N,\lambda) = 1- \left[ \sqrt{\lambda \frac{i}{N}}+\sqrt{\left(1-\lambda\right) \left(1-\frac{i}{N}\right)}\right]^2,$$ which leads to the fidelity mean error $$\begin{aligned} \bar{c}_{f}(N,\lambda) & = & \sum_{i=0}^{N} { N \choose i}\lambda^{i} \left( 1-\lambda\right)^{N-i}c_{f}(N,\lambda) \nonumber \\ & = & 1-\frac{1}{N}\sum_{i=0}^{N} { N \choose i}\lambda^{i} \left( 1-\lambda\right)^{N-i}\left( \sqrt{\lambda i}+ \sqrt{(1-\lambda)(N-i)}\right)^2, \label{EqDepolarizingCf}\end{aligned}$$ which is shown in figure \[DepolarizingFigure\] b). Although $\bar{c}_f$ does not show a simple $1/N$–dependency, it clearly decreases for increasing values of the number of qubits $N$. For a given quality $\bar{c}_{s}$ or $\bar{c}_{f}$ one can use figure \[DepolarizingFigure\] and read off the number $N$ of needed resources. The phase damping channel ------------------------- The phase-damping channel acts only on two components of the Bloch vector, leaving the third one unchanged: $$\vec{s}_{\lambda}\,\! '= \left( (1-2\lambda)\; s_{1},(1-2\lambda)\; s_{2},s_{3} \right)^{T} .$$ Here $0\le \lambda\le 1/2$ is the damping parameter. This channel, contrarily to the depolarizing channel, has a preferred basis. In terms of density matrices, it transforms the initial state $$\rho=\left( \begin{array}{cc} \rho_{\downarrow \downarrow} \;\; \rho_{\downarrow \uparrow} \\ \rho_{\uparrow \downarrow} \;\; \rho_{\uparrow \uparrow} \end{array} \right) \label{denmat}$$ where $\rho_{\downarrow \uparrow}\equiv \langle \downarrow_{z} \mid \rho \mid \uparrow_{z} \rangle$, etc., into $$\rho'=\left( \begin{array}{cc} \rho_{\downarrow \downarrow} & (1-2\lambda)\rho_{\downarrow \uparrow} \\ (1-2\lambda)\rho_{\uparrow \downarrow} & \rho_{\uparrow \uparrow} \end{array} \right).$$ We note that here the parameter $\lambda$ only appears in the off-diagonal terms. For this reason, the phase damping channel is a good model to describe decoherence [@phda]. Indeed, a repeated application of this channel leads to a vanishing of the off–diagonal terms in $\rho'$, whereas its diagonal terms are preserved. Since the parameter $\lambda$ of the phase damping channel appears in the off-diagonal elements of $\rho'$, the protocol is the following: Alice sends $N$ qubits with the Bloch vector in the $x-y$ plane; for instance, qubits in the state $\mid \uparrow_{x} \rangle = (\mid \downarrow_{z} \rangle + \mid \uparrow_{z} \rangle )/\sqrt{2}$ can be used. This would correspond to a density operator whose matrix elements are all equal to 1/2, can be used. The density matrix of Bob’s qubit is then given by $$\rho'=\frac{1}{2}\left( \begin{array}{cc} 1 & (1-2\lambda) \\ (1-2\lambda) & 1 \end{array} \right).$$ Now he measures the spin in the $x$ direction. The theoretical probabilities are $$\begin{aligned} P(\uparrow_{x}) & = & \langle \uparrow_{x} \mid \rho' \mid \uparrow_{x} \rangle =1-\lambda, \\ P(\downarrow_{x}) & = & \langle \downarrow_{x} \mid \rho' \mid \downarrow_{x} \rangle =\lambda,\end{aligned}$$ and we denote the frequency of a spin–down measurement as $i/N$, leading to $\lambda^{\rm{est}}=i/N$. The mean statistical error has again the form $$\bar{c}_{s}(N,\lambda)=\sum_{i=0}^{N} { N \choose i} \lambda^{i} \left( 1-\lambda \right)^{N-i} \left( \lambda-\frac{i}{N} \right)^2=\frac{\lambda(1-\lambda)}{N}. \label{eq7}$$ as for the depolarizing channel. For the fidelity, equation (\[fidss1\]), we obtain $$\begin{aligned} F(\vec{s}_{\lambda}\,\! ',\vec{s}\,\, '_{\lambda^{\rm est}}) & = & 1+\left[2 \lambda \lambda^{\rm est} -\lambda -\lambda^{\rm est} \right. \\ & &\left. +2\sqrt{\lambda(1-\lambda)\lambda^{\rm est}(1-\lambda^{\rm est})} \right] \left(s_{1}^{2}+s_{2}^{2}\right)\end{aligned}$$ and after averaging over the Bloch surface, according to equation (\[fidss2\]), $${\cal F} ( {\cal C}_{\lambda}, {\cal C}_{\lambda^{\rm est}}) = 1-\frac{2}{3}\lambda-\frac{2}{3}\lambda^{\rm{est}}+ \frac{4}{3}\lambda\lambda^{\rm{est}}+\frac{4}{3} \sqrt{\lambda(1-\lambda)\lambda^{\rm{est}} (1-\lambda^{\rm{est}})}$$ Therefore the fidelity mean error $\bar{c}_{f}$ reads $$\begin{aligned} \bar{c}_{f}(N,\lambda) = \sum_{i=0}^{N} { N \choose i} \lambda^{i} \left( 1-\lambda \right)^{N-i} \frac{2}{3} \left[ \lambda +( 1-2\lambda)\frac{i}{N} -2\sqrt{\lambda (1-\lambda)\frac{i}{N}\left( 1-\frac{i}{N}\right) } \right] \nonumber \\ = \frac{4}{3}\left[ \lambda \left( 1-\lambda\right) -\frac{1}{N}\sum_{i=0}^{N} { N \choose i} \lambda^{i}\left( 1-\lambda \right)^{N-i} \sqrt{\lambda(1-\lambda)i(N-i)} \right] \label{EqPhaseCf}\end{aligned}$$ which is shown in figure \[PhaseDampingFigure\] b). This mean error is very similar to that obtained for the depolarizing channel. ![a) The average statistical mean error $\bar{c}_s (N, \lambda)$ , equation (\[eq7\]), and b) the fidelity mean error $\bar{c}_f (N,\lambda)$, equation (\[EqPhaseCf\]), for a phase damping channel with parameter $\lambda$, are shown for different values of the number of qubits N.[]{data-label="PhaseDampingFigure"}](gfig3.eps){width="12cm"} The amplitude damping channel ----------------------------- The amplitude damping channel affects all components of the Bloch vector according to $$\vec{s}_{\lambda}\, '= \left( \sqrt{1-\lambda}\; s_{1},\sqrt{1-\lambda}\; s_{2},(1-\lambda)\; s_{3}+\lambda \right)^{T} \label{eq28}$$ where $0\le \lambda\le 1$ is the damping parameter. The density matrix, equation (\[denmat\]), is transformed into $$\rho'=\left( \begin{array}{cc} \lambda+(1-\lambda)\rho_{\downarrow \downarrow} & \sqrt{1-\lambda}\: \rho_{\downarrow \uparrow} \\ \sqrt{1-\lambda}\: \rho_{\uparrow \downarrow} & (1-\lambda) \rho_{\uparrow \uparrow} \end{array} \right) .$$ This channel is a good model for spontaneous decay [@deca] from an atomic excited state $\mid \uparrow _z\rangle$ to the ground state $\mid \downarrow_z \rangle$. Repeated applications of this channel cause all elements but one of the density matrix to vanish. Now the parameter $\lambda$ appears in all the elements of $\rho'$ and the channel clearly possesses a preferred basis. If Alice and Bob know that they are using an amplitude damping channel, Alice sends all $N$ qubits in the state $\mid \uparrow_{z} \rangle$. The density operator of the qubit received by Bob is $$\rho'=\left( \begin{array}{cc} \lambda & 0 \\ 0 & 1-\lambda \end{array} \right) .$$ He measures the spin along the $z$ direction (the diagonal elements of $\rho'$ are the probabilities to find spin down and spin up, respectively). We denote the frequency of spin down measurements with $i/N$. The statistical cost function is again $$\bar{c_{s}}(N,\lambda)=\sum_{i=0}^{N} { N \choose i} \lambda^{i} (1-\lambda)^{N-i} \left( \lambda-\frac{i}{N} \right)^2=\frac{\lambda (1-\lambda)}{N} \label{eq9}$$ Using equation (\[eq28\]) we obtain $$\begin{aligned} F(\vec{s}_{\lambda}\, ',\vec{s}\,\,'_{\lambda^{\rm est}}) = \frac{1}{2} \left[ 1+\sqrt{(1-\lambda)(1-\lambda^{\rm est})}\left( s_{1}^{2}+s_{2}^{2}\right) +(1-\lambda)(1-\lambda^{\rm est})s_{3}^{2} \right. \nonumber \\ + \left[\lambda(1-\lambda^{\rm est})+\lambda^{\rm est}(1-\lambda) \right]s_{3} \nonumber \\ \left. +\lambda \lambda^{\rm est}+ (1-s_{3})^2\sqrt{\lambda(1-\lambda)} \sqrt{\lambda^{\rm est}(1-\lambda^{\rm est})} \right]\end{aligned}$$ for the fidelity cost function, equation (\[fidss1\]), and $$\begin{aligned} {\cal F} ( {\cal C}_{\lambda}, {\cal C}_{\lambda^{{\rm est}}}) & = & \frac{1}{6}\left[ 4+2\sqrt{(1-\lambda)(1-\lambda^{\rm{est}})}- \lambda-\lambda^{\rm{est}} \right. \nonumber \\ & &\left. +4\lambda\lambda^{\rm{est}} +4\sqrt{\lambda(1-\lambda)\lambda^{\rm{est}} (1-\lambda^{\rm{est}})}\right]\end{aligned}$$ for the averaged fidelity (\[fidss2\]). Thus $$\begin{aligned} \bar{c_{f}}(N,\lambda) = \sum_{i=0}^{N} { N \choose i} \lambda^{i}(1-\lambda)^{N-i} \left[ 1-\frac{1}{6} \left( 4+2\sqrt{(1-\lambda)\left( 1-\frac{i}{N}\right)} -\lambda-\frac{i}{N} \right. \right. \nonumber \\ \left. \left. +4\lambda\frac{i}{N} +4\sqrt{\lambda(1-\lambda)\frac{i}{N} \left( 1-\frac{i}{N}\right)}\: \right) \right] \nonumber \\ = \frac{1}{3} \left[ 1+\lambda(1-2\lambda)-\sqrt{\frac{1-\lambda}{N}} \sum_{i=0}^{N} { N \choose i } \lambda^{N-i}(1-\lambda)^{i}\sqrt{i} \right. \nonumber \\ \left. -\frac{2}{N}\sqrt{\lambda(1-\lambda)} \sum_{i=0}^{N} { N \choose i} \lambda^{i}(1-\lambda)^{N-i}\sqrt{i(N-i)} \right] \label{EqAmplitudeCf}\end{aligned}$$ which is illustrated by figure \[AmplitudeDampingFigure\]. ![a) The average statistical mean error $\bar{c}_s (N, \lambda)$ , equation (\[eq9\]), and b) the fidelity mean error $\bar{c}_f (N,\lambda)$, equation (\[EqAmplitudeCf\]), for the amplitude damping channel with parameter $\lambda$, are shown for different values of the number of qubits N.[]{data-label="AmplitudeDampingFigure"}](gfig4.eps){width="12cm"} We see from the figures (\[DepolarizingFigure\])–(\[AmplitudeDampingFigure\]) that for a fixed number of resources the mean error $\bar{c}_{f}$ has a local maximum for small values of the parameter $\lambda$. For the amplitude damping channel there is also a second local maximum for large values of $\lambda$. The general quantum channel --------------------------- We now come to the problem of estimating the 12 parameters $\vec{\lambda}=(\lambda_{1},\ldots,\lambda_{12})^T$ of the general quantum channel. The protocol is summarized in table 1: Alice prepares 12 sets of qubits, divided into 4 groups. Bob measures the spin of the three sets in each group along $x$, $y$, and $z$, respectively. From the measurements on each set he gets an estimate of one of the parameters $\lambda_{i}$. The parametrization (\[ssprime\]) has been chosen for this purpose. The statistical cost function for the general channel is thus a generalization of the cost functions for one–parameter channels, $$\begin{aligned} \bar{c}_{s}(N,\vec{\lambda}) & = & \sum_{i_{1}=0}^{N/12} \ldots \sum_{i_{12}=0}^{N/12} \left[ \prod_{n=1}^{12} { N/12 \choose i_{n} } \lambda_{n}^{i_{n}}(1-\lambda_{n})^{N/12-i_{n}}\right] \sum_{j=1}^{12}\left( \lambda_{j}-\lambda_{j}^{\rm{est}} \right)^2 \nonumber \\ & = & \frac{1}{N/12}\sum_{j=1}^{12}\lambda_{j}(1-\lambda_{j}).\end{aligned}$$ The mean fidelity error can be calculated numerically. However, we do not give the expression here as it gives no particular further insight. The Pauli channel for qubits {#QubitPauli} ============================ Up to now we have only considered estimation methods based on measuring single qubits sent through the quantum channel. However, we are not restricted to these estimation schemes. Instead of sending single qubits we could use entangled qubit pairs [@us]. In this section we will demonstrate the superiority of such entanglement-based estimation methods for the estimation of the so called Pauli channel by comparing both schemes (for a different use of entanglement as a powerful resource when using Pauli channels, see [@palma]). The Pauli channel is widely discussed in the literature, especially in the context of quantum error correction [@code]. Its name originates from the error operators of the channel. These error operators are the Pauli spin matrices $\sigma_x$, $\sigma_y$ and $\sigma_z$. The operators define the quantum mechanical analogue to bit errors in a classical communication channel since $\sigma_x$ causes a bit flip ($|\! \downarrow \rangle$ transforms into $| \! \uparrow \rangle$ and vice versa), $\sigma_z$ causes a phase flip ($| \! \downarrow \rangle+ | \! \uparrow \rangle$ transforms into $|\!\downarrow \rangle - |\!\uparrow \rangle $) and $\sigma_y$ results in a combined bit and phase flip. The Pauli channel is described by three probabilities $\vec{\lambda}\equiv (\lambda_{1},\lambda_{2}, \lambda_{3})^T$ for the occurrence of the three errors. If an initial quantum state $\rho$ is sent through the channel, the Pauli channel transforms $\rho$ into $$\rho' = (1-\lambda_{1}-\lambda_{2}-\lambda_{3})\rho +\lambda_{1}\sigma_{x}\rho \sigma_{x} +\lambda_{2}\sigma_{y} \rho \sigma_{y} +\lambda_{3} \sigma_{z} \rho \sigma_{z}. \label{flip} \label{pau}$$ Thus we see that the density operator $\rho$ remains unchanged with probability $1-\lambda_{1}-\lambda_{2}-\lambda_{3}$, whereas with probability $\lambda_{1}$ the qubit undergoes a bit flip, with probability $\lambda_{3}$ there occurs a phase flip, and with probability $\lambda_{2}$ both a bit flip and a phase flip take place. Estimation using single qubits ------------------------------ We will first describe the single qubit estimation scheme for the Pauli channel, before we switch to the entanglement-based one in the next subsection. Following the general estimation scheme presented in section \[GeneralDescript\] the protocol to estimate the parameters of the Pauli channel, equation (\[pau\]), requires the preparation of three different quantum states with spin along three orthogonal directions. Alice sends (i) $M=N/3$ qubits in the state $| \uparrow_{z} \rangle$, (ii) $M$ qubits in the state $| \uparrow_{x} \rangle$, and (iii) $M$ qubits in the state $| \uparrow_{y} \rangle$. Bob measures their spins along the direction of spin-preparation, namely the $z$, $x$, and $y$ axes, respectively. The measurement probabilities of spin down along those three directions are then given by $$\begin{aligned} P(\downarrow_{z}) & = & \lambda_{1}+\lambda_{2}, \\ P(\downarrow_{x}) & = & \lambda_{2}+\lambda_{3}, \\ P(\downarrow_{y}) & = & \lambda_{1}+\lambda_{3}.\end{aligned}$$ The estimated parameter values can be calculated via $$\begin{aligned} \lambda_{1}^{\rm{est}} & = & \frac{1}{2}\left[ \frac{i_{3}}{M}-\frac{i_{1}}{M} +\frac{i_{2}}{M}\right] \\ \lambda_{2}^{\rm{est}} & = & \frac{1}{2}\left[ \frac{i_{1}}{M}-\frac{i_{2}}{M} +\frac{i_{3}}{M}\right] \\ \lambda_{3}^{\rm{est}} & = & \frac{1}{2}\left[ \frac{i_{2}}{M}-\frac{i_{3}}{M} +\frac{i_{1}}{M}\right]\end{aligned}$$ where $i_{k}/M$, $k=1,2,3$, denote the frequencies of spin down results along the directions $x$, $y$ and $z$. We note that, although the probabilities $\lambda_{i}$ are positive or vanish, their estimated values $\lambda_{i}^{\rm{est}}$ may be negative. This occurs because in the present case the measured frequencies are not the estimates of the parameters. Nonetheless, the average cost functions can always be evaluated. For the statistical cost function we find $$\begin{aligned} \bar{c}_s^{\rm \; sep}(N,\vec{\lambda}) & = & \sum_{i_{1}=0}^{N/3}\sum_{i_{2}=0}^{N/3}\sum_{i_{3}=0}^{N/3} { N/3 \choose i_{1} } { N/3 \choose i_{2} } { N/3 \choose i_{3} } (P(\downarrow_{z}))^{i_{3}} (1-P(\downarrow_{z}))^{N/3-i_{3}} \nonumber \\ & & \times (P(\downarrow_{x}))^{i_{1}} (1-P(\downarrow_{x}))^{N/3-i_{1}} (P(\downarrow_{y}))^{i_{2}} (1-P(\downarrow_{y}))^{N/3-i_{2}} \nonumber \\ & & \times \left[ \left( \lambda_{1}-\lambda_{1}^{\rm{est}} \right)^2+ \left( \lambda_{2}-\lambda_{2}^{\rm{est}} \right)^2 +\left( \lambda_{3}-\lambda_{3}^{\rm{est}} \right)^2 \right] \nonumber \\ & = & \frac{9}{2N}\left[ \lambda_{1}(1-\lambda_{1}-\lambda_{2})+\lambda_{2}(1-\lambda_2-\lambda_3) \right. \nonumber \\ & & \left. +\lambda_{3}(1-\lambda_3-\lambda_1)\right]. \label{ave}\end{aligned}$$ for the average error $\bar{c}_s^{\rm \; sep}$ of the estimation with separable qubits [@us]. For fixed $N$ the average error has a maximum at $\lambda_{1}=\lambda_{2}= \lambda_{3}=1/4$, in which case all acting operators occur with the same probability. On the other hand, the average error vanishes when faithful transmission or one of the errors occurs with certainty. Instead of using the statistical cost function $c_s$ we can also use the fidelity based cost function $c_f$ (\[costfct\_f\]). The average error $\bar{c}_f^{\rm \; sep}$ of the estimation via separable qubits is then given by $$\begin{aligned} \bar{c}_f^{\rm \; sep}(N,\vec{\lambda}) & = & \sum_{i_{1}=0}^{N/3} \sum_{i_{2}=0}^{N/3} \sum_{i_{3}=0}^{N/3} { N/3 \choose i_{1} } { N/3 \choose i_{2} } { N/3 \choose i_{3} } (P(\downarrow_{z}))^{i_{3}} (1-P(\downarrow_{z}))^{N/3-i_{3}} \nonumber \\ & & \times (P(\downarrow_{x}))^{i_{1}} (1-P(\downarrow_{x}))^{N/3-i_{1}} \nonumber \\ & & \times (P(\downarrow_{y}))^{i_{2}} (1-P(\downarrow_{y}))^{N/3-i_{2}} c_f(N,\vec\lambda). \label{EqPauliCfsep}\end{aligned}$$ The cost function $c_f$ cannot be calculated analytically. We shall compare the results (\[ave\]) and (\[EqPauliCfsep\]) with the same mean errors for a different estimation scheme, where entangled pairs are used, that we are now going to illustrate. Estimation using entangled states --------------------------------- All the protocols that we have considered so far are based on single qubits prepared in pure states. These qubits are sent through the channel one after another. However, one can envisage estimation schemes with different features. An interesting and powerful alternative scheme is based on the use of entangled states [@us]. In this case the estimation scheme requires Alice and Bob to share entangled qubits in the $\mid \! \psi^{-}\rangle\equiv (\mid \! \downarrow \rangle \mid \! \uparrow \rangle - \mid \! \uparrow \rangle \mid \! \downarrow \rangle)/\sqrt{2}$ Bell state. Thus from $N$ initial qubits Alice and Bob can prepare $N/2$ Bell states. Alice sends her $N/2$ qubits of the entangled pairs through the Pauli channel, which transforms the entangled state into the mixed state $$\begin{aligned} \rho'=\cal{C}(| \psi^{-} \rangle \langle \psi^{-} |) &=& \lambda_{1} | \phi^{-} \rangle \langle \phi^{-} | +\lambda_{2} | \phi^{+} \rangle \langle \phi^{+} | +\lambda_{3} | \psi^{+} \rangle \langle \psi^{+} | \nonumber \\ & & +\left( 1-\lambda_{1}-\lambda_{2}-\lambda_{3}\right) | \psi^{-} \rangle \langle \psi^{-} |,\end{aligned}$$ where $| \phi^{\pm} \rangle\equiv \left( | \downarrow \rangle | \downarrow \rangle \pm | \uparrow \rangle | \uparrow \rangle \right)/\sqrt{2}$ and $| \psi^{\pm} \rangle\equiv \left( | \downarrow \rangle | \uparrow \rangle \pm | \uparrow \rangle | \downarrow \rangle \right)/\sqrt{2}$ are the four Bell states [@bell]. Bob performs $N/2$ Bell measurements, i.e., projects each of the $N/2$ pairs of qubits onto one of the four Bell states with probabilities $$\begin{aligned} P(| \phi^- \rangle )=\lambda_{1}, \;\;\; P(| \phi^+ \rangle )=\lambda_{2}, \;\;\; & & P(| \psi^+ \rangle )=\lambda_{3}, \nonumber \\ P(| \psi^- \rangle )=1-\lambda_{1}-\lambda_{2} -\lambda_{3}.\end{aligned}$$ Consequently, the estimated parameter values are directly given by $$\lambda_{1}^{\rm est}=\frac{i_{1}}{N/2},\;\;\; \lambda_{2}^{\rm est}=\frac{i_{2}}{N/2},\;\;\; \lambda_{3}^{\rm est}=\frac{i_{3}}{N/2},\;\;\;$$ We stress that now the estimates of the parameters $\lambda_{i}$ are always nonnegative, contrarily to the estimation scheme with single qubits. The average error $\bar{c}_s^{\rm \; ent}$ of the entangled estimation based on the statistical cost function $c_s$ reads [@us] $$\begin{aligned} \bar{c}_s^{\rm \; ent}(N,\vec{\lambda})&=& \sum_{i_1+i_2+i_3+i_4=N/2}\frac{(N/2)!}{i_1!i_2!i_3!i_4!}\nonumber \\ &\times & \lambda_1^{i_1}\lambda_2^{i_2}\lambda_3^{i_3} (1-\lambda_1-\lambda_2-\lambda_3)^{i_4} \sum_{k=1}^3(\lambda_k-\lambda_k^{est})^2 \nonumber \\ &=& \frac{1}{N/2}\left[\lambda_1(1-\lambda_1)+\lambda_2(1-\lambda_2) +\lambda_3(1-\lambda_3)\right].\end{aligned}$$ As we have shown already in [@us] $\bar{c}_s^{\rm \; ent}$ is always smaller or equal to $\bar{c}_s^{\rm \; sep}$, $$\begin{aligned} \Delta_{s}(N,\vec{\lambda}) & = & \bar{c}_s^{\rm \; sep}\left( N, \vec{\lambda} \right)- \bar{c}_s^{\rm \; ent}\left( N, \vec{\lambda} \right) \nonumber \\ & = & \frac{1}{2N}\left[ 5(1-\lambda_{1}-\lambda_{2}-\lambda_{3}) (\lambda_{1}+\lambda_{2}+\lambda_{3})+\lambda_{1}\lambda_{2} \right. \nonumber \\ & &\left. +\lambda_{2}\lambda_{3}+\lambda_{3}\lambda_{1}\right]\ge 0\end{aligned}$$ If we use the fidelity–based cost function $c_f$ we can also write down the average error $\bar{c}_f^{\rm \; ent}$ for the entangled estimation scheme. It reads $$\begin{aligned} \bar{c}_f^{\rm \; ent}(N,\vec{\lambda})&=& \sum_{i_1+i_2+i_3+i_4=N/2}\frac{(N/2)!}{i_1!i_2!i_3!i_4!}\nonumber \\ &\times & \lambda_1^{i_1}\lambda_2^{i_2}\lambda_3^{i_3} (1-\lambda_1-\lambda_2-\lambda_3)^{i_4} c_f(N,\vec\lambda) \label{EqPauliCfent}\end{aligned}$$ and can be evaluated numerically. In figure \[DeltaFigure\] we show the difference $$\Delta_f(N,\vec{\lambda})\equiv \bar{c}_f^{\rm \; sep}\left( N, \vec{\lambda} \right)- \bar{c}_f^{\rm \; ent}\left( N, \vec{\lambda} \right) \label{EqDelta}$$ between the average error obtained with single qubits and entangled qubits. We compare the two average errors when the same number of qubits are used. The figure shows that the use of entangled pairs always leads to an enhanced estimation and therefore we can consider entanglement as a nonclassical resource for this application. ![The difference $\Delta (N, \vec{\lambda})$ equation (\[EqDelta\]) between the fidelity mean errors $\bar{c}_f^{\rm sep}$, equation (\[EqPauliCfsep\]), and $\bar{c}_f^{\rm ent}$, equation (\[EqPauliCfent\]) for the qubit Pauli channel. $\Delta$ is plotted as a function of $\lambda_1$ and $\lambda_3$, while keeping $\lambda_2 = 0$ and $N=6$ fixed. []{data-label="DeltaFigure"}](gfig5.eps){width="8cm"} Before ending this section we want to add some comments about our findings. We have found that the mean statistical error has the form $$\bar{c}_{s}(N,\vec{\lambda})=\frac{L}{N}\sum_{i=1}^{L}\lambda_{i} (1-\lambda_{i}) \label{univ}$$ whenever the estimates of the parameters $\lambda_{i}$ are directly given by the frequencies of measurements. This occurs also for the relevant case of the entanglement–based protocol for the Pauli channel, but not for the qubit–based protocol. Although the parametrization we have used is better indicated since the $\lambda_{i}$ represent the probabilities of occurence of the logic errors, one might be tempted to use a different parametrization $\vec{\lambda}'$ that gives again the expression (\[univ\]). Indeed, this can be done and actually the errors for the $\vec{\lambda}'$ are smaller. Nonetheless, one can show that in spite of this improvement, the scheme based on entangled pairs still gives an enhanced estimation even with the new parameters $\vec{\lambda}'$. This is not the case of the one–parameter channels, where the use of entangled pairs does not lead to enhanced estimation. The generalized Pauli channel {#QuditPauli} ============================= In the previous section we have proposed an entanglement–based method for estimating the parameters which define the Pauli channel. As we shall show in this section, this method can be easily extended to the case of quantum channels defined on higher dimensional Hilbert spaces. Let us start by considering the most general possible trace preserving transformation of a quantum system described initially by a density operator $\rho$. This transformation can be written in terms of quantum operations $A_i$ as [@chan] $$\rho '= {\cal C} \rho = \sum_{i}\lambda_{i}A_{i}\rho A_{i}^{\dagger } \label{general quantum channel}$$ with $\sum_{i} \lambda_i A_{i}A_{i}^{\dagger }={\bf 1}$. According to the Stinespring theorem [@stine] this is the most general form of a completely positive linear map. The set of operators $ A_{i} $ can be interpreted as error operators which characterize the action of a given quantum channel onto a quantum system and the set of parameters $ \lambda_{i} $ as probabilities for the action of error operators $A_{i}$. In particular, we can consider the action of the quantum channel, equation (\[general quantum channel\]), onto only one, say the second, of the subsystems of a bipartite quantum system. For sake of simplicity we assume that the two subsystems are $D$–level systems. If only the second particle is affected by the quantum channel, equation (\[general quantum channel\]), then the final state is given by $$\rho '=\sum_{i}\lambda_{i}\left( {\bf 1\otimes }A_{i}\right) \rho \left( {\bf 1\otimes }A_{i}\right) ^{\dagger } \label{quantum channel local}$$ where ${\bf 1}$ is the identity operator. In the special case of an initially pure state, i. e. $\rho =\left| \psi \right\rangle \left\langle \psi \right| $, the final state $\rho '$ becomes $$\rho '=\sum_{i}\lambda_{i}\left| \psi_i \right\rangle\left\langle\psi_i \right| \label{pure initial state}$$ where we have defined $\left| \psi_i \right\rangle\equiv \left({\bf 1\otimes }A_{i}\right) \left| \psi \right\rangle $. Our aim is the estimation of the parameter values $ \lambda_{i} $ which define the quantum channel. This can be done by projecting the state of the composite quantum system onto the set of states $% \left| \psi_{i}\right\rangle $. As in the previous protocols, the probabilities $ \lambda_{i} $ can be inferred from the relative frequencies of each state $% \left| \psi_{i} \right\rangle$ and the quality of the estimation can be quantified with the help of a cost function. However, these states can be perfectly distinguished if and only if they are mutually orthogonal. Hence, we impose the condition $$\left\langle \psi_i \right| \left. \psi_j \right\rangle =\left\langle \psi \right| \left( {\bf 1\otimes }A_{i}\right) ^{\dagger }\left( {\bf % 1\otimes }A_{j}\right) \left| \psi \right\rangle =\delta _{i,j} \label{condition}$$ on the initial state $\left| \psi \right\rangle $. It means that the initial state $\left| \psi \right\rangle $ of the bipartite quantum system should be chosen in such a way that it is mapped onto a set of mutually orthogonal states $ \left| \psi_i\right\rangle $ by the error operators $A_{i}$. It is worth to remark the close analogy between this estimation strategy and quantum error correction. A non-degenerate quantum error correcting code [@knil] corresponds to a Hilbert subspace which is mapped onto mutually orthogonal subspaces under the action of the error operators. In this sense, a state $\left| \psi \right\rangle $ satisfying the condition (\[condition\]) is a one-dimensional non-degenerated error correcting code. A necessary but not sufficient condition for the existence of a state $% \left| \psi \right\rangle $ satisfying equation(\[condition\]) is given by the inequality $$n\le D' \label{bound}$$ where $n$ is the total number of error operators which describe the quantum channel and $D'$ is the dimension of the Hilbert space. This inequality shows why the use of an entangled pair enhances the estimation scheme. Although the first particle is not affected by the action of the noisy quantum channel, it enlarges the dimension of the total Hilbert space in such a way that the condition (\[condition\]) can be fulfilled. It is also clear from this inequality that if the state $\left| \psi \right\rangle $ is a separable one, a set of orthogonal states can only be designed in principle if the number of error operators $n$ is smaller than $D'$. For instance, in the case of the Pauli channel for qubits considered in the previous section, condition (\[bound\]) does not hold if we use single qubits for the estimation. The action of the Pauli channel is defined by a set of $n=4$ error operators (including the identity) acting onto qubits ($D'=2$). Let us now apply our consideration to the case of a generalized Pauli channel. The action of this channel is given by $$\rho '=\sum_{\alpha,\beta=0}^{D-1}\lambda_{\alpha,\beta} U_{\alpha,\beta} \rho \, U_{\alpha,\beta}^{\dagger}. \label{generalized pauli channel}$$ where the $D^2$ Kraus operators $U_{\alpha,\beta}$ are defined by $$U_{\alpha,\beta}\equiv (X_D)^\beta (Z_D)^\alpha . \label{U operators}$$ The operation $$Z_{D}|j\rangle \equiv e{}^{i\frac{2\pi }{D}j}|j\rangle$$ is the generalization of a phase flip in $D$ dimensions. The $D$–dimensional bit flip is given by $$X_D|j\rangle \equiv |(j-1)~{\rm mod}~D \rangle .$$ These errors occur with probabilities $\lambda_{\alpha,\beta}$, which are normalized, $\sum_{\alpha,\beta=0}^{D-1}\lambda_{\alpha,\beta}=1$. The states $|j\rangle, j=0,...,D-1$ form a $D$-dimensional basis of the Hilbert space of the quantum system. This extension of the Pauli channel to higher dimensional Hilbert spaces has been studied previously in the context of quantum error correction [@Knill; @Gottesmann], quantum cloning machines [@Cerf] and entanglement [@Fivel]. Now we will apply the estimation strategy based on condition (\[condition\]) to this particular channel. Thus, we must look for a state of the bipartite quantum system which satisfies the set of conditions (\[condition\]), for the error operators $U_{\alpha,\beta}$, i.e. $$\langle\psi|({\bf 1}\otimes U_{\alpha,\beta})^\dagger ({\bf 1}\otimes U_{\mu,\nu})|\psi\rangle=\delta_{\alpha,\mu}\delta_{\beta,\nu}.$$ It can be easily shown that such a state is $$|\psi_{0,0}\rangle\equiv \frac{1}{\sqrt{D}} \sum_{i=0}^{D-1}|i\rangle_1\otimes|i\rangle_2 \label{state for the estimation}$$ where $|i\rangle_1$ and $|i\rangle_2$ , $j=0,...,D-1$, are basis states for the two particles. In fact, the action of the operators ${\bf 1}\otimes U_{\alpha,\beta}$ onto state $|\psi_{0,0}\rangle$ generates the basis $$|\psi_{\alpha,\beta}\rangle\equiv {\bf 1}\otimes U_{\alpha,\beta}|\psi_{0,0}\rangle= \frac{1}{\sqrt{D}}\sum_{k=0}^{D-1}e^{i\frac{2\pi}{D}\alpha k} |k\rangle_1\otimes|k-\beta\rangle_2$$ of $D^2$ maximally entangled states for the total Hilbert space. Thereby, the action of the generalized Pauli channel yields $$\rho '=\sum_{\alpha,\beta=0}^{D-1}\lambda_{\alpha,\beta} |\psi_{\alpha,\beta}\rangle\langle\psi_{\alpha,\beta}|$$ when the initial state is $\rho=|\psi_{0,0}\rangle\langle\psi_{0,0}|$. The particular values of the coefficients $\lambda_{\alpha,\beta}$ can now be obtained by projecting onto the states $|\Psi_{\alpha,\beta}\rangle$. The quality of this estimation strategy according to the statistical error is given by $$\bar{c}_s(N,\vec{\lambda})=\frac{1}{N/2}\sum_{\alpha,\beta=0}^{D-1} \lambda_{\alpha,\beta}(1-\lambda_{\alpha,\beta}) (1-\delta_{\alpha,0}\delta_{\beta,0})$$ with $N/2$ the number of pairs of quantum systems used in the estimation. Conclusions {#Conclude} =========== We have examined the problem of determining the parameters that describe a noisy quantum channel with finite resources. We have given simple protocols for the determination of the parameters of several classes of quantum channels. These protocols are based on measurements made on qubits that are sent through the channel. We have also introduced two cost functions that estimate the quality of the protocols. In the most simple protocols measurements are performed on each qubit. We have also shown that more complex schemes based on entangled pairs can give a better estimate of the parameters of the Pauli channel. Our investigations stress once more the usefulness of entanglement in quantum information. Acknowledgments {#acknowledgments .unnumbered} =============== We acknowledge support by the DFG programme “Quanten-Informationsverarbeitung”, by the European Science Foundation QIT programme and by the programmes “QUBITS” and “QUEST” of the European Commission. M A C wishes to thank V Bužek for interesting discussions and remarks. [99]{} For recent books on the topic, see Deutsch D 1985 [*Proc. R. Soc. Lond. A*]{} [**400**]{} 97 Gruska J 1999 [*Quantum Computing*]{} (London: McGraw–Hill); Nielsen M A and Chuang I L 2000 [*Quantum Computation and Quantum Information*]{} (Cambridge: Cambridge University Press); Bouwmeester D, Ekert A and Zeilinger A 2000 [*The Physics of Quantum Information*]{} (Berlin: Springer); G. Alber [*et al*]{}. 2001 [*Quantum Information*]{} (Berlin: Springer) Schumacher B 1996 [*Phys. Rev. A*]{} **54 2614; Schumacher B and Nielsen M A 1996 [*Phys. Rev. A*]{} **54 2629; Bennett C H, DiVincenzo D P, Smolin J A and Wootters W K 1996 [*Phys. Rev. A*]{} **54 3824; Lloyd S 1997 [*Phys. Rev. A*]{} **55 1613; Bennett C H, DiVincenzo D P, and Smolin J A 1997 [*Phys. Rev. Lett.*]{} **78 3217; Schumacher B and Westmoreland M D 1997 [*Phys. Rev. A*]{} **56 131; Adami C and Cerf N J 1997 [*Phys. Rev. A*]{} **56 3470; Barnum H, Nielsen M A and Schumacher B 1998 [*Phys. Rev. A*]{} **57 4153; DiVincenzo D P, Shor P W and Smolin J A 1998 [*Phys. Rev. A*]{} **57 830 Mack H, Fischer D G and Freyberger M 2000 [*Phys. Rev. A*]{} [**62**]{} 042301 Fujiwara A 2001 [*Phys. Rev. A*]{} [**63**]{} 042304 Fischer D G, Mack H, Cirone M A and Freyberger M 2001 [*Phys. Rev. A*]{} [**64**]{} 022309 Shor P W 1995 [*Phys. Rev. A*]{} **52 R2493; Calderbank A R and Shor P W 1996 [*Phys. Rev. A*]{} **54 1098; Laflamme R, Miquel C, Paz J P and Zurek W H 1996 [*Phys. Rev. Lett.*]{} **77 198; Steane A M 1996 [*Phys. Rev. Lett.*]{} **77 793 Knill E and Laflamme R 1997 [*Phys. Rev. A*]{} **55 900 Kraus K 1983 [*States, Effects, and Operations*]{}, Lecture Notes in Physics Vol. 190 (Berlin: Springer) Stinespring W F 1955 [*Proc. Amer. Math. Soc.*]{} [**6**]{} 211 Jozsa R 1994 [*J. Mod. Opt.*]{} [**41**]{} 2315 Fujiwara A and Algoet P 1999 [*Phys. Rev. A*]{} [**59**]{} 3290 For recent experiments on decoherence, see Brune M, Hagley E, Dreyer J, Maitre X, Maali A, Wunderlich C, Raimond J M and Haroche S 1996 [*Phys. Rev. Lett.*]{} [**77**]{} 4887; Myatt C J, King B E, Turchette Q A, Sackett C A, Kielpinski D, Itano W M, Monroe C and Wineland D J 2000 [*Nature*]{} [**403**]{} 269; Kokorowski D A Cronin A D Roberts T D and Pritchard D E 2001 [*Phys. Rev. Lett.*]{} [**86**]{} 2191; see also Giulini D [*et al*]{}. 1996 [Decoherence and the Appearance of a Classical World in Quantum Theory]{} (Berlin: Springer) Carmichael H 1993 [*An Open System Approach to Quantum Optics*]{} (Berlin: Springer); Carmichael H J 1999 [*Statistical Methods in Quantum Optics I*]{} (Berlin: Springer); Macchiavello C and Palma G M 2001 [*preprint*]{} quant-ph/0107052 Bennett C H and Wiesner S J 1992 [*Phys. Rev. Lett.*]{} [**69**]{} 2881 Knill E 1998 [*preprint*]{} quant–ph/9808049 Gottesmann D 1998 [*preprint*]{} quant–ph/9802007 Cerf N J 2000 [*J. Mod. Opt.*]{} [**47**]{} 187 Fivel D I 1995 [*Phys. Rev. Lett.*]{} [**74**]{} 835**************************** [^1]: Since for our estimation protocol we are only sending pure states through the quantum channel we also only average over pure states.

--- abstract: | We compare UV spectra of the recent soft X–ray transients and XTEJ1859+226. The emission line strengths in strongly suggest that the accreting material has been CNO processed. We show that this system must have come into contact with a secondary star of about $1.5\msun$, and an orbital period $\sim 15$ hr, very close to the bifurcation value at which nuclear and angular momentum loss timescales are similar. Subsequent evolution to the current period of 4.1 hr was driven by angular momentum loss. In passing through a period of 7.75 hr the secondary star would have shown essentially normal surface abundances. could thus represent a slightly later evolutionary stage of A0620–00. We briefly discuss the broad Ly${\rm \alpha}$ absorption wings in . [**Key Words:**]{} accretion, accretion discs - stars: individual XTEJ1118+480 - X–rays: stars.\ author: - | C.A. Haswell$^1$, R.I. Hynes$^2$, A.R. King$^{3}$, K. Schenker$^{3}$\ $^1$Department of Physics and Astronomy, Open University, Walton Hall, Milton Keynes, MK7 6AA\ $^2$Department of Physics and Astronomy, University of Southampton, SO17 1BJ\ $^3$Theoretical Astrophysics Group, University of Leicester, Leicester, LE1 7RH\ date: Accepted Received title: 'The UV line spectrum of the soft X–ray transient XTE J1118+480: a CNO-processed core exposed' --- == == == == \#1[[ \#1]{}]{} \#1[[ \#1]{}]{} @mathgroup@group @mathgroup@normal@group[eur]{}[m]{}[n]{} @mathgroup@bold@group[eur]{}[b]{}[n]{} @mathgroup@group @mathgroup@normal@group[msa]{}[m]{}[n]{} @mathgroup@bold@group[msa]{}[m]{}[n]{} =“019 =”016 =“040 =”336 ="33E == == == == \#1[[ \#1]{}]{} \#1[[ \#1]{}]{} == == == == \[firstpage\] INTRODUCTION {#sec:intro} ============ The recently-discovered soft X–ray transient (SXT) XTE J1118+480 ( = KV UMa; Remillard et al 2000) lies at high galactic latitude, close to the Lockman Hole in the local ISM. The exceptionally low interstellar absorption permits unprecedented wavelength coverage (Mauche et al 2000, Hynes et al 2000, McClintock et al 2001b, Chaty et al 2002). A low amplitude photometric modulation with period 4.1 hr was reported by Cook et al (2000) and Uemura et al (2000). Dubus et al (2000) present spectroscopy showing an emission line radial velocity ‘S-wave’ modulation at a period close to this. This period was recently confirmed as orbital, and a mass function of $f(M) \approx 6 \msun$ was measured by McClintock et al (2001a) and Wagner et al (2001). Therefore the accretor clearly has to be a black hole. We present the UV line spectrum of , which we compare with that of another recent SXT, XTEJ1859+226. Full analyses of these HST spectra will be presented elsewhere (Hynes et al 2002a,b; Chaty et al 2002). Here we focus on the prior evolution of . OBSERVATIONS ============ We obtained [*HST*]{} spectra of XTEJ1859+226 near the peak of outburst on 1999 October 18 and 27 and November 6. At each epoch far-UV spectra were obtained using the low-resolution G140L grating and wide (0.5arcsec) slit, yielding wavelength coverage 1150–1730Å and resolution $\sim2.1$Å. We used the standard pipeline data products except that the spectral extraction was done by hand using the [iraf]{} implementation of optimal extraction. This gave a significantly cleaner removal of geocoronal emission lines (principally Ly $\alpha$ and O[i]{} 1302Å). After extracting the one-dimensional spectra (with the pipeline wavelength and flux calibrations), we resampled them onto a common wavelength grid and took an exposure-time weighted average. For XTEJ1118+480, we obtained high-resolution (echelle) spectra using the E140M grating and 0.2arcsec square aperture on 2000 April 8, 18 and 29, May 28, June 24, and July 8. The pipeline extraction was adequate, although a few high pixel values had to be replaced by hand. The extracted spectra were rebinned into 0.2Å bins and averaged with exposure-time weighting. The process is clearly not perfect. In particular, there are a number of abrupt steps in the spectrum at longer wavelengths. These artefacts, at $\la 5$per cent level, arise from inconsistencies in the relative flux calibration of adjacent echelle orders. Figure 1 gives the UV spectra of and XTE J1859+226. Both show interstellar absorption features. In these appear sharper because the spectral resolution was higher. EMISSION LINES -------------- The difference in the equivalent widths of the emission lines in the two systems is striking (Fig. 1, Table 1; Haswell et al 2000). In XTEJ1859+226 C[<span style="font-variant:small-caps;">iv</span>]{} is the strongest line, with N[<span style="font-variant:small-caps;">v</span>]{}, C[<span style="font-variant:small-caps;">iii</span>]{}, O[<span style="font-variant:small-caps;">v</span>]{}, Si[<span style="font-variant:small-caps;">iv</span>]{}, and He[<span style="font-variant:small-caps;">ii</span>]{} also strongly present. These emission line strengths are roughly as expected for an X-ray bright LMXB (cf Sco X-1, Kallman, Boroson, & Vrtilek 1998). Compared to cataclysmic variables (CVs), the N[<span style="font-variant:small-caps;">v</span>]{}/C[<span style="font-variant:small-caps;">iv</span>]{} ratio is relatively large, while Si[<span style="font-variant:small-caps;">iv</span>]{}/C[<span style="font-variant:small-caps;">iv</span>]{} is small, as expected for a high ionisation parameter (Mauche, Lee & Kallman 1997). In contrast, for XTEJ1118+480, the carbon and oxygen lines are undetectable, while the N[<span style="font-variant:small-caps;">v</span>]{} emission appears enhanced. A similar anomalous emission line spectrum is seen in the magnetic CV AE Aqr, in which the N[<span style="font-variant:small-caps;">v</span>]{} is much stronger than the C[<span style="font-variant:small-caps;">iv</span>]{} line (Jameson et al 1980, Eracleous et al 1994). Line flux (ergs$^{-1}$cm$^{-2}$) Equivalent width (Å) Gaussian FWHM (Å) -------------- ---------- ---------------------------------- ---------------------- ------------------- XTEJ1859+226 N[v]{} $(17.1 \pm 2.0)\times10^{-9}$ $ 8.0 \pm 1.4$ 13 O[v]{} $( 4.5 \pm 0.2)\times10^{-9}$ $ 1.9 \pm 0.1$ 15 Si[iv]{} $( 4.4 \pm 0.2)\times10^{-9}$ $ 2.0 \pm 0.1$ 17 C[iv]{} $(22.9 \pm 0.3)\times10^{-9}$ $11.8 \pm 0.2$ 11 He[ii]{} $( 4.4 \pm 0.4)\times10^{-9}$ $ 2.4 \pm 0.3$ 10 XTEJ1118+480 N[v]{} $(24.0 \pm 0.5)\times10^{-13}$ $ 5.05\pm 0.14$ 14 N[iv]{} $(\la 3 )\times10^{-14}$ $ \la 1.5 $ O[v]{} $(\la 2 )\times10^{-13}$ $ \la 0.6 $ Si[iv]{} $( 5.0 \pm 0.4)\times10^{-13}$ $ 1.29\pm 0.11$ 18 C[iv]{} $(\la 4)\times10^{-14}$ $\la 0.14 $ He[ii]{} $( 4.6 \pm 0.3)\times10^{-13}$ $ 1.74\pm 0.10$ 15 The N[<span style="font-variant:small-caps;">iv</span>]{} 1718${\rm\AA}$ line would add a valuable additional constraint on the interpretation of the line spectrum in terms of the photo-ionisation conditions, so we carefully assessed our spectra to determine whether it was present. Unfortunately our well-exposed E140M echelle spectrum stops at 1710${\rm\AA}$ and our E230M echelle spectrum suffers from shorter total exposure and the low sensitivity of the NUV MAMA in this region. 1718${\rm\AA}$ falls right on the boundary between the two STIS MAMA detectors, and consequently neither is optimised at this wavelength. We find no strong (${\rm \ga 3 \times 10^{-14} erg~s^{-1}~cm^{-2}}$) line in the spectrum, but this upper limit provides a weak constraint due to the poor quality of our data at this wavelength. ABSORPTION LINES ---------------- The Ly${\rm \alpha}$ absorption in (Fig. 1) clearly has a sharp core, which is probably interstellar, with very broad wings. The continuum slope is well constrained by the spectrum longwards of $1280{\rm\AA}$. Extrapolating this continuum is suggestive of broad Ly${\rm \alpha}$ extending from $\ga 1160{\rm\AA}$ to $\la 1280{\rm\AA}$. The combination of noise, N[v]{} emission, and absorption features hinder measurement of the Ly${\rm \alpha}$ wings, but they definitely extend at least from $\sim 1180{\rm\AA}$ to $\sim 1230{\rm\AA}$. The broad absorption may be damped Ly${\rm \alpha}$ from the H[<span style="font-variant:small-caps;">i</span>]{} in our galaxy (Bowen, priv.comm.). If we assume, instead, that it is due to absorbing gas executing Keplerian motion around the black hole, and take the largest width suggested by our data, the full width implies velocities of $\sim 0.05~c$ and absorbing material at distances as close as $R_{\rm in} \sim 200 R_{\rm Sch}$. Taking the securely determined line width would imply absorption from Keplerian material at $R_{\rm in} \sim 500 R_{\rm Sch}$. An independent line of argument based on the shape of the spectral energy distribution, and dependent on the value of the neutral hydrogen column density, $N_{\rm H}$, has been used to estimate the disc inner radius (Hynes et al 2000, McClintock et al 2001, Chaty et al 2002). Hynes et al (2000) estimate $N_{\rm H} = 0.75 \times 10^{20} {\rm cm}^{-2}$, a choice which suggests the disc is terminated at $\ga 10^{3} R_{\rm Sch}$. This is inconsistent with the Keplerian interpretation of the broad Ly${\rm \alpha}$ absorption. Alternatively, informed by the [*Chandra*]{} data, McClintock et al (2001) suggest that $N_{\rm H}$ could be as high as $1.3 \times 10^{20} {\rm cm}^{-2}$, with $ R_{\rm in} \ga 65 R_{\rm Sch}$. Chaty et al (2002) adopt $N_{\rm H} = 1.0 \times 10^{20} {\rm cm}^{-2}$, which leads to $ R_{\rm in} \sim 350 R_{\rm Sch}$. The latter two values are consistent with the Keplerian interpretation of the broad Ly${\rm \alpha}$ absorption. The feature near ${\rm 1425\AA}$ does not have an obvious identification and we believe it may be spurious, although it does not correspond to any known instrumental artefacts (Sahu 2001 priv. comm.). This issue is under investigation. IONISATION DIFFERENCES VERSUS ABUNDANCE ANOMALIES ================================================= The C[<span style="font-variant:small-caps;">iv</span>]{} 1550${\rm \AA}$ and the N[<span style="font-variant:small-caps;">v</span>]{} 1240${\rm\AA}$ lines are both resonance lines of lithium-like ions, and are produced under essentially the same physical conditions. Kallman and McCray (1982, hereafter KM82) present theoretical models of compact X-ray sources predicting the ionisation structure expected in a wide variety of astrophysical situations, including galactic X-ray binaries. KM82 present 8 distinct models; in each one the source luminosity, $L$, spectral shape, and gas density, $n$, are fixed. For each model the output includes the dependence on ionisation parameter, $$\xi = {{L} \over {nR^2}},$$ (where $R$ is the distance from the central source) of the abundances (as a fraction of the total elemental abundance) of the ions of common elements. KM82 consider (i) three species detected in both and XTEJ1859+226: He[<span style="font-variant:small-caps;">ii</span>]{}, Si[<span style="font-variant:small-caps;">iv</span>]{}, and N[<span style="font-variant:small-caps;">v</span>]{}; and (ii) three species detected only in the spectrum of XTEJ1859+226: C[<span style="font-variant:small-caps;">iv</span>]{}, C[<span style="font-variant:small-caps;">iii</span>]{}, and O[<span style="font-variant:small-caps;">v</span>]{}. Strikingly similar ionisation parameters were required for the two sets of lines in all models. In almost all cases the presence of N[<span style="font-variant:small-caps;">v</span>]{} places the tightest constraint on $\xi$, and almost invariably this is encompassed within the range producing C[<span style="font-variant:small-caps;">iv</span>]{}. In the minority of cases where there exists an ionisation parameter which produces N[<span style="font-variant:small-caps;">v</span>]{} and not C[<span style="font-variant:small-caps;">iv</span>]{}, this range also produces O[<span style="font-variant:small-caps;">v</span>]{}. Consequently, KM82 give no set of photoionisation parameters which would predict production of N[<span style="font-variant:small-caps;">v</span>]{} in while suppressing the C[<span style="font-variant:small-caps;">iv</span>]{} and O[<span style="font-variant:small-caps;">v</span>]{}. The X-ray spectrum of is clearly different from that of XTEJ1859+226: the former is an extended power-law (Hynes et al 2000, McClintock et al 2001, Frontera et al 2001, Chaty et al 2002) with a cut-off at $\sim 100$ keV indicative of the low-hard state, and often attributed to an advection-dominated accretion flow (ADAF; e.g. Esin et al 2001); while in the early stages of the outburst the latter was dominated by the soft thermal black-body disk spectrum which turned over by $\sim 4$ keV (Hynes et al 2002a), typical of the high-soft state. Ho et al (2000) note that the characteristically harder ionising spectrum of an ADAF lowers the effective ionisation parameter, and hence favours the production of lower ionisation species. Hence, if anything, the differences in the X-ray spectra should cause C[<span style="font-variant:small-caps;">iv</span>]{} to be relatively prominent (compared with N[<span style="font-variant:small-caps;">v</span>]{}) in , rather than being suppressed as we observe. Hamann and Ferland (1992) used the observed C[<span style="font-variant:small-caps;">iv</span>]{} 1550${\rm \AA}$ and N[<span style="font-variant:small-caps;">v</span>]{} 1240${\rm\AA}$ lines to estimate the N/C abundance ratio in high redshift QSOs. Their photionisation calculations show that for a broad range of ionisation parameters the N[<span style="font-variant:small-caps;">v</span>]{} 1240${\rm\AA}$ / C[<span style="font-variant:small-caps;">iv</span>]{} 1550${\rm \AA}$ line ratio is lower at lower metalicities even when the N/C abundance ratio is kept constant. This means that if anything we should expect the N[<span style="font-variant:small-caps;">v</span>]{} 1240${\rm\AA}$ line to be suppressed relative to C[<span style="font-variant:small-caps;">iv</span>]{} 1550${\rm \AA}$ in which is a halo object, and consequently might be expected to have a lower metalicity than that of XTEJ1859+226. A definitive abundance analysis for would require exact knowledge of the geometry, densities, and ionising spectrum in the regions emitting the UV lines. While Hynes et al (2000, 2002b) gives some information on the geometry, and Chaty et al. (2002) gives good coverage of the X-ray spectrum, the EUV spectrum remains open to significant uncertainty (compare the range of dereddened EUV spectra in Chaty et al 2002, McClintock et al 2001, Esin et al 2001, Hynes et al 2000, Merloni et al 2000). Furthermore the range of densities present in the UV line emitting gas is difficult to determine. Consequently a quantitative abundance determination of the type carried out in stellar photospheres, where the physical conditions are well-known, is not possible. Hence it is impossible to rule out definitively a contrived spectrum and gas distribution which would produce the set (i) lines while suppressing the set (ii) lines. However as - [ set (i) and set (ii) ions require essentially identical ranges of $\xi$ ]{} - [ the ionising spectrum of might be expected to favour C[<span style="font-variant:small-caps;">iv</span>]{} relative to N[<span style="font-variant:small-caps;">v</span>]{} ]{} - [ the lower metallicity expected for might be expected to boost the C[<span style="font-variant:small-caps;">iv</span>]{}1550${\rm \AA}$ / N[<span style="font-variant:small-caps;">v</span>]{} 1240${\rm\AA}$ line ratio ]{} by far the simplest explanation of our observed UV line spectra is that a substantial underabundance of carbon is present in the surface layers of the mass donor star in . In XTEJ1118+480 (as in AE Aqr), the underabundance of carbon compared to nitrogen strongly suggests that the material in the accretion flow is substantially CNO processed (Clayton 1983). If the CNO bi–cycle achieves equilibrium, it converts most CNO nuclei into $^{14}$N. At the lower temperatures typical in $\sim 1.5 \, \msun$ MS stars (which we shall suggest as the progenitor of the companion in XTEJ1118+480) this conversion is much more at the expense of $^{12}$C than $^{16}$O, and in fact $^{17}$O increases. Hence the oxygen abundance is not expected to decrease by much in XTEJ1118+480 and this argument does not of itself explain the observed weakness of O[<span style="font-variant:small-caps;">v</span>]{} 1371${\rm \AA}$. However C[<span style="font-variant:small-caps;">iv</span>]{}1550${\rm \AA}$, N[<span style="font-variant:small-caps;">v</span>]{} 1240${\rm\AA}$ are resonance lines, whose ratio must be regarded as a robust indicator of conditions in the line emitting gas, whereas O[<span style="font-variant:small-caps;">v</span>]{} 1371${\rm \AA}$ is a subordinate line, which may not be formed efficiently in a low density photoionized gas. We thus regard the spectrum of XTEJ1118+480 as indicating CNO processing. THE EVOLUTION OF XTE J1118+480 ============================== The spectrum shown in Figure 1 implies that the companion star in must be partially nuclear–evolved and have lost its outer layers, exposing inner layers which have been mixed with CNO–processed material from the central nuclear–burning region. Mass transfer must therefore have been initiated from a somewhat evolved and sufficiently massive donor of $M_{2i}\ga 1.5\msun$, and thus with an initial period $P_i \ga 12$ hr. The main difficulty in understanding the current status of is to explain its observed short orbital period; in general, significant nuclear evolution is associated with a period [*increase*]{}, rather than the inferred decrease. There are two ways in which this decrease could have occurred: \(i) the system came into contact with initial secondary/primary mass ratio $q_i = M_{2i}/M_{1i} \ga 1$ and underwent a phase of thermal–timescale mass transfer until $q \la 1$. At the end of this phase the period would have either decreased or not increased greatly. Subsequent orbital angular momentum loss could then pull the system in to the observed $P_{\rm orb} = 4.1$ hr. This case is similar to the likely evolution of AE Aqr (Schenker et al., 2002a). \(ii) the system came into contact with $q_i < 1$ already, and subsequent evolution was driven by angular momentum loss towards shorter periods. This case resembles that proposed for A0620–00 by de Kool et al (1983), with a severe additional constraint: the donor must be sufficiently evolved to provide the observed surface abundance ratios. This requires a near–equality of the nuclear and orbital angular momentum loss timescales $t_{\rm nuc}, t_{\rm AML}$ (cf Fig. 2). Put another way, $P_i$ must have been very close to the ‘bifurcation period’ defined by Pylyser & Savonije (1988). Case (i) above can be ruled out by the current system parameters. At the end of thermal timescale mass transfer we must have $q \la 1$, which would require a donor mass $M_{2} \sim 6\msun$ and an orbital period $\sim 15$ hr. But for such a system $t_{\rm nuc} << t_{\rm AML}$ (cf Fig. 2), implying evolution to [*longer*]{} periods, in stark contrast to the current 4.1 hr. Put another way, there is an upper limit on the binary mass $M = (1+q^{-1})M_2$ for case (i) evolution to give nuclear–evolved donors at short orbital periods. The near equality $t_{\rm nuc}\sim t_{\rm AML}$ can only realistically hold if magnetic braking dominates angular momentum loss, so we require $M_2 \la 1.5\msun$ at the end of the thermal–timescale episode. But since $q \sim 1$ there, we must have total binary mass $M \la 3\msun$ in case (i) evolutions. We are therefore left only with case (ii) above. The secondary mass when the system came into contact is now constrained to be close to $1.5\msun$: CNO processing excludes significantly lower masses, while the requirement for $t_{\rm AML} \approx t_{\rm MB} > t_{\rm nuc}$ excludes significantly higher masses. Figs. 3, 4 show the evolution of such a system: a $1.5 \, \msun$ MS star was allowed to evolve and fill its Roche lobe in a 15 hr binary with a $7 \, \msun$ BH. The predicted mass transfer rate throughout the evolution, including its current value $3 \times 10^{-10} \, \msun {\rm yr}^{-1}$, shows that the system would indeed have appeared as a soft X–ray transient, according to the irradiated–disc criterion for black–hole systems given by King, Kolb & Szuszkiewicz (1997). It should be noted that the change in C/N is mostly due to a drastic depletion of $^{12}$C, supplemented by a more modest increase in $^{14}$N. The total O abundance at the surface has only decreased very little; in fact the O/N ratio is reduced by only a factor of 3 to 5 at most, predominantly due to the increase of N. Interestingly, this model also passes through $P_{\rm orb} = 7.75 \, {\rm hr}$ at around $M_2 \approx 0.6 \, \msun$, showing much weaker N enhancements (i.e.  almost normal abundances, cf Fig. 3). Thus it can also be considered a reasonable model for A0620–00. A subsequent paper (Schenker et al., 2002b) will explore this and related evolutions in detail. Finally we can compare further properties of the secondary in our model to observations: Figure 4 shows the evolution of effective temperature with orbital period, together with a simple mapping to spectral types. The procedure is based on a conversion of observed colours to spectral types (Beuermann et al., 1998) and a set of non-grey stellar atmospheres (Hauschildt et al., 1999) providing the colours for each set of stellar boundary conditions. For solar metallicity the resulting SpT turns out to be only a function of effective temperature with a very weak dependence on surface gravity. However this mapping should only be considered a first estimate, as the evolutionary code in its current form still uses grey atmospheres. Furthermore the whole conversion is based on an observed set of unevolved stars and theoretical ZAMS models, i.e. no fully self-consistent application to a partially evolved MS star such as shown in Fig. 4 is possible for the time being. Allowing for the uncertainties described above, our model may be slightly too cool (M2 rather than K5-M1 as mentioned in McClintock et al (2001), or K7-M0 by Wagner et al (2000). Similar evolutionary tracks of strongly evolved CVs are known to get hotter again (Baraffe & Kolb, 2000) at short periods, so a confirmed secondary spectral type would provide insight as to the state of nuclear evolution in the donor star of . Figure 4 shows that in any case that the donor will currently appear to be close to the ZAMS, and thus that the distance estimate derived by McClintock et al (2000) on this basis is likely to be quite good. Conclusions =========== We have compared the UV spectra of and XTEJ1859+226. The former shows strong evidence of CNO processing, which tightly constrains the evolution of the system. must have first reached contact with the donor star having a significantly nuclear–evolved core, at a period where nuclear and angular momentum loss timescales were comparable. This in turn constrains the end point of the earlier common–envelope phase: immediately after the common–envelope phase the system had a wide enough separation that significant nuclear evolution could occur before contact was achieved. In a future paper we will investigate such evolutions systematically. Acknowledgments =============== CAH and RIH gratefully acknowledge the superb support of Tony Roman, Kailash Sahu, and all involved in the implementation of our time-critical [*HST*]{} observations. Support for proposals GO-08245 and GO-08647 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. This work was supported by the Leverhulme Trust F/00-180/A. Theoretical astrophysics research at Leicester is supported by a PPARC rolling grant. We thank Christian Knigge for discussion about the nature of the ${\rm 1425\AA}$ feature, and an anonymous referee for useful comments. Baraffe, I., Kolb, U. 2000, MNRAS, 318, 354 Barret, D., Olive, J.F., Boirin, L., Done, C., Skinner, G.K., Grindlay, J.E. 2000, ApJ, 533, 329 Beuermann, K., Baraffe, I., Kolb, U., & Weichhold, M. 1998, A&A, 339, 518 Chaty, S. et al 2002, in prep Clayton, D.D. 1983, “Principle of Stellar Evolution and Nucleosynthesis”, (Chicago: University of Chicago Press) Cook, L., Patterson, J., Buczynski, D., Fried, R. 2000, IAU Circ 7397 de Kool, M., van den Heuvel, E.P.J., Pylyser, E. 1987, A&A 183, 47 Dubus, G., Kim, R.S.J., Menou, K., Szkody, P., Bowen, D.V. 2001, ApJ, 553, 307. Eracleous, M., Horne, K., Robinson, E.L., Zhang, E.-H., Marsh, T.R., Wood, J.H. 1994, ApJ, 433, 313 Esin, A, McClintock, J.E., Drake, J.J., Garcia, M.R., Haswell, C.A., Hynes, R.I., Muno, M.P. 2001, ApJ 555, 483. Frontera et al 2001, ApJ, 546, 1027 Hamann, F. and Ferland, G. 1992, ApJLett, 391, 53 Haswell, C.A., Hynes, R.I., King, A.R., Stull, J., Ioannou, Z., Webb, N.A. 2000, IAU Circ 7407 Hauschildt, P.H., Allard, F., Baron, E. 1999, ApJ, 512, 377 Ho, L.C., Rudnick, G., Rix, H.-W., Shields, J.C., McIntosh, D.H., Filippenko, A.V., Sargent, W.L., Eracleous, M. 2000, ApJ, 541, 120 Hynes, R.I., Haswell, C.A., Chaty, S., Cui, W., Shrader, C.R. 2000, HEAD, 32, 3111 Hynes, R.I., Mauche, C.W., Haswell, C.A., Shrader, C.R., Cui, W., Chaty, S. 2000, ApJLett, 539, L37 Hynes, R.I., Haswell, C.A., Chaty, S., Shrader, C.R., Cui, W. 2002a, MNRAS, in press, astro-ph/0111333 Hynes, R.I. et al 2002b, in prep. Jameson, R.F., King, A.R., Sherrington, M.R., 1980 MNRAS, 191, 559 Kallman, T., Boroson, B., Vrtilek, S.D. 1998, ApJ, 502, 441 Kallman, T. & McCray, R. 1982, ApJSupp, 50, 263 (KM82) King, A.R., Kolb, U., & Szuszkiewicz, E., 1997, ApJ, 488, 89 Kolb, U. 1992 PhD Thesis, LMU Munich Mauche, C.W., Hynes, R.I., Charles, P.A., Haswell, C.A. 2000, IAU Circ 7401 Mauche, C.W., Lee, Y.P., Kallman, T.R. 1997, ApJ, 477, 832 McClintock, J.E., Garcia, M.R. , Caldwell, N., Falco, E.E. , Garnavich, P.M., Zhao, P., 2001a, ApJLett, 551, L147. McClintock, J., Garcia, M. , Zhao, P., Caldwell, N., Falco, E. 2000, IAU Circ 7542 McClintock, J.E., Haswell, C.A., Garcia, M.R., Drake, J.J., Hynes, R.I., Marshall, H.L., Muno, M.P., Chaty, S. et al 2001b, ApJ, 555, 477. Morton, D.C. 1991, ApJSupp, 77, 119 Pylyser, E., Savonije, G.J. 1988, A&A 191, 57 Remillard, R., Morgan, E., Smith, D., Smith, E., 2000, IAU Circ 7389 Revnivtsev, M., Sunyaev, R., Borozdin, K. 2000, A&A, 361, L37 Schenker, K. et al 2002a,b, MNRAS, in prep Uemura M. 2000, IAU Circ 7418 Wagner, R.M., Foltz, C.B., Shahbaz, T., Casares, J., Charles, P.A., Starrfield, S.G., Hewett, P. 2001, ApJ, 556, 42. \[lastpage\]

--- abstract: 'Fingerprint Liveness detection, or presentation attacks detection (PAD), that is, the ability of detecting if a fingerprint submitted to an electronic capture device is authentic or made up of some artificial materials, boosted the attention of the scientific community and recently machine learning approaches based on deep networks opened novel scenarios. A significant step ahead was due thanks to the public availability of large sets of data; in particular, the ones released during the International Fingerprint Liveness Detection Competition (LivDet). Among others, the fifth edition carried on in 2017, challenged the participants in two more challenges which were not detailed in the official report. In this paper, we want to extend that report by focusing on them: the first one was aimed at exploring the case in which the PAD is integrated into a fingerprint verification systems, where templates of users are available too and the designer is not constrained to refer only to a generic users population for the PAD settings. The second one faces with the exploitation ability of attackers of the provided fakes, and how this ability impacts on the final performance. These two challenges together may set at which extent the fingerprint presentation attacks are an actual threat and how to exploit additional information to make the PAD more effective.' author: - | Giulia Orrù\ `giulia.orru@unica.it`\ Pierluigi Tuveri\ `pierluigi.tuveri@diee.unica.it`\ Luca Ghiani\ `luca.ghiani@diee.unica.it`\ Gian Luca Marcialis\ `marcialis@unica.it`\ \ University of Cagliari - Department of Electrical and Electronic Engineering - Italy title: 'Analysis of “User-Specific Effect” and Impact of Operator Skills on Fingerprint PAD Systems' --- Introduction ============ In the development of biometric systems, the study of the techniques necessary to preserve the systems’ integrity and thus to guarantee their security, is crucial. Fingerprint authentication systems are highly vulnerable to artificial reproductions of fingerprint made up of materials such as silicon, gelatine or latex, also called spoofs or fake fingerprints [@mats]. To counteract this possibility, Fingerprint Liveness Detection (FLD), also known as Fingerprint Presentation Attaks Detection (FPAD), is a discipline aimed to design pattern recognition-based algorithms for distinguishing between live and fake fingerprints. Detecting presentation attacks is not trivial because we have an arms-race problem, where we potentially are unaware of the materials and methods adopted to attack the system [@marasco]. In the last decade, most of the previously proposed PAD methods have focused on using handcrafted image features, for example LBP, BSIF [@bsif], WLD [@wld] or HIG [@hig]. Over the years, PAD methods have been refined and recently CNN-based methods are outnumbering local image descriptors methods [@cnn; @cnn2]. In order to assess the performance of FPAD algorithms by using a common experimental protocol and data sets, the international fingerprint liveness detection competition (LivDet) is organized since 2009 [@reviewlivdet]. Each edition is characterized by a different set of challenges that must be dealt with by the competitors. Working on the LivDet competitions since its birth, that is, developing a wide experience in providing large sets of data for assessing the state of the art on this topic, we realized that some data acquisition methods influence the FPAD systems performance. For this reason, the 2017 edition of the LivDet was focused on assessing how much the composition and the data acquisition methods influence the PAD systems. In particular, we faced here with the followings points: - the presence of different operators for creating and submitting spoofs, that is, two levels of operators which provided spoofs and attacked the submitted algorithms - can spoofing be an actual threat independently of the ability of attackers, or an expert is necessary to break a fingerprint PAD?; - when a PAD must be integrated into a fingerprint verification system, the templates of genuine users are available, and there is no good reason to neglect them even during the PAD design. We referred to the additional information coming from the enrolled user as “user-specific effect” [@userspec]. These aspects have been analyzed only marginally in the LivDet 2017 report [@livdet17] for sake of space. Moreover, the above points were not addressed by the previous editions of LivDet, whose history and results over the years are reported in [@reviewlivdet]. With regard to the “user-specific effect”, we reported in [@userspec] that some specific components or minute details are dependent on the user’s skin characteristic. Ref. [@userspec] reports a statistical study on the relationships of textural features coming from the same fingerprints of the same user and even from his/her different fingerprints. The correlation we measured suggests that the consequent “bias” introduced into a PAD can be beneficial if the PAD must be integrated into a fingerprint verification system. Moreover, it does not affect the overall PAD performance when it is used for the generic user of detecting a presentation attack. What we called “user-specific effect” helps in improving the correct classification of an alive fingerprint of a genuine user, and this is crucial into a fingerprint verification system where the rate of rejected genuine users must be kept as low as possible. From this point of view, the fusion with a non-zero error system, namely, the PAD, may put that rate in danger, as noticed in other publications as [@inaction]. The paper is organized as follows: Section \[proposed\] describes the purpose of the experimentation. The experimental methodology is presented in Section \[exprotocol\]. Experiments are shown in Section \[results\]. Conclusions are drawn in Section \[conclusion\]. **** --------------- --------------- ----------- ----------- ----------- ------------- -- **Algorithm** **Type** **1** **2** **3** **Overall** SSLFD N.A. 93.58 94.33 93.14 93.68 JLW\_A Deep Learning 95.08 94.09 **93.52** 94.23 JLW\_B Deep Learning **96.44** **95.59** **93.71** 95.25 OKIBrB20 Hand-crafted 84.97 83.31 84.00 84.09 OKIBrB30 Hand-crafted 92.49 89.33 90.64 90.82 ZYL\_1 Deep Learning 95.91 **95.13** 91.66 94.23 ZYL\_2 Deep Learning **96.26** **94.73** 93.17 94.72 SNOTA2017\_1 N.A. 95.03 91.26 91.58 92.62 SNOTA2017\_2 N.A. 94.04 86.72 86.74 89.17 ModuLAB Deep Learning 94.25 90.40 90.21 91.62 ganfp Deep Learning 95.67 93.66 **94.16** 94.50 PB\_LivDet\_1 Hand-crafted 93.85 89.97 91.85 91.89 PB\_LivDet\_2 Hand-crafted 92.86 90.43 92.60 91.96 hanulj Deep Learning **97.06** 92.34 92.04 93.81 SpoofWit Hand-crafted 93.66 88.82 89.97 90.82 LCPD Hand-crafted 89.87 88.84 86.87 88.52 PDfV Hand-crafted 92.86 93.31 N.A. N.A. : Types and accuracy of the algorithms participating in LivDet 2017. The accuracy of the three sensors used (**1** = Green Bit, **2** = Digital Persona, **3** = Orcanthus) and the overall have been reported.[]{data-label="tab:overall"} Proposed analysis {#proposed} ================= In real applications, the FPAD system works together with a recognition system in order to protect it from spoofing attacks. The integration of the PAD to the authentication system may reduce the recognition performance due to some live genuine sample’s rejection [@inaction]. In last years it was noticed that the presence of the same users both in the train set and in the test set, increased the PAD accuracy of the system. In fact, the existence of artifacts due to the human skin (person-specific) and to the particular curvature of ridges and valleys (finger-specific) can impact in PAD systems’ performance [@userspec; @userspec2] and can be exploit to improve the integrated system. In particular, the data acquired during the recognition system’s initial enrollment phase can be used to lower the PAD error rate and consequently, to improve the integrated system performance. This approach may be referred to as “user-specific", whilst the other one may be referred to as “generic user". It is important to emphasize that stand-alone PAD systems are usually considered based on “generic user" approaches. Usually, a generic user approach is claimed to avoid biased results. This was evident in some early works [@quality; @jia] where an unexpected and unexplained drop of error rates occurred when some users were present both in the train and in the test set, because the user-specific effect was not yet clear in the mind of the scholars. However, this biasing may be desired under certain circumstances. In fact, as already mentioned, the positive influence of the user-specific effect on the live/fake classification can be exploited in real applications, where the enrolment phase allows to record the user fingerprint image previously. It is therefore possible to state that the “generic-user” approach is the main resource when the final user population is unknown (for example in forensic applications or border checks); the “user-specific" approach can be used to improve security of an integrated fingerprint verification system (for example authentication on mobile or bank account), in which the attacks are finalize to break the verification step and not to simply being disguised for another person. On the other hand, common sense tells us that the system’s performance is influenced by the attacker’s ability to replicate the fingerprint. Although it is known that fakes can be produced with commonly used materials, even if these are of excellent quality, the attacker must know how to use them to spoof the system. In fact, we believe that an average skilled attacker may have the technique needed to circumvent the matcher, but more skills are needed to spoof an integrated system (both matcher and liveness detector). Although the impact of image quality has been used to classify fingerprints [@quality; @quality2], to the best of our knowledge, no one has ever analyzed the impact of the operator’s skill level. ----------------- ------ ----------- --------- ------------- ------ ---------- ------- ---------------- -- -- -- -- **Dataset** Live Wood Glue Ecoflex Body Double Live Gelatine Latex Liquid Ecoflex Green Bit 1000 400 400 400 1700 680 680 680 Orcanthus 1000 400 400 400 1700 680 658 680 Digital Persona 999 400 400 399 1700 679 670 679 ----------------- ------ ----------- --------- ------------- ------ ---------- ------- ---------------- -- -- -- -- Dataset and experimental protocol {#exprotocol} ================================= LivDet 2017 [@livdet17] consisted of data from three fingerprint sensors: Green Bit DactyScan84C, Orcanthus Certis2 Image and Digital Persona U.are.U 5160. It is composed of almost 6000 images for each scanner. Live images came from multiple acquisitions of all fingers of different subjects. The LivDet 2017 fake images were collected using the cooperative method. Each dataset consists of two parts, the train set, used to configure the algorithms, and the test set, used to evaluate the algorithms performance. Moreover, the materials used in the training set are different with respect to the test set as we can see in Table \[tab:datasetComposition\]. The test set can be partitioned into three parts, a user-specific (US) portion, composed by fingerprint belonging to users already present in the train set, and the other two partitions are generic user (GU), meaning they are composed by fingerprint belonging to user not present in the train set, in which the fakes are made by two different operators HS (High Skilled) and LS (Low Skilled), described in more detail below. The false samples of the US partition were made by the operator HS. On the basis of this subdivision we therefore had: $$TestSet = \{US (HS), GU (HS), GU (LS)\} \nonumber$$ Using the two partitions, $US (HS)$ and $GU (HS)$, separately to test the algorithms, we can compare the user-specific effects with the generic-user case. In this experiment we left out the $GU (LS)$ partition to avoid an operator bias. The second evaluation concerned the comparison between operators with different degrees of skill in replicating fingerprints. In particular, operator HS has a high experience in creating fakes because he has participated in multiple LivDet editions, while for operator LS this was the first edition in which he participated. Fingerprint replication requires a lot of experience, as the accuracy of the fakes heavily affects system performance. For this evaluation, partitions $GU (HS)$ and $GU (LS)$ were used separately as test set. The two operators used the same acquisition protocol using the same materials. The Table \[tab:overall\] shows the results of LivDet 2017 for the three sensors, the total accuracy and the type of algorithm presented, making a distinction between deep learning based or hand-crafted features based. The table shows that in 2017 in FPAD the solutions adopted are equally distributed between deep learning and hand-crafted algorithms. Furthermore, we see that the performance of the algorithms that use deep learning techniques are better than the classic ones of about 10 percentage points. -- ---------------- ------------------ ----------------- ------------------- ------------------ **High skilled** **Low skilled** **User-specific** **Generic User** $94.0(\pm1.9)$ $8.1(\pm2.9)$ $6.5(\pm2.4)$ $2.6(\pm1.4) $ $8.1(\pm2.9)$ $90.0(\pm3.1)$ $12.5(\pm4.6)$ $10.8(\pm3.7)$ $5.7(\pm3.4)$ $12.5(\pm4.6)$ -- ---------------- ------------------ ----------------- ------------------- ------------------ : Average accuracy and EERs \[%\] comparison between deeep learning and hand-crafted methods. The distinction between User-Specific and Generic User experiments and between high skilled operator and a low skilled operator is reported. The average accuracy refers to the entire test set.[]{data-label="tab:deephand"} Results ======= In this session the comparisons between User-Specific (US) and Generic User (GU) experiments and between highly skilled operator (HS) and a lowly skilled operator (LS) experiments were reported. In the previous papers the user-specific effect was highlighted only for PAD systems based on hand-crafted features. The high number of algorithms participating in LivDet 2017 based on deep learning methods allow us to have a clearer view on this effect. Table \[tab:deephand\] shows a comparison between deep-learning based methods and handcrafted features based methods, obtained by comparing the average accuracy and the EERs related to the analyzed case studies. This table shows that deep learning appears to be more competitive than hand-crafted techniques, as anticipated in Table \[tab:overall\]. Furthermore it is clear that even the standard deviation is more limited and therefore it is an index of how the deep learning performances have more or less the same performance and with smaller fluctuations with respect to the others. The table clearly confirms the existence of the user-specific effects for all the algorithms presented: on average the error for the generic user system is more than double the user-specific one. This result is confirmed by Fig. \[fig:userspec\], in which the EER difference between GU and US systems has been reported for each algorithm presented. To get a graphical view of these results, the ROCs related to the comparisons between user-specific and generic user for the average of the three best algorithms are reported in Fig. \[fig:AverageROCUser\]. These results demonstrate that the user-specific effect influences liveness detection methods regardless of their characteristics. To assess whether a person-specific or finger-specific effect is more influential, a more in-depth analysis is required. This evidence supports the possible exploitation of user-specific effects to improve the performance of FPAD when integrated with personal identity verification systems, especially when these are used by one or a few users, as in the case of mobile authentication systems. ![\[fig:userspec\]Comparison between User Specific and Generic User experiments. Each bar is the result of the EER difference between US and GU.](pic/differenceGU.png){width="11cm"} ![\[fig:operators\]Comparison between high skilled operator and low skilled operator. Each bar is the result of the EER difference between HS and LS.](pic/difference.png){width="11cm"} The other analysis carried out concerns the influence of the operator’s ability to spoof PAD systems. The results, shown in Tab. \[tab:deephand\] and Fig. \[fig:operators\]-\[fig:AverageROCOperator\], confirm the intuitive hypothesis that the attacker’s skills profoundly influence the performance of these systems for both hand-crafted and deep learning methods. In particular, from Fig. \[fig:AverageROCOperator\], which shows the comparisons between a low skilled and high skilled operator for the average of the three best algorithms, it is possible to deduce that this difference is not only due to the ability to create fakes, but also to the ability to use fakes to “cheat" the system. In fact, a skilled operator produces excellent fakes and knows how to use them to spoof the FPAD system, thus induces a higher EER. Depending on the sensor, the difference between the continuous curve (LS) and the dotted curve (HS) is more pronounced. Above all, the curves regarding the Green Bit sensor, which is the simplest to use as it has a scanner area almost double compared to the other sensors, have a similar trend. Conversely, the other two sensors, in addition to a smallest area, are much more difficult to use and a correct acquisition require a remarkable experience. Conclusions {#conclusion} =========== In this paper, we extended the report on the LivDet 2017 competition edition, in order to point out the main results about two more challenges proposed which were not detailed in the official report for sake of space. The obtained results can be viewed as a demonstration on what the state of the art on fingerprint PAD can reach on both challenges. Participants used deep learning and hand-crafted features based methods for the task, also the number of competitors was the biggest with respect to other editions. Reported results allowed us to confirm that the positive influence of the user-specific effect on the liveness classification can be exploited in integrated systems where the enrolment phase is necessary, thus making available samples of the specific user for the PAD design. Furthermore, it has been shown that the operator’s ability to create and use fake replicas heavily affects performance. This pointed out that, to being a serious threat for fingerprint verification system a minimum level of skill is required beside the ability to provide reliable replicas of the targeted user. ![\[fig:AverageROCUser\]Average ROC of the three best LivDet2017 algorithms, calculated for the three scanners. In the ROC the blue, green and black line respectively indicates the Green Bit, Digital Persona and Orcanthus data sets. In the graph the continuous line indicates the US effect, instead the dotted line indicates the trend of the GU ROC.](pic/UserSpecific2.png){width="11cm"} ![\[fig:AverageROCOperator\]Average ROC of the three best LivDet2017 algorithms, calculated for the three scanners. In the ROC the blue, green and black line respectively indicates the Green Bit, Digital Persona and Orcanthus data sets. In the graph the continuous line indicates the Low skilled operator in the spoof fabrication fingerprint, instead the dotted line indicates the trend of High skilled operator making fake samples in the ROC curve.](pic/Operator2.png){width="11cm"}

--- abstract: 'The observed enhancement of the Fe K$\alpha$ line in three gravitationally lensed QSOs (MG J0414+0534, QSO 2237+0305, H1413+117) is interpreted in terms of microlensing, even when equivalent X-ray continuum amplification is not observed. In order to interpret these observations, first we studied the effects of microlensing on quasars spectra, produced by straight fold caustic crossing over standard relativistic accretion disk. The disk emission was analyzed using the ray tracing method, considering Schwarzschild and Kerr metrics. When the emission is separated in two regions (an inner disk corresponding to the Fe K$\alpha$ line and an outer annulus corresponding to the continuum, or vice-versa) we find microlensing events which enhance the Fe K$\alpha$ line without noticeable amplification of the X-ray continuum, but only during a limited time interval. Continuum amplification is expected if a complete microlensing event is monitored. Second, we studied a more realistic case of amplification by caustic magnification pattern. In this case we could satisfactorily explain the observations if the Fe K$\alpha$ line is emitted from the innermost part of the accretion disk, while the continuum is emitted from a larger region. Also, we studied the chromatic effects of microlensing, finding that the radial distribution of temperature in the accretion disk, combined with microlensing itself, can induce wavelength dependent variability of $\sim$ 30% for microlenses with very small masses. All these results show that X-ray monitoring of gravitational lenses is a well suited method for studying of the innermost structure of AGN accretion disks.' author: - 'L. Č. Popovi'' c, P. Jovanovi'' c, E. Mediavilla, A.F. Zakharov, C. Abajas, J. A. Muñoz, G. Chartas' title: 'A study of the correlation between the amplification of the Fe K$\alpha$ line and the X-ray continuum of quasars due to microlensing' --- Introduction ============ Recent observational and theoretical studies suggest that gravitational microlensing can induce variability in the X-ray emission of lensed QSOs. Microlensing of the Fe K$\alpha$ line has been reported at least in three macrolensed QSOs: MG J0414+0534 [@Chart02a], QSO 2237+0305 [@Dai03], and H1413+117 [@Osh01; @Pop03; @Chart04]. The influence of microlensing in the X-ray emission has been also theoretically investigated. [@Min01] simulated the variation of the X-ray continuum due to microlensing showing that the flux magnifications for the X-ray and optical continuum emission regions are not significantly different during the microlensing event, while [@Yon98; @Yon99; @Tak01] found that simulated spectral variations caused by microlensing show different behaviour, depending on photon energy. Also, microlensed light curves for thin accretion disks around Schwarzschild and Kerr black holes were considered in [@Jar92] and microlensing light curves for the Fe K$\alpha$ were simulated by [@Jar02]. On the other hand, the influence of microlensing in the Fe K$\alpha$ spectral line shape was discussed in [@Popov01; @Chart02a] and [@Pop03; @Pop03a][^1]. [@Pop03; @Pop03a] showed that objects in a foreground galaxy with even relatively small masses can produce observable changes in the Fe K$\alpha$ line flux, much stronger than those expected for the UV and optical lines ([@Pop01b; @Aba02; @li04]). In the optical spectra, microlensing induced magnification of broad UV lines (e.g., CIV and SIV/OIV) was reported by Richards et al. (2004). Consequently, one can expect that microlensing of the Fe K$\alpha$ line region can be more frequent. Observations of the X-ray continuum and the Fe K$\alpha$ line in multi-imaged AGNs open new possibilities for the study of the unresolved X-ray emitting structure in QSOs, particularly for high redshifted QSOs [@Zak03; @Dai04]. However, an explanation for different behavior of line and continuum variability in the observed events should be given in context of the microlensing hypothesis. [@Chart02a] detected an increase of the Fe K$\alpha$ equivalent width in the image B of the lensed QSO J0414+0534 that was not followed by the continuum. [@Chart02a] explained the non-enhancement of the continuum emission in the spectrum of image B by proposing that the thermal emission region of the disk and the Compton up-scattered emission region of the hard X-ray source lie within smaller radii than the iron-line reprocessing region. Analyzing the X-ray variability of QSO 2237+0305A, [@Dai03] also measured amplification of the Fe K$\alpha$ line in component A of QSO 2237+0305 but not in the continuum. However, in this case the interpretation was different. [@Dai03] suggested that the larger size of the continuum emission region ($\sim 10^{14}$ cm $\sim$ 100 R$_g$ for M$_{BH}=10^7M_\odot$) with respect to the Fe K$\alpha$ emission region ($\sim$ 10 R$_g$) could explain this result. Finally, in H 1413+117 [@Chart04] found that the continuum and the Fe K$\alpha$ line were enhanced by a different factor. With the aim of discussing these results, we will model here the behavior of the X-ray continuum and the Fe K$\alpha$ line during a microlensing event for different sizes of the continuum and the Fe K$\alpha$ line emission regions. Microlensing of a compact accretion disk ======================================== The model --------- The assumption of a disk geometry for the distribution of the Fe K$\alpha$ emitters is supported by the spectral shape of this line in AGN (e.g. [@Nan97], where they have investigated the iron line properties of 18 Sy 1 galaxies). Regarding the X-ray continuum emission, it seems that it mainly arises from an accretion disk. For instance, [@Fab03] have shown that the X-ray spectral variability of MCG-6-30-15 can be modeled by a two-component model where the one varying component is a power-law and the other constant component is produced by very strong reflection from a relativistic disk. To study the effects of microlensing on a compact accretion disk we will use the ray tracing method considering only those photon trajectories that reach the sky plane at a given observer’s angle $\theta_{\rm obs}$ (see e.g. [@Pop03] and references therein). The amplified brightness with amplification $A(X,Y)$ for the continuum is given by $$I_{C} (X,Y;E_{obs}) = { {I_{P}}} (E_{obs},T(X,Y)) \cdot A(X,Y),$$ and for the Fe K$\alpha$ line by $$\begin{array}{ll} I_{L} (X,Y;E_{obs}) = & { {I_{P}}} (E_{0} \cdot g(X,Y),T(X,Y)) \cdot \\ &\delta (E_{obs}-E_{0} \cdot g(X,Y)) \cdot A(X,Y), \nonumber \end{array}$$ where $T(X,Y)$ is the temperature, $X$ and $Y$ are the impact parameters which describe the apparent position of each point of the accretion disk image on the celestial sphere as seen by an observer at infinity; $E_{0}$ is the the line transition energy ($E_{0}^{\rm Fe\ K\alpha}=6.4$ keV) and [$g(X,Y)=E_{\rm obs}/E_{\rm em}$ is the energy shift due to relativistic effects ($E_{\rm obs}$ is the observed energy and $E_{\rm em}$ is the emitted energy from the disk). Here we will not consider the cosmological redshift.]{} The emissivity of the disk is one of the important parameters which has influence on the continuum and line shapes. The observed continuum flux is very often fitted with one or two black-body components in the soft X-ray, in addition to the hard X-ray power law (see e.g. Page et al. 2004). The line shape, as well as the continuum distribution, strongly depend on emissivity law. In the standard Shakura-Sunyaev disc model [@Shakura73], accretion occurs via an optically thick and geometrically thin disc. The effective optical depth in the disc is very high and photons are close to thermal equilibrium with electrons. The surface temperature is a function of disk parameters and results in the multicolor black body spectrum. This component is thought to explain the ’blue bump’ in AGN and the soft X-ray emission in galactic black holes. Although the standard model does not predict the power-law X-ray emission observed in all sub-Eddington accreting black holes, the power law for the X-ray emissivity in AGN is usually accepted (see e.g. Nandra et al. 1999). But one can not exclude other emissivity laws, such as black-body or modified black-body emissivity laws. Therefore, we will use here black-body, modified black-body and power emissivity laws for both; the Fe K$\alpha$ and continuum emission. In the case of the black-body radiation law, the disk emissivity is given as (e.g. Jaroszyński et al. 1992): $$I_P(X,Y;E)=B[E,T_s(X,Y)],$$ where $$B\left( {E,T_s(X,Y)} \right) = {\frac{{2E ^{3}}}{{h^2c^{2}}}}{\frac{{1}}{{e^{{{E } \mathord{\left/ {\vphantom {{h\nu} {kT}}} \right. \kern-\nulldelimiterspace} {kT_s(X,Y)}}} - 1}}},$$ where $c$ is the speed of light, $h$ is the Planck constant, $k$ is the Boltzmann constant and $T_s(X, Y)$ is the surface temperature of X-ray accretion disk. In principle, one can assume different distribution of the surface temperature along disk. To obtained the X-ray continuum distribution using Eq. (3) one can assume that $T_s=const.$, taking that the black hole is powerful X-ray sources with an effective temperature 10$^7$ to 10$^8$ K. But, regarding the standard disk model it is expected that the surface temperature at least is radially dependent. Therefore, here we will accept the radial distribution of surface temperature given by [@Shakura73]: $$T_s(X, Y) \sim r^{-3/2}(X,Y)(1-r^{-1/2}(X,Y))^{4/5} \,{\rm K},$$ taking that an effective temperature is in an interval from 10$^7$ to 10$^8$ K. The distribution of the temperature along the radius of the disk used in this paper is given in Fig. 1 (top) and corresponding shape of spectral energy distribution is shown in Fig. 1 (bottom). In Eq. (4) $r$ is the dimensionless parameter defined as: $$\begin{aligned} r(X,Y)=\frac{R(X,Y)}{6 R_g}= \frac{1}{6}\frac{R(X,Y)c^2}{GM}= \frac{M_\odot}{M}\frac{R(X,Y)}{9~{\rm km}},\nonumber\end{aligned}$$ where $R(X,Y)$ is disk radius, expressed in gravitational radii $R_g$. However, in the innermost part of the accretion disk the Planck function cannot be used properly. Therefore we will use also the standard (classical) Shakura – Sunyaev approach, where the emissivity law is described by a “modified” black-body radiation law (Eqs. (3.4), (3.8) in [@Shakura73]; see also the discussion in [@Novikov73; @Shapiro83; @Straumann84]) $$I_P(E;X,Y) \propto x^3 \exp(-x),$$ where $x=E/kT(X,Y)$. [@Sha02] used similar expressions to study microlensing in the optical continuum. Taking into account that the observed hard X-ray continuum has a power-law type spectral shape, we will also assume that the time-independent intrinsic emissivity of the continuum is: $$I(E,r)\sim E^{-\Gamma}\times r^{-\alpha},$$ where, according to the investigation of observed X-ray spectra, $\Gamma$ and $\alpha$ are taken to be 1.5 and 2.5 (see e.g. Dovčiak et al. 2004). For the Fe K$\alpha$ emission in this case we used the same calculation as in Popović et al. (2003a,b). We should note here, that disk may be considered to be composed of a number of distinct parts with different physical conditions (e.g. radiation pressure dominant part, matter pressure dominant part, etc. see e.g. Shakura & Sunyaev 1973). Consequently, in general one can expect that the disk can be described by different emissivity laws in different parts (e.g. the black-body law may be applied in outer part of the disk). Taking into account a huge number of parameters which should be considered in the case of microlensed disk (see the next section), we will consider only one emissivity law for whole disk. The total observed flux for the continuum and the Fe K$\alpha$ line is given as $$F(E)=\int_{\rm image} [I_C(X,Y;E)+I_L(X,Y;E)]d\Omega ,$$ where $d\Omega$ is the solid angle subtended by the disk in the observer’s sky and the integral extends over the whole emitting region. As one can see from Eq. (7) the total observed flux is a sum of the continuum and the line fluxes, consequently, the amplification in the continuum and in the Fe K$\alpha$ line can be considered separately as two independent components. On the other hand, the amplifications will depend on the sizes and geometry of the continuum and line emitting regions. In further text we will consider amplifications in the line and in the continuum separately. We would like to point out here that the aim of the paper is not to create the perfect accretion disk model (taking into account different effects that can be present as e.g. opacity of the disk, spots in the disk etc.), but only to illustrate the influence of microlensing on the continuum and the Fe K$\alpha$ line amplification and demonstrate that this phenomenon could essentially change general conclusions. Therefore, we will use the three emissivity laws of the disk and the very important effect of strong gravitation (beaming and light-bending in Schwarzschild and Kerr metrics). Disk and microlens parameters ----------------------------- To apply the model one needs to define a number of parameters that describe the emission region and the deflector. In principle, we should find constraints for the: size of the disk emission region, disk inclination angle, mass of the black hole, accretion rate, relative amplification, the constant $\beta$ (Chartas et al. 2002, Popović et al. 2003), orientation of the caustic with respect to the rotation axis, direction of the caustic crossing and microlens mass. In the following subsections we choose and discuss the parameters used in the calculations. ### Accretion disk parameters For the disk inclination we adopt the averaged values given by [@Nan97] from the study of the Fe K$\alpha$ line profiles of 18 Seyfert 1 galaxies: $i=35^\circ$. The inner radius, $R_{in}$, can not be smaller than the radius of the marginally stable orbit, $R_{ms}$, that corresponds to $R_{ms}=6R_g$ (gravitational radii, $R_g=GM/c^2$, where $G$ is gravitational constant, $M$ is the mass of central black hole, and $c$ is the velocity of light) in the Schwarzschild metric and to $R_{ms}=1.23R_g$ in the case of the Kerr metric with angular momentum parameter $a=0.998$. To select the outer radius, $R_{out}$, we take into account previous investigations of the X-ray variability that support very compact X-ray emitting disks. In particular, [@Osh02] from the observed variation in the lensed blazar PKS 1830-211 infer a size of the X-ray continuum emission region of $\sim 3\times 10^{14}\rm cm$, that is in agreement with estimation for QSO 2237+03050 given by [@Dai03]. So, considering a range of black hole masses of $10^7-10^9\ \rm M_\odot$ we can conclude that the X-ray emission is coming from a compact region of the order of 10 to 100 $R_g$. This range of sizes is also acceptable for the Fe K$\alpha$ emission region (see e.g. [@Nan97; @Nan99]). To explore the suitability of the various hypothesis explaining the lack of adequate response of the X-ray continuum to the microlensing events detected in the Fe K$\alpha$ line (see §1), we are going to consider several combinations of disk sizes for the emitters of both the continuum and the line: (i) the inner and outer radii of both emission regions are the same, $R_{in}=R_{ms}$ and $R_{out}=20\ R_g$; (ii) the inner radius is the same, $R_{in}=R_{ms}$, but the outer radius of the X-ray continuum disk is smaller, $R_{out}=20\ R_g$, than the radius of the line emission disk, $R_{out}=80\ R_g$; (iii) the continuum emission disk has radii $R_{in}=R_{ms}$, $R_{out}=20\ R_g$ and the line emission disk $R_{in}=20\ R_g$ and $R_{out}=80\ R_g$ (the continuum emission takes place in an inner part of disk surrounded by an annulus of Fe K$\alpha$ emission); (iv) the continuum emission disk has radii $R_{in}=20R_{g}$, $R_{out}=80\ R_g$ and the line emission disk $R_{in}= R_{ms}$ and $R_{out}=20\ R_g$ (the Fe K$\alpha$ emission is located in the inner disk and the continuum emission in the outer annulus). We adopt the central object mass and accretion rate from [@BZ02]. We assume a black hole of mass $M_8=10^8 M_\odot$ and accretion rate $\mathop m\limits^.= 0.4$ in Eddington units $({\mathop {m}\limits^{.}} = {\frac{{1.578{\mathop {M}\limits^{.}} _{26} }}{{3.88M_{8}}} })$. We will use this value in order to determine the effective temperature distribution. These values are in agreement with [@Wang03] where it was found that the majority of QSOs have black hole masses in the range of $10^8-10^9\ M_\odot$, and accretion rates ranging from 0.01 to 1 in units of the Eddington accretion rate. It is difficult to discuss the validity of different emissivity laws for demonstation of the X-ray emission (in the line as well as in the continuum), but sometimes, as for example in the case of black-body emissivity law, the emissivity at X-ray wavelengths can be extremely small compared with, for example, optical wavelengths, and X-ray photons are emitted from a quite small region. In Fig. 1 (bottom), we presented the continuum shapes for different emisivity laws used in the calculation (maximum of each is normalized to one). The shapes of the continuum were calculated for different dimensions of the disk. As one can see from Fig. 1 (bottom), the shape of the continuum strongly depends not only on emissivity law, but also on disk dimensions. ### Microlens model and parameters Different types of caustics can be used to explain the observed microlensing events in quasars. Moreover, for the exact event one can model the caustic shape to obtain different parameters (see e.g. Abajas et al. 2004, Kochanek 2004 for the case of Q2237+0305). In order to apply an appropriate microlens model, first we will consider a standard microlensing magnification pattern (Figure 2, left) for the Q2237+0305A image with 16 Einstein ring radii (ERR) on a side and $\kappa=0.36$, $\lambda=0.40$ and $\kappa_c=0$. The mass of microlens is taken to be 1$M_\odot$. The simulation was made employing ray-shooting techniques that send rays from the observer through the lens to the source plane (Kayser et al. 1986; Schneider & Weiss 1987; Wambsganss et al 1990a,b). We assume a flat cosmological model with $\Omega =0.3$ and $H_{o}= 70\ \rm km\ s^{-1} Mpc^{-1}$. In Figure 2 we presented a comparison between the projected magnification map in the source plane and an accretion disk with a size of 1000 R$_g$ (presented as a circle in Figure 2, right). Taking into account the small dimensions of the X-ray emission region (several 10s R$_g$) the approximation of a straight fold caustic can be assumed for this pattern. We explored the general behavior of the total continuum and Fe K$\alpha$ flux amplification due to microlensing. First we used the straight fold caustic approximation. (see Eqs. (5)-(8) in [@Pop03]). We also considered an example assuming a caustic magnification pattern for the Q2237+0305A image produced by a population of low mass microlens in §3.3. In Table 1 we give the projected Einstein Ring Radii (ERR) for the lensed QSOs where amplification of the Fe K$\alpha$ line (MG J0414+0534, [@Chart02a]; QSO H1413+117, [@Osh01; @Chart04] and QSO 2237+0305, [@Dai03]) has been observed. The ERRs (expressed in gravitational radii) are computed for different deflector masses and for a black hole mass of $10^8\ \rm M_\odot$. We found that even deflectors with small mass have ERR with sizes from several tens to several hundreds of gravitational radii. To obtain a qualitative understanding of the influence of the microlens mass on our results we consider microlens masses that correspond to ERR values equal to 50 R$_g$ (see Figs. 3-7). Qualitatively the shape of the flux amplification will not be changed if we consider massive deflectors (e.g. ERR=2000 $R_g$, see Fig. 8). Even if we apply simple assumption of the straight-fold caustic, we should define the caustic parameters A$_0$ and $\beta$. These values can be considered to be different for different microlensing events. Higher values of A$_0$ and $\beta$ will cause higher amplification. Here we would like to demonstrate correlation between the line and flux amplification due to microlensing and we will adopt values considered in more details by Chartas et al. (2002) $A_0$=1 and $\beta$=1 (Witt et al. 1993). Here we considered three directions of caustic crossing; parallel (Y=0) and perpendicular (X=0) to the rotation axis as well as in a direction inclined 45$^\circ$ with respect to the rotation axis (X=Y). Results and Discussion ====================== Continuum and line profile variability -------------------------------------- In Figures 3 and 4 we present the variations of the total X-ray emission spectra (continuum + Fe K$\alpha$ line) during a straight fold caustic crossing ($A_0$=1, $\beta$=1 and ERR=50 $R_g$). The radial dependence of the emissivity is related to the black body radiation law (see §2). In Figure 3 the sizes of the continuum and line emission regions are the same, $R_{\rm inn}=R_{\rm ms}$ and $R_{\rm out}=20\ R_g$. In Figure 4, the Fe K$\alpha$ line emitting disk is larger: $R_{\rm out}=80\ R_g$. We consider both metrics, Schwarzschild and Kerr. We simulate the caustic crossing perpendicular to (first and second rows) and along (third and fourth rows) the rotation axis in both directions; $\kappa=\pm 1$, respectively. Integrated flux variability --------------------------- In Figures 5-8 we present the variation of the integrated flux (normalized to the integrated flux in the absence of microlensing) for both, the X-ray continuum and the Fe K$\alpha$ line during straight fold caustic crossings (considering that the amplification outside the caustic $A_0$=1, as well as $\beta$=1 and ERR=50 $R_g$, see Figure captions). We considered all three emissivity laws (see §2.1) Figures 5a,b correspond to case (ii) of §2.2.1, Figures 5c,d and 7 (left) to case (iii) and Figures 6 and 7 (right) to case (iv). (Case (i) is the same for line and continuum; it corresponds to the continuum variation in Figures 5a,b). [ In Fig. 7, we present cases iii) and iv) where a power-law emission is taken into account.]{} In Figure 8 a very favorable case of high ERR and inclination is considered for case (iii) of §2.2.1. Notice that FeK$\alpha $ microlensing events were observed in BAL QSOs which may have highly inclined accretion disks; $i\approx 75^\circ$. [ As one can see from Figure 8 (as it also was shown in Popović et al. 2003a,b) the amplified component is mainly very narrow in comparison with the undeformed line. This result is in agreement with the observations of Chartas et al. (2002, 2004) and Dai (2003) and supports the conclusions of these authors that enhancement of the Fe K$\alpha$ line observed in only one of the images of a lensed quasar was caused by microlensing. ]{} Some interesting results can be inferred from our exploratory work: (a) when both the line and continuum disk profiles have the same inner radius, the differences in outer radius cannot cause significant differences in the total line and the continuum flux variation (see Figs. 5a,b) for the considered emission laws, (b) when we separate the emission considering an inner disk contributing to the continuum and an outer annulus contributing to the Fe K$\alpha$ line (or vice-versa) we found significant differences between the continuum and the line amplification (see Figures. 5c,d, 6 – 8), (c) the results are qualitatively similar for [all considered emissivity laws]{} (see Figs. 5-8), (d) interchanging between Schwarzschild and Kerr metrics induces only slight differences in normalized flux amplification (see Figs. 5-7). Figures 5-8 were intended to explore the two scenarios suggested by [@Dai03] and [@Chart02a] to explain the non observed associated enhancement of the X-ray continuum in objects with a microlensed Fe K$\alpha$ line. This behavior can be expected in case (iii) when the microlens crosses the outer part of the disk (Figures 5c,d and the right panel of Fig. 7) and in case (iv) when the microlens crosses the inner part of the disk (see Fig. 6 and the left panel of Fig. 7). However, in none of the Figures does the continuum remain strictly constant during a complete Fe K$\alpha$ microlensing event. In the most favorable case (the inner Fe K$\alpha$ disk plus an outer continuum annulus; Fig. 6 and 7) we achieve a significant and relatively quick change of the Fe K$\alpha$ emission while the continuum experiences only a slow increase. This behavior could well approximate a non-varying continuum but only if we consider observations in a temporal window that fall on the peak of the microlensing event in the Fe K$\alpha$ line. In this case the continuum of the microlensed image experiences an (slowly changing or almost constant) amplification with respect to the continua of the other images, but practically it is indistinguishable from the amplifications due to global macrolensing. Microlensing by a caustic magnification pattern: An example for Q2237+0305A image --------------------------------------------------------------------------------- Here we consider a situation where a low mass population of microlenses (smaller than one solar mass) can form pattern structures (see Table 1) which are comparable with the size of the X-ray accretion disk. Moreover, the black hole mass of the lensed quasar may be of the order of $10^{9-10}M_\odot$, taking that $R_g\sim M_{BH}$, the pattern structure of low mass microlenses are comparable with a X-ray disk size of several dozens R$_g$. Therefore, here we consider that the black hole mass of the lensed quasar is $10^{9}M_\odot$. For modeling of the caustic magnification pattern for image Q2237+0305A we used the same values for the convergence and external shear as presented in Figure 2, but for a low mass population, taking that the mass of the deflectors are randomly distributed in an interval ranging from 0.1 $M_\odot$ to 0.6 $M_\odot$, with a mean value of $<m>=0.35\ M_\odot$. Also, the positions of the lenses were distributed randomly in a rectangular region in the lens plane, significantly larger than the considered region in the source plane. Now, $1ERR$ projected in the source plane corresponds to $$ERR(M)= ERR(M_{\odot}) \sqrt{{\frac{M}{M_{\odot}}} \frac{M_{8}}{M_{BH}}},$$ where $ERR(M_{\odot})=0.054$pc is the projected ERR for a solar mass deflector, $M$ is the mean mass of the deflectors and $M_{BH}$ is the black-hole mass. Taking the mean deflector mass as $<m>=0.35\ M_\odot$ and $R_g=9.547.10^{-6} M_{BH}/M_{8}$ pc, we modeled a caustic magnification pattern of 1ERR$\times$2ERR, that corresponds to 334.63R$_g \times$669.26$R_g$ in the source plane for a black hole mass of $M_{BH}=10^{9}M_\odot$ (Fig. 9). For numerical reasons, the microlens magnification map is given in pixels, 1000$\times$2000 (1pix=0.33463R$_g$ in source plane). As one can see from Figure 9, the microlensing pattern structures are comparable with a compact X-ray accretion disk. In our previous modeling based on the straight fold caustic approximation the lack of a correlation between the continuum and Fe K$\alpha$ line is expected only if the line and X-ray continuum region are separated. Recent investigations of the Fe K$\alpha$ line profile from active galaxies show that the line should be emitted from the innermost part of the accretion disk. In particular, Ballantyne & Fabian (2005) found that in BLRG 4C+74.26 the outer radius of the relativistic iron line should be within 10 R$_g$. Consequently, here we will assume that the Fe K$\alpha$ line is formed in the innermost part of the disk ($R_{\rm inn}=R_{\rm ms}$; $R_{\rm out}=20$ R$_{g}$) and that the continuum (emitted in the energy range between 0.1 keV and 10 keV) is mainly originated from a larger region ($R_{\rm inn}=20$ R$_{g}$; $R_{\rm out}=100$ R$_{g}$).[^2] On the other hand, from the straight fold caustic modeling we conclude that the correlation between the total line and continuum flux due to microlensing is not very different for different emissivity laws. Consequently, here we used the black-body emissivity law. A disk (Schwarzschild metric) with an inclination of 35$^{\circ}$ is considered. To explore the line and X-ray continuum variation we moved the disk center along the microlensing map as it is shown in Figure 9 (from left to the right corresponding 0 to 2000 pixels). In Figure 10 we present the corresponding total line and X-ray continuum flux variation. As one can see from Figure 10, there is a global correlation between the total line and continuum flux during the complete path. However, the total continuum flux variation is smooth and has a monotonic change, while the total line flux varies very strongly and randomly. In fact, during some portion of the microlensing of the emission regions by the magnification pattern, we found the total Fe K$\alpha$ line flux changes, while the continuum flux remains nearly constant (e.g. the position of the disk center between 1000 and 1200 pixels). This and the shapes of the line and continuum total flux amplification indicate that the observed microlensing amplification of the Fe K$\alpha$ in three lensed quasars may be explained if the line is originated in the innermost part of the disk and the X-ray continuum in a larger region. Also, it seems that the contribution of the continuum emitted from the innermost part of the disk (within 10 R$_g$) to the total continuum (in the energy interval from 0.1 to 10 keV) flux is not significant. Further observations are needed to provide more data which might be compared with our theoretical results. Wavelength dependence of the X-ray continuum amplification ---------------------------------------------------------- The influence of gravitational microlensing on the spectra of lensed QSOs was discussed in several papers (see Popović and Chartas 2005, and references therein). Mainly, the color index was calculated as an indicator of the microlensing (see e.g. Wambsganss & Paczynski 1991, Wyithe et al. 2000) as well as amplified flux behaviors (see Yonehara et al. 1998, Yonehara et al. 1999, Takahashi et al. 2001) of a disk, but the exact shape of the amplification as a function of wavelength (or energy) for a partly microlensed disk has not been calculated. Here, taking into account that the emitters at different radii in the accretion disk have different temperatures (see Fig. 1) and make different contributions to the observed continuum flux at a given wavelength, we calculated the amplification as a function of observed energies. During a caustic crossing microlensing effects would depend on the location of the emitters and, consequently, would induce a wavelength dependence in the amplification. This dependence can be clearly appreciated in the spectra of Figures 3, 4 and 8. In Figures 11ab-12ab we present the amplification as a function of the observed energies for an accretion disk with the characteristics given in §2.2.1, with inner radius $R_{in}=R_{ms}$ and outer radius $R_{out}=30\ R_{g}$, assuming caustic crossing along the rotation axis (X). We have considered the black body (Fig. 11ab) and the modified black body (Fig. 12ab) emissivity laws for both Schwarzschild and Kerr metrics. As it can be seen in Figures 11-12 the amplification is different for different observed energies. The amplification is higher for the hard X-ray continuum when the caustic crosses the central part of the disk (see Fig. 12). Depending mainly on the caustic location and on the emissivity law selected the difference of the amplification in the energy range studied by us can be significant (e.g. $\sim 20$% for very small mass microlenses, ERR=50$R_g$, see Figs. 11-12). This effect could induce a noticeable wavelength dependent variability of the X-ray continuum spectrum during a microlensing event (of even a 30%), providing a tool to study the innermost regions of accretion disks. Conclusions =========== We have developed a model of microlensing by a straight fold caustic of a standard accretion disk in order to discuss the observed enhancement of the Fe K$\alpha$ line in the absence of corresponding continuum amplification found in three lensed QSOs. Here we summarize several interesting results inferred from our straight fold caustic simulations. 1 - As expected both the Fe K$\alpha$ and the continuum may experience significant amplification by a microlensing event (even for microlenses of very small mass). Thus, the absence of adequate continuum amplification in the observed Fe K$\alpha$ microlensed QSOs should be related to the structure of the accretion disk and/or the geometry of the event. 2 - Extending the outer radius of the distribution of Fe K$\alpha$ emitters does not result in any significant changes in our results. This is due to the radial dependence of the emissivity as expressed in the standard accretion disk model [@Shakura73] that concentrates the emission near the center of the black hole making negligible the contribution from the outer parts to the integrated flux. In principle we could consider other less steep emissivity laws that make the outer parts of the disk more important, however previous studies (e.g. [@Nan99]) support the hypothesis of a strong emissivity gradient. 3 - Segregation of the emitters allows us to reproduce the Fe K$\alpha$ enhancement without equivalent continuum amplification if the continuum emission region lies interior to the Fe K$\alpha$ emission region or vice versa but only during limited time intervals. In fact, in none of the simulations does the continuum remain constant during a complete Fe K$\alpha$ microlensing event. In the case of an inner Fe K$\alpha$ disk plus an outer continuum annulus a significant change of the Fe K$\alpha$ emission is obtained while the continuum experiences only a very shallow gradient. This behavior could well approximate the detected non-varying continuum for observations covering only the peak of the Fe K$\alpha$ microlensing event. 4 - We have also studied a more realistic case of microlensing by a caustic magnification pattern assuming a population of low mass deflectors. In this case we can successfully reproduce the observed lack of correlation between the X-ray continuum and Fe K$\alpha$ emission amplification only if the line and continuum emission regions are separated. An extreme case of segregation is that of the two component model for the continuum suggested in several papers (see e.g. [@Fab03; @Pag04] if the contribution of the disk component to the X-ray continuum were weak enough. 5 - We have studied the chromatic effects of microlensing in the X-ray continuum and find that the dependence with wavelength of the amplification can induce, even for small mass micro-deflectors (ERR=50$R_g$), chromatic variability of about 30% in the observed energy range (from 0.1 keV to 10 keV) during a microlensing event. Further observations of the lensed quasars in the X-ray are needed to confirm these results. In any case monitoring of gravitational lenses may help us to understand the physics of the innermost part of the relativistic accretion disks. This work is a part of the projects: P1196 “Astrophysical Spectroscopy of Extragalactic Objects” supported by the Ministry of Science, Technologies and Development of Serbia and P6/88 “Relativistic and Theoretical Astrophysics” supported by the IAC. LČP was supported by Alexander von Humboldt Foundation through the program for foreign scholars. JAM is a [*Ramón y Cajal Fellow*]{} from the MCyT of Spain. Also, we would like to thank the anonymous referee for very useful comments. Abajas, C., Mediavilla, E.G., Muñoz, J.A., Popović, L. Č., & Oscoz A. 2002, ApJ 576, 640. Abajas, C., Mediavilla, E.G., Gil-Merino, R., Muñoz, J.A., Popović, L. Č., & Oscoz A. 2004, presented in IAU 225 Symposium “Impact of Gravitational Lensing on Cosmology”, 19-23 July, Lausanne, Switzerland. Ballantyne D.R., Fabian, A.C. 2005, ApJ, 622, L97. Bian, W., Zhao, Y. 2002, A[&]{}A, 395, 465. Chartas, G., Agol, E., Eracleous, M., Garmire, G., Bautz, M. W., Morgan, N. D. 2002, ApJ, 568, 509. Chartas, G., Eracleous, M., Agol, E., Gallagher, S. C. 2004, ApJ, 606, 78. Dai, X., Chartas, G., Agol, E., Bautz, M. W., & Garmire, G.P. 2003, ApJ, 589, 100. Dai, X., Chartas, G., Eracleous, M. & Garmire, G.P. 2004, ApJ, 605, 45. Dovčiak, M., Karas, V., Yaqoob, T. 2004, ApJS, 153, 205. Fabian, A. 2001, in Proc. of the 20th Texas Symposium on Relativistic Astrophysics, ed. J. C. Wheeler, H. Martel, AIP Conferences, [ 586]{}, (Americal Institute of Physics, Melville, New York) 643. Fabian, A.C., Vaughan S. 2003, MNRAS, 340, L28. Jaroszyński, M. 2002, Acta Astronomica, 52, 203. Jaroszyński, M., Wambsganss, J.W., Paczyński, B. 1992, ApJ, 396, L65. Kayser, R., Refsdal, S., & Stabell, R. 1986, A[&]{}A, 166, 36. Kochanek, C.S. 2004, ApJ, 605, 58. Lewis, G. F., Ibata, R.A. 2004, MNRAS, 348, 24. Mineshige, S., Yonehara, A., & Takahashi, R. 2001, PASA, 18, 186. Nandra K., George I.M., Mushotzky R.F., Turner T.J. & Yaqoob T. 1997, ApJ. 477, 602. Nandra, K., George, I. M., Mushotzky, R. F., Turner, T. J., Yaqoob, T. 1999, ApJ 523, 17. Novikov, I.D., & Thorne, K.S. 1973, Black Holes (Eds. C. De Witt, B. De Witt), Gordon & Breach, 344. Oshima, T., Mitsuda, K., Fujimoto R., Iyomoto N., Futamoto K., et al. 2001a, ApJ, 563, L103. Oshima, T., Mitsuda, K., Ota, N., Yonehara, A., Hattori, M., Mihara, T. & Sekimoto, Y. 2001b, ApJ, 551, 929. Page, K.L., Reeves, J.N., O’Brien, P.T., Turner, M.J.L., Worrall, D.M. 2004, MNRAS, 353, 133. Popović, L. Č., Chartas, G. 2005, MNRAS, 357, 135. Popovi[ć]{}, L., Č., Mediavilla, E.G., Muñoz J., Dimitrijević, M.S., & Jovanović, P. 2001a, SerAJ, 164, 73. Popovi[ć]{}, L.Č., Mediavilla, E.G., & Muñoz J. 2001b, A[&]{}A 378, 295. Popović, L.Č., Mediavilla, E.G., Jovanović, P., & Muñoz, J.A. 2003a, A[&]{}A, 398, 975. Popović, L.Č., Jovanović, P., Mediavilla, E.G., & Muñoz, J.A. 2003b, Astron. Astrophys. Transactions, 22, 719. Richards, G.T., Keeton, C.R., Pindor, B. et al. 2004, ApJ, 610, 679. Schneider, P., & Weiss, A. 1987, A[&]{}A, 171, 49. Shakura, N.I., & Sunyaev, R.A. 1973, A[&]{}A, 24, 337. Shalyapin, V., N., Goicoechea, L., J., Alcalde, D., Mediavilla E., Muñoz J.A., Gil-Merino, R. 2002, ApJ, 579, 127. Shapiro, S.L., & Teukolsky, S.A. 1983, Black Holes, White Dwarfs and Neutron Stars: Physics of Compact Objects, John Wiley & Sons, New York. Straumann, N. 1984, General Relativity and Relativistic Astrophysics, Springer-Verlag, Berlin, Heidelberg, New York. Takahashi, R., Yonehara, A., Mineshige, S. 2001, Publ. Astron. Soc. Japan 53, 387. Wang, J.-M., Ho, L. C.; Staubert, R. 2003, A[&]{}A, 409, 887. Wambsganss, J., Paczynski, B. 1991, AJ, 102, 86. Wambsganss, J., Paczynski, B., & Katz, N. 1990a, ApJ, 352, 407. Wambsganss, J., Schneider, P., & Paczynski, B. 1990b, ApJ, 358, L33. Witt H.J., Kayser R., Refsdal S., 1993, A[&]{}A 268, 501. Wyithe, J. S. B., Webster, R. L., Turner, E. L., Agol, E. 2000, MNRAS, 318, 1105. Yonehara, A., Mineshige, S., Fukue, J., Umemura, M., Turner, E.L. 1999, A[&]{}A, 343, 41. Yonehara, A., Mineshige, S., Manmoto, T., Fukue, J., Umemura, M., Turner, E.L. 1998, ApJ, 501, L41; ApJ, 511, L65. Zakharov, A.F., Kardashev, N.S., Lukash, V.N., & Repin, S.V. 2003, MNRAS, 342, 1325. Zakharov, A.F., & Repin, S.V. 2002a, Astronomy Reports, 46, 360. Zakharov, A.F., & Repin, S.V. 2002b, in Proc. of the Eleven Workshop on General Relativity and Gravitation in Japan, ed. J. Koga, T. Nakamura, K. Maeda, K. Tomita, (Waseda University, Tokyo) 68. Zakharov, A.F., & Repin, S.V. 2002c, in Proc. of the Workshop “XEUS - studying the evolution of the hot Universe”, ed. G. Hasinger, Th. Boller, A.N. Parmar, MPE Report 281, 339. Zakharov, A.F., & Repin, S.V. 2003a, Nuovo Cim., 118B, 1193. Zakharov, A.F., & Repin, S.V. 2003b, A[&]{}A, 406,7. Zakharov, A.F., Popović, L. Č., Jovanović, P. : 2004, A[&]{}A, 420, 881. {width="16cm"}  [|c|c|c|c|c|c|c|c|]{} Object & $z_s$ & $z_l$ & $1\times 10^{-4}M_\odot$ &$1\times 10^{-3}M_\odot$ &$1\times 10^{-2}M_\odot$ &$1\times 10^{-1}M_\odot$ & $1 M_\odot$\ MG J0414+0534& 2.64 & 0.96 & 20.3 & 64.2 & 203.1 & 642.3 & 2031.1\ QSO 2237+0305& 1.69 & 0.04 & 11.2 & 35.4 & 112.1 & 354.5 & 1121.0\ QSO H1413+117& 2.56 & 1.00 & 19.8 & 62.5 & 197.7 & 625.2 & 1977.0\ [^1]: Simulations of X-ray line profiles are presented in a number of papers, see, for example, [@Fabian01; @Zak_rep02; @Zak_rep02a; @Zak_rep02_xeus] and references therein, in particular [@ZKLR02] showed that information about the magnetic filed may be extracted from X-ray line-shape analysis; [@Zak_rep03a_AA; @Zak_rep03b_AA] discussed signatures of X-ray line-shapes for highly inclined accretion disks. [^2]: Note here, taking the continuum disk size from $R_{\rm inn}=20$ R$_{g}$ to $R_{\rm out}=100$ R$_{g}$, we neglected the contribution of the innermost part emission (from R$_{ms}$ to 20 R$_g$) to the total continuum flux only in the energy interval from 0.1 keV to 10 keV. It does not mean that there is no the continuum emission.

--- abstract: 'The nature of statistics, statistical mechanics and consequently the thermodynamics of stochastic systems is largely determined by how the number of states $W(N)$ depends on the size $N$ of the system. Here we propose a scaling expansion of the phasespace volume $W(N)$ of a stochastic system. The corresponding expansion coefficients (exponents) define the universality class the system belongs to. Systems within the same universality class share the same statistics and thermodynamics. For sub-exponentially growing systems such expansions have been shown to exist. By using the scaling expansion this classification can be extended to [*all*]{} stochastic systems, including correlated, constraint and super-exponential systems. The extensive entropy of these systems can be easily expressed in terms of these scaling exponents. Systems with super-exponential phasespace growth contain important systems, such as magnetic coins that combine combinatorial and structural statistics. We discuss other applications in the statistics of networks, aging, and cascading random walks.' address: - 'Section for Science of Complex Systems, CeMSIIS, Medical University of Vienna, Spitalgasse 23, 1090 Vienna, Austria' - 'Complexity Science Hub Vienna, Josefstädter Strasse 39, 1080 Vienna, Austria' - 'Section for Science of Complex Systems, CeMSIIS, Medical University of Vienna, Spitalgasse 23, 1090 Vienna, Austria' - 'Complexity Science Hub Vienna, Josefstädter Strasse 39, 1080 Vienna, Austria' - 'Section for Science of Complex Systems, CeMSIIS, Medical University of Vienna, Spitalgasse 23, 1090 Vienna, Austria' - 'Complexity Science Hub Vienna, Josefstädter Strasse 39, 1080 Vienna, Austria' - 'Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA' - 'IIASA, Schlossplatz 1, 2361 Laxenburg, Austria' author: - Jan Korbel - Rudolf Hanel - Stefan Thurner bibliography: - 'references.bib' title: 'Classification of complex systems by their sample-space scaling exponents' --- Introduction ============ Classical statistical physics typically deals with large systems composed of weakly interacting components, which can be decomposed into (practically) independent sub-systems. The phasespace volume $W$ or the number of states of such systems grows exponentially with system size $N$. For example, the number of configurations in a spin system of $N$ independent spins is $W(N) = 2^N$. For more complicated systems, however, where particles interact strongly, which are path-dependent, or whose configurations become constrained, exponential phasespace growth no-longer occurs, and things become more interesting. For example, in black holes the accessible number of states does not scale with the volume but with surface, which leads to non-standard entropies and thermodynamics [@Beckenstein1974; @Hawking1974; @Thirring1970]. A version of entropy that depends on the surface and the volume was recently suggested in [@biro]. Other examples include systems with interactions on networks, path-dependent processes, co-evolving systems, and many driven non-equilibrium systems. These systems are often non-ergodic and are referred to as [*complex systems*]{}. For these systems, in general, the classical statistical description based on Boltzmann-Gibbs statistical mechanics fails to make correct predictions with respect of the thermodynamic, the information theoretic, or the maximum entropy related aspects [@thurner_corominas_hanel17]. Often the underlying statistics is then dominated by fat-tailed distributions, and power-laws in particular. There have been considerable efforts to understand the origin of power-law statistics in complex systems. Some progress was made for systems with sub-exponentially growing phasespace. It was shown that systems whose phasespace grow as power laws, $W(N) \sim N^b$, are tightly related to so-called Tsallis statistics [@Sato2005]. The tremendous variety and richness of complex systems has led to the question whether it is possible to classify them in terms of their statistical behavior. Given such a classification, is it possible to arrive at a generalized concept of the statistical physics of complex systems, or do we have to establish the statistical physics framework for every particular system independently? For sub-exponentially growing systems such a classification was attempted by characterizing stochastic systems in terms of two scaling exponents of their extensive entropy [@hanel-thurner11a]. The first scaling exponent is recovered from the relation $\frac{S(\lambda W)}{S(W)} \sim \lambda ^c$, which is valid if the first three Shannon-Khinchin axioms (see supplementary material) are valid (the fourth, the composition axiom, can be violated), and if the entropy is of so-called trace form, which means that it can be expressed as $S=\sum_i^W g(p_i)$, where $p_i$ is the probability for state $i$, and $g$ some function. The second scaling exponent $d$ is obtained from a scaling relation that involves the re-scaling of the number of states $W \to W^a$. With these two scaling exponents $c$ and $d$ it becomes possible to classify sub-exponentially growing systems that fulfil the first three Shannon-Khinchin axioms [@hanel-thurner11a]. Further, the exponents $c$ and $d$ characterize the extensive entropy, $S_{c,d} \sim \sum \Gamma(1+d, c\log(p_i))$. Practically all entropies that were suggested within the past three decades, are special cases of this $(c,d)$-entropy, including Boltzmann-Gibbs-Shannon entropy ($c=1$, $d=1$), Tsallis entropy ($d=0$), Kaniadakis entropy ($c=1$, $d=1$) [@kaniadakis02], Anteonodo-Plastino entropy ($c=1$, $d>0$) [@Anteneodo1999], and all others that fulfil the first three Shannon-Khinchin axioms. In [@hanel-thurner11b] it was then shown that the exponents $c$ and $d$ are tightly related with phasespace growth of the underlaying systems. In fact, they can be derived from the knowledge of $W(N)$, ${1}/{(1-c)}= \lim_{N \to \infty} N W' / W$, and $d= \lim_{N \to \infty} \log W \left( W / (NW') +c-1 \right)$. For super-exponential systems such a classification is hitherto missing. These systems include important examples of stochastic complex systems that form new states as a result of the interactions of elements. These are systems that–besides their combinatorial number of states (e.g. exponential)–form additional states that emerge as [*structures*]{} from the components. The total number of states then grows super-exponentially with respect to system size, e.g. the number of elements. Stochastic systems with elements that can occupy several states (more than one) and that can form structures with other elements, are generally super-exponential systems. [It]{} was pointed out in [@Jensen16] that such systems might exhibit non-trivial thermodynamical properties. An example for such systems are magnetic coins of the following kind. Imagine a set of $N$ coins that come in two states, up and down. There are $2^N$ states. However, these coins are “magnetic”, and any two of them can stick to each other, forming a new bond state (neither up nor down). If there are $N=2$ coins, there are five states: the usual four states, uu, ud, du, dd, and a fifth state ‘bond’. If there are $N=3$ coins, there are 14 states, the $2^3$ combinatorial states, and six states involving bond states: state 9 is bond between coin 1 and 2, with the third coin up, state 10 is the same bond state with the third coin down, state 11 is a bond between 1 and 3 with the second con up, 12 the same bond with the second coin down, state 12 is a bond between 2 and 3, with the first state up, and finally, state 14 is the bond between 2 and 3 with the first coin down. It can be easily shown that the recursive formula for the number of states is, $W_{}(N+1) = 2 W_{}(N) + N W_{}(N-1)$, which, for large $N$, grows as $W_{}(N) \sim N^{N/2} e^{2 \sqrt{N}}$, see [@Jensen16]. In this paper we show that it is indeed possible to find a complete classification of complex stochastic systems, including the super-exponential case. By expanding a generic phasespace volume $W_{}(N)$ in a Poincaré expansion, we will see that for any possibility of phase space growth, there exists a sequence of unique expansion coefficients that are nothing but scaling exponents that describe systems in their large size limit. The [*set*]{} of scaling exponents gives us the full classification of complex systems in the sense that two systems belong to the same universality class, if it is possible to rescale one into the other with exactly these exponents. The framework presented here has been proposed in [@Hanel_thurner13] and generalizes the classification approach of [@hanel-thurner11a; @hanel-thurner11b]. It includes the sub-exponential systems as a special case. We show further that these exponents can be used straight forwardly to express–with a few additional requirements–the corresponding extensive entropy, which is the basis for the thermodynamic properties of the system. Finally, we see in several examples that many systems are fully characterized by a very few exponents. Technical details and auxiliary results are presented in the supplementary material. We reference to the supplementary material in the corresponding parts of the main text. However, readers may also go through the supplementary material before they continue reading. We use the following notation for applying a function $f$ for $n$ times, $f^{(n)}(x) = \underbrace{f(\dots(f(x))\dots)}_{n \ times}$. Rescaling phasespace ==================== Suppose that phasespace volume depends on system size $N$ (e.g. number of elements) as $W(N)$. We use the Poincaré asymptotic expansion for the $l+1$ th logarithm of $W$, $$\log^{(l+1)} W(N) = \sum_{j=0}^{n} c_j \phi_{j}(N) + {\cal O} (\phi_n(N)) \quad , \label{Poin}$$ where $\phi_{j}(N) = \log^{({j}+1)}(N)$ for $N \rightarrow \infty$. A uniqueness theorem (see e.g. [@copson]) states that the asymptotic expansion exists and is uniquely determined for any $W(N)$ for which $\log^{(l+1)} W(N) = {\cal O} (\phi_0(N))$, see supplementary material. To see how the exponents $c_j$ correspond to scaling exponents, let us define a sequence of re-scaling operations, $$r^{(n)}_\lambda(x) = \exp^{(n)}[\lambda \log^{(n)}(x)]\quad .$$ For example $r^{(0)}_\lambda(x) = \lambda x$, $r^{(1)}_\lambda(x) = x^\lambda$, etc. Obviously, $r^{(n)}_1(x) = x$. The scaling operations obey the composition rule $$r^{(n)}_\lambda[r^{(n)}_{\lambda'}(x)] = r^{(n)}_{\lambda \lambda'}(x) \quad . \label{eq:resc}$$ We can now investigate the scaling behavior of the phasespace volume in the thermodynamic limit, $N \gg 1$. The leading order of the scaling is given by the first rescaling $r_0$. We show in the supplementary material that the rescaling of phasespace is asymptotically described by $$\label{eq:scaling} W(r^{(0)}_\lambda(N)) \sim r^{(l)}_{\lambda^{c^{(l)}_{0}}}(W(N)) \ \Rightarrow \ \frac{\log^{(l)} W(\lambda N)}{\log^{(l)} W(N)} \sim \lambda^{c^{(l)}_{0}} \quad ,$$ where $c^{(l)}_{0} \in \mathds{R}$ is the leading exponent, and $l$ is determined from the condition that $c_0^{(l)}$ should be finite. Thus, to leading order, the sample space grows as $W(N) \sim \exp^{(l)}\left(N^{c^{(l)}_{0}}\right)$. We now identify the scaling laws for the sub-leading corrections through higher-order rescalings $W(r^{(k)}_\lambda(N))$. We get (see supplementary material) $$\label{eq:sub} \frac{\log^{(l)} W(r^{(k)}_\lambda(N))}{\log^{(l)} W(N)} \prod_{j=0}^{k-1} \left(\frac{\log^{(j)}(r^{(k)}_\lambda(N))}{\log^{(j)}(N)}\right)^{- c^{(l)}_{j}} \sim \lambda^{c^{(l)}_{k}} \quad .$$ Equivalently, one can express this relation as, $W(r^{(k)}_\lambda(N)) \sim r^{(l)}_{\sigma_k(N)}(W(N))$, where $\sigma_k(N) = \prod_{j=0}^{k} \left(\frac{\log^{(j)}(r^{(k)}_\lambda(N))}{\log^{(j)}(N)}\right)^{c^{(l)}_{j}}$. To extract $c^{(l)}_{j}$, take the derivative of Eq. (\[eq:scaling\]) w.r.t. $\lambda$, set $\lambda=1$ and [consider]{} the limit $N \rightarrow \infty$. For the leading scaling exponent we obtain $$c^{(l)}_{0} = \lim_{N \rightarrow \infty} \frac{N W'(N)}{\prod_{i=0}^l \log^{(i)}W(N)} \quad . \label{depp}$$ The scaling exponent corresponding to the $k$-th order is obtained in a similar way and reads, $$\label{eq:clk} c^{(l)}_{k} = \lim_{N \rightarrow \infty} \log^{(k)}(N) \left( \log^{(k-1)}(N) \left( \dots \left( \log(N) \left(\frac{N W'(N)}{\prod_{i=0}^l \log^{(i)}W(N)}-c^{(l)}_{0}\right) - c^{(l)}_{1}\right)\dots \right) -c^{(l)}_{(k-1)}\right)$$ This expression is not identically [equal]{} to zero, because the expression on the r.h.s. of Eq. (\[depp\]) becomes $c^{(l)}_{0}$ only in the limit. As a result, the phasespace volume grows as $$W(N) \sim \exp^{(l)}\left[\prod_{j=0}^{n} \left(\log^{(j)}(N)\right)^{c^{(l)}_{j}}\right] \quad,$$ which is nothing but the Poincaré asymptotic expansion in Eq. (\[Poin\]). In the supplementary material we show that the formulas for $c_j$, given by the theory of asymptotic expansions, correspond to the formulas for scaling exponents $c_j^{(l)}$ and therefore it is indeed possible to express [*any*]{} $W(N)$ in terms of an asymptotic expansion that is based on the sequence $\phi_n(N)$. The expansion coefficients are scaling exponents determined by the rescaling of phasespace. [Here $n$ denotes the minimal number of expansion terms. In the typical situations, only a few scaling exponents are non-zero. If all exponents are non-zero, we can truncate the expansion after a few terms and still preserve a high level of precision. In many realistic situations it is enough to consider $n=2$.]{} The estimation of the leading order exponent can be tricky, because looking for the order $l$ incorporates calculation of several infinite limits. Therefore, it is convenient to use an approach based on the corresponding extensive entropy. The extensive entropy ===================== The extensive entropy can be obtained by following an idea exposed in [@hanel-thurner11a; @hanel-thurner11b]. Let’s assume a so-called [trace-form]{} entropy for some probability distribution $P = (p_1,\dots,p_W)$ $$\label{eq:trace} S_g(p) = \sum_{i=1}^W g(p_i) \quad ,$$ where $g$ is some function. The aim is to find such a function $g$, for which the entropy functional $S_g$ is [*extensive*]{} for a given $W(N)$. Assuming that no prior information about the system is given, we consider uniform probabilities $p_i = 1/W$. The extensivity condition can be expressed by an equation for $g$, which is [@hanel-thurner11b] $$\label{eq:ext} S_{g}(W(N)) = W(N)\, g(1/W(N)) \sim N \quad \mathrm{for} \ N \gg 1 \quad .$$ [Alternatively, it is possible to define the extensive entropy as the solution of Euler’s differential equation, see also [@biro], $$N\, \frac{\mathrm{d} S(W(N))}{\mathrm{d}N} = S(W(N)).$$]{} The question now is, how the scaling exponents of $W(N)$ are related to scaling exponents of $S_g(W)$. We begin with the first scaling operation $r^{(0)}$. One can show that for $N \gg 1$, we have $$\label{eq:ent} S_g(r_\lambda^{(0)}(W)) \sim r^{(0)}_{\lambda^{d_0}}(S_g(W)) \Rightarrow \lambda \frac{g\left(\frac{1}{\lambda W(N)}\right)}{g\left(\frac{1}{W(N)}\right)} \sim \lambda^{d_0} \quad .$$ Thus, $g(x) \sim (1/x)^{d_0-1}$ for $x \rightarrow 0$. Again, it is possible to determine the relation for the $n$ th scaling exponent $$\label{eq:entsub} \frac{g(1/r_\lambda^{(n)}(W))\, r_\lambda^{(n)}(W)}{g(1/W) \, W} \prod_{j=0}^{k-1} \left(\frac{\log^{(j)} (r_\lambda^{(n)}(W))}{\log^{(j)}(W)}\right)^{-d_j} \sim \lambda^{d_n} \quad ,$$ or equivalently, $S_g(r_\lambda^{(n)}(W)) \sim r_{\rho_n(W)}^{(0)}(S_g(W))$, where $\rho_n(W) = \prod_{j=0}^{n} \left(\frac{\log^{(j)} (\lambda^{(k)}(W))}{\log^{(j)}(W)}\right)^{d_j}$. We can extract the scaling exponents $d_n$ by the same procedure as for $c^{(l)}_{k}$ by taking the derivative w.r.t. $\lambda$, setting $\lambda=1$ and performing the limit. For the first exponent we get $$d_0 = \lim_{W \rightarrow \infty} \left(1 - \frac{g'(1/W)}{W g(1/W)}\right) \quad .$$ De L’Hospital’s rule and applying the extensivity condition of Eq. (\[eq:ext\]) gives $g'(W(N)) \sim N$, and $$d_0 = \lim_{N \rightarrow \infty} \frac{W(N)}{N W'(N)} \quad .$$ We mentioned this result already above. The $n$ th term can be found analogously to be $$\label{eq:dn} d_n = \lim_{N \rightarrow \infty} \log^{(n)}(W) \left(\log^{(n-1)}(W) \left(\dots \left( \log(W) \left(\frac{W(N)}{N W'(N)}-d_0\right) -d_1 \right) \dots \right) -d_{n-1}\right)\quad .$$ We can now relate the scaling exponents $c^{(l)}_{k}$ and $d_n$ by comparing Eqs. (\[eq:clk\]) and (\[eq:dn\]). For this we use a similar notation as for the exponents $c^{(l)}_k$ and assign $d^{(l)}_0 \equiv d_l$ to the first non-zero exponent, $d_l \neq 0$. All higher terms are denoted by $d^{(l)}_k = d_{l+k}$. Using the fact that $N \sim (\log^{(l)}W)^{1/c^{(l)}_0}$, we finally obtain $$\begin{aligned} \begin{array}{ll} d^{(l)}_0 = \frac{1}{c^{(l)}_{0}} \\[.3cm] d^{(l)}_k = -\frac{c^{(l)}_{k}}{c^{(l)}_{0}}, & k = 1,2,\dots \quad .\\ \end{array}\end{aligned}$$ The corresponding extensive entropy can now be characterized by the function $g(x)$, which scales as $$\label{eq:g} g^{{(l,n)}}(x) \sim x \prod_{j=0}^{n} \left(\log^{(j+l)} \frac{1}{x} \right)^{d^{(l)}_j} \quad \mathrm{for} \ x \rightarrow 0\, .$$ the corresponding entropy scales as $$S^{{(l,n)}}_g(W) \sim \prod_{j=0}^{n} \left(\log^{(j+l)}W \right)^{d^{(l)}_j} \quad. \label{scal}$$ This equation is nothing but the asymptotic expansion of $\log S_g$ in terms of $\phi_{n+l}(N) = \log^{(n+l+1)}(N)$; the coefficients are again the scaling exponents that correspond to the rescaling of the entropy. Note that the entropy approach allows us to obtain additional restrictions for the scaling exponents if further information about the system is available. For example, many systems fulfil the first three of the four Shannon-Khinchin ([SK]{}) axioms, see supplementary material. There we also show that it is possible to find a representation of the entropy that obeys the three axioms and the scaling in Eq. (\[scal\]). In this case $g(x)$ can be expressed as $$\label{eq:entropy} g^{{(l,n)}}_{\left(d^{(l)}_0,\dots,d^{(l)}_n\right)}(x) = \int_0^{x} \prod_{i=0}^{n} \left[a_i + [1+\log]^{(i+l)}\left(\frac{1}{y}\right)\right]^{d^{(l)}_i} \mathrm{d} y \quad,$$ where $a_i$ are constants. One possible choice for those is $$\label{eq:ai} a_i = \max\left\{ -1 - \frac{d^{(l)}_i}{(n-l)d^{(l)}_0},0\right\} \quad .$$ The axioms impose restrictions on the range of scaling exponents. ([SK2]{}) requires that $d^{(l)}_0 > 0$; ([SK3]{}) requires that $d^{(0)}_0 \equiv d_0 < 1$. The resulting entropy can be expressed by Eq. (\[eq:ent\]). One can trivially adjust the entropy minimal value, such that for the totally ordered state, $\mathcal{S}_g(1) = 0$. This is obtained by rescaling $$\mathcal{S}_g(P) = r^{(-1)}_\lambda(S_g(P)) = \left(\sum_{i=1}^W g(p_i)\right) - g(1) \quad ,$$ where $\lambda = \exp(g(1))$. Note that the form of the entropy in Eq (\[eq:entropy\]) is equivalent to $(c,d)$-entropy for $c=1-d_0$ and $d=d_1$, and $d_j = 0$ for all $j \geq 2$. Examples ======== We conclude with several examples of systems that are characterized by different sets of scaling exponents. [*Exponential growth: the random walk*]{}. Imagine the ordinary random walk with two possibilities at any timestep–a step to the left, or to the right. The number of possible configurations (i.e. possible paths) after $N$ steps is $$W_{}(N+1) = 2 W_{}(N) \quad,$$ which means exponential phasespace growth, $W_{}(N) = 2^N$. We obtain $l=1$, $c^{(1)}_0=1$ and $c^{(1) }_j = 0$, for $j \geq 1$, and for the exponents of the entropy $d_0 = 0$, $d_1\equiv d^{(1)}_0=1$ and $d_j = 0$, for $j \geq 2$. This set of exponents belongs to the class of $(c,d)$-entropies described in [@hanel-thurner11a] for $c=1-d_0 = 1$, and $d=d_1=1$. They correspond to the scaling exponents of the Shannon entropy: from (\[eq:g\]) we obtain that $g(x) \sim x \log x$ and from (\[scal\]) we get $S(W) \sim \log W$, which is Boltzmann entropy. It is not immediately apparent what the entropy of a random walk should be. However, the random walk is equivalent to spin system of $N$ independent spins, the $2^N$ different paths correspond one-to-one to the $2^N$ configurations in the spin model, where the role entropy of it is clear. Obviously, for the random walk, (SK 1-3) are applicable. [*Sub-exponential growth: the aging random walk*]{}. In this variation of the random walk we impose correlations on the walk. After the first random choice (left or right) the walker goes one step in that direction. The second random choice is followed by two steps in the same direction, the next step is followed by three steps in the same direction, etc. For $k$ independent choices, one has to make [$N = \sum_{i=1}^{k-1} i = 1/2 k(k-1)$ steps. For this walk, we get that the number of possible paths is $$W_{}\left(N+k\right) = 2 W_{}\left(N\right) \quad ,$$ which leads to $W(N) = 2^{N/k} \sim 2^{k/2}$. For $N \gg 1$, we have $k \approx \sqrt{N}$, and we obtain a stretched exponential (sub-exponential) asymptotic behavior, $W_{}(N) \sim 2^{\sqrt{N}}$.]{} The order is again $l=1$ and the exponents are $c^{(1)}_0 = 1/2$ and $c^{(1)}_j = 0$, for $j \geq 1$. In terms of the $d$ exponents we have $d_0=0$ and $d_1 \equiv d^{(1)}_0 =2$. Therefore, the three SK axioms are applicable and the resulting extensive entropy belongs to the class of entropies characterized by the Anteodo-Plastino entropy, since we have $g(x) \sim x (\log x)^2$ and $S(W) \sim (\log W)^2$. This entropy is the special case of the $(c,d)$-entropy for $c=1$ and $d=2$, see [@hanel-thurner11a]. [*Super-exponential growth: magnetic coins*]{}. Consider $N$ coins with two states (up or down). These coins are magnetic, so that any two can stick to each other to create a pair which is a third state obtained by interactions of elements (one possible configuration). As mentioned before, in [@Jensen16] it is shown that the phasespace volume can be obtained recursively $$W_{}(N+1) = 2 W_{}(N) + N W_{}(N-1) \quad .$$ For $N \gg 1$, we get $W_{}(N) \sim N^{N/2} e^{2 \sqrt{N}}$, which yields $l=1$, and the scaling exponents $c^{(1)}_0 = 1$, $c^{(1)}_1 = 1$ and $c^{(1)}_j = 0$, for $j \geq 2$. The scaling exponents of the entropy are $d_0=0$, $d_1 \equiv d^{(1)}_0=1$, and $d_2 \equiv d^{(1)}_1=-1$. For the entropy this means, that $g(x) \sim x \log x/ \log \log x$ and $S(W) \sim \log W/\log \log W$. This case is [*not*]{} contained in the class of $(c,d)$-entropies, because the third exponent, corresponding to the doubly-logarithmic correction, is not zero. Actually we obtain $c=1$ and $d=1$, which would naively indicate Shannon entropy. However, the correction makes the system clearly super-exponential. The SK axioms are still applicable, the class of accessible entropy formulas is restricted by ([SK2]{}). For example, for the representative entropy Eq. (\[eq:entropy\]) we find that $a_0 \geq 0$ and $a_1 \geq 0$, see supplementary material. [*Super-exponential growth: random networks*]{}. Imagine a random network with $N$ nodes. When a new node is added, there emerge $N$ new possible links, which gives us $2^N$ new possible configurations for each configuration of the network with $N$ links. We obtain the recursive growth equation $$W_{}(N+1) = 2^N W_{}(N) \quad ,$$ which leads to $W_{}(N) = 2^{{N}\choose{2}}$, as expected. For this phasespace growth, we obtain $l=1$, $c_0^{(1)} = 2$ and $c_j^{(1)} = 0$ for $j \geq 1$, and $d_0=0$ and $d_1 \equiv d^{(1)}_0 =\frac{1}{2}$. The corresponding entropy can be expressed by $g(x) \sim x (\log x)^{1/2}$, and $S(W) \sim (\log W)^{1/2}$. The entropy corresponds to the class of compressed exponentials, which are super-exponential, however, the entropy belongs to the class of $(c,d)$-entropies for $c=1$ and $d=1/2$. Because all exponents are positive the entropy observes the SK axioms. [*Super-exponential growth: the cascading random walk*]{}. Consider a generalization of the random walk, where a walker can take a left or right step, but it can also split into two walkers, one of which then goes left, the other to the right. Each walker can then go left, right, or split again (multiple walkers can occupy the same position). The number of possible paths after $N$ steps is $$W_{}(N+1) = 2 W_{}(N)+W_{}(N)^2 \quad ,$$ where the first term reflects the left/right decisions, the second the splittings. We have $W_{}(N) = 2^{(2^{N-1})}-1$, and find that $l=2$, $c_0^{(2)}=1$ and $c_j^{(2)}=0$, for $j \geq 1$, and $d_0 =0$, $d_1=0$ and $d_2 \equiv d^{(2)}_0 = 1$. The corresponding extensive entropy is $g(x) \sim x \, \log \log(x)$ and scales as $S(W) \sim \log \log W$. Because the coefficients are not negative, SK axioms are applicable. However, even though all correction scaling exponents are zero, the system cannot be described in terms of $(c,d)$-entropies, because $l=2$. We would naively obtain that $c=1$ and $d=0$, which would wrongly correspond to Tsallis entropy. Alternatively, we can think of an example of a spin system with the same scaling exponents. In this case, $N$ would not describe the size of a system, but its dimension. For $N=1$, we would have two particles on the line, for $N=2$ we have 4 particles forming a square, for $N=3$ we have a cube with 8 particles in its vertices, etc. In general, we can think of a spin system of particles sitting on the vertices of a $N$-dimensional hypercube. The number of particles is naturally $2^N$ and for two possible spins we obtain $W(N) = 2^{(2^N)}$. Conclusions =========== We introduced a comprehensive classification of complex systems in the thermodynamic limit based on the rescaling properties of their phasespace volume. From a scaling-expansion of the phasespace growth with system size, we obtain a set of scaling exponents, which uniquely characterize the statistical structure of the given system. Restrictions on the scaling exponents can be obtained with further information about the system. In this context we discuss the first three Shannon-Khinchin axioms, which are valid for many complex systems. The set of exponents further determine the scaling exponents of the corresponding extensive entropy, which plays a central role in the thermodynamics of statistical systems. Thermodynamics is not the only context where entropy appears. As was shown in [@thurner_corominas_hanel17] for many complex systems the functional expressions for entropy depend on the context, in particular if one talks about the thermodynamic (extensive) entropy, the information theoretic entropy, or the entropy that appears in the maximum entropy principle. It remains to be seen if for super-exponential systems there exists an underlying relation between the scaling exponents of the extensive entropy, and the exponents obtained from a information theoretic, or maximum entropy description of the same complex systems. Acknowledgements {#acknowledgements .unnumbered} ================ We thank the participants of CSH workshop for helpful initial discussion, in particular Henrik Jeldtoft Jensen, Tamás Sándor Biró, Piergiulio Tempesta and Jan Naudts. This work was supported by the Austrian Science Fund (FWF) under project I3073. References {#references .unnumbered} ========== Supplementary material ====================== Shannon-Khinchin axioms {#shannon-khinchin-axioms .unnumbered} ----------------------- The Shannon-Khinchin axioms read: - (SK1) Entropy is a continuous function of the probabilities $p_i$ only, and should not explicitly depend on any other parameters. - (SK2) Entropy is maximal for the equi-distribution $p_i=1/W$. - (SK3) Adding a state $W+1$ to a system with $p_{W+1}=0$ does not change the entropy of the system. - (SK4) Entropy of a system composed of 2 sub-systems $A$ and $B$, is $S(A+B)=S(A)+S(B |A)$. They state requirements that must be fulfilled by any entropy. For ergodic systems all four axioms hold. For non-ergodic ones the composition axiom (SK4) is explicitly violated, and only the first three (SK1-SK3) hold. If all four axioms hold the entropy is uniquely determined to be Shannon’s; if only the first three axioms hold, the entropy is given by the $(c,d)$-entropy [@hanel-thurner11a; @hanel-thurner11b]. The SK axioms were formulated in the context of information theory but are also sensible for many physical and complex systems. Given a trace form of the entropy as in Eq. (\[eq:trace\]), the SK axioms imply the restrictions on $g(x)$: [(SK1)]{} implies that $g$ is a continuous function, [(SK2)]{} means that $g(x)$ is concave, and [(SK3)]{} that $g(0)=0$. For details, see [@hanel-thurner11a]. Rescaling in the thermodynamic limit {#rescaling-in-the-thermodynamic-limit .unnumbered} ------------------------------------ We first prove a theorem which determines the general form of rescaling relations in the thermodynamic limit for any general function. Let $g(x)$ be a positive, continuous function on $\mathds{R}^+$. Let us define the function $z(\lambda): \mathds{R}^+ \rightarrow \mathds{R}^+$ $$z(\lambda) := \lim_{x \rightarrow \infty} \frac{g(r^{(n)}_\lambda(x))}{g(x)}.$$ Then, $z(\lambda) = \lambda^c$ for some $c \in \mathds{R}$. From the definition of $z(\lambda)$, it is straightforward to show that $z(\lambda \lambda') = z(\lambda) z(\lambda')$, because $$\begin{aligned} z(\lambda \lambda') = \lim_{x \rightarrow \infty} \frac{g(r^{(n)}_{\lambda \lambda'}(x))}{g(x)} = \lim_{x \rightarrow \infty} \frac{g(r^{(n)}_{\lambda \lambda'}(x))}{g(r^{(n)}_{\lambda}(x))} \frac{g(r^{(n)}_{\lambda}(x))}{g(x)}\\ = \lim_{r^{(n)}_\lambda(x) \rightarrow \infty } \frac{g(r^{(n)}_{\lambda'}[r^{(n)}_{\lambda}(x)])}{g(r^{(n)}_{\lambda}(x))} \lim_{x \rightarrow \infty} \frac{g(r^{(n)}_\lambda(x))}{g(x)} = z(\lambda') z(\lambda) \quad .\end{aligned}$$ For the computation we used the group property of rescaling in Eq. (\[eq:resc\]) and the continuity of $g$. The only class of functions satisfying the functional equation above are power functions, $z(\lambda) = \lambda^c$. Let us take the first scaling relation of the sample space $W(r^{(0)}_\lambda(N)) = W(\lambda N)$. From the previous theorem we obtain $$\frac{W(\lambda N)}{W(N)} \sim \lambda^{c_{0}} \ \Rightarrow \ W(r^{(0)}_\lambda(N)) \sim r^{(0)}_{\lambda^{c_{0}}}(W(N)) \quad .$$ It may happen that $c_{0}$ is infinite. Thus, we may need to use higher-order scaling for the sample space, i.e., $r^{(l)}_{\lambda^{c_{0}}}(W(N))$, as shown in the main text. $l$ is determined by the condition that the scaling exponent should be finite. The first correction term is given by the scaling $W(r^{(1)}_\lambda(N)) = W(N^\lambda)$. To obtain the sub-leading correction, we have to factor out the leading growth term. This means that the scaling relation for the first sub-leading correction looks like $$\frac{(\log^{(l)} W(N^\lambda))/N^{c^{(l)}_{0} \lambda}}{(\log^{(l)} W(N))/N^{c^{(l)}_{0}}} \sim \lambda^{c^{(l)}_{1}} \quad ,$$ which is again a consequence of the above theorem. To obtain the corresponding scaling relations for higher-order scaling exponents for the sample space (\[eq:sub\]), we need to factor out all previous terms corresponding to lower-order scalings, so the scaling relation looks like $$\label{eq:sub} \frac{\log^{(l)} W(r^{(k)}_\lambda(N))}{\log^{(l)} W(N)} \prod_{j=0}^{k-1} \left(\frac{\log^{(j)}(r^{(k)}_\lambda(N))}{\log^{(j)}(N)}\right)^{- c^{(l)}_{j}} \sim \lambda^{c^{(l)}_{k}}$$ Because the left-hand side of this relation has the form of the function $z$ appearing in the theorem, the validity of the relation is satisfied for $N \rightarrow \infty$. Similarly, we can deduce the relations for scaling exponents that are associated with the extensive entropy. Asymptotic expansion in terms of nested logarithms {#asymptotic-expansion-in-terms-of-nested-logarithms .unnumbered} -------------------------------------------------- The asymptotic representation of $W(N)$ is obtained by the rescaling that corresponds to the Poincaré asymptotic expansion [@copson] of $\log^{(l+1)}(W)$ in terms of $\phi_n(N) = \log^{(n+1)}(N)$ for $N \rightarrow \infty$. Let us consider a function $f(x)$ with a singular point at $x_0$. It is possible to express its asymptotic properties in the neighborhood of $x_0$ in terms of the asymptotic series of functions $\phi_n(x)$, if $f(x) = {\cal O} (\phi_0(x))$ and $\phi_{n+1}(x) = {\cal O} (\phi_n(x))$. The series is given as $$f(x) = \sum_{j=0}^{k} c_j \phi_j(x) + {\cal O} (\phi_j(x)) \quad .$$ The coefficients can be calculated from the formulas in [@copson] $$c_k = \lim_{x \rightarrow x_0} \frac{f(x) - \sum_{j=0}^{k-1}c_j \phi_j(x)}{\phi_k(x)} \quad .$$ In our case, i.e., for $N \rightarrow \infty$ and $\phi_n(N) = \log^{(n+1)}(N)$ the function $\log^{(l+1)}(W)$ can be expressed (for appropriate $l$) in terms of this series, and the coefficients $c_k^{(l)}$ are given by $$\begin{aligned} c_k^{(l)} &=& \lim_{N \rightarrow \infty} \frac{\log^{(l+1)}(W) - \sum_{j=0}^{k-1}c_j^{(l)} \log^{(j+1)}(N)}{\log^{(k+1)}(N)}\nonumber\\ &=& \lim_{N \rightarrow \infty} \frac{\log\left(\log^{(l)}(W)/\prod_{j=0}^{k-1}\log^{(j)}(N)^{c_j^{(l)}}\right)}{\log^{(k+1)}(N)}\nonumber \quad .\end{aligned}$$ Using L’Hospital’s rule and the derivative of the nested logarithm $$\frac{\mathrm{d} \log^{(n)}(x)}{\mathrm{d} x} = \frac{1}{\prod_{j=0}^{n-1} \log^{(j)}(x)} \quad ,$$ a straightforward calculation yields Eq. (\[eq:clk\]). Derivation of $g^{{(l,n)}}_{\left(d^{(l)}_0,\dots,d^{(l)}_n\right)}$ {#derivation-of-gln_leftdl_0dotsdl_nright .unnumbered} -------------------------------------------------------------------- Which entropy functional that fulfills axioms ([SK]{} 1-3)? The choice is not unique, but a concrete entropy functional serves as a representative of the class in the thermodynamic limit. The requirements imposed by the first three [SK]{} axioms are: $g(x)$ is continuous, $g(x)$ is concave, and $g(0) = 0$. From Eq. (\[eq:g\]) we have, $g(x) \sim x \prod_{j=0}^{n} [\log^{(j+l)}\left(\frac{1}{x}\right)]^{d^{(l)}_j}$ for $x \rightarrow 0$, which gives us the scaling for the values around zero. Unfortunately, the presented form cannot be extended to the full interval $[0,1]$, because the domain of $\log^{(n)}(1/x)$ is $(0,1/\exp^{(n-2)}(1))$. This can be fixed by replacing $\log^{(n)}$ by $[1+\log]^{(n)} = 1+ \log(1+ \log(\dots))$, which is defined on the whole domain $(0,1]$, where $\lim_{x \rightarrow 0} [1+\log]^{(n)}(1/x) = +\infty$ and $[1+\log]^{(n)}(1) = 1$. The scaling remains unchanged for $x \rightarrow 0$. The second problem is that in general the function is not concave. For this we introduce the transformation $$f^\star(x) = \int_0^x \frac{f(y)}{y} \mathrm{d} y\, .$$ The original function can be obtained by $$f(x) = x \frac{\mathrm{d} f^\star(x)}{\mathrm{d} x}.$$ This transform turns an increasing/decreasing function to a convex/concave function, while the scaling for $x \rightarrow 0$ remains unchanged. Let us write the function $g$ in the form of the transform $$g(x) \sim \int_0^{x} \prod_{j=0}^{n} \left[ [1+\log^{(n)}]\left(\frac{1}{y}\right)\right]^{d_j} \, \mathrm{d} y \quad .$$ Axiom ([SK3]{}) means $g(0) = 0$. This requires that the integrand should not diverge faster than $1/x$ for $x \rightarrow 0$. This can be fulfilled for $d_0 \equiv d_0^{(0)} < 1$. Because $[1+\log]^{(n)}(1/x)$ is a decreasing function, $g(x)$ is automatically concave if $d_n \geq 0$, since a product of positive, decreasing functions is also decreasing. However, for $d_n < 0$, $[1+\log]^{(n)}(1/x)^{d_n}$ is an increasing function from zero to one and the whole product may not be decreasing. In order to solve this issue, we introduce a set of constants $a_i$ and write $g(x)$ in the form $$g(x) = \int_0^{x} \prod_{j=0}^{n} \left[a_j + [1+\log]^{(n)}\left(\frac{1}{y}\right)\right]^{d_j} \, \mathrm{d} y \quad .$$ The constants $a_i$ can be chosen to ensure that the integrand is a decreasing function. We assume $a_i \geq -1$ to avoid problems with powers of negative numbers. The second derivative of $g(x)$, i.e., the first derivative of the integrand is an increasing function and $\frac{\mathrm{d}^2 g(x)}{\mathrm{d} x^2}|_{x \rightarrow 0^+} = - \infty$ for $d_l > 0$. For $d_l < 0$, the entropy cannot be concave, so $d_l > 0$ is the restriction given by ([SK2]{}). To obtain a negative second derivative on the whole domain $[0,1]$, it is therefore enough to investigate $\frac{\mathrm{d}^2 g(x)}{\mathrm{d} x^2}|_{x = 1}$, which leads to the condition $$\label{eq:2d} \left(\prod_{j=l}^{n} (1+a_j)^{d_j-1} \right) \left(- \sum_{j=l}^{n} \frac{d_j}{1+a_j}\right) \leq 0 \quad .$$ Because $d_0^{(l)} \equiv d_l > 0$, we can choose $a_l = 0$. In the following terms, i.e., for $i > l$, $d_i$ can be both positive and negative. Positive $d_i$ pose no problem, because the term corresponding to $d_i$, i.e. $-d_i/(1+a_i)$ is negative, so we can choose $a_i=0$. When all $d_i$ are negative we can compensate the positive contribution of the negative terms by diminishing them through choice of appropriate $a_i$. If we choose $$1+a_i = - \frac{d^{(l)}_i}{n d^{(l)}_0} \quad ,$$ then Eq. (\[eq:2d\]) becomes zero. If this is given together with previous results and summarize it as $$a_i = \max\left\{ -1 - \frac{d^{(l)}_i}{n d^{l}_0},0\right\} \quad ,$$ which has been presented as in Eq. (\[eq:ai\]) in the main text. Clearly, this is not the only possible choice. Note that for all $d^{(l)}_i > 0$, one may even choose $a_i = - 1$. On the other hand, for the case of magnetic coin model, one obtains that for $a_0 = 0$, $a_1 = 0$ as well. Finally, let us show the connection to $(c,d)$-entropy derived in [@hanel-thurner11a]. In this case, we assume only $d_0$ and $d_1$ can be non-zero, which leads to $$g^{{(0,1)}}_{\left(d_0,d_1\right)}(x) = \int_0^{x} (1/y)^{d_0} (1+a_1+\log(1/x))^{d_1} \, \mathrm{d} y \quad .$$ By the choice $a_1 = -1+\frac{1}{1-d_0}$, we get $$g_{(c,d)}(x) = \frac{e}{c^{d+1}} \Gamma(1+d,1-c \log x) \quad ,$$ for $c = 1-d_0$ and $d = d_1$, which is nothing else than the Gamma entropy of [@hanel-thurner11a]. Ordering of processes and classes of equivalence {#ordering-of-processes-and-classes-of-equivalence .unnumbered} ------------------------------------------------ The set of scaling exponents form natural classes of equivalence with natural ordering. Consider two discrete random processes $X(N)$ and $Y(N)$ with sample spaces $W_X(N)$ and $W_Y(N)$, respectively. The corresponding sets of scaling exponents are denoted by $\mathcal{C}_X =\{c^{(l)}_0,c^{(l)}_1,\dots\}$, and $\mathcal{C}_Y =\{\tilde{c}^{(\tilde{l})}_0,\tilde{c}^{(\tilde{l})}_1,\dots\}$. One can introduce an ordering based on the scaling exponents. We write $$X \prec Y \ (\ \mathcal{C}_X \prec \mathcal{C}_Y) \ \mathrm{if} \ \left\{ \begin{array}{ll} l < l'\\ l = l', c^{(l)}_0 < \tilde{c}^{(\tilde{l})}_0\\ l = l', c^{(l)}_0 = \tilde{c}^{(\tilde{l})}_0, c^{(l)}_1 < \tilde{c}^{(\tilde{l})}_1\\ \mathrm{etc.} \end{array} \right. \quad .$$ This is equivalent to lexicographic ordering. One can also introduce an ordering, which takes into account only certain a number of correcting terms. So, for example $$X \prec_0 Y \ (\ \mathcal{C}_X \prec_0 \mathcal{C}_Y) \ \mathrm{if} \ \left\{ \begin{array}{ll} l < l'\\ l = l', c^{(l)}_0 < \tilde{c}^{(\tilde{l})}_0\\ \end{array} \right. \quad .$$ Similarly, one can define $\prec_k$, which takes into account only $k$ correction terms. Additionally, it is possible to introduce an equivalence relation $$X \sim Y \ \mathrm{if} \ \mathcal{C}_X \equiv \mathcal{C}_Y \ \Rightarrow \ l = l'; \ c^{(l)}_i = \tilde{c}^{(\tilde{l})}_i \ \forall i \quad .$$ and also equivalence up to certain correction $$X \sim_k Y \ \mathrm{if} \ \mathcal{C}_X \equiv \mathcal{C}_Y \ \Rightarrow \ l = l'; \ c^{(l)}_i = \tilde{c}^{(\tilde{l})}_i \ \forall i \leq k \quad .$$ As an example, for magnetic coin model and random walk we have that $X_{\rm MC} \sim_0 X_{\rm RW}$, but $X_{\rm MC} \not\sim X_{\rm RW}$. Construction of a “representative process” {#construction-of-a-representative-process .unnumbered} ------------------------------------------ To understand the mechanism of how the scaling exponents correspond to the structure of a random process, let us discuss a simple procedure to generally obtain processes with given scaling exponents $c^{(l)}_k$. We start with a random variable $X_0$ with $N$ possible outcomes, so that $W_{X_0}(N) = \{1,\dots,N\}$. The scaling exponents of this process are naturally $c^{(0)}_0 = 1$ and $c^{(0)}_k = 0$ for $k \geq 1$. Let us construct a new variable by choosing subsets of $W_{X_0}(N)$. First we can create all possible subsets of $W_{X_0}(N)$. This defines a new variable $X_1$ with $W_{X_1}(N) = 2^{W_{X_0}(N)}$, and we get $c^{(1)}_0 = 1$. Generally, the transform $$\mathfrak{2}: X \rightarrow \mathfrak{2}^{X} \quad ,$$ where $\mathfrak{2}^X$ denotes a variable on all subsets of $X$. One can easily show that this results in a shift of scaling exponents $c^{(l)}_k \rightarrow c^{(l+1)}_k$, and $d^{(l)}_k \rightarrow d^{(l+1)}_k$, because $W_{\mathfrak{2}^{X}}(N) = 2^{W_X(N)}$. The interpretation of this transformation is the following: Consider an ordinary random walk with two possible steps. If $X_0(N)$ denotes a number of steps of a random walker, then $X_1(N) = \mathfrak{2}^{X_0(N)}$ denotes the number of possible paths. When we apply the transform again, we obtain $X_2(N) = \mathfrak{2}^{X_1(N)}$. This denotes the number of possible configurations of a random walk cascade, etc. As a result, by more applications of $\mathfrak{2}$, we obtain processes with more complicated structure of the respective phasespace. To construct processes with arbitrary exponents, let us think about a procedure, where we create only partial subsets, which number $p(N)$ can be between $N$ (no partitioning) and $2^N$ (full partitioning) We denote this procedure by $\mathfrak{P}$. This process can be understood as process corresponding to a correlated random walk. This means that not every step of the walk is independent, but some steps can be determined by the previous steps, which diminishes the number of possible configurations when compared to the uncorrelated random walk. The resulting random process is obtained as the composition of $l$ uncorrelated random walks (full partitioning) and a correlated random walk $$X = \mathfrak{2}^{(l)}[\mathfrak{P}(X_0)] \quad .$$ Let us now focus on the construction of correlated random walk with a pre-determined number of states given by $p(N)$. First we consider the full set of subsets of $N$ elements with natural ordering, $$W_{\mathfrak{2}^X} =\left\{\{\},\{1\},\{2\},\dots,\{1,\dots,n\}\right\} \quad .$$ The correlations can be represented by merging subsets to $p(N)$ sequences of length $\{s(1),\dots,s(p(N))\}$, i.e., $$W_{\mathfrak{P}(X)} =\left\{ \underbrace{\{ \{\},\{1\},\{2\},\dots\}}_{s(1)},\dots,\underbrace{ \left\{ \dots, \left\{ 1,\dots,n \right\} \right\} }_{s(p(N))}\right\} \quad .$$ This means that after one independent step, there are $s(1)-1$ dependent steps, after the second independent step, there are $s(2)-1$ dependent steps, etc. Let us determine the form of function $s$ for given $p(N)$. The function $s$ can be obtained from $$\sum_{i=1}^{p(N)} s(i) = 2^N \quad .$$ In the limit of large $N$ we can assume that the function $s$ does not depend on $N$, i.e., is a priori given by the scaling exponents of the system. Let us also assume, without loss of generality, that $s$ is an increasing function (we can neglect the last cell, because its size is determined by the size of previous cells). For $N \gg 1$, we approximate the sum by the integral and obtain $$\int_0^{p(N)} s(i) \mathrm{d} i \sim 2^N \quad .$$ Denoting $S(m) = \int_0^m s(y) \mathrm{d} y$, and substituting $x=p(N)$, we recast the previous equation as, $S(x) = 2^{p^{-1}(x)}$, where $p^{-1}$ denotes the inverse function of $p$. The function $s(x)$ can be therefore determined as $$s(x) = \frac{\mathrm{d} \, 2^{p^{-1}(x)}}{\mathrm{d} x} = \frac{2^{p^{-1}(x)}}{p'(p^{-1}(x))}\quad .$$ Some examples for $s(x)$ for a corresponding $p(N)$ are - $p(N) = 2^N$, i.e., full partitioning corresponding to uncorrelated random walk. In this case, we obtain that $s(x) = const.$, as expected. - $p(N) = N$, i.e., no partitioning to maximally correlated random walk. We obtain that $s(x) \sim 2^x$, which can be seen from the relation $\sum_{i}^{N} 2^i \sim 2^N$. - $p(N) = N \log N$, which corresponds to the correction in the magnetic coin model. In this case, $s(x) \sim 2^{W(x)}/\log(W(x))$, where $W(x)$ is the Lambert W-function.

--- abstract: 'We propose an analytical ansatz, using which the ordering temperature of a quasi-one-dimensional (quasi-1D) antiferromagnetic (AF) system (weakly coupled quantum spin-1/2 chains) in the presence of the external magnetic field is calculated. The field dependence of the critical exponents for correlation functions of 1D subsystems plays a very important role. It determines the region of possible re-entrant phase transition, governed by the field. It is shown how the quantum critical point between two phases of the 1D subsystem, caused by spin-frustrating next-nearest neighbor (NNN) and multi-spin ring-like exchanges, affects the field dependence of the ordering temperature. Our results qualitatively agree with the features, observed in experiments on quasi-1D AF systems.' author: - 'A. A. Zvyagin' title: '**Magnetic phase diagram of a quasi-one-dimensional quantum spin system**' --- Introduction ============ The progress in preparation of quantum spin substances with well defined 1D subsystems has motivated the interest in studies of them during last years. Another reason for the investigation of properties of quasi-1D spin systems is the relatively rare possibility of comparison experimental data with results of exact theories for many-body models. According to the Mermin-Wagner theorem, [@MW] totally 1D spin systems with isotropic spin-spin interactions cannot have a magnetic ordering at nonzero temperatures. However, for quasi-1D spin systems, which 1D subsystems have gapless spectrum of low-lying excitations, the magnetic susceptibility, specific heat, and muon spin relaxation often manifest peculiarities, characteristic for phase transitions to magnetically ordered states. The critical temperature of the ordering of a quasi-1D AF Heisenberg spin system was first calculated in Ref.  in the absence of the external magnetic field. However, for quasi-1D spin systems, in which (the largest) exchange constants along the distinguished direction are relatively small ($\sim 1-20$ K), the transition to the ordered state can be governed by the external magnetic field. Nowadays in low-temperature experiments high values of the magnetic field (about 20 T for stationary fields and about 60 T for pulse fields) can be used. Therefore, for many quasi-1D spin systems it is possible to investigate experimentally how the external magnetic field affects the Néel ordering, i.e. to determine the H-T phase diagram. This is why, the objection of the present study is to construct an analytical theory (convenient for comparison with experiments), which has to show how the external magnetic field affects the magnetic ordering in a system of weakly coupled AF Heisenberg spin chains. To calculate the ordering temperature of the quasi-1D system we use the mean field approximation for a weak inter-chain interactions. In this approach 1D subsystems are considered as clusters (of infinite size), for the description of which we can use non-perturbative results. [@Zb] As a result, we propose a relatively simple analytical ansatz for the magnetic field dependence for the Néel temperature of a quasi-1D Heisenberg AF system. Mean-field approximation ======================== Consider a three-dimensional spin-1/2 system with AF interactions between spins, which form a hyper-cubic lattice. In the quasi-1D situation the Hamiltonians of the 1D subsystems are Heisenberg Hamiltonians of AF spin chains $${\cal H}_{1D} = J\sum_n ({\bf S}_n\cdot {\bf S}_{n+1}) -H\sum_n S_n^z \ , \label{H1}$$ where $J >0$ is the AF exchange coupling between nearest neighbor spins in the chain, $H= g\mu_B B$, $B$ is the magnetic field, $g$ is the $g$-factor of magnetic ions, and $\mu_B$ is Bohr’s magneton. Denote by $J'\ll J$ the weak inter-chain coupling between spins belonging to different 1D subsystems of the quasi-1D system. If the system is AF-ordered, we can write the magnetization of the $n$-th site of the system as $${\bf M}_n = M{\bf e}_z + (-1)^n m_N {\bf e}_x \ , \label{M}$$ where ${\bf e}_{x,z}$ are the unit vectors in the $x$- or $z$ directions, $M$ is the average magnetization, and $m_N$ is the staggered magnetization in the direction, perpendicular to the external field (the order parameter in the considered case). The inter-chain interaction can be taken into account in the mean field approximation. In that approximation in the AF phase we write the Hamiltonian of the total system as $$\begin{aligned} &&{\cal H}_{mf} = {\cal H}_{1D} + zJ'M\sum_n S^z_n \nonumber \\ &&- h_N\sum_n (-1)^n S_n^x + {\rm const} \ , \label{meanf}\end{aligned}$$ where $h_N =zJ'm_N$, and $z$ is the coordination number. The order parameter $m_N$ (or $h_N$) has to be determined self-consistently. The self-consistency equation reads $m_N = M_{N}(H,h_{N},T)$, where $M_{N}(H,h_{N},T)$ is the magnetization per site of the 1D subsystem in the effective field $H - MzJ'$ at the temperature $T$. In other words, the susceptibility of the quasi-1D system can be written in the mean field approximation as $$\chi_{q1D} = {\chi_N\over 1-zJ'\chi_N} \ , \label{chiq1D}$$ and the ordering takes place at the values of the temperature and the field, at which the denominator becomes zero. Then the transition temperature to the ordered state has to be determined from the equation $$\begin{aligned} &&1 = zJ'\chi_{N} \ , \nonumber \\ &&\chi_{N} = ( \partial M_{N}(H,h_{N},T)/ \partial h_{N})_{h_{N} \to 0} \ . \label{selfcons}\end{aligned}$$ Notice that $\chi_N$ is exponentially small for the situation with gapped low-energy eigenstates of the spin chain. It takes place, e.g., for the Heisenberg spin chain for $H > H_s$ in the ground state ($H_s =2J$ is the critical value of the magnetic field, at which the spin chain undergoes a quantum phase transition to the spin-saturated phase). In that case weak couplings $J'$ cannot yield a magnetically ordered state of a quasi-1D system. Therefore, in the following we consider only the case with gapless low-energy eigenstates of the spin chain. Susceptibility of the one-dimensional subsystem =============================================== The non-uniform static susceptibility of the 1D subsystem at low temperatures can be written as $$\chi_{\alpha}(q,T) = -i \sum_n \int dt e^{-iqn} \Theta (t)\langle [S^{\alpha}(n,t), S^{\alpha}(0,0)]\rangle_{T} \ , \label{susc}$$ where $q$ is the wave vector, $\alpha =x,y,z$, and $\langle ... \rangle_T$ denotes the thermal average at the temperature $T$. Asymptotic behavior of correlation functions for an integrable spin chain for the gapless case can be obtained in the conformal field theory limit, [@Zb] and it is possible to write the staggered part of the correlation function for the transverse to the magnetic field components in the ground state (related to $\chi_N$) as $$\langle S^x_n(t) S^x_0(0) \rangle \approx (-1)^n {C\over [n^2-(vt)^2]^{\eta/2}} + \dots \ , \label{corr}$$ where $v$ is the Fermi velocity of low-energy excitations, $\eta = 1/2Z^2$ is the correlation function exponent, $Z$ is the dressed charge of low-lying excitations (low-energy eigenstates), and $C$ is a non-universal constant. [@LZ] Asymptotic behavior can be extended for weak nonzero temperatures using the conformal mapping $(n \pm vt) \to (v/\pi T)\sinh [\pi T(n \pm vt)/v]$. Then, we can calculate susceptibilities for $q=\pi$ (we use the main approximation), Fourier transforming of the conformal mapping of Eq. (\[corr\]) at low temperatures as $$\chi_N = {C\over v} \left({2\pi T\over v}\right)^{2-\eta} B^2\left({\eta\over4},{2-\eta\over 2}\right) \ , \label{chiN}$$ where $B(x,y)=\Gamma(x)\Gamma(y)/\Gamma(x+y)$ is the Euler’s beta function. Finally, we obtain the expression for the Néel temperature below which the magnetic ordering takes place $$T_N \approx {v\over 2\pi} \biggl[ C {zJ'\over v}\sin \left({\pi \eta\over 2}\right) B^2\biggl({\eta\over 4}, {2-\eta\over 2}\biggr)\biggr]^{1\over 2-\eta} \ . \label{TN}$$ Bethe ansatz approach ===================== Fermi velocity and the critical exponent $\eta$ can be calculated exactly using the Bethe ansatz. [@Zb] In the ground state phase with gapless low-energy eigenstates at $H < H_s$, we can write $Z=\xi(A)$, $v=\varepsilon(A)/2\pi \sigma(A)$, where $\xi(x)$ and $\sigma(x)$ and $\varepsilon(A)$ are determined from the solution of the Fredholm integral equations of the second kind $$\begin{aligned} \nonumber \\ &&\sigma(x) +{1\over 2\pi}\int_{-A}^{A} dy {4\sigma(y)\over (x-y)^2+4} = {1\over \pi(1+x^2)} \ , \nonumber \\ &&\rho(x) +{1\over 2\pi}\int_{-A}^{A} dy {4\rho(y)\over (x-y)^2+4} = -{4x\over 2\pi[1+(x-A)^2]} \ , \nonumber \\ &&\xi(x) + {1\over 2\pi}\int_{-A}^{A} dy {4\xi(y)\over (x-y)^2+4} =1 \ , \label{int}\end{aligned}$$ and $$\varepsilon(A) ={4A\over (1+A^2)^2} + \int_{-A}^{A} dx\rho(x) \left(H -{2J\over x^2+1} \right) \ . \label{en}$$ The boundaries of integrations are related to the value of the magnetic field ($0\le H \le H_s$ for the phase with gapless excitations, while for $H > H_s$ the spin chain is in the spin-saturated phase with gapped excitations) via $H=2\pi J\sigma(A)/\xi(A)$. Equations (\[int\]) can be solved analytically only in some limiting cases, and numerically in other cases. In the absence of interactions between $z$-components of neighboring spins (so-called XY model, the Hamiltonian of which can be exactly mapped to the one of the non-interacting fermion model using the Jordan-Wigner transformation), the dressed charge is equal to 1. It is also equal to unity for the isotropic Heisenberg chain at $H=H_s$, where $A=0$. In the absence of the magnetic field we have $A=\infty$, and the solution of integral equations can be obtained by the Fourier transformation, which yields $v=\pi J/2$, and $Z=1/\sqrt{2}$. The numerical solution for intermediate values of $A$ shows that the dressed charge as a function of $H$ grows from $1/\sqrt{2}$ to 1 for $0\le H \le H_s$, see, e.g., Ref. , i.e. $\eta$ decreases from 1 to 1/2 in this domain of field values. Similarly, the velocity of low-energy excitations decreases with the growth of the field from $\pi J/2$ to zero in the domain $0 \le H \le H_s$. Numerical solution for the Néel temperature was given, e.g., in Ref. . Simple analytic ansatz ====================== It is not convenient, however, from the viewpoint of application of the results for comparisons with experimental data to use numerical solutions. This is why, we propose the simple ansatz for the magnetic field behavior of the the velocity $v$ and correlation function exponent $\eta$, valid in the interval $0 \le H \le H_s$: $$\begin{aligned} &&v = {\pi J\over 2} \sqrt{[1-(H/H_s)][1-(H/H_s)+(2H/\pi J)]} \ , \nonumber \\ &&\eta = {\sqrt{4f^2 - 3H^2}\over 2f} \ , \ f=\pi J\left(1 - {H\over H_s}\right) + H\ . \label{ans}\end{aligned}$$ The non-universal constant is equal to 0.18 at $H=0$ and near the saturation it behaves approximately as $C\sim 0.18\sqrt{1-2M}$, where $M$ is the average spin moment per site, cf. Ref. , which leads to the field dependence $C\approx 0.18[2(H_s-H)/2\pi H_s]^{1/4}$ in the vicinity of the critical saturation point $H_s$. Our ansatz is exact at the points $H=0$ and $H=H_s$ and in the vicinity of $H=H_s$. The main deviations of our ansatz from exact results take place for intermediate field values, between zero and $H_s$. In the above expression for the Néel temperature (and in the ones for the velocity and the critical exponent) we did not take into account logarithmic corrections, which exist for the characteristics of the isotropic Heisenberg AF spin-1/2 chain near $H=0$, see, e.g., Ref. . Those corrections can be taken into account, which yield for the susceptibility [@BETG] $$\chi_N \to \chi_N {\sqrt{\ln (24.27 J/T)}\over (2\pi)^{3/2}} \ . \label{renormchi}$$ Then for the Néel temperature we can modify our Eq. (\[TN\]) as (cf. [@LZ]): $$C \to (2\pi)^{-7/4}\sqrt{\sqrt{2(H_s-H)/H_s}\ln([48.54 \pi J/v)]} \ . \label{ren}$$ Equations (\[TN\]), (\[ans\]), and (\[ren\]) are the main result of our work. Results for the magnetic phase diagram ====================================== In Fig. \[fig1\] we present the Néel temperature of a quasi-1D spin-1/2 AF system for $z=4$ and $J'=0.1J$ as a function of the external magnetic field $H$, i.e. the $H-T$ phase diagram of the system (our results qualitatively agree with the results of numerical calculations for Bethe ansatz equations, see, e.g., Ref. ). The quasi-1D system is in the magnetically ordered state in the interval of fields and temperatures, limited by the line of the second order phase transition. ![The $H-T$ phase diagram of the quasi-1D Heisenberg spin-1/2 system with AF interactions between nearest neighbors (the value of the spin saturation field $H_s=2J$). The dashed line represents the result without logarithmic corrections, while the solid one is related to the case with taken into account logarithmic corrections.[]{data-label="fig1"}](TNA.eps){width="35.00000%"} Logarithmic corrections do not change the qualitative behavior of the Néel temperature as a function of the field. However, the values of the critical temperatures become smaller due to logarithmic corrections. The Néel temperature as a function of the external field first grows, and then goes to zero at the critical field $H=H_s$. Hence, there exists a (narrow) interval of temperatures, at which one can observe a re-entrant phase transition. In this domain of temperatures, if we enlarge the value of the field, a quasi-1D spin system is first in the paramagnetic short-range phase. Then the system undergoes a phase transition to the magnetically ordered phase, and then, for larger values of the field, it returns to the paramagnetic phase. It would be interesting to observe such a re-entrant phase transition in real quasi-1D spin systems. Such a behavior is the consequence of the field dependence of the critical exponent. If the critical exponent does not depend on the magnetic field (e.g., in the XY chain), the Néel temperature as a function of the field only decreases (following the field dependence of the velocity). Exact calculation of the Néel temperature for quasi-1D spin system with the Dzyaloshinskii-Moriya (DM) interaction [@BETG] also revealed similar to Fig. 1 behavior, i.e. the maximum in the field dependence of the critical temperature. Our calculations for spin systems with DM interactions or with the “easy-plane” magnetic anisotropy show that such magnetically anisotropic interactions produce the reduction of the maximum in the field dependence of the critical temperature. Notice that the critical temperature for weakly coupled XY spin chains is higher than for Heisenberg chains due to field-independent exponent. Also, if the symmetry of the lattice of the total system is lower than hyper-cubic, the ordering can take place not at $q=\pi$, but for some values of $q$, which depend on the lattice structure and relativistic interactions (such a case can be analyzed in the random phase approximation). In that case the Néel temperature also becomes smaller than in the hyper-cubic situation, cf. Ref. . Effect of next-nearest neighbor and multi-spin exchange couplings ================================================================= In real quasi-1D spin systems additional intra-chain exchange interactions between NNN spins exist very often. [@expfr]. To take into account such interactions we can consider the modified 1D Hamiltonian $${\cal H}_{NNN} = J_1\sum_n ({\bf S}_n\cdot {\bf S}_{n+1}) +J_2\sum_n ({\bf S}_n\cdot {\bf S}_{n+2}) -H\sum_n S_n^z \ , \label{H2}$$ where $J_2$ is the exchange integral for next-nearest neighbor couplings. For $J_2 >0$ such a spin chain reveals a spin frustration. Unfortunately, for this Hamiltonian an exact solution cannot be obtained analytically for any values of $J_{1,2}$. Nevertheless, approximate bosonization studies and numerical calculations suggest that for $J_2 > 0.24...J_1$ the spin gap is opened for low-energy excitations. [@ON] For the above mentioned reasons a system of weakly coupled chains with gapped excitations cannot be ordered magnetically. However, as follows from Ref. , despite the fact that for most of studied compounds exchange constants satisfy the condition $J_2 > 0.24...J_1$, the spin gap was not confirmed experimentally. To describe theoretically quasi-1D spin systems with spin frustration due to intra-chain interactions without spin gap and with a weak inter-chain coupling, we consider another model, the Hamiltonian of which is ${\cal H}_{NNN}$ with additional terms, describing multi-spin ring-like interactions. The advantage of that model is its exact integrability: The model permits an exact Bethe ansatz solution. We do not state, naturally, that the model describes all features of the experiments. [@expfr] However, many properties of the model are similar to what was observed in Ref. , at least, for this model low-energy eigenstates are gapless. Hence, from this viewpoint, it qualitatively agrees with the data of experiments, [@expfr] unlike the model with the Hamiltonian ${\cal H}_{NNN}$. Multi-spin ring exchange interactions are often present in oxides of transition metals, where a direct exchange between magnetic ions is complimented by a super-exchange between magnetic ions via nonmagnetic ones. [@exp] The modified Hamiltonian of such 1D subsystem has the form $$\begin{aligned} &&{\cal H}_{mod} = {\cal H}_{NNN} + J_4\sum_n ( ({\bf S}_{n-1}{\bf S}_{n+1}) ({\bf S}_{n}{\bf S}_{n+2}) \nonumber \\ &&- ({\bf S}_{n-1}{\bf S}_{n+2}) ({\bf S}_{n}{\bf S}_{n+1}) ) \ . \label{H4} \end{aligned}$$ Notice that multi-spin interactions are less relevant from the renormalization group viewpoint than two-spin interactions. Quantum properties of the model can be seen from the exact solution, [@MT] for the parametrization of coupling constants $J_1=J(1-y)$, $J_2 =Jy/2$, $J_4=2Jy$ for any $J$ and $y$ (in what follows we consider $J>0$, $y \ge 0$). This exactly solvable model, while being formally less realistic than the model with the Hamiltonian ${\cal H}_{NNN}$, reveals features, more similar to the properties of experimentally studied quasi-1D systems with spin frustration. [@expfr] For $y=0$ the model describes the Heisenberg spin-1/2 chain. The ground state of the model depends on values of the parameter $y$ and an external magnetic field. [@MT] At $T=0$ for large values of the magnetic field the model is in the spin-saturated phase, divided from other phases by the line of the second order quantum phase transition. For low values of $y$ and $H$ the model is in the phase, which properties are similar to the phase of the Heisenberg spin-1/2 chain in a weak magnetic field (Luttinger liquid). [@MT] The model is in this phase for $y < y_{cr} = 4/\pi^2$ at $H=0$ and for $y < y_{cr}(H)$ for nonzero fields. The point $y_{cr}$ is the quantum critical one. For $y > y_{cr}$ the model is in an incommensurate phase with nonzero spontaneous magnetization at $H=0$. Last two phases are divided from each other by the line of the second order quantum phase transition. Quantum phase transitions can be observed in the temperature behavior of thermodynamic characteristics of the model, like the magnetic susceptibility and the specific heat, that were also calculated exactly. [@MT] This model permits us to know, how NNN interactions (together with the multi-spin ones), which cause the quantum phase transition, can modify the $H-T$ phase diagram, obtained above for the case of only nearest neighbor couplings in chains. In this situation the expressions for the velocity and the critical exponent can be modified using the substitution $\pi J \to \pi J(1- x)$ (cf. Ref. ), where $x=y/y_{cr}$. We concentrate on the case $y < y_{cr}$ ($0 \le x \le 1$) for the 1D subsystem, which has Luttinger liquid properties. In this case a quasi-1D spin system undergoes the transition to the AF ordered state. [@ZD] Ordering in the incommensurate phase was studied in Ref. . The results of our analysis are presented in Fig. 2. The ordered phase is inside the region, limited by the surface, at which the second-order phase transition takes place. ![The phase diagram for a quasi-1D spin-1/2 chain with spin frustration, caused by nearest, NNN interactions and the ring exchange. The Néel temperature is a function of the parameter $x$, which shows how close the quantum critical point (caused by spin-frustrating interactions) $x=1$ is, and the external magnetic field.[]{data-label="fig2"}](TnnnA.eps){width="50.00000%"} The maximum in the field dependence of the critical temperature, cf. Fig. 1, is shifted towards low values of the field with the growth of $x$, i.e. spin-frustrating NNN and multi-spin couplings can reduce the domain of temperatures, at which re-entrant phase transition can take place. We believe, that while the considered model seems less realistic, the mentioned feature has the generic nature for quasi-1D spin systems. Phase diagrams, similar to the ones, presented in Figs. 1 and 2, were obtained experimentally for real quasi-1D compounds [@faz]. Phase $H-T$ diagrams in those compounds show maxima of the field dependencies of critical temperatures. Namely, the ordering temperature in studied quasi-1D copper oxides first increases with the growth of the value of the field, reaches its maximum, and then decreases to zero at the value of the field, where the spin chain has the spin saturation. Notice, that in one of those compounds measurements reveal different values of the ordering temperatures for different directions of the external field, which can be caused by the weak magnetic anisotropy of the intra-chain exchange interactions. Conclusions =========== In summary, we have used a simple analytical ansatz to calculate the ordering temperature of a quasi-1D system, consisting of weakly interacting quantum spin-1/2 chains with AF couplings in the presence of the external magnetic field, when the weak inter-chain coupling is taken into account in the mean field approximation, and the characteristics of spin chains are obtained non-perturbatively. Our results show that the field dependence of the critical exponents for correlation functions of 1D subsystems plays a very important role. In particular, that dependence determines the region of possible re-entrant phase transition, governed by the field. We have shown also how a quantum critical point between two phases of the 1D subsystem, caused by spin-frustrating NNN and multi-spin ring-like exchanges, affects the field dependence of the ordering temperature. Our results qualitatively agree with the features, observed in experiments on quasi-1D AF systems. We expect that our results are generic for quasi-1D systems and that they can be helpful for experimentalists, who study magnetic properties of such systems, especially due to the recent progress in obtaining high values of the magnetic field in experiments. I thank the Deutsche Forschungsgemeinschaft for the financial support via the Mercator Program. The support from the Institute of Chemistry of the V. Karazin Kharkov National University is acknowledged. [99]{} N. D. Mermin and H. Wagner, Phys. Rev. Lett. [**17**]{}, 1133 (1966). H.J. Schulz, Phys. Rev. Lett. [**77**]{}, 2790 (1996). See, e.g., A.A. Zvyagin [*Finite Size Effects in Correlated Electron Models: Exact Results*]{}, Imperial College Press, London, 2005. S. Wessel and S. Haas, Phys. Rev. B [**62**]{}, 316 (2000). S. Lukyanov and A. Zamolodchikov, Nucl. Phys. B [**493**]{}, 571 (1997); S. Lukyanov, Phys. Rev. B [**59**]{}, 11163 (1999); V. Barzykin, Phys. Rev. B [**63**]{}, 140412(R) (2001); T. Hikihara and A. Furusaki, Phys. Rev. B [**69**]{}, 064427 (2004). M. Bocquet, F.H.L. Essler, A.M. Tsvelik, and A.O. Gogolin, Phys. Rev. B [**64**]{}, 094425 (2001). M. Matsuda and K. Katsumata, J. Magn. Magn. Mater. [**140-145**]{}, 1671 (1995); I.A. Zaliznyak, C. Broholm, M. Kibune, M. Nohara, and H. Takagi, Phys. Rev. Lett. [**83**]{}, 5370 (1999); N. Maeshima, M. Hagiwara, Y. Narumi, K. Kindo, T. C. Kobayasi, and K. Okunishi, J. Phys.: Condensed Matter [**15**]{}, 3607 (2003); M. Hase, K. Ozawa, and N. Shinya, Phys. Rev. B [**68**]{}, 214421 (2003); T. Masuda, A. Zheludev, A. Bush, M. Markina, and A. Vasiliev, Phys. Rev. Lett. [**92**]{}, 177201 (2004); A.A. Gippius, E.N. Morozova, A.S. Moskvin, A.V. Zalessky, A.A. Bush, M. Baenitz, H. Rosner, and S.-L. Drechsler, Phys. Rev. B [**70**]{}, 020406(R) (2004); H.-A. Krug von Nidda, L.E. Svistov, M.V. Eremin, R.M. Eremina, A. Loidl, V. Kataev, A. Validov, A. Prokofiev, and W. Aßmus, Phys. Rev. B [**65**]{}, 134445 (2002); S.-L. Drechsler, O. Volkova, A.N. Vasiliev, N. Tristan, J. Richter, M. Schmitt, H. Rosner, J. Malek, R. Klingeler, A.A. Zvyagin and B. Büchner, Phys. Rev. Lett. [**98**]{}, 077202 (2007). K. Okamoto and K. Nomura, Phys. Lett. A [**169**]{}, 433 (1992). A. Zheludev, M. Kenzelmann, S. Raymond, E. Ressouche, T. Masuda, K. Kakurai, S. Maslov, I. Tsukada, K. Uchinokura, and A. Wildes, Phys. Rev. Lett. [**85**]{}, 4799 (2000); M. Kohgi, K. Iwasa, J.M. Mignot, B. Fak, P. Gegenwart, M. Lang, A. Ochiai, H. Aoki, and T. Suzuki, Phys. Rev. Lett. [**86**]{}, 2439 (2001). N. Muramoto and M. Takahashi, J. Phys. Soc. Jpn. [**68**]{}, 2098 (1999); A.A. Zvyagin, J. Phys. A [**34**]{}, R21 (2001); A.A. Zvyagin and A. Klümper, Phys. Rev. B [**68**]{}, 144426 (2003). A.A. Zvyagin and S.-L. Drechsler, Phys. Rev. B [**78**]{}, 014429 (2008). A.N. Vasil’ev, L.A. Ponomarenko, H. Manaka, I. Yamada, M. Isobe, and I. Ueda, Phys. Rev. B [**64**]{}, 024419 (2001); M. Poirier, M. Castonguay, A. Reycolevschi, and G. Dhalenne, [*ibid.*]{} [**66**]{}, 054402 (2002).

--- abstract: 'Conventional methods for object detection typically require a substantial amount of training data and preparing such high-quality training data is very labor-intensive. In this paper, we propose a novel few-shot object detection network that aims at detecting objects of unseen categories with only a few annotated examples. Central to our method are our Attention-RPN, Multi-Relation Detector and Contrastive Training strategy, which exploit the similarity between the few shot support set and query set to detect novel objects while suppressing false detection in the background. To train our network, we contribute a new dataset that contains 1000 categories of various objects with high-quality annotations. To the best of our knowledge, this is one of the first datasets specifically designed for few-shot object detection. Once our few-shot network is trained, it can detect objects of unseen categories without further training or fine-tuning. Our method is general and has a wide range of potential applications. We produce a new state-of-the-art performance on different datasets in the few-shot setting. The dataset link is <https://github.com/fanq15/Few-Shot-Object-Detection-Dataset>. [^1]' author: - | Qi Fan[^2]  \ HKUST\ [qfanaa@cse.ust.hk]{} - | Wei Zhuo[ ^fnsymbol[1]{}^]{}\ Tencent\ [wei.zhuowx@gmail.com]{} - | Chi-Keung Tang\ HKUST\ [cktang@cse.ust.hk]{} - | Yu-Wing Tai\ Tencent\ [yuwingtai@tencent.com]{} bibliography: - 'main.bib' title: 'Few-Shot Object Detection with Attention-RPN and Multi-Relation Detector' --- Introduction ============ Existing object detection methods typically rely heavily on a huge amount of annotated data and require long training time. This has motivated the recent development of few-shot object detection. Few-shot learning is challenging given large variance of illumination, shape, texture, etc, in real-world objects. While significant research and progress have been made [@snell2017prototypical; @Sachin2017; @santoro2016meta; @vinyals2016matching; @Finn2017ModelAgnosticMF; @cai2018memory; @gidaris2018dynamic; @yang2018learning], all of these methods focus on image classification rarely tapping into the problem of few-shot object detection, most probably because transferring from few-shot classification to few-shot object detection is a non-trivial task. Central to object detection given only a few shots is how to localize an unseen object in a cluttered background, which in hindsight is a general problem of object localization from a few annotated examples in novel categories. Potential bounding boxes can easily miss unseen objects, or else many false detections in the background can be produced. We believe this is caused by the inappropriate low scores of good bounding boxes output from a region proposal network (RPN) making a novel object hard to be detected. This makes the few-shot object detection intrinsically different from few-shot classification. Recent works for few-shot object detection [@Chen2018LSTDAL; @kang2018few; @karlinsky2019repmet; @yan2019metarcnn] on the other hand all require fine-tuning and thus cannot be directly applied on novel categories. ![Given different objects as supports (top corners above), our approach can detect all objects in the same categories in the given query image.[]{data-label="fig:intro"}](intro_v6.jpg){width="7.5cm"} In this paper, we address the problem of few-shot object detection: given a few support images of novel target object, our goal is to detect all foreground objects in the test set that belong to the target object category, as shown in Fig. \[fig:intro\]. To this end, we propose two main contributions: First, we propose a general few-shot object detection model that can be applied to detect novel objects without re-training and fine-tuning. With our carefully designed contrastive training strategy, attention module on RPN and detector, our method exploits matching relationship between object pairs in a weight-shared network at multiple network stages. This enables our model to perform online detection on objects of novel categories requiring [*no*]{} fine-training or further network adaptation. Experiments show that our model can benefit from the attention module at the early stage where the proposal quality is significantly enhanced, and from the multi-relation detector module at the later stage which suppresses and filters out false detection in the confusing background. Our model achieves new state-of-the-art performance on the ImageNet Detection dataset and MS COCO dataset in the few-shot setting. The second contribution consists of a large well-annotated dataset with 1000 categories with only a few examples for each category. Overall, our method achieves significantly better performance by utilizing this dataset than existing large-scale datasets, COCO [@lin2014microsoft]. To the best of our knowledge, this is one of the first few-shot object detection datasets with an unprecedented number of object categories (1000). Using this dataset, our model achieves better performance on different datasets even without any fine-tuning. Related Works ============= [**General Object Detection.**]{} Object detection is a classical problem in computer vision. In early years, object detection was usually formulated as a sliding window classification problem using handcrafted features [@dalal2005histograms; @felzenszwalb2010object; @vioda2001rapid]. With the rise of deep learning [@NIPS2012_4824], CNN-based methods have become the dominant object detection solution. Most of the methods can be further divided into two general approaches: proposal-free detectors and proposal-based detectors. The first line of work follows a one-stage training strategy and does not explicitly generate proposal boxes [@redmon2016you; @redmon2017yolo9000; @liu2016ssd; @lin2017focal; @liu2018receptive]. On the other hand, the second line, pioneered by R-CNN [@girshick2014rich], first extracts class-agnostic region proposals of the potential objects from a given image. These boxes are then further refined and classified into different categories by a specific module [@girshick2015fast; @ren2015faster; @he2017mask; @singh2018sniper]. An advantage of this strategy is that it can filter out many negative locations by the RPN module which facilitates the detector task next. For this sake, RPN-based methods usually perform better than proposal-free methods with state-of-the-art results [@singh2018sniper] for the detection task. The methods mentioned above, however, work in an intensive supervision manner and are hard to extend to novel categories with only several examples. [**Few-shot learning.**]{} Few-shot learning in a classical setting [@thrun1996learning] is challenging for traditional machine learning algorithms to learn from just a few training examples. Earlier works attempted to learn a general prior [@fei2006one; @lake2011one; @lake2013one; @lake2015human; @wong2015one], such as hand-designed strokes or parts which can be shared across categories. Some works [@snell2017prototypical; @oreshkin2018tadam; @triantafillou2017few; @hariharan2017low] focus on metric learning in manually designing a distance formulation among different categories. A more recent trend is to design a general agent/strategy that can guide supervised learning within each task; by accumulating knowledge the network can capture the structure variety across different tasks. This research direction is named meta-learning in general [@Sachin2017; @Finn2017ModelAgnosticMF; @koch2015siamese; @munkhdalai2017meta; @Munkhdalai2018RapidAW]. In this area, a siamese network was proposed in [@koch2015siamese] that consists of twin networks sharing weights, where each network is respectively fed with a support image and a query. The distance between the query and its support is naturally learned by a logistic regression. This matching strategy captures inherent variety between support and query regardless of their categories. In the realm of matching framework, subsequent works [@santoro2016meta; @vinyals2016matching; @cai2018memory; @yang2018learning; @kang2018few; @wang2018low] had focused on enhancing feature embedding, where one direction is to build memory modules to capture global contexts among the supports. A number of works [@li2019DN4; @lifchitz2019dense] exploit local descriptors to reap additional knowledge from limited data. In [@kim2019egnn; @gidaris2019generating] the authors introduced Graph Neural Network (GNN) to model relationship between different categories. In [@li2019finding] the given entire support set was traversed to identify task-relevant features and to make metric learning in high-dimensional space more effective. Other works, such as [@Sachin2017; @finn2017model], dedicate to learning a general agent to guide parameter optimization. Until now, few-shot learning has not achieved groundbreaking progress, which has mostly focused on the classification task but rarely on other important computer vision tasks such as semantic segmentation [@dong2018few; @Michaelis2018OneShotSI; @Hu2019AttentionbasedMG], human motion prediction [@Gui2018FewShotHM] and object detection [@Chen2018LSTDAL]. In [@dong2018pami] unlabeled data was used and multiple modules were optimized alternately on images without box. However, the method may be misled by incorrect detection in weak supervision and requires re-training for a new category. In LSTD [@Chen2018LSTDAL] the authors proposed a novel few-shot object detection framework that can transfer knowledge from one large dataset to another smaller dataset, by minimizing the gap of classifying posterior probability between the source domain and the target domain. This method, however, strongly depends on the source domain and is hard to extend to very different scenarios. Recently, several other works for few-shot detection [@Chen2018LSTDAL; @kang2018few; @karlinsky2019repmet; @yan2019metarcnn] have been proposed but they learn category-specific embeddings and require to be fine-tuned for novel categories. Our work is motivated by the research line pioneered by the matching network [@koch2015siamese]. We propose a general few-shot object detection network that learns the matching metric between image pairs based on the Faster R-CNN framework equipped with our novel attention RPN and multi-relation detector trained using our contrastive training strategy. FSOD: A Highly-Diverse Few-Shot Object Detection Dataset {#section:dataset} ======================================================== {width="13.5cm"} The key to few-shot learning lies in the generalization ability of the pertinent model when presented with novel categories. Thus, a high-diversity dataset with a large number of object categories is necessary for training a general model that can detect unseen objects and for performing convincing evaluation as well. However, existing datasets [@lin2014microsoft; @Geiger2012CVPR; @everingham2010pascal; @OpenImages; @krishna2017visual] contain very limited categories and they are not designed in the few-shot evaluation setting. Thus we build a new few-shot object detection dataset. ![The dataset statistics of FSOD. The category image number are distributed almost averagely. Most classes (above 90%) has small or moderate amount of images (in \[22, 108\]), and the most frequent class still has no more than 208 images.[]{data-label="fig:dataset_hist"}](dataset.png){width="7cm"} [**Dataset Construction.**]{} We build our dataset from existing large-scale object detection datasets for supervised learning  [@OpenImages; @deng2009imagenet]. These datasets, however, cannot be used directly, due to 1) the label system of different datasets are inconsistent where some objects with the same semantics are annotated with different words in the datasets; 2) large portion of the existing annotations are noisy due to inaccurate and missing labels, duplicate boxes, objects being too large; 3) their train/test split contains the same categories, while for the few-shot setting we want the train/test sets to contain different categories in order to evaluate its generality on unseen categories. To start building the dataset, we first summarize a label system from [@OpenImages; @deng2009imagenet]. We merge the leaf labels in their original label trees, by grouping those in the same semantics (e.g., ice bear and polar bear) into one category, and removing semantics that do not belong to any leaf categories. Then, we remove the images with bad label quality and those with boxes of improper size. Specifically, removed images have boxes smaller than 0.05% of the image size which are usually in bad visual quality and unsuitable to serve as support examples. Next, we follow the few-shot learning setting to split our data into training set and test set without overlapping categories. We construct the training set with categories in MS COCO dataset [@lin2014microsoft] in case researchers prefer a pretraining stage. We then split the test set which contains 200 categories by choosing those with the largest distance with existing training categories, where the distance is the shortest path that connects the meaning of two phrases in the is-a taxonomy [@miller1995wordnet]. The remaining categories are merged into the training set that in total contains 800 categories. In all, we construct a dataset of 1000 categories with unambiguous category split for training and testing, where 531 categories are from ImageNet dataset [@deng2009imagenet] and 469 from Open Image dataset [@OpenImages]. [**Dataset Analysis.**]{} Our dataset is specifically designed for few-shot learning and for evaluating the generality of a model on novel categories, which contains 1000 categories with 800/200 split for training and test set respectively, around 66,000 images and 182,000 bounding boxes in total. Detailed statistics are shown in Table \[dataset\_summery\_table\] and Fig. \[fig:dataset\_hist\]. Our dataset has the following properties: [*High diversity in categories:*]{} Our dataset contains 83 parent semantics, such as mammal, clothing, weapon, etc, which are further split to 1000 leaf categories. Our label tree is shown in Fig. \[fig:dataset\_tree\]. Due to our strict dataset split, our train/test sets contain images of very different semantic categories thus presenting challenges to models to be evaluated. [*Challenging setting:*]{} Our dataset contains objects with large variance on box size and aspect ratios, consisting of 26.5% images with no less than three objects in the test set. Our test set contains a large number of boxes of categories [*not*]{} included in our label system, thus presenting great challenges for a few-shot model. {width="0.83\linewidth"} Train Test ------------------- ---------------- ------------------ No. Class 800 200 No. Image 52350 14152 No. Box 147489 35102 Avg No. Box / Img 2.82 2.48 Min No. Img / Cls 22 30 Max No. Img / Cls 208 199 Avg No. Img / Cls 75.65 74.31 Box Size \[6, 6828\] \[13, 4605\] Box Area Ratio \[0.0009, 1\] \[0.0009, 1\] Box W/H Ratio \[0.0216, 89\] \[0.0199, 51.5\] : Dataset Summary. Our dataset is diverse with large variance in box size and aspect ratio.[]{data-label="dataset_summery_table"} Although our dataset has a large number of categories, the number of training images and boxes are much less than other large-scale benchmark datasets such as MS COCO dataset, which contains 123,287 images and around 886,000 bounding boxes. Our dataset is designed to be compact while effective for few-shot learning. Our Methodology =============== In this section, we first define our task of few-shot detection, followed by a detailed description of our novel few-shot object detection network. Problem Definition ------------------ Given a support image $s_{c}$ with a close-up of the target object and a query image $q_{c}$ which potentially contains objects of the support category $c$, the task is to find all the target objects belonging to the support category in the query and label them with tight bounding boxes. If the support set contains $N$ categories and $K$ examples for each category, the problem is dubbed $N$-way $K$-shot detection. Deep Attentioned Few-Shot Detection ----------------------------------- We propose a novel attention network that learns a general matching relationship between the support set and queries on both the RPN module and the detector. Fig. \[fig:framework\] shows the overall architecture of our network. Specifically, we build a weight-shared framework that consists of multiple branches, where one branch is for the query set and the others are for the support set (for simplicity, we only show one support branch in the figure). The query branch of the weight-shared framework is a Faster R-CNN network, which contains RPN and detector. We utilize this framework to train the matching relationship between support and query features, in order to make the network learn general knowledge among the same categories. Based on the framework, we introduce a novel attention RPN and detector with multi-relation modules to produce an accurate parsing between support and potential boxes in the query. ### Attention-Based Region Proposal Network In few-shot object detection, RPN is useful in producing potentially relevant boxes for facilitating the following task of detection. Specifically, the RPN should not only distinguish between objects and non-objects but also filter out negative objects not belonging to the support category. However, without any support image information, the RPN will be aimlessly active in every potential object with high objectness score even though they do not belong to the support category, thus burdening the subsequent classification task of the detector with a large number of irrelevant objects. To address this problem, we propose the attention RPN (Fig. \[fig:rpn\]) which uses support information to enable filtering out most background boxes and those in non-matching categories. Thus a smaller and more precise set of candidate proposals is generated with high potential containing target objects. ![Attention RPN. The support feature is average pooled to a $1 \times 1 \times C$ vector. Then the depth-wise cross correlation with the query feature is computed whose output is used as attention feature to be fed into RPN for generating proposals.[]{data-label="fig:rpn"}](rpn-2.png){width="0.81\linewidth"} We introduce support information to RPN through the attention mechanism to guide the RPN to produce relevant proposals while suppressing proposals in other categories. Specifically, we compute the similarity between the feature map of support and that of the query in a depth-wise manner. The similarity map then is utilized to build the proposal generation. In particular, we denote the support features as $X\in t^{S \times S \times C}$ and feature map of the query as $Y\in t^{H\times W \times C}$, the similarity is defined as $$\begin{split} &\mathbf{G}_{h,w,c} = \sum_{i,j} X_{i,j,c} \cdot Y_{h+i-1,w+j-1,c}, \quad i,j\in \{1,...,S\} \end{split} \label{equ:attentionrpn}$$ where ${\mathbf G}$ is the resultant attention feature map. Here the support features $X$ is used as the kernel to slide on the query feature map [@bertinetto2016fully; @lu2018class] in a depth-wise cross correlation manner [@li2018siamrpn++]. In our work, we adopt the features of top layers to the RPN model, the res4\_6 in ResNet50. We find that a kernel size of $S=1$ performs well in our case. This fact is consistent with [@ren2015faster] that global feature can provide a good object prior for objectness classification. In our case, the kernel is calculated by averaging on the support feature map. The attention map is processed by a $3\times 3$ convolution followed by the objectiveness classification layer and box regression layer. The attention RPN with loss $L_{rpn}$ is trained jointly with the network as in [@ren2015faster]. ![Multi-Relation Detector. Different relation heads model different relationships between the query and support image. The global relation head uses global representation to match images; local relation head captures pixel-to-pixel matching relationship; patch relation head models one-to-many pixel relationship.[]{data-label="fig:head"}](head.png){width="5.8cm"} ### Multi-Relation Detector In an R-CNN framework, an RPN module will be followed by a detector whose important role is re-scoring proposals and class recognition. Therefore, we want a detector to have a strong discriminative ability to distinguish different categories. To this end, we propose a novel multi-relation detector to effectively measure the similarity between proposal boxes from the query and the support objects, see Fig. \[fig:head\]. The detector includes three attention modules, which are respectively the [**global-relation head**]{} to learn a deep embedding for global matching, the [**local-correlation head**]{} to learn the pixel-wise and depth-wise cross correlation between support and query proposals and the [**patch-relation head**]{} to learn a deep non-linear metric for patch matching. We experimentally show that the three matching modules can complement each other to produce higher performance. Refer to the supplemental material for implementation details of the three heads. [**Which relation heads do we need?**]{} We follow the $N$-way $K$-shot evaluation protocol proposed in RepMet [@schwartz2018repmet] to evaluate our relation heads and other components. Table \[table:head\] shows the ablation study of our proposed multi-relation detector under the naive 1-way 1-shot training strategy and 5-way 5-shot evaluation on the FSOD dataset. We use the same evaluation setting hereafter for all ablation studies on the FSOD dataset. For individual heads, the local-relation head performs best on both $AP_{50}$ and $AP_{75}$ evaluations. Surprisingly, the patch-relation head performs worse than other relation heads, although it models more complicated relationship between images. We believe that the complicated relation head makes the model difficult to learn. When combining any two types of relation head, we obtain better performance than that of individual head. By combining all relation heads, we obtain the full multi-relation detector and achieve the best performance, showing that the three proposed relation heads are complementary to each other for better differentiation of targets from non-matching objects. All the following experiments thus adopt the full multi-relation detector. Global R Local R Patch R $AP_{50}$ $AP_{75}$ ---------- --------- --------- -------------- -------------- 47.7 34.0 50.5 35.9 45.1 32.8 49.6 35.9 53.8 38.0 54.6 38.9 [**55.0**]{} [**39.1**]{} : Experimental results for different relation head combinations in the 1-way 1-shot training strategy.[]{data-label="table:head"} Two-way Contrastive Training Strategy ------------------------------------- A naive training strategy is matching the same category objects by constructing a training pair ($q_{c}$, $s_{c}$) where the query image $q_{c}$ and support image $s_{c}$ are both in the same $c$-th category object. However a good model should not only match the same category objects but also distinguish different categories. For this reason, we propose a novel 2-way contrastive training strategy. According to the different matching results in Fig. \[fig:training\], we propose the 2-way contrastive training to match the same category while distinguishing different categories. We randomly choose one query image $q_{c}$, one support image $s_{c}$ containing the same $c$-th category object and one other support image $s_{n}$ containing a different $n$-th category object, to construct the training triplet ($q_{c}$, $s_{c}$, $s_{n}$), where $c \neq n$. In the training triplet, only the $c$-th category objects in the query image are labeled as foreground while all other objects are treated as background. During training, the model learns to match every proposal generated by the attention RPN in the query image with the object in the support image. Thus the model learns to not only match the same category objects between ($q_{c}$, $s_{c}$) but also distinguish objects in different categories between ($q_{c}$, $s_{n}$). However, there are a massive amount of background proposals which usually dominate the training, especially with negative support images. For this reason, we balance the ratio of these matching pairs between query proposals and supports. We keep the ratio as 1:2:1 for the foreground proposal and positive support pairs $(p_f, s_p)$, background proposal and positive support pairs $(p_b, s_p)$, and proposal (foreground or background) and negative support pairs $(p, s_n)$. We pick all $N$ $(p_f, s_p)$ pairs and select top $2N$ $(p_b, s_p)$ pairs and top $N$ $(p, s_n)$ pairs respectively according to their matching scores and calculate the matching loss on the selected pairs. During training, we use the multi-task loss on each sampled proposal as $L = L_{matching} + L_{box}$ with the bounding-box loss $L_{box}$ as defined in [@girshick2015fast] and the matching loss being the binary cross-entropy. ![The 2-way contrastive training triplet and different matching results. Only the positive support has the same category with the target ground truth in the query image. The matching pair consists of the positive support and foreground proposal, and the non-matching pair has three categories: (1) positive support and background proposal, (2) negative support and foreground proposal and (3) negative support and background proposal.[]{data-label="fig:training"}](training.png){width="0.9\linewidth"} [**Which training strategy is better?**]{} Refer to Table \[table:training\]. We train our model with the 2-way 1-shot contrastive training strategy and obtain 7.9% $AP_{50}$ improvement compared with the naive 1-way 1-shot training strategy, which indicates the importance in learning how to distinguish different categories during training. With 5-shot training, we achieve further improvement which was also verified in [@snell2017prototypical] that few-shot training is beneficial to few-shot testing. It is straightforward to extend our 2-way training strategy to multi-way training strategy. However, from Table \[table:training\], the 5-way training strategy does not produce better performance than the 2-way training strategy. We believe that only one negative support category suffices in training the model for distinguishing different categories. Our full model thus adopts the 2-way 5-shot contrastive training strategy. [**Which RPN is better?**]{} We evaluate our attention RPN on different evaluation metrics. To evaluate the proposal quality, we first evaluate the recall on top 100 proposals over 0.5 IoU threshold of the regular RPN and our proposed attention RPN. Our attention RPN exhibits better recall performance than the regular RPN (0.9130 0.8804). We then evaluate the average best overlap ratio (ABO [@uijlings2013selective]) across ground truth boxes for these two RPNs. The ABO of attention RPN is 0.7282 while the same metric of regular RPN is 0.7127. These results indicate that the attention RPN can generate more high-quality proposals. Table \[table:training\] further compares models with attention RPN and those with the regular RPN in different training strategies. The model with attention RPN consistently performs better than the regular RPN on both $AP_{50}$ and $AP_{75}$ evaluation. The attention RPN produces 0.9%/2.0% gain in the 1-way 1-shot training strategy and 2.0%/2.1% gain in the 2-way 5-shot training strategy on the $AP_{50}$/$AP_{75}$ evaluation. These results confirm that our attention RPN generates better proposals and benefits the final detection prediction. The attention RPN is thus adopted in our full model. Experiments =========== In the experiments, we compare our approach with state-of-the-art (SOTA) methods on different datasets. We typically train our full model on FSOD training set and directly evaluate on these datasets. For fair comparison with other methods, we may discard training on FSOD and adopt the same train/test setting as these methods. In these cases, we use a multi-way[^3] few-shot training in the fine-tuning stage with more details to be described. Training Strategy Attention RPN $AP_{50}$ $AP_{75}$ ------------------- --------------- -------------- -------------- -- 1-way 1-shot 55.0 39.1 1-way 1-shot 55.9 41.1 2-way 1-shot 63.8 42.9 2-way 5-shot 65.4 43.7 2-way 5-shot [**67.5**]{} [**46.2**]{} 5-way 5-shot 66.9 45.6 : Experimental results for training strategy and attention RPN with the multi-relation detector.[]{data-label="table:training"} Training Details ---------------- Our model is trained end-to-end on 4 Tesla P40 GPUs using SGD with a batch size of 4 (for query images). The learning rate is 0.002 for the first 56000 iterations and 0.0002 for later 4000 iterations. We observe that pre-training on ImageNet [@deng2009imagenet] and MS COCO [@lin2014microsoft] can provide stable low-level features and lead to a better converge point. Given this, we by default train our model from the pre-trained ResNet50 on [@lin2014microsoft; @deng2009imagenet] unless otherwise stated. During training, we find that more training iterations may damage performance, where too many training iterations make the model over-fit to the training set. We fix the weights of Res1-3 blocks and only train high-level layers to utilize low-level basic features and avoid over-fitting. The shorter side of the query image is resized to 600 pixels; the longer side is capped at 1000. The support image is cropped around the target object with 16-pixel image context, zero-padded and then resized to a square image of $320 \times 320$. For few-shot training and testing, we fuse feature by averaging the object features with the same category and then feed them to the attention RPN and the multi-relation detector. We adopt the typical metrics [@lin2017focal], i.e. *AP*, *AP$_{50}$* and *AP$_{75}$* for evaluation. Comparison with State-of-the-Art Methods ---------------------------------------- ### ImageNet Detection dataset In Table \[table:imagenet\], we compare our results with those of LSTD [@Chen2018LSTDAL] and RepMet [@schwartz2018repmet] on the challenging ImageNet based 50-way 5-shot detection scenario. For fair comparison, we use their evaluation protocol and testing dataset and we use the same MS COCO training set to train our model. We also use soft-NMS [@Bodla2017Soft] as RepMet during evaluation. Our approach produces 1.7% performance gain compared to the state-of-the-art (SOTA) on the $AP_{50}$ evaluation. To show the generalization ability of our approach, we directly apply our model trained on FSOD dataset on the test set and we obtain 41.7% on the $AP_{50}$ evaluation which is surprisingly better than our fine-tuned model (Table \[table:imagenet\]). It should be noted that our model trained on FSOD dataset can be directly applied on the test set without fine-tuning to achieve SOTA performance. Furthermore, although our model trained on FSOD dataset has a slightly better $AP_{50}$ performance than our fine-tuned model on the MS COCO dataset, our model surpasses the fine-tuned model by 6.4% on the $AP_{75}$ evaluation, which shows that our proposed FSOD dataset significantly benefits few-shot object detection. With further fine-tuning our FSOD trained model on the test set, our model achieves the best performance, while noting that our method without fine-tuning already works best compared with SOTA. Method dataset fine-tune $AP_{50}$ $AP_{75}$ ------------------------------- ------------- --------------- -------------- -------------- LSTD [@Chen2018LSTDAL] COCO $^{ImageNet}$ 37.4 - RepMet [@karlinsky2019repmet] COCO $^{ImageNet}$ 39.6 - Ours COCO $^{ImageNet}$ 41.3 21.9 Ours FSOD$^\dag$ 41.7 28.3 Ours FSOD$^\dag$ $^{ImageNet}$ [**44.1**]{} [**31.0**]{} : Experimental results on ImageNet Detection dataset for 50 novel categories with 5 supports. $^\dagger$ means that the testing categories are removed from FSOD training dataset. $^{ImageNet}$ means the model is fine-tuned on ImageNet Detection dataset.[]{data-label="table:imagenet"} Method dataset fine-tune $AP$ $AP_{50}$ $AP_{75}$ ------------------------- ------------- ----------- -------------- -------------- -------------- FR [@kang2018few] COCO $^{coco}$ 5.6 12.3 4.6 Meta [@yan2019metarcnn] COCO $^{coco}$ 8.7 19.1 6.6 Ours COCO $^{coco}$ 11.1 20.4 10.6 Ours FSOD$^\dag$ [**16.6**]{} [**31.3**]{} [**16.1**]{} : Experimental results on MS COCO minival set for 20 novel categories with 10 supports. $^\dagger$ means that the testing categories are removed from FSOD training dataset. $^{coco}$ means the model is fine-tuned on MS COCO dataset.[]{data-label="table:coco"} ### MS COCO dataset {#coco} In Table \[table:coco\], we compare our approach[^4] with Feature Reweighting [@kang2018few] and Meta R-CNN [@yan2019metarcnn] on MS COCO minival set. We follow their data split and use the same evaluation protocol: we set the 20 categories included in PASCAL VOC as novel categories for evaluation, and use the rest 60 categories in MS COCO as training categories. Our fine-tuned model with the same MS COCO training dataset outperforms Meta R-CNN by 2.4%/1.3%/4.0% on $AP$/$AP_{50}$/$AP_{75}$ metrics. This demonstrates the strong learning and generalization ability of our model, as well as that, in the few-shot scenario, learning general matching relationship is more promising than the attempt to learn category-specific embeddings [@kang2018few; @yan2019metarcnn]. Our model trained on FSOD achieves more significant improvement of 7.9%/12.2%/9.5% on $AP$/$AP_{50}$/$AP_{75}$ metrics. Note that our model trained on FSOD dataset are directly applied on the novel categories without any further fine-tuning while all other methods use 10 supports for fine-tuning to adapt to the novel categories. Again, without fine-tuning our FSOD-trained model already works the best among SOTAs. {width="0.82\linewidth"} \[fig:vis\] {width="0.83\linewidth"} Realistic Applications {#applications} ---------------------- We apply our approach in different real-world application scenarios to demonstrate its generalization capability. Fig. \[fig:vis\] shows qualitative 1-shot object detection results on novel categories in our test set. We further apply our approach on the wild penguin detection [@Arteta16] and show sample qualitative 5-shot object detection results in Fig. \[fig:penguin\]. Method FSOD pretrain fine-tune $AP_{50}$ $AP_{75}$ ------------------------ ------------------ ----------- -------------- -------------- -- -- -- FRCNN [@ren2015faster] $^{fsod}$ 11.8 6.7 FRCNN [@ren2015faster] $^{fsod}$ 23.0 12.9 LSTD [@Chen2018LSTDAL] $^{fsod}$ 24.2 13.5 Ours trained directly [**27.5**]{} [**19.4**]{} : Experimental results on FSOD test set for 200 novel categories with 5 supports evaluated in novel category detection. $^{fsod}$ means the model is fine-tuned on FSOD dataset.[]{data-label="table:fsod"} [**Novel Category Detection.**]{} Consider this common real-world application scenario: given a massive number of images in a photo album or TV drama series without any labels, the task is to annotate a novel target object (e.g., a rocket) in the given massive collection without knowing which images contain the target object, which can be in different sizes and locations if present. In order to reduce manual labor, one solution is to manually find a small number of images containing the target object, annotate them, and then apply our method to automatically annotate the rest in the image collection. Following this setting, we perform the evaluation as follows: We mix all test images of FSOD dataset, and for each object category, we pick 5 images that contain the target object to perform this novel category object detection in the entire test set. Note that different from the standard object detection evaluation, in this evaluation, the model evaluates every category separately and has no knowledge of the complete categories. We compare with LSTD [@Chen2018LSTDAL] which needs to be trained on novel categories by transferring knowledge from the source to target domain. Our method, however, can be applied to detect object in novel categories **without any further re-training or fine-tuning**, which is fundamentally different from LSTD. To compare empirically, we adjust LSTD to base on Faster R-CNN and re-train it on 5 fixed supports for each test category separately in a fair configuration. Results are shown in Table \[table:fsod\]. Our method outperforms LSTD by 3.3%/5.9% and its backbone Faster R-CNN by 4.5%/6.5% on all 200 testing categories on $AP_{50}/AP_{75}$ metrics. More specifically, without pre-training on our dataset, the performance of Faster R-CNN significantly drops. Note that because the model only knows the support category, the fine-tuning based models need to train every category separately which is time-consuming. [**Wild Car Detection.**]{} We apply our method[^5] to wild car detection on KITTI [@Geiger2012CVPR] and Cityscapes [@Cordts2016Cityscapes] datasets which are urban scene datasets for driving applications, where the images are captured by car-mounted video cameras. We evaluate the performance of *Car* category on KITTI training set with 7481 images and Cityscapes validation set with 500 images. DA Faster R-CNN [@chen2018domain] uses massively annotated data from source domains (KITTI/Cityscapes) and unlabeled data from target domains (Cityscapes/KITTI) to train the domain adaptive Faster R-CNN, and evaluated the performance on target domains. Without any further re-training or fine-tuning, our model with 10-shot supports obtains comparable or even better $AP_{50}$ performance (37.0% 38.5% on Cityscapes and 67.4% 64.1% on KITTI) on the wild car detection task. Note that DA Faster R-CNN are specifically designed for the wild car detection task and they use much more training data in similar domains. More Categories [****]{} More Samples? {#fsod_result} -------------------------------------- Our proposed dataset has a large number of object categories but with few image samples in each category, which we claim is beneficial to few-shot object detection. To confirm this benefit, we train our model on MS COCO dataset, which has more than 115,000 images with only 80 categories. Then we train our model on FSOD dataset with different category numbers while keeping similar number of training image. Table \[table:category\] summarizes the experimental results, where we find that although MS COCO has the most training images but its model performance turns out to be the worst, while models trained on FSOD dataset have better performance as the number of categories incrementally increases while keeping similar number of training images, indicating that a limited number of categories with too many images can actually impede few-shot object detection, while large number of categories can consistently benefit the task. Thus, we conclude that category diversity is essential to few-shot object detection. Dataset No. Class No. Image $AP_{50}$ $AP_{75}$ -------------------------- ----------- ----------- -------------- -------------- COCO [@lin2014microsoft] 80 115k 49.1 28.9 FSOD 300 26k 60.3 39.1 FSOD 500 26k 62.7 41.9 FSOD 800 27k [**64.7**]{} [**42.6**]{} : Experimental results of our model on FSOD test set with different numbers of training categories and images in the 5-way 5-shot evaluation.[]{data-label="table:category"} Conclusion ========== We introduce a novel few-shot object detection network with Attention-RPN, Multi-Relation Detectors and Contrastive Training strategy. We contribute a new FSOD which contains 1000 categories of various objects with high-quality annotations. Our model trained on FSOD can detect objects of novel categories requiring no pre-training or further network adaptation. Our model has been validated by extensive quantitative and qualitative results on different datasets. This paper contributes to few-shot object detection and we believe worthwhile and related future work can be spawn from our large-scale FSOD dataset and detection network with the above technical contributions. Appendix A: Implementation Details of Multi-Relation Detector {#appendix-a-implementation-details-of-multi-relation-detector .unnumbered} ============================================================= Given the support feature $f_s$ and query proposal feature $f_q$ with the size of $ 7 \times 7 \times C$, our multi-relation detector is implemented as follows. We use the sum of all matching scores from the three heads as the final matching scores. #### Global-Relation Head We concatenate $f_s$ and $f_q$ to the concatenated feature $f_c$ with the size of $ 7 \times 7 \times 2C$. Then we average pool $f_c$ to a $1\times 1\times 2C$ vector. We then use an MLP with two fully connected (fc) layers with ReLU and a final fc layer to process $f_c$ and generate matching scores. #### Local-Relation Head We first use a weight-shared $1\times 1\times C$ convolution to process $f_s$ and $f_q$ separately. Then we calculate the depth-wise similarity using the equation in Section 4.2.1 of the main paper with $S=H=W=7$. Then we use a fc layer to generate matching scores. #### Patch-Relation Head We first concatenate $f_s$ and $f_q$ to the concatenated feature $f_c$ with the size of $ 7 \times 7 \times 2C$. Then $f_c$ is fed into the patch-relation module, whose structure is shown in Table \[table:patch\_relation\]. All the convolution layers followed by ReLU and pooling layers in this module have zero padding to reduce the feature map size from $7\times7$ to $1\times1$. Then we use a fc layer to generate matching scores and a separate fc layer to generate bounding box predictions. Appendix B: More Implementation Details {#appendix-b-more-implementation-details .unnumbered} ======================================= B.1. Training and Fine-tuning details {#b.1.-training-and-fine-tuning-details .unnumbered} ------------------------------------- Here we show more details for the experiments in Section 5.2 of the main paper. In Section 5.2, we follow other methods to train our model on MS COCO dataset [@lin2014microsoft] and fine-tune on the target datasets. When we train our model on MS COCO, we remove the images with boxes smaller than the size of $32 \times 32$. Those boxes are usually in bad visual quality and hurt the training when they serve as support examples. When we fine-tune our model on the target datasets, we follow the same setting of other methods[@Chen2018LSTDAL; @kang2018few; @karlinsky2019repmet; @yan2019metarcnn] for fair comparison. Specifically, LSTD [@Chen2018LSTDAL] and RepMet [@karlinsky2019repmet] use 5 support images per category where each image contains one or more object instances, and the Feature Reweighting [@kang2018few] and Meta R-CNN [@yan2019metarcnn] use a strict rule to adopt 10 object instances per category for fine-tuning. B.2. Evaluation details {#b.2.-evaluation-details .unnumbered} ----------------------- There are two evaluation settings in the main paper. [**Evaluation setting 1:**]{} The ablation experiments adopt the episode-based evaluation protocol defined in RepMet [@karlinsky2019repmet], where the setting is borrowed from the few-shot classification task [@Sachin2017; @vinyals2016matching]. There are 600 random evaluation episodes in total, which guarantee every image in the test set can be evaluated in a high probability. In each episode, for $N$-way $K$-shot evaluation, there are $K$ support images for each of the $N$ categories, and there are 10 query images for each category where each query image containing at least one instance belonging to this category. So there are $K \times N$ supports and $10 \times N$ query images in each episode. Note that all these categories and images are randomly chosen in each episode. [**Evaluation setting 2:**]{} Other comparison experiments with baselines adopt the standard object detection evaluation protocol, which is a full-way, N-shot evaluation. During evaluation, the support branches in our model can be discarded once the support features are attained, then the support features serve as model weights for the forward process. Type Filter Shape Stride/Padding ---------- -------------- ---------------- Avg Pool 3x3x4096 s1/p0 Conv 1x1x512 s1/p0 Conv 3x3x512 s1/p0 Conv 1x1x2048 s1/p0 Avg Pool 3x3x2048 s1/p0 : Architecture of the patch-relation module.[]{data-label="table:patch_relation"} Appendix D: FSOD Dataset Class Split {#appendix-d-fsod-dataset-class-split .unnumbered} ==================================== Here we describe the training/testing class split in our proposed FSOD Dataset. This split was used in our experiments. Training Class Split {#training-class-split .unnumbered} -------------------- lipstick, sandal, crocodile, football helmet, umbrella, houseplant, antelope, woodpecker, palm tree, box, swan, miniskirt, monkey, cookie, scissors, snowboard, hedgehog, penguin, barrel, wall clock, strawberry, window blind, butterfly, television, cake, punching bag, picture frame, face powder, jaguar, tomato, isopod, balloon, vase, shirt, waffle, carrot, candle, flute, bagel, orange, wheelchair, golf ball, unicycle, surfboard, cattle, parachute, candy, turkey, pillow, jacket, dumbbell, dagger, wine glass, guitar, shrimp, worm, hamburger, cucumber, radish, alpaca, bicycle wheel, shelf, pancake, helicopter, perfume, sword, ipod, goose, pretzel, coin, broccoli, mule, cabbage, sheep, apple, flag, horse, duck, salad, lemon, handgun, backpack, printer, mug, snowmobile, boot, bowl, book, tin can, football, human leg, countertop, elephant, ladybug, curtain, wine, van, envelope, pen, doll, bus, flying disc, microwave oven, stethoscope, burrito, mushroom, teddy bear, nail, bottle, raccoon, rifle, peach, laptop, centipede, tiger, watch, cat, ladder, sparrow, coffee table, plastic bag, brown bear, frog, jeans, harp, accordion, pig, porcupine, dolphin, owl, flowerpot, motorcycle, calculator, tap, kangaroo, lavender, tennis ball, jellyfish, bust, dice, wok, roller skates, mango, bread, computer monitor, sombrero, desk, cheetah, ice cream, tart, doughnut, grapefruit, paddle, pear, kite, eagle, towel, coffee, deer, whale, cello, lion, taxi, shark, human arm, trumpet, french fries, syringe, lobster, rose, human hand, lamp, bat, ostrich, trombone, swim cap, human beard, hot dog, chicken, leopard, alarm clock, drum, taco, digital clock, starfish, train, belt, refrigerator, dog bed, bell pepper, loveseat, infant bed, training bench, milk, mixing bowl, knife, cutting board, ring binder, studio couch, filing cabinet, bee, caterpillar, sofa bed, violin, traffic light, airplane, closet, canary, toilet paper, canoe, spoon, fox, tennis racket, red panda, cannon, stool, zucchini, rugby ball, polar bear, bench, pizza, fork, barge, bow and arrow, kettle, goldfish, mirror, snail, poster, drill, tie, gondola, scale, falcon, bull, remote control, horn, hamster, volleyball, stationary bicycle, dishwasher, limousine, shorts, toothbrush, bookcase, baseball glove, computer mouse, otter, computer keyboard, shower, teapot, human foot, parking meter, ski, beaker, castle, mobile phone, suitcase, sock, cupboard, crab, common fig, missile, swimwear, saucer, popcorn, coat, plate, stairs, pineapple, parrot, fountain, binoculars, tent, pencil case, mouse, sewing machine, magpie, handbag, saxophone, panda, flashlight, baseball bat, golf cart, banana, billiard table, tower, washing machine, lizard, brassiere, ant, crown, oven, sea lion, pitcher, chest of drawers, crutch, hippopotamus, artichoke, seat belt, microphone, lynx, camel, rabbit, rocket, toilet, spider, camera, pomegranate, bathtub, jug, goat, cowboy hat, wrench, stretcher, balance beam, necklace, scoreboard, horizontal bar, stop sign, sushi, gas stove, tank, armadillo, snake, tripod, cocktail, zebra, toaster, frying pan, pasta, truck, blue jay, sink, lighthouse, skateboard, cricket ball, dragonfly, snowplow, screwdriver, organ, giraffe, submarine, scorpion, honeycomb, cream, cart, koala, guacamole, raven, drawer, diaper, fire hydrant, potato, porch, banjo, hammer, paper towel, wardrobe, soap dispenser, asparagus, skunk, chainsaw, spatula, ambulance, submarine sandwich, axe, ruler, measuring cup, scarf, squirrel, tea, whisk, food processor, tick, stapler, oboe, hartebeest, modem, shower cap, mask, handkerchief, falafel, clipper, croquette, house finch, butterfly fish, lesser scaup, barbell, hair slide, arabian camel, pill bottle, springbok, camper, basketball player, bumper car, wisent, hip, wicket, medicine ball, sweet orange, snowshoe, column, king charles spaniel, crane, scoter, slide rule, steel drum, sports car, go kart, gearing, tostada, french loaf, granny smith, sorrel, ibex, rain barrel, quail, rhodesian ridgeback, mongoose, red backed sandpiper, penlight, samoyed, pay phone, barber chair, wool, ballplayer, malamute, reel, mountain goat, tusker, longwool, shopping cart, marble, shuttlecock, red breasted merganser, shutter, stamp, letter opener, canopic jar, warthog, oil filter, petri dish, bubble, african crocodile, bikini, brambling, siamang, bison, snorkel, loafer, kite balloon, wallet, laundry cart, sausage dog, king penguin, diver, rake, drake, bald eagle, retriever, slot, switchblade, orangutan, chacma, guenon, car wheel, dandie dinmont, guanaco, corn, hen, african hunting dog, pajama, hay, dingo, meat loaf, kid, whistle, tank car, dungeness crab, pop bottle, oar, yellow lady’s slipper, mountain sheep, zebu, crossword puzzle, daisy, kimono, basenji, solar dish, bell, gazelle, agaric, meatball, patas, swing, dutch oven, military uniform, vestment, cavy, mustang, standard poodle, chesapeake bay retriever, coffee mug, gorilla, bearskin, safety pin, sulphur crested cockatoo, flamingo, eider, picket fence, dhole, spaghetti squash, african elephant, coral fungus, pelican, anchovy pear, oystercatcher, gyromitra, african grey, knee pad, hatchet, elk, squash racket, mallet, greyhound, ram, racer, morel, drumstick, bovine, bullet train, bernese mountain dog, motor scooter, vervet, quince, blenheim spaniel, snipe, marmoset, dodo, cowboy boot, buckeye, prairie chicken, siberian husky, ballpoint, mountain tent, jockey, border collie, ice skate, button, stuffed tomato, lovebird, jinrikisha, pony, killer whale, indian elephant, acorn squash, macaw, bolete, fiddler crab, mobile home, dressing table, chimpanzee, jack o’ lantern, toast, nipple, entlebucher, groom, sarong, cauliflower, apiary, english foxhound, deck chair, car door, labrador retriever, wallaby, acorn, short pants, standard schnauzer, lampshade, hog, male horse, martin, loudspeaker, plum, bale, partridge, water jug, shoji, shield, american lobster, nailfile, poodle, jackfruit, heifer, whippet, mitten, eggnog, weimaraner, twin bed, english springer, dowitcher, rhesus, norwich terrier, sail, custard apple, wassail, bib, bullet, bartlett, brace, pick, carthorse, ruminant, clog, screw, burro, mountain bike, sunscreen, packet, madagascar cat, radio telescope, wild sheep, stuffed peppers, okapi, bighorn, grizzly, jar, rambutan, mortarboard, raspberry, gar, andiron, paintbrush, running shoe, turnstile, leonberg, red wine, open face sandwich, metal screw, west highland white terrier, boxer, lorikeet, interceptor, ruddy turnstone, colobus, pan, white stork, stinkhorn, american coot, trailer truck, bride, afghan hound, motorboat, bassoon, quesadilla, goblet, llama, folding chair, spoonbill, workhorse, pimento, anemone fish, ewe, megalith, pool ball, macaque, kit fox, oryx, sleeve, plug, battery, black stork, saluki, bath towel, bee eater, baboon, dairy cattle, sleeping bag, panpipe, gemsbok, albatross, comb, snow goose, cetacean, bucket, packhorse, palm, vending machine, butternut squash, loupe, ox, celandine, appenzeller, vulture, crampon, backboard, european gallinule, parsnip, jersey, slide, guava, cardoon, scuba diver, broom, giant schnauzer, gordon setter, staffordshire bullterrier, conch, cherry, jam, salmon, matchstick, black swan, sailboat, assault rifle, thatch, hook, wild boar, ski pole, armchair, lab coat, goldfinch, guinea pig, pinwheel, water buffalo, chain, ocarina, impala, swallow, mailbox, langur, cock, hyena, marimba, hound, knot, saw, eskimo dog, pembroke, sealyham terrier, italian greyhound, shih tzu, scotch terrier, yawl, lighter, dung beetle, dugong, academic gown, blanket, timber wolf, minibus, joystick, speedboat, flagpole, honey, chessman, club sandwich, gown, crate, peg, aquarium, whooping crane, headboard, okra, trench coat, avocado, cayuse, large yellow lady’s slipper, ski mask, dough, bassarisk, bridal gown, terrapin, yacht, saddle, redbone, shower curtain, jennet, school bus, otterhound, irish terrier, carton, abaya, window shade, wooden spoon, yurt, flat coated retriever, bull mastiff, cardigan, river boat, irish wolfhound, oxygen mask, propeller, earthstar, black footed ferret, rocking chair, beach wagon, litchi, pigeon.\ Testing Class Split {#testing-class-split .unnumbered} ------------------- beer, musical keyboard, maple, christmas tree, hiking equipment, bicycle helmet, goggles, tortoise, whiteboard, lantern, convenience store, lifejacket, squid, watermelon, sunflower, muffin, mixer, bronze sculpture, skyscraper, drinking straw, segway, sun hat, harbor seal, cat furniture, fedora, kitchen knife, hand dryer, tree house, earrings, power plugs and sockets, waste container, blender, briefcase, street light, shotgun, sports uniform, wood burning stove, billboard, vehicle registration plate, ceiling fan, cassette deck, table tennis racket, bidet, pumpkin, tablet computer, rhinoceros, cheese, jacuzzi, door handle, swimming pool, rays and skates, chopsticks, oyster, office building, ratchet, salt and pepper shakers, juice, bowling equipment, skull, nightstand, light bulb, high heels, picnic basket, platter, cantaloupe, croissant, dinosaur, adhesive tape, mechanical fan, winter melon, egg, beehive, lily, cake stand, treadmill, kitchen & dining room table, headphones, wine rack, harpsichord, corded phone, snowman, jet ski, fireplace, spice rack, coconut, coffeemaker, seahorse, tiara, light switch, serving tray, bathroom cabinet, slow cooker, jalapeno, cartwheel, laelia, cattleya, bran muffin, caribou, buskin, turban, chalk, cider vinegar, bannock, persimmon, wing tip, shin guard, baby shoe, euphonium, popover, pulley, walking shoe, fancy dress, clam, mozzarella, peccary, spinning rod, khimar, soap dish, hot air balloon, windmill, manometer, gnu, earphone, double hung window, conserve, claymore, scone, bouquet, ski boot, welsh poppy, puffball, sambuca, truffle, calla lily, hard hat, elephant seal, peanut, hind, jelly fungus, pirogi, recycling bin, in line skate, bialy, shelf bracket, bowling shoe, ferris wheel, stanhopea, cowrie, adjustable wrench, date bread, o ring, caryatid, leaf spring, french bread, sergeant major, daiquiri, sweet roll, polypore, face veil, support hose, chinese lantern, triangle, mulberry, quick bread, optical disk, egg yolk, shallot, strawflower, cue, blue columbine, silo, mascara, cherry tomato, box wrench, flipper, bathrobe, gill fungus, blackboard, thumbtack, longhorn, pacific walrus, streptocarpus, addax, fly orchid, blackberry, kob, car tire, sassaby, fishing rod, baguet, trowel, cornbread, disa, tuning fork, virginia spring beauty, samosa, chigetai, blue poppy, scimitar, shirt button. {width="1.0\linewidth"} {width="1.0\linewidth"} {width="0.99\linewidth"} {width="0.99\linewidth"} [^1]: This research is supported in part by Tencent and the Research Grant Council of the Hong Kong SAR under grant no. 1620818. [^2]: Both authors contributed equally. [^3]: The fine-tuning stage benefits from more ways during the multi-way training, so we use as many ways as possible to fill up the GPU memory. [^4]: Since Feature Reweighting and Meta R-CNN are evaluated on MS COCO, in this subsection we discard pre-training on [@lin2014microsoft] for fair comparison to follow the same experimental setting as described. [^5]: We also discard the MS COCO pretraining in this experiment.

--- abstract: 'Radiative association cross sections and rates are computed, using a quantum approach, for the formation of C$_2$ molecules (dicarbon) during the collision of two ground state C($^3$P) atoms. We find that transitions originating in the C$\;^1\Pi_g$, d$\;^3\Pi_g$, and 1$\;^5\Pi_u$ states are the main contributors to the process. The results are compared and contrasted with previous results obtained from a semi-classical approximation. New *ab initio* potential curves and transition dipole moment functions have been obtained for the present work using the multi-reference configuration interaction approach with the Davidson correction (MRCI+Q) and aug-cc-pCV5Z basis sets, substantially increasing the available molecular data on dicarbon. Applications of the current computations to various astrophysical environments and laboratory studies are briefly discussed focusing on these rates.' author: - 'James F. Babb' - 'R. T. Smyth' - 'B. M. McLaughlin' date: Accepted 2019 March 14 title: Dicarbon formation in collisions of two carbon atoms --- \[firstpage\] Introduction {#sec:introduction} ============ Dicarbon (C$_2$) was first observed spectroscopically in flames and arcs and continues to be a useful diagnostic there, and in carbon plasmas for other laboratory and industrial applications [@nemes11]. The molecule has been observed in a host of extraterrestrial sources such as comets, carbon stars, protoplanetary nebulae, and molecular clouds. Interstellar dicarbon has been detected at optical wavelengths in diffuse [@ChaLut78; @ewine93; @gredel01] and translucent [@ewine89; @iglesias-groth11] molecular clouds. The optical detection of C$_2$ in comets is an element of their classification into “typical” and “depleted” [@ahearn95; @cochran12]. Dicarbon is present in solar [@lambert78] and model stellar atmospheres, including the pioneering work of @tsuji64 and @lord65, and seen, for example, in solar optical [@grevesse73] and infrared spectra [@brault82] and in infrared spectra of carbon-rich giant stars [@goebel83; @loidl01]. The presence of the Swan bands of dicarbon (optical wavelengths) is an important element in the classification scheme of carbon stars [@keenan93; @green13]. The mechanisms of formation of dicarbon vary, depending on the operative chemistries: For example, in diffuse molecular clouds dissociative recombination of $\mbox{CH}_2^+$ leads to $\mbox{C}_2$ [@black77; @federman89], while in comets a chemistry starting with photodissociation of $\mathrm{C}_2\mathrm{H}_2$ or $\mathrm{C}_2\mathrm{H}$ may be operative [@jackson76]. Of particular interest for the present work is the formation of carbonaceous dust in the ejecta of core-collapse supernovae, where the formation of dicarbon through radiative association enters chemical models [@liu92; @cherchneff09; @clayton18; @sluder18] and is an initial step in models of formation of larger carbon clusters through condensation [@clayton99; @clayton01; @clayton13] or nucleation [@lazzati16]. \[Later, in Sec. \[sec:discussion\], we discuss in more detail laboratory experiments on carbon vapors generated by laser radiation. We note at this point that evidence of associative collisions of two ground state carbon atoms was found by @monchicourt91 in light emission from laser-induced expansion of carbon vapor from a graphite rod.\] ![Experimentally observed C$_2$ band systems connecting eleven singlet, triplet, and quintet electronic states dissociating to ground state carbon atoms; schematic illustration of electronic state term energies $T_e$ in $\mbox{cm}^{-1}$ calculated in the present work \[after @messerle67 [@tanabashi07; @bornhauser15; @macrae16; @furtenbacher16]\].[]{data-label="fig:bands"}](Fig1){width="\textwidth"} Fig. \[fig:bands\] illustrates a sample of the experimentally observed bands [@tanabashi07; @bornhauser15; @macrae16; @furtenbacher16] connecting eleven singlet, triplet, and quintet states of the C$_2$ molecule that contribute to the overall radiative association rate coefficient for this molecule. In the ejecta of SN1987A and other core-collapse supernovae, CO and SiO were detected, see [@cherchneff11; @sarangi18], through fundamental $(\Delta\nu=1)$ bands[^1] of ground molecular electronic states allowing observational tests of molecular formation models [@liu92; @liu95; @cherchneff11; @sarangi13; @rho18]—dicarbon, however, lacks a permanent electric dipole moment and analogous vibrational transitions (fundamental bands) do not exist, making reliable theoretical predictions of rate coefficients imperative. Moreover, recent three-dimensional mapping of CO and SiO in the SN 1987A core ejecta with the Atacama Large Millimeter/submillimeter Array (ALMA) shows a clumpy mixed structure calling for improvements beyond one-dimension in hydrodynamical and chemical modeling of molecular formation [@abellan17]; a reliable description of dicarbide formation might improve such future calculations. Finally, understanding the origins of cosmic dust and the roles played by supernovae in contributing to extragalactic dust depends on progress in modeling dust formation [@sarangi18; @sluder18]. The radiative association (RA) rate was originally estimated to have a rate coefficient $k_{C_2} \approx 1 \times 10^{-17}$ cm$^3$/s [@prasad80; @millar91] for theoretical models of interstellar clouds and subsequent semi-classical calculations [@andreazza97] found comparable values with a weak temperature dependence increasing from $3.07 \times 10^{-18}$ cm$^3$/s at 300 K to $1.65 \times 10^{-17}$ cm$^3$/s at $14,700$ K. In recent studies using a quantum approach on systems such as SiP [@golubev13], SiO [@forrey16; @forrey17] and CS [@patillo18; @forrey18], it was found that the semi-classical calculations [@andreazza97; @andreazza06] underestimated the cross sections and rates, particularly at low temperatures. In the present study we obtain results from a quantum approach to estimate the cross sections and rate coefficients for C$_2$ formation by radiative association using new highly accurate ab initio molecular data for the potential energy curves (PEC’s) and transition dipole moments (TDM’s) coupling the states of interest. Results from our present quantum approach are compared with the previous semi-classical results of @andreazza97 and conclusions are drawn. The layout of this paper is as follows. An overview of how the molecular data is obtained for our dynamical calculations is presented in . In , a brief overview of the radiative association cross section and rates are outlined. The computed radiative association cross sections, and rates are presented in and are compared with the previous semi-classical work of @andreazza97 in . Finally in conclusions are drawn from our work. Atomic units are used throughout unless otherwise specified. Theory and Calculations {#sec:theory} ======================= Potential Curves and Transition Dipole Moments {#sec:pecstdms} ---------------------------------------------- In a similar manner to our recent all electron molecular structure and resulting dynamical studies on diatomic systems such as: SiO [@forrey16; @forrey17], CS [@patillo18; @forrey18], HeC$^+$ [@babb17a], SH$^+$ [@shen15; @mcmillan16], CH$^+$ [@babb17b], and HeAr$^+$ [@babb18a], the potential energy curves (PECs) and transition dipole moments (TDMs) for the eighteen singlet, triplet and quintet electronic states are calculated within an MRCI+Q approximation for the approach of ground state carbon atoms. That is, we use a state-averaged-multi-configuration-self-consistent-field (SA-MCSCF) approach, followed by multi-reference configuration interaction (MRCI) calculations together with the Davidson correction (MRCI+Q) [@Helgaker2000]. The SA-MCSCF method is used as the reference wave function for the MRCI calculations. Low-lying singlet, triplet and quintet electronic states and the transition dipole matrix elements connecting these molecular states are calculated and used in the present dynamical calculations for the radiative association process. The literature on the molecular properties of dicarbon is extensive; sources with comprehensive bibliographies include @martin92 [@nemes11; @zhang11; @boschen14; @macrae16; @furtenbacher16; @yurchenko18; @varandas18]. Potential energy curves and transition dipole moments as a function of internuclear distance $R$ are calculated starting from a bond separation of $R = 1.5$ Bohr extending out to $R=20$ Bohr. The basis sets used in the present work are the augmented correlation consistent polarized core valence quintuplet \[aug-cc-pcV5Z (ACV5Z)\] Gaussian basis sets. The use of such large basis sets is well known to recover 98% of the electron correlation effects in molecular structure calculations [@Helgaker2000]. All the PEC and TDM calculations for the C$_2$ molecule were performed with the quantum chemistry program package <span style="font-variant:small-caps;">molpro</span> 2015.1 [@molpro2015], running on parallel architectures. For molecules with degenerate symmetry, an Abelian subgroup is required to be used in <span style="font-variant:small-caps;">molpro</span>. So for a diatomic molecule like C$_2$ with D$_{{\infty}h}$ symmetry, it will be substituted by D$_{2h}$ symmetry with the order of irreducible representations being ($A_g$, $B_{3g}$, $B_{2g}$, $B_{1g}$, $B_{1u}$, $B_{2u}$, $B_{3u}$, $A_u$). When symmetry is reduced from D$_{{\infty}h}$ to D$_{2h}$, [@herz50] the correlating relationships are $\sigma_g \rightarrow a_g$, $\sigma_u \rightarrow a_u$, $\pi_g \rightarrow$ ($b_{2g}$, $b_{3g}$) , $\pi_u \rightarrow$ ($b_{2u}$, $b_{3u}$) , $\delta_g \rightarrow$ ($a_g$, $b_{1g}$), and $\delta_u \rightarrow$ ($a_u$, $b_{1u}$). In order to take account of short-range interactions, we employed the non-relativistic state-averaged complete active-space-self-consistent-field (SA-CASSCF)/MRCI method available within the <span style="font-variant:small-caps;">molpro</span> [@molpro2012; @molpro2015] quantum chemistry suite of codes. For the C$_2$ molecule, molecular orbitals (MOs) are put into the active space, including (3$a_g$, 1$b_{3u}$, 1$b_{2u}$, 0$b_{1g}$, 3$b_{1u}$, 1$b_{2g}$, 1$b_{3g}$, 0$a_u$), symmetry MOs. The molecular orbitals for the MRCI procedure were obtained using the SA-MCSF method, for singlet and triplet spin symmetries, we carried out the averaging processes on the two lowest states of the symmetries; ($A_g$, $B_{3u}$, $B_{1g}$, $B_{1u}$) and the lowest states of the symmetries; ($B_{2u}$, $B_{3g}$, $B_{2g}$ and $A_u$). A similar approach was also used for the quintet states. This approach provides an accurate representation of the singlet, triplet and quintet states of interest as the molecule dissociated. At bond separations beyond $R = 14$ Bohr, the PECs are smoothly fitted to functions of the form $$\label{eq:longrange} V(R) = \frac{C_5}{R^5}-\frac{C_6}{R^6}\,,$$ where for the particular electronic state, $C_5$ is the quadrupole-quadrupole electrostatic interaction [@Knipp38; @chang67] and $C_6$ is the dipole-dipole dispersion (van der Waals) coefficient (we use atomic units unless otherwise specified). For $R < 1.5$ Bohr, short-range interaction potentials of the form $V(R) = A \exp(-BR)+C$ are fitted to the *ab initio* potential curves. Estimates of the values of the quadrupole-quadrupole coefficients $C_5$ were given by @Knipp38, and by @boggio-pasqua00 (for singlet and triplet $\Sigma$, $\Pi$ and $\Delta$ electronic states, which suffices to determine those for quintet states by symmetry). The long range dispersion coefficient $C_6$ (averaged over the possible fine structure levels of two carbon atoms) was calculated to be $40.9 \pm 4.4$ using many-body perturbation theory by @MilKel72 and estimated to be 46.29 using the London formula by @chang67. In fitting the long-range form Eq. (\[eq:longrange\]) to the calculated potential energy data, we adjusted the values of $C_5$ and $C_6$, as necessary to match the slopes of the potential energy curves. The adopted values are given in Table \[table:states\]. We began with estimates of $C_5$ from @Knipp38 and @boggio-pasqua00 and limited our adjustment of $C_6$ to either the value of @MilKel72 or that of @chang67. For the B$^{\prime}{}^1\Sigma_g^+$ state, there is a barrier in the potential energy curve (0.0086 eV or 69 $\textrm{cm}^{-1}$ at $R=7.12$), reflected in the positive value of $C_5$ which fit the data. This is in good accord with a value found by @varandas08, 43 $\textrm{cm}^{-1}$ at $R=8.37$, in an extensive study of diabatic representations of the X$^1\Sigma_g^+$ and B$^{\prime}{}^1\Sigma_g^+$ states. (This barrier energy is too low to appreciably affect the dynamics calculations presented below.) [c@[$\,$]{}lDDDD]{}\[ht!\] X&$^1\Sigma_g^+$ & 0.0 & 21.81 & 5 & 40.9\ A&$^1\Pi_u$ & 8312.4 & 0. & 0. & 46.29\ B&$^1\Delta_g$ & 11895. & 3.635 & 3.635 & 46.29\ B$^{\prime}$&$^1\Sigma_g^+$ & 15205. & 0. & 16. & 40.29\ C&$^1\Pi_g$ & 34231. & $-$14.54 & $-$13.49 & 40.9\ 1&$^1\Sigma_u^-$ & 39500. & 0. & . & .\ a&$^3\Pi_u$ & 678.74 & $-$14.54 & $-$14.54 & 46.29\ b&$^3\Sigma_g^-$ & 6254.9 & 0. & 0. & 40.9\ c&$^3\Sigma_u^+$ & 9157.5 & 21.81 & 5. & 40.9\ d&$^3\Pi_g$ & 20031. & 0. & 0. & 46.29\ 2&$^3\Sigma_u^+$ & 40042. & 0. & 0. & 40.9\ 1&$^3\Delta_u$ & 41889. & 3.635 & 8. & 46.29\ 1&$^5\Pi_g$ & 30165. & $-$14.54 & $-$13.49 & 40.9\ 1&$^5\Sigma_g^+$ & 40197. & 0. & 0. & 40.9\ 1&$^5\Pi_u$ & 51897. & 0. & 0. & 46.29\ 2&$^5\Sigma_g^+$ & 64088. & 21.81 & 12. & 40.29\ 1&$^5\Delta_g$ & 49900. & 3.635 & . & .\ 1&$^5\Sigma_u^-$ & . & 0. & . & .\ As a consequence of fitting the potentials to Eq. (\[eq:longrange\]), the calculated potentials (as output from <span style="font-variant:small-caps;">molpro</span>) were shifted. In Table \[table:states\] we list the final values of $T_e$ (in $\textrm{cm}^{-1})$ relative to the minimum of the $\textrm{X}^1\Sigma_g^+$ potential energy curve and the term energies are plotted schematically in Fig. \[fig:bands\] for experimentally observed bands. Our calculated values may be compared with the recent experimental fits in Table 4, column 3, of @furtenbacher16, and we agree to within 100 $\textrm{cm}^{-1}$ for the a$^3\Pi_u$, A$^1\Pi_u$, c$^3\Sigma_u^+$, and B$^1\Delta_g$ states, and to within 205 $\textrm{cm}^{-1}$ for the b$^3\Sigma_g^-$ and B$^{\prime}{}^1\Sigma_g^+$ states. For the 1$^5\Pi_g$ state, our value of $T_e$ is within 9 $\textrm{cm}^{-1}$ of the *ab initio* value given by @schmidt11 (expressed with respect to the minimum of the a$^3\Pi_u$ curve and calculated using the aug-cc-pV5Z basis sets). Our calculated 1$^5\Pi_u$ state supports a shallow well in agreement with previous calculations [@bruna01; @bornhauser15; @Visser2019], deepening the shelf-like form of the curve found in the earlier calculations of @kirby79. The accuracy of the present potential energy curves is very satisfactory for the purposes of the present study. The potential curves for C$_2$ singlet, triplet and quintet states are shown in Fig. \[fig:pots\]. Structures due to nonadiabatic couplings are apparent in the c$^3\Sigma_u^+$ and 2$^3\Sigma_u^+$ curves due to their mutual interaction and interactions with higher states (see Fig. 3 of @kirby79). As we will show below, the calculated cross sections involving the c$^3\Sigma_u^+$ and 2$^3\Sigma_u^+$ states are several orders of magnitude smaller than those from the leading transitions contributing to the total radiative association cross sections. Similar structures in the quintet states [@bruna01] occur at energies at about 1 eV corresponding to kinetic temperatures above the range of the present study. Thus, we ignore nonadiabatic couplings and expect that they will have at most a minor effect on the net total cross sections for radiative association. The TDMs for the C$_2$ molecule are similarly extended to long- and short-range internuclear distances. For $R > 14$ a functional fit of the form $D(R) = a \exp(-bR) + c$ is applied, while in the short range $R < 1.5$ a quadratic fit of the form $D(R) = a' R^2 + b' R + c'$ is adopted. The TDMs for singlet transitions are shown in Fig. \[fig:tdmsing\], for triplet transitions in Fig. \[fig:tdmtrip\], and for quintet transitions in Fig. \[fig:tdmquint\]. As shown in Figs. \[fig:tdmsing\] and \[fig:tdmtrip\] the results are in satisfactory agreement with previous calculations for the Phillips, Swan, Ballik-Ramsay, and Schmidt-Kable (Duck) bands [@oneil87; @langhoff90; @kokkin07; @brooke13]. In addition, our results for the Bernath B$^{\prime}\Sigma_g^+$-A$^1\Pi_u$ and Deslandres-d’Azambuja bands are provided over a substantially larger range of internuclear distances compared to the earlier MRDCI calculations of @chabalowski83. (The band 1$^5\Sigma_g^+$-1$^5\Pi_u$ has not yet been observed [@bornhauser15]; we point out that 2$^3\Sigma_u^+$-d$^3\Pi_g$, 1$^3\Delta_u$-d$^3\Pi_g$, and 1$^5\Delta_g$-1$^5\Pi_u$ bands may exist. Additionally, we observe that the 2$^3\Sigma_u^+$-d$^3\Pi_g$ band might contribute at wave numbers where significant spectral congestion is seen in dicarbon [@tanabashi07].) Cross Sections {#sec:results} ============== The quantum mechanical cross section for the radiative association process $\sigma^{\textrm{QM}}_{i \rightarrow f}(E)$, where the initial $i$ and final $f$ electronic states are labeled by their molecular states (e.g. d$^3\Pi_g$) can be calculated using perturbation theory (see, for example, [@Babb1995; @Franco1996] and [@Babb1998]). The result is $$\sigma^{QM}_{i \rightarrow f} (E) = P_{i}\sum_{v^{\prime}J^{\prime}}^{}\sum_{J}^{} \frac{64}{3} \frac{\pi^5}{137.036^3} \frac{\nu^3}{2\mu E} \mathcal{S}_{J J^{\prime}} |M_{i E J, f v^{\prime} J^{\prime}}|^2, \label{quantum}$$ where the sum is over initial partial waves with angular momenta $J$ and final vibrational $v^{\prime}$ and rotational $J^{\prime}$ quantum numbers, $\mathcal{S}_{J,J^{\prime}}$ are the appropriate line strengths [@Cowan1981; @Curtis2003] or Hönl-London factors [@Watson2008], $137.036$ is the speed of light in atomic units, $\mu$ is the reduced mass of the collision system, and $M_{i E J, f v^{\prime} J^{\prime}}$ is given by the integral $${ M_{i E J, f v^{\prime} J^{\prime}}} =\int_{0}^{\infty} F_{i E J}(R) D_{if}(R) \Phi_{f v^{\prime} J^{\prime}} (R)dR. \label{matrix}$$ The wave function $\Phi_{\Lambda^{\prime} v^{\prime} J^{\prime}} (R)$ is a bound state wave function of the final electronic state, $F_{\Lambda EJ} (R)$ is an energy-normalized continuum wave function of the initial electronic state, and $D_{if}(R)$ is an electric dipole transition dipole moment between $i$ and $f$. Due to presence of identical nuclei and the absence of nuclear spin in $^{12}\mbox{C}_2$, the rotational quantum numbers of the $^1\Sigma_g^+$ states are even and for any given value of $\Lambda=1$ or 2, only one lambda-doubling level is populated [@amiot83]. Thus, the statistical weight factor $P_i$ is given by $$\label{eq:P} P_i = (2S_i+1)/81,$$ where $S_i$ is the total spin of the initial molecular electronic state (here 1, 3, or 5), and there are for two C$(^3P)$ atoms $3^4=81$ molecular states labeled by $\Lambda$ and $S$. Thus, for example, for the $^{12}\mbox{C}_2$ molecule considered here, $P_i=\frac{1}{81}$, $\frac{3}{81}$ or $\frac{5}{81}$, respectively, for $i=A^1\Pi_u$, $b^3\Sigma_g^+$, or $1^5\Pi_g$. [c@[$~$]{}Lc@[$~$]{}Lc]{}\[ht!\] C&$^1\Pi_g$ & A&$^1\Pi_u$ & Deslandres-d’Azambuja\ d&$^3\Pi_g$ & a&$^3\Pi_u$ & Swan\ 1&$^5\Pi_u$ & 1&$^5\Pi_g$ & Radi-Bornhauser\ 2&$^3\Sigma_u^+$ & d&$^3\Pi_g$ &\ 1&$^3\Delta_u$ & d&$^3\Pi_g$ &\ b&$^3\Sigma_g^-$ & a&$^3\Pi_u$ & Ballik-Ramsey\ d&$^3\Pi_g$ & c&$^3\Sigma_u^+$ & Schmidt-Kable\ A&$^1\Pi_u$ & X&$^1\Sigma_g^+$ & Phillips\ B$^{\prime}$&$^1\Sigma_g^+$&A &$^1\Pi_u$ & Bernath B$^\prime$\ B&$^1\Delta_g$ & A&$^1\Pi_u$ & Bernath B\ 2&$^5\Sigma_g^+$ & 1&$^5\Pi_u$ &\ The bound and continuum state wave functions may be computed from their respective Schrödinger equations using the grid-based approach of the Numerov method [@Cooley61; @johnson77], where we used step sizes of up to 0.001 Bohr. For example, for the $\text{X}\ ^1\Sigma_g^+$ state, we find 55 vibrational levels and $J=0,2,...,174$, though not all levels will contribute to the cross sections. A finely spaced energy grid is required to account for resonances. ![Radiative association cross sections (units of cm$^2$) as a function of the collision energy $E$ (eV) for the collision of two $^{12}\mbox{C}(^3P)$ atoms. Results are shown for the singlet, triplet, and quintet transitions listed in Table \[table:details\]. In order to gauge the cross sections, all data are plotted with the statistical factor $P_{i}=1$. We see that the Deslandres-d’Azambuja (C$^1\Pi_g$-A$^1\Pi_u$), Swan (d$^3\Pi_g$-a$^3\Pi_u$) and Radi-Bornhauser (1$^5\Pi_u$-1$^5\Pi_g$) bands have the largest cross sections.[]{data-label="fig:cross-sections"}](Fig6){width="\textwidth"} In Fig. \[fig:cross-sections\], results are shown for the radiative association cross section as a function of energy for the singlet, triplet, and quintet transitions listed in Table \[table:details\]. We plot the cross sections with the statistical factor $P_i$ set equal to unity for all states. Numerous shape resonances are visible. For the 1$^3\Delta_u$-d$^3\Pi_g$ and 1$^5\Pi_u$-1$^5\Pi_g$ cross sections, resonance tunneling features are visible, corresponding to potential barriers (local maxima) in the entrance channels. The local maxima from the potential energy curves are $0.015$ eV at $R=6$ for 1$^3\Delta_u$, $0.37$ eV at $R=3.6$ for 2$^5\Sigma_g^+$, and $0.42$ eV at $R=3.5$ for 1$^5\Pi_u$ and as seen in Fig. \[fig:cross-sections\] the corresponding cross sections sharply diminish for collision energies below these values. Note, the cross sections for Deslandres-d’Azambuja transitions (C$^1\Pi_g$-A$^1\Pi_u$) dominate the other cross sections for all collision energies, followed by the cross sections for the Swan (d$^3\Pi_g$-a$^3\Pi_u$) transitions. The cross sections for Radi-Bornhauser transitions (1$^5\Pi_u$-1$^5\Pi_g$) rise sharply as the relative energy increases. The other cross sections are at least an order of magnitude weaker than those corresponding to Deslandres-d’Azambuja, Swan, and Radi-Bornhauser transitions and will not contribute significantly to the total cross section. While the $1^5\Pi_u$ state does support some bound levels [@bornhauser17], the calculated cross sections from the 2$^5\Sigma_g^+$-$1^5\Pi_u$ state are negligible because of the steeply repulsive tail of the initial 2$^5\Sigma_g^+$ electronic state. Rate Constant ------------- The potential energy curves and transition dipole moments were then used to calculate the cross sections and rates for radiative association in the C$_2$ molecule. The thermal rate constant (in $\mbox{cm}^3 \mbox{s}^{-1}$) at a given temperature $T$ to form a molecule by radiative association is given by $$k_{i\rightarrow f} ={\left( \frac{8}{\mu\pi}\right )}^{1/2} {\left( \frac{1}{k_B T}\right)}^{3/2} \int_{0}^{\infty} E\; \sigma_{i\rightarrow f} (E)\; e^{-E/k_B T} dE\ , \label{rateconstant}$$ where $k_B$ is Boltzmann’s constant. The complicated resonance structures make it challenging to calculate accurately the rate coefficient using numerical integration [@Bennett2003; @Gustafsson2012]. As a guide, the relationship $v\sigma^\textrm{QM}_{i\rightarrow f}$, where $v=\sqrt{2E / \mu}$, is used to designate an effective energy-dependent rate coefficient, where $R(E)$ for a transition from state $i$ to state $f$ is given by, $$R(E) = \sqrt{ 2E / \mu }\, \sigma^\textrm{QM}_{i\rightarrow f} (E)$$ This form is often used to define a quasi-rate coefficient rather than one averaged over a Maxwellian distribution and was utilized in ultra-cold collisional studies [@krems10; @bmcl14]. ![Radiative association quasi-energy dependent rates $R$ (cm$^3$/s) as a function of effective kinetic temperature (K) for the C$_2$ molecule. The appropriate weighting factors, Eq. (\[eq:P\]), are included. The Deslandres-d’Azambuja (C$^1\Pi_g$-A$^1\Pi_u$), and the Swan (d$^3\Pi_g$-a$^3\Pi_u$) bands are the major contributors to the total rate, though the Radi-Bornhauser (1$\,^5\Pi_u$-1$\,^5\Pi_g$) band contributes for $T> 5000$ K.[]{data-label="fig:ratesv"}](Fig7){width="\textwidth"} In Fig. \[fig:ratesv\] results are shown for the radiative association rates $R(E)$ (in cm$^3$/s) as a function of energy expressed in temperature units (K) for the cross sections calculated for the C$_2$ molecule. The Deslandres-d’Azambuja (C$^1\Pi_g$-A$^1\Pi_u$) transitions are the main contributors for low temperature and the Radi-Bornhauser (1$\,^5\Pi_u$-1$\,^5\Pi_g$) bands are the main contributors for high temperatures. The Swan (d$^3\Pi_g$-a$^3\Pi_u$) bands also contribute to the total rate. ![Maxwellian averaged radiative association rates (cm$^3$/s) as a function of temperature (Kelvin) for the C$_2$ molecule. Results are shown for the dominant singlet, triplet, and quintet transitions with their appropriate statistical factor included. The total quantal rate (brown line) is seen to lie above the previous total semiclassical rate [@andreazza97] (dashed black line) at all but the highest temperatures. []{data-label="fig:ratesm"}](Fig8){width="\textwidth"} In Fig. \[fig:ratesm\] we compare our Maxwellian averaged quantal rates for the dominant Deslandres-d’Azambuja, Swan, and Radi-Bornhauser transitions and their sum with those determined from the previous semi-classical approximation by [@andreazza97] over the temperature range 100-10,000 Kelvin. The quantal rates have the appropriate statistical population included so a comparison could be made directly with the previous semiclassical results of @andreazza97. The total rate coefficient is fit to better than 6 percent by the function $$\begin{aligned} \label{eq:fit} \alpha(T) &=& 5.031 \times 10^{-18} + 1.501\times10^{-16} T^{-1}+ 2.517\times 10^{-21} T -1.89\times 10^{-25} T^2\quad \textrm{cm}^3/\textrm{s},\nonumber\\ & & \quad 100\leq T\leq 10,000~K\end{aligned}$$ Contemporary discussions of resonances in radiative association cross sections were given by, for example, @Bennett2003 [@BarvanHem06; @BenDicLei08; @augustovicova12; @mrugala13; @golubev13], though earlier researchers also considered the problem in detail [@GiuSuzRou76; @FloRou79; @GraMosRou83], with considerations of radiative and tunneling contributions to resonance widths. Our procedures for calculating the cross sections, e.g. Eq. (\[quantum\]), and the corresponding rate coefficients, as shown in Fig. \[fig:ratesm\], include shape and resonance tunneling resonances. Procedures for the precision treatment of the effects of radiative decay have been developed: *i)* excise certain resonances from the cross sections, recalculate them including the sum of the partial widths for radiative decay and tunneling, and then insert them back; *ii)* add the separately calculated resonance cross sections to the semi-classical cross sections; or *iii)* add the separately calculated resonance cross sections to a background smooth base line derived from the quantum cross sections. @franz11 and @antipov13 examined radiative association of carbon and oxygen to form CO ($\mbox{A}^1\Pi$-$\mbox{X}^1\Sigma^+$), where there is a local maximum in the $\mbox{A}^1\Pi$ state of 0.079 eV (900 K effective temperature). Previous calculations of rate coefficients corresponding to semi-classical cross sections (using different molecular data) were available for comparison [@dalgarno90; @singh99]. Using the quantum-mechanical theory \[our Eq. (\[quantum\])\], using semi-classical methods, and with specific treatment of resonances, @franz11 and @antipov13 found that for the rate coefficients corresponding to quantum cross sections with Eq. (\[quantum\]) [@franz11] or to hybrid calculations using procedure *ii)* (semi-classical theory combined with additive quantum treatment of resonances) [@antipov13] that resonances contribute below about 900 K. However, the rate coefficients for the quantum calculation, Fig. 2 of [@franz11] and those for the hybrid calculations, Fig. 4a of [@antipov13], do not deviate until the temperature is below 100 K. Precision treatment of radiative decay for the numerous resonances in our cross sections might yield enhanced values at lower temperatures, but given the application envisioned, namely to calculate corresponding rate coefficients for applications to the chemical models of supernovae ejecta, where many chemical reactions enter and few reaction rate coefficients are known precisely and where the temperatures of interest are perhaps of order 1000-2000 K, the present procedures are satisfactory. Discussion {#sec:discussion} ========== We see that our present quantal rates are larger than those from the previous semiclassical results of @andreazza97 at temperatures below 7000 K, and our quantal rates persist with a value of about $7\times 10^{-18}$ $\mbox{cm}^3\mbox{s}^{-1}$ as temperatures approach 100 K. @andreazza97 listed the Swan and Deslandres-d’Azambuja transitions (in that order) as the leading contributors to the radiative association process. Our results indicate that the Deslandres-d’Azambuja transitions dominate the Swan transitions for all temperatures. @andreazza97 used a potential energy function for the C$\;^1\Pi_g$ state with a barrier of $0.002$ eV; they did not list the cross sections, but in the semiclassical method the cross sections for C$\;^1\Pi_g$-A$\;^1\Pi_u$ transitions would be negligible for relative energies less than effective collisional temperature of the barrier, about 25 K, so this would not affect the comparison at thermal temperatures. We note that @andreazza97 did not include quintet states (for which no experimental data were available at that time) and except for the trend towards lower temperatures, our total rate coefficient is in reasonable agreement with their calculation. We note that the 1$\;^5\Pi_u$-1$\;^5\Pi_g$ transition is the main contributor (as seen from Fig. \[fig:ratesm\]) to the rate coefficient for temperatures above 5000 K. As mentioned above, in laboratory carbon vapors generated by laser radiation, the presence of dicarbon is confirmed by, primarily, radiation from the Swan bands, but also from the Deslandres-d’Azambuja bands [@savastenko11]. In light emission from laser-induced expansion of carbon vapor from a graphite rod, @monchicourt91 found evidence of associative collisions of two ground state carbon atoms. However, the operative source of dicarbon depends on factors such as temperature and distance from the graphite substrate and there is evidence that dicarbon is formed by dissociation from the graphite [@iida94] or recombination through three-body reactions [@savastenko11]. Recent modeling at certain densities of laser-induced plasmas using local thermal equilibrium and equations of states indicates C$_2$ may form in plasmas of Si, N, or Ar and C at characteristic temperatures roughly less than 5000 K [@shabanov15], corresponding to the relatively low temperature region of the plasma [@degiacomo17]. That dicarbon is formed by a recombination process involving two carbon atoms in laser-induced plasma chemistry was shown experimentally using carbon isotopes by @dong13. The dominance of the 1$\;^5\Pi_u$-1$\;^5\Pi_g$ transitions in our calculations is interesting in light of theories of dicarbon formation in laser plasmas. In particular, @little87 theorized that the d$\;^3\Pi_g$ $(v=6)$ band of dicarbon is populated by recombination collisions of atomic carbon, possibly in the presence of a third body, through the 1$\;^5\Pi_g$ state (cf. @caubet94). Subsequently, @bornhauser11 experimentally verified that the metastable 1$\;^5\Pi_g$ state perturbs the d$\;^3\Pi_g$ state [@bornhauser15]; population of the 1$\;^5\Pi_g$ state leads to Swan band ($v=6$) fluorescence. Subsequent experiments demonstrated the existence of the 1$\;^5\Pi_u$ state [@bornhauser15]. Depending on the densities and applicable chemistries of laser generated carbon vapors, the radiative association process 1$\;^5\Pi_u$-1$\;^5\Pi_g$ might contribute to the production of dicarbon in the 1$\;^5\Pi_g$ state enhancing the mechanism of @little87. For models of carbonaceous dust production in core-collapse supernovae, the present results provide improved rates for an initial step in carbon chain production. Our rates may be compared to rates from a kinetic theory based nucleation model [@lazzati16], which are a factor of $10^6$ larger, and can be applied to molecular nucleation models [@sluder18] or chemical reaction network models [@yu13; @clayton18]. Conclusions {#sec:conclusions} =========== Accurate cross sections and rates for the formation of dicarbon by the radiative association process were computed for transitions from several excited electronic states using new *ab initio* potentials and transition dipole moment functions. Where calculated values exist in the literature, our TDM functions are in good agreement with previous results, but our calculations are presented over a more extensive range of internuclear distances. We also present TDM functions for the B$^1\Delta_g$-A$^1\Pi_u$ d$^3\Pi_g$-1$^3\Delta_u$, d$^3\Pi_g$-2$^3\Sigma^+_u$, 1$\;^5\Pi_u$-1$\;^5\Pi_g$ (Radi-Bornhauser band), $1\;^5\Sigma_g^+$-$1\;^5\Pi_u$, $2\;^5\Sigma_g^+$-$1\;^5\Pi_u$, and $1\;^5\Delta_g$-$1\;^5\Pi_u$ transitions, substantially extending the available TDM data for dicarbon. We found that at the highest temperatures the quintet state Radi-Bornhauser ($1\;^5\Pi_u$-$1\;^5\Pi_g$) transitions are the dominant channel for radiative association, followed by the Deslandres-d’Azambuja (C$^1\Pi_g$-A$^1\Pi_u$) and Swan (d$^3\Pi_g$-a$^3\Pi_u$) transitions, while at lower temperatures the Deslandres-d’Azambuja transitions are dominant. The computed cross sections and rates for C$_2$ are suitable for applicability in a variety of interstellar environments including diffuse and translucent clouds and ejecta of core-collapse supernovae. In addition, our calculations do not contradict evidence that in laser ablated vapors dicarbon formation proceeds through the $1\;^5\Pi_g$ state. We concur with @furtenbacher16 that further experimental studies on the Deslandres-d’Azumbuja transitions are desirable. This work was supported by a Smithsonian Scholarly Studies grant. BMMcL acknowledges support by the US National Science Foundation through a grant to ITAMP at the Center for Astrophysics  Harvard & Smithsonian under the visitor’s program, the University of Georgia at Athens for the award of an adjunct professorship, and Queen’s University Belfast for a visiting research fellowship (VRF). We thank Captain Thomas J. Lavery, USN, Ret., for his constructive comments that enhanced the quality of this manuscript. The authors acknowledge this research used grants of computing time at the National Energy Research Scientific Computing Centre (NERSC), which is supported by the Office of Science of the U.S. Department of Energy (DOE) under Contract No. DE-AC02-05CH11231. The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-center.eu) for funding this project by providing computing time on the GCS Supercomputer HAZEL HEN at Höchstleistungsrechenzentrum Stuttgart (www.hlrs.de). ITAMP is supported in part by NSF Grant No. PHY-1607396. natexlab\#1[\#1]{}\[1\][[\#1](#1)]{} \[1\][doi: [](http://doi.org/#1)]{} \[1\][[](http://ascl.net/#1)]{} \[1\][[](https://arxiv.org/abs/#1)]{} Abell[á]{}n, F. J., Indebetouw, R., Marcaide, J. M., [et al.]{} 2017, , 842, L24, A’Hearn, M. F., Millis, R. C., Schleicher, D. O., Osip, D. J., & Birch, P. V. 1995, , 118, 223 , , C. 1983, , 52, 329, Andreazza, C. M., Marinho, E. P., & Singh, P. D. 2006, , 372, 1653, Andreazza, C. M., & Singh, P. D. 1997, MNRAS, 287, 287, Antipov, S. V., Gustafsson, M., & Nyman, G. 2013, , 430, 946, Augustovi[č]{}ov[á]{}, L., [Š]{}pirko, V., Kraemer, W. P., & Sold[á]{}n, P. 2012, Chem. Phys. Lett., 531, 59, Babb, J. F., & Dalgarno, A. 1995, , 51, 3021, Babb, J. F., & Kirby, K. P. 1998, in [Molecular Astrophysics of Stars and Galaxies]{}, ed. [T W Hartquist and D A Williams]{} (Oxford, UK: Clarendon Press), 11 Babb, J. F., & McLaughlin, B. M. 2017, J. Phys. B: At. Mol. Opt. Phys., 50, 044003, —. 2017, , 468, 2052, —. 2018, , 860, 151, Barinovs, [Ğ]{}., & van Hemert, M. C. 2006, ApJ, 636, 923, Bennett, O. J., Dickinson, A. S., Leininger, T., & Gad[é]{}a, F. X. 2008, MNRAS, 384, 1743, Bennett, O. J., Dickinson, A. S., Leininger, T., & Gad[é]{}a, X. 2003, MNRAS, 341, 361, Black, J. H., & Dalgarno, A. 1977, , 34, 405, Boggio-Pasqua, M., Voronin, A., Halvick, P., & Rayez, J.-C. 2000, J. Molec. Struct.: THEOCHEM, 531, 159, Bornhauser, P., Marquardt, R., Gourlaouen, C., [et al.]{} 2015, , 142, 094313, Bornhauser, P., Sych, Y., Knopp, G., Gerber, T., & Radi, P. P. 2011, , 134, 044302, Bornhauser, P., Visser, B., Beck, M., [et al.]{} 2017, , 146, 114309, Boschen, J. S., Theis, D., Ruedenberg, K., & Windus, T. L. 2014, Theor. Chem. Acc., 133, 1425, , J. W., [Testerman]{}, L., [Grevesse]{}, N., [et al.]{} 1982, , 108, 201 Brooke, J. S., Bernath, P. F., Schmidt, T. W., & Bacskay, G. B. 2013, J. Quant. Spectrosc. Rad. Trans., 124, 11, Bruna, P. J., & Grein, F. 2001, Can. J. Phys., 79, 653, Cairnie, M., Forrey, R. C., Babb, J. F., Stancil, P. C., & McLaughlin, B. M. 2017, , 471, 2481, Caubet, P., & Dorthe, G. 1994, Chem. Phys. Lett., 218, 529, Chabalowski, C. F., Peyerimhoff, S. D., & Buenker, R. J. 1983, Chem. Phys., 81, 57, , F. H., J., & [Lutz]{}, B. L. 1978, , 221, L91, Chang, T. Y. 1967, , 39, 911, Cherchneff, I., & Dwek, E. 2009, , 703, 642 , I., & [Sarangi]{}, A. 2011, Proc. IAU, 7 (S280), 228, Clayton, D. D. 2013, , 762, 5, , D. D., [Deneault]{}, E. A. N., & [Meyer]{}, B. S. 2001, , 562, 480, Clayton, D. D., Liu, W., & Dalgarno, A. 1999, Science, 283, 1290, , D. D., & [Meyer]{}, B. S. 2018, Geochimica et Cosmochimica Acta, 221, 47, , A. L., [Barker]{}, E. S., & [Gray]{}, C. L. 2012, , 218, 144, Cooley, J. W. 1961, Math. Comp., 15, 363, Cowan, R. D. 1981, [The Theory of Atomic Structure and Spectra]{} (Berkeley, California, USA: University of California Press) Curtis, L. J. 2003, [Atomic Structure and Lifetimes: A Conceptual Approach]{} (Cambridge, UK: Cambridge University Press) Dalgarno, A., Du, M. L., & You, J. H. 1990, , 349, 675, De Giacomo, A., & Hermann, J. 2017, J. Phys. D: Appl. Phys., 50, 183002, Dong, M., Mao, X., Gonzalez, J. J., Lu, J., & Russo, R. E. 2013, Anal. Chem., 85, 2899, , S. R., & [Huntress]{}, W. T., J. 1989, , 338, 140, Flower, D. R., & Roueff, E. 1979, A&A, 72, 361 Forrey, R. C., Babb, J. F., Stancil, P. C., & McLaughlin, B. M. 2016, J. Phys. B: At. Mol. Opt. Phys., 49, 184002, —. 2018, , 479, 4727, Franz, J., Gustafsson, M., & Nyman, G. 2011, , 414, 3547, Furtenbacher, T., Szab[ó]{}, I., Cs[á]{}sz[á]{}r, A. G., [et al.]{} 2016, , 224, 44, Gianturco, F. A., & Gori Giorgi, P. 1996, , 54, 4073, , A., [Roueff]{}, E., & [van Regemorter]{}, H. 1976, J. Phys. B: At. Mol. Opt. Phys., 9, 1021, , J. H., [Bregman]{}, J. D., [Cooper]{}, D. M., [et al.]{} 1983, , 270, 190, Golubev, N. V., Bezrukov, D. S., Gustafsson, M., Nyman, G., & Antipov, S. V. 2013, J. Phys. Chem. A, 117, 8184, , M. M., [Moseley]{}, J. T., & [Roueff]{}, E. 1983, , 269, 796, , R., [Black]{}, J. H., & [Yan]{}, M. 2001, , 375, 553, , P. 2013, , 765, 12, , N., & [Sauval]{}, A. J. 1973, , 27, 29 Gustafsson, M., Antipov, S. V., Franz, J., & Nyman, G. 2012, , 137, 104301, Helgaker, T., J[ø]{}rgesen, P., & Olsen, J. 2000, [Molecular Electronic-Structure Theory]{} (New York, USA: Wiley) Herzberg, G. 1950, [Spectra of Diatomic Molecules]{} (New York, USA: Van Nostrand) , S. 2011, , 411, 1857, Iida, Y., & Yeung, E. S. 1994, Appl. Spectrosc., 48, 945, Jackson, W. 1976, J. Photochem., 5, 107, Johnson, B. R. 1977, , 67, 4086, Keenan, P. C. 1993, , 105, 905, Kirby, K., & Liu, B. 1979, , 70, 893, , J. K. 1938, Phys. Rev., 53, 734, Kokkin, D. L., Bacskay, G. B., & Schmidt, T. W. 2007, , 126, 084302, Krems, R., Friedrich, B., & Stwalley, W. C., eds. 2010, Cold Molecules: Theory, Experiment, Applications (Boca Raton, FL: CRC Press) , D. L. 1978, , 182, 249, Langhoff, S. R., Bauschlicher, C. W., Rendell, A. P., & Komornicki, A. 1990, , 92, 3000, Lazzati, D., & Heger, A. 2016, , 817, 134, Little, C., & Browne, P. 1987, Chem. Phys. Lett., 134, 560, , W., & [Dalgarno]{}, A. 1995, , 454, 472, , W., [Dalgarno]{}, A., & [Lepp]{}, S. 1992, , 396, 679, , R., [Lan[ç]{}on]{}, A., & [J[ø]{}rgensen]{}, U. G. 2001, , 371, 1065, Lord, III, H. C. 1965, Icarus, 4, 279, Macrae, R. M. 2016, Sci. Prog., 99, 1, Martin, M. 1992, J. Photochem. Photobio. A: Chem., 66, 263, , B. M., [Lamb]{}, H. D. L., [Lane]{}, I. C., & [McCann]{}, J. F. 2014, J. Phys. B: At. Mol. Opt. Phys., 47, 145201, , E. C., [Shen]{}, G., [McCann]{}, J. F., [McLaughlin]{}, B. M., & [Stancil]{}, P. C. 2016, J. Phys. B: At. Mol. Opt. Phys., 49, 084001, , G., & [Krauss]{}, L. 1967, Z. Natur. A, 22, 2015, , T. J., [Bennett]{}, A., [Rawlings]{}, J. M. C., [Brown]{}, P. D., & [Charnley]{}, S. B. 1991, , 87, 585 Miller, J. H., & Kelly, H. P. 1972, Phys. Rev. A, 5, 516, Monchicourt, P. 1991, , 66, 1430, Mruga[ł]{}a, F., & Kraemer, W. P. 2013, , 138, 104315, Nemes, L., & Irle, S., eds. 2011, Spectroscopy, Dynamics and Molecular Theory of Carbon Plasmas and Vapors (Singapore: World Scientific) ONeil, S. V., Rosmus, P., & Werner, H. 1987, , 87, 2847, Pattillo, R. J., Cieszewski, R., Stancil, P. C., [et al.]{} 2018, , 858, 10, Prasad, S. S., & Huntress, W. T. 1980, , 43, 1, , J., [Geballe]{}, T. R., [Banerjee]{}, D. P. K., [et al.]{} 2018, , 864, L20, Sarangi, A., & Cherchneff, I. 2013, , 776, 107, Sarangi, A., Matsuura, M., & Micelotta, E. R. 2018, , 214, 63, Savastenko, N. A., & Tarasenko, N. V. 2011, in Spectroscopy, Dynamics and Molecular Theory of Carbon Plasmas and Vapors, ed. L. Nemes & S. Irle (Singapore: World Scientific), 167–198 Schmidt, T. W., & Bacskay, G. B. 2011, , 134, 224311, Shabanov, S. V., & Gornushkin, I. B. 2015, Appl. Phys. A, 121, 1087, Shen, G., Stancil, P. C., Wang, J. G., McCann, J. F., & McLaughlin, B. M. 2015, J. Phys. B: At. Mol. Opt. Phys., 48, 105203, Singh, P. D., Sanzovo, G. C., Borin, A. C., & Ornellas, F. R. 2002, , 303, 235, , A., [Milosavljevi[ć]{}]{}, M., & [Montgomery]{}, M. H. 2018, , 480, 5580, , A., [Hirao]{}, T., [Amano]{}, T., & [Bernath]{}, P. F. 2007, , 169, 472, , T. 1964, Ann. Tokyo Astron. Observ., 2nd. Ser., 9, 1 , E. F., & [Black]{}, J. H. 1989, , 340, 273, , E. F., [Blake]{}, G. A., [Draine]{}, B. T., & [Lunine]{}, J. I. 1993, in Protostars and Planets III, ed. E. H. [Levy]{} & J. I. [Lunine]{} (Tucson: University of Arizona Press), 163 Varandas, A. J. C. 2008, , 129, 234103, Varandas, A. J. C., & Rocha, C. M. R. 2018, Philos. Trans. Royal Soc. A, 376, 20170145, Visser, B., Beck, M., Bornhauser, P., [et al.]{} 2019, Molec. Phys., online Jan. 22, 2019, Watson, J. K. G. 2008, J. Mol. Spec., 253, 5, Werner, H.-J., Knowles, P. J., Knizia, G., Manby, F. R., & Sch[ü]{}tz, M. 2012, WIREs Comput. Mol. Sci., 2, 242, Werner, H.-J., Knowles, P. J., Knizia, G., [et al.]{} 2015, <span style="font-variant:small-caps;">molpro</span>, version 2015.1, a package of *ab initio* programs. <http://www.molpro.net> Yu, T., Meyer, B. S., & Clayton, D. D. 2013, , 769, 38, Yurchenko, S. N., Szab[ó]{}, I., Pyatenko, E., & Tennyson, J. 2018, , 480, 3397, Zhang, X.-N., Shi, D.-H., Sun, J.-F., & Zhu, Z.-L. 2011, Chin. Phys. B, 20, 043105, \[lastpage\] [^1]: CO was also detected in the first overtone band $(\Delta\nu =2)$. In SN 1987A individual rotational lines of CO and of SiO were detected at late epoch, see @abellan17 [@sarangi18].

--- abstract: 'We provide a constructive method designed in order to control the stability of a given periodic orbit of a general completely integrable system. The method consists of a specific type of perturbation, such that the resulting perturbed system becomes a codimension-one dissipative dynamical system which also admits that orbit as a periodic orbit, but whose stability can be a-priori prescribed. The main results are illustrated in the case of a three dimensional dissipative perturbation of the harmonic oscillator, and respectively Euler’s equations form the free rigid body dynamics.' author: - 'Răzvan M. Tudoran' title: '[**Controlling the stability of periodic orbits of completely integrable systems**]{}' --- \[section\] \[theorem\][Definition]{} \[theorem\][Lemma]{} \[theorem\][Remark]{} \[theorem\][Proposition]{} \[theorem\][Corollary]{} \[theorem\][Example]{} **AMS 2000**: 37C27; 37C75; 34C25; 37J35. **Keywords**: dissipative dynamics; periodic orbits; characteristic multipliers; stability. Introduction {#section:one} ============ The main purpose of this work is to provide a constructive method of controlling the stability of periodic orbits of completely integrable systems. The method consists of a specific type of perturbation, such that the resulting perturbed system becomes a codimension-one dissipative dynamical system. The controllability procedure is based on an explicit formula for the characteristic multipliers of a given periodic orbit of a general codimension-one dissipative dynamical system. Even if this explicit formula is not the main purpose of the paper, it can be considered as the core of the article, since all main results are actually based on it. Recall that the explicit knowledge of the characteristic multipliers of a periodic orbit, is extremely useful for the study of its stability (for details regarding the stability of periodic orbits see e.g., [@hartman], [@verhulst]). Because of the local nature of the main results, one can suppose that we work on an open subset $U\subseteq \mathbb{R}^n$. More precisely, let $\dot x =X(x)$, $X\in\mathfrak{X}(U)$, be a given codimension-one dissipative dynamical system, i.e., there exists $k,p\in\mathbb{N}$ such that $k+p=n-1$, and some smooth functions $I_1,\dots,I_k, D_1,\dots, D_p$, $h_1$, $\dots$, $h_p \in\mathcal{C}^{\infty}(U,\mathbb{R})$ such that the vector field $X$ conserves $I_1,\dots,I_k$, and dissipates $D_1,\dots, D_p$ with associated dissipation rates $h_1 D_1, \dots, h_p D_p$. Suppose that $\Gamma:=\{\gamma(t)\subset U : 0\leq t\leq T \}$ is a $T-$periodic orbit of $\dot x =X(x)$ such that $\Gamma\subset ID^{-1}(\{0\})$, and moreover, $0 \in \mathbb{R}^{n-1}$ is a regular value of the map $ID:=(I_1,\dots,I_k,D_1,\dots,D_p):U\subseteq \mathbb{R}^{n}\rightarrow \mathbb{R}^{n-1}$. In the above hypothesis, the first result states that, if $$\nabla I_1(\gamma(t)),\dots,\nabla I_k(\gamma(t)),\nabla D_1(\gamma(t)),\dots, \nabla D_p (\gamma(t)), X(\gamma(t))$$ are linearly independent for each $0\leq t \leq T$, then the characteristic multipliers of the periodic orbit $\Gamma$ are, $\underbrace{1,\dots,1}_{k+1 \ \mathrm{times}}, \exp\left(\int_{0}^{T}h_1 (\gamma(s))ds\right),\dots, \exp\left(\int_{0}^{T}h_p (\gamma(s))ds\right)$. The second result of this work is a consequence of the explicit computation of the characteristic multipliers of a given periodic orbit, and consists of two stability results. More precisely, if there exists $i_0 \in\{1,\dots, p\}$ such that $\int_{0}^{T}h_{i_0} (\gamma(s))ds>0$, then the periodic orbit $\Gamma$ is unstable. On the other hand (supposing that $0\in \mathbb{R}^{k}$ is a regular value of the map $I:=(I_1,\dots,I_k)$), if $\int_{0}^{T}h_1 (\gamma(s))ds<0$, $\dots$, $\int_{0}^{T}h_p (\gamma(s))ds<0$, then the periodic orbit $\Gamma$ is orbitally phase asymptotically stable, with respect to perturbations along the invariant manifold $I^{-1}(\{0\})$. The third result of this article, which is also the main result, provides a method to partially stabilize a given periodic orbit of a completely integrable dynamical system. More precisely, to a given completely integrable system, and respectively a given periodic orbit, we explicitly associate a dissipative dynamical system admitting the same periodic orbit, and moreover, this periodic orbit is orbitally phase asymptotically stable, relative to a certain dynamically invariant set. Note that dissipative perturbations were also used in stabilization procedures of equilibrium states of Hamiltonian systems, see e.g., [@rtud], [@bc]. The structure of the paper is the following. In the second section, one recalls a characterization of codimension-one dissipative dynamical systems, that will be used in the next sections. The third section is dedicated to the explicit computation of the characteristic multipliers of a given periodic orbit of a general codimension-one dissipative dynamical system. The fourth section uses the results from the previous section, in order to provide sufficient conditions to guarantee the partial stability, and respectively the instability of periodic orbits of codimension-one dissipative dynamical systems. The last section contains an explicit method to stabilize (relatively to a certain dynamically invariant set) a given periodic orbit of a completely integrable dynamical system. Codimension-one dissipative dynamical systems ============================================= In this short section we recall some results concerning the codimension-one dissipative dynamical systems. For more details regarding the characterization of general dissipative dynamical systems see e.g., [@rtudoran]. Recall that by a codimension-one dissipative dynamical system (defined eventually on an open subset $U\subseteq \mathbb{R}^{n}$), we mean a dynamical system $\dot x =X(x)$, $X\in\mathfrak{X}(U)$, for which there exist $k,p\in\mathbb{N}$, with $k+p=n-1$, and some smooth functions $I_1,\dots,I_k, D_1,\dots, D_p$, $h_1$, $\dots$, $h_p \in\mathcal{C}^{\infty}(U,\mathbb{R})$, such that the vector field $X$ conserves $I_1,\dots,I_k$, and dissipates $D_1,\dots, D_p$, with associated dissipation rates $h_1 D_1, \dots, h_p D_p$, i.e., $\mathcal{L}_{X}I_1 =\dots= \mathcal{L}_{X}I_k =0$, and respectively $\mathcal{L}_{X}D_1 =h_1 D_1$, $\dots$, $\mathcal{L}_{X}D_p =h_p D_p$, where the notation $\mathcal{L}_{X}$ stands for the Lie derivative along the vector field $X$. Note that $\mathcal{L}_{X} f =\langle X,\nabla f \rangle$, where $f\in\mathcal{C}^{\infty}(U,\mathbb{R})$ is an arbitrary smooth function, and $\nabla$ states for the gradient operator associated to the standard inner product on $\mathbb{R}^n$, namely $\langle\cdot,\cdot\rangle$. Let us recall now a result from [@rtudoran] regarding the local structure of a codimension-one dissipative dynamical system. \[DSC\] Let $k,p\in \mathbb{N}$ be two natural numbers such that $k+p = n-1$, and let $I_1,\dots, I_k, D_1,\dots, D_p, h_1, \dots, h_p \in \mathcal{C}^{\infty}(U,\mathbb{R})$ be a given set of smooth functions defined on an open subset $U\subseteq \mathbb{R}^{n}$, such that $\{\nabla I_1 ,\dots,\nabla I_k , \nabla D_1 , \dots,\nabla D_p\}\subset \mathfrak{X}(U)$ forms a set of pointwise linearly independent vector fields on $U$. Then the smooth vector fields $X\in \mathfrak{X}(U)$ which verify simultaneously the conditions $$\label{edr} \left\{\begin{array}{l} \mathcal{L}_{X}I_1= \dots = \mathcal{L}_{X}I_k = 0,\\ \mathcal{L}_{X}D_1 = h_1 D_1, \dots, \mathcal{L}_{X}D_p = h_p D_p,\\ \end{array}\right.$$ are characterized as follows $$X = X_0 + \nu \left[\star\left(\bigwedge_{j=1}^{p} \nabla D_j\wedge\bigwedge_{l=1}^{k} \nabla I_l \right)\right],$$ where $\nu\in\mathcal{C}^{\infty}(U,\mathbb{R})$ is an arbitrary rescaling function, $$X_{0}=\left\| \bigwedge_{i=1}^{p} \nabla D_i\wedge\bigwedge_{j=1}^{k} \nabla I_j \right\|_{n-1}^{-2}\cdot\sum_{i=1}^{p}(-1)^{n-i}h_i D_i \Theta_i,$$ $$\Theta_i = \star\left[ \bigwedge_{j=1, j\neq i}^{p} \nabla D_j \wedge \bigwedge_{l=1}^{k} \nabla I_l \wedge\star\left(\bigwedge_{j=1}^{p} \nabla D_j\wedge\bigwedge_{l=1}^{k} \nabla I_l \right)\right],$$ and “$\star$” stands for the Hodge star operator for multivector fields. \[ret\] The vector field $X_0$ is itself a solution of the system , while the vector field $\star\left(\bigwedge_{j=1}^{p} \nabla D_j\wedge\bigwedge_{l=1}^{k} \nabla I_l \right)$ is a solution of the homogeneous system $$\left\{\begin{array}{l} \mathcal{L}_{X}I_1= \dots = \mathcal{L}_{X}I_k = 0,\\ \mathcal{L}_{X}D_1 = \dots =\mathcal{L}_{X}D_p = 0.\\ \end{array}\right.$$ Let us now recall from [@rtudoran] a consequence of the above theorem, which gives a local characterization of completely integrable systems. \[CIS\] In the case when $p=0$ (and consequently $k=n-1$), the dynamical system generated by the vector field $X$ will be completely integrable, and the conclusion of Theorem becomes: The smooth vector fields $X\in \mathfrak{X}(U)$ which verify the conditions $$\mathcal{L}_{X}I_1= \dots = \mathcal{L}_{X}I_{n-1} = 0,\\$$ are given by $$X = \nu \left[\star\left(\nabla I_1 \wedge\dots\wedge\nabla I_{n-1} \right)\right],$$ where $\nu\in\mathcal{C}^{\infty}(U,\mathbb{R})$ is a smooth arbitrary function. The characteristic multipliers of periodic orbits of codimension-one dissipative dynamical systems ================================================================================================== In this section we compute explicitly the characteristic multipliers of periodic orbits of codimension-one dissipative dynamical systems. Let us recall first that for a general dynamical system $\dot x=X(x)$, generated by a smooth vector field $X\in \mathfrak{X}(U)$, defined on an open subset $U\subseteq \mathbb{R}^{n}$, and respectively a given $T-$periodic orbit $\Gamma:=\{\gamma(t)\subset U : 0\leq t\leq T \}$, the characteristic multipliers of $\Gamma$ are the eigenvalues of the fundamental matrix $u(T)$, where $u$ is the solution of the variational equation $$\dfrac{du}{dt}=DX(\gamma(t))u(t), \ u(0)=I_{n,n},$$ and $I_{n,n}$ stands for the identity matrix of dimensions $n\times n$. Recall that since $\Gamma$ is a periodic orbit, $1$ will be always a characteristic multiplier of $\Gamma$, (see e.g., [@moser]). Taking into account the complexity of the variational equation, the computation of characteristic multipliers in general is almost impossible, since there exist no general methods to solve explicitly the variational equation. One of the main results of this paper is to complete this task for the class of codimension-one dissipative dynamical systems, if one knows an explicit parameterization of the periodic orbit to be analyzed. In order to do that we will use the following result from [@gasul]. \[TR\] Let $\Gamma=\{\gamma(t)\subset U : 0\leq t\leq T \}$ be a $T-$periodic orbit of a dynamical system $\dot x=X(x)$. Consider a smooth function $f:U\subseteq \mathbb{R}^{n}\rightarrow \mathbb{R}^{n-1}$, $f=(f_1,\dots,f_{n-1})^{\top}$, such that: - $\Gamma$ is contained in $\bigcap_{i=1}^{n-1}\{f_i (x)=0\}$, - the crossing of all the manifolds $\{f_i (x)=0\}$ for $i\in\{1,\dots,n-1\}$ are transversal over $\Gamma$, - there exists a $(n-1)\times (n-1)$ matrix $k(x)$ of real functions satisfying: $$\label{ED} Df(x)X(x)=k(x)f(x).$$ Let $v(t)$ be the $(n-1)\times (n-1)$ fundamental matrix solution of $$\label{EFD} \dfrac{dv}{dt}=k(\gamma(t))v(t), \ v(0)=I_{n-1,n-1},$$ where $I_{n-1,n-1}$ stands for the identity matrix of dimensions $(n-1)\times (n-1)$. Then the characteristic multipliers of $\Gamma$ are $\{1\}\cup \sigma(v(T))$, where $\sigma(v(T))$ stands for the spectrum of $v(T)$. Using the above result, we will compute explicitly the characteristic multipliers of a given periodic orbit of a codimension-one dissipative dynamical system. Let us state now the main result of this section. \[MT\] Let $\dot x= X(x)$ be a codimension-one dissipative dynamical system generated by a smooth vector field $X\in\mathfrak{X}(U)$ defined eventually on an open subset $U\subseteq \mathbb{R}^{n}$, such that there exist $k,p\in\mathbb{N}$, $k+p=n-1$, and respectively $I_1,\dots,I_k, D_1,\dots, D_p$, $h_1$, $\dots$, $h_p \in\mathcal{C}^{\infty}(U,\mathbb{R})$ such that $\mathcal{L}_{X}I_1 =\dots= \mathcal{L}_{X}I_k =0$, and $\mathcal{L}_{X}D_1 =h_1 D_1$, $\dots$, $\mathcal{L}_{X}D_p =h_p D_p$. Suppose that $\Gamma=\{\gamma(t)\subset U : 0\leq t\leq T \}$ is a $T-$periodic orbit of $\dot x= X(x)$, such that the following conditions hold true: - $\Gamma\subset ID^{-1}(\{0\})$, and $0 \in \mathbb{R}^{n-1}$ is a regular value of the map $$ID=(I_1,\dots,I_k,D_1,\dots,D_p):U\subseteq \mathbb{R}^{n}\rightarrow \mathbb{R}^{n-1},$$ - $\nabla I_1(\gamma(t)),\dots,\nabla I_k(\gamma(t)),\nabla D_1(\gamma(t)),\dots, \nabla D_p (\gamma(t)), X(\gamma(t))$ are linearly\ independent for each $0\leq t \leq T$. Then, the characteristic multipliers of the periodic orbit $\Gamma$ are $$\underbrace{1,\dots,1}_{k+1 \ \mathrm{times}}, \exp\left(\int_{0}^{T}h_1 (\gamma(s))ds\right), \dots, \exp\left(\int_{0}^{T}h_p (\gamma(s))ds\right).$$ The first two conditions from the hypothesis of Theorem obviously imply the first two conditions of this theorem. Let us construct now a matrix $(n-1)\times (n-1)$, $k(x)$, which verifies the equivalent of equation , namely the equation $$\label{EDD} D(ID)^{\top}(x)X(x)=k(x)(ID)^{\top}(x),$$ where by $(ID)^{\top}$ we mean the transpose of the matrix $ID=(I_1,\dots,I_k,D_1,\dots,D_p)$. If we define $ k=\left[ {\begin{array}{*{20}c} O_{k,k} & O_{k,p} \\ O_{p,k} & H_{p,p} \\ \end{array}} \right], $ where $O_{k,k},O_{k,p},O_{p,k}$ stands for the null matrices of dimensions $k \times k, k \times p, p \times k,$ and $H_{p,p}=\operatorname{diag}[h_1,\dots,h_p ]$, then the equation can be equivalently written as follows $$\label{split} \left\{\begin{array}{l} \langle\nabla I_1,X\rangle=\dots=\langle\nabla I_k,X\rangle=0,\\ \langle\nabla D_1,X\rangle=h_1 D_1,\dots, \langle\nabla D_p,X\rangle=h_p D_p.\\ \end{array}\right.$$ Hence, the matrix $k$ is a solution of equation , since the system is obviously equivalent to $$\left\{\begin{array}{l} \mathcal{L}_{X}I_1= \dots = \mathcal{L}_{X}I_k = 0,\\ \mathcal{L}_{X}D_1 = h_1 D_1, \dots, \mathcal{L}_{X}D_p = h_p D_p,\\ \end{array}\right.$$ which is by hypothesis verified by the vector field $X$. Let us solve now the equation associated to the matrix $k$. In order to solve the equation, let us split first the $(n-1)\times (n-1)$ matrix $v$ as follows $ v=\left[ {\begin{array}{*{20}c} v_{k,k} & v_{k,p} \\ v_{p,k} & v_{p,p} \\ \end{array}} \right], $ where the blocks $v_{k,k},v_{k,p},v_{p,k},v_{p,p}$ have dimension $k\times k$, $k\times p$, $p\times k$, and respectively $p\times p$. Using the same type of splitting, the initial condition matrix $v(0)=I_{n-1,n-1}$ splits as follows $$v(0)=\left[ {\begin{array}{*{20}c} v_{k,k}(0) & v_{k,p}(0) \\ v_{p,k}(0) & v_{p,p}(0) \\ \end{array}} \right] =\left[ {\begin{array}{*{20}c} I_{k,k }& O_{k,p} \\ O_{p,k} & I_{p,p} \\ \end{array}} \right],$$ where $I_{k,k}, I_{p,p}$ stands for the identity matrices of dimensions $k\times k$, and respectively $p\times p$. Consequently, the equation becomes $$\left\{\begin{array}{l} \left[ {\begin{array}{*{20}c} \dot{v}_{k,k}(t) & \dot{v}_{k,p}(t) \\ \dot{v}_{p,k}(t) & \dot{v}_{p,p}(t) \\ \end{array}} \right] =\left[ {\begin{array}{*{20}c} O_{k,k} & O_{k,p} \\ O_{p,k} & H_{p,p}(\gamma(t)) \\ \end{array}} \right]\cdot\left[ {\begin{array}{*{20}c} v_{k,k}(t) & v_{k,p}(t) \\ v_{p,k}(t) & v_{p,p}(t) \\ \end{array}} \right], \\ \left[ {\begin{array}{*{20}c} v_{k,k}(0) & v_{k,p}(0) \\ v_{p,k}(0) & v_{p,p}(0) \\ \end{array}} \right]=\left[ {\begin{array}{*{20}c} I_{k,k} & O_{k,p} \\ O_{p,k} & I_{p,p} \\ \end{array}} \right], \end{array}\right.$$ or equivalently $$\left\{\begin{array}{l} \dot{v}_{k,k}(t)=O_{k,k},\\ \dot{v}_{k,p}(t)=O_{k,p},\\ \dot{v}_{p,k}(t)=H_{p,p}(\gamma(t)) v_{p,k}(t),\\ \dot{v}_{p,p}(t)=H_{p,p}(\gamma(t)) v_{p,p}(t),\\ v_{k,k}(0)=I_{k,k},\\ v_{k,p}(0)=O_{k,p},\\ v_{p,k}(0)=O_{p,k},\\ v_{p,p}(0)=I_{p,p}, \end{array}\right.$$ where $H_{p,p}(\gamma(t))=\operatorname{diag}\left[h_1(\gamma(t)),\dots,h_p(\gamma(t))\right]$, and $t\in[0,T]$. Using standard ODE techniques, one obtains the unique solution $$v:[0,T]\rightarrow GL(n-1,\mathbb{{{\mathbb R}}}),$$ $$v(t)=\left[ {\begin{array}{*{20}c} I_{k,k} & O_{k,p} \\ O_{p,k} & v_{p,p}(t) \\ \end{array}} \right],$$ where $$v_{p,p}(t)=\operatorname{diag}\left[\exp\left(\int_{0}^{t}h_1(\gamma(s))ds\right),\dots,\exp\left(\int_{0}^{t}h_p(\gamma(s))ds\right)\right].$$ Since from Theorem , the characteristic multipliers of the periodic orbit $\Gamma$ are given by $\{1\}\cup \sigma(v(T))$, we obtain the conclusion. Note that if $p=0$, we recover a classical result concerning the characteristic multipliers of periodic orbits of completely integrable systems. More precisely, if $p=0$, then the dynamical system $\dot x=X(x)$ from Theorem becomes completely integrable, and using the conclusion of Theorem we get that the characteristic multipliers of any periodic orbit of a completely integrable system, are all equal to one. Some stability results regarding the periodic orbits of codimension-one dissipative dynamical systems ===================================================================================================== This section has two main purposes, namely, the first purpose is to provide sufficient conditions to guarantee the partial orbital asymptotic stability with asymptotic phase of periodic orbits of a codimension-one dissipative dynamical system, whereas the second purpose is to give sufficient conditions to guarantee the instability of periodic orbits of a codimension-one dissipative dynamical system. Let us start by recalling some definitions and also some general results concerning the stability of the periodic orbits of a general dynamical system. In order to do that, let $\dot x =X(x)$ be a dynamical system generated by a smooth vector field $X\in\mathfrak{X}(U)$, defined eventually on an open subset $U\subseteq \mathbb{R}^{n}$. Suppose $\Gamma =\{\gamma(t)\subset U : 0\leq t\leq T \}$ is a $T-$periodic orbit of $\dot x =X(x)$. - The periodic orbit $\Gamma$ is called **orbitally stable** if, given $\varepsilon >0$ there exists a $\delta >0$ such that $\operatorname{dist}(x(t,x_0),\Gamma)<\varepsilon $ for all $t>0$ and for all $x_0 \in U$ such that $\operatorname{dist}(x_0,\Gamma)<\delta $. - The periodic orbit $\Gamma$ is called **unstable** if it is not orbitally stable. - The periodic orbit $\Gamma$ is called **orbitally asymptotically stable** if it is orbitally stable and (by choosing $\delta$ smaller if necessary), $\operatorname{dist}(x(t,x_0),\Gamma)\rightarrow 0$ as $t\rightarrow \infty$. - The periodic orbit $\Gamma$ is called **orbitally phase asymptotically stable**, if it is asymptotically orbitally stable and there is a $\delta >0$ such that for each $x_0 \in U$ with $\operatorname{dist}(x_0,\Gamma)<\delta $, there exists $\theta_{0}=\theta_0 (x_0)$ such that $$\lim_{t\rightarrow \infty}\|x(t,x_0)-\gamma(t+\theta_{0})\|=0.$$ Let us now recall a classical result which gives some sufficient conditions to guarantee the stability/instability of a periodic orbit in terms of its characteristic multipliers. For more details regarding these results see e.g., [@hartman], [@verhulst]. \[AW\] Suppose $\Gamma =\{\gamma(t)\subset U : 0\leq t\leq T \}$ is a $T-$periodic orbit of the dynamical system $\dot x =X(x)$ generated by a smooth vector field $X\in\mathfrak{X}(U)$, defined eventually on an open subset $U\subseteq \mathbb{R}^{n}$. - If the characteristic multiplier one is simple (has multiplicity one) and the rest of the characteristic multipliers of the periodic orbit $\Gamma$ have all of them the modulus strictly less then one, then the periodic orbit $\Gamma$ is asymptotically orbitally stable with asymptotic phase. - If there exists at least one characteristic multiplier of the periodic orbit $\Gamma$, whose modulus is strictly greater then one, then the periodic orbit $\Gamma$ is unstable. Let us now state the main result of this section, which is a generalization of the above result in the case when the characteristic multiplier one is not simple. \[SPO\] Let $\dot x= X(x)$ be a codimension-one dissipative dynamical system generated by a smooth vector field $X\in\mathfrak{X}(U)$ defined eventually on an open subset $U\subseteq \mathbb{R}^{n}$, such that there exists $k,p\in\mathbb{N}$ with $p>0$, $k+p=n-1$, and respectively $I_1,\dots,I_k, D_1,\dots, D_p$, $h_1$, $\dots$, $h_p \in\mathcal{C}^{\infty}(U,\mathbb{R})$ such that $\mathcal{L}_{X}I_1 =\dots= \mathcal{L}_{X}I_k =0$, and $\mathcal{L}_{X}D_1 =h_1 D_1$, $\dots$, $\mathcal{L}_{X}D_p =h_p D_p$. Suppose $\Gamma=\{\gamma(t)\subset U : 0\leq t\leq T \}$ is a $T-$periodic orbit of $\dot x= X(x)$, such that the following conditions hold true: - $\Gamma\subset ID^{-1}(\{0\})$, and $0 \in \mathbb{R}^{n-1}$ is a regular value of the map $$ID=(I_1,\dots,I_k,D_1,\dots,D_p):U\subseteq \mathbb{R}^{n}\rightarrow \mathbb{R}^{n-1},$$ - $\nabla I_1(\gamma(t)),\dots,\nabla I_k(\gamma(t)),\nabla D_1(\gamma(t)),\dots, \nabla D_p (\gamma(t)), X(\gamma(t))$ are linearly\ independent for each $0\leq t \leq T$. Then, if moreover $0 \in \mathbb{R}^{k}$ is a regular value of the map $I=(I_1,\dots,I_k):U\subseteq \mathbb{R}^{n}\rightarrow \mathbb{R}^{k},$ and if $$\int_{0}^{T}h_1 (\gamma(s))ds<0, \dots, \int_{0}^{T}h_p (\gamma(s))ds<0,$$ then the periodic orbit $\Gamma$ is orbitally phase asymptotically stable, with respect to perturbations along the invariant manifold $I^{-1}(\{0\})$. On the other hand, if there exists $i_0 \in \{1,\dots,p\}$ such that $\int_{0}^{T}h_{i_0} (\gamma(s))ds>0$, then the periodic orbit $\Gamma$ is unstable. Recall from Theorem that the characteristic multipliers of the periodic orbit $\Gamma$ of the vector field $X$ are $$\underbrace{1,\dots,1}_{k+1 \ \mathrm{times}}, \exp\left(\int_{0}^{T}h_1 (\gamma(s))ds\right), \dots, \exp\left(\int_{0}^{T}h_p (\gamma(s))ds\right).$$ By a classical result concerning the properties of characteristic multipliers in the presence of first integrals (see e.g., [@moser]) we have that, if the common level set of the first integrals $I_1,\dots,I_k$, containing $\Gamma$, is a smooth manifold, then the characteristic multipliers of $\Gamma$ as a periodic orbit of the restriction of the vector field $X$ to this dynamically invariant manifold are the following: $1$ (due to the fact that $\Gamma$ is a periodic orbit also for the restriction of $X$), and respectively the rest of $n-k-1$ characteristic multipliers of $\Gamma$ as a periodic orbit of $X$. Recall that $\Gamma$ as a periodic orbit of $X$ has $k+1$ characteristic multipliers equal to one ($k$ of them associated to the first integrals $I_1,\dots,I_k$, and one due to the fact that $\Gamma$ is a periodic orbit), and respectively some other $n-k-1$ characteristic multipliers (possible some of them also being equal to one). Consequently, if $0\in \mathbb{R}^{k}$ is a regular value of the map $I:=(I_1,\dots,I_k)$, then the dynamical system $\dot x = X|_{I^{-1}(\{0\})}(x),$ given by the restriction of the vector field $X$ to the dynamically invariant manifold $I^{-1}(\{0\})$, admits $\Gamma$ as periodic orbit (by dynamical invariance), and the associated characteristic multipliers are $$1, \exp\left(\int_{0}^{T}h_1 (\gamma(s))ds\right),\dots,\exp\left(\int_{0}^{T}h_p (\gamma(s))ds\right).$$ Hence, one can apply the Theorem for the dynamical system generated by the vector field $X|_{I^{-1}(\{0\})}$ and respectively for the periodic orbit $\Gamma$, and conclude the corresponding stability/instability results. Consequently, by dynamical invariance, the same conclusions hold true also for the periodic orbit $\Gamma$ associated to the original vector field $X$ with respect to perturbations along the invariant manifold $I^{-1}(\{0\})$. More precisely, if $\int_{0}^{T}h_1 (\gamma(s))ds<0$, $\dots$, $\int_{0}^{T}h_p (\gamma(s))ds<0$, then the characteristic multipliers of the periodic orbit $\Gamma$ of the vector field $X|_{I^{-1}(\{0\})}$ have the following properties: the characteristic multiplier one is simple (its multiplicity is one), and the rest of characteristic multipliers have modulus strictly less then one, and hence the periodic orbit is orbitally phase asymptotically stable. Hence, because of dynamical invariance, the same conclusion holds in the case of the vector field $X$ with respect to perturbations along the invariant manifold $I^{-1}(\{0\})$. On the other hand (even if $0\in \mathbb{R}^{k}$ it is not a regular value of the map $I:=(I_1,\dots,I_k)$), if there exists $i_0 \in\{1,\dots, p\}$ such that $\int_{0}^{T}h_{i_0} (\gamma(s))ds>0$, we obtain directly from Theorem that the periodic orbit $\Gamma$ of the vector field $X$, it is unstable. Let us illustrate now the results of the above theorem in the case of a three dimensional dissipative perturbation of the harmonic oscillator. \[DHO\] Let $\dot{\mathbf{x}}=X(\mathbf{x})$, $\mathbf{x}=(x,y,z)\in\mathbb{R}^{3}$, be the dynamical system generated by the smooth vector field $$X(x,y,z)=y\partial_{x}-x\partial_{y}+zh(x,y,z)\partial_{z},$$ where $h\in\mathcal{C}^{\infty}(\mathbb{R}^{3},\mathbb{R})$ is an arbitrary given smooth real function. Then, the set $\Gamma=\{\gamma(t)=(\sin t,\cos t,0): 0\leq t\leq 2\pi\}$ is a $2\pi-$periodic orbit of the dynamical system $\dot{\mathbf{x}}=X(\mathbf{x})$. Note that the above defined dynamical system is a codimension-one dissipative system, associated with the following data - $I:\mathbb{R}^{3}\rightarrow \mathbb{R}$, $I(x,y,z)=x^2 +y^2 -1$, - $D:\mathbb{R}^{3}\rightarrow \mathbb{R}$, $D(x,y,z)=z$, - $h:\mathbb{R}^{3}\rightarrow \mathbb{R}$, since, $\mathcal{L}_{X}I=0$ and $\mathcal{L}_{X}D=hD$. The hypothesis of the Theorem are verified since - $\Gamma \subset ID^{-1}(\{(0,0)\})$, - $(0,0)$ is a regular value of the map $ID:\mathbb{R}^{3}\rightarrow \mathbb{R}^{2}$, $ID(x,y,z)=(x^2 +y^2 -1,z)$, - $\nabla I(\sin t,\cos t,0),\nabla D(\sin t,\cos t,0),X(\sin t,\cos t,0)$ are linearly independent\ for each $t\in[0,2\pi]$, - $0$ is a regular value of the map $I$. Hence, by Theorem we obtain the following conclusions: - if $\int_{0}^{2\pi}h(\sin t,\cos t,0)dt<0$, then the periodic orbit $\Gamma$ is orbitally phase asymptotically stable, with respect to perturbations along the cylinder $I^{-1}(\{0\})$, - if $\int_{0}^{2\pi}h(\sin t,\cos t,0)dt>0$, then the periodic orbit $\Gamma$ is unstable. Orbitally asymptotically stabilizing the periodic orbits of completely integrable dynamical systems =================================================================================================== The purpose of this section is to apply the results from the previous section in order to partially orbitally asymptotically stabilize, a given periodic orbit of a completely integrable dynamical system. In order to do that, let us consider a completely integrable dynamical system $\dot x=X(x)$, $X\in\mathfrak{X}(U)$, defined eventually on an open subset $U\subseteq \mathbb{R}^{n}$ (i.e., it admits a set of $n-1$ first integrals, $I_1,\dots,I_k, D_1,\dots, D_p\in\mathcal{C}^{\infty}(U,\mathbb{R})$, independent at least on an open subset $V \subseteq U$). Suppose that $\Gamma=\{\gamma(t)\subset V : 0\leq t\leq T \}$ is a $T-$periodic orbit of the system $\dot x=X(x)$. The idea for the stabilization procedure is to perturb the completely integrable system $\dot x=X(x)$, in such a way that the perturbed dynamical system becomes a dissipative dynamical system on $V$, which admits also $\Gamma$ as a periodic orbit, and moreover verifies the hypothesis of Theorem . Note that using classical perturbation methods, the persistence of periodic orbits after perturbations, follows as a consequence of the implicit function theorem. The method introduced in this section, provide for the class of completely integrable dynamical system, an explicit perturbation which preserve (under reasonable conditions) an a-priori given periodic orbit. \[OST\] Let $\dot x= X(x)$ be a completely integrable dynamical system generated by a smooth vector field $X\in\mathfrak{X}(U)$ defined eventually on an open subset $U\subseteq \mathbb{R}^{n}$, and let $k,p\in\mathbb{N}$ be two natural numbers, with $k+p=n-1$, such that there exist $n-1$ first integrals $I_1,\dots,I_k, D_1,\dots, D_p\in\mathcal{C}^{\infty}(U,\mathbb{R})$, independent on an open subset $V \subseteq U$. Suppose the system $\dot x= X(x)$ admits a $T-$periodic orbit $\Gamma=\{\gamma(t)\subset V : 0\leq t\leq T \}$ such that: - $\Gamma\subset ID^{-1}(\{0\})$, and $0 \in \mathbb{R}^{n-1}$ is a regular value of the map $$ID=(I_1,\dots,I_k,D_1,\dots,D_p):V\subseteq \mathbb{R}^{n}\rightarrow \mathbb{R}^{n-1},$$ - $\nabla I_1(\gamma(t)),\dots,\nabla I_k(\gamma(t)),\nabla D_1(\gamma(t)),\dots, \nabla D_p (\gamma(t)), X(\gamma(t))$ are linearly\ independent for each $0\leq t \leq T$. If moreover, $0 \in \mathbb{R}^{k}$ is a regular value of the map $I=(I_1,\dots,I_k):V \subseteq \mathbb{R}^{n}\rightarrow \mathbb{R}^{k},$ then for any choice of smooth functions $h_1,\dots, h_p \in\mathcal{C}^{\infty}(V,\mathbb{R})$ such that $$\int_{0}^{T}h_1 (\gamma(s))ds<0, \dots, \int_{0}^{T}h_p (\gamma(s))ds<0,$$ $\Gamma$, as a periodic orbit of the dissipative dynamical system $\dot x =X(x)+X_{0}(x)$, $x\in V$, $$X_{0}=\left\| \bigwedge_{i=1}^{p} \nabla D_i\wedge\bigwedge_{j=1}^{k} \nabla I_j \right\|_{n-1}^{-2}\cdot\sum_{i=1}^{p}(-1)^{n-i}h_i D_i \Theta_i,$$ $$\Theta_i = \star\left[ \bigwedge_{j=1, j\neq i}^{p} \nabla D_j \wedge \bigwedge_{l=1}^{k} \nabla I_l \wedge\star\left(\bigwedge_{j=1}^{p} \nabla D_j\wedge\bigwedge_{l=1}^{k} \nabla I_l \right)\right],$$ is orbitally phase asymptotically stable, with respect to perturbations along the invariant manifold $I^{-1}(\{0\})$. On the other hand, for any choice of smooth functions $k_1,\dots, k_p \in\mathcal{C}^{\infty}(V,\mathbb{R}),$ such that there exists $i_0 \in \{1,\dots,p\}$ for which $$\int_{0}^{T}k_{i_0} (\gamma(s))ds>0,$$ $\Gamma$, as a periodic orbit of the dissipative dynamical system $\dot x =X(x)+X_{0}(x)$, $x\in V$, $$X_{0}=\left\| \bigwedge_{i=1}^{p} \nabla D_i\wedge\bigwedge_{j=1}^{k} \nabla I_j \right\|_{n-1}^{-2}\cdot\sum_{i=1}^{p}(-1)^{n-i}k_i D_i \Theta_i,$$ $$\Theta_i = \star\left[ \bigwedge_{j=1, j\neq i}^{p} \nabla D_j \wedge \bigwedge_{l=1}^{k} \nabla I_l \wedge\star\left(\bigwedge_{j=1}^{p} \nabla D_j\wedge\bigwedge_{l=1}^{k} \nabla I_l \right)\right],$$ is an unstable periodic orbit. Let us show first that the perturbed system $\dot x = X(x)+ X_0 (x)$, is a codimension-one dissipative dynamical system. In order to do that, will be enough to prove that the vector field $X+X_0$ conserves $I_1,\dots,I_k$ and dissipates $D_1,\dots, D_p$. For a unified notation, let us denote for each $j\in\{1,\dots,p\}$, $u_j = h_j$, for the fist hypothesis of Theorem , and respectively $u_j = k_j$, for the second hypothesis of Theorem . Recall from Remark that: $$\left\{\begin{array}{l} \mathcal{L}_{X_0}I_1= \dots = \mathcal{L}_{X_0}I_k = 0,\\ \mathcal{L}_{X_0}D_1 = u_1 D_1, \dots, \mathcal{L}_{X_0}D_p = u_p D_p.\\ \end{array}\right.$$ Hence, for each $i\in\{1,\dots,k\}$ and respectively $j\in\{1,\dots,p\}$, we obtain $$\left\{\begin{array}{l} \mathcal{L}_{X+X_0}I_i= \mathcal{L}_{X}I_i + \mathcal{L}_{X_0}I_i=0+0=0,\\ \mathcal{L}_{X+X_0}D_j = \mathcal{L}_{X}D_j+\mathcal{L}_{X_0}D_j= 0+ u_j D_j = u_j D_j,\\ \end{array}\right.$$ and consequently, $\dot x = X(x)+X_{0}(x)$, $x\in V$, is a codimension-one dissipative dynamical system. Recall that $\Gamma=\{\gamma(t)\subset V : 0\leq t\leq T \}$ is a periodic orbit of the dynamical system $\dot x = X(x)+X_{0}(x)$ too, since the hypothesis $\Gamma\subset ID^{-1}(\{0\})$ implies that $D_i \circ \gamma =0$, for every $i\in\{1,\dots, p\}$, and consequently $X_{0} (\gamma(t))=0$, for every $t\in[0,T]$. Note that since $(X+X_0)(\gamma(t))=X(\gamma(t))$, for every $t\in[0,T]$, the condition that $\nabla I_1(\gamma(t)),\dots,\nabla I_k(\gamma(t)),\nabla D_1(\gamma(t)),\dots, \nabla D_p (\gamma(t)), X(\gamma(t))$ are linearly independent for each $0\leq t \leq T$, is obviously equivalent with the condition that $$\nabla I_1(\gamma(t)),\dots,\nabla I_k(\gamma(t)),\nabla D_1(\gamma(t)),\dots, \nabla D_p (\gamma(t)), (X+X_0)(\gamma(t))$$ are linearly independent for each $0\leq t \leq T$. Now the conclusions of Theorem follow by applying the Theorem for the codimension-one dissipative dynamical system $\dot x = X(x)+ X_0 (x)$, $x\in V$, and respectively the $T-$periodic orbit $\Gamma=\{\gamma(t)\subset V : 0\leq t\leq T \}$. \[remimp\] In the hypothesis of the Theorem , note that: - a consequence of the Remark is that the set of points $x\in U$ such that $$\left\| \bigwedge_{i=1}^{p} \nabla D_i (x)\wedge\bigwedge_{j=1}^{k} \nabla I_j (x)\right\|_{n-1}=0,$$ is a subset of the equilibrium points of the completely integrable vector field $X$. - the condition $\Gamma\subset ID^{-1}(\{0\})$ implies that **for any choice** of smooth functions $h_1,\dots, h_p \in\mathcal{C}^{\infty}(V,\mathbb{R})$, the control vector field $X_0 \in \mathfrak{X}(V)$, given by $$X_{0}=\left\| \bigwedge_{i=1}^{p} \nabla D_i\wedge\bigwedge_{j=1}^{k} \nabla I_j \right\|_{n-1}^{-2}\cdot\sum_{i=1}^{p}(-1)^{n-i}h_i D_i \Theta_i,$$ $$\Theta_i = \star\left[ \bigwedge_{j=1, j\neq i}^{p} \nabla D_j \wedge \bigwedge_{l=1}^{k} \nabla I_l \wedge\star\left(\bigwedge_{j=1}^{p} \nabla D_j\wedge\bigwedge_{l=1}^{k} \nabla I_l \right)\right],$$ verifies that $X_{0} (\gamma(t))=0$, for every $t\in[0,T]$; - each of the smooth functions $h_1,\dots, h_p \in\mathcal{C}^{\infty}(V,\mathbb{R})$ might be chosen of the type e.g., $h(x)=-(\psi^2 (x)+ c)$, $x\in V$, with $\psi\in\mathcal{C}^{\infty}(V,\mathbb{R})$ and $c\in (0,\infty)$, since $$\int_{0}^{T}h (\gamma(s))ds=-\int_{0}^{T}\psi^2 (\gamma(s))ds -Tc \leq -Tc<0;$$ - the smooth function $k_{i_0} \in\mathcal{C}^{\infty}(V,\mathbb{R})$ might be chosen of type e.g., $k(x)=\phi^2 (x)+ c$, $x\in V$, with $\phi\in\mathcal{C}^{\infty}(V,\mathbb{R})$ and $c\in (0,\infty)$, since $$\int_{0}^{T}k (\gamma(s))ds=\int_{0}^{T}\phi^2 (\gamma(s))ds +Tc \geq Tc>0.$$ Let us now illustrate the above stabilization result in the case of the harmonic oscillator. We consider this simple example in order to point out that different choices of $I's$ and $D's$ may generate different domains of definition for the perturbed vector field. Let us consider the family of harmonic oscillators, described by the three dimensional vector field $$X(x,y,z)=y \partial_{x}-x \partial_{y} \in\mathfrak{X}(\mathbb{R}^3).$$ The induced dynamical system, $$\label{HO}\dot{\mathbf{x}}=X(\mathbf{x}), \ \mathbf{x}=(x,y,z)\in\mathbb{R}^{3},$$ admits a $2\pi-$periodic orbit given by $\Gamma=\{\gamma(t)=(\sin t,\cos t,0): 0\leq t\leq 2\pi\}$. Moreover, the system is completely integrable, since it has two independent first integrals, namely $$I_1 (x,y,z)=x^2 +y^2 -1, \ I_2 (x,y,z)=z.$$ In order to apply the Theorem , the candidates for the functions $I$ and $D$ are the first integrals $I_1$ and respectively $I_2$. Consequently, we have two cases, namely $I=I_1$ and $D=I_2$, and respectively $I=I_2$ and $D=I_1$. $\diamond$ **Let us now analyze the first case, namely $I=I_1$ and $D=I_2$.** By straightforward computations we obtain that the vector field $X_0$ from Theorem , in this case has the expression $$X_0 (x,y,z)=z u(x,y,z)\partial_{z}, \ (x,y,z)\in\mathbb{R}^{3},$$ and consequently it verifies the condition $X_0 \circ\gamma=0$, for any smooth real function $u\in\mathcal{C}^{\infty}(\mathbb{R}^{3},\mathbb{R})$. Consequently, the perturbed system $$\label{PSTR} \dot{\mathbf{x}}=X(\mathbf{x})+X_{0}(\mathbf{x}), \ \mathbf{x}=(x,y,z)\in\mathbb{R}^{3},$$ is a codimension-one dissipative dynamical system associated to $I,D,u\in\mathcal{C}^{\infty}(\mathbb{R}^{3},\mathbb{R})$, i.e., $\mathcal{L}_{X+X_{0}}I=0$, and respectively $\mathcal{L}_{X+X_{0}}D=uD$. Since $X_0 \circ\gamma=0$, for any smooth real function $u\in\mathcal{C}^{\infty}(\mathbb{R}^{3},\mathbb{R})$, we obtain that $\Gamma$ is a periodic orbit of the dissipative system $\dot{\mathbf{x}}=X(\mathbf{x})+X_{0}(\mathbf{x})$, for any smooth real function $u\in\mathcal{C}^{\infty}(\mathbb{R}^{3},\mathbb{R})$. Moreover, the rest of hypothesis of Theorem are similar to those of Theorem , and were already been verified in Example for the vector field $$X(x,y,z)+X_0 (x,y,z)= y \partial_{x}-x \partial_{y}+ zu(x,y,z)\partial_{z}.$$ Hence, by Theorem , we obtain the following conclusions: - for any smooth function $u\in \mathcal{C}^{\infty}(\mathbb{R}^{3},\mathbb{R})$ such that $\int_{0}^{2\pi}u(\sin t,\cos t,0)dt<0$, the periodic orbit $\Gamma$ of the associated perturbed system is orbitally phase asymptotically stable, with respect to perturbations along the cylinder $I^{-1}(\{0\})$; - for any smooth function $u\in \mathcal{C}^{\infty}(\mathbb{R}^{3},\mathbb{R})$ such that $\int_{0}^{2\pi}u(\sin t,\cos t,0)dt>0$, the periodic orbit $\Gamma$ of the associated perturbed system is unstable. $\diamond$ $\diamond$ **Let us now analyze the second case, namely $I=I_2$ and $D=I_1$.** By straightforward computations we obtain that the vector field $X_0$ from Theorem , in this case has the expression $$X_0 (x,y,z)=\dfrac{u(x,y,z)(x^2 +y^2 -1)}{2(x^2 +y^2)}\left(x\partial_{x}+y\partial_{y}\right), \ (x,y,z)\in V:=\mathbb{R}^{3}\setminus\{(0,0,z): z\in\mathbb{R}\},$$ and consequently it verifies the condition $X_0 \circ\gamma=0$, for any smooth real function $u\in\mathcal{C}^{\infty}(V,\mathbb{R})$. Consequently, the perturbed system $$\label{PSTR1} \dot{\mathbf{x}}=X(\mathbf{x})+X_{0}(\mathbf{x}), \ \mathbf{x}=(x,y,z)\in V,$$ is a codimension-one dissipative dynamical system associated to $I,D,u$, i.e., $\mathcal{L}_{X+X_{0}}I=0$, and respectively $\mathcal{L}_{X+X_{0}}D=uD$. Since $X_0 \circ\gamma=0$, for any smooth real function $u\in\mathcal{C}^{\infty}(V,\mathbb{R})$, we obtain that $\Gamma$ is a periodic orbit of the dissipative system $\dot{\mathbf{x}}=X(\mathbf{x})+X_{0}(\mathbf{x})$, for any smooth real function $u\in\mathcal{C}^{\infty}(V,\mathbb{R})$. Moreover, the rest of hypothesis of Theorem are obviously verified, since they are similar with those from the previous case. Note that in this case, the perturbed vector field is given by $$\begin{aligned} X(x,y,z)+X_0 (x,y,z)&= \left[ \dfrac{u(x,y,z)x(x^2 +y^2 -1)}{2(x^2 +y^2)}+y\right]\partial_{x}\\ &+\left[ \dfrac{u(x,y,z)y(x^2 +y^2 -1)}{2(x^2 +y^2)}-x \right]\partial_{y}.\end{aligned}$$ Hence, by Theorem , we obtain the following conclusions: - for any smooth function $u\in \mathcal{C}^{\infty}(V,\mathbb{R})$ such that $\int_{0}^{2\pi}u(\sin t,\cos t,0)dt<0$, the periodic orbit $\Gamma$ of the associated perturbed system is orbitally phase asymptotically stable, with respect to perturbations along the plane $I^{-1}(\{0\})$; - for any smooth function $u\in \mathcal{C}^{\infty}(V,\mathbb{R})$ such that $\int_{0}^{2\pi}u(\sin t,\cos t,0)dt>0$, the periodic orbit $\Gamma$ of the associated perturbed system is unstable. Let us now illustrate the main result in the case of a mechanical dynamical system, namely Euler’s equations from the free rigid body dynamics. Let us recall that Euler’s equations in terms of the rigid body angular momenta are generated by the vector field $$X(x,y,z)=\left( \dfrac{1}{I_3}-\dfrac{1}{I_2}\right)yz \partial_{x}+ \left( \dfrac{1}{I_1}-\dfrac{1}{I_3}\right)zx\partial_{y}+ \left( \dfrac{1}{I_2}-\dfrac{1}{I_1}\right)xy\partial_{z}\in\mathfrak{X}(\mathbb{R}^3),$$ where $I_1, I_2, I_3$ are the moments of inertia in the principal axis frame of the rigid body. We will suppose in the following that $I_1 > I_2 > I_3 > 0$. The induced dynamical system, $$\label{HOO}\dot{\mathbf{x}}=X(\mathbf{x}), \ \mathbf{x}=(x,y,z)\in\mathbb{R}^{3},$$ is completely integrable, since it has two independent first integrals, namely $$F_1 (x,y,z)=\dfrac{1}{2}\left(\dfrac{x^2}{I_1} +\dfrac{y^2}{I_2} +\dfrac{z^2}{I_3} \right), \ F_2 (x,y,z)=\dfrac{1}{2}\left(x^2 + y^2 + z^2 \right).$$ Recall that there exists an open and dense subset of the image of the map $(F_1,F_2):\mathbb{R}^3 \rightarrow \mathbb{R}^2$, such that each fiber of any element $(h,c)$ from this set, corresponds to periodic orbits of Euler’s equations. Moreover, any such element is a regular value of $(F_1,F_2)$, as well as its components for the corresponding maps, $F_1$ and respectively $F_2$. For more details see, e.g., [@dgt]. Let $(h,c)\in \mathbb{R}^2$ belongs to the above mention set, and let us denote $$J_1 (x,y,z):=F_1(x,y,z)-h, \quad J_2 (x,y,z):= F_2 (x,y,z)- c.$$ In the above conditions, the set $$\Gamma=\left\{\gamma(t)=(\gamma_1 (t),\gamma_2 (t),\gamma_3 (t)): 0\leq t\leq \dfrac{4K\sqrt{I_1 I_2 I_3}}{\sqrt{2(I_2 - I_3)(h I_1 - c)}}\right\},$$ where $$\begin{aligned} \gamma_1 (t)&=\sqrt{\dfrac{2I_1 (c-h I_3)}{I_1 - I_3}} \cdot \operatorname{cn}\left(\sqrt{\dfrac{2(I_2 - I_3)(h I_1 - c)}{I_1 I_2 I_3}}\cdot t;k\right),\\ \gamma_2 (t)&=\sqrt{\dfrac{2I_2 (c-h I_3)}{I_2 - I_3}} \cdot \operatorname{sn}\left(\sqrt{\dfrac{2(I_2 - I_3)(h I_1 - c)}{I_1 I_2 I_3}}\cdot t;k\right),\\ \gamma_3 (t)&=-\sqrt{\dfrac{2I_3 (-c+h I_1)}{I_1 - I_3}} \cdot \operatorname{dn}\left(\sqrt{\dfrac{2(I_2 - I_3)(h I_1 - c)}{I_1 I_2 I_3}}\cdot t;k\right),\\ k&=\sqrt{\dfrac{(h I_3 - c)(I_1 - I_2)}{(h I_1 - c)(I_3 - I_2)}},\\ K&=\int_{0}^{1}\dfrac{dt}{\sqrt{(1-t^2)(1-k^2 t^2)}},\end{aligned}$$ is a $\dfrac{4K\sqrt{I_1 I_2 I_3}}{\sqrt{2(I_2 - I_3)(h I_1 - c)}}-$periodic orbit of the dynamical system , which belongs to the common zero level set of the first integrals $J_1$ and $J_2$. See for details, e.g., [@dgt]. Recall that the above parameterization of $\Gamma$ is given in terms of Jacobi elliptic functions. By straightforward computations we get that $\nabla J_1 (\mathbf{x}), \nabla J_2 (\mathbf{x}), X (\mathbf{x})$, are linearly dependent vectors if and only if $\mathbf{x}$ is an equilibrium point of the dynamical system . Hence, the vectors $\nabla J_1 (\gamma(t)), \nabla J_2 (\gamma(t)), X (\gamma(t))$, are linearly independent for each $t\in \left[0, \dfrac{4K\sqrt{I_1 I_2 I_3}}{\sqrt{2(I_2 - I_3)(h I_1 - c)}}\right].$ In order to apply the Theorem , the candidates for the functions $I$ and $D$ are the first integrals $J_1$ and respectively $J_2$. Consequently, we have two cases, namely $I=J_1$ and $D=J_2$, and respectively $I=J_2$ and $D=J_1$. $\diamond$ **Let us now analyze the first case, namely $I=J_1$ and $D=J_2$.** By straightforward computations we obtain that the vector field $X_0$ from Theorem , in this case has the expression $$\begin{aligned} X_0 (x,y,z)&=\dfrac{u(x,y,z)\left[\dfrac{1}{2}(x^2 +y^2 +z^2)-c \right]}{\left[xy\left(\dfrac{1}{I_1}-\dfrac{1}{I_2}\right)\right]^2 +\left[yz\left(\dfrac{1}{I_2}-\dfrac{1}{I_3}\right)\right]^2+\left[xz\left(\dfrac{1}{I_1}-\dfrac{1}{I_3}\right)\right]^2}\\ &\cdot \{ x\left[ \dfrac{1}{I_2}\left(\dfrac{1}{I_2}-\dfrac{1}{I_1}\right) y^2 + \dfrac{1}{I_3}\left(\dfrac{1}{I_3}-\dfrac{1}{I_1}\right) z^2\right]\partial_{x} \\ &+ y\left[ \dfrac{1}{I_1}\left(\dfrac{1}{I_1}-\dfrac{1}{I_2}\right) x^2 + \dfrac{1}{I_3}\left(\dfrac{1}{I_3}-\dfrac{1}{I_2}\right) z^2\right]\partial_{y}\\ &+ z\left[ \dfrac{1}{I_1}\left(\dfrac{1}{I_1}-\dfrac{1}{I_3}\right) x^2 + \dfrac{1}{I_2}\left(\dfrac{1}{I_2}-\dfrac{1}{I_3}\right) y^2\right]\partial_{z} \} , \ (x,y,z)\in V,\end{aligned}$$ where $V:=\mathbb{R}^3 \setminus\{\{(x,0,0): x\in\mathbb{R}\}\cup \{(0,y,0): y\in\mathbb{R}\} \cup \{(0,0,z): z\in\mathbb{R}\}\}$ (note that $V$ is exactly the complement of the set of equilibrium states of Euler’s equations). Recall that the vector field $X_0$ verifies the condition $X_0 \circ\gamma=0$, for any smooth real function $u\in\mathcal{C}^{\infty}(V,\mathbb{R})$, since $\Gamma$ belongs to the zero level set of the first integral $J_2$. Consequently, the perturbed system $$\label{PSTRA} \dot{\mathbf{x}}=X(\mathbf{x})+X_{0}(\mathbf{x}), \ \mathbf{x}=(x,y,z)\in\mathbb{R}^{3},$$ is a codimension-one dissipative dynamical system associated to $I,D,u$, i.e., $\mathcal{L}_{X+X_{0}}I=0$, and respectively $\mathcal{L}_{X+X_{0}}D=uD$. Since $X_0 \circ\gamma=0$, for any smooth real function $u\in\mathcal{C}^{\infty}(V,\mathbb{R})$, we obtain that $\Gamma$ is a periodic orbit of the dissipative system $\dot{\mathbf{x}}=X(\mathbf{x})+X_{0}(\mathbf{x})$, for any smooth real function $u\in\mathcal{C}^{\infty}(V,\mathbb{R})$. Since all the hypothesis of Theorem are verified, we obtain the following conclusions. If one denotes $T:=\dfrac{4K\sqrt{I_1 I_2 I_3}}{\sqrt{2(I_2 - I_3)(h I_1 - c)}}$, then - for any smooth function $u\in \mathcal{C}^{\infty}(V,\mathbb{R})$ such that $\int_{0}^{T}u(\gamma_1 (t),\gamma_2 (t),\gamma_3 (t))dt<0$, the periodic orbit $\Gamma$ of the associated perturbed system is orbitally phase asymptotically stable, with respect to perturbations along the ellipsoid $$I^{-1}(\{0\})=\left\{(x,y,z): \ \dfrac{1}{2}\left(\dfrac{x^2}{I_1} +\dfrac{y^2}{I_2} +\dfrac{z^2}{I_3} \right)-h=0 \right\};$$ - for any smooth function $u\in \mathcal{C}^{\infty}(V,\mathbb{R})$ such that $\int_{0}^{T}u(\gamma_1 (t),\gamma_2 (t),\gamma_3 (t))dt>0$, the periodic orbit $\Gamma$ of the associated perturbed system is unstable. $\diamond$ $\diamond$ **Let us now analyze the second case, namely $I=J_2$ and $D=J_1$.** By straightforward computations we obtain that the vector field $X_0$ from Theorem , in this case has the expression $$\begin{aligned} X_0 (x,y,z)&=\dfrac{u(x,y,z)\left[\dfrac{1}{2}\left(\dfrac{x^2}{I_1} +\dfrac{y^2}{I_2} +\dfrac{z^2}{I_3}\right)-h \right]}{\left[xy\left(\dfrac{1}{I_1}-\dfrac{1}{I_2}\right)\right]^2 +\left[yz\left(\dfrac{1}{I_2}-\dfrac{1}{I_3}\right)\right]^2+\left[xz\left(\dfrac{1}{I_1}-\dfrac{1}{I_3}\right)\right]^2}\\ &\cdot \{ x\left[ \left(\dfrac{1}{I_1}-\dfrac{1}{I_2}\right) y^2 + \left(\dfrac{1}{I_1}-\dfrac{1}{I_3}\right) z^2\right]\partial_{x} \\ &+ y\left[ \left(\dfrac{1}{I_2}-\dfrac{1}{I_1}\right) x^2 + \left(\dfrac{1}{I_2}-\dfrac{1}{I_3}\right) z^2\right]\partial_{y}\\ &+ z\left[ \left(\dfrac{1}{I_3}-\dfrac{1}{I_1}\right) x^2 + \left(\dfrac{1}{I_3}-\dfrac{1}{I_2}\right) y^2\right]\partial_{z} \} , \ (x,y,z)\in V,\end{aligned}$$ where $V:=\mathbb{R}^3 \setminus\{\{(x,0,0): x\in\mathbb{R}\}\cup \{(0,y,0): y\in\mathbb{R}\} \cup \{(0,0,z): z\in\mathbb{R}\}\}$ (note that $V$ is exactly the complement of the set of equilibrium states of Euler’s equations). Recall that the vector field $X_0$ verifies the condition $X_0 \circ\gamma=0$, for any smooth real function $u\in\mathcal{C}^{\infty}(V,\mathbb{R})$, since $\Gamma$ belongs to the zero level set of the first integral $J_1$. Consequently, the perturbed system $$\label{PSTRB} \dot{\mathbf{x}}=X(\mathbf{x})+X_{0}(\mathbf{x}), \ \mathbf{x}=(x,y,z)\in\mathbb{R}^{3},$$ is a codimension-one dissipative dynamical system associated to $I,D,u$, i.e., $\mathcal{L}_{X+X_{0}}I=0$, and respectively $\mathcal{L}_{X+X_{0}}D=uD$. Since $X_0 \circ\gamma=0$, for any smooth real function $u\in\mathcal{C}^{\infty}(V,\mathbb{R})$, we obtain that $\Gamma$ is a periodic orbit of the dissipative system $\dot{\mathbf{x}}=X(\mathbf{x})+X_{0}(\mathbf{x})$, for any smooth real function $u\in\mathcal{C}^{\infty}(V,\mathbb{R})$. Since all the hypothesis of Theorem are verified, we obtain the following conclusions. If one denotes $T:=\dfrac{4K\sqrt{I_1 I_2 I_3}}{\sqrt{2(I_2 - I_3)(h I_1 - c)}}$, then - for any smooth function $u\in \mathcal{C}^{\infty}(V,\mathbb{R})$ such that $\int_{0}^{T}u(\gamma_1 (t),\gamma_2 (t),\gamma_3 (t))dt<0$, the periodic orbit $\Gamma$ of the associated perturbed system is orbitally phase asymptotically stable, with respect to perturbations along the sphere $$I^{-1}(\{0\})=\left\{(x,y,z): \ \dfrac{1}{2}\left(x^2 + y^2 + z^2 \right)-c =0 \right\};$$ - for any smooth function $u\in \mathcal{C}^{\infty}(V,\mathbb{R})$ such that $\int_{0}^{T}u(\gamma_1 (t),\gamma_2 (t),\gamma_3 (t))dt>0$, the periodic orbit $\Gamma$ of the associated perturbed system is unstable. Acknowledgment {#acknowledgment .unnumbered} -------------- This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-RU-TE-2011-3-0103. [99]{} [R.M. Tudoran]{}\ West University of Timişoara\ Faculty of Mathematics and Computer Science\ Department of Mathematics\ Blvd. Vasile Pârvan, No. 4\ 300223 - Timişoara, România.\ E-mail: [tudoran@math.uvt.ro]{}\

--- abstract: | In this paper we present an interacting-agent model of speculative activity explaining bubbles and crashes in stock markets. We describe stock markets through an infinite-range Ising model to formulate the tendency of traders getting influenced by the investment attitude of other traders. Bubbles and crashes are understood and described qualitatively and quantitatively in terms of the classical phase transitions. When the interactions among traders become stronger and reach some critical values, a second-order phase transition and critical behaviour can be observed, and a bull market phase and a bear market phase appear. When the system stays at the bull market phase, speculative bubbles occur in the stock market. For a certain range of the external field that we shall call the investment environment, multistability and hysteresis phenomena are observed. When the investment environment reaches some critical values, the rapid changes in the distribution of investment attitude are caused. The first-order phase transition from a bull market phase to a bear market phase is considered as a stock market crash. Furthermore we estimate the parameters of the model using the actual financial data. As an example of large crashes we analyse Japan crisis (the bubble and the subsequent crash in the Japanese stock market in 1987-1992), and show that the good quality of the fits, as well as the consistency of the values of the parameters are obtained from Japan crisis. The results of the empirical study demonstrate that Japan crisis can be explained quite naturally by the model that bubbles and crashes have their origin in the collective crowd behaviour of many interacting agents.\ [**keyword**]{} Speculative Bubbles; Stock market crash; Phase transition; Mean field approximation; Japan crisis author: - | Taisei Kaizoji\ Department of Economics, University of Kiel [^1] title: 'Speculative Bubbles and Crashes in Stock Markets: An Interacting-Agent Model of Speculative Activity' --- Introduction ============ The booms and the market crashes in financial markets have been an object of study in economics and a history of economy for a long time. Economists [@Keynes], and Economic historians [@Mackay], [@Kindlberger], [@Galbraith] have often suggested the importance of psychological factors and irrational factors in explaining historical financial euphoria. As Keynes [@Keynes], a famous economist and outstandingly successful investor, acutely pointed out in his book, [*The General theory of Employment, Interest and Money*]{}, stock price changes have their origin in the collective crowd behaviour of many interacting agents rather than fundamental values which can be derived from careful analysis of present conditions and future prospects of firms. In a recent paper published in the Economic Journal, Lux [@Lux1] modelled the idea explicitly and proposed a new theoretical framework to explain bubbles and subsequent crashes which links market crashes to the phase transitions studied in statistical physics. In his model the emergence of bubbles and crashes is formalised as a self-organising process of infections among heterogeneous traders [^2]. In recent independent works, several groups of physicists [@SJ1], [@SJ2], [@SJ3], [@JS1], [@JS2], [@JS3], [@F1], [@F2], [@V1], [@V2], [@Gluzman] proposed and demonstrated empirically that large stock market crashes, such as the 1929 and the 1987 crashes, are analogous to critical points. They have claimed that financial crashes can be predicted using the idea of log-periodic oscillations or by other methods inspired by the physics of critical phenomena [^3]. In this paper we present an interacting-agent model of speculative activity explaining bubbles and crashes in stock markets. We describe stock markets through an infinite-range Ising model to formulate the tendency of traders getting influenced by the investment attitude of other traders. Bubbles and crashes are understood and described qualitatively and quantitatively in terms of the classical phase transitions [^4]. Although the interacting-agent hypothesis [@Lux2] is advocated as an alternative approach to the efficient market hypothesis (or rational expectation hypothesis) [@Fama], little attention has been given to the point how probabilistic rules, that agents switch their investment attitude, are connected with their decision-making or their expectation formations. Our interacting-agent model follows the line of Lux [@Lux1], but differs from his work in the respect that we model speculative activity here from a viewpoint of traders’ decision-making. The decision-making of interacting-agents will be formalised by [*minimum energy principle*]{}, and the stationary probability distribution on traders’ investment attitudes will be derived. Next, the stationary states of the system and speculative dynamics are analysed by using the mean field approximation. It is suggested that the mean field approximation can be considered as a mathematical formularisation of [*Keynes’ beauty contest*]{}. There are three basic stationary states in the system: a bull market equilibrium, a bear market equilibrium, and a fundamental equilibrium. We show that the variation of parameters like the bandwagon effect or the investment environment, which corresponds to the external field, can change the size of cluster of traders’ investment attitude or make the system jump to another market phase. When the bandwagon effect reaches some critical value, a second-order phase transition and critical behaviour can be observed. There is a symmetry breaking at the fundamental equilibrium, and two stable equilibria, the bull market equilibrium and bear market equilibrium appear. When the system stays in the bull market equilibrium, speculative bubble occurs in the stock market. For a certain range of the investment environment multistability and hysteresis phenomena are observed. When the investment environment reaches some critical values, the rapid changes (the first-order phase transitions) in the distribution of investment attitude are caused. The phase transition from a bull market phase to a bear market phase is considered as a stock market crash. Then, we estimate the parameters of the interacting-agent model using an actual financial data. As an example of large crashes we will analyse the Japan crisis (bubble and crash in Japanese stock market in 1987-1992). The estimated equation attempts to explain Japan crisis over 6 year period 1987-1992, and was constructed using monthly adjusted data for the first difference of TOPIX and the investment environment which is defined below. Results of estimation suggest that the traders in the Japanese stock market stayed the bull market equilibria, so that the speculative bubbles were caused by the strong bandwagon effect and betterment of the investment environment in 3 year period 1987-1989, but a turn for the worse of the investment environment in 1990 gave cause to the first-order phase transition from a bull market phase to a bear market phase. We will demonstrate that the market-phase transition occurred in March 1990. In Section 2 we construct the model. In Section 3 we investigate the relationship between crashes and the phase transitions. We implement an empirical study of Japan crisis in Section 4. We give some concluding remarks in Section 5. An Interacting-Agent Model of Speculative Activity ================================================== We think of the stock market that large numbers of traders participate in trading. The stock market consists of N traders (members of a trader group). Traders are indexed by $ j = 1, 2, ........, N $. We assume that each of them can share one of two investment attitudes, buyer or seller, and buy or sell a fixed amount of stock (q) in a period. $ x_{i} $ denotes the investment attitude of trader $ i $ at a period. The investment attitude $ x_{i} $ is defined as follows: if trader $ i $ is the buyer of the stock at a period, then $ x_{i} = + 1 $. If trader $ i $, in contrast, is the seller of the stock at a period, then $ x_{i} = - 1 $. Decision-Making of traders -------------------------- In the stock market the price changes are subject to the law of demand and supply, that the price rises when there is excess demand, and the price falls when there is excess supply. It seems natural to assume that the price raises if the number of the buyer exceeds the number of the seller because there may be excess demand, and the price falls if the number of seller exceeds the number of the seller because there may be excess supply. Thus a trader, who expects a certain exchange profit through trading, will predict every other traders’ behaviour, and will choose the same behaviour as the other traders’ behaviour as thoroughly as possible he could. The decision-making of traders will be also influenced by changes of the firm’s fundamental value, which can be derived from analysis of present conditions and future prospects of the firm, and the return on the alternative asset (e.g. bonds). For simplicity of an empirical analysis we will use the ratio of ordinary profits to total capital that is a typical measure of investment, as a proxy for changes of the fundamental value, and the long-term interest rate as a proxy for changes of the return on the alternative asset. Furthermore we define the investment environment as [*investment environment = ratio of ordinary profits to total capital - long-term interest rate*]{}. When the investment environment increases (decreases) a trader may think that now is the time for him to buy (sell) the stock. Formally let us assume that the investment attitude of trader $ i $ is determined by minimisation of the following [*disagreement*]{} function $ e_i(x) $, $$e_i(x) = - \frac{1}{2} \sum^N_{j=1} a_{ij} x_i x_j - b_i s x_i. \label{eqn:a1}$$ where $ a_{ij} $ denotes the strength of trader $ j $’s influence on trader $ i $, and $ b_i $ denotes the strength of the reaction of trader $ i $ upon the change of the investment environment $ s $ which may be interpreted as an external field, and $ x $ denotes the vector of investment attitude $ x = (x_1, x_2,......x_N) $. The optimisation problem that should be solved for every trader to achieve minimisation of their disagreement functions $ e_i(x) $ at the same time is formalised by $$\min E(x) = - \frac{1}{2} \sum^N_{i=1} \sum^N_{j=1} a_{ij} x_i x_j - \sum^N_{i=1} b_i s x_i. \label{eqn:a2}$$ Now let us assume that trader’s decision making is subject to a probabilistic rule. The summation over all possible configurations of agents’ investment attitude $ x = (x_1,.....,x_N) $ is computationally explosive with size of the number of trader $ N $.Therefore under the circumstance that a large number of traders participates into trading, a probabilistic setting may be one of best means to analyse the collective behaviour of the many interacting traders. Let us introduce a random variable $ x^k = (x^k_1, x^k_2,......, x^k_N) $, $ k = 1, 2,....., K $. The state of the agents’ investment attitude $ x^k $ occur with probability $ P(x^k) = \mbox{Prob}(x^k) $ with the requirement $ 0 < P(x^k) < 1 $ and $ \sum^K_{k=1} P(x^k) = 1 $. We define the amount of uncertainty before the occurrence of the state $ x^k $ with probability $ P(x^k) $ as the logarithmic function: $ I(x^k) = - \log P(x^k) $. Under these assumptions the above optimisation problem is formalised by $$\min \langle E(x) \rangle = \sum^N_{k=1} P(x^k) E(x^k) \label{eqn:a3}$$ subject to $ H = - \sum^N_{k=1} P(x^k) \log P(x^k), \quad \sum^N_{k=-N} P(x^k) = 1 $, where $ E(x^k) = \frac{1}{2} \sum^N_{i=1} E_i(x^k) $. $ x^k $ is a state, and $ H $ is information entropy. $ P(x^k) $ is the relative frequency the occurrence of the state $ x^k $. The well-known solutions of the above optimisation problem is $$P(x^k) = \frac{1}{Z} \exp(- \mu E(x^k)), \quad Z = \sum^K_{k=1} \exp(- \mu E(x^k)) \quad k = 1, 2,....., K. \label{eqn:a4}$$ where the parameter $ \mu $ may be interested as a [*market temperature*]{} describing a degree of randomness in the behaviour of traders . The probability distribution $ P(x^k) $ is called the [*Boltzmann distribution*]{} where $ P(x^k) $ is the probability that the traders’ investment attitude is in the state $ k $ with the function $ E(x^k) $, and $ Z $ is the partition function. We call the optimising behaviour of the traders with interaction among the other traders a [*relative expectation formation*]{}. The volume of investment ------------------------ The trading volume should depends upon the investment attitudes of all traders. Since traders are supposed to either buy or sell a fixed amount of stock (q) in a period, the aggregate excess demand for stock at a period is given by $ q \sum^N_{i=1} x_i $. The price adjustment processes ------------------------------ We assume the existence of a market-maker whose function is to adjust the price. If the excess demand $ q x $ is positive (negative), the market maker raises (reduces) the stock price for the next period. Precisely, the new price is calculated as the previous price $ y_t $ plus some fraction of the excess demand of the previous period according: $ \Delta y_{t+1} = \lambda q \sum^N_{i=1} x_{it} $, where $ x_{it} $ denotes the investment attitude of trader $ i $ at period $ t $, $ \Delta y_{t+1} $ the price change from the current period to the next period, i.e. $ \Delta y_{t+1} = y_{t+1} - y_{t} $, and the parameter $ \lambda $ represents the speed of adjustment of the market price. At the equilibrium prices that clear the market there should exist an equal number of buyers or sellers, i.e.: $ \sum^N_{i=1} x_{it} = 0 $. Using the Boltzmann distribution (\[eqn:a4\]) the mean value of the price changes $ \Delta \bar{y} $ is given by $ \Delta \bar{y} = \sum^N_{k=1} P(x^k) q \sum^N_{i=1} x^k_i $. Mean-Field Approximation : Keynes’ Beauty Contest ------------------------------------------------- We can derive analytically the stationary states of the traders’ investment attitude using a mean field approximation which is a well known technique in statistical physics. Let us replace the discrete summation of the investment attitude with the mean field variable, $ \langle x \rangle = \langle (1/N) \sum^N_{i=1} x_i \rangle $. We additionally assume that the parameter $ a_{ij} $ is equal to $ a $ $ a_{ij} = a $ and $ b_i = b $ for every trader. In this case our interacting-agent model reduces to the Ising model with long-range interactions. Then the function (\[eqn:a2\]) is approximated by $ E(x) \approx - (1/2) \sum^N_{i=1} \sum^N_{j=1} a N \langle x \rangle x_i - \sum^N_{i=1} b s x_i $. Since the mean value of the stationary distribution is given by $ \langle E(x) \rangle = - \partial \log Z /\partial \mu $, the mean field $ \langle x \rangle $ is $$\langle x \rangle = \tanh(\mu a N \langle x \rangle + \mu b s). \label{eqn:a5}$$ The first term of the right side of Eq. (\[eqn:a5\]) represents that traders tend to adopt the same investment attitude as prediction of the average investment attitude and the second term represents the influence of the change of investment environment to traders’ investment attitude. The first term may be interpreted as a mathematical formularisation of [*Keynes’ beauty contest*]{}. Keynes [@Keynes] argued that stock prices are not only determined by the firm’s fundamental value, but in addition mass psychology and investors expectations influence financial markets significantly. It was his opinion that professional investors prefer to devote their energies not to estimating fundamental values, but rather to analysing how the crowd of investors is likely to behave in the future. As a result, he said, most persons are [*largely concerned, not with making superior long-term forecasts of the probable yield of an investment over its whole life, but with foreseeing changes in the conventional basis of valuation a short time ahead of the general public*]{}. Keynes used his famous [*beauty contest*]{} as a parable to stock markets. In order to predict the winner of a beauty contest, objective beauty is not much important, but knowledge or prediction of others’ predictions of beauty is much more relevant. In Keynes view, the optimal strategy is not to pick those faces the player thinks prettiest, but those the other players are likely to be about what the average opinion will be, or to proceed even further along this sequence. Speculative Price Dynamics -------------------------- We assume that the traders’ investment attitude changes simultaneously (synchronous dynamics) in discrete time steps. We represent the dynamics of the investment attitude using a straightforward iteration of Equation (\[eqn:a5\]), such that: $$\langle x \rangle_{t+1} = \tanh(\alpha \langle x \rangle_t + \beta s_t), \quad \alpha = \mu a N, \beta = \mu b. \label{eqn:a6}$$ We call $ \alpha $ the bandwagon coefficient and $ \beta $ the investment environment coefficient below because $ \alpha $ may be interpreted as a parameter that denotes the strength that traders chase the price trend. Next, let us approximate the adjustment process of the price using the mean field $ \langle x \rangle_t $. We can get the following speculative price dynamics, $$\langle \Delta y \rangle_{t+1} = \lambda N \langle x \rangle_t. \label{eqn:a7}$$ Inserting the mean field of the price change, $ \langle \Delta y \rangle_{t} $ into eq. \[eqn:a6\], the dynamics of the mean field of $ \langle x \rangle_t $ can be rewritten as $ \langle x \rangle_{t+1} = \tanh(\frac{\mu a}{\lambda} \langle \Delta y \rangle_{t+1} + \mu b s_t) $. This equation shows that the traders result in basing their trading decisions on an analysis of the price trend $ \langle \Delta y \rangle_{t+1} $ as well as the change of the investment environment $ s_t $. Speculative Bubbles and Crashes =============================== The stationary states of the mean field $ \langle x \rangle $ satisfy the following $$\langle x \rangle_t = \tanh (\alpha \langle x \rangle_t + \beta s) \equiv f(\langle x \rangle_t , s). \label{eqn:a8}$$ Eq. \[eqn:a8\] has a unique solution $ \langle x \rangle $ for arbitrary $ \beta s $, when $ \alpha $ is less than $ 1 $. The stationary state $ \langle x \rangle^* $ with $ \beta s = 0 $ is called as the [*fundamental equilibrium*]{} which is stable and corresponds to the maximum of the stationary distribution of the investment attitude $ P(x^*) $. In this stable equilibrium there is equal numbers of traders sharing both investment attitudes on average. Moving in the parameter space $ (\alpha , \beta) $ and starting from different configurations one have several possible scenarios of market-phase transitions. To begin with, let us consider the case that the investment environment $ s $ is equal to zero. In theory, when the bandwagon coefficient $ \alpha $ is increased starting from the phase at $ 0 < \alpha < 1 $, there is a symmetry breaking at critical point, $ \alpha = 1 $, and two different market phases appear, that is, the [*bear market*]{} and the [*bull market*]{}. In analogy to physical systems the transitions may be called the second-order phase transition. Fig. 1 illustrates the two graphical solutions of the Eq. (\[eqn:a8\]) for $ \alpha > 1 $ and $ s = 0 $ and for $ 0 < \alpha < 1 $ and $ s = 0 $. The figure shows that for $ 0 < \alpha < 1 $ and $ s = 0 $ the fundamental equilibrium is unique and stable, but for $ \alpha > 1 $ and $ s = 0 $ the fundamental equilibrium $ 0 $ becomes unstable, and the two new equilibria, the bull market equilibrium $ a $ and the bear market equilibrium $ b $ are stable. At the bull market equilibrium more than half number of traders is buyer, so that the speculative bubbles occur in the stock market. Then, let us consider the effect of changes of the investment environment $ s $ in the case with a weak bandwagon coefficient, $ 0 < \alpha < 1 $ and a positive investment environment coefficient, $ \beta > 0 $. Fig. 2 shows that as the investment environment $ s $ changes for the better ($ s > 0 $), the system shifts to the bull market phase. By contrast, as the investment environment $ s $ changes for the worse ($ s < 0 $), the system shifts to the bear market phase. Therefore when the bandwagon effect is weak, the stock price go up or down slowly according rises or falls in the investment environment. Finally, let us consider the effect of changes of $ \beta s $ in the case with a strong bandwagon coefficient $ \alpha > 1 $. In this case multistability and hysteresis phenomena in the distribution of investment attitude, as well as market-phase transitions are observed. The system has three equilibria, when $ \alpha > 1 $ and $ |s| < s^* $, where $ s^* $ is determined by the equation, $$\cosh^2 [\beta s \pm \sqrt{\alpha (\alpha - 1)}] = \alpha. \label{eqn:a9}$$ Two maximum of the stationary distribution on the investment attitude are found at $ \langle x \rangle^- $ and $ \langle x \rangle^+ $ and one minimum at $ \langle x \rangle^* $. For $ |s| = s^* $ and $ \alpha > 1 $, two of the three equilibria coincide at $$\langle x \rangle_c = \sqrt{(\alpha - 1)/\alpha}. \label{eqn:a10}$$ For $ |s| > s^* $ and $ \alpha > 1 $ the two of three equilibria vanishes and only one equilibrium remains. In analogy to physical systems the transitions may be called first-order phase transitions. Fig. 3 illustrates the effect of changes in the investment environment in the case with a strong bandwagon effect $ \alpha > 1 $. In this case speculative bubbles and market crashes occur. When the investment environment changes for the better (worse), the curve that denotes $ f(\langle x \rangle_t , s) $, shifts upward (downward). As the investment environment $ s $ keeps on rising, and when it reach a critical value, that is, $ |s| = s^* $, the bear market equilibrium and the fundamental equilibrium vanish, and the bull market equilibrium becomes an unique equilibrium of the system, so that the speculative bubble occurs in the stock market. Even though the investment environment changes for the worse, the hysteresis phenomena are observed in a range of the investment environment that can be calculated by solving Eq. (\[eqn:a9\]). In other words the speculative bubble continues in a range of the investment environment. However when the negative impact of the investment environment reaches a critical value, that is, $ |s| = s^* $, the bull market vanish. Further decrease of $ s $ cause the market-phase transition from a bull market to a bear market. In a bear market more than half number of trader is seller on average, so that the stock price continues to fall on average. Thus this market phase transition may be considered as the stock market crash. Empirical Analysis : Japan Crisis ================================= Perhaps the most spectacular boom and bust of the late twentieth century involved the Japan’s stock markets in 1987-1992. Stock prices increased from 1982 to 1989. At their peak in December 1989, Japanese stocks had a total market value of about 4 trillion, almost 1.5 times the value of all U.S. equities and close to 45 percent of the world’s equity market capitalisation. The fall was almost as extreme as the U.S. stock-market crash from the end of 1929 to mid-1932. The Japanese (Nikkei) stock-market index reached a high of almost 40,000 yen in the end of 1989. By mid-August 1992, the index had declined to 14,309 yen, a drop of about 63 percent. In this section we will estimate the parameter vector $ (\alpha , \beta) $ for the Japan’s stock market in 1987-1992. We use monthly adjusted data for the first difference of Tokyo Stock Price Index (TOPIX) and the investment environment over the period 1988 to 1992. In order to get the mean value of the stock price changes, TOPIX is adjusted through the use of a simple centred 12 point moving average, and are normalised into the range of $ -1 $ to $ +1 $ by dividing by the maximum value of the absolute value of the price changes. The normalised price change is defined as $ \langle \Delta p \rangle_t $. Fig. 4 and Fig. 5 show the normalised stock price changes and the investment environment respectively. This normalised stock price change $ \langle \Delta p \rangle_t $ is used as a proxy of the mean field $ \langle x \rangle_t $. Inserting $ \langle \Delta p \rangle_t $ for Eq. (\[eqn:a8\]), we get $ f(\langle \Delta p \rangle_t , s_t) = \tanh(\alpha \langle \Delta p \rangle_t + \beta s_t) $. Model estimation: The gradient-descent algorithm ------------------------------------------------ The bandwagon coefficient $ \alpha $ and the investment environment coefficient $ \beta $ should be estimated using any one of estimation technique. Since we cannot get the analytical solution because the function $ f(\langle \Delta p \rangle_t , s_t) $ is non-linear, we use the [*gradient-descent algorithm*]{} for the parameter estimation [^5]. The gradient-descent algorithm is a stable and robust procedure for minimising the following one-step-prediction error function $$E(\alpha_k, \beta_k) = \frac{1}{2} \sum^n_{t=1} [\langle \Delta p \rangle_{t} - f(\langle \Delta p \rangle_{t-1}, s_{t-1})]^2. \label{eqn:a33}$$ More specifically, the gradient-descent algorithm changes the parameter vector $ (\alpha_k , \beta_k) $ to satisfy the following condition: $$\Delta E(\alpha_k , \beta_k) = \frac{\partial E(\alpha_k, \beta_k)} {\partial \alpha_k} \Delta \alpha_k + \frac{\partial E(\alpha_k, \beta_k)} {\partial \beta_k} \Delta \beta_k < 0, \label{eqn:a34}$$ where $ \Delta E(\alpha_k , \beta_k) = E(\alpha_{k} , \beta_{k}) - E(\alpha_{k-1} , \beta_{k-1}) $, $ \Delta \alpha_k = \alpha_{k} - \alpha_{k-1} $, and $ \Delta \beta_k = \beta_{k} - \beta_{k-1} $. To accomplish this the gradient-descent algorithm adjusts each parameter $ \alpha_k $ and $ \beta_k $ by amounts $ \Delta \alpha_{k} $ and $ \Delta \beta_{k} $ proportional to the negative of the gradient of $ E(\alpha_k , \beta_k) $ at the current location: $$\alpha_{k+1} = \alpha_k - \eta \frac{\partial E(\alpha_k, \beta_k)} {\partial \alpha_k}, \quad \beta_{k+1} = \beta_k - \eta \frac{\partial E(\alpha_k, \beta_k)} {\partial \beta_k} \label{eqn:a35}$$ where $ \eta $ is a learning rate. The gradient-descent rule necessarily decreases the error with a small value of $ \eta $. If the error function (\[eqn:a33\]), thus, has a single minimum at $ E(\alpha_k , \beta_k) = 0 $, then the parameters $ (\alpha_k , \beta_k) $ approaches the optimal values with enough iterations. The error function $ f(\alpha_k , \beta_k) $, however, is non-linear, and hence it is possible that the error function (\[eqn:a33\]) may have [*local minima*]{} besides the global minimum at $ E(\alpha_k , \beta_k) = 0 $. In this case the gradient-descent rule may become stuck at such a local minimum. To check convergence of the gradient-descent method to the global minimum we estimate the parameters for a variety of alternative start up parameters $ (\alpha_0 , \beta_0) $ with $ \eta = 0.01 $. As a consequence the gradient-descent rule estimated the same values of the parameters, $ (\alpha^* , \beta^*) = (1.04 , 0.5) $ with enough iterations. Results ------- The fit of the estimated equation can be seen graphically in Fig. (6), which compare the actual and forecasted series over one period, respectively. The correlation coefficient that indicates goodness of fit is $ 0.994 $. The capacity of the model to forecast adequately in comparison with competing models is an important element in the evaluation of its overall performance. Let us consider the linear regression model $ \langle \Delta{y} \rangle_{t+1} = c \langle \Delta{y} \rangle_t + d s_t $. The regression results are $ \langle \Delta{p} \rangle_{t+1} = 0.855 \langle \Delta{p} \rangle_t + 3.81 s_t $. The correlation coefficient of the linear regression model is equal to 0.93, and is lower that that of the interacting-agent model. Root mean squared errors (RMSEs) of the interacting-agent model is equal to 0.05, and that of the linear regression model is equal to 0.148 over the period 1987-1992. The interacting-agent model has a smaller RMSE than that of the linear regression model. By the traditional RMSE criterion the interacting-agent model is, thus, superior to the linear regression model in term of their forecasting performance. In conclusion we can say that the model not only explains the Japan crisis but also correctly forecasts changing the stock price over the period 1987-1992. Given that the set of parameters ($ \alpha , \beta $) is equal to (1.04, 0.5), the critical point of the investment environment at which the phase transition are caused from the bull market to bear market are calculated from the theoretical results in the preceding section. One can say that the phase transition from the bull market to the bear market occurs when the investment environment is below $ -0.007 $ under $ (\alpha , \beta) = (1.04, 0.5) $. By contrast the phase transition from the bear market to the bull market occurs when the investment environment is beyond $ 0.007 $ under $ (\alpha , \beta) = (1.04, 0.5) $. Period $ \langle \Delta p \rangle $ Predicted value $ s_t $ --------- ------------------------------ ----------------- --------- 1989.09 0.4591 0.4487 0.0182 1989.10 0.4468 0.4361 0.0122 1989.11 0.4531 0.4408 0.011 1989.12 0.4369 0.4258 0.0075 1990.01 0.2272 0.2294 -0.002 1990.02 0.0985 0.0986 -0.0055 1990.03 -0.2013 -0.2097 -0.0099 1990.04 -0.2354 -0.2415 -0.0066 1990.05 -0.0845 -0.0871 -0.0004 1990.06 -0.0883 -0.0924 -0.0029 1990.07 -0.3136 -0.3159 -0.0067 1990.08 -0.5245 -0.4992 -0.0136 : The stock market crash in the Japanese stock market[]{data-label="tab1"} From theoretical view point one may say that bursting speculative bubbles will begin when the investment environment is below $ -0.007 $, and the process of the collapse of the stock market continues till the investment environment is beyond $ 0.007 $. In the real world the investment environment became below $ - 0.007 $ at March 1990 for the first time and continued to be negative values since then over the period 1987-1992. The fall of the actual price in the Japan’s stock market began at March 1990, and the stock prices continued to fall over the period 1990-1992. On these grounds we have come to the conclusion that theory and practice are in a perfect harmony. (See Table \[tab1\].) Conclusion ========== This paper presents an interacting-agent model of speculative activity explaining the bubbles and crashes in stock markets in terms of a mean field approximation. We show theoretically that these phenomena are easy to understand as the market- phase transitions. Bubbles and crashes are corresponding to the second-order transitions and the first-order transitions respectively. Furthermore we estimate the parameters of the model for the Japanese stock market in 1987-1992. The empirical results demonstrate that theory and practice are in a perfect harmony. This fact justifies our model that bubbles and crashes have their origin in the collective crowd behaviour of many interacting agents. Acknowledgements ================ I would like to thank Masao Mori and Thomas Lux for helpful comments and suggestions to an earlier version of this paper. [99]{} J. M. Keynes, [*The general theory of Employment, Interest and Money* ]{}. Harcourt, 1934. C. Mackay, [*Extraordinary Popular Delusions and the Madness of Crowds* ]{} The Noonday Press, 1932. C. P. Kindlberger, [*Manias, Panics, and Crashes: A History of Financial Crises* ]{}, MacMillan, London, 1989. J. K. Galbraith, [*A Short History of Financial Euphoria* ]{}. Whittle Direct Books, Tennessee, U.S.A., 1990. T. Lux,[*The Economic Journal* ]{}, [**105**]{} (1995) 881-896. T. Kaizoji, [*Simulation* ]{} (in Japanese), 12, 2 (1993) 67-74. D. Sornette, A. Johansen, J.P. Bouchaud, [*J. Phys. I. France* ]{} [**6**]{} (1996) 167-175. D. Sornette, A. Johansen, [*Physica A* ]{} [**245**]{} (1997) 411. D. Sornette and A. Johansen, [*Physica A* ]{} [**261**]{} (Nos. 3-4) (1998) A. Johansen, D. Sornette, [*Eur.Phys.J. B* ]{} [**1**]{} (1998) 141-143. A. Johansen, D. Sornette, [*Int. J. Mod. Phys. C* ]{} [**10**]{} (1999) 563-575. A. Johansen, D. Sornette, [*Int. J. Mod. Phys. C* ]{} [**11**]{} (2000) 359-364. J. A. Feigenbaum, P.G.O. Freund, [*Int. J. Mod. Phys. B* ]{} [**10**]{} (1996) 3737-3745. J. A. Feigenbaum, P.G.O. Freund, [*Int. J. Mod. Phys. B* ]{} [**12**]{} (1998) 57-60. N. Vandewalle, Ph. Boveroux, A. Minguet, M. Ausloos, [*Physica A*]{} [**225**]{} (1998) 201-210. N. Vandewalle, M. Ausloos, Ph. Boveroux, A. Minguet, [*Eur. Phys. J. B*]{} [**4**]{} (1998) 139-141. S. Gluzman, and V. I. Yukalov, [*Mod. Phys. Lett. B* ]{} [**12**]{} (1998) 75-84. L. Laloux, M. Potters, R. Cont, J.-P. Aguilar and J.-P. Bouchaud, [*Europhysics Letters* ]{}, [**45**]{}, 1 (1999) 1-5. D. Chowhury, D. Stauffer [*Eur. Phys. J. B* ]{}, [**8**]{}, (1999) 477. J. A. Holyst, K. Kacperski, F. Schweitzer, submetted in [*Physica A* ]{}, (2000). T. Lux, and M. Marchesi, [*Nature* ]{}, [**297**]{} (1999) 498-500. E. Fama, [*Journal of Finance* ]{} [**46**]{} (1970) 1575-1618. D. E. Rumelhart, G. E. Hinton, R. J. Williams, in: [*Parallel Distributed Processing* ]{} [**1**]{}, chap. 8, D. E. Rumelhart, J. L. McClelland, et al., eds. Cambridge, MA: MIT Press, 1986. Figure Captions =============== Figure 1. The second-order phase transition. Numerical solution of Eq.(\[eqn:a8\]) for $ s = 0 $. The curve is the RHS of \[eqn:a8\] plotted for different values of $ \alpha $: the AA line - $ \alpha = 0.9 $, the BB line - $ \alpha = 2 $.\ Figure 2. Numerical solution of Eq.(\[eqn:a8\]) for $ \alpha = 0.8 $ and $ \beta = 1 $. $ f(\langle x \rangle_t , s) $ is the RHS of (8) plotted for different values of $ \alpha $: the AA line - $ s = 0.3 $, the BB line - $ s = -0.3 $.\ Figure 3. The first-order transition: Speculative bubble and crash. Numerical solution of Eq. (\[eqn:a8\]) for $ \alpha = 1.8 $ and $ \beta = 1 $. The curves is $ f(\langle x \rangle_t , s) $ plotted for different values of the investment environment: the AA line - $ s = 0.41 $, the BB line - $ s = -0.41 $.\ Figure 4. The adjusted stock price change, $ \langle \Delta p \rangle $: January 1987 - December 1992.\ Figure 5. The Japan’s investment environment $ s_t $: January 1987 - December 1992.\ Figure 6. One-step-forecast of the stock price changes: January 1987 - December 1992. [^1]: Olshausenstr. 40, 24118 Kiel, Germany. On leave from Division of Social Sciences, International Christian University, Osawa, Mitaka, Tokyo, 181-8585, Japan. E-mail: kaiyoji@icu.ac.jp [^2]: For a similar study see also Kaizoji [@Kaizoji]. [^3]: See also a critical review on this literature [@Laloux]. [^4]: A similar idea has been developed in the Cont-Bouchaud model with an Ising modification [@Chowdhury] from another point of view. For a related study see also [@Holyst]. They study phase transitions in the social Ising models of opinion formation. [^5]: The gradient-descent method is often used for training multilayer feedforward networks. It is called the back- propagation learning algorithm which is one of the most important historical developments in neural networks \[Rumelhart et al. [@Rumelhart].

--- abstract: 'We study entanglement distillability of bipartite mixed spin states under Wigner rotations induced by Lorentz transformations. We define weak and strong criteria for relativistic [*isoentangled*]{} and [*isodistillable*]{} states to characterize relative and invariant behavior of entanglement and distillability. We exemplify these criteria in the context of Werner states, where fully analytical methods can be achieved and all relevant cases presented.' author: - 'L. Lamata' - 'M. A. Martin-Delgado' - 'E. Solano' title: Relativity and Lorentz Invariance of Entanglement Distillability --- \#1[[**[\#1]{}**]{}]{} \#1 \#1[\#1|]{} \#1[|\#1]{} \#1\#2[\#1|\#2]{} \#1\#2[\#1,\#2]{} \#1[\#1]{} §[[S]{}]{} ß Entanglement is a quantum property that played a fundamental role in the debate on completeness of quantum mechanics. Nowadays, entanglement is considered a basic resource in present and future applications of quantum information, communication, and technology [@NielsenChuang; @rmp]. However, entangled states are fragile, and interactions with the environment destroy their coherence, thus degrading this precious resource. Fortunately, entanglement can still be recovered from a certain class of states which share the property of being distillable. This means that even in a decoherence scenario, entanglement can be extracted through purification processes that restore their quantum correlations [@Bennett1; @generaldistill]. An entangled state can be defined as a quantum state that is not separable, and a separable state can always be expressed as a convex sum of product density operators [@Werner]. In particular, a bipartite separable state can be written as $\rho=\sum_iC_i\rho^{(\rm a)}_i\otimes\rho^{(\rm b)}_i$, where $C_i\geq0$, $\sum_i C_i=1$, and $\rho^{(\rm a)}_i$ and $\rho^{(\rm b)}_i$ are density operators associated to subsystems $\rm A$ and $\rm B$. In quantum field theory, special relativity (SR) [@Rindler; @VerchWerner] and quantum mechanics are described in a unified manner. From a fundamental point of view, in addition, it is relevant to study the implications of SR on the modern quantum information theory (QIT) [@SRQIT]. Recently, Peres [*et al.*]{} [@SRQIT1] have observed that the reduced spin density matrix of a single spin $1/2$ particle is not a relativistic invariant, given that Wigner rotations [@Wigner] entangle the spin with the particle momentum distribution when observed in a moving referential. This astonishing result, intrinsic and unavoidable, shows that entanglement theory must be reconsidered from a relativistic point of view [@BartlettTerno]. On the other hand, the fundamental implications of relativity on quantum mechanics could be stronger than what is commonly believed. For example, Wigner rotations induce also decoherence on two entangled spins [@PachosSolano; @AlsingMilburn; @GingrichAdami]. However, they have not been studied yet in the context of mixed states and distillable entanglement [@Peres; @Horodeckis1]. A typical situation in SR pertains to a couple of observers: one is stationary in an inertial frame $\cal{S}$ and the other is also stationary in an inertial frame $\cal{S}'$ that moves with velocity $\vec{v}$ with respect to $\cal{S}$. The problems addressed in SR consider the relation between different measurements of physical properties, like velocities, time intervals, and space intervals, of objects as seen by observers in $\cal{S}$ and $\cal{S}'$. However, in QIT, it is assumed that the measurements always take place in a proper reference frame, either $\cal{S}$ or $\cal{S}'$. To see the effects of SR on QIT [@SRQIT], we need to enlarge the typical situations where quantum descriptions and measurements take place. In order to analyze the new possibilities that SR offers, we introduce the following concepts *i) Weak isoentangled state* $\rho^{\rm WIE}$: A state that is entangled in all considered reference frames. This property is independent of the chosen entanglement measure ${\cal E}$. *ii) Strong isoentangled state* $\rho^{\rm SIE}_{\cal E}$: A state that is entangled in all considered reference frames, while having a constant value associated with a given entanglement measure ${\cal E}$. This concept depends on the ${\cal E}$ chosen. *iii) Weak isodistillable state* $\rho^{\rm WID}$: A state that is distillable in all considered reference frames. This implies that the state is entangled for these observers. *iv) Strong isodistillable state* $\rho^{\rm SID}_{\cal E}$: A state that is distillable in all considered reference frames, while having a constant value associated with a given entanglement measure ${\cal E}$. This concept depends on the ${\cal E}$ chosen. In general, the following hierarchy of sets holds (see Fig. \[grafreldist2\] for a pictorial representation) $$\{\rho^{\rm WIE}\}\supset\{\rho^{\rm SIE}_{\cal E}\}\supset\{\rho^{\rm SID}_{\cal E}\}\subset\{\rho^{\rm WID}\}\subset\{\rho^{\rm WIE}\} .$$ ![(Color online) Hierarchy for the sets of states WIE, SIE, WID, and SID.\[grafreldist2\]](Fig1.eps){height="4"} To illustrate the relative character of distillability, let us consider the specific situation in which Alice (A) and Bob (B) share a bipartite mixed state of Werner type with respect to an inertial frame $\cal{S}$. Moreover, in order to complete the SR+QIT scenario, we also consider another inertial frame $\cal{S}'$, where relatives A’ and B’ of A and B are moving with relative velocity $\vec{v}$ with respect to $\cal{S}$. Using the picture of Einstein’s trains, we may think that A and B are at the station platform sharing a set of mixed states, while their relatives A’ and B’ are travelling in a train sharing another couple of entangled particles of the same characteristics. The mixed state is made up of two particles, say electrons with mass $m$, having two types of degrees of freedom: momentum $\vec{p}$ and spin $s=\half$. The former is a continuous variable while the latter is a discrete one. By [*definition*]{}, we consider our logical or computational qubit to be the spin degree of freedom. Each particle is assumed to be localized, as in a box, and its momentum $\vec{p}$ will be described by the same Gaussian distribution. We assume that the spin degrees of freedom of particles $A$ and $B$ are decoupled from their respective momentum distributions and form the state $$\begin{aligned} \rho^{AB}_{\cal{S}}:= F | \Psi^{-}_{\vec{q}} \rangle \langle \Psi^{-}_{\vec{q}} | && \!\!\!\!\!\! + \frac{1-F}{3} \bigg( | \Psi^{+}_{\vec{q}} \rangle \langle \Psi^{+}_{\vec{q}} | \nonumber \\ && + | \Phi^{-}_{\vec{q}} \rangle \langle \Phi^{-}_{\vec{q}} | + | \Phi^{+}_{\vec{q}} \rangle \langle \Phi^{+}_{\vec{q}} | \bigg) . \label{Wernerrela1}\end{aligned}$$ Here, $F$ is a parameter such that $0 \leq F \leq 1$, $$\begin{aligned} \!\!\!\!\!\!\!\! | \Psi^{\pm}_{\vec{q}} \rangle \!\! & := & \!\! \frac{1}{\sqrt{2}}\lbrack \Psi_1^{(\rm a)} (\vec{q_a}) \Psi_2^{(\rm b)} (\vec{q_b}) \pm\Psi_2^{(\rm a)} (\vec{q_a}) \Psi_1^{(\rm b)} (\vec{q_b})\rbrack , \nonumber \\ \!\!\!\!\!\! | \Phi^{\pm}_{\vec{q}} \rangle \!\!\! & := & \!\!\! \frac{1}{\sqrt{2}} \lbrack \Psi_1^{(\rm a)} (\vec{q_a}) \Psi_1^{(\rm b)} (\vec{q_b}) \pm\Psi_2^{(\rm a)} (\vec{q_a}) \Psi_2^{(\rm b)} (\vec{q_b})\rbrack , \label{Bellrela}\end{aligned}$$ where $\vec{q_a}$ and $\vec{q_b}$ are the corresponding momentum vectors of particles $A$ and $B$, as seen in $\cal{S}$, and $$\begin{aligned} \Psi_1^{(\rm a)} (\vec{q_a}) &:=& {\cal G} ( \vec{q_a} ) |\!\!\uparrow \rangle = \left( \begin{array}{cccc} {\cal G} ( \vec{q_a} ) \\ 0 \\ \end{array} \right) \nonumber \\ \Psi_2^{(\rm a)} (\vec{q_a}) &:=& {\cal G} ( \vec{q_a} ) |\!\!\downarrow \rangle = \left( \begin{array}{cccc} 0 \\ {\cal G} ( \vec{q_a} ) \\ \end{array} \right) \nonumber \\ \Psi_1^{(\rm b)} (\vec{q_b}) &:=& {\cal G} ( \vec{q_b} ) |\!\!\uparrow \rangle = \left( \begin{array}{cccc} {\cal G} ( \vec{q_b} ) \\ 0 \\ \end{array} \right) \nonumber \\ \Psi_2^{(\rm b)} (\vec{q_b}) &:=& {\cal G} ( \vec{q_b} ) |\!\!\downarrow \rangle = \left( \begin{array}{cccc} 0 \\ {\cal G} ( \vec{q_b} ) \\ \end{array} \right) , \label{Skets}\end{aligned}$$ with Gaussian momentum distributions ${\cal G} ( \vec{q} ):= \pi^{-3/4}w^{-3/2} \exp ( - {\rm q}^2 /2 w^2 )$, being ${ \rm q} := |\vec{q}|$. $|\!\!\uparrow \rangle$ and $|\!\!\downarrow \rangle$ represent spin vectors pointing up and down along the $z$-axis, respectively. If we trace momentum degrees of freedom in Eq. (\[Bellrela\]), we obtain the usual spin Bell states, $\{ | \Psi^{-} \rangle , | \Psi^{+} \rangle, | \Phi^{-} \rangle, | \Phi^{+} \rangle \}$. If we do the same in Eq. (\[Wernerrela1\]), we remain with the usual spin Werner state [@Werner] $$\begin{aligned} \left( \begin{array}{cccc} \frac{1-F}{3} & 0 & 0 & 0 \\ 0 & \frac{2F+1}{6} & \frac{1-4F}{6} & 0 \\ 0 & \frac{1-4F}{6} & \frac{2F+1}{6} & 0 \\ 0 & 0 & 0 & \frac{1-F}{3} \\ \end{array} \right) , \label{Werner1}\end{aligned}$$ written in matrix form, out of which Bell state $| \Psi^{-} \rangle$ can be distilled if, and only if, $F > 1 / 2$. We consider also another pair of similar particles, $A'$ and $B'$, with the same state as $A$ and $B$, $\rho^{A' B'}_{\cal{S}'} = \rho^{A B}_{\cal{S}}$, but seen in another reference frame $\cal{S}'$. The frame $\cal{S}'$ moves with velocity $\vec{v}$ along the $x$-axis with respect to the frame $\cal{S}$. When we want to describe the state of $A'$ and $B'$ as observed from frame $\cal{S}$, rotations on the spin variables, conditioned to the value of the momentum of each particle, have to be introduced. These conditional spin rotations, considered first by Wigner [@Wigner], are a natural consequence of Lorentz transformations. In general, Wigner rotations entangle spin and momentum degrees of freedom for each particle. We want to encode quantum information in the two qubits determined by the spin degrees of freedom of our two spin-$1/2$ systems. However, the reduced two-spin state, after a Lorentz transformation, increases its entropy and reduces its initial degree of entanglement. If we consider the velocities of the particles as having only non-zero components in the $z$-axis, each state vector of $A'$ and $B'$ in Eq. (\[Skets\]) transforms as $$\begin{aligned} &&\Psi_1(\vec{q})= \left( \begin{array}{cccc} {\cal G} ( \vec{q} ) \\ 0 \\ \end{array} \right) \rightarrow \Lambda[\Psi_1(\vec{q})]=\left( \begin{array}{cccc} \cos \theta_{\vec{q}} \\ \sin \theta_{\vec{q}} \\ \end{array} \right) {\cal G} ( \vec{q} ) \nonumber \\ && \Psi_2(\vec{q})=\left( \begin{array}{cccc} 0 \\ {\cal G} ( \vec{q} ) \\ \end{array} \right) \rightarrow \Lambda[\Psi_2(\vec{q})]=\left( \begin{array}{cccc} -\sin \theta_{\vec{q}} \\ \,\,\,\,\, \cos \theta_{\vec{q}} \\ \end{array} \right) {\cal G} ( \vec{q} ) ,\nonumber\\ \label{transfrules}\end{aligned}$$ where $\cos \theta_{\vec{q}}$ and $\sin \theta_{\vec{q}}$ express Wigner rotations conditioned to the value of the momentum vector. The most general bipartite density matrix in the rest frame for arbitrary spin-1/2 states and Gaussian product states in momentum, is spanned by the tensor products of $\Psi^{(\rm a)}_1$, $\Psi^{(\rm a)}_2$, $\Psi^{(\rm b)}_1$, and $\Psi^{(\rm b)}_2$, and can be expressed as $$\rho=\sum_{ijkl=1,2}C_{ijkl}\Psi^{(\rm a)}_i( \vec{q_a} )\otimes\Psi^{(\rm b)}_j( \vec{q_b} ) [\Psi^{(\rm a)}_k( \vec{q_a'} )\otimes\Psi^{(\rm b)}_l( \vec{q_b'} )]^{\dag}.\label{densitygeneral}$$ Under a boost, Eq. (\[densitygeneral\]) will transform into $$\begin{aligned} \Lambda\rho\Lambda^{\dag}\!\!\! & = &\!\!\!\!\! \sum_{ijkl=1,2}C_{ijkl}\Lambda^{(\rm a)}[\Psi^{(\rm a)}_i( \vec{q_a} )]\otimes \Lambda^{(\rm b)}[\Psi^{(\rm b)}_j( \vec{q_b} )]\nonumber\\ &&\times\{\Lambda^{(\rm a)}[\Psi^{(\rm a)}_k( \vec{q_a'} )]\otimes\Lambda^{(\rm b)}[\Psi^{(\rm b)}_l( \vec{q_b'} )]\}^{\dag}.\label{densityboost}\end{aligned}$$ Tracing out the momentum degrees of freedom, we obtain $$\begin{aligned} & \mathrm{Tr}_{\rm \vec{q_a}, \vec{q_b}} & (\Lambda\rho\Lambda^{\dag})\nonumber \\ & = & \!\!\!\!\!\!\!\! \sum_{ijkl=1,2}C_{ijkl}\mathrm{Tr}_{\rm \vec{q_a}}(\Lambda^{(\rm a)}[\Psi^{(\rm a)}_i( \vec{q_a} )]\{\Lambda^{(\rm a)}[\Psi^{(\rm a)}_k( \vec{q_a})]\}^{\dag}) \nonumber \\ && \otimes \mathrm{Tr}_{\rm \vec{q_b}}(\Lambda^{(\rm b)}[\Psi^{(\rm b)}_j( \vec{q_b} )]\{\Lambda^{(\rm b)}[\Psi^{(\rm b)}_l( \vec{q_b})]\}^{\dag}).\label{densityboosttr}\end{aligned}$$ Following Peres [*et al.*]{} [@SRQIT1], we compute the Lorentz transformed density matrix of state $\Psi_1$, after tracing out the momentum. The expression, to first order in $w/m$, reads $$\mathrm{Tr}_{\rm \vec{q}}[\Lambda\Psi_1(\Lambda\Psi_1)^{\dag}]=\frac{1}{2} \left(\begin{array}{cc}1+n_z' & 0\\0 & 1-n_z'\end{array}\right),\label{peresmatriz1}$$ where $n'_z:=1-\left(\frac{w}{2m}\tanh \frac{\alpha}{2}\right)^2$ and $\cosh\alpha:=\gamma=(1-\beta^2)^{-1/2}$. Larger values of $w / m$ are possible and mathematically correct [@GingrichAdami], though not necessarily physically consistent. First, the Newton-Wigner localization problem [@Sakurai] prevents us from considering momentum distributions with $w \lesssim m$. In that case, particle creation would manifest and our model, relying on a bipartite state of the Fock space, would break down. Second, $w \sim m$ would produce fast wave-packet spreading, yielding an undesired particle delocalization. This can be generalized to the other three tensor products involving $\Psi_1$ and $\Psi_2$, $$\begin{aligned} \mathrm{Tr}_{\rm \vec{q}}[\Lambda\Psi_2(\Lambda\Psi_2)^{\dag}]& = &\frac{1}{2}\left(\begin{array}{cc}1-n_z' & 0\\0 & 1+n_z'\end{array}\right),\label{peresmatriz2}\\ \mathrm{Tr}_{\rm \vec{q}}[\Lambda\Psi_1(\Lambda\Psi_2)^{\dag}]&= &\frac{1}{2}\left(\begin{array}{cc}0 & 1+n'_z\\-(1-n_z') & 0\end{array}\right),\label{peresmatriz3}\\ \mathrm{Tr}_{\rm \vec{q}}[\Lambda\Psi_2(\Lambda\Psi_1)^{\dag}]&=&\frac{1}{2} \left(\begin{array}{cc}0 & -(1-n'_z)\\1+n_z' & 0\end{array}\right).\label{peresmatriz4}\end{aligned}$$ With the help of Eqs. (\[densityboosttr\]-\[peresmatriz4\]), it is possible to compute the effects of the Lorentz transformation, associated with a boost in the $x$-direction, on any density matrix of two spin-1/2 particles with factorized Gaussian momentum distributions. In particular, Eq. (\[Wernerrela1\]) is reduced to $$\left( \!\!\! \begin{array}{cccc} \frac{1}{4} + c_F {n'_z}^2 & 0 & 0 & c_F ({n'_z}^2-1) \\ 0 & \frac{1}{4} - c_F {n'_z}^2 & c_F({n'_z}^2+1) & 0 \\ 0 & c_F ({n'_z}^2+1) & \frac{1}{4} - c_F {n'_z}^2 & 0 \\ c_F ({n'_z}^2-1) & 0 & 0 & \frac{1}{4} + c_F {n'_z}^2 \\ \end{array} \!\!\! \right) , \\ \label{Werner2}$$ where $c_F:=\frac{1-4F}{12}$. We can apply now the positive partial transpose (PPT) criterion [@Peres; @Horodeckis1] to know whether this state is entangled and distillable. Due to the box-inside-box structure of Eq. (\[Werner2\]), it is possible to diagonalize its partial transpose in a simple way, finding the eigenvalues $$\begin{aligned} x_1 = \frac{2F+1}{6} \,\, , \hspace*{0.5cm} x_2 = \frac{1-F}{3} + \frac{1-4F}{6} {n'_z}^2 \,\, , \nonumber \\ [0.1cm] x_3 = \frac{1-F}{3} -\frac{1-4F}{6} {n'_z}^2 \,\, , \hspace*{0.5cm} x_4 = \frac{2F+1}{6} \,\, .\end{aligned}$$ Given that $F > 0$, $x_1$ and $x_4$ are always positive, and also $x_3$ for $0 < {n'_z} < 1$. The eigenvalue $x_2$ is negative if, and only if, $F > N'_z$, where $N'_z:=(2 + {n'_z}^2)/(2 + 4{n'_z}^2)$. The latter implies that in the interval $$\begin{aligned} \frac{1}{2} < F < N'_z \label{eqNz}\end{aligned}$$ distillability of state $| \Psi^{-} \rangle$ is possible for the spin state in A and B, but impossible for the spin state in A’ and B’, both described in frame $\cal{S}$. We plot in Fig. \[grafreldist\] the behavior of $N'_z$ as a function of the rapidity $\alpha$. The region below the curve (ND) corresponds to the $F$ values for which distillation is not possible in the Lorentz transformed frame. On the other hand, the region above the curve (D), corresponds to states which are distillable for the corresponding values of $n'_z$. Notice that there are values of $F$ for which the Werner states are weak isodistillable and weak isoentangled, corresponding to the states in the region D above the curve for the considered range of $n'_z$. On the other hand, there are states that will change from distillable (entangled) into separable for a certain value of $n'_z$, showing the relativity of distillability and separability. ![$N'_z$ of Eq. (\[eqNz\]) vs. the rapidity $\alpha$, for $w/2m=0.1$. \[grafreldist\]](Fig2.eps){height="4.5cm"} The study of strongly isoentangled and strongly isodistillable two-spin states is a much harder task that will depend on the entanglement measure we choose. We believe that these cases impose demanding conditions and, probably, this kind of states does not exist. However we would like to give a plausibility argument to justify this conjecture. Our argument is based on two mathematical points: (i) analytic continuation is a mathematical tool that allows to extend the analytic behavior of a function to a region where it was not initially defined, and (ii) an analytic function is either constant or it changes along all its interval of definition. Point (i) will allow us to extend analytically our calculation to $n'_z=0$, an unphysical but mathematically convenient limit. Point (ii) will be applied to any well-behaved entanglement measure. We consider then a general spin density matrix $$\begin{aligned} \hspace*{-0.6cm} \rho:=\left( \!\!\! \begin{array}{cccc} a_1 & b_1 & b_2 & b_3 \\ b_1^* & a_2 & c_1 & c_2 \\ b_2^* & c_1^* & a_3 & d\\ b_3^* & c_2^* & d^* & a_4 \\ \end{array} \!\!\! \right), \label{generaldensity}\end{aligned}$$ where $a_1$, $a_2$, $a_3$, and $a_4$ are real, and $\sum_i a_i=1$. The analytic continuation of the Lorentz transformed state, according to Eqs. (\[densityboosttr\]-\[peresmatriz4\]), in the limit $n'_z\rightarrow 0$, is $$\begin{aligned} \hspace*{-0.6cm} \left( \!\!\! \begin{array}{cccc} 1/4 & \frac{i(\mathrm{\Im} b_1+\mathrm{\Im} d)}{2} & \frac{i(\mathrm{\Im} b_2+\mathrm{\Im} c_2)}{2} & \frac{(\mathrm{\Re} b_3-\mathrm{\Re} c_1)}{2} \\ \frac{-i(\mathrm{\Im} b_1+\mathrm{\Im} d)}{2} & 1/4 & \frac{(-\mathrm{\Re} b_3+\mathrm{\Re} c_1)}{2} & \frac{i(\mathrm{\Im} b_2+\mathrm{\Im} c_2)}{2} \\ \frac{-i(\mathrm{\Im} b_2+\mathrm{\Im} c_2)}{2} & \frac{(-\mathrm{\Re} b_3+\mathrm{\Re} c_1)}{2} & 1/4 & \frac{i(\mathrm{\Im} b_1+\mathrm{\Im} d)}{2}\\ \frac{(\mathrm{\Re} b_3-\mathrm{\Re} c_1)}{2} & \frac{-i(\mathrm{\Im} b_2+\mathrm{\Im} c_2)}{2} & \frac{-i(\mathrm{\Im} b_1+\mathrm{\Im} d)}{2} & 1/4 \\ \end{array} \!\!\! \right), \label{separable}\end{aligned}$$ where $\Re$ and $\Im$ denote the real and imaginary parts. This state is separable because its eigenvalues, given by $$\begin{aligned} \lambda_{1,2}&=&\frac{1}{4}[1-2 \mathrm{\Re}(b_3-c_1)\pm 2\mathrm{\Im}(b_1+b_2+c_2+d)]\nonumber\\ \lambda_{3,4}&=&\frac{1}{4}[1+2 \mathrm{\Re}(b_3-c_1)\pm 2\mathrm{\Im}(b_1-b_2-c_2+d)]\nonumber\\ \,\,\,\,\,\,\,\,\,\, \label{autovisoentang}\end{aligned}$$ coincide with the corresponding ones for the partial transpose matrix. In this case, $\lambda_1\leftrightarrow\lambda_4$, and $\lambda_2\leftrightarrow\lambda_3$. So, according to the PPT criterion, the analytic continuation of the Lorentz transformed density matrix of all two spin-1/2 states, with factorized Gaussian momentum distributions, converges to a separable state in the limit of $n'_z\rightarrow 0$ [@comment]. Our analytic calculation holds for $n'_z \lesssim 1$, leaving out of reach the case $n'_z=0$. However, any analytic measure of entanglement, due to this behavior of the analytic continuation at $n'_z=0$, is forced to change with $n'_z$ for $n'_z \lesssim 1$, except for states separable in all frames. In this way, we give evidence of the non-existence of strong isoentangled and isodistillable states, for variations of the parameter $n'_z$ under the present assumptions. From a broader perspective, our analysis considered the invariance of entanglement and distillability of a two spin-$1/2$ system under a particular completely positive (CP) map, the one determined by the local Lorentz-Wigner transformations. The study of similar properties in the context of general CP maps is an important problem that, to our knowledge, has not received much attention in QIT, and that will require a separate and more abstract analysis. Moreover, for higher dimensional spaces, like a two spin-$1$ system (qutrits), the notion of relativity of bound entanglement will also arise [@Horodeckis2]. In summary, the concepts of weak and strong isoentantangled and isodistillable states were introduced, which should help to understand the relationship between special relativity and quantum information theory. The study of Werner states allowed us to show that distillability is a relative concept, depending on the frame in which it is observed. We have proven the existence of weak isoentangled and weak isodistillable states in our range of validity of the parameter $n'_z$. We also conjectured the non-existence of strong isoentangled and isodistillable two-spin states. We give evidence for this result relying on the analytic continuation of the Lorentz transformed spin density matrix for a general two spin-1/2 particle state with factorized momentum distributions. L.L. acknowledges financial support from Spanish MEC through FPU grant AP2003-0014, CSIC 2004 5 0E 271 and FIS2005-05304 projects. M.A.M.-D. thanks DGS for grant under contract BFM 2003-05316-C02-01. E.S. acknowledges support from SFB 631, EU RESQ and EuroSQIP projects. [99]{} M. Nielsen and I. Chuang, [*Quantum Computation and Quantum Information*]{} (Cambridge University Press, Cambridge, England, 2000). A. Galindo and M. A. Martín-Delgado, Rev. Mod. Phys. [**74**]{}, 347 (2002). C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, Phys. Rev. Lett. [**76**]{}, 722 (1996). H. Bombín and M. A. Martín-Delgado, Phys. Rev. A [**72**]{}, 032313 (2005). R. F. Werner, Phys. Rev. A [**40**]{}, 4277 (1989). W. Rindler, [*Relativity*]{} (Oxford University Press, New York, 2001). R. Verch and R.F. Werner, Rev. Math. Phys. [**17**]{}, 545 (2005). A. Peres and D. R. Terno, Rev. Mod. Phys. [**76**]{}, 93 (2004). A. Peres, P. F. Scudo, and D. R. Terno, Phys. Rev. Lett. [**88**]{}, 230402 (2002). E. Wigner, Annals of Mathematics, [**40**]{}, 39 (1939). S.D. Bartlett and D.R. Terno, Phys. Rev. A [**71**]{}, 012302 (2005). P. M. Alsing and G. J. Milburn, Quantum Inf. Comput. [**2**]{}, 487 (2002). J. Pachos and E. Solano, Quantum Inf. Comput. [**3**]{}, 115 (2003). R. M. Gingrich and C. Adami Phys. Rev. Lett. [**89**]{}, 270402 (2002). A. Peres, Phys. Rev. Lett. [**77**]{}, 1413 (1996). M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A [**223**]{}, 1 (1996). J.J. Sakurai, [*Advanced Quantum Mechanics*]{} (Addison-Wesley, New York, 1967), p. 119. We consider $n'_z \rightarrow 0$ to support our conjecture, though we know from previous arguments that it is unphysical. M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. Lett. [**80**]{}, 5239 (1998).

--- abstract: | Let $M$ be a compact, pseudoconvex-oriented, $(2n+1)$-dimensional, abstract CR manifold of hypersurface type, $n\geq 2$. We prove the following: (i) If $M$ admits a strictly CR-plurisubharmonic function on $(0,q_0)$-forms, then the complex Green operator $G_q$ exists and is continuous on $L^2_{0,q}(M)$ for degrees $q_0\le q\le n-q_0$. In the case that $q_0=1$, we also establish continuity for $G_0$ and $G_n$. Additionally, the ${\bar\partial_b}$-equation on $M$ can be solved in $C^\infty(M)$. (ii) If $M$ satisfies “a weak compactness property" on $(0,q_0)$-forms, then $G_q$ is a continuous operator on $H^s_{0,q}(M)$ and is therefore globally regular on $M$ for degrees $q_0\le q\le n-q_0$; and also for the top degrees $q=0$ and $q=n$ in the case $q_0=1$. We also introduce the notion of a “plurisubharmonic CR manifold" and show that it generalizes the notion of “plurisubharmonic defining function" for a a domain in $\C^N$ and implies that $M$ satisfies the weak compactness property. address: - 'T. V. Khanh' - 'School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia' - 'A. Raich' - 'Department of Mathematical Sciences, SCEN 327, 1 University of Arkansas, Fayetteville, AR 72701, USA' author: - Tran Vu Khanh and Andrew Raich title: 'The Kohn-Laplace equation on abstract CR manifolds: Global regularity' --- [^1] [^2] Introduction ============ Let $M$ be an abstract compact smooth CR manifold of real dimension $2n+1$ equipped with a Cauchy-Riemann structure $T^{1,0}M$. The tangential Cauchy-Riemann operator ${\bar\partial_b}$ is well-defined on smooth $(p,q)$-forms and our interest is to understand the regularity of the canonical solution $u$ to the ${\bar\partial_b}$-equation, ${\bar\partial_b}u ={\varphi}$, i.e., the solution of the ${\bar\partial_b}$-equation that is orthogonal to the kernel of ${\bar\partial}_b$. Our approach is via $L^2$-methods and we obtain the canonical solution to the ${\bar\partial_b}$-equation by solving the related ${\Box_b}$-equation ${\Box_b}u=f$ where the Kohn Laplacian ${\Box_b}= {\bar\partial_b}{\bar\partial^*_b}+{\bar\partial^*_b}{\bar\partial_b}$ and ${\bar\partial^*_b}$ is the $L^2$-adjoint of ${\bar\partial_b}$ in an appropriate $L^2$-space. We provide the technical details below. The operator ${\Box_b}$ maps $(0,q)$-forms to $(0,q)$-forms and its inverse on $(0,q)$-forms (when it exists) is called the *complex Green operator* and denote $G_q$. Our primary goal in this paper to find sufficient conditions for the global regularity and exact regularity of $G_q$ and related operators. Global regularity means that $G_q$ maps smooth forms to smooth forms, and exact regularity means that $G_q$ is continuous on all of the $L^2$-Sobolev spaces $H^s_{0,q}(M)$, $s\geq 0$. Before we can investigate the exact and global regularity questions, we must generalize the existing $L^2$-theory for solving the ${\Box_b}$ and ${\bar\partial_b}$-equations in $L^2_{0,q}(M)$. Additionally, we define a family of weighted Kohn Laplacians and show that for a given index $s \geq 0$, there are weighted complex Green operators whose inverses are continuous on $H^s_{0,q}(M)$. History – the $L^2$-theory of ${\bar\partial_b}$ on CR manifolds ---------------------------------------------------------------- A classical result of Hörmander is that solvability of an operator (e.g., ${\bar\partial_b}$, ${\Box_b}$, etc.) is equivalent to it having closed range. Solvability is closely linked to geometric and potential theoretic conditions on the manifold $M$. Curvature, for example, is measured by the Levi form. The Levi form on boundaries of domains in $\C^{n+1}$ is essentially the complex analysis analog of the second fundamental form, and when it is nonnegative, $M$ is called pseudoconvex. The first results establishing closed range of ${\bar\partial_b}$ in $L^2_{0,q}(M)$ occur when $M\subset\C^{n+1}$, $n\ge 1$, is pseudoconvex. Shaw [@Sha85] proves that ${\bar\partial_b}$ has closed range for $0 \leq q \leq n-1$ when $M$ is the boundary of a pseudoconvex domain. With Boas, Shaw supplements her earlier result and establishes the result in the top degree $q=n$ [@BoSh86]. Using a microlocal argument, and in fact, developing the microlocal machinery, Kohn proves that if $M$ is compact, pseudoconvex, and the boundary of a smooth complex manifold which admits a strictly plurisubharmonic function defined in a neighborhood of $M$, then $\dib_b$ has closed range [@Koh86]. In [@Nic06] Nicoara extends the Kohn microlocal machine and proves that on an embedded, compact, pseudoconvex-oriented CR manifold of dimension $(2n+1)\geq5$, ${\bar\partial_b}$ has closed range and can be solved in $C^\infty$. When pseudoconvexity is relaxed, Harrington and Raich prove closed range and related estimates at the level of $(0,q)$-forms for a fixed $q$, $1 \leq q \leq n-1$, by developing a weak $Y(q)$ condition that is much weaker than pseudoconvexity but still suffices [@HaRa11; @HaRa15]. They prove similar results to those of Nicoara [@Nic06]. Harrington and Raich work on embedded manifolds in [@HaRa11] and in a Stein manifold in [@HaRa15]. In [@HaRa15], they adopt Shaw’s techniques to work on manifolds of minimal smoothness. In all of the above cases, a crucial property of the manifolds is the existence a strictly CR plurisubharmonic function in a neighborhood of $M$. Indeed, our hypotheses in Theorem \[thm:L\^2-theory\] below include the existence of such a function, and our main task in proving Theorem \[thm:L\^2-theory\] is to show that the arguments from the previous works hold in the generality in which we work. The main focus of this paper, however, is the global and exact regularity statements. History – global regularity for $G_q$ ------------------------------------- An operator is *globally regular* if it preserves $C^\infty$. Determining necessary and sufficient conditions for the global regularity of the complex Green operator (and ${\bar\partial}$-Neumann operator) is a one of the major questions in the $L^2$-theory of the tangential Cauchy-Riemann operators. On compact manifolds, it is clear that if $G_q$ preserve $C^\infty$ locally, then it will preserve $C^\infty$ globally. However, local regularity of $G_q$ requires strong hypotheses, e.g., subelliptic estimates [@Koh85], subelliptic estimates with multipliers [@Koh00; @BaPiZa15], or superlogarithmic estimates [@Koh02; @KhZa11]. In parallel with this work [@Kha18], the first author introduces a new potential theoretical condition named the $\sigma$-superlogarithmic property to prove the local regularity of $G_q$. However, this condition is far stronger than necessary to imply global regularity. The techniques to prove global regularity do so by showing that $G_q$ is *exactly regular*, meaning that $G_q$ is a bounded operator from $H^s_{0,q}(M)$ to itself. There are several known conditions that suffice to prove exact (and hence global) regularity. The first is that $M$ is the boundary of a domain that admits a plurisubharmonic defining function [@BoSt91]. The second is when $G_q$ is a compact operator. $G_q$ is known to be a compact operator by Raich [@Rai10] when $M$ is a smooth, orientable, pseudoconvex CR manifold of hypersurface type of real dimension at least five that satisfies a pair of potential-theoretical conditions that he names properties $(CR\T-P_q)$ and $(CR\T-P_{n-1-q})$. Straube [@Str12] improves Raich’s result by shedding the orientability hypothesis and showing that $(CR\T-P_q)$ is equivalent to $(P_q)$ (see [@Str08] for details and background on $(P_q)$). Pinton, Zampieri, and Khanh further reduce the hypotheses by removing the embeddability requirement so that $M$ is an abstract CR manifold [@KhPiZa12a]. They also prove a (real) dimension 3 compactness result under the additional assumption that ${\bar\partial_b}$ has closed range on functions. The third known condition for global and exact regularity of $G_q$ is the existence of special 1-form that is exact on the null-space of the Levi form. Straube and Zeytuncu prove global regularity for $G_q$ on a smooth, oriented, compact, pseudoconvex CR submanifold in $\C^n$ under this hypothesis [@StZe15]. The genesis of the three known sufficient conditions is that they, or their analogs, are sufficient conditions for the global/exact regularity of the ${\bar\partial}$-Neumann operator on domains in $\C^n$. See Boas and Straube [@BoSt99] for a discussion of the early history of the global regularity question and Straube [@Str08] or Harrington [@Har11] for recent work. Main Goal and Major Results --------------------------- The main goal of this paper is to prove a global regularity result which encompasses and improves on all of the existing work. In particular, our condition is weaker than Straube and Zeytuncu’s in two respects: 1) we do not require exactness of the special 1-form $\alpha$, only a bound in the spirit of [@Str08], and 2) we no longer require $M$ to be embedded. We state the main result below but defer the technical definitions to Section \[sec:definitions\], though we do need the form $\gamma$ that is dual to the “bad" direction $T$, that is, the dual to the vector that is orthogonal to the CR structure of the manifold. Also $d\gamma$ is connected to the Levi form and $\L_{\lambda}$ is the analog of the complex Hessian for the function $\lambda$. \[thm:global regularity\] Let $M$ be a smooth, compact, pseudoconvex-oriented CR manifold of dimension $2n+1$ which admits a CR plurisubharmonic function on $(0,q_0)$-forms, $n\geq 2$. Assume that for every ${\epsilon}>0$ there exist a $C^\infty$ real valued function $\lambda_{\epsilon}$, a purely imaginary vector field $T_{\epsilon}$, and a constant $A_{\epsilon}>0$ so that $|\lambda_{\epsilon}|$, $\gamma(T_{\epsilon})$ are uniformly bounded, $\gamma(T_{\epsilon})$ is bounded away from zero, and $$\label{eqn:1/eps est} \la(\L_{\lambda_{\epsilon}}+A_{\epsilon}d\gamma)\lrcorner u,\bar u\ra\ge \frac{1}{{\epsilon}}|\alpha_{\epsilon}|^2|u|^2$$ for any $(0,q_0)$-forms $u$. The form $\alpha_{\epsilon}$ is real and defined by $\alpha_{\epsilon}=-\T{\{Lie\}}_{T_{\epsilon}}(\gamma).$ If $q_0\le q\le n-q_0$, then the operators $G_q$, $\dib_bG_q$, $G_q\dib_b$, $\dib_b^*G_q$, $G_q\dib_b^*$, $I-\dib_b^*\dib_bG_q$, $I-\dib_b^*G_q\dib_b$, $I-\dib_b^*\dib_bG_q$, $I-\dib_bG_q\dib_b^*$, $\dib_b^*G^2_q\dib_b$ and $\dib_bG_q^2\dib_b^*$ are both globally regular and exactly regular in the $L^2$-Sobolev space $H^s$, $s\geq 0$. In the case $q=1$, the operators $G_0 = {\bar\partial^*_b}G_1^2 {\bar\partial_b}$ and $G_n = {\bar\partial_b}G_{n-1}^2 {\bar\partial^*_b}$ and hence $G_0$, $\dib_bG_0$, $G_n$, $\dib_b^*G_n$ are both globally regular and exactly regular in $H^s$ as well, $s\geq 0$. The pseudoconvex-oriented is a necessary condition for working on global abstract CR manifolds containing open Levi flat sets in the sense that on such a Levi flat set we can choose local contact forms whose Levi forms are locally pseudoconvex at every point at which they are defined, but there does not exist a global contact form whose Levi form is globally pseudoconvex. The hypothesis of Theorem \[thm:global regularity\] is weaker than property $(CR\T-P_q)$ (introduced by Raich [@Rai10]). Furthermore, if $\alpha=:-\T{\{Lie\}}_{T}(\gamma)$ is exact on the null-space of the Levi form then Straube and Zeytuncu [@StZe15 Proposition 1] prove that for each ${\epsilon}>0$ there exists $T_{\epsilon}$ such that $\alpha_{\epsilon}= -\T{\{Lie\}}_{T_{\epsilon}}(\gamma)$ satisfies $|\alpha_{\epsilon}|\le{\epsilon}$. So if we choose $\lambda_{\epsilon}:=\lambda$ is a strictly CR-plurisubharmonic with associated constant $A_{\epsilon}=1$ as in below, then is satisfied for all ${\epsilon}>0$, and hence the hypotheses of Theorem \[thm:global regularity\] holds. The following two corollaries are proven using the arguments of [@StZe15]. \[cor:CRpsh on (0,1)\] Let $M$ be a smooth, compact, pseudoconvex-oriented CR manifold of dimension $2n+1$ which admits a CR plurisubharmonic function on $(0,1)$-forms, $n\geq 2$. If the real 1-form $\alpha=-\{\T{Lie}\}_{T}(\gamma)$ is exact on the null space of the Levi form, then the hypothesis in Theorem \[thm:global regularity\] holds for $(0,1)$-forms. Let $M$ be a smooth, compact, pseudoconvex-oriented CR manifold of dimension $2n+1$ which admits a CR plurisubharmonic function on $(0,1)$-forms, $n\geq 2$. Let $S$ be the set of non-strictly pseudoconvex points. Suppose that at each point of $S$, the (real) tangent space is contained in the null space of the Levi form at the point. If the first de Rham cohomology of $S$ is trivial, then $\alpha$ is exact on the null space of the Levi form (this happens if $S$ is simply connected). Consequently, the hypotheses of Corollary \[cor:CRpsh on (0,1)\] hold. The proof follows exactly from [@StZe15 Theorem 3]. Let $\N_x$ be the null space of the Levi form at $x$. Analogously to Lemma [@BoSt93 Lemma on p. 230], we have $(d\alpha|_x)(X{\wedge}Y)=0$ if $X,Y\in \N_x\oplus \overline{\N_x}$. Thus, $d\alpha=0$ on $S$. Since the first DeRham cohomology of $S$ is trivial, there exists $h$ on $C^\infty(\overline{S})$ such that such that $\alpha=dh$ on $S$. This mean, $\alpha$ is exact on $\N_x$. Also in [@StZe15], the authors show that if the defining functions of $M$ are plurisubharmonic in some neighborhood of $M$ then $\alpha$ is exact on the null space of the Levi form. Their proof strongly relies on the fact that $M$ is embedded in $\C^n$, however, we are able to remove the embeddability assumption. \[thm:CRpsh (0,1)\] Let $M$ be a smooth, compact, plurisubharmonic-oriented CR manifold of dimension $2n+1$ that admits a CR plurisubharmonic function on $(0,1)$-forms. Then for any ${\epsilon}>0$ there exists a vector $T_{\epsilon}$ whose length is bounded and bounded away from $0$ with associated form $\alpha_{\epsilon}= -\T{\{Lie\}}_{T_{\epsilon}}(\gamma)$ satisfying $|\alpha_{\epsilon}|<{\epsilon}$. Consequently, the conclusion of Theorem \[thm:global regularity\] holds for all $0 \leq q \leq n$. The secondary goal of this paper is to restate in a slightly more general form the $L^2$-theory for ${\bar\partial_b}$ without the embeddability assumption. \[thm:L\^2-theory\] Let $M$ be a smooth, compact, pseudoconvex-oriented CR manifold of dimension $2n+1$ that admits a strictly CR-plurisubharmonic function on $(0,q_0)$-forms. If $q_0 \leq q \leq n-q_0$ then the following hold: 1. The $L^2$ basic estimate $$\label{L2basic} {\|{u}\|}_{L^2}^2\le c({\|{\dib_bu}\|}_{L^2}^2+{\|{\dib_b^*u}\|}_{L^2}^2)$$ holds for all $u\in \T{Dom}(\dib_b)\cap\T{Dom}(\dib_b)\cap {\mathcal H}^{\perp}_{0,q}(M)$. 2. The operators ${\bar\partial_b}: L^2_{0,\tilde q}(M)\to L^2_{0,\tilde q+1}(M)$ and ${\bar\partial^*_b}: L^2_{0,\tilde q+1}(M)\to L^2_{0,\tilde q}(M)$ have closed range when $\tilde q = q$ or $q-1$. Additionally, $\Box_b:L^2_{0,q}(M)\to L^2_{0,q}(M)$ has closed range; 3. The operators $G_q$, ${\bar\partial^*_b}G_q$, $G_q{\bar\partial^*_b}$, ${\bar\partial_b}G_q$, $G_q{\bar\partial_b}$, $I - {\bar\partial^*_b}{\bar\partial_b}G_q$, $I - {\bar\partial^*_b}G_q {\bar\partial_b}$, $I - {\bar\partial_b}{\bar\partial^*_b}G_q$, $I - {\bar\partial_b}G_q {\bar\partial^*_b}$ are $L^2$ bounded. In the case $q=1$, the operators $G_0 = {\bar\partial^*_b}G_1^2 {\bar\partial_b}$ and $G_n = {\bar\partial_b}G_{n-1}^2 {\bar\partial^*_b}$ and hence $G_0$, $\dib_bG_0$, $G_n$, $\dib_b^*G_n$ are continuous on $L^2$. 4. The $\dib_b$-equation $\dib_bu=\varphi$ has a solution $u\in C^\infty_{0,\tilde q-1}(M)$ for $\tilde q=q$ or $q+1$ if $\varphi$ is a $\dib_b$-closed, $C^\infty$-smooth $(0,\tilde q)$-form. 5. The space of harmonic forms $\mathcal H_{0,q}(M)$ is finite dimensional. There is little new work to be done to prove Theorem \[thm:L\^2-theory\]. Our hypotheses are exactly what Nicoara [@Nic06] and Harrington and Raich [@HaRa11] use. In their work, embeddability is only used to establish the existence of a strictly CR plurisubharmonic function. We will simply highlight aspects of the earlier proofs that we need to prove Theorem \[thm:L\^2-theory\]. Our main contribution is the development of an elliptic regularization that does not require $M$ to be embedded, whereas the previous methods strongly used the embeddedness. The outline of the rest of the paper is as follows. The technical preliminaries are given in Section \[sec:definitions\]. The establishment of a basic estimate comprises the beginning of Section \[sec:basic estimate\]. The remainder of the section develops the microlocal framework that we use to prove the main theorem. We prove Theorem \[thm:L\^2-theory\] in Section \[sec:proving L\^2 theory\]. Its proof follows the argument of [@HaRa11] (and in [@Nic06]), and like [@HaRa11], the microlocal argument proves an auxiliary result on a carefully constructed weighted spaces, Theorem \[thm:main theorem for weighted spaces\]. It is through the weighted spaces that we can solve ${\bar\partial_b}$ in $C^\infty$. We conclude the paper in Section \[sec:hypoellipticity\] with proofs of Theorem \[thm:L\^2-theory\] and Theorem \[thm:CRpsh (0,1)\]. Definitions {#sec:definitions} =========== Let $M$ be a real smooth manifold of dimension $2n+1$, $n\ge 1$. Let $\C TM$ be the complexified tangent bundle over $M$, and $T^{1,0}M$ be a subbundle of $\C TM$. We say that $M$ is a *CR manifold of hypersurface type* equipped with CR structure $T^{1,0}M$ (or CR manifold for short) if the following conditions are satisfied: 1. $\dim_\C T^{1,0}M=n$, 2. $T^{1,0}M\cap T^{0,1}M=\{0\}$, where $T^{0,1}M=\overline{T^{1,0}M}$, 3. for any $L, L'\in \Gamma(U, T^{1,0}M)$, the Lie bracket $[L,L']$ is still in $\Gamma(U, T^{1,0}M)$, where $U$ is any open set of $M$ and $\Gamma(U, T^{1,0}M)$ denotes the space of smooth sections of $T^{1,0}M$ over $U$ (this condition is nonexistent when $n=1$). On $M$, we choose a Riemann metric $\la\cdot,\cdot\ra$ which induces a Hermitian metric on $T^{1,0}M\oplus T^{0,1}M$ so that $\la L,\bar L'\ra=0$ for any $L,L'\in T^{1,0}M$. Given the metric $\la\cdot,\cdot\ra$, we choose a local frame $\{L_1,\dots, L_n\}$ of $T^{1,0}M$ and a purely imaginary vector field $T$ that is orthogonal (and hence transversal) to $T^{1,0}M\oplus T^{0,1}M$ so that $\{L_1,\dots,L_n,\bar L_1,\dots,\bar L_n, T\}$ forms an orthonormal basis of $\C TM$. See [@Bog91] for details. Denote by $\omega_1,...,\omega_{n},\bar {\omega}_1,...,\bar{\omega}_{n},\gamma$ the dual basis of $1$-forms for $L_1,...,L_{n}$, $\bar L_1,...,\bar L_{n}$, and $T$. We call any open set $U \subset M$ a [*local patch*]{} if it admits such vectors and forms. Our interest is in global solvability, so we need a suitable partition of unity. We call a cover $\{U_\mu\}$ of $M$ a *good cover* if each $U_\alpha$ is a local patch. We also let $\{\eta_\mu\}$ be a partition of unity subordinate to $\{U_\mu\}$.\ For a $C^2$ function $\phi$ on $M$, we call the alternating $(1,1)$-form $\L_\phi=\frac{1}{2}\big(\di_b\dib_b-\dib_b\di_b\big)\phi$ on $T^{1,0}M\times T^{0,1}M$ the *Levi form of $\phi$*. The Levi form $\mathcal L_M$ of $M$ at $x$ is the Hermitian form given by $d\gamma (L_x \wedge \bar L_x') = \gamma([\bar L_x',L_x])$ for $L,\,L'\in T^{1,0}_x$. We say that $M$ is *pseudoconvex at $x\in M$* if the Levi form is positive semidefinite in a neighborhood of $x$, i.e., $d\gamma(L\wedge\bar L)\geq 0$ for all $L\in T^{1,0}_x$. $M$ is *orientable* if there exists is a global 1-form $\tilde \gamma$. We say that $M$ is *pseudoconvex-oriented* if the $1$-form $\gamma \in \C TM^*$ is globally defined and the Levi form is positive semidefinite for all $x\in M$. All of the manifolds that we consider in this paper will be pseudoconvex-oriented. We express $d\gamma$ in a local patch $U$ as $$\label{dgamma} d\gamma=-\left(\sum_{i,j=1}^nc_{ij}{\omega}_i{\wedge}{\bar{\omega}}_j+\sum_{j=1}^nc_{0j}\gamma{\wedge}{\omega}_j+\sum_{j=1}^n\bar c_{0j}\gamma{\wedge}{\bar{\omega}}_j\right),$$ where the integrability condition of CR structure and the Cartan formula forces the coefficients of ${\omega}_i{\wedge}{\omega}_j$ and ${\bar{\omega}}_j{\wedge}{\bar{\omega}}_j$ to be zero. The pseudoconvex-oriented condition is equivalent to the Levi matrix $\L_{M}:=\{c_{ij}\}_{ij=1}^n$ being positive semidefinite at every $x\in M$. We extend the $n\times n$ Levi matrix to a $(n+1)\times (n+1)$ matrix $\{c_{ij}\}_{ij=0}^n$ with entries $c_{0j}$ for $j=1,\dots,n$, $c_{j0}=\overline{c_{0j}}$, and $c_{00}$ to be chosen. We say that $M$ is *plurisubharmonic* at $x$ if there exists $c_{00}$ such that the extended Levi matrix $\{c_{ij}\}_{ij=0}^n\ge 0$ in a neighborhood of $x$ and $M$ is [*plurisubharmonic-oriented*]{} if it is pseudoconvex-oriented and plurisubharmonic at every $x\in M$. It is obvious that if $M$ is embedded in a Stein manifold $X$ and admits a plurisubharmonic defining function $r$ then $M$ is plurisubharmonic-oriented. Indeed, in this case the plurisubharmonic-oriented condition is fulfilled if we choose $\gamma=\frac{i}{2}(\di r-\dib r)$ and $c_{00}$ is the $T:=L_0-\bar L_0$ component of $[L_0,\bar L_0]$ where $L_0\in T^{1,0}X$ is the dual of $\di r$. Let $\alpha=\sum_{j=1}^n \left(c_{0j}{\omega}_j+ \bar c_{0j}{\bar{\omega}}_j\right)$ and observe that $\alpha = -\{\T{Lie}\}_T(\gamma)$. In [@StZe15], the $(1,0)$-form $\alpha$ is called [*exact on the null space of the Levi form*]{} if there exists a smooth function $h$, defined in a neighborhood of $K$, the set of weakly pseudoconvex points of $M$, such that $$dh(L_z)(z) = \alpha(L_z)(z),\quad L_z\in \mathcal N_z\cap T^{1,0}_z, \quad z\in K,$$ where $\mathcal N_z$ is the null space of the Levi form at $z\in K$. Below, we will show that if $M$ is plurisubharmonic-oriented then $\alpha$ is exact on the null space of the Levi form.\ Denote by $dV$ the element of volume on $M$, the induced $L_2$-inner product and norm on $ C^\infty_{p,q}(M)$ is defined by $$(u,v)=\int_M\la u, \bar v\ra dV,\qquad {\|{u}\|}_{L^2}^2=(u,u).$$ The function space $L_{p,q}^2(M)$ is the Hilbert space obtained by completing $ C^\infty_{p,q}(M)$ under the $L^2$-norm. The Sobolev spaces $H^s_{p,q}(M)$ are obtained by completing $ C^\infty_{p,q}(M)$ under the usual $H^s(M)$ norm, ${\|{\cdot}\|}_{H^s}$, applied componentwise. We now want to define ${\bar\partial_b}$ on $(p,q)$-forms, extending our definition from functions. Define the operator $\dib_b: C^\infty_{p,q}(M)\to C^\infty_{p,q+1}(M)$ to be the projection of the de Rham exterior differential operator $d$ to $ C^\infty_{p,q}(M)$. We denote by $\dib^*_b:\, C^\infty_{p,q+1}(M)\to C^\infty_{p,q}(M)$ the $L^2$-adjoint of $\dib_b$ and define the Kohn-Laplacian by $$\Box_b:=\dib_b\dib_b^*+\dib_b^*\dib_b: C^\infty_{p,q}(M) \to C^\infty_{p,q}(M).$$ The space of harmonic $(p,q)$-forms $\mathcal H_{p,q}(M):=\ker\Box_{b}$ coincides with $\ker \dib_b\cap\ker\dib^*_b$. Our result will be to find a sufficient condition so that the estimate $$\label{1.1} {\|{u}\|}_{L^2}\leq c({\|{\dib_b u}\|}_{L^2}+{\|{\dib^*_bu}\|}_{L^2})\quad \T{for }u\in \T{Dom}(\dib_b)\cap Dom(\dib^*_b)\cap \mathcal H^\perp_{p,q},$$ holds and $\mathcal H_{p,q}(M)$ is finite dimensional. A consequence of this estimate is that $\Box_{b}$ has a bounded inverse on ${\mathcal H}^\perp_{p,q}(M)$. In this case, we extend the inverse to be identically $0$ on ${\mathcal H}_{p,q}(M)$. In the case that ${\Box_b}$ is an invertible operator on $L^2_{p,q}(M)\cap{\mathcal H}^\perp_{p,q}(M)$, we denote the inverse by $G_{p,q}$ and call it the *complex Green operator*. The existence of a strictly CR plurisubharmonic function means that the curvature of $M$ does not play a factor in the existence of $G_{p,q}$, so it suffices to take $p=0$, and we denote $G_{0,q}$ by $G_q$. If $M$ is not embedded in a Stein manifold, we can define strictly CR plurisubharmonic functions as follows. Let $M$ be a pseudoconvex CR manifold. A $C^\infty$ real-valued function $\lambda$ defined on $M$ is *strictly CR plurisubharmonic on $(0,q)$-forms* if there exists a constant $a>0$ so that so that $$\begin{aligned} \label{CRpluri} \la \left(\L_\lambda+ d\gamma\right)\lrcorner u,\bar u\ra\ge a|u|^2,\end{aligned}$$ for any $u\in C^\infty_{0,q}(M)$. For $u$ defined on $U$, the contraction operator $\lrcorner$ is defined by $$\theta\lrcorner u=\sum_{I\in\I_{q-1}} \left(\sum_{j=1}^n \theta_j u_{jI}\right){\bar{\omega}}_I\quad\T{if $\theta=\sum_j\theta_j\,{\omega}_j$ is a $(1,0)$-form on $U$};$$ and $$\quad\theta\lrcorner u=\sum_{I\in\I_{q-1}}\sum_{j=1}^n \left(\sum_{i=1}^n \theta_{ij} u_{iI}\right){\bar{\omega}}_j{\wedge}{\bar{\omega}}_{I}\quad\T{if $\theta=\sum_{i,j=1}^n \theta_{ij}\,{\omega}_i{\wedge}{\bar{\omega}}_j$ is a $(1,1)$-form on $U$}.$$ Thus, the Levi form $d\gamma$ and $\L_\phi$ acting on $(0,q)$-forms $u,v$ defined in $U$ can be expressed as $$\la d\gamma\lrcorner u,\bar v\ra=\sum_{I\in\I_{q-1}}\sum_{i,j=1}^nc_{ij}u_{iI}\overline{v_{jI}}\quad \T{and } \la \L_\phi\lrcorner u,\bar v\ra=\sum_{I\in\I_{q-1}}\sum_{i,j=1}^n\phi_{ij}u_{iI}\overline{v_{jI}}.$$ Strictly CR-plurisubharmonic functions always exist if $M$ is strictly pseudoconvex or embedded into a Stein manifold. They do not, however, always exist on abstract CR manifolds. See, for example, Grauert’s example [@Gra63]. Working in local coordinates {#subsec:prelims for the basic est} ---------------------------- Let $U$ be a local patch of $M$ with its associated basis of tangential vector fields $L_1,...,L_{n},\bar L_1,...,\bar L_{n},T$ and dual basis $\omega_1,...,\omega_{n},\bar\omega_1,...,\bar\omega_{n}, \gamma$. For the moment, we work locally on $U$. The condition of pseudoconvexity of $M$ at $x$ is equivalent to the Levi matrix $\{c_{ij}\}_{i,j=1}^n\ge 0$ in a neighborhood of $x$. We recall that $M$ is [*pseudoconvex-oriented*]{} (resp., [*plurisubharmonic-oriented*]{}) if there exist a global 1-form section $\gamma$ (resp., a global 1-form section $\gamma$ and a smooth function $c_{00}$) such that the Levi matrix $\{c_{ij}\}_{ij=1}^n$ (resp. the extended Levi matrix $\{c_{ij}\}_{ij=0}^n$) is positive semidefinite for every $x\in M$. We further define $c_{ij}^k$ to be the $L_k$-component of $[L_i,\bar L_j]$. Since $d$ applied to a $(1,0)$-form can produce a $(2,0)$-form and a $(1,1)$-form [@Bog91 §8.2, Lemma 1], it follows from the definition of ${\bar\partial_b}$ that $$\Label{31} \begin{split} c_{ij}^k:= {\omega}_k([L_i,\bar L_j]) \underset{\T{Cartan}}=- {\bar\partial_b}{\omega}_k(L_i\wedge\bar L_j). \end{split}$$ Therefore, $$\Label{32} {\bar\partial_b}{\omega}_k=-\sum_{i,j=1}^nc_{ij}^k\, {\omega}_i\wedge{\bar{\omega}}_j.$$ and conjugating yields $$\Label{33} {\partial}_b{\bar{\omega}}_k=\sum_{i,j=1}^n\overline{c_{ji}^k}\, {\omega}_i\wedge{\bar{\omega}}_j.$$ Using Cartan’s formula again, we conclude that $-\overline{c_{ji}^k}$ coincides with the $\bar L_k$-component of $[L_i,\bar L_j]$. Thus the full commutator is expressed by $$\Label{34} [L_i,\bar L_j]=c_{ij}T+\sum_{k=1}^n c_{ij}^kL_k - \sum_{k=1}^n\overline{ c_{ji}^k}\bar L_k.$$ For a smooth function $\phi$ on $U$, we want to describe the matrix $(\phi_{ij})$ of the Hermitian form $\frac{1}{2}\big(\di_b\dib_b-\dib_b\di_b\big)\phi$. Now, $\dib_b\phi=\sum_{k=1}^n\bar L_k(\phi){\bar{\omega}}_k$ and therefore $$\Label{35} \begin{split} \di_b\dib_b\phi&=\di_b\Big(\sum_{k=1}^n\bar L_k(\phi){\bar{\omega}}_k\Big) \\&\underset{\T{by \eqref{33}}}=\sum_{i,j=1}^n\left(L_i\bar L_j(\phi)+\sum_{k=1}^n\overline{c_{ji}^k}\bar L_k(\phi)\right){\omega}_i\wedge{\bar{\omega}}_j. \end{split}$$ Similarly, $$\Label{36} \begin{split} \dib_b\di_b\phi&=\dib_b(\sum_{k=1}^nL_k(\phi){\omega}_k) \\ &\underset{\T{by \eqref{32}}}=\sum_{i,j=1}^n \left(-\bar L_jL_i(\phi)-\sum_{k=1}^nc_{ij}^kL_k(\phi)\right){\omega}_i\wedge {\bar{\omega}}_j. \end{split}$$ Combining with we get $$\Label{37} \begin{split} \phi_{ij}&= \frac{1}{2}\Big(\di_b\dib_b\phi-\dib_b\di_b\phi\Big)\big(L_i\wedge\bar L_j\big) \\ &=\frac12\left(L_i\bar L_j(\phi)+\bar L_jL_i(\phi)+\sum_{k=1}^n\left(\overline{c_{ji}^k}\bar L_k(\phi)+c_{ij}^kL_k(\phi)\right)\right) \\ &=\bar L_jL_i(\phi)+\frac12\left([L_i,\bar L_j](\phi)+\sum_{k=1}^n\left(\overline{c_{ji}^k}\bar L_k(\phi)+c_{ij}^kL_k(\phi)\right)\right) \\ &\underset{\T{by \eqref{34}}}=\bar L_jL_i(\phi)+\frac12c_{ij}T(\phi)+\sum_{k=1}^nc_{ij}^kL_k(\phi). \end{split}$$ To express a form in local coordinates, let $\I_q = \{(j_1,\dots,j_q) \in \N^q : 1 \leq j_1 < \cdots < j_q \leq n\}$, and for $J\in \I_q$, $I\in\I_{q-1}$, and $j\in\mathbb N$, ${\epsilon}^{jI}_J$ be the sign of the permutation $\{j,I\}\to J$ if $\{j\} \cup I = J$ as sets, and $0$ otherwise. If $u\in C^\infty_{0,q}(M)$, then $u$ is expressed locally as a combination $$u= \sum_{J\in\I_q} u_J\, \bar\omega_J,$$ of basis forms $\bar\omega_J=\bar\omega_1\wedge...\wedge\bar\omega_{j_q}$ where $J = (j_1,\dots,j_q)$ and $C^\infty$-coefficients $u_J$. We can also express the operator $\dib_b : C^\infty_{0,q}(M)\to C^\infty_{0,q+1}(M)$ and its $L_2$ adjoint $\dib^*_b : C^\infty_{0,q+1}(M)\to C^\infty_{0,q}(M)$ in the local basis as follows: $$\begin{aligned} \label{dib} \dib_b u&= \sum_{{\genfrac{}{}{0pt}{2}{J\in\I_q}{K\in\I_{q+1}}}} \sum_{k=1}^n {\epsilon}^{kJ}_K \bar L_k u_J\, {\bar{\omega}}_K + \sum_{{\genfrac{}{}{0pt}{2}{J\in\I_q}{K\in\I_{q+1}}}} b_{JK} u_J\, {\bar{\omega}}_K $$ and $$\begin{aligned} \Label{dib*} \dib_b^*v=- \sum_{J\in\I_q}\left(\sum_{j=1}^n L_j v_{jK}+ \sum_{K\in\I_{q+1}}a_{JK} v_K\right){\bar{\omega}}_J\end{aligned}$$ where $b_{JK}, a_{JK}\in C^\infty(U)$.\ The basic estimate on CR manifolds {#sec:basic estimate} ================================== In this section, we will work with the weighted $L_\phi^2$-norm defined by $${\|{u}\|}_{L^2_\phi}^2=(u,u)_\phi:={\|{ue^{-\frac{\phi}{2}}}\|}_{L^2}^2=\int_M\la u,\bar u\ra e^{-\phi}\,dV.$$ Let $\dib^*_{b,\phi}$ be the $L_\phi^2$-adjoint of $\dib_b$. It is easy to see that for forms $u\in C^\infty_{0,q+1}(M)$ supported on $U_\mu$ $$\Label{dib*phi} \dib_{b,\phi}^*u=- \sum_{J\in\I_q}\left(\sum_{j=1}^n \sum_{K\in\I_{q+1}} \delta_j^\phi u_{jK}+ \sum_{K\in\I_{q+1}}a_{JK}u_K\right){\bar{\omega}}_J$$ where $\delta^\phi_j \varphi:=e^\phi L_j(e^{-\phi}\varphi)$ and $a_{JK}\in C^\infty(U)$. For such $u$, $$\di_b(\phi)\lrcorner u= - [\dib_b^*,\phi]u=[{\bar\partial_b},\phi]^*u=\sum_{I \in \I_{q-1}} \sum_{j=1}^n L_j(\phi)u_{jI}{\bar{\omega}}_I,$$ and hence $\dib_{b,\phi}^*u=\dib^*_bu-\di_b(\phi)\lrcorner u$. Furthermore, $$\begin{aligned} [\delta^\phi_i,\bar L_j]&=\bar L_jL_i(\phi)+[L_i,\bar L_j] {\nonumber}\\ \underset{\T{by \eqref{37} and \eqref{34}}}=&\phi_{ij}-\frac12c_{ij}T(\phi)-\sum_{k=1}^nc_{ij}^kL_k(\phi)+c_{ij}T-\sum_{k=1}^n\overline{c_{ji}^k}\bar L_k+\sum_{k=1}^n c_{ij}^kL_k {\nonumber}\\ &=\phi_{ij}+c_{ij}T-\sum_{k=1}^n\overline{c_{ji}^k}\bar L_k+\sum_{k=1}^nc_{ij}^k\delta^\phi_k -\frac12c_{ij}T(\phi). \label{38}\end{aligned}$$ The equalities and lead us to Kohn-Morrey-Hörmander inequality or basic estimate for CR manifolds. It does not function quite in the same manner as the Kohn-Morrey-Hörmander inequality on domains because it cannot be applied directly to prove closed range estimates. The terms involving $T$ require significant effort to estimate. In fact, estimating the $T$ terms are the heart of the proof of Theorem \[thm:L\^2-theory\]. Equations similar in spirit to Theorem \[kmh\] have appeared before (e.g., [@HaRa11 Equation (12) and (10)]) but the earlier versions do not apply to as wide of a class of CR manifolds as we consider here. We will not need it here, but we could write Theorem \[kmh\] even more generally by using the weak $Y(q)$ technology (namely, the form $\Upsilon$) from [@HaRa15]. We do not do that here for expositional clarity. \[kmh\]Let $M$ be a CR manifold and $U$ a local patch. Let $\phi$ be a real $C^2$ function and $q_0$ be an integer with $0\le q_0\le n$. There exists a constant $C$ (independent of $\phi$) such that for any $u\in C^\infty_{0,q}(M)$ with support in $U$, $$\begin{gathered} {\|{\bar{\partial}_b u}\|}^2_{L^2_\phi}+{\|{\bar{\partial}^*_{b,\phi} u}\|}^2_{L^2_\phi}+C{\|{u}\|}^2_{L^2_\phi} \geq \frac{1}{2}\Big(\sum^{q_0}_{j=1}{\|{\delta_j^{\phi} u}\|}^2_{L^2_\phi}+\sum^{n}_{j=q_0+1}{\|{\bar{L}_ju}\|}^2_{L^2_\phi} \Big) {\nonumber}\\ + \sum_{I\in\I_{q-1}}\sum_{i,j=1}^n (\phi_{ij}u_{iI},u_{jI})_\phi-\sum_{J\in\I_q}\sum_{j=1}^{q_0}(\phi_{jj}u_J,u_J)_\phi \\ +\operatorname{Re}\bigg\{\sum_{I\in\I_{q-1}}\sum_{i,j=1}^n (c_{ij}Tu_{iI},u_{jI})_\phi-\sum_{J\in\I_q}\sum_{j=1}^{q_0}(c_{jj}Tu_J,u_J)_\phi \bigg\}. {\nonumber}\end{gathered}$$ We use and compute $$\begin{gathered} \|{\bar\partial_b}u\|_{L^2_\phi}^2 = \sum_{{\genfrac{}{}{0pt}{2}{K\in\I_{q+1}}{J,J'\in\I_q}}} \sum_{k,k'=1}^n {\epsilon}^{kJ}_{k'J'} \big(\bar L_k u_J, \bar L_{k'} u_{J'}\big)_\phi + \sum_{{\genfrac{}{}{0pt}{2}{K\in\I_{q+1}}{J,J'\in\I_q}}} \big(b_{JK}u_J,b_{J'K}u_{J'}\big)_\phi \\ + 2\operatorname{Re}\bigg[ \sum_{{\genfrac{}{}{0pt}{2}{K\in\I_{q+1}}{J,J'\in\I_q}}} \sum_{k=1}^n {\epsilon}^{kJ}_K \big(\bar L_k u_J, a_{J'K} u_{J'}\big)_\phi\bigg].\end{gathered}$$ If ${\epsilon}^{kJ}_{k'J'} \neq 0$, then either $k = k'$ and $J=J'$ or $J = \{k\} \cup I$ and $J' = \{k'\} \cup I$ for some $I \in \I_{q-1}$. In the latter case ${\epsilon}^{kJ}_{k'J'} = {\epsilon}^{kk'I}_{k'kI}=-1$. We also use the notation $(\bar L u, u)_\phi$ to denote any term of the form $(a \bar L_k u_J, u_{J'})_\phi$ or its conjugate where $a\in C^\infty(U_\mu)$. We use $(\delta^\phi u, u)_\phi$ to denote any term of the same form with $\delta^\phi_j$ replacing $\bar L_k$. It therefore follows that $$\begin{aligned} \|{\bar\partial_b}u\|_{L^2_\phi}^2 &= \sum_{J\in\I_q}\sum_{k\not\in J} \|\bar L_k u_J\|_{L^2_\phi}^2 - \sum_{I\in\I_{q-1}}\sum_{{\genfrac{}{}{0pt}{2}{k,k'=1}{k\neq k'}}}^n \big( \bar L_k u_{k' I}, \bar L_{k'} u_{kI}\big)_\phi +(\bar L u, u)_\phi + O(\|u\|_{L^2_\phi}^2) \\ &= \sum_{J\in\I_q}\sum_{k=1}^n \|\bar L_k u_J\|_{L^2_\phi}^2 - \sum_{I\in\I_{q-1}}\sum_{k,k'=1}^n \big( \bar L_k u_{k' I}, \bar L_{k'} u_{kI}\big)_\phi +(\bar L u, u)_\phi + O(\|u\|_{L^2_\phi}^2).\end{aligned}$$ A similar (but simpler) calculation shows that $$\|{\bar\partial^*}_{b,\phi} u\|_{L^2_\phi}^2 = \sum_{I\in\I_{q-1}} \sum_{j,j'=1}^n \big(\delta^\phi_j u_{jI}, \delta^\phi_{j'} u_{j'I}\big)_\phi +(\delta^\phi u, u)_\phi + O(\|u\|_{L^2_\phi}^2)$$ To proceed next, we integrate by parts and observe that $$\label{eqn:Lu to delta u} \big(\bar L_k u_J, \bar L_{k'} u_{J'}\big)_\phi = \big(\delta^\phi_{k'} u_J, \delta^\phi_k u_{J'}\big)_\phi - \big([\delta^\phi_{k'},\bar L_k] u_J, u_{J'}\big)_\phi + (\bar L u,u)_\phi + (\delta^\phi u, u)_\phi+O({\|{u}\|}^2_{L^2_\phi}).$$ An immediate consequence of this equality is that $$-\big(\bar L_k u_{jI}, \bar L_j u_{kI}\big)_\phi + \big(\delta^\phi_j u_{jI}, \delta^\phi_k u_{kI}\big)_\phi = \big([\delta^\phi_j,\bar L_k] u_{jI},u_{kI}\big)_\phi + (\bar L u, u)_\phi + (\delta^\phi u,u)_\phi+O({\|{u}\|}^2_{L^2_\phi})$$ and therefore $$\begin{aligned} \label{eqn:dbarb + dbarbs proto} \|{\bar\partial_b}u\|_{L^2_\phi}^2 + \|{\bar\partial^*}_{b,\phi}\|_{L^2_\phi}^2 =& \sum_{J\in\I_q}\sum_{k=1}^n \|\bar L_k u_J\|_{L^2_\phi}^2 + \operatorname{Re}\bigg\{ \sum_{I,I'\in\I_{q-1}}\sum_{j,k=1}^n \big([\delta^\phi_j,\bar L_k] u_{jI},u_{kI}\big)_\phi \bigg\}\\ &+ (\bar L u, u)_\phi + (\delta^\phi u,u)_\phi+ O({\|{u}\|}^2_{L^2_\phi}). {\nonumber}\end{aligned}$$ Finishing the proof requires four observations. First, using on the terms $([\delta^\phi_j,\bar L_k] u_{jI},u_{kI})_\phi$ produces the off-diagonal terms involving $\phi_{jk}$ and $c_{jk}T$. Second, the on-diagonal terms appear when (\[eqn:Lu to delta u\]) is applied to $\|L_j u_J\|_{L^2_\phi}^2$ for $1 \leq j \leq q_0$. Third, we must control $(\bar L u, u)_\phi$ and $(\delta^\phi u,u)_\phi$, but this is a simple matter of recognizing that $(\bar L u,u)_\phi = (\delta^\phi u, u)_\phi + O(\|u\|_{L^2_\phi}^2)$ so we can absorb all of these terms using a small constant/large constant argument where we pay the price of reducing the coefficient of the “gradient" terms to $1/2$ and increasing $O(\|u\|_{L^2_\phi}^2)$. We have the result, except that it is not yet clear that the $O(\|u\|_{L^2_\phi}^2)$ term is independent of $\phi$ because the term $c_{jk} T(\phi)$ appears in (\[38\]). However, $\{c_{jk}\}_{j,k=1}^n$ is a positive semidefinite matrix and hence has real eigenvalues, and $T$ is a purely imaginary operator. This means $$\operatorname{Re}\Big\{\sum_{j,k=1}^n \big(c_{jk}T(\phi) u_{jI},u_{kI}\big)_\phi\Big\} =0.$$ The $T(\phi)$ terms that appear from the integration by parts in the second observation (the one regarding the on-diagonal terms) are handled identically. The difference between the Kohn-Morrey-Hörmander estimate for domains (see, e.g., [@Str10] or [@ChSh01]) and Theorem \[kmh\] is the presence of $T$ instead of a boundary integral. It is for estimating the $T$ term that we use a microlocal argument. Specifically, the estimate for $T$ uses a consequence of the sharp G[å]{}rding inequality. Recall the formulation from [@Rai10]. \[prop:Garding 1\] Let $R$ be a first order pseudodifferential operator such that $\sigma(R)\geq\kappa$ where $\kappa$ is some positive constant and $(h_{jk})$ an $n\times n$ hermitian matrix (that does not depend on $\xi$). Then there exists a constant $C$ such that if the sum of any $q$ eigenvalues of $(h_{jk})$ is nonnegative, then $$\operatorname{Re}\Big\{\sum_{I\in\I_{q-1}}\sum_{j,k=1}^{n}\big(h_{jk}R u_{jI}, u_{kI}\big) \Big \} \geq \kappa \operatorname{Re}\sum_{I\in\I_{q-1}}\sum_{j,k=1}^{n} \big(h_{jk} u_{jI}, u_{kI}\big) -C \|u\|_{L^2}^2,$$ and if the the sum of any collection of $(n-q)$ eigenvalues of $(h_{jk})$ is nonnegative, then $$\begin{gathered} \operatorname{Re}\Big\{\sum_{J\in\I_q}\sum_{j=1}^n \big(h_{jj}R u_J, u_J\big) -\sum_{H\in\I_{q-1}}\sum_{j,k=1}^n\big(h_{jk}R u_{jH}, u_{kH}\big)\Big \} \\ \geq \kappa \operatorname{Re}\Big\{\sum_{J\in\I_q}\sum_{j=1}^{n} \big(h_{jj} u_J, u_J\big) -\sum_{H\in\I_{q-1}}\sum_{j,k=1}^n \big(h_{jk} u_{jH}, u_{kH}\big) \Big \} -C \|u\|_{L^2}^2 .\end{gathered}$$ Note that $(h_{jk})$ may be a matrix-valued function in $z$ but may not depend on $\xi$. Microlocal analysis – the setup {#s3} ------------------------------- To bound the terms from Theorem \[kmh\] that involve $T$, we continue to work on smooth forms that are supported in a small neighborhood $U\subset M$. Our approach is microlocal and we adopt the familiar setup introduced by Kohn [@Koh86; @Koh02]. See also Nicoara [@Nic06] and Raich [@Rai10]. Denote the coordinates $\R^{2n+1}$ by $x=(x',x_{2n+1})=(x_1,...x_{2n},x_{2n+1})$ with the origin at some $x_0\in U$. We can arrange the coordinates so that if $z_j=x_j+\sqrt{-1}x_{j+n}$ for $j=1,\dots,n$, then $L_j|_{z_0}=\frac{\di}{\di z_j}|_{z_0}$ for $j=1,\dots,n$, and $T=-\sqrt{-1}\frac{\di}{\di x_{2n+1}}$. Let $\xi=$ $(\xi_1,...,\xi_{2n+1})$ $=(\xi',\xi_{2n+1})$ be the dual coordinates to $x$ in Fourier space. Let $\mathcal C^+,\mathcal C^-,\mathcal C^0$ be a covering of $\R^{2n+1}$ so that $$\begin{aligned} \begin{split} \mathcal C^+=&\big\{\xi : \xi_{2n+1}> \frac{1}{4} |\xi'| \big\}\cap \{\xi : |\xi|\ge 1\};\\ \mathcal C^-=&\{\xi : \xi\in \mathcal C^+\};\\ \mathcal C^0=&\{\xi : | \xi_{2n+1}|<\frac{3}{4}|\xi'| \}\cup \{\xi :|\xi|<3\}. \end{split}\end{aligned}$$ For the remainder of this paper, let $\psi$ be a smooth function so that $\psi \equiv 1$ on $\{\xi : \xi_{2n+1} > \frac 13 |\xi'|\} \cap \{\xi : |\xi| \geq 2\}$ and $\operatorname{supp}\psi \subset {\mathcal C}^+$. It follows from the definitions of ${\mathcal C}^+$, ${\mathcal C}^0$, and $\psi$, that $\operatorname{supp}d\psi \subset {\mathcal C}^0$. Define $$\psi^+(\xi):=\psi(\xi), \qquad \psi^-(\xi):=\psi(-\xi), \qquad \psi^0(\xi):=\sqrt{1-(\psi^+(\xi))^2-(\psi^-(\xi))^2}.$$ Let $\tilde\psi^0$ be a smooth function that dominates $\psi^0$ in the sense that $\operatorname{supp}\tilde\psi^0\subset \mathcal C^0$ and $\tilde\psi^0=1$ on a neighborhood of $\operatorname{supp}\psi^0\cup\operatorname{supp}(d\psi^+) \cup \operatorname{supp}(d\psi^-)$. Associated to the smooth function $\psi$ is a pseudodifferential operator $\Psi$ whose symbol $\sigma(\Psi)=\psi$. This means that if $\varphi\in C^\infty_c(U)$, then $$\widehat{\Psi \varphi}(\xi)=\psi(\xi)\hat{\varphi}(\xi),$$ where $\hat{}$ denotes the Fourier transform. The operators $\Psi^+$, $\Psi^-$, $\Psi^0$, and $\tilde\Psi^0$ are defined analogously, with symbols $\psi^+$, $\psi^-$, $\psi^0$, and $\tilde\psi^0$, respectively. By construction, $(\psi^+)^2 + (\psi^-)^2 + (\psi^0)^2 =1$, from which it follows immediately that $(\Psi^+)^*\Psi^+ + (\Psi^-)^*\Psi^- + (\Psi^0)^*\Psi^0 = Id$, the identity operator. For the proof of Theorem \[thm:L\^2-theory\], we will need dilated versions $\Psi^\bullet$ and $\tilde\Psi^0$ where the superscript $\bullet$ means $+$, $-$, or $0$. Let $A\ge1$ (chosen later). Let $\Psi^\bullet_A$ and $\tilde\Psi^0_A$ be the pseudodifferential operators with symbol $\psi^\bullet_A(\xi) = \psi^\bullet(\xi/A)$ and $\tilde\psi^\bullet_A(\xi) = \tilde\psi^\bullet(\xi/A)$, respectively. We say that a cutoff function $\zeta$ dominates a cutoff function $\zeta'$ and denote by $\zeta'\prec\zeta$ if $\zeta \equiv 1$ on $\operatorname{supp}\zeta'$. We write the next several results for a generic $A$ but will use $A:=t$ in the proof of Theorem \[thm:global regularity\] and $A:=A_{\epsilon}$ in the proof of Theorem \[thm:CRpsh (0,1)\]. The next result follows immediately from Proposition \[prop:Garding 1\] and the arguments of Lemma 4.6 and Lemma 4.7 in [@Rai10]. Let $M$ be a pseudoconvex CR manifold and $U$ be a local patch of $M$. For $\phi\in C^\infty(M)$, $u\in C^\infty_{0,q}(M)$, and cutoff functions $\zeta\prec\tilde\zeta\prec \zeta'$ on $U$, we have $$\begin{aligned} \label{Psi+} \begin{split} (i) \qquad\T{\normalfont Re}\Big((d\gamma\lrcorner T\tilde\zeta \Psi^+_A\zeta u,\tilde\zeta \Psi^+_A\zeta u)_\phi\Big) \ge &A(d\gamma\lrcorner \tilde\zeta \Psi^+_A\zeta u,\tilde\zeta \Psi^+_A\zeta u)_\phi\\ -&c{\|{\tilde\zeta \Psi^+_A\zeta u}\|}_{L^2_\phi}^2-c_{A,\phi}{\|{ \zeta'\tilde\Psi^0_A\zeta u}\|}_{L^2}^2 \end{split} \end{aligned}$$ for any $q=1,\dots, n$; and $$\begin{aligned} \label{Psi-} \begin{split} (ii) \qquad\T{\normalfont Re}&\Big((d\gamma\lrcorner T\tilde\zeta \Psi^-_A\zeta u,\tilde\zeta \Psi^-_A\zeta u)_{-\phi}-(\operatorname{Tr}(d\gamma) T\tilde\zeta \Psi^-_A\zeta u,\tilde\zeta \Psi^-_A\zeta u)_{-\phi}\Big)\\ \ge &A\Big(\operatorname{Tr}(d\gamma) \tilde\zeta \Psi^-_A\zeta u,\tilde\zeta \Psi^-_A\zeta u)_{-\phi}-(d\gamma\lrcorner \tilde\zeta \Psi^-_A\zeta u,\tilde\zeta \Psi^-_A\zeta u)_{-\phi}\Big)\\ & -c{\|{\tilde\zeta \Psi^-_A\zeta u}\|}^2_{L^2_{-\phi}}-c_{A,\phi}{\|{ \zeta'\tilde\Psi^0_A\zeta u}\|}_{L^2}^2 \end{split} \end{aligned}$$ for any $q=0,1,\dots, n-1$. Here, $c$ (resp. $c_{A,\phi}$) is a positive constant independent (resp. dependent) of $\phi$. In combination with the Kohn-Morrey-Hömander inequality, Lemma \[l3.2\] yields \[c3.3\] Let $M$ be a pseudoconvex CR manifold and $U$ be a local patch of $M$. For $\phi\in C^\infty(M)$, $u\in C^\infty_{0,q}(M)$, and cutoff function $\zeta\prec\tilde\zeta\prec \zeta'$ on $U$, then we have $$\begin{aligned} \Label{nova2} \begin{split} c_{A,\phi}{\|{\zeta'\tilde\Psi_A^0\zeta u}\|}_{L^2}^2+&c\left({\|{\tilde \zeta\Psi_A^+\zeta u}\|}_{L^2_\phi}^2+{\|{\bar{\partial}_b \tilde \zeta\Psi_A^+\zeta u}\|}_{L^2_\phi}^2+{\|{\bar{\partial}^*_{b,\phi}\tilde \zeta\Psi_A^+\zeta u}\|}_{L^2_\phi}^2\right)\\ \ge &\left((\L_\phi+Ad\gamma)\lrcorner \tilde \zeta\Psi_A^+\zeta u,\tilde \zeta\Psi_A^+\zeta u\right)_\phi \end{split} \end{aligned}$$ for any $n=1,\dots, n$; and $$\begin{aligned} \Label{nova3} \begin{split} c_{A,\phi}{\|{\zeta'\tilde\Psi_A^0\zeta u}\|}_{L^2}^2+&c\left({\|{\tilde \zeta\Psi_A^-\zeta u}\|}^2_{L^2_{-\phi}}+{\|{\bar{\partial}_b \tilde \zeta\Psi_A^-\zeta u}\|}^2_{L^2_{-\phi}}+{\|{\bar{\partial}^*_{b,-\phi}\tilde \zeta\Psi_A^-\zeta u}\|}^2_{L^2_{-\phi}}\right)\\ \ge &\Big([\operatorname{Tr}(\L_\phi)+A\operatorname{Tr}(d\gamma)]\times \tilde\zeta \Psi^-_A\zeta u,\tilde\zeta \Psi^-_A\zeta u\Big)_{-\phi}\\ &- \left([\L_\phi+Ad\gamma]\lrcorner \tilde \zeta\Psi_A^{-}\zeta u,\tilde \zeta\Psi_A^-\zeta u\right)_{-\phi} \end{split} \end{aligned}$$ for any $q=0,\dots,n-1$. The $L^2$-Sobolev Theory for ${\Box_b}$ and the proof of Theorem \[thm:L\^2-theory\] {#sec:proving L^2 theory} ==================================================================================== The existence to the solution of the $\Box_b$ when $n\geq 2$ and $1\leq q \leq n-1$. ------------------------------------------------------------------------------------ We now assume that $M$ is endowed a with a smooth function $\lambda$ that is strictly CR-plurisubharmonic on $(0,q_0)$-forms whose defining inequality is given by (\[CRpluri\]). The function $\lambda$ is $q_0$-compatible in the language of Harrington and Raich [@HaRa11], and we can follow their argument nearly verbatim to establish a weighted $L^2$-theory (compare with the proof of [@HaRa11 Theorem 1.2]). Observe that if the inequality (\[CRpluri\]) holds for $q_0$, then it holds for any $q\ge q_0$. Additionally, $$\begin{aligned} \label{new2}\begin{split} \la[\operatorname{Tr}(\L_{\lambda})+\operatorname{Tr}(d\gamma)] v,\bar v\ra -\la(\L_{\lambda}+d\gamma)\lrcorner v, \bar v\ra \ge a|v|^2\end{split}\end{aligned}$$ for all $(0,q)$-forms $v$ with $q\le n-q_0$. For each $t\ge 1$, we use Corollary \[c3.3\] with $\phi=t\lambda$ to obtain $$\begin{aligned} \Label{local1} \begin{split} c_{t}{\|{\zeta'\tilde\Psi_t^0\zeta u}\|}_{L^2}^2+&c\left({\|{\tilde \zeta\Psi_t^+\zeta u}\|}^2_{L^2_{t\lambda}}+{\|{\bar{\partial}_b \tilde \zeta\Psi_t^+\zeta u}\|}^2_{L^2_{t\lambda}}+{\|{\bar{\partial}^*_{b,t\lambda}\tilde \zeta\Psi_t^+\zeta u}\|}^2_{L^2_{t\lambda}}\right)\\ \ge &\left((\L_{t\lambda}+td\gamma)\lrcorner \tilde \zeta\Psi_t^+\zeta u,\tilde \zeta\Psi_t^+\zeta u\right)_{t\lambda}\\ \ge & at{\|{\tilde \zeta\Psi_t^+\zeta u}\|}^2_{L^2_{t\lambda}} \end{split} \end{aligned}$$ for any $u\in C^\infty_{0,q}(M)$ with $q\ge q_0$. Analogously, we also have $$\begin{aligned} \Label{local2} \begin{split} c_{t}{\|{\zeta'\tilde\Psi_t^0\zeta u}\|}_{L^2}^2+& c\left({\|{\tilde \zeta\Psi_t^-\zeta u}\|}^2_{L^2_{-t\lambda}}+ {\|{\bar{\partial}_b \tilde \zeta\Psi_t^-\zeta u}\|}^2_{L^2_{-t\lambda}}+{\|{\bar{\partial}^*_{b,-t\lambda}\tilde \zeta\Psi_t^-\zeta u}\|}^2_{L^2_{-t\lambda}}\right)\\ \ge &\, a t{\|{\tilde \zeta\Psi_t^-\zeta u}\|}^2_{L^2_{-t\lambda}} \end{split} \end{aligned}$$ for any $u\in C^\infty_{0,q}(M)$ with $ q\le n-q_0$. This estimate holds for cutoff functions $\zeta,\tilde\zeta,\zeta'$ having compact support on a local patch $U$ of $M$. In order to prove a global estimate, we let $\{U_\nu\}$ be a cover of $M$ and $\{\zeta_\nu\}$ be a partition of unity subordinate to $\{U_\nu\}$. Supported on each $U_\nu$ are the pseudodifferential operators $\Psi^{\cdot,\nu}_t$ and $\tilde\Psi^{0,\nu}_t$ where $\cdot$ represents $+$, $-$, or $0$. For each $\zeta_\nu$, let $\tilde\zeta_\nu$ be a cutoff function that dominates $\zeta_\nu$. We define an inner product and norm that are well-suited to estimates using microlocal analysis. Set $$\begin{gathered} \la| u,v |\ra_{t{\lambda}} = \sum_\nu \Big[ ( \tilde\zeta_\nu\Psi^+_t\zeta_\nu u^\nu, \tilde\zeta_\nu\Psi^+_t\zeta_\nu v^\nu )_{L^2_{t\lambda}} + (\tilde\zeta_\nu \Psi^0_t \zeta_\nu u^\nu ,\tilde\zeta_\nu \Psi^0_t \zeta_\nu v^\nu ) + (\tilde\zeta_\nu \Psi^-_t \zeta_\nu u^\nu, \tilde\zeta_\nu \Psi^-_t \zeta_\nu v^\nu )_{-t\lambda} \Big]\end{gathered}$$ and $$\la| u |\ra_{t\lambda} ^2 = \sum_{\nu} \Big[ \| \tilde\zeta_\nu \Psi^+_t \zeta_\nu u^\nu \|_{L^2_{t\lambda}}^2 + \|\tilde\zeta_\nu \Psi^0_t \zeta_\nu u^\nu \|_{L^2}^2 + \|\tilde\zeta_\nu \Psi^-_t \zeta_\nu u^\nu \|_{L^2_{-t\lambda}}^2 \Big],$$ where $u^\nu$ is the form $u$ expressed in the local coordinates on $U_\nu$. The superscript $\nu$ will often be omitted. We denote the adjoint of ${\bar\partial_b}$ with respect to this norm by ${\bar\partial_b}^{*,t}$ and the associated quadratic form $$Q_{b,t\lambda}\la| u,v|\ra = \la| {\bar\partial}u, {\bar\partial}v |\ra_{t\lambda} + \la| {\bar\partial_b}^{*,t} u, {\bar\partial_b}^{*,t} v |\ra_{t\lambda}.$$ The space of $t\lambda$-harmonic forms ${\mathcal H}^q_{t\lambda}(M)$ is $${\mathcal H}^q_{t\lambda}(M) = \{ u \in L^2_{0,q}(M) : Q_{b,t\lambda}\la|u,u|\ra =0\}.$$ By using the pseudoconvex-oriented hypothesis, the estimates - for $U_\nu$, and the well-known elliptic estimate for $\Psi^0$, it follows that there exists $T_0>0$ such that for any $t\ge t_0$ the estimate $$\begin{aligned} \label{L2} \la| u |\ra_{t\lambda} ^2 \le \frac{c}{t}Q_{b,t\lambda}\la|u,u|\ra+c_t{\|{u}\|}_{H^{-1}}^2\end{aligned}$$ holds for all $u\in C^\infty_{0,q}(M)$ with $q_0\le q\le n-q_0$. See [@Nic06; @HaRa11] for details. For a form $u$ on $M$, the Sobolev norm of order $s$ is given by the following: $$\|u\|_{H^s}^2 = \sum_\nu \|\tilde\zeta_\nu\Lambda^s\zeta_\nu {\varphi}^\nu \|_{L^2}^2$$ where $\Lambda$ is defined to be the pseudodifferential operator with symbol $(1+|\xi|^2)^{1/2}$. As in [@HaRa11], we can also bring the estimate to higher order Sobolev indices: for each $s\ge0$, there exists $T_s>0$ such that for any $t\ge T_s$, $$\begin{aligned} \label{Hs} \la|\Lambda^s u |\ra_{t\lambda} ^2 \le \frac{c}{t}\left(\la|\Lambda^s\dib_bu|\ra_{t\lambda}^2+\la|\Lambda^s\dib_b^{*,t}u|\ra_{t\lambda}^2\right)+c_t{\|{u}\|}_{H^{s-1}}^2 \end{aligned}$$ holds for all $u\in C^\infty_{0,q}(M)$ with $q_0\le q\le n-q_0$. In [@Nic06], it is shown that there exist constants $c_t$ and $C_t$ so that $$\label{eqn:norm equivalence} c_t \| u\|_{L^2}^2 \leq \la| u|\ra_{t\lambda}^2 \leq C_t \|u\|_{L^2}^2$$ where $c_t$ and $C_t$ depend on $\max_{M}|\lambda|$. We thus have closed range estimates for ${\bar\partial_b}:H^s_{0,q}(M)\to H^s_{0,q+1}(M)$ and ${\bar\partial_b}^{*,t}:H^s_{0,q}(M)\to H^s_{0,q-1}(M)$. The following theorem now follows from the arguments of [@HaRa11]. \[thm:main theorem for weighted spaces\] Let $M^{2n+1}$ be an abstract CR manifold that is pseudoconvex-oriented and admits a smooth function $\lambda$ that is strictly CR plurisubharmonic on $(0,q_0)$-forms for some $1 \leq q_0 \leq \frac n2$. Then for all $q_0 \leq q \leq n-q_0$ and $s\geq 0$, there exists $T_s\geq 0$ so that the following hold: 1. The operators ${\bar\partial_b}: L^2_{0,q}(M)\to L^2_{0,q+1}(M)$ and ${\bar\partial_b}: L^2_{0,q-1}(M)\to L^2_{0,q}(M)$ have closed range with respect to $\la|\cdot|\ra_{t{\lambda}}$. Additionally, for any $s>0$ if $t\geq T_s$, then ${\bar\partial_b}: H^s_{0,q}(M)\to H^s_{0,q+1}(M)$ and ${\bar\partial_b}: H^s_{0,q-1}(M)\to H^s_{0,q}(M)$ have closed range with respect to $\la|\Lambda^s\cdot|\ra_{t{\lambda}}$; 2. The operators ${\bar\partial_b}^{*,t}: L^2_{0,q+1}(M)\to L^2_{0,q}(M)$ and ${\bar\partial_b}^{*,t}: L^2_{0,q}(M)\to L^2_{0,q-1}(M)$ have closed range with respect to $\la|\cdot|\ra_{t{\lambda}}$. Additionally, if $t\geq T_s$, then ${\bar\partial_b}^{*,t}: H^s_{0,q+1}(M)\to H^s_{0,q}(M)$ and ${\bar\partial_b}^{*,t}: H^s_{0,q}(M)\to H^s_{0,q-1}(M)$ have closed range with respect to $\la|\Lambda^s\cdot|\ra_{t{\lambda}}$; 3. The Kohn Laplacian defined by ${\Box_b}^t = {\bar\partial_b}{\bar\partial_b}^{*,t} + {\bar\partial_b}^{*,t}{\bar\partial_b}$ has closed range on $L^2_{0,q}(M)$ (with respect to $\la|\cdot|\ra_{t{\lambda}}$) and also on $H^s_{0,q}(M)$ (with respect to $\la|\Lambda^s\cdot|\ra_{t{\lambda}}$) if $t\geq T_s$; 4. The space of harmonic forms ${\mathcal H}^q_t(M)$, defined to be the $(0,q)$-forms annihilated by ${\bar\partial_b}$ and ${\bar\partial_b}^{*,t}$ is finite dimensional; 5. The complex Green operator $G_{q,t}$ is continuous on $L^2_{0,q}(M)$ (with respect to $\la|\cdot|\ra_{t{\lambda}}$) and also on $H^s_{0,q}(M)$ (with respect to $\la|\Lambda^s\cdot|\ra_{t{\lambda}}$) if $t\geq T_s$; 6. The canonical solution operators for ${\bar\partial_b}$, ${\bar\partial_b}^{*,t} G_{q,t}:L^2_{0,q}(M)\to L^2_{0,q-1}(M)$ and $G_{q,t}{\bar\partial_b}^{*,t} : L^2_{0,q+1}(M)\to L^2_{0,q}(M)$ are continuous (with respect to $\la|\cdot|\ra_{t{\lambda}}$). Additionally,\ ${\bar\partial_b}^{*,t} G_{q,t}:H^s_{0,q}(M)\to H^s_{0,q-1}(M)$ and $G_{q,t}{\bar\partial_b}^{*,t} : H^s_{0,q+1}(M)\to H^s_{0,q}(M)$ are continuous (with respect to $\la|\Lambda^s\cdot|\ra_{t{\lambda}}$) if $t\geq T_s$. 7. The canonical solution operators for ${\bar\partial_b}^{*,t}$, ${\bar\partial_b}G_{q,t}:L^2_{0,q}(M)\to L^2_{0,q+1}(M)$ and $G_{q,t}{\bar\partial_b}: L^2_{0,q-1}(M)\to L^2_{0,q}(M)$ are continuous (with respect to $\la|\cdot|\ra_{t{\lambda}}$). Additionally,\ ${\bar\partial_b}G_{q,t}:H^s_{0,q}(M)\to H^s_{0,q+1}(M)$ and $G_{q,t}{\bar\partial_b}: H^s_{0,q-1}(M)\to H^s_{0,q}(M)$ are continuous (with respect to $\la|\Lambda^s\cdot|\ra_{t{\lambda}}$) if $t\geq T_s$. 8. The Szegö projections $S_{q,t} = I - {\bar\partial_b}^{*,t}{\bar\partial_b}G_{q,t}$ and $S_{q-1,t} = I - {\bar\partial_b}^{*,t} G_{q,t} {\bar\partial_b}$ are continuous on $L^2_{0,q}(M)$ and $L^2_{0,q-1}(M)$, respectively and with respect to $\la|\cdot|\ra_{t{\lambda}}$. Additionally, if $t\geq T_s$, then $S_{q,t}$ and $S_{q-1,t}$ are continuous on $H^s_{0,q}$ and $H^s_{0,q-1}$ (with respect to $\la|\Lambda^s\cdot|\ra_{t{\lambda}}$), respectively. 9. If $\tilde q = q$ or $q+1$ and $\alpha\in H^s_{0,q}(M)$ so that ${\bar\partial_b}\alpha =0$ and $\alpha \perp {\mathcal H}^{\tilde q}_t$ (with respect to $\la|\cdot|\ra_{t{\lambda}}$), then there exists $u\in H^s_{0,\tilde q-1}(M)$ so that $${\bar\partial_b}u = \alpha;$$ 10. If $\tilde q = q$ or $q+1$ and $\alpha\in C^\infty_{0,\tilde q}(M)$ satisfies ${\bar\partial_b}\alpha=0$ and $\alpha \perp {\mathcal H}^{\tilde q}_t$ (with respect to $\la| \cdot, \cdot |\ra_{t{\lambda}}$), then there exists $u\in C^\infty_{0,\tilde q-1}(M)$ so that $${\bar\partial_b}u = \alpha.$$ Turning to the proof of Theorem \[thm:L\^2-theory\], a consequence of Theorem \[thm:main theorem for weighted spaces\] and is that ${\bar\partial_b}:L^2_{0,\tilde q}(M)\to L^2_{0,\tilde q+1}(M)$ has closed range, $\tilde q = q$ or $q-1$. Functional analysis shows that the $L^2$-adjoint operators ${\bar\partial^*_b}:L^2_{0,\tilde q+1}(M) \to L^2_{0,\tilde q}(M)$ also has closed range [@Hor65 Theorem 1.1.1]. Additionally, the finite dimensionality of $\mathcal H^q_t(M)$ combined with the closed range of ${\bar\partial_b}$ on $L^2_{0,\tilde q}(M)$, $\tilde q = q,q-1$ implies the finite dimensionality of the unweighted space of harmonic forms $\mathcal H_{0,q}(M)$. While this fact is likely well-known, Straube and Raich give a proof in [@RaSt08 p.772]. The cases $q=0$ and $q=n$ (when $n\geq 2$) follow easily from the formulas $G_0 = {\bar\partial^*_b}G_1^2 {\bar\partial_b}$ and $G_n = {\bar\partial}G_{n-1}^2 {\bar\partial^*_b}$ and the already proven parts of the theorem. This concludes the proof of Theorem \[thm:L\^2-theory\]. Global hypoellipticity of $\Box_b$ and the Proof of Theorem \[thm:CRpsh (0,1)\] {#sec:hypoellipticity} =============================================================================== A weak compactness estimate for $\Box_b$ {#s6} ---------------------------------------- In this section, we assume: i) for any ${\epsilon}>0$ there exist a vector $T_{\epsilon}$ transversal to $T^{1,0}M\oplus T^{0,1}M$ such that $0<c_1<\gamma(T_{\epsilon})<c_2$ uniformly in ${\epsilon}$ and ii) there exists a covering $\{U_\eta\}$ by local patches such that on each $U:=U_\eta$ there exists $\lambda_{\epsilon}:=\lambda^\eta_{\epsilon}$ so that $\lambda_{\epsilon}$ is uniformly bounded and $$\la(\L_{\lambda_{\epsilon}}+A_{\epsilon}d\gamma)\lrcorner u,\bar u\ra\ge \frac{1}{{\epsilon}}|\alpha_{\epsilon}|^2|u|^2$$ holds on $U$ for all $(0,q_0)$-forms $u\in C^\infty_{0,q}(M)$. Here, $\alpha_{\epsilon}=-\{\T{Lie}\}_{T_{\epsilon}}(\gamma)$. In Section \[sec:proving L\^2 theory\], we proved estimates for weighted operators in Sobolev spaces. Now, under our stronger assumption of the existence of $T_{\epsilon}$, we will prove estimates in Sobolev spaces for the unweighted system $(\dib_b,\dib_b^*)$. In order to do that, we use the composition weight $\phi=\chi(\lambda_{\epsilon})$ for a smooth function $\chi:\R\to \R$ chosen later but satisfying $\dot\chi,\ddot{\chi}> 0$. By the definition of the Levi form , it follows that $$\L_{\chi(\lambda_{\epsilon})}=\dot\chi \L_{\lambda_{\epsilon}}+\ddot\chi \di_b\lambda_{\epsilon}{\wedge}\dib_b\lambda_{\epsilon},$$ and hence $$\begin{aligned} \label{notice1}\begin{split} \la (\L_{\chi(\lambda_{\epsilon})}+\dot\chi A_{\epsilon}d\gamma)\lrcorner u,\bar u\ra =&\dot\chi\la (\L_{\lambda_{\epsilon}}+A_{\epsilon}d\gamma)\lrcorner u,\bar u\ra +\ddot\chi|\di_b\lambda_{\epsilon}\lrcorner u|^2\\ \ge&\frac{1}{{\epsilon}}|\sqrt{\dot{\chi}}\alpha_{\epsilon}|^2|u|^2+\ddot\chi|\di_b\lambda_{\epsilon}\lrcorner u|^2. \end{split}\end{aligned}$$ We also notice that $$\begin{aligned} \label{notice2}|\dib^*_{b,\chi(\phi^{\epsilon})}u|^2 \le 2 |\dib_b^*u|^2+2\dot\chi^2|\di_b\lambda_{\epsilon}\lrcorner u|^2.\end{aligned}$$ Now we use Corollary, \[c3.3\](i) for $\phi:=\chi(\lambda_{\epsilon})$, $A:=\dot\chi(\lambda_{\epsilon}) A_{\epsilon}$ and plug in and into to obtain $$\begin{aligned} \label{400} \begin{split} c_{{\epsilon}}{\|{\zeta'\tilde\Psi_{A}^0\zeta u}\|}_{L^2}^2+&c\left({\|{\tilde \zeta\Psi_{A}^+\zeta u}\|}_{L^2_\phi}^2+2{\|{\dot\chi\di_b\lambda_{\epsilon}\lrcorner \tilde\zeta\Psi_{A}^+\zeta u}\|}_{L^2_\phi}^2+{\|{\bar{\partial}_b \tilde \zeta\Psi_{A}^+\zeta u}\|}_{L^2_\phi}^2+{\|{\bar{\partial}^*_{b,\phi}\tilde \zeta\Psi_{A}^+\zeta u}\|}_{L^2_\phi}^2\right)\\ &\ge \frac{1}{{\epsilon}}{\|{\sqrt{\dot\chi}|\alpha_{\epsilon}|\tilde \zeta\Psi_{A}^+\zeta u}\|}_{L^2_\phi}^2+{\|{\sqrt{\ddot{\chi}}\di_b\lambda_{\epsilon}\lrcorner \tilde \zeta\Psi_{A}^+\zeta u}\|}_{L^2_\phi}^2 \end{split} \end{aligned}$$ for any $u\in C^\infty_{0,q}(M)$ with $q=q_0,\dots,n$. The function $\lambda_{\epsilon}$ is uniformly bounded, so we may assume that $|\lambda_{\epsilon}|\le 1$. Thus, if we choose $\chi(t)=\frac{1}{2c}e^{t-1}$ then $\ddot\chi(t)\ge 2c \dot\chi^2(t)$ for $|t|\le 1$, and hence we can absorb $2c{\|{\dot\chi\di_b\lambda_{\epsilon}\lrcorner \tilde\zeta\Psi_{A_{\epsilon}}^+\zeta u}\|}^2_{\chi(\lambda_{\epsilon})}$ by the RHS. By this choice of $\chi$, we also get a uniform bound for $e^{-\chi}$ and $\dot{\chi}\ge \frac{1}{2e^2c}$. Consequently, we can remove the weight from both sides of and obtain $$\begin{aligned} \label{400b} \begin{split} c_{{\epsilon}}{\|{\zeta'\tilde\Psi_{A}^0\zeta u}\|}_{L^2}^2+&c\left({\|{\tilde \zeta\Psi_{A}^+\zeta u}\|}_{L^2}^2+{\|{\bar{\partial}_b \tilde \zeta\Psi_{A}^+\zeta u}\|}_{L^2}^2+{\|{\bar{\partial}^*_{b}\tilde \zeta\Psi_{A}^+\zeta u}\|}_{L^2}^2\right)\\ &\ge \frac{1}{{\epsilon}}{\|{|\alpha_{\epsilon}|\tilde \zeta\Psi_{A}^+\zeta u}\|}_{L^2}^2 \end{split} \end{aligned}$$ for any $u\in C^\infty_{0,q}(M)$ with $q\ge q_0$.\ To bound the $\Psi^-$ terms, we cannot use an analogous argument. The problem is that there is no $|\di_b\lambda_{\epsilon}\lrcorner u|^2$ term to absorb unwanted terms. Indeed, $$\begin{aligned} \label{no211}\begin{split} &\la [\operatorname{Tr}(\L_{\chi(\lambda_{\epsilon})})+\dot\chi A\operatorname{Tr}(d\gamma)]\times u,\bar u\ra-\la (\L_{\chi(\lambda_{\epsilon})}+\dot\chi A_{\epsilon}d\gamma)\lrcorner u,\bar u\ra \\ &=\dot\chi\left(\la [\operatorname{Tr}(\L_{\lambda_{\epsilon}})+A_{\epsilon}\operatorname{Tr}(d\gamma)]\times u,\bar u\ra-\la (\L_{\lambda_{\epsilon}}+A_{\epsilon}d\gamma)\lrcorner u,\bar u\ra\right)\\ &= \ddot\chi\left(\la\operatorname{Tr}(\di_b\lambda_{\epsilon}{\wedge}\dib_b\lambda_{\epsilon})\times u,\bar u\ra -|\di_b\lambda_{\epsilon}\lrcorner u|^2\right)\\ &\ge \frac{1}{{\epsilon}}|\sqrt{\dot{\chi}}\alpha_{\epsilon}|^2|u|^2+\ddot\chi|\dib_b\lambda_{\epsilon}{\wedge}u|^2; \end{split}\end{aligned}$$ and the $|\dib_b\lambda_{\epsilon}{\wedge}u|$ cannot absorb $|\di_b\lambda_{\epsilon}\lrcorner u|$ in general. Instead, we can obtain the estimate for $\Psi^-$ by a Hodge-$*$ argument (see [@Koh02; @Kha16b]). Indeed, using the ideas in [@Kha16b Theorem 5], it follows that is equivalent to $$\begin{aligned} \label{401} \begin{split} c_{{\epsilon}}{\|{\zeta'\tilde\Psi_{A}^0\zeta u}\|}_{L^2}^2+&c\left({\|{\tilde \zeta\Psi_{A}^-\zeta u}\|}_{L^2}^2+{\|{\bar{\partial}_b \tilde \zeta\Psi_{A}^-\zeta u}\|}_{L^2}^2+{\|{\bar{\partial}^*_{b}\tilde \zeta\Psi_{A}^-\zeta u}\|}_{L^2}^2\right)\\ &\ge \frac{1}{{\epsilon}}{\|{|\alpha_{\epsilon}|\tilde \zeta\Psi_{A}^-\zeta u}\|}_{L^2}^2 \end{split} \end{aligned}$$ for any $u\in C^\infty_{0,q}(M)$ with $q\le n-q_0$.\ To obtain our global estimates, we use and on each local patch $U_\eta$ of the covering $\{U_\eta\}$, together with the elliptic estimate of the $\Psi^0$-terms and $$\sum_{\eta}\left({\|{[\bar{\partial}_b, \tilde \zeta\Psi_{A}^\pm\zeta] u}\|}_{L^2}^2+{\|{[\bar{\partial}_b, \tilde \zeta\Psi_{A}^\pm\zeta ]u}\|}_{L^2}^2\right) \le c\left({\|{u}\|}_{L^2}^2+{\|{\dib_bu}\|}_{L^2}^2+{\|{\dib_b^*u}\|}_{L^2}^2\right)+c_{\epsilon}{\|{u}\|}^2_{H^{-1}}.$$ We then see $$\begin{aligned} \begin{split} & \frac{1}{{\epsilon}}{\|{|\alpha_{\epsilon}|u}\|}_{L^2}^2\le c\left({\|{\dib_bu}\|}_{L^2}^2+{\|{\dib_b^*u}\|}_{L^2}^2+{\|{u}\|}_{L^2}^2\right)+c_{\epsilon}{\|{u}\|}^2_{H^{-1}} \end{split} \end{aligned}$$ for any $u\in C^\infty_{0,q}(M)$ with $q_0\le q\le n-q_0$. If $M$ admits a strictly CR-plurisubharmonic function on $(0,q_0)$-forms, we have already proved that $${\|{u}\|}_{L^2}^2\le c\left({\|{\dib_bu}\|}_{L^2}^2+{\|{\dib_b^{ *}u}\|}_{L^2}^2\right)+c'{\|{u}\|}^2_{H^{-1}}$$ for any $u\in C^\infty_{0,q}(M)$ with $q_0\le q\le n-q_0$. Thus, we have the following theorem. Assume that the hypothesis of Theorem \[thm:global regularity\] holds. Then for any $\epsilon>0$ there exist a vector $T_\epsilon$ and a constants $c_\epsilon>0$ such that $c_1\le \gamma(T_\epsilon)\le c_2$ uniformly in $\epsilon$ and $$\begin{aligned} \label{Theta+} \frac{1}{{\epsilon}}\big\||\alpha_{\epsilon}|u\big\|_{L^2}^2+{\|{u}\|}_{L^2}^2\le c \left({\|{\dib_b u}\|}_{L^2}^2+{\|{\dib_b^*u}\|}_{L^2}^2\right)+c_\epsilon {\|{u}\|}^2_{H^{-1}} \end{aligned}$$ holds for all $u\in C^\infty_{0,q}(M)$ with $q_0\le q\le n-q_0$ and $\alpha_{\epsilon}=-\{\T{Lie}\}_{T_{\epsilon}}(\gamma)$. Global hypoellipticity for $\Box_b$ ----------------------------------- Let $s\geq 0$ be an integer and $D^s$ denote a differential operator of order $s$, $\nabla_b=(\nabla_b^{1,0},\nabla_b^{0,1})$ where $\nabla_b^{1,0}=(L_1,\dots, L_n)$ and $\nabla_b^{0,1}=(\bar L_1,\dots,\bar L_n)$. By the Kohn-Morrey-Hörmander inequality, for $s\ge 1$, $\epsilon>0$ there exists $c_{\epsilon,s}>0$ such that $$\begin{aligned} \label{421} \begin{split}{\|{\nabla_bu}\|}_{H^{s-1}}^2\le& c_s\left({\|{\nabla_b D^{s-1}u}\|}_{L^2}^2+{\|{u}\|}_{H^{s-1}}^2\right)\\ \le& c_{s}\left( {\|{\dib_b D^{s-1}u}\|}_{L^2}^2+{\|{\dib_b^* D^{s-1}u}\|}_{L^2}^2+{\|{u}\|}^2_{H^{s-1}}+{\|{u}\|}_{H^s}{\|{u}\|}_{H^{s-1}}\right)\\ \le& c_{s}\left({\|{\dib_bu}\|}^2_{H^{s-1}}+{\|{\dib_b^*u}\|}^2_{H^{s-1}}\right)+c_{s,\epsilon}{\|{u}\|}_{H^{s-1}}^2+{\epsilon}{\|{u}\|}^2_{H^s}, \end{split}\end{aligned}$$ Let $T_{\epsilon}$ be a global, purely imaginary vector field transversal to $T^{1,0}M\oplus T^{0,1}M$ such that $0<c_1\le\gamma(T_{\epsilon})\le c_2$. There exist a function $b_\epsilon$ and a vector field $X_\epsilon\in T^{1,0}M\oplus T^{0,1}M$ such that $0<c_1\le b_{\epsilon}\le c_2$ and $$T=b_{\epsilon}T_{\epsilon}+X_{\epsilon}$$ This implies that for any $s\ge 1$, there exist constants $c>0$ and $c_{\epsilon,s}>0$ such that $$\begin{aligned} \label{423} {\|{T^su}\|}_{L^2}^2\le c{\|{T^s_{\epsilon}u}\|}_{L^2}^2+c_{{\epsilon}, s}\left({\|{\nabla_b u}\|}^2_{H^{s-1}}+{\|{u}\|}_{H^{s-1}}^2\right) + {\epsilon}\|u\|_s^2\end{aligned}$$ From , and ${\|{u}\|}^2_{H^s}\le c_s{\|{\nabla_bu}\|}_{L^2}^2+c{\|{T^su}\|}_{L^2}^2$, we get the reduction from $D^s$ to $T^s_\epsilon$ by the inequality $$\begin{aligned} \label{424} {\|{u}\|}_{H^s}^2\le c_{s,{\epsilon}}\left({\|{\dib_bu}\|}^2_{H^{s-1}}+{\|{\dib_b^*u}\|}^2_{H^{s-1}}+{\|{u}\|}_{H^{s-1}}^2\right)+c{\|{T_\epsilon^su}\|}_{L^2}^2. \end{aligned}$$ Moreover, since $${\|{\dib_bu}\|}^2_{H^{s-1}}+{\|{\dib_b^*u}\|}^2_{H^{s-1}}\le c{\|{\Box_bu}\|}_{H^{s-2}}\|u\|_{H^s}+c_s{\|{u}\|}^2_{H^{s-1}}$$ it follows that $$\begin{aligned} \label{424b} {\|{u}\|}_{H^s}^2\le c_{s,{\epsilon}}\left({\|{\Box_bu}\|}^2_{H^{s-2}}+{\|{u}\|}_{H^{s-1}}^2\right)+c{\|{T_\epsilon^su}\|}_{L^2}^2. \end{aligned}$$ Denote by $\alpha_{\epsilon}^{1,0}$ and $\alpha_{\epsilon}^{0,1}$ the $(1,0)$-part and $(0,1)$-part of the real $1$-form $\alpha_{\epsilon}:=-\{\T{Lie}\}_{T_{\epsilon}}(\gamma)$. Now we express $\alpha_{\epsilon}^{1,0}$ and $\alpha_{\epsilon}^{0,1}$ in a local basis. Let $\alpha_{{\epsilon},j}$ be the $T$-component of the commutator $[T_{\epsilon},L_j]$. Then $$\begin{aligned} \label{425} \begin{split} \alpha^{1,0}_{\epsilon}(L_j)=-\left(\{\T{Lie}\}_{T_{\epsilon}}(\gamma)\right)(L_j)=-\left(T_{\epsilon}\gamma(L_j)-\gamma([T_{\epsilon},L_j])\right)=\gamma([T_{\epsilon},L_j])=\alpha_{{\epsilon},j} \end{split} \end{aligned}$$ and hence $\alpha^{1,0}_{\epsilon}=\sum_{j=1}^n\alpha_{{\epsilon},j}{\omega}_j$ and $\alpha^{0,1}_{\epsilon}=\overline{\alpha^{1,0}_{\epsilon}}=\sum_{j=1}^n\overline{\alpha_{{\epsilon},j}}{\bar{\omega}}_j$. The commutators $[\dib_b,T_{\epsilon}]$, $[\dib_b^*,T_{\epsilon}]$ and the forms $\alpha^{1,0}_{\epsilon}$ and $\alpha^{0,1}_{\epsilon}$ are related by $$[\dib_b,T_{\epsilon}]=\alpha^{0,1}_{\epsilon}{\wedge}T+\tilde{\mathcal X}_{\epsilon}=b_{\epsilon}\alpha^{0,1}_{\epsilon}{\wedge}T_{\epsilon}+{\mathcal X}_{\epsilon};$$ $$[\dib_b^*,T_{\epsilon}]=-\alpha^{1,0}_{\epsilon}\lrcorner T+\tilde{\mathcal Y}_{\epsilon}=-b_{\epsilon}\alpha^{1,0}_{\epsilon}\lrcorner T_{\epsilon}+{\mathcal Y}_{\epsilon}.$$ Here, $\mathcal X_{\epsilon}:C^\infty_{0,q}(M)\to C^\infty_{0,q+1}(M)$ and $\mathcal Y_{\epsilon}:C^\infty_{0,q}(M)\to C^\infty_{0,q-1}(M)$ so that ${\|{\mathcal X_{\epsilon}u}\|}_{L^2}\le c_{\epsilon}{\|{\nabla_b u}\|}_{L^2}$ and ${\|{\mathcal Y_{\epsilon}u}\|}_{L^2}\le c_{\epsilon}{\|{\nabla_b u}\|}_{L^2}$. In general, for any $s\ge 1$, there exist $\mathcal X_{s,{\epsilon}}:C_{q}^\infty(M)\to C_{q+1}^\infty(M)$ and $\mathcal Y_{s,{\epsilon}}:C_{q}^\infty(M)\to C_{q-1}^\infty(M)$ such that $$\begin{aligned} \label{comTs}\begin{split} [\dib_b,T^s_{\epsilon}]=&sb_{\epsilon}\alpha^{0,1}_{\epsilon}{\wedge}T^s_{\epsilon}+\mathcal X_{s,{\epsilon}};\\ [\dib_b^*,T^s_{\epsilon}]=&-s b_{\epsilon}\alpha^{1,0}_{\epsilon}\lrcorner T^s_{\epsilon}+\mathcal Y_{s,{\epsilon}}, \end{split}\end{aligned}$$ and ${\|{\mathcal X_{s,{\epsilon}} u}\|}_{L^2}+{\|{\mathcal Y_{s,{\epsilon}}u}\|}_{L^2}\le c_{s,{\epsilon}}{\|{\nabla_b u}\|}_{H^{s-1}}$. Now we are ready to prove *a priori* estimates and the estimates for elliptic regulation. Assume that for any $\epsilon>0$ there exist a vector $T_\epsilon$ and a constants $c,c_1,c_2,c_\epsilon>0$ such that $c_1\le \gamma(T_\epsilon)\le c_2$ uniformly in $\epsilon$ and $$\begin{aligned} \label{weakcompact} \frac{1}{{\epsilon}}{\|{|\{\T{Lie}\}_{T_{\epsilon}}(\gamma)|u}\|}_{L^2}^2+{\|{u}\|}_{L^2}^2\le c \left({\|{\dib_b u}\|}_{L^2}^2+{\|{\dib_b^*u}\|}_{L^2}^2\right)+c_\epsilon {\|{u}\|}^2_{H^{-1}} \end{aligned}$$ holds for all $u\in C^\infty_{0,q}(M)$ with $q_0\le q\le n-q_0$. Then $$\begin{aligned} \label{priori} {\|{u}\|}^2_{H^s}&\le &c_s\left({\|{\dib_bu}\|}^2_{H^s}+{\|{\dib_b^*u}\|}^2_{H^s}+{\|{u}\|}_{L^2}^2\right)\\ {\|{\dib_bu}\|}^2_{H^s}+{\|{\dib_b^*u}\|}^2_{H^s}&\le &c_s\left({\|{\Box_bu}\|}^2_{H^s}+{\|{u}\|}_{L^2}^2\right)\\ {\|{\dib_b\dib_b^*u}\|}^2_{H^s}+{\|{\dib_b^*\dib_bu}\|}^2_{H^s}&\le& c_s\left({\|{\Box_bu}\|}^2_{H^s}+{\|{u}\|}_{L^2}^2\right)\end{aligned}$$ for all $u\in C^\infty_{0,q}(M)$ and nonnegative $s\in\Z$. Furthermore, for any $s\in \mathbb N$, there exists $ \delta_s>0$ such that $$\begin{aligned} \label{eqn:elliptic reg est} {\|{u}\|}_{H^s}^2\le c_s({\|{\Box_b^\delta u}\|}_{H^s}^2+{\|{u}\|}^2)\end{aligned}$$ holds for all $u\in C^\infty_{0,q}(M)$ uniformly in $\delta\in(0,\delta_s)$. The operator $\Box_b^\delta=\Box_b+\delta \left(T^*T+\T{Id}\right)$. We prove by inducting in $s$. It is easy to see that holds for $s=0$. We now assume that holds for $s-1$ with $s\ge 1$ and we are going to prove this estimate still holds on level $s$. We first fix ${\epsilon}>0$ independent of $s$. We start with with $u$ replaced by $T^s_{\epsilon}u$ and use the equality $$|\alpha_{\epsilon}|^2|u|^2=2\left(|\alpha^{0,1}_{\epsilon}{\wedge}u|^2+|\alpha^{1,0}_{\epsilon}\lrcorner u|^2\right)$$ to see $$\begin{aligned} \label{t420} \begin{split} {\|{T^s_{\epsilon}u}\|}_{L^2}^2+&\frac{1}{{\epsilon}}\left({\|{\alpha^{0,1}_{\epsilon}{\wedge}T^s_{\epsilon}u}\|}_{L^2}^2+{\|{\alpha_{\epsilon}^{1,0}\lrcorner T^s_{\epsilon}u}\|}_{L^2}^2\right) \\ \le &c\left( {\|{\dib_bT^s_{\epsilon}u}\|}_{L^2}^2+{\|{\dib_b T^s_{\epsilon}u}\|}_{L^2}^2\right)+c_{\epsilon}{\|{T^s_{\epsilon}u}\|}_{H^{-1}}^2. \end{split} \end{aligned}$$ From , we have $$\begin{aligned} \label{t421a} \begin{split} {\|{\dib_bT^s_{\epsilon}u}\|}_{L^2}^2\le&\, 3{\|{T^s_{\epsilon}\dib_bu}\|}_{L^2}^2+cs^2{\|{\alpha_{\epsilon}^{0,1} {\wedge}T^s_{\epsilon}u}\|}_{L^2}^2+c_{{\epsilon},s}{\|{\nabla_b u}\|}^2_{H^{s-1}},\\\ {\|{\dib_b^*T^s_{\epsilon}u}\|}_{L^2}^2\le& \, 3{\|{T^s_{\epsilon}\dib_b^*u}\|}_{L^2}^2+cs^2{\|{\alpha^{1,0}_{\epsilon}\lrcorner T^s_{\epsilon}u}\|}_{L^2}^2+c_{{\epsilon},s}{\|{\nabla_b u}\|}^2_{H^{s-1}}, \end{split}\end{aligned}$$ where we have used that $b_{\epsilon}$ is uniformly bounded. However, for each $s$, there exists ${\epsilon}_s$ such that for any ${\epsilon}<{\epsilon}_s$ the expression $s^2\left({\|{\alpha^{0,1}_{\epsilon}{\wedge}T^s_{\epsilon}u}\|}_{L^2}^2+{\|{\alpha^{1,0}_{\epsilon}\lrcorner T^s_{\epsilon}u}\|}_{L^2}^2\right)$ can be absorbed. Thus $$\begin{aligned} \label{t422a} \begin{split} {\|{T^s_{\epsilon}u}\|}_{L^2}^2+&\frac{1}{{\epsilon}}\left({\|{\alpha^{0,1}_{\epsilon}{\wedge}T^s_{\epsilon}u}\|}_{L^2}^2+{\|{\alpha^{1,0}_{\epsilon}\lrcorner T^s_{\epsilon}u}\|}_{L^2}^2\right)\\ \le c&\left( {\|{T^s_{{\epsilon}}\dib_b u}\|}_{L^2}^2+{\|{T^s_{{\epsilon}}\dib^*_b u}\|}_{L^2}^2\right)+c_{s,{\epsilon}}\left({\|{\nabla_b u}\|}^2_{H^{s-1}}+{\|{u}\|}_{H^{s-1}}^2\right). \end{split} \end{aligned}$$ Combining with and , it follows $$\begin{aligned} \label{t422} \begin{split} {\|{u}\|}_{H^s}^2 \le c\left( {\|{T^s_{{\epsilon}}\dib_b u}\|}_{L^2}^2+{\|{T^s_{{\epsilon}}\dib^*_b u}\|}_{L^2}^2\right)+c_{s,{\epsilon}}\left({\|{\dib_b u}\|}^2_{H^{s-1}}+{\|{\dib_b^* u}\|}^2_{H^{s-1}}+{\|{u}\|}_{H^{s-1}}^2\right)\\ \end{split} \end{aligned}$$ Thus the first *a priori* estimate follows by using the inductive hypothesis for ${\|{u}\|}^2_{H^{s-1}}$.\ For the second *a priori* estimate, it follows from that $$\begin{aligned} \label{t424a} \begin{split} {\|{T^s_{\epsilon}\dib_b u}\|}_{L^2}^2=&(T^s_{\epsilon}\dib_b^*\dib_bu,T^s_{\epsilon}u)+([\dib_b^*,T^s]\dib_bu,T^s_{\epsilon}u)+(T^s_{\epsilon}\dib_b u, [T^s_{\epsilon},\dib_b]u)\\ =&(T^s_{\epsilon}\dib_b^*\dib_bu,T^s_{\epsilon}u)-s(b_{\epsilon}\alpha^{1,0}_{\epsilon}\lrcorner T^s_{\epsilon}\dib_bu,T^s_{\epsilon}u)+(\mathcal Y_{s,{\epsilon}}\dib_bu,T^s_{\epsilon}u)\\ &+s(T^s_{\epsilon}\dib_b u, b_{\epsilon}\alpha^{0,1}_{\epsilon}{\wedge}T^s_{\epsilon}u)+(T^s_{\epsilon}\dib_b u, \mathcal X_{s,{\epsilon}} u); \end{split} \end{aligned}$$ and similarly, $$\begin{aligned} \label{t424b} \begin{split} {\|{T^s_{\epsilon}\dib_b^* u}\|}_{L^2}^2 =&(T^s_{\epsilon}\dib_b\dib_b^*u,T^s_{\epsilon}u)+s(b_{\epsilon}\alpha_{\epsilon}^{0,1}{\wedge}T^s_{\epsilon}\dib_b^*u,T^s_{\epsilon}u)+(\mathcal X_{s,{\epsilon}}\dib_b^*u,T^s_{\epsilon}u)\\ &-s(T^s_{\epsilon}\dib_b^* u, b_{\epsilon}\alpha_{\epsilon}^{1,0}\lrcorner T^s_{\epsilon}u)+(T^s_{\epsilon}\dib_b^* u, \mathcal Y_{s,{\epsilon}} u), \end{split} \end{aligned}$$ Next, we sum and use the equality $(\alpha^{0,1}_{\epsilon}{\wedge}u,v)=( u,\alpha^{1,0}_{\epsilon}\lrcorner v)$, the $(sc)-(lc)$ inequality, and the uniform boundedness of $b_{\epsilon}$ to obtain $$\begin{aligned} \label{t424c} \begin{split} &{\|{T^s_{\epsilon}\dib_b u}\|}_{L^2}^2+ {\|{T^s_{\epsilon}\dib_b^* u}\|}_{L^2}^2\le (T^s_{\epsilon}\Box_bu,T^s_{\epsilon}u)\\ &+c_{s,{\epsilon}}\left({\|{\nabla_b\dib_bu}\|}^2_{H^{s-1}}+{\|{\nabla_b\dib_b^* u}\|}^2_{H^{s-1}}+{\|{\nabla_bu}\|}^2_{H^{s-1}}+{\|{\dib_bu}\|}^2_{H^{s-1}}+{\|{\dib_b^* u}\|}^2_{H^{s-1}}+{\|{u}\|}^2_{H^{s-1}}\right)\\ &+\T{sc}\left({\|{T^s_{\epsilon}\dib_bu}\|}_{L^2}^2+{\|{T^s_{\epsilon}\dib_b^* u}\|}_{L^2}^2+{\|{T^s_{\epsilon}u}\|}_{L^2}^2\right)+\T{lc}s^2\left({\|{\alpha_{\epsilon}^{0,1}{\wedge}T^s_{\epsilon}u}\|}_{L^2}^2+{\|{\alpha_{\epsilon}^{1,0}\lrcorner T^s_{\epsilon}u}\|}_{L^2}^2\right)\\ \end{split} \end{aligned}$$ By , we may absorb the term the last line by choosing ${\epsilon}<{\epsilon}_s$ sufficiently small. We can bound $\|u\|_s^2$ with (\[t422\]) and observe $$\begin{aligned} \label{t423} \begin{split} {\|{u}\|}^2_{H^s}+{\|{\dib_b u}\|}_{H^s}^2+{\|{\dib_b^* u}\|}_{H^s}^2 \le &\ c(T^s_{\epsilon}\Box_b u,T^s_{\epsilon}u )+I \end{split}\end{aligned}$$ where $$\begin{aligned} \label{t423b} \begin{split}I=&c_{s,{\epsilon}}\left({\|{\nabla_b\dib_bu}\|}^2_{H^{s-1}}+{\|{\nabla_b\dib_b^* u}\|}^2_{H^{s-1}}+{\|{\nabla_bu}\|}^2_{H^{s-1}}+{\|{\dib_bu}\|}^2_{H^{s-1}}+{\|{\dib_b^* u}\|}^2_{H^{s-1}}+{\|{u}\|}^2_{H^{s-1}}\right)\\ \le& c_{s,{\epsilon}}\Big({\|{\dib_bu}\|}_{s}{\|{\dib_bu}\|}_{H^{s-1}}+{\|{\dib_b^*u}\|}_{s}{\|{\dib_b^*u}\|}_{H^{s-1}}+{\|{u}\|}_{H^s}{\|{u}\|}_{H^{s-1}}\\ & +{\|{\dib_b\dib_b^*u}\|}^2_{H^{s-1}}+{\|{\dib_b^*\dib_bu}\|}^2_{H^{s-1}}+{\|{\dib_bu}\|}^2_{H^{s-1}}+{\|{\dib_b^* u}\|}^2_{H^{s-1}}+{\|{u}\|}^2_{H^{s-1}}\Big)\\ \le& \T{sc}\left({\|{u}\|}^2_{H^s}+{\|{\dib_b u}\|}_{H^s}^2+{\|{\dib_b^* u}\|}_{H^s}^2\right)+c_{{\epsilon},s}\T{lc}\left({\|{\Box_bu}\|}^2_{H^{s-1}}+{\|{u}\|}_{L^2}^2\right) \end{split}\end{aligned}$$ here the second inequality follows by the middle line of and the last inequality follows by the inductive hypothesis. Thus, $$\begin{aligned} \label{t42c} \begin{split} {\|{u}\|}^2_{H^s}+{\|{\dib_b u}\|}_{H^s}^2+{\|{\dib_b^* u}\|}_{H^s}^2 \le &\ c(T^s_{\epsilon}\Box_b u,T^s_{\epsilon}u )+c_{{\epsilon},s}\left({\|{\Box_bu}\|}^2_{H^{s-1}}+{\|{u}\|}_{L^2}^2\right) \end{split}\end{aligned}$$ The second *a priori* estimate now follows immediately by . The last *a priori* estimate follows by the second *a priori* estimate and the inequality $$\begin{aligned} \nonumber\label{t424} \begin{split} &{\|{\dib_b\dib_b^*u}\|}^2_{H^s}+{\|{\dib_b\dib_b^*u}\|}^2_{H^s}={\|{\Box_bu}\|}^2_{H^s}-2\Re(\Lambda^s \dib_b\dib_b^*u,\Lambda^s \dib_b^*\dib_bu)\\ &= {\|{\Box_bu}\|}_{{H^s}}^2-2\Re\Big((\Lambda^s \dib_b\dib_b\dib_b^*u,\Lambda^s \dib_bu)+([\dib_b,\Lambda^s]\dib_b\dib_b^*u,\Lambda^s \dib_bu)+(\Lambda^s \dib_b\dib_b^*u,[\Lambda^s, \dib_b^*]\dib_bu)\Big)\\ &\le{\|{\Box_bu}\|}^2_{{H^s}}+\T{sc}{\|{\dib_b\dib_b^*u}\|}^2_{H^s}+\T{lc}{\|{\dib_bu}\|}^2_{H^s}. \end{split}\end{aligned}$$ We now prove (\[eqn:elliptic reg est\]), the estimate that allows us to use the method of elliptic regularization. We first show $$\label{t45}{\|{u}\|}^2_{H^{s+1}}\le c(T^s_{\epsilon}T^* T u,T^s_{\epsilon}u)+c_{{\epsilon},s}\left({\|{\Box_bu}\|}^2_{H^{s-1}}+{\|{u}\|}^2_{H^s}\right).$$ This estimate follows quickly by combining , $${\|{u}\|}^2_{H^{s+1}}\le c{\|{TT^{s}_{\epsilon}u}\|}_{L^2}^2+c_{{\epsilon},s}\left({\|{\Box_bu}\|}^2_{H^{s-1}}+{\|{u}\|}^2_{H^s}\right),$$ and $$\begin{aligned} \begin{split}{\|{TT^{s}_{\epsilon}u}\|}^2 =&(T^s_{\epsilon}T^* T u,T^s_{\epsilon}u)+([T^*T, T^{s}_{\epsilon}] u,T^s_{\epsilon}u))\\ \le&(T^s_{\epsilon}T^*Tu,T^s_{\epsilon}u)+\T{sc}{\|{u}\|}_{H^{s+1}}^2+c_{{\epsilon},s}\T{lc}{\|{u}\|}^2_{H^s}. \end{split}\end{aligned}$$ But and , we get $$\begin{aligned} \label{t45b} \begin{split} {\|{u}\|}^2_{H^s}+&{\|{\dib_b u}\|}_{H^s}^2+{\|{\dib_b^* u}\|}_{H^s}^2+\delta {\|{u}\|}^2_{H^{s+1}} \\ \le&\, c(T^s_{\epsilon}\left((\Box_b+\delta (T^*_{{\epsilon}}T_{\epsilon}+I)) u,T^s_{\epsilon}u \right) +c_{{\epsilon},s}\left({\|{\Box_bu}\|}^2_{H^{s-1}}+\delta {\|{u}\|}^2_{H^s}+{\|{u}\|}_{L^2}^2\right)\\ \le &\, c_{{\epsilon},s}\left({\|{\Box_b^\delta u}\|}^2_{H^s}+{\|{\Box_b^\delta u^2}\|}_{H^{s-1}}+{\|{u}\|}_{L^2}^2\right)\\ &\, +\delta c_{{\epsilon},s}{\|{u}\|}^2_{H^s}+\delta^2c_{{\epsilon},s}{\|{u}\|}^2_{H^{s+1}}, \end{split}\end{aligned}$$ where we have used that ${\|{\Box_bu}\|}^2_{H^{s-1}}\le 2{\|{\Box_b^\delta u}\|}^2_{H^{s-1}}+\delta^2{\|{u}\|}^2_{H^{s+1}}$. Now we fix ${\epsilon}$ depending on $s$ so the above estimates hold, and choose $\delta_s$ such that $\delta c_{{\epsilon},s}$ and $\delta^2 c_{{\epsilon},s}$ are small for any $\delta\le \delta_s$. Thus, we may absorb the last line and obtain an inequality stronger than the desired inequality. Proof of Theorem \[thm:global regularity\] ------------------------------------------ [*(i) Proof of the global regularity of $G_q$.*]{} For a given $\varphi\in C^\infty_{0,q}(M)$, we first prove that $G_q\varphi\in C^\infty_{0,q}(M)$ by elliptic regularization using the elliptic perturbation $\Box_b^\delta:=\Box_b+\delta(T^*T+\T{Id})$. First, though, we make one short remark. Since $\mathcal H_{0,q}(M)$ is finite dimensional, and all norms on finite dimensional vector spaces are equivalent, it follows that $\|u\|_{L^2} \approx \|u\|_{H^{-1}(M)}$ for any $u\in\mathcal H_{0,q}(M)$. From this equivalence and the density of smooth forms, we may conclude that harmonic forms are smooth and $\|u\|_{H^s} \approx \|u\|_{L^2}$ where the equivalence depends on $s$ but is independent of the harmonic $(0,q)$-form $u$. Let $Q^{\delta}_b(\cdot,\cdot)$ be the quadratic form on $H^1_{0,q}(M)$ defined by $$Q^{\delta}_b(u,v)=Q_b(u,v)+\delta((Tu,Tv)+(u,v))=(\Box_b^\delta u,v)$$ By (\[424\]), we have ${\|{u}\|}^2_1\le c_\delta Q_b^\delta(u,u)$ for any $u\in H^1_{0,q}(M)$. Consequently, $\Box_b^\delta$ is a self-adjoint, elliptic operator with inverse $G_q^\delta$. By elliptic theory, we know that if $\varphi\in C^\infty_{0,q}(M)$, then $G_q^\delta \varphi\in C^\infty_{0,q}(M)$. We can therefore use (\[eqn:elliptic reg est\]) with $u=G^\delta_q\varphi$ and estimate $${\|{G^\delta_q\varphi}\|}^2_{H^s}\le c_s\left( {\|{\Box_b^\delta G_q^\delta\varphi}\|}^2_{H^s}+{\|{G_q^\delta u}\|}_{L^2}^2\right)=c_s\left( {\|{\varphi}\|}^2_{H^s}+{\|{G_q^\delta \varphi}\|}_{L^2}^2\right)$$ where the equality follows from the identity $\Box_b^\delta G_q^\delta=Id$ (since $\T{Ker}(\Box_b^\delta)=\{0\}$). By Lemma \[lem:lem below\], ${\|{G_q^\delta \varphi}\|}_{L^2}\le c{\|{\varphi}\|}_{L^2}$ uniformly in $\delta$ when $1 \leq q_0\le q\le n-q_0$. Thus, ${\|{G_q^\delta \varphi}\|}_{H^s}$ is uniformly bounded and hence there exists a subsequence $\delta_k$ and $\tilde u\in H^s_{0,q}(M)$ such that $G^{\delta_k}_q \varphi\to \tilde u$ weakly in $H^s_{0,q}(M)$. Consequently, $G^{\delta_k}_q \varphi\to \tilde u$ weakly in the $Q_b$-norm, which means that if $v\in H^2_{0,q}(M)$, then $$\lim_{\delta_k\to 0}Q_b(G^{\delta_k}_q\varphi,v)=Q_b(\tilde u,v).$$ On the other hand, $$Q_b(G_q\varphi,v)=(\varphi,v)=Q^\delta_b(G^\delta_q \varphi,v)=Q_b(G^\delta_q \varphi,v)+\delta\left((TG^\delta_q \varphi,Tv)+(G^\delta_q \varphi,v)\right)$$ for all $v\in H^2_{0,q}(M)$. It follows that $$|Q_b((G^\delta_q \varphi-G_q\varphi),v)|\le \delta {\|{G^\delta_q\varphi}\|}_{L^2}{\|{v}\|}_{2}\le c\delta {\|{\varphi}\|}_{L^2}{\|{v}\|}_2\to 0\quad \T{as $\delta\to 0$}$$ where we have again used the inequality ${\|{G_q^\delta \varphi}\|}_{L^2}\le c{\|{\varphi}\|}_{L^2}$ uniformly in $\delta$. We therefore have $G_q\varphi=\tilde u\in H^s_{0,q}(M)$. This holds for arbitrary $s\in \mathbb N$, so the Sobolev Lemma implies that $G_q\varphi\in C^\infty_{0,q}(M)$.\ [*(ii) Proof of the exact regularity of $G_q$, $\dib_bG_q$, $\dib_b^*G_q$, $I-\dib_b^*\dib_bG_q$ and $I-\dib_b\dib_b^*G_q$.*]{} For $\varphi\in C^\infty_{0,q}(M)$, we use the estimates in Theorem \[comTs\] with $u=G_q\varphi\in C^\infty_{0,q}(M)$ and observe $$\begin{aligned} \label{Cinfty} \begin{split} {\|{G_q\varphi}\|}_{H^s}^2+&{\|{\dib_bG_q\varphi}\|}_{H^s}^2+{\|{\dib_b^*G_q\varphi}\|}_{H^s}^2+{\|{\dib_b^*\dib_bG_q\varphi}\|}_{H^s}^2+{\|{\dib_b\dib_b^*G_q\varphi}\|}_{H^s}^2\\ \le &c_s({\|{\Box_bG_q\varphi}\|}^2_{H^s}+{\|{G_q\varphi}\|}_{L^2}^2)=c_s({\|{(I-H_q)\varphi}\|}^2_{H^s}+{\|{G_q\varphi}\|}_{L^2}^2) \\ \le&c_s({\|{\varphi}\|}^2_{H^s}+{\|{H_q\varphi}\|}^2_{H^s}+{\|{G_q\varphi}\|}_{L^2}^2)\\ \le& c_s({\|{\varphi}\|}^2_{H^s}+{\|{\varphi}\|}_{L^2})\le c_s{\|{\varphi}\|}_{H^s}^2. \end{split}\end{aligned}$$ We have shown the Sobolev estimate holds for $\varphi\in C^\infty_{0,q}(M)$, and this space is dense in $H^s_{0,q}(M)$. Consequently, since $G_q$ is continuous on $L^2_{0,q}(M)$, so that the Sobolev estimate carries over to $\varphi\in H^s_{0,q}(M)$. This means $G_q$, $\dib_bG_q$, $\dib_b^*G_q$, $I-\dib_b^*\dib_bG_q$, and $I-\dib_b\dib_b^*G_q$ are exactly regular.\ For $G_q\dib_b$ and $I-\dib_b^*G_q\dib_b$, let $\varphi\in C^\infty_{0,q-1}(M)$. By using the estimate with $u=G_q\dib_b\varphi\in C^\infty_{0,q}(M)$ we obtain $$\begin{aligned} \begin{split} {\|{G_q\dib_b\varphi}\|}_{H^s}^2+{\|{\dib_b^*G_q\dib_b\varphi}\|}_{H^s}^2\le &c(T^s_{\epsilon}\Box_b G_q\dib_b \varphi, T^s_{\epsilon}G_q\dib_b\varphi)+c_{{\epsilon},s}\left({\|{\Box_bG_q\dib_b\varphi}\|}^2_{H^{s-1}}+{\|{G_q\dib_b\varphi}\|}^2_{L^2}\right)\\ \le &c(T^s_{\epsilon}(I-H_q) \dib_b \varphi, T^s_{\epsilon}G_q\dib_b\varphi)+c_{{\epsilon},s}\left({\|{(I-H_q)\dib_b\varphi}\|}^2_{H^{s-1}}+{\|{\varphi}\|}^2_{L^2}\right)\\ \le &c\Big((T^s_{\epsilon}\varphi, T^s_{\epsilon}\dib_b^*G_q\dib_b\varphi)+([T^s_{\epsilon},\dib_b]\varphi, T^s_{\epsilon}G_q\dib_b\varphi)+(T^s_{\epsilon}\varphi,[\dib_b^*,T^s_{\epsilon}] G_q\dib_b\varphi)\\ &+(T^*_{\epsilon}T^{s}_{\epsilon}H_q\varphi, T^{s-1}_{\epsilon}G_q\dib_b\varphi) \Big)+c_{{\epsilon},s}\left({\|{\dib_b\varphi}\|}^2_{H^{s-1}}+{\|{\varphi}\|}^2_{L^2}\right)\\ \le &\T{sc}\left({\|{G_q\dib_b\varphi}\|}_{H^s}^2+{\|{\dib_b^*G_q\dib_b\varphi}\|}_{H^s}^2\right)+c_{\epsilon}\T{lc}\left({\|{\varphi}\|}_{H^s}^2+{\|{H_q\varphi}\|}_{H^{s+1}}^2\right) \end{split}\end{aligned}$$ Absorbing the sc term by the LHS and using the fact that ${\|{H_q\dib_b\varphi}\|}_{H^{s+1}}\le c{\|{\dib_b\varphi}\|}_{L^2}\le c{\|{\varphi}\|}_1$, we conclude that $${\|{G_q\dib_b\varphi}\|}_{H^s}^2+{\|{(I-\dib_b^*G_q\dib_b)\varphi}\|}_{H^s}^2\le c_s{\|{\varphi}\|}_{H^s}^2$$ for all $\varphi\in C^\infty_{0,q-1}(M)$. As in the above argument, this Sobolev estimate also holds for $\varphi\in H^s_{0,q-1}(M)$. We may then prove the exact regularity of $G_q\dib_b^*$ and $(I-\dib_bG_q\dib_b^*)$ for forms of degree $(0,q+1)$ similarly.\ Finally, exact regularity of $\dib_b^*G_q$, $G_q\dib_b$, $\dib_bG_q$, $G_q\dib_b^*$ implies that $\dib_b^*G_q^2\dib_b$ and $\dib_bG_q^2\dib_b^*$ are also exactly regular. It is known that on the top degrees the Green operators $G_0$ and $G_n$ are given by $\dib_b^*G_1^2\dib_b$ and $\dib_bG_{n-1}^2\dib_b^*$, respectively. Moreover, $\dib_bG_0=G_1\dib_b$, $\dib_b^*G_n=G_{n-1}\dib_b^*$. Therefore, if $q=1$ then $G_0, \dib_bG_0$, $G_n$, $\dib_b^*G_n$ are exactly regular.\ The proof of Theorem  \[thm:global regularity\] is complete, pending the following technical lemma. \[lem:lem below\] Fix $1\le q\le n-1$. Let $M^{2n+1}$ be an abstract CR manifold that the $L^2$ basic estimate $$\label{L2basic1} {\|{u}\|}_{L^2}^2\le c({\|{\dib_bu}\|}_{L^2}^2+{\|{\dib_b^*u}\|}_{L^2}^2) + C \|u\|_{H^{-1}}^2$$ holds for all $u\in \T{Dom}(\dib_b)\cap\T{Dom}(\dib_b)$. Then ${\|{G_q^\delta \varphi}\|}\le c {\|{\varphi}\|}$ uniformly in $\delta$ for $\varphi\in L^2_{0,q}(M)$. Fix $\delta>0$. It suffices to show that $$\label{n1} {\|{u}\|}_{L^2}^2\le c Q_b^\delta(u,u)$$ holds for some constant $c>0$ that is independent of $\delta>0$ and $u\in H^1_{0,q}(M)$. The basic estimate certainly implies that $${\|{u}\|}_{L^2}^2\le cQ_b^\delta(u,u) +c'{\|{u}\|}^2_{H^{-1}},$$ uniformly in $\delta$ for all $u\in H^1_{0,q}(M)$. Assume that fails. Then there exists $u_k$ with ${\|{u_k}\|}_{L^2}^2=1$ so that $$\label{n2} {\|{u_k}\|}_{L^2}^2\ge k Q_b^\delta(u_k,u_k).$$ For $k$ sufficiently large, we can use and absorb $Q_b^\delta(u_k,u_k)$ by ${\|{u_k}\|}^2$ and to prove $$\label{n3} {\|{u_k}\|}_{L^2}^2\le 2c'{\|{u_k}\|}^2_{H^{-1}}.$$ Since $L^2_{0,q}(M)$ is compact in $H^{-1}_{0,q}(M)$, there exists a subsequence $u_{k_j}$ that converges in $H^{-1}_{0,q}(M)$. Thus, forces $u_{k_j}$ to converge in $L^2_{0,q}(M)$; and forces $u_{k_j}$ to converge in the $Q_b^\delta(\cdot,\cdot)$-norm as well. The limit $u$ satisfies ${\|{u}\|}_{L^2}=1$. However, a consequence of is that ${\|{u}\|}_{L^2}=0$ since $Q^\delta_b(u,u)\ge \delta {\|{u}\|}_{L^2}^2$. This is a contradiction and holds. Proof of Theorem \[thm:CRpsh (0,1)\] ------------------------------------ Assume that there exists a global contact form $\tilde \gamma$ and a smooth function $\tilde{c}_{00}$ such that the extended Levi matrix $\mathcal{\tilde M}:=\{\tilde c_{ij}\}_{i,j=0}^n$ is positive semidefinite. Thus we can use the Schwarz inequality for the two vectors $u=(0,u_1,\dots, u_n)$, $v=(1,0,\dots,0)$ in $\C^{n+1}$ and get $$\begin{aligned} \label{Schwarz}\left|\sum_{j=1}^n\tilde c_{0j}u_j\right|^2=\left|\mathcal{\tilde M}(u,v)\right|^2\le \mathcal{\tilde M}(u,u)\mathcal{\tilde M}(v,v)=\left|\sum_{i,j=1}^n\tilde c_{ij}u_i\bar u_j\right||\tilde{c}_{00}|. \end{aligned}$$ On the other hand, there exists a smooth function $h$ in $M$ such that $\tilde \gamma=e^{-h}\gamma$. Thus, $$\begin{aligned} \label{dtildegamma}\begin{split} d\tilde\gamma=&\, e^{-h}\,d\gamma-e^{-h}\,dh{\wedge}\gamma\\ =&-e^{-h}\left(\sum_{i,j=1}^nc_{ij}\,{\omega}_i{\wedge}{\bar{\omega}}_j+\sum_{j=1}^nc_{0j}\,\gamma{\wedge}{\omega}_j +\sum_{j=1}^n\overline{c_{0j}}\,\gamma{\wedge}{\bar{\omega}}_j\right)+e^{-h}\sum_{j=1}^n\left(L_j(h)\,\gamma{\wedge}{\omega}_j+\bar L_j(h)\,\gamma{\wedge}{\bar{\omega}}_j\right)\\ =&-\sum_{i,j=1}^ne^{-h}c_{ij}\,{\omega}_i{\wedge}{\bar{\omega}}_j-\sum_{j=1}^n\left(e^{-h}(c_{0j}-L_j(h))\,\gamma{\wedge}{\omega}_j+e^{-h}\overline{(c_{0j}-L_j(h))}\,\gamma{\wedge}{\bar{\omega}}_j\right)\\ =&-\left(\sum_{i,j=1}^n\tilde c_{ij}\,{\omega}_i{\wedge}{\bar{\omega}}_j+\sum_{j=1}^n\tilde c_{0j}\,\tilde \gamma{\wedge}{\omega}_j+\sum_{j=1}^n\overline{ \tilde c_{0j}}\,\tilde \gamma{\wedge}{\bar{\omega}}_j \right), \end{split} \end{aligned}$$ where $\tilde{c}_{ij}=e^{-h}c_{ij}$ for $i,j=1,\dots,n$ and $\tilde c_{0j}=c_{0j}-L_j(h)$. Substituting these $\tilde c_{ij}$ into , we get $$\begin{aligned} \label{new1}\left|\sum_{j=1}^n(c_{0j}-L_j(h))u_j\right|^2\le |\tilde c_{00}|\left|\sum_{i,j=1}^n \tilde c_{ij}u_i\bar u_j\right| = |\tilde c_{00}|e^{-h} \sum_{i,j=1}^n c_{ij}u_i\bar u_j \end{aligned}$$ holds at any $x\in M$ and any vector $u=(u_1,\dots,u_n)\in \C^n$. Recall that $$\alpha=-\{\T{Lie}\}_{T}(\gamma)=-\sum_{j=1}^n\left(d\gamma(T,L_j){\omega}_j+d\gamma(T,\bar L_j){\bar{\omega}}_j\right)=\sum_{j=1}^n\left(c_{0j}{\omega}_j+\bar c_{0j}{\bar{\omega}}_j\right),$$ (where the last inequality follows by ). If $L=\sum_ju_jL_j\in T^{1,0}M$ then we rewrite as $$|\alpha(L)-dh(L)|\le c d\gamma(L{\wedge}\bar L).$$ This calculation implies that $\alpha$ is exact on the null space of Levi form. The argument from [@StZe15 Proposition 1] then finishes the proof. Straube and Zeytuncu’s work shows that if $\alpha$ is exact on the null space of the Levi form then for any ${\epsilon}$, then we can find a vector $T_{\epsilon}$ traversal to $T^{0,1}M\oplus T^{0,1}M$ such that the $T$-component of $[T_{\epsilon},L]$ is less than $\epsilon$ for any unit vector field $L\in T^{1,0}$. Their result is stated for embedding manifolds in $\C^N$ but a careful examination of the proof reveals that embeddedness is an unnecessary assumption with their proof. [KPZ12]{} A. Boggess. . Studies in Advanced Mathematics. CRC Press, Boca Raton, Florida, 1991. L. Baracco, S. Pinton, and G. Zampieri. Hypoellipticity of the [K]{}ohn-[L]{}aplacian [$\square_b$]{} and of the [$\overline\partial$]{}-[N]{}eumann problem by means of subelliptic multipliers. , 362(3-4):887–901, 2015. http://dx.doi.org/10.1007/s00208-014-1144-1. H. Boas and M.-C. Shaw. Sobolev estimates for the [L]{}ewy operator on weakly pseudoconvex boundaries. , 274:221–231, 1986. H. Boas and E. Straube. Sobolev estimates for the complex [G]{}reen operator on a class of weakly pseudoconvex boundaries. , 16:1573–1582, 1991. H. Boas and E. Straube. de [R]{}ham cohomology of manifolds containing the points of infinite type, and [S]{}obolev estimates for the [$\overline\partial$]{}-[N]{}eumann problem. , 3(3):225–235, 1993. H. Boas and E. Straube. Global regularity of the $\overline\partial$-[N]{}eumann problem: a survey of the ${L}\sp 2$-[S]{}obolev theory. In [*Several [C]{}omplex [V]{}ariables ([B]{}erkeley, [CA]{}, 1995–1996)*]{}, Mat. Sci. Res. Inst. Publ., 37, pages 79–111. Cambridge Univ. Press, Cambridge, 1999. S.-C. Chen and M.-C. Shaw. , volume 19 of [*Studies in Advanced Mathematics*]{}. American Mathematical Society, 2001. H. Grauert. Bemerkenswerte pseudokonvexe [M]{}annigfaltigkeiten. , 81:377–391, 1963. P. Harrington. Global regularity for the [$\overline\partial$]{}-[N]{}eumann operator and bounded plurisubharmonic exhaustion functions. , 228(4):2522–2551, 2011. http://dx.doi.org/10.1016/j.aim.2011.07.008. L. H[ö]{}rmander. ${L}^{2}$ estimates and existence theorems for the $\bar \partial $ operator. , 113:89–152, 1965. P. Harrington and A. Raich. Regularity results for $\bar\partial_b$ on [CR]{}-manifolds of hypersurface type. , 36(1):134–161, 2011. P. Harrington and A. Raich. Closed range for $\bar\partial$ and $\bar\partial_b$ on bounded hypersurfaces in [S]{}tein manifolds. , 65(4):1711–1754, 2015. T.V. Khanh. The Kohn-Laplace equation on abstract cr manifolds: local regularity. . T.V. Khanh. Equivalence of estimates on a domain and its boundary. , 44(1):29–48, 2016. J.J. Kohn. Estimates for $\bar\partial_b$ on compact pseudoconvex [CR]{} manifolds. In [*Proceedings of Symposia in Pure Mathematics*]{}, volume 43, pages 207–217. American Mathematical Society, 1985. J.J. Kohn. The range of the tangential [C]{}auchy-[R]{}iemann operator. , 53:525–545, 1986. J. J. Kohn. Hypoellipticity at points of infinite type. In [*Analysis, geometry, number theory: the mathematics of [L]{}eon [E]{}hrenpreis ([P]{}hiladelphia, [PA]{}, 1998)*]{}, volume 251 of [*Contemp. Math.*]{}, pages 393–398. Amer. Math. Soc., Providence, RI, 2000. J.J. Kohn. Superlogarithmic estimates on pseudoconvex domains and [CR]{} manifolds. , 156:213–248, 2002. T.V. Khanh, S. Pinton, and G. Zampieri. Compactness estimates for [$\square\sb {b}$]{} on a [CR]{} manifold. , 140(9):3229–3236, 2012. T.V. Khanh and G. Zampieri. Estimates for regularity of the tangential [$\overline{\partial}$]{}-system. , 284(17-18):2212–2224, 2011. A. Nicoara. Global regularity for $\bar\partial_b$ on weakly pseudoconvex [CR]{} manifolds. , 199:356–447, 2006. A. Raich. Compactness of the complex [G]{}reen operator on [CR]{}-manifolds of hypersurface type. , 348(1):81–117, 2010. A. Raich and E. Straube. Compactness of the complex [G]{}reen operator. , 15(4):761–778, 2008. M.-C. Shaw. ${L}\sp 2$-estimates and existence theorems for the tangential [C]{}auchy-[R]{}iemann complex. , 82:133–150, 1985. E. Straube. A sufficient condition for global regularity of the $\overline\partial$-[N]{}eumann operator. , 217(3):1072–1095, 2008. E. Straube. . ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS), Z[ü]{}rich, 2010. E. J. Straube. The complex [Green]{} operator on [CR]{}-submanifolds of $\mathbb{C}^n$ of hypersurface type: compactness. , 364(8):4107–4125, 2012. E. Straube and Y. Zeytuncu. Sobolev estimates for the complex [G]{}reen operator on [CR]{} submanifolds of hypersurface type. , 201(3):1073–1095, 2015. [^1]: The first author was supported by ARC grant DE160100173. The second author was partially supported by NSF grant DMS-1405100. This work was done in part while the authors were visiting members at the Vietnam Institute for Advanced Study in Mathematics (VIASM). They would like to thank the institution for its hospitality and support. [^2]:

--- abstract: 'We provide explicit conditions, in terms of the transition kernel of its driving particle, for a Markov branching process to admit a scaling limit toward a self-similar growth-fragmentation with negative index. We also derive a scaling limit for the genealogical embedding considered as a compact real tree.' author: - 'Benjamin Dadoun[^1]' title: | <span style="font-variant:small-caps;">Self-similar growth-fragmentations\ as scaling limits of Markov branching processes</span> --- [*[**Keywords:**]{} Growth-fragmentation, scaling limit, Markov branching tree*]{} [*[**Classification:**]{} 60F17, 60J80*]{} Introduction {#sec:Newton} ============ Imagine a bin containing $n$ balls which is repeatedly subject to random (binary) divisions at discrete times, until every ball has been isolated. There is a natural random (binary) tree with $n$ leaves associated with this partitioning process, where the subtrees above a given height $k\ge0$ represent the different subcollections of all $n$ balls at time $k$, and the number of leaves of each subtree matches the number of balls in the corresponding subcollection. The habitual [*Markov branching property*]{} stipulates that these subtrees must be independent conditionally on their respective size. In the literature on random trees, a central question is the approximation of so called continuum random trees (CRT) as the size of the discrete trees tends to infinity. We mention especially the works of Aldous [@Aldous91a; @Aldous91b; @Aldous93] and Haas, Miermont, et al. [@Haas04; @Miermont05; @Haas08; @Haas11; @Haas12]. Concerning the above example, Haas and Miermont [@Haas12] obtained, under some natural assumption on the splitting laws, distributional scaling limits regarded in the Gromov–Hausdorff–Prokhorov topology. In the Gromov–Hausdorff sense where trees are considered as compact metric spaces, they especially identified the so called [*self-similar fragmentation trees*]{} as the scaling limits. The latter describe the genealogy of [*self-similar fragmentation processes*]{}, which, reciprocally, are known to record the size of the components of a (continuous) fragmentation tree above a given height [@Haas04], and thus correspond to scaling limits for partition sequences of balls as their number $n$ tends to infinity. One key tool in the work of Haas and Miermont [@Haas12] is provided by some non-increasing integer-valued Markov chain which, roughly speaking, depicts the size of the subcollection containing a randomly tagged ball. This Markov chain essentially captures the dynamics of the whole fragmentation and, by their previous work [@Haas11], itself possesses a scaling limit. (0,0) circle (.8) (0,0) circle (.1) (45:.5) circle (.1) (135:.5) circle (.1) (225:.5) circle (.1) (315:.5) circle (.1) (20:.8) – (20:3) (20:3.8) circle (.8) ($(30:.4)+(-20:3.8)$) circle (.1) ($(150:.4)+(-20:3.8)$) circle (.1) ($(-90:.4)+(-20:3.8)$) circle (.1) (-20:.8) – (-20:3) (-20:3.8) circle (.8) ($(135:.4)+(20:3.8)$) circle (.1) ($(-45:.4)+(20:3.8)$) circle (.1) (20:7.6) – (20:9.2) (20:10) circle (.8) ($(30:.4)+(20:10)$) circle (.1) ($(150:.4)+(20:10)$) circle (.1) ($(-90:.4)+(20:10)$) circle (.1) ($(20:6.8)+(-20:0.8)$) – ($(20:6.8)+(-16:2.35)$) ($(20:6.8)+(-16:3.15)$) circle (.8) ($(20:6.8)+(-16:3.15)$) circle (.1) ($(-20:3.8)+(20:.8)$) – ($(-20:3.8)+(20:2.2)$) ($(-20:3.8)+(18:3)$) circle (.8) ($(135:.4)+(-20:3.8)+(18:3)$) circle (.1) ($(-45:.4)+(-20:3.8)+(18:3)$) circle (.1) (-20:4.6) – (-20:6) (-20:6.8) circle (.8) (-20:6.8) circle (.1); (20:4.6) – (20:6) (20:6.8) circle (.8) ($(45:.4)+(20:6.8)$) circle (.1) ($(135:.4)+(20:6.8)$) circle (.1) ($(-45:.4)+(20:6.8)$) circle (.1) ($(-135:.4)+(20:6.8)$) circle (.1) ($(-20:3.8)+(18:3)+(0:.8)$) – ($(-20:3.8)+(18:3)+(0:2.2)$) ($(-20:3.8)+(18:3)+(0:3)$) circle (.8) ($(45:.4)+(-20:3.8)+(18:3)+(0:3)$) circle (.1) ($(135:.4)+(-20:3.8)+(18:3)+(0:3)$) circle (.1) ($(-45:.4)+(-20:3.8)+(18:3)+(0:3)$) circle (.1) ($(-135:.4)+(-20:3.8)+(18:3)+(0:3)$) circle (.1) ($(-20:3.8)+(18:3)+(0:3)$) circle (.1); ($(20:10)+(0:1)$) – ($(20:10)+(0:2)$) ($(-20:3.8)+(20:3)+(-4:3)+(0:1)$) – ($(-20:3.8)+(20:3)+(-4:3)+(0:2)$); The purpose of the present work is to study more general dynamics which incorporate *growth*, that is the addition of new balls in the system (see \[fig:Archimedes\]). One example of recent interest lies in the exploration of random planar maps [@Kortchemski15; @Curien16], which exhibits “holes” (the yet unexplored areas) that split or grow depending on whether the new edges being discovered belong to an already known face or not. We thus consider a Markov branching system in discrete time and space where at each step, every particle is replaced by either one particle with a bigger size (growth) or by two smaller particles in a conservative way (fragmentation). We condition the system to start from a single particle with size $n$ (we use the superscript $\cdot^{(n)}$ in this respect) and we are interested in its behaviour as $n\to\infty$. Namely, we are looking for: 1. A functional scaling limit for the process in time $({\mathds X}(k)\colon k\ge0)$ of all particle sizes: $$\left(\frac{{\mathds X}^{(n)}(\lfloor{a}_nt\rfloor)}{n}\colon t\ge0\right) {\xrightarrow[n\to\infty]{(d)}}\,\bigl({{\boldsymbol{\mathbf{Y}}}}(t)\colon t\ge0\bigr),$$ in some sequence space, where the ${a}_n$ are positive (deterministic) numbers; 2. A scaling limit for the system’s genealogical tree, seen as a random metric space $({\chi}^{(n)},{d_n})$: $$\left({\chi}^{(n)},\frac{{d_n}}{{a}_n}\right) {\xrightarrow[n\to\infty]{(d)}}\,{\mathcal Y},$$ in the Gromov–Hausdorff topology. Like in the pure-fragmentation setting we may single out some specific integer-valued Markov chain, but which of course is no longer non-increasing. To derive a scaling limit for this chain, a first idea is to apply, as a substitution to [@Haas11], the more general criterion of Bertoin and Kortchemski [@Kortchemski16] in terms of the asymptotic behaviour of its transition kernel at large states. However, this criterion is clearly insufficient for the convergences stated above as it provides no control on the “microscopic” particles. To circumvent this issue, we choose to “prune” the system by freezing the particles below a (large but fixed) threshold. That is to say, we let the system evolve from a large size $n$ but stop every individual as soon as it is no longer bigger than some threshold ${M}>0$ which will be independent of $n$, and we rather study the modifications ${{\boldsymbol{\mathbf{X}}}}^{(n)}$ and ${\mathcal X}^{(n)}$ of the process and the genealogical tree that are induced by this procedure. The limits ${{\boldsymbol{\mathbf{Y}}}}$ and ${\mathcal Y}$ that we aim at are, respectively, a [*self-similar growth-fragmentation process*]{} and its associated genealogical structure. Indeed, the scaling limits of integer-valued Markov chains investigated in [@Kortchemski16], which we build our work upon, belong to the class of so called [*positive self-similar Markov processes*]{} (pssMp), and these processes constitute the cornerstone of Bertoin’s self-similar growth-fragmentations [@Bertoin17; @Curien16]. Besides, in the context of random planar maps [@Kortchemski15; @Curien16], they have already been identified as scaling limits for the sequences of perimeters of the separating cycles that arise in the exploration of large Boltzmann triangulations. Informally, a self-similar growth-fragmentation ${{\boldsymbol{\mathbf{Y}}}}$ depicts a system of particles which all evolve according to a given pssMp and whose each negative jump $-y<0$ begets a new independent particle with initial size $y$. In our setting, the self-similarity property has a negative index and makes the small particles split at higher rates, in such a way that the system becomes eventually extinct [@Bertoin17 Corollary 3]. The genealogical embedding ${\mathcal Y}$ is thus a compact real tree; its formal construction is presented in [@Rembart16]. Because of growth, one main difference with the conservative case is, of course, that the mass of a particle at a given time no longer equals the size (number of leaves) of the corresponding genealogical subtree. In a similar vein, choosing the uniform distribution to mark a ball at random will appear less relevant than a size-biased pick from an appropriate (nondegenerate) supermartingale. This will highlight a Markov chain admitting a self-similar scaling limit (thanks to the criterion [@Kortchemski16]), and which we can plug into a many-to-one formula. Under an assumption preventing an explosive production of relatively small particles, we will then be able to establish our first desired convergence. Concerning the convergence of the (rescaled) trees ${\mathcal X}^{(n)}$, we shall employ a Foster–Lyapunov argument to obtain an uniform control on their heights, which are nothing else than the exctinction times of the processes ${{\boldsymbol{\mathbf{X}}}}^{(n)}$. Contrary to what one would first expect, it turns out that a good enough Lyapunov function is not simply a power of the size, but merely depends on the scaling sequence $({a}_n)$. This brings a tightness property that, together with the convergence of “finite-dimensional marginals”, will allow us to conclude. In the next section we set up the notation and the assumptions more precisely, and state our main two results. Assumptions and results {#sec:Gauss} ======================= Our basic data are probability transitions ${p}_{n,m},\,m\ge n/2$ and $n\in{\mathbb N}$ “sufficiently large”, with which we associate a Markov chain, generically denoted ${X}$, that governs the law of the particle system ${\mathds X}$: at each time $k\in{\mathbb N}$ and with probability ${p}_{n,m}$, every particle with size $n$ either grows up to a size $m>n$, or fragmentates into two independent particles with sizes $m\in\{\lceil n/2\rceil,\ldots,n-1\}$ and $n-m$. That is to say, ${X}^{(n)}(0)=n$ is the size of the initial particle in ${\mathds X}^{(n)}$, and given ${X}(k)$ for some $k\ge0$, ${X}(k+1)$ is the largest among the (one or two) particles replacing ${X}(k)$. We must emphasize that the transitions ${p}_{n,m}$ from $n$ “small” are irrelevant since our assumptions shall only rest upon the asymptotic behavior of ${p}_{n,m}$ as $n$ tends to infinity. Indeed, for the reason alluded in the that we explain further below, we rather study the pruned version ${{\boldsymbol{\mathbf{X}}}}$ where particles are frozen (possibly at birth) when they become not bigger than a thresold parameter ${M}>0$, which we will fix later on. Keeping the same notation, this means that ${X}$ is a Markov chain stopped upon hitting $\{1,2,\ldots,{M}\}$. For convenience, *we omit to write the dependency in ${M}$*, and set ${p}_{n,n}{\coloneqq}1$ for $n\le{M}$. In turn, the law of the genealogical tree ${\mathcal X}$ can be defined inductively as follows (we give a more rigorous construction in \[sec:Euler\]). Let $1\le k_1\le\cdots\le k_p$ enumerate the instants during the lifetime ${\zeta}^{(n)}$ of ${X}^{(n)}$ when $n_i{\coloneqq}{X}^{(n)}(k_i-1)-{X}^{(n)}(k_i)>0$. Then ${\mathcal X}^{(n)}$ consists in a branch with length ${\zeta}^{(n)}$ to which are respectively attached, at positions $k_i$ from the root, independent trees distributed like ${\mathcal X}^{(n_i)}$ (agreeing that ${\mathcal X}^{(n)}$ degenerates into a single vertex for $n\le{M}$). We view ${\mathcal X}^{(n)}$ as a metric space with metric denoted by ${d_n}$. Suppose $({a}_n)_{n\in{\mathbb N}}$ is a sequence of positive real numbers which is regularly varying with index ${\gamma}>0$, in the sense that for every $x>0$, $$\lim_{n\to\infty}\frac{{a}_{\lfloor nx\rfloor}}{{a}_n}=x^{\gamma}. \label{eq:Riemann}$$ Our starting requirement will be the convergence in distribution $$\left(\frac{{X}^{(n)}(\lfloor{a}_nt\rfloor)}n\colon t\ge0\right) {\xrightarrow[n\to\infty]{(d)}}\,\bigl({Y}(t)\colon t\ge0\bigr), \label{eq:Poincare}$$ in the space ${\mathbb D}({\mathopen{[}0,\infty\mathclose{)}},{\mathbb R})$ of càdlàg functions on ${\mathopen{[}0,\infty\mathclose{)}}$ (endowed with Skorokhod’s J~1~ topology), towards a positive strong Markov process $({Y}(t)\colon t\ge0)$, continuously absorbed at $0$ in an almost surely finite time ${\upzeta}$, and with the following self-similarity property: &&\[eq:Lagrange\] Since the seminal work of Lamperti [@Lamperti72], this simply means that $$\log{Y}(t)={\xi}\!\left(\int_0^t{Y}(s)^{-{\gamma}}\,{\mathrm d}s\right)\!,\qquad t\ge0, \label{eq:Euclid}$$ where ${\xi}$ a Lévy process which drifts to $-\infty$ as $t\to\infty$. We denote by ${\Psi}$ the characteristic exponent of ${\xi}$ (so there is the Lévy–Khinchine formula $E[\exp(q{\xi}(t))]=\exp(t{\Psi}(q))$ for every $t\ge0$ and every $q\in{\mathbb C}$, wherever this makes sense), and by ${\Lambda}$ the Lévy measure of its jumps (that is a measure on ${\mathbb R}\setminus\{0\}$ with $\int(1\wedge y^2)\,{\Lambda}({\mathrm d}y)<\infty$). In order to state precisely our assumptions, we need to introduce some more notation. We define $$\begin{gathered} {\kappa}(q){\coloneqq}{\Psi}(q)+\int_{{\mathopen{(}-\infty,0\mathclose{)}}}\bigl(1-e^y\bigr)^q \,{\Lambda}({\mathrm d}y), \intertext{and, for every~$n\in{\mathbb N}$, the discrete exponents} {\Psi}_n(q){\coloneqq}{a}_n\sum_{m=1}^\infty{p}_{n,m} \left[\left(\frac mn\right)^{\!q}-1\right]\!,\quad\text{and}\quad {\kappa}_n(q){\coloneqq}{\Psi}_n(q)+{a}_n\sum_{m=1}^{n-1}{p}_{n,m} \left(1-\frac mn\right)^{\!q}.\end{gathered}$$ Finally, we fix some parameter ${{q^*}}>0$. After [@Kortchemski16 Theorem 2], convergence  holds under the following two assumptions: \[assump:H1\] For every $t\in{\mathbb R}$, $$\lim_{n\to\infty}{\Psi}_n({\mathrm i}t)={\Psi}({\mathrm i}t).$$ \[assump:H2\] We have $$\limsup_{n\to\infty}\; {a}_n\sum_{m=2n}^\infty{p}_{n,m}\left(\frac mn\right)^{\!{{q^*}}} <\,\infty.$$ Indeed, by [@Kallenberg02 Theorem 15.14 & 15.17], \[assump:H1\] is essentially equivalent to (A1)&(A2) of [@Kortchemski16], while  rephrases Assumption (A3) there. We now introduce the new assumption: \[assump:H3\] We have either ${\kappa}({{q^*}})<0$, or ${\kappa}({{q^*}})=0$ and ${\kappa}'({{q^*}})>0$. Moreover, for some ${\varepsilon}>0$, $$\lim_{n\to\infty} {a}_n\sum_{m=1}^{n-1}{p}_{n,m}\left(1-\frac mn\right)^{\!{{q^*}}-{\varepsilon}} =\,\int_{{\mathopen{(}-\infty,0\mathclose{)}}}\bigl(1-e^y\bigr)^{{{q^*}}-{\varepsilon}}\,{\Lambda}({\mathrm d}y). \label{eq:H3}$$ Postponing the description of the limits, we can already state our two convergence results formally: [thm:Hilbert]{} Suppose . Then we can fix a freezing threshold ${M}$ sufficiently large so that, for every $q\ge1\vee{{q^*}}$, the convergence in distribution $$\left(\frac{{{\boldsymbol{\mathbf{X}}}}^{(n)}(\lfloor{a}_nt\rfloor)}n\colon t\ge0\right) {\xrightarrow[n\to\infty]{(d)}}\,\bigl({{\boldsymbol{\mathbf{Y}}}}(t)\colon t\ge0\bigr),$$ holds in the space ${\mathbb D}({\mathopen{[}0,\infty\mathclose{)}},{\ell^{q\downarrow}})$, where ${{\boldsymbol{\mathbf{Y}}}}$ is the self-similar growth-fragmentation driven by ${Y}$, and $${\ell^{q\downarrow}}{\coloneqq}\left\{{\boldsymbol{\mathbf{x}}}{\coloneqq}(x_1\ge x_2\ge\cdots\ge0)\colon \sum_{i=1}^\infty(x_i)^q<\infty\right\}$$ (that is, the family of particles at a given time is always ranked in the non-increasing order). [thm:Grothendieck]{} Suppose , and ${{q^*}}>{\gamma}$. Then we can fix a freezing threshold ${M}$ sufficiently large so that there is the convergence in distribution $$\left({\mathcal X}^{(n)},\frac{{d_n}}{{a}_n}\right){\xrightarrow[n\to\infty]{(d)}}{\mathcal Y},$$ in the Gromov–Hausdorff topology, where ${\mathcal Y}$ is the random compact real tree related to ${{\boldsymbol{\mathbf{Y}}}}$’s genealogy. #### Description of the limits. As explained in the , the process ${Y}$ portrays the size of particles in the self-similar growth-fragmentation process ${{\boldsymbol{\mathbf{Y}}}}{\coloneqq}({{\boldsymbol{\mathbf{Y}}}}(t)\colon t\ge0)$, whose construction we briefly recall (referring to [@Bertoin17; @Curien16] for more details): The Eve particle ${Y}_{\varnothing}$ is distributed like ${Y}$. We rank the negative jumps of a particle ${Y}_u$ in the decreasing order of their absolute sizes (and chronologically in case of ex aequo). When this particle makes its $j$^th^ negative jump, say with size $-y<0$, then a daughter particle ${Y}_{uj}$ is born at this time and evolves, independently of its siblings, according to the law of ${Y}$ started from $y$. (Recall that ${Y}$ is eventually absorbed at $0$, so we can indeed rank the negative jumps in this way; for definiteness, we set ${b}_{uj}{\coloneqq}\infty$ and ${Y}_{uj}{\mathrel{\rlap{\raisebox{0.3ex}{$\cdot$}} \raisebox{-0.3ex}{$\cdot$}}\equiv}0$ if ${Y}_u$ makes less than $j$ negative jumps during its lifetime.) Particles are here labeled on the Harris–Ulam tree ${\mathbb U}$, the set of finite words over ${\mathbb N}$. Write ${b}_u$ for the birth time of ${Y}_u$. Then $${{\boldsymbol{\mathbf{Y}}}}(t)=\Bigl( {Y}_u(t-b_u)\colon\,u\in{\mathbb U},\,{b}_u\le t \Bigr), \qquad t\ge0.$$ After [@Bertoin17; @Curien16], this process is self-similar with index $-{\gamma}$. Roughly speaking, this means that a particle with size $x>0$ evolves $x^{-{\gamma}}$ times “faster” than a particle with size $1$. Since here $-{\gamma}<0$, there is the snowball effect that particles get rapidly absorbed toward $0$ as time passes, and it has been shown [@Bertoin17 Corollary 3] that such a growth-fragmentation becomes eventually extinct, namely that ${\epsilon}{\coloneqq}\inf\{t\ge0\colon{{\boldsymbol{\mathbf{Y}}}}(t)=\emptyset\}$ is almost surely finite. The extinction time ${\epsilon}$ is also the height of the genealogical structure ${\mathcal Y}$ seen as a compact real tree. Referring to [@Rembart16] for details, we shall just sketch the construction. Let ${\mathcal Y}_{u,0}$ consists in a segment with length ${\upzeta}_u{\coloneqq}\inf\{t\ge0\colon{Y}_u(t)=0\}$ rooted at a vertex $u$. Recursively, define ${\mathcal Y}_{u,h+1}$ by attaching to the segment ${\mathcal Y}_{u,0}$ the trees ${\mathcal Y}_{uj,k}$ at respective distances ${b}_{uj}-{b}_u$, for each born particle $uj$, $j\le h+1$. The limiting tree ${\mathcal Y}{\coloneqq}{\mathop{\mathrm{lim}\!\!\uparrow}}_{h\to\infty}{\mathcal Y}_h$ fulfills a so called [*recursive distributional equation*]{}. Namely, by [@Rembart16 Corollary 4.2], given the sequence of negative jump times and sizes $({b}_j,y_j)$ of ${Y}$ and an independent sequence ${\mathcal Y}_1,{\mathcal Y}_2,\ldots$ of copies of ${\mathcal Y}$, the action of grafting, on a branch with length ${\upzeta}{\coloneqq}\inf\{t\ge0\colon{Y}(t)=0\}$ and at distances ${b}_j$ from the root, the trees ${\mathcal Y}_j$ rescaled by the multiplicative factor $y_j^{{\gamma}}$, yields a tree with the same law as ${\mathcal Y}$. With this connection, Rembart and Winkel [@Rembart16 Corollary 4.4] proved that ${\epsilon}$ admits moments up to the order $\sup\{q>0\colon{\kappa}(q)<0\}/{\gamma}$. When particles do not undergo sudden positive growth (i.e., ${\Lambda}({\mathopen{(}0,\infty\mathclose{)}})=0$), Bertoin et al. [@Curien16 Corollary 4.5] more precisely exhibited a polynomial tail behavior of this order for the law of ${\epsilon}$. #### Discussion on the assumptions. Observe that  implies when the Lévy measure ${\Lambda}$ of ${\xi}$ is bounded from above (in particular, when ${\xi}$ has no positive jumps). By analyticity, \[assump:H1,assump:H2\] imply that ${\Psi}_n(z)\to{\Psi}(z)$ as $n\to\infty$, for $0\le\Re z\le{{q^*}}$. Further, since ${\Psi}<{\kappa}$, the first condition in  implies that ${\Psi}({{q^*}})<0$. Besides, assuming that ${\kappa}(q)\le0$ for some $q>0$ is necessary (and sufficient) [@Stephenson16] to prevent local explosion of the growth-fragmentation ${{\boldsymbol{\mathbf{Y}}}}$ (a phenomenon which would not allow us to view ${{\boldsymbol{\mathbf{Y}}}}$ in some $\ell^q$-space). Under  and , the condition  in  yields the convergence ${\kappa}_n(z)\to{\kappa}(z)$ for $\Re z$ in a left-neighbourhood of ${{q^*}}$. Thus, while  states the convergence to a characteristic exponent related to ${Y}$, \[assump:H3\] adds the convergence to a cumulant function, which constitute a key feature of self-similar growth-fragmentations [@Shi17]. We stress that our assumptions do not provide any control on the “small particles” ($n\le{M}$). Therefore we need to “freeze” these particles (meaning that they no longer grow or beget children), since otherwise their number could become quickly very high and make the system explode, as we illustrate in the example below. Basically, ${M}$ will be taken so that ${\kappa}_n(q)\le0$ for some $q$ and all $n$, which can be seen as a discrete-level analogue of the non-explosion condition. [exm:cex]{} Suppose that a particle with size $n$ increases to size $n+1$ with probability $p<1/2$ and, at least when $n$ is small, splits into two particles with sizes $1$ and $n-1$ with probability $1-p$. Thus, at small sizes, the unstopped Markov chain essentially behaves like a simple random walk. On the one hand, we know from Cramér’s theorem that for every ${\varepsilon}>0$ sufficiently small, $${\mathbb P}\Bigl({X}^{(1)}(k)>(1-2p+{\varepsilon})k\Bigr),\qquad k\ge0,$$ decreases exponentially at a rate $c_p({\varepsilon})>0$. On the other hand, the number of particles with size $1$ is bounded from below by $Z^{[1]}$, where ${\boldsymbol{\mathbf{Z}}}{\coloneqq}(Z^{[1]},Z^{[2]})$ is a $2$-type Galton–Watson process whose mean-matrix $$\begin{pmatrix}0&1\\2(1-p)&0\end{pmatrix}$$ has spectral radius $r_p{\coloneqq}\sqrt{2(1-p)}$, so that by the Kesten–Stigum theorem the number of particles with size $1$ at time $k\to\infty$ is of order at least $r_p^k$, almost surely. Consequently the expected number of particles which are above $(1-2p+{\varepsilon})k$ at time $2k$ is of exponential order at least $m_p({\varepsilon}){\coloneqq}\log r_p-c_p({\varepsilon})$. It is easily checked that this quantity may be positive (e.g., $m_{1/4}(1/4)\ge0.16$). Thus, without any “local” assumption on the reproduction law at small sizes, the number of small particles may grow exponentially and we cannot in general expect ${\mathds X}^{(n)}(\lfloor{a}_n\cdot\rfloor)/n$ to be tight in ${\ell^{q\downarrow}}$, for some $q>0$. However, this happens to be the case for the perimeters of the cycles in the branching peeling process of random Boltzmann triangulations [@Kortchemski15], where versions of \[thm:Hilbert,thm:Grothendieck\] hold for ${\gamma}=1/2$, ${{q^*}}=3$, and ${M}=0$, although ${\kappa}_n(3)\le0$ seems fulfilled only for ${M}\ge3$ (which should mean that the holes with perimeter $1$ or $2$ do not contribute to a substantial part of the triangulation). We start with the relatively easy convergence of finite-dimensional marginals (\[sec:Euler\]). Then, we develop a few key results (\[sec:Fermat21\]) that will be helpful to complete the proofs of \[thm:Hilbert\] (\[sec:Hilbert\]) and \[thm:Grothendieck\] (\[sec:Grothendieck\]). Convergence of finite-dimensional marginals {#sec:Euler} =========================================== In this section, we prove finite-dimensional convergences for both the particle process ${{\boldsymbol{\mathbf{X}}}}$ and its genealogical structure ${\mathcal X}$. (We mention that the freezing procedure is of no relevance here as it will be only useful in the next section to establish tightness results; in particular the freezing threshold ${M}$ will be fixed later.) We start by adopting a representation of the particle system ${{\boldsymbol{\mathbf{X}}}}$ that better matches that of ${{\boldsymbol{\mathbf{Y}}}}$ given above. We define, for every word $u{\coloneqq}u_1\cdots u_i\in{\mathbb N}^i$, the [*$u$-locally largest particle*]{}[^2] $({X}_u(k)\colon k\ge0)$ by induction on $i=0,1,\ldots$ Initially, for $i=0$, there is a single particle ${X}_{\varnothing}$ labeled by $u={\varnothing}$, born at time ${\beta}_{\varnothing}{\coloneqq}0$ and distributed like ${X}$. Then, we enumerate the sequence $({\beta}_{u1},n_1),({\beta}_{u2},n_2),\ldots$ of the negative jump times and sizes of ${X}_u$ so that $n_1\ge n_2\ge\cdots$ and ${\beta}_{uj}<{\beta}_{u(j+1)}$ whenever $n_j=n_{j+1}$. The processes ${X}_{uj},\,j=1,2,\ldots$ are independent and distributed like ${X}^{(n_j)}$ respectively (for definiteness, we set ${\beta}_{uj}{\coloneqq}\infty$ and ${X}_{uj}{\mathrel{\rlap{\raisebox{0.3ex}{$\cdot$}} \raisebox{-0.3ex}{$\cdot$}}\equiv}0$ if ${X}_u$ makes less than $j$ negative jumps during its lifetime), and we have $$\bigl({{\boldsymbol{\mathbf{X}}}}(k)\bigr)_{k\ge0} {\stackrel d=}\Bigl({X}_u(k-{\beta}_u)\colon u\in{\mathbb U},\, {\beta}_u\le k\Bigr)_{k\ge0}.$$ Recall the notation $\cdot^{(n)}$ to stress that the system is started from a particle with size ${X}_{\varnothing}(0)=n$. [lem:Euler]{} Suppose . Then for every finite subset $U\subset{\mathbb U}$, there is the convergence in ${\mathbb D}({\mathopen{[}0,\infty\mathclose{)}},{\mathbb R}^U)$: $$\left(\frac{{X}_u^{(n)}(\lfloor{a}_nt\rfloor-{\beta}^{(n)}_u)}n \colon u\in U\right)_{t\ge0} {\xrightarrow[n\to\infty]{(d)}}\,\Bigl({Y}_u(t-{b}_u)\colon u\in U\Bigr)_{t\ge0}. \label{eq:Poincare2}$$ [prf:Euler]{} We detail the argument used to prove the second part of [@Kortchemski15 Lemma 17]. For $h\ge0$, let ${\mathbb U}_h{\coloneqq}\{u\in{\mathbb U}\colon|u|\le h\}$ be the set of vertices with height at most $h$ in the tree ${\mathbb U}$. It suffices to show &(I\_h) &,&&&&&&&&&& by induction on $h$. The statement $(\mathscr I_0)$ is given by . Now, if $U$ is a finite subset of ${\mathbb U}_{h+1}$ and $F_u,\,u\in U,$ are continuous bounded functions from ${\mathbb D}({\mathopen{[}0,\infty\mathclose{)}},{\mathbb R})$ to ${\mathbb R}$, then the branching property entails that, for ${\mathrm X}_u{\coloneqq}{X}_u(\lfloor{a}_n\cdot\rfloor-{\beta}_u)/n$, $${\mathbb E}\!\left[\prod_{u\in U}\! F^{\vphantom{(n)}}_u\bigl({\mathrm X}^{(n)}_u\bigr)\;\bigg|\; \bigl({X}_u^{(n)}\colon u\in{\mathbb U}_h\bigr)\right] \,\,=\!\!\prod_{u\in U\cap\,{\mathbb U}_h}\!\!\!\!\! F^{\vphantom{(n)}}_u\bigl({\mathrm X}^{(n)}_u\bigr) \;\cdot\!\! \prod_{\substack{u\in U\\|u|=h+1}}\!\!\!\! E_{{\mathrm X}^{(n)}_u(0)}^{(n)}\bigl[F_u\bigr],$$ where $E_x^{(n)}$ stands for expectation under the law $P_x^{(n)}$ of ${\mathrm X}_{\varnothing}$ started from $x$, which by , and , converges weakly as $n\to\infty$ to the law $P_x$ of ${Y}$ started from $x$. The values ${\mathrm X}^{(n)}_u(0)$ for $|u|=h+1$ correspond to (rescaled) negative jump sizes of particles at height $h$. With [@Jacod87 Corollary VI.2.8] and our convention of ranking the jump sizes in the non-increasing order, the convergence in distribution $({\mathrm X}^{(n)}_u(0)\colon u\in U,\,|u|=h+1) \to({Y}_u(0)\colon u\in U,\,|u|=h+1)$ as $n\to\infty$ thus holds jointly with $(\mathscr I_h)$. Further, thanks to the Feller property [@Lamperti72 Lemma 2.1] of ${Y}$, its distribution is weakly continuous in its starting point. By the continuous mapping theorem we therefore obtain, applying back the branching property, that $${\mathbb E}\!\left[\prod_{u\in U}\!F^{\vphantom{(n)}}_u \bigl({\mathrm X}^{(n)}_u\bigr)\right] {\xrightarrow[n\to\infty]{}}\,{\mathbb E}\!\left[\prod_{u\in U}\! F_u\bigl({Y}_u(\cdot-{b}_u)\bigr)\right]\!.$$ A priori, this establishes the convergence in distribution $({\mathrm X}^{(n)}_u\colon u\in U) \to({Y}_u(\cdot-{b}_u)\colon u\in U)$ only in the product space ${\mathbb D}({\mathopen{[}0,\infty\mathclose{)}},{\mathbb R})^U$. By [@Jacod87 Proposition 2.2] it will also hold in ${\mathbb D}({\mathopen{[}0,\infty\mathclose{)}},{\mathbb R}^U)$ provided that the processes ${Y}_u,\,u\in U,$ almost surely never jump simultaneously. But this is plain since particles evolve independently and the jumps of ${Y}$ are totally inaccessible. Thus $(\mathscr I_h)\implies(\mathscr I_{h+1})$. Next, we proceed to the convergence of the finite-dimensional marginals of ${\mathcal X}$, which we shall first formally construct. For each $u\in{\mathbb U}$ with ${\beta}_u<\infty$, let ${\zeta}_u$ denote the lifetime of the stopped Markov chain ${X}_u$. Recall the definition in \[sec:Gauss\] of the trees ${\mathcal Y},{\mathcal Y}_h,\,h\ge0$, related to ${{\boldsymbol{\mathbf{Y}}}}$’s genealogy, that echoes Rembart and Winkel’s construction [@Rembart16]. Similarly, let ${\mathcal X}_{u,0}$ simply consist of an edge with length ${\zeta}_u$, rooted at a vertex $u$. Recursively, define ${\mathcal X}_{u,h+1}$ by attaching to the edge ${\mathcal X}_{u,0}$ the trees ${\mathcal X}_{uj,k}$ at a distance ${\beta}_{uj}-{\beta}_u$ from the root $u$, respectively, for each born particle $uj,\,j\le h+1,$ descending from $u$. The tree ${\mathcal X}_h{\coloneqq}{\mathcal X}_{{\varnothing},h}$ is a finite tree whose vertices are labeled by the set ${{\mathbb U}^{(h)}}$ of words over $\{1,\ldots,h\}$ with length at most $h$. Plainly, the sequence ${\mathcal X}_h,\,h\ge0,$ is consistent, in that ${\mathcal X}_h$ is the subtree of ${\mathcal X}_{h+1}$ with vertex set ${{\mathbb U}^{(h)}}$, and we may consider the inductive limit ${\mathcal X}{\coloneqq}{\mathop{\mathrm{lim}\!\!\uparrow}}_{h\to\infty}{\mathcal X}_h$. We write ${d_n}(v,v')$ for the length of the unique path between $v$ and $v'$ in ${\mathcal X}^{(n)}$. All these trees belong to the space ${\mathscr T}$ of (equivalence classes of) compact, rooted, real trees and can be embedded as subspaces of a large metric space (such as, for instance, the space $\ell^1({\mathbb N})$ of summable sequences [@Aldous93 Section 2.2]). Irrespectively of the embedding, they can be compared one with each other through the so called Gromov–Hausdorff metric ${\mathrm{d}_{\mathrm{GH}}}$ on ${\mathscr T}$. We forward the reader to [@LeGall06; @Evans08] and references therein. [lem:Galois]{} Suppose . Then for all $h\in{\mathbb N}$, there is the convergence in $({\mathscr T},{\mathrm{d}_{\mathrm{GH}}})$: $$\left({\mathcal X}^{(n)}_h,\frac{{d_n}}{{a}_n}\right){\xrightarrow[n\to\infty]{(d)}}\,{\mathcal Y}_h.$$ It suffices to show the joint convergence of all branches. The branch going through the vertex $u\in{{\mathbb U}^{(h)}}$ has length ${\mathcal E}^{(n)}_u{\coloneqq}{\beta}^{(n)}_u+{\zeta}^{(n)}_u$ in $({\mathcal X}^{(n)}_h,{d_n})$, and length ${\epsilon}_u{\coloneqq}{b}_u+{\upzeta}_u$ in ${\mathcal Y}_h$. Recall that conditionally on $\{{X}_u(0)=n\}$, the random variable ${\zeta}_u$ has the same distribution as ${\zeta}^{(n)}{\coloneqq}{\zeta}^{(n)}_{\varnothing}$. By [@Kortchemski16 Theorem 3.(i)], the convergence $$\frac{{\zeta}^{(n)}}{{a}_n}{\xrightarrow[n\to\infty]{(d)}}{\upzeta}{\coloneqq}\inf\{t\ge0\colon{Y}(t)=0\}$$ holds jointly with . Adapting the proof of \[lem:Euler\], we can more generally check that for every finite subset $U\subset{\mathbb U}$, we have, jointly with , $$\left(\frac{{\mathcal E}^{(n)}_u}{{a}_n}\colon u\in U\right) {\xrightarrow[n\to\infty]{(d)}}\bigl({\epsilon}_u\colon u\in U\bigr).$$ In particular, this is true for $U{\coloneqq}{{\mathbb U}^{(h)}}$. To conclude this section, we restate an observation of Bertoin, Curien, and Kortchemski [@Kortchemski15 Lemma 21] which results from the convergence of finite-dimensional marginals (\[lem:Euler\]): with high probability as $h\to\infty$, “non-negligible” particles have their labels in ${{\mathbb U}^{(h)}}$. Specifically, say that an individual $u\in{\mathbb U}$ is $(n,{\varepsilon})$-good, and write $u\in{\mathcal G(n,{\varepsilon})}$, if the particles ${X}_v$ labeled by each ancestor $v$ of $u$ (including $u$ itself) have size at birth at least $n{\varepsilon}$. Then: [lem:Neumann]{} We have $$\mathop{\vphantom{\limsup}\lim}_{h\to\infty} \limsup_{\vphantom{h}n\to\infty}\; {\mathbb P}^{(n)}\!\left({\mathcal G(n,{\varepsilon})}\not\subseteq{{\mathbb U}^{(h)}}\right)=\,0.$$ A size-biased particle and a many-to-one formula {#sec:Fermat21} ================================================ We now introduce a “size-biased particle” and relate it to a many-to-one formula. This will help us derive tightness estimates in \[sec:Hilbert,sec:Grothendieck\], and thus complement the finite-dimensional convergence results of the preceding section. Recall from \[assump:H1,assump:H2,assump:H3\] that there exists ${{q_*}}<{{q^*}}$ such that, as $n\to\infty$, ${\kappa}_n(q)\to{\kappa}(q)<0$ for every $q\in{\mathopen{[}{{q_*}},{{q^*}}\mathclose{)}}$. Consequently, *we may and will suppose for the remaining of this section that the freezing threshold ${M}$ is taken sufficiently large so that ${\kappa}_n({{q_*}})\le0$ for every $n>{M}$*. (Note that ${\kappa}_n({{q_*}})=0$ for $n\le{M}$, by our convention ${p}_{n,n}{\coloneqq}1$.) [lem:Abel]{} For every $n\in{\mathbb N}$, $${\mathbb E}^{(n)}\!\left[\bigl({X}(1)\bigr)^{{q_*}}+\bigl(n-{X}(1)\bigr)_+^{{q_*}}\right]\le\,n^{{q_*}}. \label{eq:Abel}$$ Therefore, the process[^3] $$\sum_{u\in{\mathbb U}} \bigl({X}_u(k-{\beta}_u)\bigr)^{{q_*}},\qquad k\ge0,$$ is a supermartingale under ${\mathbb P}^{(n)}$. [prf:Abel]{} The left-hand side of  is $$n^{{q_*}}\sum_{m=0}^\infty{p}_{n,m} \left[\left(\frac mn\right)^{\!{{q_*}}} +\left(1-\frac mn\right)_{\!+}^{\!{{q_*}}}\right] =\,n^{{q_*}}\left(1+\frac{{\kappa}_n({{q_*}})}{{a}_n}\right)\!,$$ where ${\kappa}_n({{q_*}})\le0$. Hence the first part of the statement. The second part follows by applying the branching property at any given time $k\ge0$: [E]{}\^[(n)]{}&=\_[iI]{}[E]{}\^[(x\_i)]{}\ &\_[iI]{}(x\_i)\^[[q\_\*]{}]{}\ &=\_[u[U]{}]{} ([X]{}\_u(k-\_u))\^[[q\_\*]{}]{}. Put differently, the condition “${\kappa}_n({{q_*}})\le0$” formulates that $n\mapsto n^{{q_*}}$ is superharmonic with respect to the “fragmentation operator”. This map plays the same role as the function $f$ in [@Kortchemski15], where it takes the form of a cubic polynomial (${{q_*}}=3$) and $$\sum_{u\in{\mathbb U}}f\bigl({X}_u(k-{\beta}_u)\bigr),\qquad k\ge0,$$ is actually a martingale. To a broader extent, the map $n\mapsto n^{{q_*}}$ could be replaced by any regularly-varying sequence with index ${{q_*}}$, but probably at the cost of heavier notation. \[lem:Abel\] allows us to introduce a (defective) Markov chain $({\bar X}(k)\colon k\ge0)$ on ${\mathbb N}$, to which we add $0$ as cemetery state, with transition $${\mathbb E}^{(n)}\!\left[f\bigl({\bar X}(1)\bigr);\,{\bar X}(1)\neq0\right] =\sum_{m=1}^\infty{p}_{n,m}\left[\left(\frac mn\right)^{\!{{q_*}}} f(m) +\left(1-\frac mn\right)_{\!+}^{\!{{q_*}}} f(n-m)\right]\!. \label{eq:Weierstrass}$$ We let ${\bar\zeta}{\coloneqq}\inf\{k\ge0\colon{\bar X}(k)=0\}$ denote its lifetime. Up to a change of probability measure, ${\bar X}$ follows the trajectory of a randomly selected particle in ${{\boldsymbol{\mathbf{X}}}}$. It admits the following scaling limit (which could also be seen as a randomly selected particle in ${{\boldsymbol{\mathbf{Y}}}}$; see [@Curien16 Section 4]): [pro:Descartes]{} There is the convergence in distribution $$\left(\frac{{\bar X}^{(n)}(\lfloor{a}_nt\rfloor)}n\colon t\ge0\right) {\xrightarrow[n\to\infty]{(d)}}\,\left({\bar Y}(t)\colon t\ge0\right)\!, \label{eq:Descartes}$$ in ${\mathbb D}({\mathopen{[}0,\infty\mathclose{)}},{\mathbb R})$, where the limit ${\bar Y}$ fulfills the same identity  as ${Y}$, but for a (killed) Lévy process ${\bar\xi}$ with characteristic exponent ${\bar{\kappa}}(q){\coloneqq}{\kappa}({{q_*}}+q)$. Further, if ${\bar\upzeta}$ denotes the lifetime of ${\bar Y}$, then the convergence $$\frac{{\bar\zeta}^{(n)}}{{a}_n}{\xrightarrow[n\to\infty]{(d)}}{\bar\upzeta}$$ holds jointly with . [prf:Descartes]{} Write ${\bar\Lambda}_n$ for the law of $\log({\bar X}^{(n)}(1)/n)$, with the convention $\log0{\coloneqq}-\infty$. We see from  that ${a}_n\,{\mathbb P}({\bar X}(1)=0)=-{\kappa}_n({{q_*}})$, and, for every $0\le q\le{{q^*}}-{{q_*}}$, $$\begin{aligned} \int_{\mathbb R}(e^{qy}-1)\,{a}_n{\bar\Lambda}_n({\mathrm d}y) &={a}_n\sum_{m=0}^\infty{p}_{n,m}\left[\left(\frac mn\right)^{\!{{q_*}}} \left(\!\left(\frac mn\right)^{\!q}-1\right) +\left(1-\frac mn\right)_{\!+}^{\!{{q_*}}} \left(\!\left(1-\frac mn\right)^{\!q}-1\right)\right]\\ &={\kappa}_n({{q_*}}+q)-{\kappa}_n({{q_*}}).\end{aligned}$$ Hence $$-{a}_n{\bar\Lambda}_n(\{-\infty\})+\int_{\mathbb R}(e^{qy}-1)\,{a}_n{\bar\Lambda}_n({\mathrm d}y) ={\kappa}_n({{q_*}}+q){\xrightarrow[n\to\infty]{}}{\bar{\kappa}}(q).$$ Furthermore, by , $$\limsup_{n\to\infty}\;{a}_n\int_1^\infty e^{({{q^*}}-{{q_*}})y} \,{\bar\Lambda}_n({\mathrm d}y) \le\limsup_{n\to\infty}\;{a}_n\sum_{m=2n}^\infty{p}_{n,m} \left(\frac mn\right)^{\!{{q^*}}}<\,\infty.$$ In other words, assumptions (A1), (A2) and (A3) of [@Kortchemski16] are satisfied (w.r.t the Markov chain ${\bar X}$ and the limiting process ${\bar Y}$). Our statement thus follows from Theorems 1 and 2 there[^4]. Heading now toward pathwise and optional many-to-one formulae, we first set up some notation. Let ${\mathtt A}\subseteq{\mathbb N}$ be a fixed subset of states, and let $\ell\in\partial{\mathbb U}$ refer to an infinite word over ${\mathbb N}$, which we see as a branch of ${\mathbb U}$. For every $u\in{\mathbb U}\cup\partial{\mathbb U}$ and every $k\ge0$, set $${\widetilde X}_u(k){\coloneqq}{X}_{u[k]}(k-{\beta}_{u[k]}),$$ where $u[k]$ is the youngest ancestor $v$ of $u$ with ${\beta}_v\le k$, and write ${\tau^{{\mathtt A}}}_u{\coloneqq}\inf\{k\ge0\colon{\widetilde X}_u(k)\in{\mathtt A}\}$ for the first hitting time of ${\mathtt A}$ by ${\widetilde X}_u$. Let also ${\bar\tau^{{\mathtt A}}}{\coloneqq}\inf\{k\ge0\colon{\bar X}(k)\in{\mathtt A}\}$. Now, imagine that once a particle hits ${\mathtt A}$, it is stopped and thus has no further progeny. The state when all particles have hit ${\mathtt A}$ in finite time is ${x^{{\mathtt A}}}_u{\coloneqq}{\widetilde X}_u({\tau^{{\mathtt A}}}_u),\,u\in{\mathbb U}_{\mathtt A}$, where ${\mathbb U}_{\mathtt A}{\coloneqq}\{u\in{\mathbb U}\colon\ell[{\tau^{{\mathtt A}}}_\ell]=u \text{ for some }\ell\in\partial{\mathbb U}\text{ with } {\tau^{{\mathtt A}}}_\ell<\infty\}$. [lem:Dirichlet]{} 1. \[itm:many-to-one\] For every $n\in{\mathbb N}$, every $k\ge0$, and every $f\colon{\mathbb N}^{k+1}\to{\mathbb R}_+$, $${\mathbb E}^{(n)}\!\left[\sum_{u\in{\mathbb U}} \bigl({X}_u(k-{\beta}_u)\bigr)^{{q_*}}f\!\left({\widetilde X}_u(i)\colon i\le k\right) \right] =\,n^{{q_*}}\, {\mathbb E}^{(n)}\!\left[f\!\left({\bar X}(i)\colon i\le k\right)\!;\, {\bar\zeta}>k\right]\!.$$ 2. For every $n\in{\mathbb N}$, every ${\mathtt A}\subseteq{\mathbb N}$, and every $f\colon{\mathbb Z}_+\times{\mathbb N}\to{\mathbb R}_+$, $${\mathbb E}^{(n)}\!\left[\sum_{u\in{\mathbb U}_{\mathtt A}}\!\bigl({x^{{\mathtt A}}}_u\bigr)^{{q_*}}f\bigl({\tau^{{\mathtt A}}}_u,{x^{{\mathtt A}}}_u\bigr)\right] =\,n^{{q_*}}\, {\mathbb E}^{(n)}\!\left[f\bigl({\bar\tau^{{\mathtt A}}},{\bar X}\bigl({\bar\tau^{{\mathtt A}}}\bigl)\bigr);\, {\bar\zeta}>{\bar\tau^{{\mathtt A}}}\right]\!.$$ [prf:Dirichlet]{} (i)The proof is classical (see e.g.  [@Shi15 Theorem 1.1] and proceeds by induction on $k$. The identity clearly holds for $k=0$. Using  together with the branching property at time $k$, &[E]{}\^[(n)]{}&&\ &=\_[u[U]{}]{} \_[m=0]{}\^[p]{}\_[x\_[u,k]{},m]{}(m\^[[q\_\*]{}]{} f(x\_[u,0]{},…,x\_[u,k]{},m) +(x\_[u,k]{}-m)\_+\^[[q\_\*]{}]{} f(x\_[u,0]{},…,x\_[u,k]{},x\_[u,k]{}-m))\ &=\_[u[U]{}]{}(x\_[u,k]{})\^[[q\_\*]{}]{} [E]{}\^[(x\_[u,k)]{}]{}. &[E]{}\^[(n)]{}=n\^[[q\_\*]{}]{} [E]{}\^[(n)]{}&\ & =n\^[[q\_\*]{}]{} [E]{}\^[(n)]{}.& (ii)For every $k\ge0$ and every $x_0,\ldots,x_k\in{\mathbb N}$, we set $f^{\mathtt A}_k(x_0,\ldots,x_k){\coloneqq}{\mathds 1}_{\{x_0\notin{\mathtt A},\ldots,x_{k-1}\notin{\mathtt A},x_k\in{\mathtt A}\}}f(k,x_k)$. Then $$\begin{aligned} {\mathbb E}^{(n)}\!\left[\sum_{u\in{\mathbb U}_{\mathtt A}}\!\bigl({x^{{\mathtt A}}}_u\bigr)^{{q_*}}f\bigl({\tau^{{\mathtt A}}}_u,{x^{{\mathtt A}}}_u\bigr)\right] &=\,\sum_{k=0}^\infty {\mathbb E}^{(n)}\!\left[\sum_{u\in{\mathbb U}}\bigl({X}_u(k-b_u)\bigr)^{{q_*}}f^{\mathtt A}_k\!\left({\widetilde X}_u(i)\colon i\le k\right) \right]\\[.4em] &=\,n^{{q_*}}\,\sum_{k=0}^\infty {\mathbb E}^{(n)}\!\left[f\bigl(k,{\bar X}(k)\bigr);\,{\bar\tau^{{\mathtt A}}}=k;\, {\bar\zeta}>k\right]\\[.4em] &=\,n^{{q_*}}\, {\mathbb E}^{(n)}\!\left[f\bigl({\bar\tau^{{\mathtt A}}},{\bar X}\bigl({\bar\tau^{{\mathtt A}}}\bigl)\bigr);\, {\bar\zeta}>{\bar\tau^{{\mathtt A}}}\right]\!,\end{aligned}$$ by  and the monotone convergence theorem. We now combine \[pro:Descartes\] and \[lem:Dirichlet\] to derive the following counterpart of [@Kortchemski15 Lemma 14] that we will apply in the next two sections. Consider the hitting set ${\mathtt A}{\coloneqq}\{1,\ldots,\lfloor n{\varepsilon}\rfloor\}$ and denote by ${x^{\le n{\varepsilon}}}_u{\coloneqq}{x^{{\mathtt A}}}_u,\,u\in{\mathbb U}^{\le n{\varepsilon}}{\coloneqq}{\mathbb U}_{\mathtt A},$ the population of particles stopped below $n{\varepsilon}$. [cor:Ramanujan]{} We have $$\mathop{\vphantom{\limsup}\lim}_{{\varepsilon}\to0}\, \limsup_{\vphantom{{\varepsilon}}n\to\infty}\; n^{-{{q_*}}}\, {\mathbb E}^{(n)}\!\left[\sum_{u\in{\mathbb U}^{\le n{\varepsilon}}}\!\!\!\! \bigl({x^{\le n{\varepsilon}}}_u\bigr)^{{q_*}}\right] =\,0.$$ [prf:Ramanujan]{} By \[lem:Dirichlet\], $$\begin{aligned} n^{-{{q_*}}}\, {\mathbb E}^{(n)}\!\left[\sum_{u\in{\mathbb U}^{\le n{\varepsilon}}}\!\!\!\! \bigl({x^{\le n{\varepsilon}}}_u\bigr)^{{q_*}}\right] &=\,{\mathbb P}^{(n)}\!\left({\bar\zeta}>{\bar\tau^{\le n{\varepsilon}}}\right)\!, \intertext{where ${\bar\tau^{\le n{\varepsilon}}}{\coloneqq}\inf\{k\ge0\colon{\bar X}(k)\le n{\varepsilon}\}$. Thus, if~${\bar\upzeta}$ is the lifetime of~${\bar Y}$ and ${\bar\uptau^{\le{\varepsilon}}}{\coloneqq}\inf\{t\ge0\colon{\bar Y}(t)\le{\varepsilon}\}$, then by \cref{pro:Descartes} and the continuous mapping theorem,} \limsup_{n\to\infty}\; n^{-{{q_*}}}\, {\mathbb E}^{(n)}\!\left[\sum_{u\in{\mathbb U}^{\le n{\varepsilon}}}\!\!\! \bigl({x^{\le n{\varepsilon}}}_u\bigr)^{{q_*}}\right] &\le\,{\mathbb P}\!\left({\bar\upzeta}>{\bar\uptau^{\le{\varepsilon}}}\right)\!, \end{aligned}$$ which tends to $0$ as ${\varepsilon}\to0$. Proof of \[thm:Hilbert\] {#sec:Hilbert} ======================== We prove \[thm:Hilbert\] by combining \[lem:Euler\] with the next two “tightness” properties. We suppose that \[assump:H1,assump:H2,assump:H3\] hold and recall that ${{\mathbb U}^{(h)}}\subset{\mathbb U}$ refers to the set of words over $\{1,\ldots,h\}$ with length at most $h$. [lem:Jacobi1]{} For every $\delta>0$, $$\mathop{\vphantom{\limsup}\lim}_{h\to\infty}\; {\mathbb P}\!\left(\sup_{\vphantom{h}t\ge0} \sum_{u\in{\mathbb U}\setminus{{\mathbb U}^{(h)}}}\!\!\!\! \bigl({Y}_u(t-{b}_u)\bigr)^{{q^*}}\!>\delta\right)=\;0.$$ [lem:Jacobi1]{} This was already derived in [@Kortchemski15 Lemma 20], and results from the following fact [@Bertoin17 Corollary 4]: $${\mathbb E}\!\left[\sum_{u\in{\mathbb U}}\sup_{t\ge0}\; \bigl({Y}_u(t-{b}_u)\bigr)^q\right]<\,\infty \quad\text{for}\enspace{\kappa}(q)<0.$$ [lem:Jacobi2]{} If ${M}$ is sufficiently large, then for every $\delta>0$, $$\mathop{\vphantom{\limsup}\lim}_{h\to\infty} \limsup_{\vphantom{h}n\to\infty}\; {\mathbb P}^{(n)}\!\left(\sup_{\vphantom{h}k\ge0} \sum_{u\in{\mathbb U}\setminus{{\mathbb U}^{(h)}}}\!\!\!\! \bigl({X}_u(k-{\beta}_u)\bigr)^{{q^*}}>\delta n^{{q^*}}\right) =\;0.$$ [lem:Jacobi2]{} Let us first take ${{q_*}}<{{q^*}}$ and ${M}$ as in \[sec:Fermat21\]. As in the proof of [@Kortchemski15 Lemma 22] and by definition of ${\mathcal G(n,{\varepsilon})}$ in \[sec:Euler\], we claim that each particle in $\{{X}_u(k-{\beta}_u)\colon u\in{\mathbb U}\setminus{\mathcal G(n,{\varepsilon})}\}$ has an ancestor with size at birth smaller than $n{\varepsilon}$. Thanks to the branching property, we may therefore consider that these particles derive from a system that has first been “frozen” below the level $n{\varepsilon}$, that is, with the notations of \[sec:Fermat21\], from a particle system having ${x^{\le n{\varepsilon}}}_u,\,u\in{\mathbb U}^{\le n{\varepsilon}},$ as initial population. Hence, by \[lem:Abel\] and Doob’s maximal inequality, $$\begin{aligned} {\mathbb P}^{(n)}\!\left(\sup_{k\ge0} \sum_{u\in{\mathbb U}\setminus{\mathcal G(n,{\varepsilon})}}\!\!\!\! \bigl({X}_u(k-{\beta}_u)\bigr)^{{q^*}}>\delta n^{{q^*}}\right) &\le\,\frac1{\delta^{{{q_*}}/{{q^*}}}\,n^{{q_*}}}\, {\mathbb E}^{(n)}\!\left[\sum_{u\in{\mathbb U}^{\le n{\varepsilon}}}\!\!\! \bigl({x^{\le n{\varepsilon}}}_u\bigr)^{{q_*}}\right] \end{aligned}$$ (bounding from above the ${\ell^{{{q^*}}}}$-norm by the ${\ell^{{{q_*}}}}$-norm). We conclude by \[cor:Ramanujan\] and \[lem:Neumann\]. [thm:Hilbert]{} From \[lem:Euler,lem:Jacobi1,lem:Jacobi2\], we deduce the convergence in distribution $$\left(\frac{{X}_u^{(n)}(\lfloor{a}_nt\rfloor-{\beta}^{(n)}_u)}n\colon u\in{\mathbb U}\right)_{t\ge0} {\xrightarrow[n\to\infty]{(d)}}\,\bigl(Y_u(t-{b}_u)\colon u\in{\mathbb U}\bigr)_{t\ge0},$$ in the space ${\mathbb D}({\mathopen{[}0,\infty\mathclose{)}},{\ell^{{{q^*}}}}({\mathbb U}))$ of ${\ell^{{{q^*}}}}({\mathbb U})$-valued càdlàg functions on ${\mathopen{[}0,\infty\mathclose{)}}$, where $${\ell^{{{q^*}}}}({\mathbb U}){\coloneqq}\left\{{\boldsymbol{\mathbf{x}}}{\coloneqq}(x_u\colon u\in{\mathbb U})\colon\sum_{u\in{\mathbb U}}(x_u)^{{q^*}}<\infty\right\}\!.$$ Since for $q\ge1$, rearranging sequences in the non-increasing order does not increase their $q$-distance [@Lieb01 Theorem 3.5], the convergence in ${\ell^{{{q^*}}}}({\mathbb U})$ implies that in ${\ell^{q\downarrow}},\,q\ge1\vee{{q^*}}$. Proof of \[thm:Grothendieck\] {#sec:Grothendieck} ============================= Similarly to the previous section, by \[lem:Galois\] the proof of \[thm:Grothendieck\] is complete once we have established that $$\begin{gathered} \lim_{h\to\infty}{\mathbb P}\Bigl({\mathrm{d}_{\mathrm{GH}}}\bigl({\mathcal Y},{\mathcal Y}_h)>\delta\Bigr) \,=\,0,\notag \shortintertext{and} \mathop{\vphantom{\limsup}\lim}_{h\to\infty} \limsup_{\vphantom{h}n\to\infty}\; {\mathbb P}^{(n)}\!\left({\mathrm{d}_{\mathrm{GH}}}\Bigl({\mathcal X}^{(n)}, {\mathcal X}^{(n)}_h\Bigr)>\delta{a}_n\right) =\,0, \label{eq:Ramanujan}\end{gathered}$$ for all $\delta>0$. The first display is clear since the tree ${\mathcal Y}$ is compact. The second is a consequence of the following counterpart of [@Kortchemski15 Conjecture 1]: [lem:Cantor]{} Suppose , and ${{q^*}}>{\gamma}$. Then for every $q<{{q^*}}$, and for ${M}$ sufficiently large, $$\sup_{n\in{\mathbb N}}\;{\mathbb E}\!\left[ \left(\frac{\operatorname{ht}\bigl({\mathcal X}^{(n)}\bigr)}{ {a}_n}\right)^{\!q/{\gamma}\,}\right] <\,\infty,$$ where $\operatorname{ht}\bigl({\mathcal X}^{(n)}\bigr){\coloneqq}\sup_{x\in{\mathcal X}^{(n)}}{d_n}({\varnothing},x)$ is the height of the tree ${\mathcal X}^{(n)}$. The proof of \[lem:Cantor\] involves martingale arguments. Let us formerly derive . [eq:Ramanujan]{} We start as in the proof of \[thm:Hilbert\]: thanks to \[lem:Neumann\] and the branching property, with high probability as $h\to\infty$, the connected components of ${\mathcal X}^{(n)}\setminus{\mathcal X}^{(n)}_h$ are included in independent copies of ${\mathcal X}$ stemming from the population ${x^{\le n{\varepsilon}}}_u,\,u\in{\mathbb U}^{\le n{\varepsilon}},$ of particles frozen below $n{\varepsilon}$. Specifically, $${\mathbb P}^{(n)}\!\left({\mathrm{d}_{\mathrm{GH}}}\Bigl({\mathcal X}^{(n)}, {\mathcal X}^{(n)}_h\Bigr)>\delta{a}_n\right) \le\,{\mathbb P}^{(n)}\!\left({\mathcal G(n,{\varepsilon})}\not\subseteq{{\mathbb U}^{(h)}}\right) +{\mathbb E}^{(n)}\!\left[ \sum_{u\in{\mathbb U}^{\le n{\varepsilon}}}\!\!\! {\mathbb P}^{({x^{\le n{\varepsilon}}}_u)}\Bigl(\operatorname{ht}({\mathcal X})>\delta{a}_n\Bigr)\right]\!.$$ Now, take ${{q_*}}<q<{{q^*}}$ and ${M}$ large enough so that both \[lem:Cantor\] and the results of \[sec:Fermat21\] hold. So, there exists a constant $C>0$ such that $$\begin{aligned} {\mathbb E}^{(n)}\!\left[ \sum_{u\in{\mathbb U}^{\le n{\varepsilon}}}\!\!\! {\mathbb P}^{({x^{\le n{\varepsilon}}}_u)}\Bigl(\operatorname{ht}({\mathcal X})>\delta{a}_n\Bigr)\right] &=\,{\mathbb E}^{(n)}\!\left[\sum_{u\in{\mathbb U}^{\le n{\varepsilon}}}\!\!\! {\mathbb P}^{({x^{\le n{\varepsilon}}}_u)}\!\left(\operatorname{ht}({\mathcal X})>\delta{a}_{{x^{\le n{\varepsilon}}}_u} \frac{{a}_n}{{a}_{{x^{\le n{\varepsilon}}}_u}}\right)\right]\\[.4em] &\le\,C\,{\mathbb E}^{(n)}\!\left[\sum_{u\in{\mathbb U}^{\le n{\varepsilon}}}\!\! \left(\frac{{a}_{{x^{\le n{\varepsilon}}}_u}}{{a}_n}\right)^{\!q/{\gamma}\,}\right]\!.\end{aligned}$$ But we know by Potter’s bounds [@Bingham87 Theorem 1.5.6] that we may find $c>0$ such that $$\left(\frac{{a}_m}{{a}_n} \right)^{q/{\gamma}}\le c\left(\frac mn\right)^{\!{{q_*}}\,}\!,$$ whenever $n$ is sufficiently large and $m\le n$. Since ${x^{\le n{\varepsilon}}}_u\le n$ (for $0<{\varepsilon}<1$), we can again conclude by \[cor:Ramanujan\] and \[lem:Neumann\]. Prior to prove \[lem:Cantor\], we need a preparatory lemma. Let us define $${\widetilde{\kappa}}_n(q){\coloneqq}{a}_n\sum_{m=1}^\infty{p}_{n,m} \left[\left(\frac{{a}_m}{{a}_n}\right)^{\!q/{\gamma}}-1 +\left(\frac{{a}_{n-m}}{{a}_n}\right)^{\!q/{\gamma}\,} \right]\!,$$ which slightly differs from ${\kappa}_n(q)$ to the extent that we have replaced the map $m\mapsto m^q$ by the $q$-regularly-varying sequence ${A_{q}}(m){\coloneqq}{a}_m^{q/{\gamma}},\,m\in{\mathbb N}$ (for convenience, we have set ${a}_m{\coloneqq}0,\,m\le0$). Of course, ${\widetilde{\kappa}}_n={\kappa}_n$ if ${a}_m=m^{\gamma}$ for every $m\in{\mathbb N}$. [lem:Cauchy]{} There exists ${{q_*}}<{{q^*}}$ such that, for every $q\in{\mathopen{[}{{q_*}},{{q^*}}\mathclose{)}}$, $$\lim_{n\to\infty}{\widetilde{\kappa}}_n(q)={\kappa}(q)<0.$$ [prf:Cauchy]{} We will more generally show that for every $q$-regularly-varying sequence $(r_n)$, $$\left|{a}_n\sum_{m=1}^\infty{p}_{n,m} \left[\frac{r_m}{r_n}-\left(\frac mn\right)^{\!q\,}\right]\right| +\left|{a}_n\sum_{m=1}^{n-1}{p}_{n,m}\left[\frac{r_{n-m}}{r_n} -\left(1-\frac mn\right)^{\!q\,}\right]\right| {\xrightarrow[n\to\infty]{}}\,0,$$ provided $q<{{q^*}}$ is close enough to ${{q^*}}$. Denoting by ${\Lambda}_n$ the law of $\log({X}^{(n)}(1)/n)$, we observe that $${a}_n\sum_{m=1}^\infty{p}_{n,m} \left[\frac{r_m}{r_n}-\left(\frac mn\right)^{\!q\,}\right] ={a}_n\int_{-\infty}^\infty \left[\left(\frac{r_{ne^x}}{r_n}\right) -e^{qx}\right]{\Lambda}_n({\mathrm d}x),$$ which, by repeating the arguments in [@Kortchemski16 Proof of Lemma 4.9], tends to $0$ as $n\to\infty$. Next, another application of Potter’s bounds shows that for every $c>1$ and $\delta>0$ arbitrary small, $$\frac1c\left(\frac mn\right)^{\!q+\delta} \le\frac{r_m}{r_n} \le\,c\left(\frac mn\right)^{\!q-\delta}$$ whenever $m<n$ are sufficiently large. Thus, recalling that ${\Psi}_n(q)\to{\Psi}(q)$ and ${\kappa}_n(q)\to{\kappa}(q)$ for every $q$ in some left-neighbourhood of ${{q^*}}$, we have $$\begin{aligned} \liminf_{n\to\infty}\;{a}_n\sum_{m=1}^{n-1}{p}_{n,m}\left[ \frac{r_{n-m}}{r_n}-\left(1-\frac mn\right)^{\!q\,}\right] &\ge\,\frac1c\,\Bigl({\kappa}(q+\delta)-{\Psi}(q+\delta)\Bigr) -\Bigl({\kappa}(q)-{\Psi}(q)\Bigr),\\ \intertext{and} \limsup_{n\to\infty}\;{a}_n\sum_{m=1}^{n-1}{p}_{n,m}\left[ \frac{r_{n-m}}{r_n}-\left(1-\frac mn\right)^{\!q\,}\right] &\le\,\frac1c\,\Bigl({\kappa}(q-\delta)-{\Psi}(q-\delta)\Bigr) -\Bigl({\kappa}(q)-{\Psi}(q)\Bigr). \end{aligned}$$ We conclude by letting $c\to1$ and $\delta\to0$. We finally proceed to the proof of \[lem:Cantor\]. [lem:Cantor]{} We shall rely on a Foster–type technique close to the machinery developed in [@Aspandiiarov98]; see in particular the proof of Theorem 2’ there. First, observe that $\operatorname{ht}({\mathcal X})$ is distributed like the extinction time ${\mathcal E}$ of ${{\boldsymbol{\mathbf{X}}}}$: $$\operatorname{ht}\bigl({\mathcal X}^{(n)}\bigr) {\stackrel d=}\,\sup_{u\in{\mathbb U}}\;{\mathcal E}^{(n)}_u\, {\eqqcolon}\,{\mathcal E}^{(n)}.$$ Fix $q\in{\mathopen{(}{\gamma},{{q^*}}\mathclose{)}}$ arbitrary close to ${{q^*}}$ and set $r{\coloneqq}q/{\gamma}$. By \[lem:Cauchy\], suppose ${M}$ large enough so that ${\widetilde{\kappa}}_m(q)<0$ for every $m>{M}$. It is easy to see as in the proof of \[lem:Abel\] that the process $$\Gamma(k){\coloneqq}\sum_{u\in{\mathbb U}} {A_{q}}\bigl({X}_u(k-{\beta}_u)\bigr),\qquad k\ge0,$$ is a supermartingale under ${\mathbb P}^{(n)}$ (with respect to the natural filtration $(\mathcal F_k)_{k\ge0}$ of ${{\boldsymbol{\mathbf{X}}}}$): indeed, for ${{\boldsymbol{\mathbf{X}}}}(k)=(x_i\colon i\in I)$, $${\mathbb E}^{(n)}\!\left[\Gamma(k+1)-\Gamma(k)\;\Big|\; \mathcal F_k\right] =\,\sum_{i\in I}{\widetilde{\kappa}}_{x_i}(q)\,{A_{q-{\gamma}}}(x_i),$$ where the right-hand side is (strictly) negative on the event $\{{\mathcal E}>k\}=\{\exists i\in I\colon x_i>{M}\}$. We will more precisely show the existence of $\eta>0$ sufficiently small such that the process $$G(k){\coloneqq}\left(\Gamma(k)^{1/r}+\eta\,\bigl({\mathcal E}\wedge k \bigr)\right)^{\!r\!},\qquad k\ge0,$$ is a $(\mathcal F_k)_{k\ge0}$-supermartingale under ${\mathbb P}^{(n)}$, for any $n\in{\mathbb N}$. Then, the result will be readily obtained from $\eta^r\,{\mathbb E}^{(n)}[({\mathcal E}\wedge k)^r]\le{\mathbb E}^{(n)}[G(k)]\le{\mathbb E}^{(n)}[G(0)] ={A_{q}}(n)={a}_n^r$ and an appeal to Fatou’s lemma. On the one hand, we have $$\sigma{\coloneqq}\sum_{i\in I}{A_{q-{\gamma}}}(x_i) \ge\left(\sum_{i\in I}{A_{q}}(x_i)\right)^{\!1-{\gamma}/q}$$ because $$\frac{{A_{q}}(x_i)}{\sigma^{q/(q-{\gamma})}}\,= \left(\frac{{A_{q-{\gamma}}}(x_i)}\sigma\right)^{\!q/(q-{\gamma})} \le\,\frac{{A_{q-{\gamma}}}(x_i)}{\sigma},$$ where $q/(q-{\gamma})>1$ and the right-hand side sums to $1$ as $i$ ranges over $I$. Then, if we let $\eta>0$ sufficiently small such that ${\widetilde{\kappa}}_m(q)\le -r\eta$ for every $m>{M}$, we deduce that $${\mathbb E}^{(n)}\!\left[\Gamma(k+1)\;\Big|\;\mathcal F_k\right] \le\,\Gamma(k)\left(1-r\eta\,\Gamma(k)^{-{\gamma}/q} \,{\mathds 1}_{\{{\mathcal E}>k\}}\right)\!.$$ Raising this to the power $1/r={\gamma}/q$ yields $${\mathbb E}^{(n)}\!\left[\Gamma(k+1)\;\Big|\;\mathcal F_k\right]^{1/r} \le\,\Gamma(k)^{1/r}\left(1-\eta\,\Gamma(k)^{{\gamma}/(q-{\gamma})} \,{\mathds 1}_{\{{\mathcal E}>k\}}\right) =\,\Gamma(k)^{1/r}-\eta\,{\mathds 1}_{\{{\mathcal E}>k\}}, \label{eq:Weyl}$$ by concavity of $x\mapsto x^{1/r}$. On the other hand, the supermartingale property also implies that $(\Gamma(k+1)^{1/r}+a)^r$ is integrable for every constant $a>0$; we may thus apply the generalized triangle inequality [@Aspandiiarov98 Lemma 1] with the positive, convex increasing function $x\mapsto x^r$, the positive random variable $\Gamma(k+1)^{1/r}$, and the probability ${\mathbb P}^{(n)}(\;\cdot\mid\mathcal F_k)$ (under which ${\mathcal E}\wedge(k+1)$ can be seen as a positive constant): $${\mathbb E}^{(n)}\!\left[\left(\Gamma(k+1)^{1/r}+\eta\,\bigl({\mathcal E}\wedge(k+1) \bigr)\right)^r\;\Big|\;\mathcal F_k\right]^{1/r} \le\;{\mathbb E}^{(n)}\bigl[\Gamma(k+1)\;\big|\;\mathcal F_k\bigr]^{1/r} +\eta\,\bigl({\mathcal E}\wedge(k+1)\bigr).$$ Reporting  shows as desired that $(G(k)\colon k\ge0)$ is a supermartingale. Acknowledgments {#acknowledgments .unnumbered} =============== The author would like to thank Jean Bertoin for his constant suggestions and guidance, and also Bastien Mallein for helpful discussions and comments. [10]{} D. Aldous, *The continuum random tree. [I]{}*, Ann. Probab. **19** (1991), no. 1, 1–28. [<span style="font-variant:small-caps;">mr</span>: [](http://www.ams.org/mathscinet-getitem?mr=1085326)]{} [to3em]{}, *The continuum random tree. [II]{}. [A]{}n overview*, Stochastic analysis ([D]{}urham, 1990), London Math. Soc. Lecture Note Ser., vol. 167, Cambridge Univ. Press, Cambridge, 1991, pp. 23–70. [<span style="font-variant:small-caps;">mr</span>: [](http://www.ams.org/mathscinet-getitem?mr=1166406)]{} [to3em]{}, *The continuum random tree. [III]{}*, Ann. Probab. **21** (1993), no. 1, 248–289. [<span style="font-variant:small-caps;">mr</span>: [](http://www.ams.org/mathscinet-getitem?mr=1207226)]{} S. Aspandiiarov and R. Iasnogorodski, *General criteria of integrability of functions of passage-times for non-negative stochastic processes and their applications*, Teor. Veroyatnost. i Primenen. **43** (1998), no. 3, 509–539. [<span style="font-variant:small-caps;">mr</span>: [](http://www.ams.org/mathscinet-getitem?mr=1681072)]{} J. Bertoin, *Markovian growth-fragmentation processes*, Bernoulli **23** (2017), no. 2, 1082–1101. [<span style="font-variant:small-caps;">mr</span>: [](http://www.ams.org/mathscinet-getitem?mr=3606760)]{} J. Bertoin, T. Budd, N. Curien, and I. Kortchemski, *Martingales in self-similar growth-fragmentations and their connections with random planar maps*, 2016. [<span style="font-variant:small-caps;">arXiv</span>: [](http://arXiv.org/abs/1605.00581)]{} J. Bertoin, N. Curien, and I. Kortchemski, *Random planar maps & growth-fragmentations*, 2015, To appear in *Ann. Probab.* [<span style="font-variant:small-caps;">arXiv</span>: [](http://arXiv.org/abs/1507.02265)]{} J. Bertoin and I. Kortchemski, *Self-similar scaling limits of [M]{}arkov chains on the positive integers*, Ann. Appl. Probab. **26** (2016), no. 4, 2556–2595. [<span style="font-variant:small-caps;">mr</span>: [](http://www.ams.org/mathscinet-getitem?mr=3543905)]{} J. Bertoin and R. Stephenson, *Local explosion in self-similar growth-fragmentation processes*, Electron. Commun. Probab. **21** (2016), Paper No. 66,12. [<span style="font-variant:small-caps;">mr</span>: [](http://www.ams.org/mathscinet-getitem?mr=3548778)]{} N. H. Bingham, C. M. Goldie, and J. L. Teugels, *Regular variation*, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1987. [<span style="font-variant:small-caps;">mr</span>: [](http://www.ams.org/mathscinet-getitem?mr=898871)]{} S. N. Evans, *Probability and real trees*, Lecture Notes in Mathematics, vol. 1920, Springer, Berlin, 2008, Lectures from the 35th Summer School on Probability Theory held in Saint-Flour, July 6–23, 2005. [<span style="font-variant:small-caps;">mr</span>: [](http://www.ams.org/mathscinet-getitem?mr=2351587)]{} B. Haas and G. Miermont, *The genealogy of self-similar fragmentations with negative index as a continuum random tree*, Electron. J. Probab. **9** (2004), no. 4,57–97. [<span style="font-variant:small-caps;">mr</span>: [](http://www.ams.org/mathscinet-getitem?mr=2041829)]{} [to3em]{}, *Self-similar scaling limits of non-increasing [M]{}arkov chains*, Bernoulli **17** (2011), no. 4, 1217–1247. [<span style="font-variant:small-caps;">mr</span>: [](http://www.ams.org/mathscinet-getitem?mr=2854770)]{} [to3em]{}, *Scaling limits of [M]{}arkov branching trees with applications to [G]{}alton-[W]{}atson and random unordered trees*, Ann. Probab. **40** (2012), no. 6, 2589–2666. [<span style="font-variant:small-caps;">mr</span>: [](http://www.ams.org/mathscinet-getitem?mr=3050512)]{} B. Haas, G. Miermont, J. Pitman, and M. Winkel, *Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models*, Ann. Probab. **36** (2008), no. 5, 1790–1837. [<span style="font-variant:small-caps;">mr</span>: [](http://www.ams.org/mathscinet-getitem?mr=2440924)]{} J. Jacod and A. N. Shiryaev, *Limit theorems for stochastic processes*, Grundlehren der Mathematischen Wissenschaften \[Fundamental Principles of Mathematical Sciences\], vol. 288, Springer-Verlag, Berlin, 1987. [<span style="font-variant:small-caps;">mr</span>: [](http://www.ams.org/mathscinet-getitem?mr=959133)]{} O. Kallenberg, *Foundations of modern probability*, second ed., Probability and its Applications (New York), Springer-Verlag, New York, 2002. [<span style="font-variant:small-caps;">mr</span>: [](http://www.ams.org/mathscinet-getitem?mr=1876169)]{} J. Lamperti, *Semi-stable [M]{}arkov processes. [I]{}*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete **22** (1972), 205–225. [<span style="font-variant:small-caps;">mr</span>: [](http://www.ams.org/mathscinet-getitem?mr=0307358)]{} J.-F. Le Gall, *Random real trees*, Ann. Fac. Sci. Toulouse Math. (6) **15** (2006), no. 1, 35–62. [<span style="font-variant:small-caps;">mr</span>: [](http://www.ams.org/mathscinet-getitem?mr=2225746)]{} E. H. Lieb and M. Loss, *Analysis*, second ed., Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. [<span style="font-variant:small-caps;">mr</span>: [](http://www.ams.org/mathscinet-getitem?mr=1817225)]{} G. Miermont, *Self-similar fragmentations derived from the stable tree. [II]{}. [S]{}plitting at nodes*, Probab. Theory Related Fields **131** (2005), no. 3, 341–375. [<span style="font-variant:small-caps;">mr</span>: [](http://www.ams.org/mathscinet-getitem?mr=2123249)]{} F. [Rembart]{} and M. [Winkel]{}, *[Recursive construction of continuum random trees]{}*, 2016. [<span style="font-variant:small-caps;">arXiv</span>: [](http://arXiv.org/abs/1607.05323)]{} Q. Shi, *Growth-fragmentation processes and bifurcators*, Electron. J. Probab. **22** (2017), Paper No. 15, 25. [<span style="font-variant:small-caps;">mr</span>: [](http://www.ams.org/mathscinet-getitem?mr=3622885)]{} Z. Shi, *Branching random walks*, Lecture Notes in Mathematics, vol. 2151, Springer, Cham, 2015, Lecture notes from the 42nd Probability Summer School held in Saint Flour, 2012, École d’Été de Probabilités de Saint-Flour. \[Saint-Flour Probability Summer School\]. [<span style="font-variant:small-caps;">mr</span>: [](http://www.ams.org/mathscinet-getitem?mr=3444654)]{} [^1]: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland. Email: benjamin.dadoun@math.uzh.ch [^2]: In the peeling of random Boltzmann maps [@Kortchemski15], the locally-largest cycles are called [*left-twigs*]{}. [^3]: We dropped the indicator ${\mathds 1}_{\{{\beta}_u\le \cdot\}}$ to not burden the notation. [^4]: Strictly speaking, the results are only stated when there is no killing, that is ${\kappa}({{q_*}})=0$, but as mentioned by the authors [@Kortchemski16 p. 2562, §2], they can be extended using the same techniques to the case where some killing is involved.

--- abstract: 'We report the first demonstration of cooling by three-body losses in a Bose gas. We use a harmonically confined one-dimensional (1D) Bose gas in the quasi-condensate regime and, as the atom number decreases under the effect of three-body losses, the temperature $T$ drops up to a factor four. The ratio $k_B T /(m c^2)$ stays close to 0.64, where $m$ is the atomic mass and $c$ the sound speed in the trap center. The dimensionless 1D interaction parameter $\gamma$, evaluated at the trap center, spans more than two order of magnitudes over the different sets of data. We present a theoretical analysis for a homogeneous 1D gas in the quasi-condensate regime, which predicts that the ratio $k_B T/(mc^2)$ converges towards 0.6 under the effect of three-body losses. More sophisticated theoretical predictions that take into account the longitudinal harmonic confinement and transverse effects are in agreement within 30% with experimental data.' author: - Max Schemmer - Isabelle Bouchoule title: 'Cooling a Bose gas by three-body losses' --- The identification and understanding of cooling processes, both on the theoretical and the experimental side, is crucial to the development of cold atom physics [@dalibard_laser_1989; @anderson_observation_1995]. It can help to elaborate strategies to enter new regimes and it can also improve the control over state preparation in experiments where cold atoms are used as quantum simulators of many body systems. Ultra-cold atom gases are metastable systems, their ground state being a solid phase. They are thus plagued with intrinsic recombination processes, that in practice limit their lifetime. Such process are mainly three-body collisions during which a strongly bound dimer is formed. It amounts to three-body losses because the dimer is typically no longer trapped and the remaining atom escapes because of its large kinetic energy. These losses are known to produce an undesired heating in cold gases. In the case of a thermal gas, since they occur predominantly in the regions of high atomic density, where the potential energy is low, these losses increase the energy per remaining particle, leading to an anti-evaporation process [@weber_three-body_2003]. In Bose-Einstein condensates (BEC) confined in deep traps, it was predicted that three-body collisions produce a heating of the BEC through secondary collisions with high energy excitations formed by the loss process [@guery-odelin_excitation-assisted_1999]. This paper constitutes a breakthrough since for the first time we identify a cooling associated to three-body losses in a cold Bose gas. The effect of losses has been investigated for 1D Bose gases in the quasicondensate regime [^1] in the case of one-body losses [@rauer_cooling_2016; @grisins_degenerate_2016; @johnson_long-lived_2017; @schemmer_monte_2017]. This work was recently extended [@bouchoule_cooling_2018-1] to any j-body loss process, for Bose gases in the BEC or quasicondensate regime, in any dimension $d$, and for homogeneous gases as well as gases confined in a smooth potential. Theses studies focus on the effect of losses on low energy excitations in the gas, the phononic modes, which correspond to density waves propagating in the condensate. On the one hand, the energy in these modes is reduced by losses since the amplitude of density modulations is decreased, removing interaction energy from the mode. On the other hand, the discrete nature of the loss process comes with accompanying shot noise which induces density fluctuations, increasing the energy per mode. It has been shown that the competition between these processes leads to a stationary value of the ratio $k_B T / (m c^2)$ where $m$ is the atom mass and $c$ the speed of sound. This value, of the order of one, depends on $j$, $d$, and on the confining potential [@bouchoule_cooling_2018-1]. For three-body losses in a 1D quasicondensates ($j=3$, $d=1$) confined in a harmonic potential one expects $k_B T/(mc_p^2)$ to converge to 0.70 [@bouchoule_cooling_2018-1], where $c_p$ is evaluated at the peak density. In this paper, we show for the first time experimentally that three-body losses induce a cooling and we identify the stationary value of $k_B T/(mc_p^2)$ associated to the three-body process. More precisely, investigating the time evolution of a 1D quasi-condensate, we observe a decrease of the temperature as the atom number decreases under the effect of three-body losses. Moreover, on the whole observed time-interval, the ratio $k_BT/(mc_p^2)$ stays about constant, at a value close to $0.64$, which indicates that the stationary value of $k_B T/(mc_p^2)$ imposed by the loss process is reached. We took several data sets for different parameters. In terms of the 1D dimensionless parameter $\gamma$ [@lieb_exact_1963] characterizing the strength of the interactions [^2], our data span more than two orders of magnitude. We compare the experimental data with numerical calculations based on the results of [@bouchoule_cooling_2018-1], which take into account the harmonic longitudinal confinement of the gas and the swelling of the transverse wave function under the effect of interactions. The experimental results are close to those predictions. In order to present the underlying physics, we derive in this paper the evolution of the temperature under three-body losses, in the more simple case of a homogeneous purely 1D quasicondensate. The experiment uses an atom-chip set up [^3] where $^{87}$Rb atoms are magnetically confined using current-carrying micro-wires. An elongated atomic cloud is prepared using radio frequency (RF) forced evaporative cooling in a trap of transverse frequency $\omega_{\perp}$. Depending on the data set, $\omega_{\perp}/(2\pi)$ varies between and and the atomic peak linear densities $n$ vary between $\unit[22]{\mu m^{-1}}$ and $\unit[257]{\mu m^{-1}}$. The temperature fulfills $k_BT<\hbar\omega_\perp$ and the gas mostly behaves as a 1D Bose gas [@armijo_mapping_2011]. It moreover lies in the quasicondensate regime [@kheruntsyan_pair_2003], characterized by weak correlations between atoms, as in a Bose-Einstein condensates [^4], and in particular small density fluctuations  [^5]. As long as the atoms are in the ground state of the transverse potential, interactions between atoms are well described by a 1D effective coupling constant $g=2\hbar\omega_\perp a$, where $a=$ is the 3D scattering length [^6], and the chemical potential is given by $\mu=gn$. This is valid only as long as $\mu\ll \hbar \omega_\perp$, which requires $na\ll 1$. In the presented data $na$ takes values as large as 1.3 and the broadening of the transverse wavefunction due to interactions has to be taken into account for quantitative analysis. In particular, the equation of state becomes $\mu = \hbar \omega_{\perp} (\sqrt{1+4na} - 1)$ [@fuchs_hydrodynamic_2003]. The quasi-condensates is confined in the longitudinal direction with a harmonic potential $V(z)$ of trapping frequency $\omega_z/(2\pi)=\unit[8.5]{Hz}$, weak enough so that the longitudinal profile $n_0(z)$ is well described by the Local Density Approximation (LDA), with a local chemical potential $\mu(z)=\mu_p-V(z)$, where $\mu_p$ is the peak chemical potential. It extends over $2R$ where the Thomas-Fermi radius $R$ fulfills $V(R)=\mu_p$. Once the quasicondensate is prepared, we increase the frequency of the radio-frequency field, by several kHz, a value sufficient so that it no longer induces losses. We then investigate the evolution during the waiting time $t$. Five different data set are investigated, differing in the value of the transverse confinement and the initial temperature and peak density. ![Peak density, in log scale, versus the waiting time $t$, for the five different data-sets. Solid lines are ab-initio calculations of the effect of three-body losses, for initial peak densities equal to that of the experimental data.[]{data-label="fig.n_p"}](n_p.pdf){width="\linewidth"} Using absorption images we record the density profile of the gas, from which we extract the peak density $n_p$. Fig. \[fig.n\_p\] shows evolution of $n_p$ with the waiting time $t$ for the different data sets. We observe a decrease of $n_p$ whose origin is three-body recombinations, as justified by calculations presented below. In a three-body recombination, a molecule (a dimer) is formed and its binding energy is released in the form of kinetic energy of the molecule and the remaining atom. They both leave the trap since their energy is typically much larger than the trap depth, limited by the radio-frequency field. Thus, the effect of three-body process is to decrease the gas density according to ${\rm d}\rho/{\rm d}t= - \rho^3g^{(3)}(0)\kappa$, where $\rho$ is the three dimensional atomic density, $g^{(3)}(0)$ is the normalized three-body correlation function at zero distance, and $\kappa = \unit[(1.8\pm 0.5)\times 10^{-41}]{m^6/s}$ is the three-body loss rate for $^{87}$Rb [@soding_three-body_1999]. In a quasi-condensate, correlations between atoms are small and $g^{(3)}(0)\simeq 1$ [^7]. Moreover, integrating ${\rm d}\rho/{\rm d}t$ over the transverse shape of the cloud, we obtain a one-dimensional rate of density decrease $ {\rm d}n_0(t)/{\rm d}t = - K n_0(t)^3$, where $K = \kappa/n_0^3 \iint {\rm d} x {\rm d} y \, \rho(x,y)^3 $. Taking into account the transverse broadening of the wavefunction using the Gaussian ansatz results of [@salasnich_effective_2002], we obtain $K=K^0/(1+2n_0a)$, where $K^0=\kappa m^2 \omega_{\perp}^2/(3 \pi^2 \hbar^2 ) $ [^8]. Finally, the rate of variation of the total atom number $N$ is $${\rm d} N/{\rm d}t = -\int_{-R}^{R}{\rm d}z K(z) n_0(z)^3. \label{eq.dNdt}$$ At any time, the measured profile is very close to an equilibrium profile, which indicates the loss rate is small enough to ensure adiabatic following of $n_0(z)$. Then $N$ and $n_0(z)$ are completely determined by $n_p$ and Eq. (\[eq.dNdt\]) can be transformed into a differential equation for $n_p$. We solve it numerically for the parameters of the experimental data, namely the frequency $\omega_\perp$ and the initial peak density, using the LDA to rely $N$ and $n_0(z)$ to $n_p$. Calculations, shown in Fig. \[fig.n\_p\], are in good agreement with the experimental data, which confirms that losses are largely dominated by three-body losses. ![Evolution of the temperature for the five data sets (same color code as in Fig. \[fig.n\_p\]). Inset: density ripples power spectrum corresponding to the encircled point, with the fit in solid line yielding the temperature. []{data-label="fig.T_T0"}](T_T0){width="\linewidth"} The temperature of the gas is determined analyzing the large density ripples that appear after a time of flight $t_f$ [@imambekov_density_2009; @dettmer_observation_2001; @manz_two-point_2010; @rauer_cooling_2016; @schemmer_monitoring_2017]. Interactions are effectively quickly turned off by the transverse expansion of the gas and the subsequent free evolution transforms longitudinal phase fluctuations into density fluctuations. Using an ensemble of images taken in the same experimental condition, we extract the density ripple power spectrum $$\langle |\rho(q)|^2\rangle = \left \langle \left | \int {\rm d} z \left(n(z,t_{f})-\langle n(z,t_{f})\rangle \right) e^{iqz} \right |^2 \right \rangle.$$ We choose $t_f$ small enough so that the density ripples occurring near the position $z$ are produced by atoms which where initially in a small portion of the cloud, located near $z$. We can thus use, within a LDA, the analytic predictions for homogeneous gases to compute the expected power spectrum of the trapped gas [@schemmer_monitoring_2017]. We take into account the finite resolution of the imaging system modeling its impulse response function by a Gaussian of rms wifth $\sigma_{\rm{res}}$. For a given data set the density ripple power spectrum recorded at $t=0$ is fitted with the temperature $T$ and $\sigma_{\rm{res}}$, the latter depending on the transverse width of the cloud and thus on $\omega_\perp$. We then fit $\langle |\rho_q|^2\rangle$ at larger values of $t$ with $T$ as a single parameter (see inset Fig. \[fig.T\_T0\]). The time evolution of $T$ is shown in Fig. \[fig.T\_T0\] for the five different data sets investigated in this paper. The temperature decreases with $t$, which indicates a cooling mechanism associated to the three-body losses. Note that this thermometry probes phononic collective modes since the experimentally accessible wavevectors are much smaller than the inverse healing length $\xi^{-1} = \sqrt{m g n_0}/\hbar$. ![Evolution of the ratio $k_B T /(mc_p^2)$, in the course of the three-body loss process, for the five data sets (same color code as in Fig. \[fig.n\_p\]). Solid (resp. dashed) lines: asymptotic ratio for a 1D homogeneous (resp. harmonically confined) gas. Dotted lines: numerical calculation, that take into account the transverse swelling, for two different initial situations closed to that of experimental data. []{data-label="fig.k_BT_mc2"}](kBT_mc2){width="\linewidth"} Fig. \[fig.k\_BT\_mc2\] shows the same data, with the temperature normalized to $mc_p^2$, where $c_p=\sqrt{n_p \partial_n\mu|_{n_p}/m}$ is the sound velocity at the center of the cloud, shown versus the peak density $n_p$. While $n_p$ explore more than one order of magnitude, remarkably $k_BT/(mc_p^2)$ shows small dispersion and is close to its mean value $0.64$, the standard deviation being $0.02$ [^9]. The absolute linear density is however not the most relevant quantity. A 1D gas at thermal equilibrium is characterised by the dimensionless quantities $\gamma=mg/(\hbar^2 n)$ and $t_{\rm{YY}}= \hbar^2 k_B T/(m g^2)$ [@kheruntsyan_pair_2003]. In particular the quantum degeneracy condition corresponds to the line $t_{\rm{YY}} \gamma \simeq 1$. Moreover, the crossover between the ideal Bose gas regime and the quasicondensate regime occurs, within the region $\gamma \ll 1$, along the line $t\gamma^{3/2}\simeq 1$. Finally, within the quasi-condensate regime, the line $t_{\rm{YY}}\gamma \simeq 1$ separates the high temperature regime, where the zero distance two-body correlation function $g^{(2)}(0)$ is dominated by thermal fluctuations and $g^{(2)}(0)>1$, from the low temperature regime, where $g^{(2)}(0)$ is dominated by quantum fluctuations and is smaller than 1 [^10]. Here we generalize these 1D parameters to quasi-1D gases introducing $\tilde{t}= \hbar^2 k_B T n^2/(m^3c^4)$ and $\tilde{\gamma}=m^2c^2/(\hbar^2n^2)$. For a harmonically confined gas, we refer in the following to the values of $\tilde{t}$ and $\tilde{\gamma}$ evaluated at the trap center. The evolution of the state of the gas during the three-body loss process is shown in Fig. \[fig.phase\_diagram\] in the ($\tilde{t},\tilde{\gamma}$) space. All data collapse on the line $\tilde{t}\tilde{\gamma}= k_B T/(m c^2_p)= 0.7$, with a maximum deviation of 36%, while $\tilde{t}$ explore more than 2 order of magnitude. ![The data collapse on the line $\tilde{t}_{YY} \tilde \gamma = 0.7$. The lower right corner corresponds to the strongly interacting Tonks-Girardeau regime. The data sets and color codes are the same as in all other figures.[]{data-label="fig.phase_diagram"}](phase_diag_data){width="\linewidth"} The physics at the origin of the observed behavior can be understood by considering the simple case of a pure 1D homogeneous quasicondensate. We give here a simplified analysis and refer the reader to [@bouchoule_cooling_2018-1] for a more complete study. At first, let us solely consider the effect of three-body losses, during a time interval $dt$, in a small cell of the gas of length $\Delta$. The density is $n=n_0+\delta n$, where $n_0$ is the mean density and $\delta n\ll n_0$ since we consider a quasicondensate. The density evolves according to $d n = - K n^3 dt + d\eta $, where $d \eta$ is a random variable of vanishing mean value reflecting the stochastic nature of the loss process. During $dt$ the loss process is close to poissonian and $\langle d \eta^2\rangle = 3 K n^3/\Delta d t\simeq 3 K n_0^3/\Delta dt$, where the factor 3 comes from the fact that each loss event amounts to the loss of 3 atoms. To first order in $\delta n$, the mean density evolves according to $d n_0 = - K n_0^3 dt $, and expansion of $dn$ yields $$\label{eq.deltan} {\rm d} \delta n = - 3 K n_0^2 \delta n {\rm d}t + {\rm d} \eta.$$ The two terms of the r.h.s. correspond to the two competing effects of losses. The first term, a drift term, reduces the density fluctuations: it thus decreases the interaction energy, leading to a cooling. The second term, a stocastic term due to the discrete nature of the atom losses, increases the density fluctuations and thus induces a heating. Going to the continuous limit, one has $\langle d\eta(z)d\eta(z')\rangle=3K n_0^3 dt\delta(z-z')$. Let us now consider the intrinsic dynamics of the gas. Within the Bogoliubov approximation, valid in the quasicondensate regime, one identifies independent collective modes and, up to a constant term, the Hamiltonian of the gas writes $H = \sum_k H_k$, where $$H_k = A_k \delta n_k^2 + B_k \theta_k^2$$ is the Hamiltonian of the collective mode of wave vector $k$ [@schemmer_monte_2017]. Here the conjugate quadratures $\delta n_k$ and $\theta_k$ are the Fourier components of $\delta n$ and $\theta$, $B_k = \hbar^2 k^2 n_0/(2m)$ and, as long as phononic modes are considered, $A_k = g/2$. At thermal equilibrium the energy is equally distributed between the quadratures so that $\langle H_k\rangle/2 = A_k \langle \delta n_k^2 \rangle = B_k \langle\theta_k^2\rangle $. Let us compute the evolution of $\langle H_k\rangle$ under the effect of losses, assuming the loss rate is small compared to the mode frequency $\omega_k$ such that the equipartition holds for all times. First the Hamiltonian parameter $B_k$ changes according to $d B_k = -K n_0^2 B_k d t$. Second, according to Eq. \[eq.deltan\], the losses modifies the distribution on the quadrature $\delta n_k$ and we obtain $d\langle \delta n_k^2\rangle/dt= -6Kn_0^2 \langle\delta n_k^2\rangle + 3K n_0^3$ [^11]. Summing this two contributions leads to $$\frac{d \langle H_k\rangle}{ dt} = - 7/2 Kn_0^2 \langle H_k\rangle + 3/2 K n_0^3 g.$$ From this equation, and using $dn_0/dt=-Kn_0^3$, we derive the evolution of the ratio $y=\langle H_k\rangle/(mc^2)$, where $c=\sqrt{gn_0/m}$ is the speed of sound. We find that $y$ converges at long times towards the stationary value $y_\infty=0.6$. Phononic modes typically have large occupation numbers for values of $y$ of the order of or larger than 1 so that $\langle H_k \rangle \simeq k_B T$, where $T$ is the mode temperature, and $y = k_B T / (m c^2)$. In presence of a harmonic longitudinal potential, calculations which assume that the loss rate is small enough to neglect non-adiabatic coupling between modes, predict a stationary value of the ratio $k_B T/(mc_p^2)$ equal to $y_\infty=0.70(1)$ [@bouchoule_cooling_2018-1], a value close to experimental data. For a more precise comparison of data with theory, we compute the time-evolution of $y$ according to formula derived in [@bouchoule_cooling_2018-1], that take into account the transverse swelling of the wavefunction which occurs in our data at large $na$. The results, shown in Fig. \[fig.k\_BT\_mc2\] for two different initial situations, is close to experimental data. Even at the beginning of the observed time-evolution, the ratio $k_BT/(mc_p^2)$ in our gases is close to its asymptotic value. Data are taken only for gases that were sufficiently cooled by evaporative cooling to be in the quasicondensate regime, where both our thermometry and the theoretical description of the effect of losses are applicable. It occurs that, in our experiment, when the gas enters the quasicondensate regime the ratio $k_BT/(mc_p^2)$ is already close to 0.7. In conclusion, we showed in this paper that, under a three-body losses process, the temperature of a quasicondensate in the quasi-1D regime decreases in time. The ratio $k_BT/(mc_p^2)$ stays close to the predicted stationary value, which results from the competition between the cooling effect of losses and the heating due to the stochastic nature of losses. This work raises many different questions. First the cooling mechanism presented in this paper is not restricted to 1D quasicondensates and it would be interesting to investigate it in other regimes and dimensions, in particular as one approaches the Tonks regime of 1D gases. Second, while results presented in this paper concern only the phononic modes, it would be interesting to study the effect of losses on higher energy modes. They might reach higher temperatures than phononic modes, as predicted for one-body losses [@johnson_long-lived_2017], and the stability of such a non thermal situation might be particular to the case of 1D gases. Finally, it is interesting to compare the three-body losses cooling to the commonly used evaporative cooling mechanism, which occurs via the removal of atoms whose energy energy is larger than the trap depth. Its efficiency drops drastically for temperatures lower than $mc_p^2/k_B$: the relevant excitations are then phonons, which do not extend beyond the condensate, and are thus very difficult to “evaporate”. Thus, obtaining, by means of evaporative cooling, temperatures lower than the asymptotic temperature imposed by three-body losses is not guaranteed. This work was supported by Région Île de France (DIM NanoK, Atocirc project). The authors thank Dr Sophie Bouchoule of C2N (centre nanosciences et nanotechnologies, CNRS / UPSUD, Marcoussis, France) for the development and microfabrication of the atom chip. Alan Durnez and Abdelmounaim Harouri of C2N are acknowledged for their technical support. C2N laboratory is a member of RENATECH, the French national network of large facilities for micronanotechnology. M. Schemmer acknowledges support by the Studienstiftung des Deutschen Volkes. [^1]: Quasi-condensates are characteristic of weakly interacting 1D Bose gases at low enough temperature: repulsive interactions prevent large density fluctuations such that the gas resembles locally a BEC, although it does not sustain true long range order [@petrov_regimes_2000]. [^2]: $\gamma=mg/(\hbar^2n)$ where $g$ is the interaction coupling constant, $n$ the linear density, $m$ the atomic mass and $\hbar$ the Planck constant. We evaluate it using the linear density at the trap center. [^3]: The experiment is described in more detail in [@jacqmin_momentum_2012]. [^4]: Thermally activated phonons however prevent the establishment of a well defined phase. [^5]: Density fluctuations are considered only in a coarse grained approximation, valid for lengths much larger than the interparticle distance. [^6]: We are far from confinement-induced resonances predicted in [@olshanii_atomic_1998]. [^7]: According to 1D Bogoliubov calculations, $g^{(3)}(0)-1=3(g^{(2)}(0)-1)=\sqrt{\gamma}f(k_BT/(gn))$, where $f$ is a dimensionaless function wich takes the value $f(0.64)\simeq -1.5$. We then find [*a posteriori*]{} that $|g^{(3)}(0)-1|$ spans the interval $[0.02,0.25]$ for our data. [^8]: We check the Gaussian ansatz gives correct results up to 20% for our parameters by comparing with numerical solution of the Gross-Pitaevskii equation. [^9]: These values have been obtained on the data-points satisfying $n_p a < 0.2$ such that the effect of transverse swelling is small. [^10]: $g^{(2)}(0)=1$ for $t_{\rm{YY}}\gamma = 1.5(1)$. [^11]: Losses are also responsible for an increase of the spread along the quadrature $\theta_k$. However, the associated heating is negligible for phononic modes [@bouchoule_cooling_2018-1].

--- address: | $^{\star}$Institute of Physics, Cracow University of Technology, ulica Podchorażych 1, PL-30084 Cracow, Poland\ E-mail: mduras@riad.usk.pk.edu.pl\ AD 2000 July 19 author: - 'MACIEJ. M. DURAS$^{\star}$' title: | STATISTICS OF COMPLEX DISSIPATIVE SYSTEMS\ “Space Time Chaos: Characterization, Control and Synchronization”; June 19, 2000 - June 23, 2000; Department of Physics and Applied Mathematics and Institute of Physics, University of Navarra, Pamplona, Navarra, Spain --- =cmr8 1.5pt \#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{} Summary {#sec-Summary} ======= A complex quantum system with energy dissipation is considered. The quantum Hamiltonians $H$ belong the complex Ginibre ensemble. The complex-valued eigenenergies $Z_{i}$ are random variables. The second differences $\Delta^{1} Z_{i}$ are also complex-valued random variables. The second differences have their real and imaginary parts and also radii (moduli) and main arguments (angles). For $N$=3 dimensional Ginibre ensemble the distributions of above random variables are provided whereas for generic $N$- dimensional Ginibre ensemble second difference’s, radius’s and angle’s distributions are analytically calculated. The law of homogenization of eigenergies is formulated. The analogy of Wigner and Dyson of Coulomb gas of electric charges is studied. The stabilisation of system of electric charges is dealt with. Introduction {#sec-introduction} ============ We study generic quantum statistical systems with energy dissipation. The quantum Hamiltonian operator $H$ is in given basis of Hilbert’s space a matrix with random elements $H_{ij}$ [@Haake; @1990; @Guhr; @1998; @Mehta; @1990; @0]. The Hamiltonian $H$ is not Hermitean operator, thus its eigenenergies $Z_{i}$ are complex-valued random variables. We assume that distribution of $H_{ij}$ is governed by Ginibre ensemble [@Haake; @1990; @Guhr; @1998; @Ginibre; @1965; @Mehta; @1990; @1]. $H$ belongs to general linear Lie group GL($N$, [**C**]{}), where $N$ is dimension and [**C**]{} is complex numbers field. Since $H$ is not Hermitean, therefore quantum system is dissipative system. Ginibre ensemble of random matrices is one of many Gaussian Random Matrix ensembles GRME. The above approach is an example of Random Matrix theory RMT [@Haake; @1990; @Guhr; @1998; @Mehta; @1990; @0]. The other RMT ensembles are for example Gaussian orthogonal ensemble GOE, unitary GUE, symplectic GSE, as well as circular ensembles: orthogonal COE, unitary CUE, and symplectic CSE. The distributions of the eigenenergies $Z_{1}, ..., Z_{N}$ for $N \times N$ Hamiltonian matrices is given by Jean Ginibre’s formula [@Haake; @1990; @Guhr; @1998; @Ginibre; @1965; @Mehta; @1990; @1]: $$\begin{aligned} & & P(z_{1}, ..., z_{N})= \label{Ginibre-joint-pdf-eigenvalues} \\ & & =\prod _{j=1}^{N} \frac{1}{\pi \cdot j!} \cdot \prod _{i<j}^{N} \vert z_{i} - z_{j} \vert^{2} \cdot \exp (- \sum _{j=1}^{N} \vert z_{j}\vert^{2}), \nonumber\end{aligned}$$ where $z_{i}$ are complex-valued sample points ($z_{i} \in {\bf C}$). For Ginibre ensemble we define complex-valued spacings $\Delta^{1} Z_{i}$ and second differences $\Delta^{2} Z_{i}$: $$\Delta^{1} Z_{i}=Z_{i+1}-Z_{i}, i=1, ..., (N-1), \label{first-diff-def}$$ $$\Delta ^{2} Z_{i}=Z_{i+2} - 2Z_{i+1} + Z_{i}, i=1, ..., (N-2). \label{Ginibre-second-difference-def}$$ The $\Delta^{2} Z_{i}$ are extensions of real-valued second differences $$\Delta^{2} E_{i}=E_{i+2}-2E_{i+1}+E_{i}, i=1, ..., (N-2), \label{second-diff-def}$$ of adjacent ordered increasingly real-valued energies $E_{i}$ defined for GOE, GUE, GSE, and Poisson ensemble PE (where Poisson ensemble is composed of uncorrelated randomly distributed eigenenergies) [@Duras; @1996; @PRE; @Duras; @1996; @thesis; @Duras; @1999; @Phys; @Duras; @1999; @Nap; @Duras; @1996; @APPB; @Duras; @1997; @APPB]. There is an analogy of Coulomb gas of unit electric charges pointed out by Eugene Wigner and Freeman Dyson. A Coulomb gas of $N$ unit charges moving on complex plane (Gauss’s plane) [**C**]{} is considered. The vectors of positions of charges are $z_{i}$ and potential energy of the system is: $$U(z_{1}, ...,z_{N})= - \sum_{i<j} \ln \vert z_{i} - z_{j} \vert + \frac{1}{2} \sum_{i} \vert z_{i}^{2} \vert. \label{Coulomb-potential-energy}$$ If gas is in thermodynamical equilibrium at temperature $T= \frac{1}{2 k_{B}}$ ($\beta= \frac{1}{k_{B}T}=2$, $k_{B}$ is Boltzmann’s constant), then probability density function of vectors of positions is $P(z_{1}, ..., z_{N})$ Eq. (\[Ginibre-joint-pdf-eigenvalues\]). Complex eigenenergies $Z_{i}$ of quantum system are analogous to vectors of positions of charges of Coulomb gas. Moreover, complex-valued spacings $\Delta^{1} Z_{i}$ are analogous to vectors of relative positions of electric charges. Finally, complex-valued second differences $\Delta^{2} Z_{i}$ are analogous to vectors of relative positions of vectors of relative positions of electric charges. The $\Delta ^{2} Z_{i}$ have their real parts ${\rm Re} \Delta ^{2} Z_{i}$, and imaginary parts ${\rm Im} \Delta ^{2} Z_{i}$, as well as radii (moduli) $\vert \Delta ^{2} Z_{i} \vert$, and main arguments (angles) ${\rm Arg} \Delta ^{2} Z_{i}$. Second Difference Distributions {#sec-second-difference-pdf} =============================== We define following random variables for $N$=3 dimensional Ginibre ensemble: $$Y_{1}=\Delta ^{2} Z_{1}, A_{1}= {\rm Re} Y_{1}, B_{1}= {\rm Im} Y_{1}, \label{Ginibre-Y1A1B1-def}$$ $$R_{1} = \vert Y_{1} \vert, \Phi_{1}= {\rm Arg} Y_{1}, \label{Ginibre-polar-second-diff-def}$$ and for the generic $N$-dimensional Ginibre ensemble [@Duras; @2000; @JOptB]: $$W_{1}=\Delta ^{2} Z_{1}, P_{1} = \vert W_{1} \vert, \Psi_{1}= {\rm Arg} W_{1}. \label{Ginibre-W1P1Psi1-def-N-dim}$$ Their distributions for are given by following formulae respectively [@Duras; @2000; @JOptB]: $$\begin{aligned} & & f_{Y_{1}}(y_{1})=f_{(A_{1}, B_{1})}(a_{1}, b_{1})= \label{Ginibre-marginal-pdf-Y1-def} \\ & & =\frac{1}{576 \pi} [ (a_{1}^{2} + b_{1}^{2})^{2} + 24] \cdot \exp (- \frac{1}{6} (a_{1}^{2}+a_{2}^{2})), \nonumber\end{aligned}$$ is second difference distribution for 3-dimensional Ginibre ensemble. $$f_{A_{1}}(a_{1})= \frac{\sqrt{6}}{576 \sqrt{\pi}} (a_{1}^{4}+6a_{1}^{2}+ 51) \cdot \exp (- \frac{1}{6} a_{1}^{2}), \label{Ginibre-marginal-pdf-Y1Re-def}$$ $$f_{B_{1}}(b_{1})= \frac{\sqrt{6}}{576 \sqrt{\pi}} (b_{1}^{4}+6b_{1}^{2}+ 51) \cdot \exp (- \frac{1}{6} b_{1}^{2}), \label{Ginibre-marginal-pdf-Y1Im-def}$$ $$\begin{aligned} & & f_{R_{1}}(r_{1})= \label{Ginibre-polar-second-diff-result} \\ & & \Theta(r_{1}) \frac{1}{288}r_{1}(r_{1}^{4}+24) \cdot \exp(- \frac{1}{6} r_{1}^{2}), \nonumber \\ & & f_{\Phi_{1}}(\phi_{1})= \frac{1}{2 \pi}, \phi_{1} \in [0, 2 \pi], \nonumber\end{aligned}$$ are real part’s, imaginary part’s, modulus’s and angle’s probability distributions for 3-dimensional Ginibre ensemble, where $\Theta(r_{1})$ is Heaviside (step) function. Notice, that the angle $\Phi_{1}$ has uniform distribution. Next, second difference’s distribution for $N$-dimensional Ginibre ensemble reads $P_{3}(w_{1})$ [@Duras; @2000; @JOptB]: $$\begin{aligned} & & P_{3}(w_{1})= \label{W1-pdf-I-result} \\ & & = \pi^{-3} \sum_{j_{1}=0}^{N-1} \sum_{j_{2}=0}^{N-1} \sum_{j_{3}=0}^{N-1} \frac{1}{j_{1}!j_{2}!j_{3}!}I_{j_{1}j_{2}j_{3}}(w_{1}), \nonumber \\ & & I_{j_{1}j_{2}j_{3}}(w_{1})= \label{W1-pdf-I-F} \\ & & = 2^{-2j_{2}} \frac{\partial^{j_{1}+j_{2}+j_{3}}} {\partial^{j_{1}} \lambda_{1} \partial^{j_{2}} \lambda_{2} \partial^{j_{3}} \lambda_{3}} F(w_{1},\lambda_{1},\lambda_{2},\lambda_{3}) \vert _{\lambda_{i}=0}, \nonumber\end{aligned}$$ $$\begin{aligned} & & F(w_{1},\lambda_{1},\lambda_{2},\lambda_{3})= \label{W1-pdf-I-F-final} \\ & & = A(\lambda_{1},\lambda_{2},\lambda_{3}) \exp[-B(\lambda_{1},\lambda_{2},\lambda_{3}) \vert w_{1} \vert^{2}], \nonumber\end{aligned}$$ $$\begin{aligned} & & A(\lambda_{1},\lambda_{2},\lambda_{3})= \label{W1-pdf-I-A} \\ & & =\frac{(2\pi)^{2}} {(\lambda_{1}+\lambda_{2}-\frac{5}{4}) \cdot (\lambda_{1}+\lambda_{3}-\frac{5}{4})-(\lambda_{1}-1)^{2}}, \nonumber \\ & & B(\lambda_{1},\lambda_{2},\lambda_{3})= \label{W1-pdf-I-B} \\ & & =(\lambda_{1}-1) \cdot \frac{2 \lambda_{1}-\lambda_{2}-\lambda_{3}+\frac{1}{2}} {2 \lambda_{1}+\lambda_{2}+\lambda_{3}-\frac{9}{2}}. \nonumber\end{aligned}$$ Finally, $$\begin{aligned} & & f_{P_{1}}(\varrho_{1}) =2 \pi P_{3}(\varrho_{1})= \label{Ginibre-polar-second-diff-result-N-dim} \\ & & = 2 \pi^{-2} \sum_{j_{1}=0}^{N-1} \sum_{j_{2}=0}^{N-1} \sum_{j_{3}=0}^{N-1} \frac{1}{j_{1}!j_{2}!j_{3}!}I_{j_{1}j_{2}j_{3}}(\varrho_{1}), \nonumber \\ & & f_{\Psi_{1}}(\psi_{1})= \frac{1}{2 \pi}, \psi_{1} \in [0, 2 \pi], \nonumber\end{aligned}$$ are modulus’s and angle’s probability distributions for $N$-dimensional Ginibre ensemble. Notice, that again the angle $\Psi_{1}$ has uniform distribution. Conclusions {#sect-conclusions} =========== We compare second difference distributions for different ensembles by defining following dimensionless second differences: $$C_{\beta} = \frac{\Delta^{2} E_{1}}{<S_{\beta}>}, \label{rescaled-second-diff-GOE-GUE-GSE-PE}$$ $$X_{1}=\frac{A_{1}}{<R_{1}>}, \label{Ginibre-X1-def}$$ where $<S_{\beta}>$ are the mean values of spacings for GOE(3) ($\beta=1$), for GUE(3) ($\beta=2$), for GSE(3) ($\beta=4$), for PE ($\beta=0$) [@Duras; @1996; @PRE; @Duras; @1996; @thesis; @Duras; @1999; @Phys; @Duras; @1999; @Nap; @Duras; @1996; @APPB; @Duras; @1997; @APPB], and $<R_{1}>$ is mean value of radius $R_{1}$ for $N$=3 dimensional Ginibre ensemble [@Duras; @2000; @JOptB]. On the basis of comparison of results for Gaussian ensembles, Poisson ensemble, and Ginibre ensemble we formulate [@Duras; @1996; @PRE; @Duras; @1996; @thesis; @Duras; @1999; @Phys; @Duras; @1999; @Nap; @Duras; @1996; @APPB; @Duras; @1997; @APPB; @Duras; @2000; @JOptB]: [**Homogenization Law:**]{} [*Random eigenenergies of statistical systems governed by Hamiltonians belonging to Gaussian orthogonal ensemble, to Gaussian unitary ensemble, to Gaussian symplectic ensemble, to Poisson ensemble, and finally to Ginibre ensemble tend to be homogeneously distributed.*]{} It can be restated mathematically as follows: [*If $H \in$ [GOE, GUE, GSE, PE, Ginibre ensemble]{}, then [Prob]{}($D$)=[max]{}, where D is random event corresponding to vanishing of second difference’s probability distributions at the origin.*]{} Both of above formulation follow from the fact that the second differences’ distributions assume global maxima at origin for above ensembles [@Duras; @1996; @PRE; @Duras; @1996; @thesis; @Duras; @1999; @Phys; @Duras; @1999; @Nap; @Duras; @1996; @APPB; @Duras; @1997; @APPB; @Duras; @2000; @JOptB]. For Coulomb gas’s analogy the vectors of relative positions of vectors of relative positions of charges statistically most probably vanish. It means that the vectors of relative positions tend to be equal to each other. Thus, the relative distances of electric charges are most probably equal. We call such situation stabilisation of structure of system of electric charges on complex plane. Acknowledgements {#sect-acknowledgements .unnumbered} ================ It is my pleasure to most deeply thank Professor Jakub Zakrzewski for formulating the problem. I also thank Professor Antoni Ostoja-Gajewski for creating optimal environment for scientific work. References {#references .unnumbered} ========== [99]{} F. Haake, [*Quantum Signatures of Chaos*]{} (Berlin Heidelberg New York, Springer-Verlag, 1990), Chapters 1, 3, 4, 8 pp 1-11, 33-77, 202-213. T. Guhr T, A. Müller-Groeling, and H. Weidenmüller, [*Phys. Rept.*]{} [**299**]{}, 189 (1998). M. L. Mehta, [*Random matrices*]{} (Boston, Academic Press, 1990), Chapters 1, 2, 9 pp 1-54, 182-193. J. Ginibre, [*J. Math. Phys.*]{} [**6**]{}, 440 (1965). M. L. Mehta, [*Random matrices*]{} (Boston, Academic Press, 1990), Chapter 15 pp 294-310. M. M. Duras, and K. Sokalski, [*Phys. Rev.*]{} E [**54**]{}, 3142 (1996). M. M. Duras, [*Finite difference and finite element distributions in statistical theory of energy levels in quantum systems*]{} (PhD thesis, Jagellonian University, Cracow, July 1996). M. M. Duras, and K. Sokalski, [*Physica*]{} [**D125**]{}, 260 (1999). M M. Duras, [*Proceedings of the Sixth International Conference on Squeezed States and Uncertainty Relations, 24 May-29 May 1999, Naples, Italy*]{} (Greenbelt, Maryland: NASA), at press. M. M. Duras, and K. Sokalski, [*Acta Phys. Pol.*]{} [**B27**]{}, 2027 (1996). M. M. Duras, K. Sokalski, and P. Su[ł]{}kowski, [*Acta Phys. Pol.*]{} [**B28**]{} 1023 (1997). M. M. Duras, [*J. Opt. B: Quantum Semiclass. Opt.*]{} [**2**]{}, 287 (2000).

--- author: - 'J. Fritz' - 'B. M. Poggianti' - 'A. Cava' - 'T. Valentinuzzi' - 'A. Moretti' - 'D. Bettoni' - 'A. Bressan' - 'W. J. Couch' - 'M. D’Onofrio' - 'A. Dressler' - 'G. Fasano' - 'P. Kjærgaard' - 'M. Moles' - 'A. Omizzolo' - 'J. Varela' date: 'Received ...; accepted ...' title: | WINGS-SPE II:\ A catalog of stellar ages and star formation histories, stellar masses and dust extinction values for local clusters galaxies --- [The WIde-field Nearby Galaxy clusters Survey () is a project whose primary goal is to study the galaxy populations in clusters in the local universe ($z<0.07$) and of the influence of environment on their stellar populations. This survey has provided the astronomical community with a high quality set of photometric and spectroscopic data for $77$ and $48$ nearby galaxy clusters, respectively.]{} [In this paper we present the catalog containing the properties of galaxies observed by the  SPEctroscopic ([wings-spe]{}) survey, which were derived using stellar populations synthesis modelling approach. We also check the consistency of our results with other data in the literature.]{} [Using a spectrophotometric model that reproduces the main features of observed spectra by summing the theoretical spectra of simple stellar populations of different ages, we derive the stellar masses, star formation histories, average age and dust attenuation of galaxies in our sample.]{} [$\sim 5300$ spectra were analyzed with spectrophotometric techniques, and this allowed us to derive the star formation history, stellar masses and ages, and extinction for the  spectroscopic sample that we present in this paper.]{} [The comparison with the total mass values of the same galaxies derived by other authors based on  data, confirms the reliability of the adopted methods and data.]{} INTRODUCTION ============ One of the best places to study the influence of dense environments on galaxy evolution are galaxy clusters. The fact that early-type galaxies are more common in clusters, while spirals are preferentially found in the field, is a manifestation of the so-called morphology-density relation, which was discovered to be a common pattern over a wide range of environmental densities, from local groups of galaxies to distant clusters [see, for example, @postman84; @dressler97; @postman05]. Not only the morphology, but also the stellar content of galaxies is influenced by the galaxy environment, and clusters host galaxies with the oldest stellar populations. Dense environments are capable of altering the star formation history of a galaxy, quenching its star formation activity as it falls to the cluster, as a results of phenomena such as gas stripping, tidal interactions, and/or gas starvation. While studies of the stellar populations are already available for distant clusters [see, e.g., @poggianti99; @poggianti08], a similar analysis on a homogeneous and complete set of data at low redshift has been lacking until now.  was conceived as a survey to serve as a local comparison for the distant clusters studies. Thanks to its deep and high-quality optical imaging and its large sample of cluster galaxy spectra, it enables us to study in detail the link between galaxy morphology and star formation history. Optical spectra are nowadays widely exploited to derive the properties of the stellar population content of galaxies, by means of spectral synthesis techniques. In this paper we present the results of the spectrophotometric analysis performed on the spectra of a sample of local clusters galaxies from the  survey, describing how stellar masses, star formation histories, dust attenuation and average age are obtained. The project [see @fasano06] is providing the largest set of homogeneous spectroscopic data for galaxies belonging to nearby clusters. Originally designed as a B and V band photometric survey,  has widened its database to also include near-infrared bands [J and K, see @valentinuzzi09] and ultraviolet photometry (Omizzolo et al., in prep.), H$\alpha$ imaging (Vilchez et al., in preparation) and optical spectroscopy [@cava09]. With such a wealth of data,  has a considerable legacy value for the astronomical community, becoming the local benchmark with which the properties of galaxies in high redshift clusters can be compared with. In this paper, we present the catalogs that we are providing as on-line databases and give a full description of all the measurements and stellar population properties that are given. In order to do so, we will summarize the main features of our spectrophotometric model that are used to derive such quantities, already described in detail in previous work [@fritz07 F07 hereafter]. Furthermore, in order to make all the potential users of the databases more confident with the quantities that we derive, we present a detailed and careful validation of our results, by comparing the values obtained on a subsample of  galaxies that are in common with the Sloan Digital Sky Survey. The paper outline is as follows: after describing the  spectroscopic dataset in [**§2**]{}, in [**§3**]{} we give a brief review of the adopted spectrophotometric model and recall the characteristics of the theoretical spectra that are used; in [**§4**]{} we describe the properties of the stellar populations that are derived and how they are computed, while in [**§5**]{} we present a validation of our results by comparing them with other literature data and, finally, in [**§6**]{}, we describe the items that will be provided in the final catalog, and give an example. We remind that the  project assumes a standard $\Lambda$ cold dark matter ($\Lambda$CDM) cosmology with H$_0=70$, $\Omega_\Lambda=0.70$ and $\Omega_M=0.30$.  SPECTROSCOPIC DATASET {#sec:data} ====================== Out of the 77 cluster fields imaged by the  photometric survey [@varela09], 48 were also observed spectroscopically. While the reader should refer to [@cava09] for a complete description of the spectroscopic sample, (including completeness analysis and quality check), here we will briefly summarize the features that are more relevant for this work’s purposes.\ Medium resolution spectra for $\sim 6000$ galaxies were obtained during several runs at the 4.2m William Herschel Telescope (WHT) and at the 3.9m Anglo Australian Telescope (AAT) with multifiber spectrographs (WYFFOS and 2dF, respectively), yielding reliable redshift measurements. The fiber apertures were $1.6''$ and $2''$, respectively, and the spectral resolution $\sim 6$ and $\sim 9$ Å FWHM for the WHT and AAT spectra, respectively. The wavelength coverage ranges from $\sim 3590$ to $\sim 6800$ Å for the WHT observations, while spectra taken at the AAT covered the $\sim 3600$ to $\sim 8000$ Å domain. Note also that, for just one observing run at the WHT (in which 3 clusters were observed), the spectral resolution was $\sim 3$ Å FWHM, with the spectral coverage ranging from $\sim 3600$ to $\sim 6890$ Å. THE METHOD ========== We derive stellar masses, star formation histories, extinction values and average stellar ages of galaxies by analysing their integrated spectra by means of spectral synthesis techniques. The model that is used for this analysis has already been described in detail in F07, but here we will briefly and schematically recall its main features and parameters. The fitting technique {#sec:fit} --------------------- The model reproduces the most important features of an observed spectrum with a theoretical one, which is obtained by summing the spectra of Single Stellar Population (SSP, hereafter) models of different stellar ages and a fixed, common value of the metallicity. Before being added together, each SSP spectrum is weighted with a proper value of the stellar mass and dust extinction by an amount which, in general, depends on the SSP age itself. The best fit model parameters are obtained by calculating the differences between the observed and model spectra, and evaluating them by means of a standard $\chi^2$ function: $$\label{eqn:chi2} \chi^2=\sum_{i=1}^{N} \left(\frac{M_i-O_i}{\sigma_i}\right)^2$$ where $M_i$ and $O_i$ denote the quantities measured from the model and observed spectra, respectively (i.e. continuum fluxes and equivalent widths of spectral lines), with $\sigma_i$ being the observed uncertainties and $N$ being the total number of observed constraints. The observed errors on the flux are computed by taking into account the local spectral signal-to-noise ration, while uncertainties on the equivalent widths are derived mainly from the measurement method (see section 2.2 in F07, and Fritz et al. 2010b, in prep., for further details). The observed features that are used to compare the likelihood between the model and the observed spectra are chosen from the most significant emission and absorption lines and continuum flux intervals. In particular, we compare, when measurable. the equivalent widths of H$\alpha$, H$\beta$, H$\delta$, H$\epsilon$+Ca[ii (h)]{}, Ca[ii (k)]{}, H$\eta$ and . Other lines, even though prominent, were only measured but not used to constrain the model parameters. Key examples are the line at 5007 Å, because it is too sensitive to the physical conditions of the gas and of the ionizing source, and the Na and Mg lines at $\sim$5890 and 5177 Å respectively, because they are strongly affected by the enhancement of $\alpha$-elements, which is not taken into account by our SSPs). The continuum flux is measured in specific wavelength ranges, chosen to avoid any important spectral line, so to sample as best as possible the shape of the spectral continuum. Particular emphasis is given to the 4000 Åbreak, D4000, defined by [@bruzual83], as it is considered a good indicator of the stellar age. As already mentioned, the amount of dust extinction is let free to vary as a function of the SSP age. Treating extinction in this way is equivalent, in some sense, to taking into account the fact that the youngest stars are expected to be still embedded within the dusty molecular clouds where they formed, while as they become older, they progressively emerge from them. In this picture, the spectra of SSPs of different ages are supposed to be dust-reddened by different amounts; dust is assumed to be distributed so to simulate a uniform layer in front of the stars, and the Galactic extinction curve [@cardelli89] is adopted. Building a self-consistent chemical model, that would take into account changes in the metal content of a galaxy and its chemical evolution as a function of mass and star formation history, was far beyond the scope of this work. This is why we adopted a homogeneous value for the metallicity for our theoretical spectra, and left it to the model free to choose between three different sets of metallicity, namely Z=0.05, Z=0.02 and Z=0.004 (super-solar, solar and sub-solar, respectively). Fitting an observed spectrum with a single value of the metallicity is equivalent to assuming that this value belongs to the stellar population that is dominating its light. However, as described in F07, acceptable fits are obtained for most of the spectra adopting different metallicities, which means that this kind of analysis is often not able to provide a unique value for the metallicity. It is clear that, assuming a unique value for the SSP’s metallicity when reproducing an observed spectrum is a simplifying assumption since, in practice, the stellar populations of a galaxy span a range in metallicity values. One could hence question the reliability of the mass and of the SFH determination done by using one single metallicity value. To better understand this possible bias due to the mix of metallicities that is expected in galaxies, we repeated the check already performed in F07: we built template synthetic spectra with 26 different SFHs as in F07, but with values of the metallicity varying as a function of stellar age, to roughly simulate a chemical evolution, and we analyzed them by means of our spectrophotometric fitting code. The results clearly show that the way we deal with the metallicity does not introduce any bias in the recovered total stellar mass or SFH. SSP parameters {#sec:ssp} -------------- All of the stellar population properties that are derived are strictly related to the theoretical models that we use in our fitting algorithm. It is hence of foundamental importance to give all the details of the physics and of the parameters that were used to build them. First of all, WE make use of the Padova evolutionary tracks [@bertelli94] and use a standard [@salpeter55] initial mass function (IMF), with masses in the range 0.15-120 M$_\odot$. Our optical spectra were obtained using two different sets of observed stellar atmospheres: for ages younger than $10^9$ years we used [@jacoby84], while for older SSPs we used spectra from the MILES library [@Psanchez04; @Psanchez06] and both sets were degraded in spectral resolution, in order to match that of our observed spectra (namely 3, 6 and 9 Å of FWHM, see Sect.\[sec:data\] for details). Using the theoretical libraries of Kurucz, the SSP spectra were extended to the ultra-violet and infrared, widening, in this way, the wavelength range down to 90 and up to $\sim 10^9$ Å (note that in these intervals the spectral resolution is much lower, being $\sim 20$ Å, but in any case outside the range of interest for the spectra used for our analysis). Gas emission, whose effect is visible through emission lines, was also computed –and included in the theoretical spectra– by means of the photoionisation code [cloudy]{} [@ferland96]. The optical spectra of SSPs younger than $\sim 2\times 10^7$ display, in this way, both permitted and forbidden lines (typically, hydrogen, , , and ). This nebular component was computed assuming [*case B recombination*]{} [see @osterbrock89], an electron temperature of $10^4$ K, and an electron density of $100$ cm$^{-3}$. The radius of the ionizing star cluster was assumed to be 15 pc, and its mass $10^4$ M$_\odot$. Finally, emission from the circumstellar envelopes of AGB stars was computed and added as described in [@bressan98]. The initial set of SSPs was composed of 108 theoretical spectra referring to stellar ages ranging from $10^5$ to $20\times 10^9$ years, for each one of the three afore-mentioned values of the metallicity. Determining the age of stellar populations from an integrated optical spectrum with such a high temporal resolution is well beyond the capabilities of any spectral analysis. Hence, as a first step, we reduced the stellar age resolution by binning the spectra. This was done by taking into account both the characteristics of the evolutionary phases of stars, and the trends in spectral features as a function of the SSP age (see both section 2.1.1 and Fig. 1 in F07). After combining the spectra at this first stage, we ended up with 13 stellar age bins. -- -- -- -- As we describe in F07, this set of theoretical spectra originally included also a SSP whose age, namely $\sim 17.5$ Gyr, is older than the universe age. The use of this SSP was merely statistical: since the only appreciable difference between the three oldest SSPs of our set is, actually, the mass-to-luminosity ratio, using such an old SSP would prevent our random search of the best fit model to be systematically biased towards the youngest of the old SSPs. Nevertheless, the adoption of such an approach can lead some models to be dominated by this very old stellar population yielding, in this way, mass values that are too high, due to the higher mass-to-light ratio. To overcome this issue we decide to avoid the use of the oldest stellar populations, limiting ourselves to stellar populations whose ages are consistent with that of the universe. We will hence refer, from now on, to these 12 SSPs. The best fit search ------------------- Finding the best combination of the parameters that minimizes the differences between the observed and the model spectrum, is a non-linear problem, due to the presence of extinction. Furthermore, it is also underdetermined, which means that the number of constraints is lower than the number of parameters. In fact, in our case, we are using SSPs of 12 different ages, so that our task turns into finding the combination of 12 mass and extinction values that better fits the observed spectrum. To find the set of 24 parameters that will yield the best fit model, we use the Adaptive Simulated Annealing algorithm, which randomly explores the parameters space, searching for an absolute minimum in the $\chi^2$ function. This method is particulary suited to such problems, where the function to minimize has lots of local minima: once a promising zone for a minimum, in the parameter space, is found, the algorithm not only refines the search of the local minimum, but also checks for the presence of other, deeper minima, outside the local “low-$\chi^2$ valley”. Uncertainties ------------- All the physical parameters that are derived from the the spectral analysis, refer to a best fit model for an observed spectrum. The limited wavelength range under analysis, the well known age-metallicity degeneracy, and the non-linearity of the problem, together with the fact that it is underdetermined, makes the solution non-unique. This means that models with different characteristics may equally well reproduce the observed spectral features. To account for this, we give error-bars related to mass, extinction and age values. To compute such uncertainties, we exploit the characteristics of the minimisation algorithm: the path towards the best fit model (or the minimum $\chi^2$) depends on the starting points so, in general, starting from different initial positions can lead to different minimum points, i.e. to best fit models with different parameters. We hence perform 11 optimisations, each time starting from a different point in the parameters space. In this way we end up with 11 best fit models that we verified are well representative of the space of the solutions. We take, as a reference, the model with the median total mass among these 11. All the errorbars are computed as the average difference between the values of the models with the highest and lowest total stellar mass. The quality of the fits ----------------------- The similarity between an observed spectrum and its best fit model is measured, as explained in § \[sec:fit\], by means of a $\chi^2$ function taking into account both spectral continuum fluxes and the equivalent widths of significant lines. Our choice to use a wide range both in metallicity and SSP ages, and to let both extinction and mass vary freely, are the key ingredients that allow us to satisfactorily reproduce any galactic spectrum, at least in principle. In practice, low quality spectra due to low S/N, bad flux calibration, bad subtraction of sky or telluric lines, can give rise to a bad fit. To demonstrate that there are no systematic failures of any of the observed features that are used as constraints, in Fig.\[fig:rms\] we show the difference between the values calculated for the model and for the observed spectrum, averaged over all the  sample. In the left-hand panel we show, plotted as red squares, the average values of the difference for the flux in the spectral continuum, together with the rms (red errorbars), and the average values of the observed errors (blue errorbars). The plot in the right-hand panel of the same figure shows the differences for the equivalent widths of the spectral lines. The line is the one that shows the highest displacement with respect to the zero-difference line, due to the fact that this line is in the spectral region with the highest noise. This makes it also more difficult to measure, and it also explains why its observed value has the average largest error. Overall all the features are well reproduced, with no systematic failure. THE PROPERTIES OF STELLAR POPULATIONS ===================================== In this section we describe the properties of the stellar populations that are derived from our spectrophotometric synthesis, that are now publicly available. Fitting the main features of an optical spectrum allows us to derive the characteristics of the stellar populations whose light we see in the integrated spectrum: total mass, mass of stars as a function of age, the metallicity and dust extinction are typical quantities that can be obtained. As already pointed out, using this particular technique, it is almost impossible to recover a unique value for the stellar metallicity due to both the degeneracy issues such as the age-metallicity and age-extinction and to the fact that we do not consider SSP models with $\alpha$-element enhancements. In fact, in the vast majority of cases at least two values of the metallicity are found to provide equally good fits. Stellar masses {#sec:mass} -------------- When stellar masses are derived by means of spectrophotometric techniques, it is important to clearly state which definition of mass is used. As already made clear by Longhetti & Saracco [2009, but see also @renzini06], the use of spectral synthesis techniques leads to three different definitions of the stellar mass, namely: 1. the initial mass of the SSP, at age zero; this is nothing but the mass of gas turned into stars 2. the mass locked into stars, both those which are still in the nuclear-burning phase, and remnants such as white dwarfs, neutron stars and stellar black holes 3. the mass of stars that are still shining, i.e. in a nuclear-burning phase. The difference between the three definitions is a function of the stellar age and, in particular, it can be up to a factor of 2 between mass definition 1) and 3), in the oldest stellar populations. We will provide the user with masses calculated using all of the afore mentioned definitions, following the same enumeration. To compute the values of stellar mass, we exploit the fact that the theoretical spectra are given in luminosity per unit of solar mass. Once the model spectrum is converted to flux by accounting for the luminosity distance factor, the K-correction is naturally performed by fitting the spectra at their observed redshifts. All of the observed spectra are normalised by means of their observed V-band magnitude within the fiber aperture. Obviously, in order to obtain a stellar mass value referring to the whole galaxy (that we will dub “total stellar mass”, from now on), one should use a spectrum representative of the whole galaxy, which is not at our disposal. Since we have both aperture and total photometry for all the objects of our spectroscopic sample, we use the total V magnitude to rescale the model spectrum: in this way we are assuming that the colour gradient of the aperture-to-total magnitude is negligible (this assumption is made by several authors: see e.g. @kauffmann03). When speaking of “total magnitude” here, we refer to the `MAG_AUTO` value [see @varela09 for further details], that is the `SExtractor` magnitude computed within the Kron aperture.\ In Fig. \[fig:colorgrad\] we show the comparison between the observed $(B-V)$ colour computed using the magnitudes within a 5 kpc aperture (x axis) an the one computed using the spectroscopic fiber aperture (y axis), with the black line being the 1:1 relation. We consider the color within 5 kpc a good approximation of the total color (in fact, it closely follows the color derived using AUTO magnitudes), and is a good aperture compromise for both large and small galaxies in our sample. The average difference between the fibre and 5 kpc colours is $\sim 0.1$ mag, due to the presence of bluer (thus probably younger) stars in the outskirts of the galaxies. We will provide values of the stellar mass referring to both apertures, and a colour term which can be used to correct the total mass to account for radial gradients in the stellar populations content, as described below. ![The comparison between values of the $(B-V)$ colour as computed from a 5 kpc aperture (x axis) and the fiber aperture (y axis) magnitudes. The solid line represents the 1:1 relation that highlights a systematic, off-set of $\sim 0.1$ mag: the 5 kpc colour is bluer as expected since the total magnitude is sampling, on average, younger populations in the outskirts of the galaxies.[]{data-label="fig:colorgrad"}](color_gradient.eps){height="50.00000%"} Colour corrections {#sec:colcor} ------------------ To correct total masses for colour gradients, we exploit the [@bdj01] prescription, which was derived in order to compute stellar masses in galaxies by means of photometric data. According to their work, the M/L ratio of a galaxy can be expressed by the following: $$\label{eqn:bdj} \log_{10}\left( \frac{M}{L_\lambda}\right)=a_\lambda+b_\lambda\cdot COL$$ where $L_\lambda$ is the luminosity in a given band (denoted by $\lambda$) while the $a_\lambda$ and $b_\lambda$ coefficients depend on the band that is used, and on the population synthesis models (including IMF, isochrones, etc.), and $COL$ is the colour term. Table 4 in [@bdj01] presents a list of such coefficients for various bands, models and two metallicities (subsolar —Z=0.008— and solar —Z=0.02—). For the calculations that follow, we will use V and B band data, and assume the [@kodama97] models, that use a Salpeter IMF and a solar metallicity value, which yields $a_V=-0.18$ and $b_V=1.00$. Note that, using [@bruzual03] or [pegase]{} [@fioc97] models, will not substantially affect the results. As already mentioned above, when going from the stellar mass calculated over the fiber magnitude to the one referring to the whole galaxy, there is the implicit assumption that the colour calculated within the fiber aperture is the same as the one calculated with the total magnitudes (here we assume a $\sim 5$ kpc aperture), while this is not true for most cases. Starting from Eq.\[eqn:bdj\] and after some algebra, we derive a colour-correction term as follows: $$\label{eqn:ccol} C_{corr}= b_V\cdot [(B_5-V_5)-(B_f-V_f)]$$ where the term $(B_5-V_5)$ is the colour computed from 5 kpc aperture magnitudes and $(B_f-V_f)$ is the observed colour within the fiber. This factor, which is given in our final catalogs, must be added to the total mass value in order to account for colour gradients. As a consistency check for the values of the total stellar mass computed by means of our models, we compare them to the values that can be obtained by means of Eq.\[eqn:bdj\], which yields the following: $$\label{eqn:bdj2} \log_{10}\frac{M}{M_\odot}=-0.4\cdot(V-V_\odot)+a_V+b_V\cdot (B^k_5-V^k_5)$$ where $B^k_5-V^k_5$ are the K-corrected (i.e. rest-frame) magnitudes extracted from a 5 kpc aperture, $V$ is the total absolute magnitude (obtained from V `MAG_AUTO`, K-corrected) and $V_\odot=4.82$ is the absolute magnitude of the sun in the V band. K-corrections were taken from [@poggianti97]. ![A comparison of the total stellar mass of galaxies in the  sample, as computed by means of B and V band photometry, on the $x$-axis, assuming the prescriptions given in [@bdj01] (see text for details), and by means of our spectral fitting. The solid line represents the 1:1 relation. A colour correction term, computed as explained in the text, was applied to the spectroscopic-derived values, while the photometric values were corrected to account for the difference in the IMF mass limits.[]{data-label="fig:bdj"}](bdj_comp.eps){height=".5\textwidth"} In Fig.\[fig:bdj\] we show the comparison between total stellar masses computed by means of our spectral fitting (on the $y$-axis) and those obtained by means of the [@bdj01] prescription (i.e. by adopting Eq.\[eqn:bdj2\]). We applied a 0.064 dex correction to account for the differences in the adopted IMF (@bdj01 use a Salpeter IMF with masses in the 0.1–100 M$_\odot$ range, while we use 0.15–120 M$_\odot$), and we added the colour correction term to the spectroscopic-derived mass values, as explained above. The agreement between the two different methods is, on average, always better than 0.1 dex. A similar comparison between stellar masses obtained from spectral fitting and from photometry, calculated using aperture magnitudes instead of the total ones, shows an equally good agreement between the two methods. The [@bdj01] mass photometric values are also provided in our final catalogs. The star formation history {#sec:sfh} -------------------------- As we describe in F07 and summarize in Sect. \[sec:ssp\], our search for the best fit-model is performed using 12 SSPs of different ages, obtained, in turn, by binning a much higher age-resolution stellar age grid. Still, we verified that it is not possible to recover the star formation as a function of stellar age with the relatively high temporal resolution provided by the 12 SSPs. After performing accurate tests on template spectra that were built in order to match the spectral features of  spectra in terms of both spectral resolution, signal-to-noise ratio and wavelength coverage, we found that it is possible to properly recover the star formation history (hereafter, SFH) in 4 main stellar age bins. The details of the choice are explained in F07; here we just recall their ranges that are, respectively: $0-2\times 10^7$, $2\times 10^7-6\times 10^8$, $6\times 10^8-5.6\times 10^9$ and $5.6\times 10^9-14\times 10^9$ years. The SFH is given in our catalogs in two different forms: 1) percentage of the stellar mass and 2) star formation rate (SFR) in the four bins. The first is computed according to the following: $$\label{eqn:massbin} M_{bin}=\sum_{i=1}^{N_{bin}}\left(C_i\times M^\star_i \right) / \sum_{i=1}^{N_{SSP}}\left(C_i\times M^\star_i \right)$$ where $N_{bin}$ is the number of SSPs contained in a given age bin; $C _i$ is the normalisation constant of each SSP of that bin, i.e. the stellar mass at each age according to definition 1; $M^\star_i$ is the factor, which is a function of the stellar age, that converts the SSP initial mass (definition 1.) into either the mass locked into stars (mass definition 2.) or into mass of nuclear burning stars (definition 3.), while the sum at the denominator is the total stellar mass (according to definitions 2 and 3, respectively). The star formation rate as a function of the stellar age is computed by dividing the stellar mass of a given age bin by its duration. Definition 1 of the mass was applied in this calculation [see also equation 1 in @longhetti09]. The current SFR value, i.e. the one calculated within the youngest age bin, deserves a particular attention, since it is calculated by fitting the equivalent width of emission lines, namely Hydrogen (H$\alpha$ and H$\beta$) and Oxygen ( at 3727 Å). The lines’ luminosity is entirely attributed to star formation processes neglecting other mechanisms that can produce ionizing flux. In this way we are overestimating the current SFR in both LINERS and AGNs. In a forthcoming work, we will present an analysis of standard diagnostic diagrams such as those by [@veilleux87], with the lines’ intensities accurately measured by subtracting stellar templates from the observed spectrum (Marziani et al., in prep.). This work will enable the distinction between “pure” star forming systems and those where other mechanisms might be co-responsible for line emission. Dust extinction --------------- According to the “selective extinction” hypothesis [@calzetti94], which we fully consider in our modelling, each SSP has its own value of the dust attenuation. We compute an age-averaged value of dust extinction, as it is derived by the model, by using Eq.\[eqn:av\]: $$\label{eqn:av} A_V=-2.5\times \log_{10}\left[\frac{L_{tot}^M(5550)}{L_{unext}^M(5550)} \right]$$ where $L_{tot}^M$ and $L_{unext}^M$ are, respectively, the model spectrum and the model non-attenuated spectrum (i.e. the model with the same SFH as $L_{tot}^M$ but with $A_V=0$ for each stellar population). We calculate two distinct values: we first take into account only stellar populations that are younger than $\sim 2\times 10^7$, i.e. those that are responsible for nebular emission; this value is comparable with extinction that is computed from emission lines ratio. Secondly, we use all stellar populations providing, in this way, an extinction value which is averaged over SSP of all ages. Average ages ------------ Exploiting the information derived by our analysis, we are able to provide an estimate of the average age of a galaxy, weighted on the stellar populations that compose its spectrum. Given that the mass-to-light ratio changes as a function of the age, there are two different definitions that can be given: the mass-weighted and the luminosity-weighted age [see also @fernandes03]. The latter is the most commonly given, since it is directly derived from the spectrum, being weighted in this way towards the age of the stellar populations that dominate the light, while the first definition requires the knowledge of the mass distribution as a function of stellar age, i.e. the SFH. We can compute the logarithm of these two quantities as follows: $$\label{eqn:lwage} \langle \log(T)\rangle_L=\frac{1}{L_{tot} (V)}\times \sum_{i=1}^{N_{SSP}}L_i(V)\times \log(t_i)$$ for the logarithm of the luminosity weighted age, where $L_i(V)$ and $L_{tot}(V)$ are the restframe luminosities of the [*i-th*]{} SSP and of the total spectrum, respectively, in the V-band, and $t_i$ the age of the $i-th$ SSP. The mass-weighted age is computed in a similar way as: $$\label{eqn:mwage} \langle \log(T)\rangle_M=\frac{1}{M_{tot}}\times \sum_{i=1}^{N_{SSP}}M_i\times \log(t_i)$$ and, similarly, $M_{tot}$ and $M_i$ are the total mass and the mass of the [*i-th*]{} SSP, respectively. Hence, while the luminosity-weighted age gives an estimate of the age of stars that dominate the optical spectrum, being in this way more sensitive to the presence of young stars, the mass-weighted value is more representative of the actual average age of a galaxy’s stellar populations. Note that to compute these values, we use the finest age grid, averaging over the 12 stellar populations. We provide both the luminosity-weighted age computed from the V-band, and the one computed from the bolometric luminosity. The two values are, anyway, very similar. Absolute magnitude computation and prediction {#sec:mag} --------------------------------------------- The fact that the theoretical SSP spectra that we use for our modeling cover a wide range in wavelengths, allows us to compute absolute magnitudes in various bands that are not covered by the observed spectra, without having to assume any K-correction. To compute the absolute magnitude of a galaxy, we take the best-fit model spectrum, compute its flux as if it was observed at 10 pc and convolve it with the proper filter transmission curve: $$M_b=\frac{\int_{\lambda_0}^{\lambda_1} F_{d=10pc}^M(\lambda)\times T_b(\lambda) \; d\lambda}{\int_{\lambda_0}^{\lambda_1} T_b(\lambda) \; d\lambda}$$ where $T_b(\lambda)$ is the transmission curve of the filter for the band $b$ and $F_{d=10pc}^M(\lambda)$ is the model spectrum calculated at a distance of 10 pc. For the sake of clearity, in Table \[tab:f0\] we provide the zero-point fluxes, expressed in erg/s/cm$^2$/Å that were used to compute all of the magnitudes. $f_0$ $\lambda_{eff}$ Band ----------- ----------------- ------ \[Å\] 4.217e-09 3605 U 6.600e-09 4413 B 3.440e-09 5512 V 1.749e-09 6586 R 8.396e-10 8060 I 3.076e-10 12370 J 1.259e-10 16460 H 4.000e-11 22100 K 8.604e-09 3521 u 4.676e-09 4804 g 2.777e-09 6253 r 1.849e-09 7668 i 1.315e-09 9115 z : Zero-point fluxes that are used to calculate observed expected magnitudes and absolute magnitudes, together with their effective lambda. Johnson and   filters characteristics were taken from the Asiago Database of Photometric Systems [@moro00].[]{data-label="tab:f0"} $UBVRIJHK$ magnitudes are computed according to the Johnson system, while $ugriz$ magnitudes are calculated in order to match the Sloan system. VALIDATION ========== In order to compare with the widely used  masses, we performed a comparison of the stellar mass values for a subsample of  galaxies that has been spectroscopically observed also by the . As a reference for masses from the SLOAN survey we used those derived by [@gallazzi05], using the Data Release 4 (DR4)[^1], and those obtained from the photometry exploiting Data Release 7 (DR7)[^2]. In this way, we built two sub-samples of galaxies observed by both surveys, namely 395 in the -DR4 sample, and 606 in the -DR7 sample. \[!t\] ------------------------------------------------------- ------------------------------------------------------------ {height=".495\textwidth"} {height=".495\textwidth"} ------------------------------------------------------- ------------------------------------------------------------ We performed a double check: as a first step, we exploited our spectrophotometric model to derive, using  spectra and $g$ (model-)band magnitudes of the -DR4 sample, the same quantities that were inferred for  galaxies. In this way, comparing the results obtained with the same () data but with different methods, we can demonstrate the reliability of our technique. As a second step, we compare total stellar mass values obtained with our model and data, to those of the  DR4 and DR7, respectively. To ensure this comparison is significant, we have to consider the details of the models used to derive such quantities. In particular, we have to take into account the differences in the IMFs that are assumed, i.e. [@salpeter55] for  (we recall here that the mass limits that we have adopted are 0.15 and 120 M$_\odot$, respectively), [@chabrier03] for masses derived by [@gallazzi05], and [@kroupa01] for , DR7, respectively. We have determined that the difference between Salpeter’s and Kroupa’s IMF is a factor of $\sim1.33$ (0.125 dex), the Salpeter IMF yielding the highest values of masses, while Chabrier’s IMF yields stellar masses that are 1.1 (0.04 dex) times lower with respect to Kroupa’s [see, e.g., @cimatti08]. For the sake of homogeneity, and only for the purposes of these sanity checks, we will rescale all the mass values to the [@kroupa01] IMF. Note that all the mass values we refer to are calculated according to definition 2. (see Sect.\[sec:mass\]). In Fig.\[fig:mass\_comp\] we show how the different methods compare, exploiting both DR4 and DR7 data. In the left-hand panel, we plot mass values derived using our model against those obtained by [@gallazzi05], both from  DR4 data. Our mass determination was obtained by fitting the spectrum —which was normalized to the total model $g$ band magnitude— in the same way as done for  data. The agreement between the two methods overall is good, with an rms of $\sim 0.21$. In the right-hand panel of Fig.\[fig:mass\_comp\] we show the comparison between the masses we derived from the  spectroscopy scaled to the $g$ band fiber magnitude and the fiber-aperture photometric masses from the DR7. Hence, we are comparing two mass estimates within the same fibre aperture, obtained using either the  spectroscopy+photometry or only  photometry, and we do not have to deal with aperture effects. The rms is $\sim 0.17$, but it is worth noting that, the data displays a $\sim 0.15$dex systematic offset, in the sense that the DR7 yields slightly lower masses. This is in contrast to the DR4 comparison which shows remarkable agreement, even though there is some dispersion with respect to the 1:1 relation. A small offset in the same direction is present also when comparing DR4 and DR7 masses for galaxies in common, as shown in Fig.\[fig:sdss2\]. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ![The comparison between total mass values from the  DR4, as calculated by [@gallazzi05], and those obtained from DR7 photometry. Both masses are rescaled to a Kroupa (2001) IMF.[]{data-label="fig:sdss2"}](dr4_7_bis.eps "fig:"){height=".48\textwidth"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- In Fig.\[fig:mass\_comp2\] we move to the comparison of the total stellar masses we derived from the  data, corrected for color gradients, and total masses given by Gallazzi et al. (2004) and DR7, always considering galaxies of the subsamples in common. The scatter around the 1:1 relation is slightly larger in these cases, (0.23 and 0.22 for DR4 and DR7, respectively) and for the DR7-derived masses the average difference is negligible at low masses and tends to increase with mass. For this comparison, in addition to the different mass estimate methods, the data are also different:  spectra are taken within an aperture of $\sim 2''$ while Sloan fibers cover a $\sim 3''$ aperture, centered on a position that can be, in general, different. Also, source and flux extraction techniques, and the spectral resolution of the two surveys are different. The general agreement, however, is satisfactory, and the scatter is similar to the $\sim 0.2$dex accuracy expected with these methods (e.g. Cimatti et al. 2008). --------------------------------------------------------- --------------------------------------------------------- {height=".48\textwidth"} {height=".48\textwidth"} --------------------------------------------------------- --------------------------------------------------------- We conclude that, despite the substantial differences in the fitting approach, in the adopted theoretical libraries and in the characteristics of the datasets themselves, our total mass values are in overall agreement with those of a considerable number of objects that the  has in common with . THE CATALOGS ============ In this section we briefly describe the most relevant quantities given in the catalogs we are releasing to the astronomical community. About 70% of the observed spectra have been fitted with a $\chi^2 \leq 3$ and we consider fits with such values to be reliable. For higher values, a visual inspection is recommended to asses the reliability of the spectral fit. For each spectrum that has been analyzed we give the following: - the reduced $\chi^2$ for the fits obtained for the three values of metallicity. Note that we take as a reference model the one with the value of the metallicity that yields the lowest $\chi^2$ value, regardless of the fact that other values of the metallicity are also providing acceptable fits. These values are also useful to flag potentially unreliable fits. A $\chi^2 \leq 3$ can be used as a discriminant for blindly accepting a result; - extinction in the V band, in magnitudes, computed from the model spectrum both averaging on young stellar populations (i.e. with age$\leq 2\times 10^7$ years) and on all ages, including uncertainties on both quantities; - SFR in the four main age bins as defined in Sect.\[sec:sfh\], with related uncertainties, all expressed in $M_\odot/yr$; note that these SFR only refer to values normalized to the fiber-aperture magnitude. In order to compute the global value, one should multiply the fiber-SFR by a factor $\wp=10^{-0.4\cdot (V_{tot}-V_{fib})}$, that is the ratio of total and aperture fluxes; - percentage of the stellar mass in the 4 main age bins, with related uncertainties, calculated for the different mass definitions - the logarithm of total stellar mass, expressed in $M_\odot$, within the fiber aperture, according to the 3 definitions explained in section \[sec:mass\], together with the related uncertainties (expressed in logarithm of solar masses as well), which are computed for the definition 3. mass value - the logarithm of total stellar mass, expressed in $M_\odot$, computed by rescaling the fiber spectrum to the total V magnitude (see section \[sec:mass\]), and the related uncertainties (in logarithm of the solar mass), uncorrected for color gradients; - the logarithm of the stellar mass calculated from the B and V band photometry, according to the [@bdj01] prescription, for both total and fiber magnitudes; - the colour-correction term, described in §4.2, to be added to the total mass to account for color gradients; - the logarithm of the luminosity-weighted age computed both using the luminosity in the V band, and the bolometric emission, and the related uncertainties: the latter are computed only with respect to the bolometric luminosity-weighted age; - the logarithm of the mass-weighted age, and the related uncertainty - Galactic extinction-corrected observed B and V magnitude referring to both the fiber and the total aperture; we report these magnitudes even though they are actually measured values [see @varela09], because these are the values used to rescale the observed spectrum and, hence, to derive the total mass. Values of extinction within our Galaxy for each of the clusters were taken from NED [see also @schlegel98]; - absolute V and B magnitudes calculated from the observed spectrum, derived from both aperture and total magnitudes; - Johnson (UBVIRJHK) and Sloan (ugriz) expected observed magnitudes calculated from our best model spectrum, both within our fiber aperture and total; - Johnson (UBVIRJHK) and Sloan (ugriz) absolute magnitudes calculated from our best model spectrum, both within our fiber aperture and total. Whenever one of the above listed quantities is not available, this is flagged with a 99.99. All the data and physical quantities described in this paper will be available by querying the  database at the following web address:\ `http://web.oapd.inaf.it/wings/`. In Table \[tab:sfh\] we give an example of how the full set of information will look like, reporting data for 5 galaxies of the sample. A description of each item, together with their units, can be found in table \[tab:header\].\ [**ACKNOWLEDGEMENTS**]{}\ This paper took great advantage from discussions with Anna Gallazzi and Jarle Brinchmann, who kindly provided us with all the details of their stellar masses calculations using DR4 and DR7, respectively.\ Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England.\ The SDSS Web Site is: `http://www.sdss.org/`.\ We are grateful to the anonymous referee, whose comments and remarks helped us to improve the quality and the readability of this work. Bell, E. F., & de Jong, R. S. 2001, , 550, 212 Bertelli, G., Bressan, A., Chiosi, C., Fagotto, F., & Nasi, E. 1994, , 106, 275 Bressan, A., Granato, G. L., & Silva, L. 1998, , 332, 135 Bruzual A., G. 1983, , 273, 105 Bruzual, G., & Charlot, S. 2003, , 344, 1000 Calzetti, D., Kinney, A. L., & Storchi-Bergmann, T. 1994, , 429, 582 Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, , 345, 245 Cava, A., et al. 2009, , 495, 707 Chabrier, G. 2003, , 586, L133 Cimatti, A., et al. 2008, , 482, 21 Dressler, A., et al. 1997, , 490, 577 Fasano, G., et al. 2006, , 445, 805 Ferland, G. J. 1996, Hazy, a Brief Introduction to CLOUDY, in University of Kentucky, Department of Physics and Astronomy Internal Report Fernandes, R. C., Le[ã]{}o, J. R. S., & Lacerda, R. R. 2003, , 340, 29 Fioc, M., & Rocca-Volmerange, B. 1997, , 326, 950 Fritz, J., et al. 2007, , 470, 137 Fukugita, M., Ichikawa, T., Gunn, J. E., Doi, M., Shimasaku, K., & Schneider, D. P. 1996, , 111, 1748 Gallazzi, A., Charlot, S., Brinchmann, J., White, S. D. M., & Tremonti, C. A. 2005, , 362, 41 Jacoby G.H., Hunter D.A., Christian C.A.  1984, , 56, 257 Kauffmann, G., Heckman, T. M., White, S. D. M., et al. 2003, , 341, 33 Kodama, T., & Arimoto, N. 1997, , 320, 41 Kroupa, P. 2001, , 322, 231 Longhetti, M., & Saracco, P. 2009, , 394, 774 Moro, D., & Munari, U. 2000, , 147, 361 Osterbrock, D. E. 1989, Research supported by the University of California, John Simon Guggenheim Memorial Foundation, University of Minnesota, et al. Mill Valley, CA, University Science Books, 1989, 422 p. Poggianti, B. M. 1997, , 122, 399 Poggianti, B. M., Smail, I., Dressler, A., Couch, W. J., Barger, A. J., Butcher, H., Ellis, R. S., & Oemler, A., Jr. 1999, , 518, 576 Poggianti, B. M., et al. 2008, , 684, 888 Postman, M., & Geller, M. J. 1984, , 281, 95 Postman, M., et al. 2005, , 623, 721 Renzini, A. 2006, , 44, 141 Salpeter, E. E. 1955, , 121, 161 S[á]{}nchez-Bl[á]{}zquez, P. 2004, Ph.D. Thesis, S[á]{}nchez-Bl[á]{}zquez, P., et al. 2006, , 371, 703 Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, , 500, 525 Valentinuzzi, T., et al. 2009, , 501, 851 Varela, J., et al. 2009, , 497, 667 Veilleux, S., & Osterbrock, D. E. 1987, , 63, 295 \[tab:sfh\] -------------------------- ---------------- -------- -------- -------- -------- ------- ------- ------- ------- ------- -------- -------- -------- -------- WINGSJ103833.76-085623.3 A1069\_12\_129 234.64 7.193 7.108 7.215 0.020 0.426 0.504 0.241 0.184 0.0527 0.0377 0.0328 0.0226 WINGSJ103834.09-085719.2 A1069\_12\_144 247.74 11.378 11.296 11.214 0.004 0.931 0.252 0.342 0.107 0.0058 0.0190 0.0320 0.0068 WINGSJ103835.89-085031.5 A1069\_11\_163 856.69 1.568 1.624 9.674 0.050 1.045 0.320 0.200 0.059 0.3143 0.2649 0.1988 0.0918 WINGSJ103842.69-084611.6 A1069\_11\_132 908.99 9.000 8.351 8.407 0.020 0.621 0.240 0.165 0.079 0.3780 0.1027 1.1693 0.1998 WINGSJ103843.03-085602.8 A1069\_11\_171 250.29 1.548 1.395 1.832 0.020 1.299 0.241 0.400 0.134 1.7552 1.2031 0.0721 0.0566 -------------------------- ---------------- -------- -------- -------- -------- ------- ------- ------- ------- ------- -------- -------- -------- -------- : \ ----- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -- -- -- ... 0.0320 0.0134 0.0073 0.0156 0.0044 0.0054 0.0059 0.0050 0.0787 0.0877 0.0941 0.0774 0.6617 0.6560 0.6559 0.2682 0.2553 0.2509 ... 0.0014 0.0028 0.0080 0.0047 0.0012 0.0015 0.0018 0.0073 0.1979 0.2167 0.2397 0.0900 0.0745 0.0732 0.0754 0.1497 0.7263 0.7086 ... 4.2464 3.6629 8.1438 6.6400 0.0001 0.0001 0.0001 0.0001 0.0013 0.0014 0.0015 0.0006 0.2344 0.2343 0.2408 0.3196 0.7642 0.7643 ... 0.0828 0.0992 0.4542 0.3301 0.0015 0.0020 0.0022 0.0008 0.1367 0.1462 0.1573 0.0737 0.0835 0.0847 0.0889 0.0995 0.7783 0.7672 ... 0.6712 0.4775 1.3432 0.6312 0.0024 0.0030 0.0034 0.0026 0.0028 0.0032 0.0036 0.0031 0.2260 0.2264 0.2318 0.1758 0.7688 0.7674 ----- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -- -- -- : \ ----- -------- -------- --------- --------- --------- --------- --------- --------- --------- --------- --------- --------- --------- --------- --------- ... 0.2440 0.2206 8.3839 8.2619 8.2169 7.7948 9.0159 8.8939 8.8489 8.4267 8.4184 9.0504 0.1210 8.41915 8.14517 ... 0.6832 0.1753 7.9725 7.8784 7.8131 7.2090 8.6765 8.5824 8.5171 7.9129 8.2291 8.9330 -0.0390 8.34588 8.32951 ... 0.7576 0.3193 10.9570 10.8295 10.7690 10.0297 11.4170 11.2895 11.2290 10.4895 10.8701 11.3301 -0.1890 9.86304 9.88039 ... 0.7516 0.1461 9.6955 9.5724 9.5191 8.9408 10.0235 9.9004 9.8471 9.2687 9.6927 10.0207 -0.0190 8.63437 8.27387 ... 0.7612 0.1768 10.1717 10.0432 9.9849 9.1670 10.6597 10.5312 10.4729 9.6546 10.0673 10.5553 0.0610 9.31117 9.29801 ----- -------- -------- --------- --------- --------- --------- --------- --------- --------- --------- --------- --------- --------- --------- --------- : \ ----- --------- ---------- --------- -------- -------- -------- -------- --------- --------- --------- --------- -------- ... 7.65407 9.62516 9.33765 20.984 21.353 19.404 19.833 -15.845 -15.432 -17.426 -17.013 21.040 ... 7.62478 9.70064 9.22101 21.474 21.923 19.714 20.073 -15.424 -15.112 -17.184 -16.872 21.598 ... 9.08572 10.00135 9.37758 19.764 21.263 18.614 19.883 99.990 -19.176 99.990 -20.325 21.850 ... 7.43700 9.83851 9.36962 20.944 21.433 20.124 20.553 99.990 -18.331 99.990 -19.151 21.676 ... 8.71111 10.00002 9.24891 18.294 18.873 17.074 17.683 -18.774 -17.898 -19.993 -19.117 19.217 ----- --------- ---------- --------- -------- -------- -------- -------- --------- --------- --------- --------- -------- : \ ----- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- ... 21.467 20.999 20.511 20.006 19.425 18.837 18.441 19.460 19.887 19.419 18.931 18.426 ... 21.922 21.442 21.050 20.630 20.277 19.829 19.480 19.838 20.162 19.682 19.290 18.870 ... 21.415 19.737 18.833 18.022 16.974 16.268 15.528 20.700 20.265 18.587 17.683 16.872 ... 21.995 21.033 20.436 19.950 19.511 19.004 18.332 20.856 21.175 20.213 19.616 19.130 ... 19.259 18.225 17.535 16.817 15.980 15.326 14.894 17.997 18.039 17.005 16.315 15.597 ----- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- : \ ----- -------- -------- -------- --------- --------- --------- --------- --------- --------- --------- --------- --------- --------- --------- ... 17.845 17.257 16.861 -15.870 -15.416 -15.823 -16.329 -16.841 -17.408 -17.991 -18.261 -17.450 -16.996 -17.403 ... 18.517 18.069 17.720 -15.463 -15.094 -15.458 -15.889 -16.307 -16.650 -17.085 -17.329 -17.223 -16.854 -17.218 ... 15.824 15.118 14.378 -18.647 -19.199 -20.276 -21.043 -21.818 -22.763 -23.449 -23.753 -19.797 -20.349 -21.426 ... 18.691 18.184 17.512 -18.420 -18.212 -18.730 -19.237 -19.650 -20.201 -20.725 -21.017 -19.240 -19.032 -19.550 ... 14.760 14.106 13.674 -17.910 -17.921 -18.790 -19.509 -20.222 -21.013 -21.653 -21.943 -19.130 -19.141 -20.010 ----- -------- -------- -------- --------- --------- --------- --------- --------- --------- --------- --------- --------- --------- --------- : \ ----- --------- --------- --------- --------- --------- -------- -------- -------- -------- -------- -------- -------- -------- ... -17.909 -18.421 -18.988 -19.571 -19.841 21.795 21.152 20.952 20.787 20.635 20.215 19.572 19.372 ... -17.649 -18.067 -18.410 -18.845 -19.089 22.355 21.584 21.478 21.384 21.317 20.595 19.824 19.718 ... -22.193 -22.968 -23.913 -24.599 -24.903 22.758 20.581 19.350 18.866 18.508 21.608 19.431 18.200 ... -20.057 -20.470 -21.021 -21.545 -21.837 22.454 21.508 20.884 20.749 20.748 21.634 20.688 20.064 ... -20.729 -21.442 -22.233 -22.873 -23.163 20.021 18.691 18.031 17.641 17.341 18.801 17.471 16.811 ----- --------- --------- --------- --------- --------- -------- -------- -------- -------- -------- -------- -------- -------- : \ ----- -------- -------- --------- --------- --------- --------- --------- --------- --------- --------- --------- --------- ... 19.207 19.055 -15.083 -15.695 -15.895 -16.043 -16.223 -16.663 -17.275 -17.475 -17.623 -17.803 ... 19.624 19.557 -14.658 -15.371 -15.471 -15.534 -15.638 -16.418 -17.131 -17.231 -17.294 -17.398 ... 17.716 17.358 -17.777 -19.777 -20.548 -20.961 -21.310 -18.927 -20.927 -21.698 -22.111 -22.460 ... 19.929 19.928 -17.603 -18.599 -18.827 -18.861 -19.062 -18.423 -19.419 -19.647 -19.681 -19.882 ... 16.421 16.121 -17.097 -18.416 -19.032 -19.375 -19.697 -18.317 -19.636 -20.252 -20.595 -20.917 ----- -------- -------- --------- --------- --------- --------- --------- --------- --------- --------- --------- --------- : [^1]: Stellar masses computed by [@gallazzi05], by means of DR4 data are publicly available at this website: [^2]: Stellar mass values for the DR7 data release were taken from the following  website:

--- abstract: | Multiuser Multi-Packet Transmission (MPT) from an Access Point (AP) equipped with multiple antennas to multiple single-antenna nodes can be achieved by exploiting the spatial dimension of the channel. In this paper we present a queueing model to analytically study such systems from the link-layer perspective, in presence of random packet arrivals, heterogeneous channel conditions and packet errors. The analysis relies on a blind estimation of the number of different destinations among the packets waiting in the queue, which allows for building a simple, but general model for MPT systems with per-node First-In First-Out (FIFO) packet scheduling. Simulation results validate the accuracy of the analytical model and provide further insights on the cross-relations between the channel state, the number of antennas, and the number of active users, as well as how they affect the system performance. The simplicity and accuracy of the model makes it suitable for the evaluation of Medium Access Control (MAC) protocols for Ad-Hoc or Wireless Local Area Networks supporting multiuser MPT in non-saturation conditions, where the queueing dynamics play an important role on the achieved performance, and simple user selection algorithms are required. **Keywords**: Multiuser MPT, SDMA, MAC protocols, Queueing model, Wireless Networks, Performance Evaluation author: - | Boris Bellalta$^{(1)}$, Azadeh Faridi, Jaume Barcelo, Vanesa Daza, Miquel Oliver\ Department of Information and Communication Technologies.\ Universitat Pompeu Fabra, Barcelona, Spain\ (1) Corresponding author, e-mail: boris.bellalta@upf.edu bibliography: - 'SMACMA.bib' title: 'Performance Analysis of a Multiuser Multi-Packet Transmission System for WLANs in Non-Saturation Conditions' --- Introduction {#Sec:Intro} ============ In packet-based wireless networks, the use of spatial multiplexing allows for simultaneous transmission of multiple packets, directed to a single or multiple destinations. In this paper, we focus on a scenario where an AP equipped with multiple antennas is able to simultaneously transmit multiple packets, each directed to a different single-antenna node. This scenario is known as Multiuser Multi-Packet Transmission (MPT), a packet-based extension of Downlink Space Division Multiple Access (DL-SDMA). Research on Multiuser MPT has mainly been focused on the design of efficient joint precoding and user selection strategies, as a trade-off between computational complexity and the ability to maximize the system capacity, i.e., the number of bits per Hertz of available bandwidth that can be successfully transmitted over the channel. [A comprehensive survey of such works has been presented in [@mietzner2009multiple]]{}. In these works, it is usually assumed that the transmitter has separate per-user queues that are always saturated. This reduces the problem to finding the set of users that maximize the system sum-rate capacity, based on the current state of the channel. Specific schemes for user selection range from random selection to greedy schedulers that benefit from the existing multiuser diversity [@gesbert2007shifting]. The main advantage of random schedulers is their simplicity, as the specific channel conditions of the destinations are not considered for selecting the set of destinations at each transmission. Therefore, they do not need to have recent channel information from all potential destinations, but only from the ones selected for transmission, reducing the overhead required to obtain and keep this information updated. In addition, it provides a fair channel access to the competing users, as all users are selected only based on their traffic load, regardless of their instantaneous channel conditions. For these reasons, they are specially suitable for Ad-Hoc Networks or WLANs, where on the one hand, fairness and simplicity are design requirements and, on the other hand, the sporadic and bursty traffic patterns may not allow for taking full advantage of the existence of Channel State Information (CSI) at the transmitter from all potential destinations. This approach has been widely considered in the design of MAC protocols for WLANs supporting MPT by extending the RTS/CTS mechanism [@cai2008distributed; @li2010multi]. There are still very few works that address spatial multiplexing from the link-layer perspective, and even fewer that focus on the queueing dynamics. In [@zhou2008queuing], the authors present a point-to-point MIMO system, considering both transmit diversity (STBC) and spatial multiplexing (BLAST) schemes for a single-user MPT. A similar work that overcomes the approximation done in [@zhou2008queuing] for the BLAST scenario is presented in [@zhao2012performance], although the presented results are only valid for two antennas. To the best of our knowledge, [@rashid2009cross] is the only work where a detailed queueing model for Multiuser MPT systems is presented. In [@rashid2009cross], the nodes are assumed to be equipped with at least as many antennas as the AP. Therefore, the AP can use its multiple antennas to send multiple packets to a single or multiple users simultaneously, following the independent stream scheduler approach [@heath2001multiuser]. Using the independent stream scheduler, destinations are selected based on the CSI, allowing the transmission of multiple packets to the same destination if in that way the throughput is maximized. In [@bellalta2009space], an M/G/1 batch-service queueing model was proposed to characterize the behavior of MPT systems. However, the queueing model assumes that always the maximum number of packets can be transmitted, regardless of the number of destinations represented among the buffered packets at the AP. Therefore, the model is only valid for when the nodes are equipped with at least as many antennas as the AP. The same model was then used in [@bellalta2010upper; @bellalta2012role] to study the queueing behavior in multiuser MPT systems when all the receivers have a single antenna. This implies that each transmission can contain at most one packet per destination. Comparing with the analytical results from [@bellalta2009space], the simulation results in [@bellalta2010upper] and [@bellalta2012role] show the loss in performance due to having a single antenna at each destination. In [@bellaltaapproximate], this performance loss is modeled analytically, by using an estimate of the number of different destinations represented among the packets waiting in the queue at the AP, which determines the number of packets that can be scheduled in each transmission. This approach was further validated in [@bellalta2011buffer], where the effect of the buffer size in such systems, as well as in the accuracy of the analytical model presented, is evaluated. In all those works an ideal channel is considered, and therefore, the negative effect, in terms of lower transmission rates, caused by the simultaneous transmission of multiple packets is not included in the analysis. In this paper, we extend the model and the results presented in [@bellaltaapproximate; @bellalta2011buffer] by providing a detailed analytical model that includes a realistic channel model, supports heterogeneous channel conditions, multiple transmission rates, packet errors, and a tunable scheduler based on the observed channel conditions in a scenario that consists of a multiple-antenna AP and single-antenna users. Moreover, new insights on the model accuracy in terms of the number of nodes and buffer size are provided, considering heterogeneous traffic and nodes with heterogeneous channel conditions. Hence, the presented model can be used to understand and evaluate the different interactions that exist in a Multiuser MPT system between the traffic load, the buffer size, the number of antennas at the AP, the number of different nodes, the channel characteristics, and the protocol overheads required for CSI estimation and reporting. In addition, due to the model characteristics, it can be easily coupled with other link-layer mechanisms to evaluate more complex systems. The paper is structured as follows. The scenario, together with the system model and the assumptions considered, is introduced in Section \[Sec:Model\]. Section \[Sec:Queue\] presents the Multiuser MPT queuing model. Section \[Sec:Results\] presents the results, including the validation of the queueing model. Finally, the main conclusions of this work are summarized and the future research lines are stated. System Model and Assumptions {#Sec:Model} ============================ A network consisting of a multiple-antenna AP and $N\in[1,\infty)$ single-antenna nodes located a single hop away from the AP is considered. The AP is equipped with $M$ antennas, allowing it to create up to $\min(N,M)$ simultaneous beams and transmit a different packet in each by using a multiuser beamformer. The packets included in each transmission are selected based on a per-node FIFO scheduler as detailed in Section \[Sec:PerNode\]. Multiple transmission rates are available, but only one is used at each transmission, and is picked based on the Channel State Information (CSI) provided by the selected nodes. Despite this rate selection, we assume that packets can still suffer errors due to both transmitter and receiver hardware characteristics, such as clock drifts [@han2009all]. Erroneous packets are retransmitted until they are successfully received. Packets of a constant length of $L_d$ bits destined to the $N$ single-antenna nodes arrive to the AP according to an aggregate Poisson arrival process of rate $\lambda$, containing independent and identically distributed shares of traffic per node. Given that we have to keep track of the order in which packets arrive to the AP to apply the per-node FIFO packet scheduling, a single virtual finite-buffer of size $K$ is considered, and therefore, the finite-buffer space is fully shared by all arriving packets. Moreover, as traffic differentiation between nodes is not considered, compared to the use of $N$ different queues of size $\lceil K/N \rceil$, a single shared buffer of size $K$ is optimal in terms of minimizing the packet losses due to buffer overflow [@kamoun1980analysis]. A detailed model of the AP architecture is shown in Figure \[Fig:AP\_model\], including the single virtual shared buffer. Finally, it should be noted that the considered system does not take advantage of the multiuser diversity as, at each transmission, the AP only requests the CSI of the selected users. In order to consider multiuser diversity, the AP would have to request the CSI for all the nodes with packets waiting for transmission at the AP, based on which it could select the most appropriate nodes for transmission. However, if the number of nodes is large, the time required to obtain the CSI from all nodes may be very large, specially given that the CSI information is requested and transmitted at the lowest rate. In this case, the time needed to collect and process the CSI for all nodes can be longer than the time required for both collecting the CSI and transmitting the corresponding packets for a few selected nodes, even if that is done at the lowest transmission rate. A second consideration is that with sporadic traffic, the time between two transmissions from the AP to the same node may be long, which makes it difficult to reuse previously stored CSI. Per-node FIFO Packet Scheduling {#Sec:PerNode} ------------------------------- At every transmission opportunity, the AP starts by constructing what we refer to as a *space-batch*, containing up to $M$ packets directed to different destinations. The space-batch is constructed on a FIFO basis, however, once a packet destined to a certain node is placed in the space-batch, all subsequent packets directed to that node are skipped and left for future space-batches. This is because each node is only equipped with a single antenna and cannot receive more than one packet at a time, and therefore, there can be at most one packet per destination in every space-batch. The scheduling of space-batches takes place immediately after the completion of each transmission if the queue is not empty. Otherwise, the AP will wait until a new arrival enters the queue, immediately after which a space-batch containing only one packet will be constructed. When there are two or more packets present in the queue after a transmission, there may be multiple packets among them destined for the same node. Therefore, only a subset of the packets in the queue might be eligible for transmission in a single space-batch. Let $\xi$ be the number of eligible packets (destined to distinct destinations) in the queue immediately after a given transmission. Then the size of the next space-batch scheduled after that transmission is given by: $$\label{Eq:batch_policy} m = \max(1,\min(\xi,s_{\max})) = \left \{\begin{array}{lr} 1 ~ & ~ \xi = 0 \\ \xi ~ & ~ 1 < \xi \leq s_{\max}\\ s_{\max} ~ & ~ s_{\max} < \xi \\ \end{array}\right.$$ where $s_{\max}\leq M$ is a system parameter indicating the maximum number of spatial streams allowed to be sent at each transmission. Note when $\xi=0$, the next space-batch will be scheduled as soon as a packet becomes available and will always contain exactly one packet. One of the key properties of this scheduler is that it does not discriminate between different destinations based on their channel status. Therefore, given that the buffer space is shared by all nodes, all arriving packets observe the same blocking probability, which means that destinations with a higher traffic load will have a higher chance of having packets presents in the queue and consequently, will be able to transmit more frequently. However, not choosing the destinations based on the their channel status may be detrimental in terms of the overall system performance as the system will not always transmit at the highest possible rate. To mitigate this problem, the $s_{\max}$ parameter can be used to reduce the number of parallel transmissions, as it will increase the SNR of each transmitted spatial stream and, furthermore, it will improve the performance of the multiuser beamforming, thus increasing the chances to transmit at larger rates without altering the fairness of the scheduler. Frame Structure --------------- Let $\mathcal{S}$ be the set of nodes for which a packet is included in the current space-batch. After constructing the space-batch, the AP broadcasts the identity of the nodes in $\mathcal{S}$. This is immediately followed by a series of training sequences from the AP to the selected destinations. Based on the training sequences, each node in $\mathcal{S}$ reports its estimated CSI back to the AP. The AP will use the received CSI to form the required number of beams and to choose the appropriate transmission rate, as will be detailed shortly. The general structure of a transmitted frame is depicted in Figure \[Fig:Space\_batch\]. It consists of five parts, transmitted in the order presented below: - *Preamble and space-batch information (of length $L_{\text{sb}}$ bits)*: Contains the initial preamble used to synchronize all receivers and the headers required to inform those nodes that have been selected for receiving a packet in the next space-batch. - *Training Sequences (of length $M \times L_{\text{tr}}$ bits)*: Required for estimating the CSI between each of the $M$ antennas at the AP and the receiving antenna at each selected node, and used to calculate the beamforming vectors. - *CSI feed-back (of length $m \times L_{\text{CSI}}$ bits)*: Used for nodes to report their estimated CSI to the AP. - *Data Packet (of length $L_{d}$ bits)*: Includes the space-batch data packets. - *ACKs (of length $m \times L_{\text{ACK}}$ bits)*: Used for nodes to notify the correct reception of the data packet. Channel Model and Transmission Rate Selection {#SEC:rate_select} --------------------------------------------- A quasi-static fading channel that changes from space-batch to space-batch transmission is considered. Let $\Gamma_{n,m}$ denote the instantaneous Signal-to-Noise ratio (SNR) observed by node $n \in \mathcal{S}$ when $m$ spatial streams are included in the space-batch, i.e., $m = |\mathcal{S}|$. The instantaneous SNR observed by a node is assumed to be independent of that of the other nodes. Considering that a ZF beamforming is used, and that the fadings are independent in both time and space at each transmission, the received SNR at each node is assumed to follow a $\chi^2$-distribution with $l = 2 \times (M - m + 1)$ degrees of freedom [@paulraj2003introduction]. The cumulative distribution function for [$\Gamma_{n,m}$]{} is therefore given by: $$\begin{aligned} \label{Eq:ZF_SINR} F_{\Gamma_{n,m}}(\gamma) = \Pr\{\Gamma_{n,m} \leq \gamma\} = 1-\sum_{k=0}^{M-m}{\frac{1}{k!}\left( \frac{\gamma}{\bar{\Gamma}_{n,m}} \right)^{k} e^{-\frac{\gamma} {\bigghost \bar{\Gamma}_{n,m}}}}\end{aligned}$$ where $\bar{\Gamma}_{n,m}$ is the average SNR observed by node $n$ when a space-batch of size $m$ is transmitted and depends only on the pathloss between the AP and node $n$. Note that $\bar{\Gamma}_{n,m}= \bar{\Gamma}_{n,1}/m$, since the transmission power is equally divided between the $m$ parallel streams. Upon receiving the CSI feedback, which is assumed to be ideal, the AP picks a transmission rate based on the SNR values calculated for every selected node. The transmission rates are chosen from a finite set of values $\mathcal{R}=\{r_1,\ldots,r_R\}$, where $r_1 < \cdots < r_R$. The transmission rate for the $n$-th spatial stream, $\hat{r}_n$, is chosen to be $r_i$ if $\Gamma_{n,m}$ falls in the range $(\gamma_{i},\gamma_{i+1}]$, where $\{\gamma_j\}_{j=0}^{R+1}$ are predetermined thresholds. Once the transmission rate at which each spatial stream can be transmitted is known, the AP chooses the smallest rate, $\hat{r}=\min\{\hat{r}_n\}_{n\in \mathcal{S}}$, and uses it as the transmission rate for the whole space-batch. It is assumed that the probability of a space-batch suffering channel errors is negligible if the proper $\hat{r}$ is used. However, as it will be detailed in the next subsection, errors due to hardware characteristics can still happen. It should be noted here that the control parts of the frame are all transmitted omni-directionally at $r_1$. Then, the duration of the control part of the frame is independent of the chosen transmission rate, $\hat{r}$, and is given by: $$\label{Eq:sb_duration} T_c(m)=\frac{L_{\text{sb}}+M\cdot L_{\text{tr}} + m\cdot L_{\text{CSI}} + m\cdot L_{\text{ACK}}}{r_1}$$ The duration of a frame depends on the number of antennas and the transmission rate $\hat{r}$ chosen for the space-batch transmission, hence: $$\label{Eq:sb_duration} T(m,\hat{r})= T_c(m) + \frac{L_{d}}{\hat{r}}$$ ### Packet Errors Apart from channel conditions, packet errors can be caused by other factors such as hardware characteristics [@han2009all]. Since these errors do not depend on the channel conditions, we assume them to homogeneously affect all users. Let $p_e$ be the probability that a packet suffers transmission errors. Then, the probability that $y$ of the $m$ packets included in a space-batch contain errors is given by $$\begin{aligned} \label{Eq:psi_ym} \psi_{y|m}=\binom{m}{y}{p_e^{y}(1-p_e)^{m-y}}\end{aligned}$$ An Example of the System Operation ---------------------------------- In Figure \[Fig:Temporal\], a specific example of the system operation is shown for $M=2$ antennas and a buffer of capacity $K=4$ packets, in a system with transmission rates $\mathcal{R}=\{r_1,r_2,r_3\}$. The $(i-1)$-st space-batch comprises a single packet as the transmission is scheduled as soon as a new packet arrives to the AP and it can be transmitted at rate $r_3$. During the $(i-1)$-st space-batch transmission, two packets, one directed to node $4$ and the other to node $3$, arrive to the AP and are buffered and assembled together in the $i$-th space-batch after the $(i-1)$-st space-batch transmission is completed. Based on the CSI received from nodes $3$ and $4$, rates $\hat{r}_3 = r_3$ and $\hat{r}_4 = r_1$ can be supported. The space-batch is then transmitted at rate $\hat{r} = r_1 = \min \{r_3,r_1\}$. Similarly, during the $i$-th space-batch transmission two more packets directed to node $4$ arrive to the queue, as well as one directed to node $5$, which is blocked because there is no free space in the buffer. Observe that, when the $(i+1)$-st space-batch is scheduled, there are only two packets in the transmission buffer and both are directed to node $4$. In this situation, only one packet can be transmitted, which in our example is done at rate $r_2$. Analytical Model {#Sec:Queue} ================ In this section we present the analytical model. In Table \[Tbl:nomenclature\], we introduce the main notation used to build the model. Variable Description ------------------- ---------------------------------------------------------------------------------------------- $\lambda$ Aggregate packet arrival rate $N$ Number of STAs $M$ Number of antennas at the AP $K$ AP buffer size $R$ Number of transmission rates $p_e$ Packet transmission error probability $s_{\max}$ Maximum number of packets that can be sent at each transmission $m$ Current space-batch size $y$ Number of erroneous packets in a space-batch transmission $T(m,\hat{r})$ Duration of a space-batch comprising $m$ packets and transmitted at rate $\hat{r}$ $V[T(m,\hat{r})]$ Number of packet arrivals in $T(m,\hat{r})$ $s(i)$ Maximum space-batch size when there are $i$ packets in the queue $p_{m|i}$ Probability to transmit a space-batch of $m$ packets when there are $i$ packets in the queue $\psi_{y|m}$ Probability that $y$ of the $m$ transmitted packets contain errors $\phi_{r_l|m}$ Probability that a space-batch comprising $m$ packets is transmitted at rate $r_l$ : Notation used in the analytical model.[]{data-label="Tbl:nomenclature"} Queueing Model {#Sec:QueueModel} -------------- Poisson arrivals of rate $\lambda$ and a general service time distribution are considered. The buffer has a size of $K$ packets and no extra space is considered for the packets in service, i.e., the packets included in a space-batch transmission remain stored in the queue until their transmission is completed. As explained before, the scheduling of a space-batch takes place immediately after each departure, if the queue is not empty. Here a [*departure*]{} instant refers both to the moments at which all packets from a successfully transmitted space-batch are purged from the queue and to the end of channel outage periods. If just after a departure the queue is empty, the AP will wait until there is a new arrival, at which instant a new transmission containing a single packet is scheduled. Let $q_k$ denote the queue occupancy at the end of the $k$-th interdeparture epoch. If $m_k$ is the number of transmitted packets, $y_k$ the number of packets suffering errors, and $\hat{r}(k)$ the transmission rate at which the space-batch is transmitted during the $k$-th epoch, then $q_k$ evolves according to the following recursion: $$\label{Eq:queue_recursion} q_{k} = \min\left\{V[T(m_{k},\hat{r}(k))]+ q_{k-1}-m_{k}+y_{k},K-m_{k}+y_{k}\right\}$$ where $V[T(m_{k},\hat{r}(k))]$ is the number of packet arrivals during the $k$-th inter-departure epoch. Therefore, $V[T(m_{k},\hat{r}(k))]+ q_{k-1}-m_{k}+y_{k}$ is the number of packets that would be present in the queue at the end of the $k$-th epoch if the buffer had infinite capacity, and $K-m_{k}+y_{k}$ is the maximum possible queue occupancy immediately after a departure for a finite buffer. Notice that for very small values of $K$, the queue occupancy after a departure can be lower than $s_{\max}$, which limits the size of the next space-batch by design to a lower value than its maximum. To avoid this situation, the queue size has to be at least $K\leq 2 s_{\max}$. In order to find analytical formulation for key performance metrics such as delay and throughput, the steady-state queue occupancy probabilities need to be calculated. To derive these probabilities, first, the steady-state distribution for the queue states immediately after departures, $\piD$, is derived using a discrete-time embedded Markov chain. Then the PASTA (Poisson Arrivals See Time Averages) property of the Poisson arrivals is applied to find the occupancy distribution at *arbitrary times*, $\piS$, as a function of $\piD$. In what follows, in order to make a clear distinction between the aforementioned two different steady-state probabilities, we define two different sets of states and a corresponding terminology for their probability distribution as follows: - [**queue state at departure instants:**]{} number of packets stored in the queue immediately after a departure. Hereafter, the steady-state probability distribution for these states is referred to as the [*departure distribution*]{}. This is what was denoted above by the row vector $\piD$. - [**queue state at arbitrary times:**]{} number of packets stored in the queue at any arbitrary time. The steady-state probability distribution for these states will simply be referred to as the [*steady-state distribution*]{} of the queue. This is what was denoted above by the row vector $\piS$. ### Blind Estimation of the Space-Batch Size Distribution {#Sec:prob_m} Let $p_{m|q}$ be the probability that $m$ packets are scheduled given that $q$ packets were stored in the queue at the last departure instant prior to the space-batch construction. This probability is given by $$\label{Eq:p_event_model} p_{m|q} = \left\{ \begin{array}{lr} 1,&~~~m=1, q=0 \\ \Pr\{\Xi_q =m\},&~~~ m<s_{\max}, q \geq m\\ \sum_{\xi = s_{\max}}^N \Pr\{\Xi_q =\xi\},&~~~ m = s_{\max}, q \geq m\\ 0, & \text{otherwise} \end{array} \right.$$ where $\Xi_q$ is the random variable denoting the number of packets eligible for transmission in a single space-batch, among the $q$ packets present in the queue. Notice that in the case that the queue is empty, i.e., $q=0$, the next space-batch will always contain only a single packet, i.e., the first packet arriving to the queue, as can be seen in the first line of (\[Eq:p\_event\_model\]). Each of the $q$ packets in the queue can be destined to any of the $N$ different nodes in the network, and therefore there are a total of $N^q$ different possible queue arrangements in terms of destination representation. To compute the probability $\Pr\{\Xi_q = \xi\}$, we need to find the fraction of such arrangements for which there are exactly $\xi$ different destinations represented. Consider one such favorable arrangement with exactly $\xi$ nodes represented. Let $\bar{\bm{\mu}}_\xi=(\mu_1, \ldots, \mu_\xi)$ be a vector containing the number of packets of each of the $\xi$ destinations represented in the queue. Then $\mu_i$ need to be positive integers and $\sum_{i=1}^\xi \mu_i = q$. We define $\Psi_{q,\xi}$ to be the set containing all such vectors, i.e., $\Psi_{q,\xi} = \{\bar{\bm{\mu}}_\xi \in \mathbb{Z}_{+}^\xi \, | \, \sum_{i=1}^\xi \mu_i = q \}$. Then corresponding to every vector in $\Psi_{q,\xi}$, we have $\text{PR}^q_{\bar{\bm{\mu}}_\xi}$ possible different queue arrangements, where $$\text{PR}^q_{\bar{\bm{\mu}}_\xi}= \frac{q!}{\prod_{i = 1}^{\xi} \mu_i !}$$ is the number of permutations of $q$ elements, partitioned into sets of $\mu_{1},\ldots, \mu_{\xi}$ repeated elements. The probability of having $\xi$ eligible packets in the queue is then given by: $$\label{comb_ques_sense_rate} {\Pr \left\{{\Xi_q = \xi}\right\}}=\dfrac{\binom{N}{\xi} \sum_{\bar{\bm{\mu}}_\xi \in \Psi_{q,\xi}} \text{PR}^q_{\bar{\bm{\mu}}_\xi}}{N^q}$$ where $\sum_{\bar{\bm{\mu}}_\xi \in \Psi_{q,\xi}} \text{PR}^q_{\bar{\bm{\mu}}_\xi}$ is the total number of possible queue arrangements for a fixed set of $\xi$ represented nodes and $\binom{N}{\xi}$ is the number of such sets. Notice that (\[comb\_ques\_sense\_rate\]) can be rewritten as follows $$\begin{aligned} Pr\{\Xi_q = \xi\}=\binom{N}{\xi} \xi! {\genfrac{\{}{\}}{0pt}{}{q}{1,\xi}}\end{aligned}$$ where $ {\genfrac{\{}{\}}{0pt}{}{a}{b,c}}$ denotes the generalized Stirling Numbers of the second kind [@cernuschi2001combinatorial]. It should be noted here that in our calculations above, it is assumed that, given a randomly chosen packet from the queue, the probability that it is destined to any given target node is $1/N$, which is the probability that an arriving packet is directed to that target node. However, this is not exactly true because the space-batches are constructed containing no repeated packets and every space-batch departure will reduce the diversity of the queue. This assumption greatly simplifies the analysis, however, as we will see, it does not bear any significant impact on the analytical results, which actually match the simulations quite well. Equation (\[comb\_ques\_sense\_rate\]) generalizes the results in [@mirkovic2008theoretical] by providing a single expression for any value of $M$, $N$ and $q$. In [@mirkovic2008theoretical] only a single case is provided for $q=4$, $N=4$ and $M=4$. In this case, since $M = N$, $p_{m|q}={\Pr \left\{{\Xi_q = m}\right\}}$ for $q>0$, which for $q = 4$ and different values of $m$ is given by: $$\begin{aligned} p_{1|4}&=& \frac{4}{4^{4}} \left(\frac{4!}{4!}\right)\\ p_{2|4}&=& \frac{\binom{4}{2}}{4^4}\left({\frac{4!}{1!3!}+\frac{4!}{2! 2!}+\frac{4!}{3! 1!}}\right)\\ p_{3|4}&=& \frac{\binom{4}{3}}{4^4}\left({\frac{4!}{1! 1! 2!}+\frac{4!}{1! 2! 1!}+\frac{4!}{2! 1! 1!}}\right)\\ p_{4|4}&=& \frac{1}{4^{4}}\left(\frac{4!}{1! 1! 1! 1!}\right)\end{aligned}$$ ### Distribution of the Selected Rate, $\hat{r}$ Based on the rate selection mechanism explained in Section \[SEC:rate\_select\] and the SNR distribution indicated by (\[Eq:ZF\_SINR\]), the probability that a given node $n$ has a feasible rate $r_i$ is given by $$\theta_{n,i|m}=\Pr\{\hat{r}_n = r_i~|~m\} = F_{\Gamma_{n,m}}(\gamma_{i+1})-F_{\Gamma_{n,m}}(\gamma_i)$$ where $m$ is the number of [packets]{} included in the space-batch. Given the set of destinations, $\mathcal{S}$, included in the next space-batch transmission, a rate $\hat{r} = r_i$ will be selected if, based on the CSI feedback from those nodes in $\mathcal{S}$, the smallest feasible rate is $r_i$, i.e., $\min\{\hat{r}_n\}_{n\in\mathcal{S}} = r_i$. Therefore, $\phi_{r_i}(\mathcal{S})$, the probability that for a given space-batch composition, $\mathcal{S}$, the rate $r_i$ is chosen, is given by: $$\begin{aligned} \phi_{r_i}(\mathcal{S}) &=& \Pr \left\{ {\min\{\hat{r}_n\}_{n\in\mathcal{S}} = r_i} \right\}\nonumber\\ &=&\prod_{n \in \mathcal{S}}\Pr \{ \hat{r}_n \geq r_i~|~m\}-\prod_{n \in \mathcal{S}}\Pr\{\hat{r}_n \geq r_{i+1}~|~m\}\nonumber\\ &=&\prod_{n \in \mathcal{S}}{\left(\sum_{j=i}^{R}{\theta_{n,j|m}}\right)}-\prod_{n\in\mathcal{S}}{\left(\sum_{j=i+1}^{R}{\theta_{n,j|m}}\right)}\end{aligned}$$ where the second equality is due to the assumed independence of the observed SNR values for different nodes and can be interpreted as the probability that all the nodes in $\mathcal{S}$ have rates no smaller than $r_i$ but not all of those rates are strictly larger than $r_i$. ### Distribution at departure epochs, $\piD$ The departure probability distribution, $\piD$, is computed by solving the linear system: $$\label{Eq:EMC} \piD = \piD {\mathbf{P}}$$ together with the normalization condition: $$\piD \mathbf{1}^{T} = 1$$ where ${\mathbf{P}}$ is the probability transition matrix of the embedded discrete-time Markov chain of the occupancy of the batch-service queue, immediately after departure instants, with each element $p_{i,j}$, $i,j\in[0,K]$, representing the probability to move from state $i$ to state $j$. In this chain, transitions occur at departure instants, i.e., immediately after the complete transmission of a frame or at the end of a channel outage period, and the states represent the queue occupancy immediately after a departure. The $p_{i,j}$ transition probabilities can be viewed as $$p_{i,j} = {\Pr \left\{{q_k = j|q_{k-1} = i}\right\}}$$ for any $k$, where $q_k$ and $q_{k-1}$ are related according to the recursion in (\[Eq:queue\_recursion\]). As it can be seen in the recursion, $p_{i,j}$ not only depends on the size of the transmitted space-batch, but also on the time it takes to transmit the corresponding frame, which depends on the chosen rate $\hat{r}$, which in turn depends on $\mathcal{S}$, the composition of the space-batch. Let $p_{i,j}(m,y,r)$ denote the conditional transition probability given that at state $i$ a space-batch of size $m$ packets is transmitted at rate $r$ and there are $y$ erroneous packets. This probability depends on the value of $V[T(m,r)]$, the random variable representing the number of arrivals during the transmission of the $m$ packets sent at rate $r$. For any state $i$ in the chain, the last reachable state is $j=K-m+y$. Therefore, $$\label{Eq:transition_pr_conditional} p_{i,j}(m,y,r)= \left\{ \begin{array}{lr} {\Pr \left\{{V[T(m,r)] = j - (i-m+y)}\right\}}, & ~ j < K-m+y\\ {\Pr \left\{{V[T(m,r)] \geq j - (i-m+y)}\right\}}, & ~ j = K-m+y \end{array} \right.$$ where $i\in \left[0,K\right]$, $j \geq i-m+y$, and $m\leq s(i)$, with the function $s(i)=\max(1,\min(i,s_{\max}))$ defined as the maximum possible size of next space-batch when at the end of last departure there are $i$ packets in the queue. For all other values of $i,j$, we have $p_{i,j}(m,y,r) = 0$. Note that departing at state $j = K-m+y$ means that the queue has been containing $q = K$ packets just before the departure, and therefore, some arrivals have possibly been blocked. For all other reachable states from state $i$, the queue has had room for more packets just before the departure and therefore no arrivals could have been blocked. For Poisson arrivals of rate $\lambda$, the number of arrivals during $T(m,r)$ has in general the following distribution: $$\label{Eq:arrival_per_cycle_pdf} {\Pr \left\{{V[T(m,r)] = v}\right\}}=\int_{0}^{\infty}{e^{-\lambda t}\frac{(\lambda t)^v}{v!} f_{T(m,r)}(t)}dt$$ where $f_{T(m,r)}(t)$ is the probability density function of $T(m,r)$. In our case, since given $m$ and $r$ the frame duration $T(m,r)$ is constant, it can be simplified to: $$\label{Eq:arrival_per_cycle_pdf_det} {\Pr \left\{{V[T(m,r)] = v}\right\}} = e^{-\lambda T(m,r)}\frac{(\lambda T(m,r))^v}{v!}$$ For any feasible state pair $(i,j)$, i.e., $i\in \left[0,K\right]$ and $j \in \left[i-m+y,K-m+y\right]$, from (\[Eq:transition\_pr\_conditional\]) and (\[Eq:arrival\_per\_cycle\_pdf\_det\]), we have: $$\label{Eq:p_ij} p_{i,j}(m,y,r)=\left \{\begin{array}{lr} \displaystyle{e^{-\lambda T(m,r)}\frac{(\lambda T(m,r))^v}{v!}} , & ~ j < K-m+y \\ \displaystyle{1-\sum_{z=i-m+y}^{K-m+y-1}{p_{i,j}(m,y,r)}} , & ~ j = K-m+y \end{array}\right.$$ To calculate the unconditional transition probabilities, $p_{i,j}$, we need to average $p_{i,j}(m,y,r)$ over all possible values of $m$, $y$ and $r$, i.e., $$\label{Eq:transition_probs} p_{i,j}=\sum_{m=1}^{s(i)}{p_{m|i}\sum_{y=0}^{m}{\psi_{y|m}\sum_{l=1}^{R}{\phi_{r_l|m}p_{i,j}(m,y,r_l)}}}$$ where $p_{m|i}$ is given by (\[Eq:p\_event\_model\]), with $q=i$, and $\psi_{y|m}$ by (\[Eq:psi\_ym\]). $\phi_{r_l|m}$ is the probability that the rate $r_l$ is chosen for a space-batch, given that it contains $m$ packets. This probability is given by: $$\phi_{r_l|m} =\frac{\sum_{\forall \mathcal{S},~|\mathcal{S}|=m}\phi_{r_l}(\mathcal{S})}{\binom{N}{m}}$$ where the sum is taken over all sets $\mathcal{S}$ of cardinality $m$, i.e., containing $m$ distinct destination nodes. ### Distribution at arbitrary times, $\piS$ Using the PASTA property [@gross1998fundamentals] of Poisson arrivals, the probability that at an arbitrary time in the steady-state the queue contains $q = k$ packets is equal to the probability that a random arrival observes $k$ packets in the queue. In other words, $$\label{Eq:pis1} \pi^s_k={\Pr \left\{{q_a{(t)}=k}\right\}}$$ where $\pi^s_k$ is the $k$-th element of $\piS$, and $q_a{(t)}$ is the state of the queue observed by an arrival at time $t$. The right hand side of (\[Eq:pis1\]) can be expanded by conditioning on $q_d(t)$, the state of the queue at the most recent departure before $t$, i.e., $$\label{Eq:pis2} \pi^s_k = \sum_{i=0}^{k}{{\Pr \left\{{q_a{(t)}=k|q_d{(t)}=i}\right\}}{\Pr \left\{{q_d{(t)}=i}\right\}}}$$ In order to calculate ${\Pr \left\{{q_d{(t)}=i}\right\}}$, we observe that this probability can be viewed as the probability that the arrival at time $t$ happens to occur during departure state $i$, i.e., when the node is in state $i$ of the embedded Markov chain discussed in the previous subsection. Therefore, this probability is equal to the expected fraction of time that the node spends in the departure state $i$. Let the random variable $W(i)$ denote the time spent in departure state $i$. Of this time, $T_d(i)$ seconds will be spent in transmission mode, and only if $i = 0$, an additional $I$ seconds will be spent in idle mode before entering transmission mode. Therefore: $$\label{Eq:epoch_duration} E[W(i)] = E[I] + E[T_d(i)] = \frac{1}{\lambda}\left[ 1-i \right]^+ + E[T_d(i)]$$ where the term $\frac{1}{\lambda}\left[ 1-i \right]^+$ is nonzero only when $i=0$ and is equal to the expected time needed for the queue occupancy to reach $1$ packet. Then, the expected length of an interdeparture epoch is $$\begin{aligned} \label{Eq:epoch_duration} E[W] &= \sum_{i=0}^{K}{\pi^d_i \left(\frac{1}{\lambda}\left[ 1-i \right]^+ + E[T_d(i)]\right)} \nonumber\\ &=\frac{1}{\lambda}\pi^d_0+\sum_{i=0}^{K}{\pi^d_i E[T_d(i)]}\end{aligned}$$ with $$\label{Eq:epoch_duration} E[T_d(i)] =\sum_{m=1}^{s(i)}{p_{m|i}{\sum_{l=1}^{R}{\phi_{r_l|m}T(m,r_l)}}}$$ The probability ${\Pr \left\{{q_d{(t)}=i}\right\}}$ in (\[Eq:pis2\]) can now be calculated as follows: $$\label{Eq:depart_i} {\Pr \left\{{q_d{(t)}=i}\right\}} = \frac{\pi^d_i E[W(i)]}{E[W]}$$ which can be interpreted as the fraction of total time spent in departure state $i$. The term ${{\Pr \left\{{q_a{(t)}=k|q_d{(t)}=i}\right\}}}$ in (\[Eq:pis2\]) is the probability that an arrival during the departure state $i$ observes $k$ packets in the queue. This probability can be viewed as the fraction of arrivals in departure state $i$ which observe $k$ packets in the queue at the moment of their arrival. The expected total number of arrivals in state $i$ is given by $\lambda E[W(i)]$. Of these, only one may observe $k<K$ frames in the queue, provided that there are enough arrivals. Let $q_{n+1}$ be the state at which the next departure will leave the queue. If the space-batch size for this departure is $m$, and it contains $y$ erroneous packets, then the queue occupancy just before this next departure is $q_{n+1}+m-y$. In order for an arrival to have observed $k<K$ packets in the queue, we need $q_{n+1} + m-y \geq k+1$ packets. Therefore, the probability that an arrival in state $i$ observes $k$ packets in the queue, given the space-batch size $m$ and there are $y$ erroneous packets, is given by: $$\begin{aligned} \label{Eq:prob_k_observed_given_m} \Pr\left\{q_{n+1} \geq k+1 -m+y | q_n = i\right\}=\sum_{y=1}^{m}{\psi_{y|m}{\sum_{l=1}^{R}{\phi_{r_l|m}\sum_{j = k+1-m+y}^{K-m+y} p_{i,j}(m,y,r_l)}}}\end{aligned}$$ The probability that an arrival in state $i$ observes $k$ packets in the queue is then given by: $$\begin{aligned} \label{Eq:prob_i_j_dep_} \Pr\left\{q_a{(t)}\right.\left.=k |q_d{(t)}=i\right\} = \frac{\sum_{m=1}^{s(i)}{p_{m|i} {\Pr \left\{{q_{n+1} \geq k+1-m+y | q_n = i}\right\}}}}{\lambda E[W(i)]}\end{aligned}$$ From (\[Eq:pis2\]), (\[Eq:depart\_i\]), and (\[Eq:prob\_i\_j\_dep\_\]), the steady state queue occupancy distribution, $\piS$, for states $0 \leq k \leq K-1$ can be computed as shown in (\[Eq:pis\_k\]). $$\begin{aligned} \label{Eq:pis_k} \pi^s_k=\frac{1}{\lambda E[W]}\sum_{i=0}^{k} \pi_i^d \left(\sum_{m=1}^{s(i)}{p_{m|i}\left(\sum_{y=1}^{m}{\psi_{y|m}\left(\sum_{l=1}^{R}{\phi_{r_l|m}\sum_{j = k+1-m+y}^{K-m+y} p_{i,j}(m,y,r_l)} \right)}\right)}\right)\end{aligned}$$ For $k=K$, we have $$\label{Eq:pis_k_last} \pi^s_K= 1 - \sum_{i = 0}^{K-1} \pi^s_i$$ Note that in (\[Eq:pis\_k\]), when $k=0$, for all values of $m$ and $y$ we have $$\sum_{j = k+1-m+y}^{K-m+y} {p_{i,j}(m,y,r)}=1,$$ and (\[Eq:pis\_k\]) simplifies to: $$\label{Eq:pis_k_smin} \pi^s_0=\frac{1}{\lambda E[W]} \pi_0^d$$ This is because in this case, during the departure state $i=0$, exactly one arrival will observe $0$ packets in the queue with probability $1$. Performance Metrics ------------------- Once the $\boldsymbol{\pi}^d$ and $\boldsymbol{\pi}^s$ distributions are obtained, several performance metrics can be derived from them: - [**Blocking Probability:**]{} The probability that an arriving packet to the AP is discarded because there is no space for it in the transmission queue: $P_b=\pi^s_K$ - [**System Throughput:**]{} Number of bits that can be successfully transmitted from the AP per second: $S=\lambda(1-P_b)L_d$ - [**Average Queue Occupancy:**]{} The average number of packets in the queue: $E[Q]=\sum_{q=1}^{K}{q\pi^s_q}$ - [**Average Response Delay:**]{} The average delay that a packet suffers, from its entrance to the queue until it is transmitted, computed from the average queue occupancy by applying the Little’s Law [@gross1998fundamentals]: $E[D]=\frac{E[Q]}{\lambda(1-P_b)}$ - [**Average Space-Batch Size:**]{} Average number of packets included in the transmitted space-batches: $E[s]=\sum_{q=0}^{K}{\pi^d_q\left(\sum_{m=1}^{s(q)}{m \cdot p_{m|q}}\right)}$ Results {#Sec:Results} ======= In this section, the analytical model is validated through simulations, and some insights on how the number of antennas, number of active users, channel conditions, and traffic load impact the performance of a SDMA-based Multiuser MPT system are provided. The values of the parameters used for both the simulations and the analytical model are listed in Table \[Tbl:parameters\]. A simulator of the described scenario has been built, from scratch, using the C++ language and based on the COST (Component Oriented Simulation Toolkit) libraries [@chen2001component]. The simulator accurately reproduces the system operation described in Section \[Sec:Model\]. Therefore, by comparing the results obtained from the simulator with the ones obtained from the analytical model, we can assess the accuracy of the analytical model and observe the impact of the different assumptions used to built it. For each point, a single simulation with a duration of $1000$ seconds has been run. This duration is sufficiently long for getting confidence intervals that are not graphically visible. In the results, we first focus on the impact of the number of users and queue size on the system performance. We then shift our focus to the effect of channel conditions by considering different SNR and $p_e$ values. Finally, we consider the case in which the traffic is heterogeneous. Parameter Value --------------------------- --------------------------------------------- $L_{\text{sb}}$ $256$ bits $L_{\text{tr}}$ $64$ bits $L_{\text{CSI}}$ $64$ bits $L_d$ $8000$ bits $L_{\text{ACK}}$ $64$ bits $M$ $8$ antennas $\left\{\gamma_j\right\}$ $\left\{10,~15,~20,~+\infty\right\}$ dB $\mathcal{R}$ $\left\{6,~12,~18,~24\right\}$ Mbits/second : Parameters considered for the performance evaluation[]{data-label="Tbl:parameters"} Buffer Size and Number of Users ------------------------------- Figure \[Fig:LoadKU\] shows the behavior of different performance metrics under ideal channel conditions for two values of $K$ ($K=25$ and $K=100$ packets) and four values of $N$ ($N=4$, $N=8$, $N=16$ and $N=32$ nodes) when the aggregate traffic load ($\lambda L$) increases from $40$ Mbps to $120$ Mbps. By ideal channel conditions, we refer to the case in which the AP can always transmit at the highest available transmission rate ($24$ Mbps), regardless of the number of parallel streams being transmitted, and $p_{\text{e}}=0$. Therefore, $s_{\max}$ can be set to its highest value ($s_{\max}=M=8$) to maximize the number of packets that can be included in each space-batch. We consider this scenario in order to focus on how the buffer size and the number of users affect the accuracy of the analysis. In this case, the system performance in terms of blocking probability (Figure \[Fig:LoadKU\_BP\]) and expected delay (Figure \[Fig:LoadKU\_ED\]) is only affected by the ability of the AP to schedule large space-batches (Figure \[Fig:LoadKU\_EB\]), which in turn depends on the queue occupancy (Figure \[Fig:LoadKU\_EQ\]) and the number of nodes sharing the aggregate traffic load. Obviously, in the case where there are fewer nodes than $s_{\max}$, the system performance is limited by the number of nodes. \ For $K=25$, in terms of blocking probability and delay, it can be observed that as the number of nodes increases, for any given aggregate load, the system performance improves since the increased diversity in traffic makes the construction of larger space-batches more likely. For $K=100$, the same behavior can be observed, although the performance gain achieved by increasing the number of nodes is less significant. This is because a larger buffer can store more packets, and therefore, the probability that it contains packets directed to a higher number of destinations is also higher, which allows for transmission of larger space-batches even when the number of nodes is small. However, on the downside, a larger queue size results in a longer expected delay due to increased waiting time. In terms of accuracy, the precision of the analytical model improves as the queue size grows. Since any inaccuracy between the analytical model and the simulations is due to to the blind estimation of the number of nodes represented in the queue, equation (\[Eq:p\_event\_model\]) provides a better estimation of the space-batch size when the queue occupancy is higher, and there is potentially a higher number of destinations represented in the queue. The accuracy of the analytical model is also a function of the number of nodes through (\[Eq:p\_event\_model\]). For $N=1$, as there cannot be any error in the estimation of the number of nodes represented in the queue, the model is exact. For $N\leq M$, the diversity that is removed from the queue at every transmission increases with $N$, and therefore, the accuracy decreases and reaches its minimum at $N=M$. Finally, for $N>M$, as $N$ increases, the analytical model becomes more accurate. This is because in this case the space-batch size is often smaller than the queue diversity, and therefore the departure has a less significant effect on the remaining diversity of the queue. This is why in Figure \[Fig:LoadKU\], where $M=8$, the analytical curves corresponding to $N=4$ and $N=8$ are the curves showing the highest, albeit not significant, discrepancy with their simulated counterparts. Heterogeneous Channel Conditions -------------------------------- In Figure \[Fig:HLoad\_BP\], the blocking probability for $N=16$ nodes, $K=50$ packets, and different $s_{\max}$ values is plotted against different traffic loads in heterogeneous channel conditions. The $16$ nodes are distributed in the coverage area of the AP, in a way that a first group of $5$ nodes observe an average SNR equal to $25$ dBs, a second group of $5$ nodes observe an average SNR equal to $45$ dBs, and finally, a third group of $6$ nodes observe an average SNR equal to $35$ dBs. In all cases the packet error probability is $p_{\text{e}}=0$. As can be seen in this figure, lower blocking probability can be achieved by appropriately choosing the value of $s_{\max}$, which in this specific case is $s_{\max}=6$. The optimal $s_{\max}$ value is a trade-off between the number of packets included in each space-batch and the transmission rate at which the space-batch can be transmitted. When the traffic load increases, the queue occupancy grows, and the probability to schedule larger space-batches also increases. However, as the CSI is not used for selecting neither the number nor the specific destinations to which packets are sent, the transmission rate at which space-batches are sent decreases as the number of spatial streams increases. Therefore, using the $s_{\max}$ parameter, the system can achieve better performance by trading off the maximum number of packets transmitted with the average transmission rate observed. The optimal value of $s_{\max}$ increases with the SNR observed by the nodes. Errors ------ In Figure \[Fig:HLoad\_Errors\], the blocking probability for $N=16$ nodes, $K=50$ packets, $s_{\text{smax}}=6$, and different $p_{\text{e}}$ values is plotted against different traffic loads in non-ideal and heterogeneous channel conditions. The $16$ nodes are distributed in the coverage area of the AP, in a way that a first group of $5$ nodes observe an average SNR equal to $25$ dBs, a second group of $5$ nodes observe an average SNR equal to $45$ dBs, and finally, a third group of $6$ nodes observe an average SNR equal to $35$ dBs. The results show the effect of packet errors in the blocking probability. Increasing , the number of packets successfully transmitted in each space-bath decreases, which reduces the packet departure rate from the queue, increasing the packet blocking probability. For example, for a traffic load of $60$ Mbps, the blocking probability with $p_e=0.1$ is $10^{-4}$, which moves to $10^{-1}$ for $p_e=0.3$. Heterogeneous Traffic Loads --------------------------- The goal of this subsection is to show that the presented model, even though it only supports homogeneous traffic, can be used to understand the system performance when it carries heterogeneous flows with certain patterns. In detail, by concentrating the traffic in a subset of nodes, the performance in terms of blocking probability can be lower bounded by a homogeneous network of the same size, and upper bounded by another homogeneous network of the size of the subset. The impact of having a different traffic load for each node is evaluated using the same ideal channel conditions as in the first case, considering $N=16$ users, $M=8$ antennas, $s_{\max}=8$ and $K=50$ packets. The traffic load for node $i$ is $\lambda_i=\frac{\alpha_i}{\sum_{\forall{j}}{\alpha_j}}\lambda$, with $\alpha_i$ a traffic scaling parameter that determines the fraction of traffic directed to node $i$, and $\lambda$ the aggregate packet arrival rate, which follows a Poisson process. Five different traffic profiles are considered here, as listed in Table \[Tbl:TP\]. They are designed to evaluate how the overall system performance is affected when the fraction of traffic directed to a subset of the active nodes increases. For instance, TP2 assigns eight times more traffic to the first four nodes than to the other twelve nodes. As it will be observed, the presence of heterogeneous traffic will cause a loss on the system performance, mainly because it reduces the queue diversity and consequently, the ability to schedule large space-batches. TP $\{\alpha_i,\ldots,\alpha_{16}\}$ ------ -------------------------------------------------------------- Hom. $\{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1\}$ TP1 $\{4,4,4,4,1,1,1,1,1,1,1,1,1,1,1,1\}$ TP2 $\{8,8,8,8,1,1,1,1,1,1,1,1,1,1,1,1\}$ TP3 $\{16,16,16,16,1,1,1,1,1,1,1,1,1,1,1,1\}$ TP4 $\alpha_i\sim \mathcal{U}[0,16]$, $\alpha_i \in \mathcal{R}$ : Traffic Profiles (TP) []{data-label="Tbl:TP"} Figures \[Fig:HetTraffic\_Pb\] and \[Fig:HetTraffic\_EB\] show the blocking probability and the expected space-batch size respectively for the different traffic profiles from Table \[Tbl:TP\], as well as the blocking probability calculated using the model for $N=16$ and $N=4$ nodes with homogeneous traffic. For all non-homogeneous traffic patterns, the plots are obtained using simulation. It can be observed that when the aggregate traffic load is more concentrated among the four nodes indicated in Table \[Tbl:TP\], the blocking probability increases and tends to the performance of a network consisting of only four homogeneous nodes. This is clearly observed in Figure \[Fig:HetTraffic\_Pb\] for the case where four users have $16$ times the traffic load of each one of the remaining twelve users (i.e., TP3). This loss on performance is caused by the reduction of the number of packets transmitted at each space-batch (Figure \[Fig:HetTraffic\_EB\]). Regarding TP4, where the traffic load of each user is assigned randomly, the blocking probability is not distant from the one obtained with homogeneous traffic and, on average, is better than the performance with TP1, where four users have twice the traffic load of the rest. Moreover, it can be observed how the minimum blocking probability obtained using TP4 is the same as with homogeneous traffic, and that the maximum blocking probability is only slightly higher than the one obtained using TP1. Conclusions =========== We have presented a queuing model for the performance evaluation of SDMA-based Multiuser MPT systems using per-node FIFO packet scheduling. The analysis is built around a blind estimation of the space-batch size distribution, which is only based on the distribution of the traffic load between nodes, which is assumed to be homogeneous. This approximation allows to keep the model reasonably simple but, as the results show, also very accurate. In such conditions, the model is expected to be used both alone, to evaluate the impact of the number of nodes, number of antennas, transmission rates, etc. on the system performance, or coupled with other link-layer mechanisms for the evaluation of more complex systems. For example, it can be easily combined with a model of the Distributed Coordination Function (DCF) [@zhao2011modeling; @bellalta2013performance] to evaluate the performance in non-saturation conditions of the upcoming IEEE 802.11ac amendment [@IEEE80211ac], that will support Multiuser MPT by using spatial multiplexing. The presented model can be further extended in future works to cover other aspects, e.g.: 1) to consider non-uniform traffic distribution among nodes, as well as other Markov-based arrival processes, 2) to formulate schedulers that also consider [the existing]{} multiuser diversity (i.e., schedulers that pick the packets from the buffer based on the instantaneous CSI), 3) to combine the model with packet fragmentation and aggregation techniques in order to reduce the overheads due to CSI estimation and balance the duration of all transmissions using the individual transmission rates [@bellalta2012performance], and 4) to consider different strategies to obtain and apply the CSI, including the case in which the AP is only able to use a set of pre-defined beamforming matrices, which affect the packets that can be selected for transmission, and the use of the Explicit Compressed Feedback protocol defined in the upcoming IEEE 802.11ac amendment. Acknowledgements {#acknowledgements .unnumbered} ================ This work has been partially supported by the Spanish Government under projects TEC2012-32354 (Plan Nacional I+D), TEC2009-13000 (Plan Nacional I+D), CSD2008-00010 (Consolider-Ingenio Program) and by the Catalan Government (SGR2009\#00617).

--- abstract: 'In this paper we study the property of the Arf good subsemigroups of $\mathbb{N}^n$, with $n\geq2$. We give a way to compute all the Arf semigroups with a given collection of multiplicity branches. We also deal with the problem of determining the Arf closure of a set of vectors and of a good semigroup, extending the concept of characters of an Arf numerical semigroup to Arf good semigroups.' author: - | Giuseppe Zito\ \ \ title: Arf good semigroups --- 14.2806pt Introduction {#introduction .unnumbered} ============ In this paper we study a particular class of good subsemigroups of $\mathbb{N}^n$. The concept of good semigroup was introduced in [@BDF]. Its definition depends on the properties of the value semigroups of one dimensional analytically unramfied ring (for example the local rings of an algebraic curve), but in the same paper it is shown that the class of good semigroups is bigger than the class of value semigroups. Therefore the good semigroups can be seen as a natural generalization of the numerical semigroup and can be studied without referring to the ring theory context, with a more combinatorical approach. In this paper we deal only with local good semigroups, i.e good semigroups $S \subseteq \mathbb{N}^n$ such that the only element of $S$ with zero component is the zero vector. In this paper we focus on the class of local Arf good semigroups. This is motivated by the importance of the Arf numerical semigroups in the study of the equivalence of two algebroid branches. Given an algebroid branch $R$, its multiplicity sequence is defined to be the sequence of the multiplicities of the succesive blowups $R_i$ of $R$. Two algebroid branches are equivalent if and only if they have the same multiplicity sequence (cf. [@Campi Definition 1.5.11]). In [@Arf] it is introduced the concept of Arf ring and it is shown that for each ring $R$ there is a smallest Arf overring $R'$, called the Arf closure of $R$, that has also the same multiplicity sequence of $R$. In the same paper it is proved that two algebroid branches are equivalent if and only if their Arf closure have the same value semigroup, that is a numerical Arf semigroup, i.e. a numerical semigroup $S$ such that $S(s)-s$ is a semigroup, for each $s\in S$, where $S(s)=\left\{n \in S; n\geq s \right\}$. All these facts can be generalized to algebroid curves (with more than one branch) and this naturally leads to define the Arf good semigroups of $\mathbb{N}^n$ by extending the numerical definition considering the usual partial ordering given by the components. In the numerical case an Arf semigroup $S=\left\{ s_0=0<s_1<s_2,\ldots \right\}$ is completely described by its multiplicity sequence, that is the sequence of the differences $s_{i+1}-s_i$. Extending the concept of multiplicity sequence, in [@BDF] it is also shown that to each local Arf good semigroup can be associated a multiplicity tree that characterizes the semigroup completely. A tree $T$ of vectors of $\mathbb{N}^n$ has to satisfy some properties to be a multiplicity tree of a local Arf good semigroup. For instance it must have multiplicity sequences along its branches (since the projections are Arf numerical semigroups) and each node must be able to be expressed as a sum of nodes in a subtree of $T$ rooted in it. Thus, taking in account this $1$-$1$ correspondence, the aim of this paper is to study Arf good semigroups by characterizing their multiplicity trees, finding an unambiguous way to describe them. Using this approach, we can also deal with the problem of finding the Arf closure of a good semigroup $S$, that is the smallest Arf semigroup containing $S$. The structure of the paper is the following. In Section \[section2\], given a collection of $n$ multiplicity sequences $E$, we define the set $\sigma(E)$ of all the Arf semigroups $S$ such that the $i$-th projection $S_i$ is an Arf numerical semigroup associated to the $i$-th multiplicity sequences of $E$. We define also the set $\tau(E)$ of the corresponding multiplicity trees and we describe a tree in $\tau(E)$ by an upper triangular matrix $\left(p_{i,j}\right)$, where $p_{i,j}$ is the highest level where the $i$-th and $j$-th branches are glued, and we give a way to deduce from $E$ the maximal value that can be assigned to the $p_{i,j}$. This fact let us to understand when the set $\sigma(E)$ is finite. We introduce the class of untwisted trees that are easier to study because they are completely described by the second diagonal of their matrix, and we notice that a tree can be always transformed in to an untwisted one by permuting its branches. In Section \[section3\] we address the problem of understanding when a set of vectors $G \subseteq \mathbb{N}^n$ determines uniquely an Arf semigroup of $\mathbb{N}^n$. Thus we define $\textrm{Arf}(G)$ as the minimum of the set $S(G)=\left\{ S: S\subseteq \mathbb{N}^n \textrm{ is an Arf semigroup and } G\subseteq S \right\}$ , and we find the properties that $G$ has to satisfy in order to have a good definition for $\textrm{Arf}(G)$ (cf. Theorem \[T45\]). Finally, given a $G$ satisfying these properties, we give a procedure for computing $\textrm{Arf}(G)$. In Section \[section4\] we adapt the techniques learned in the previous section to the problem of determining the Arf closure of a good semigroup. In [@DPMT], the authors solved this problem for $n=2$, leaving it open for larger dimensions. In this section we use the fact that a good semigroup $S$ can be completely described by its finite subset $ \textrm{Small}(S)=\left\{ s\in S : s\leq \delta \right\}$, where $\delta$ is the smallest element such that $\delta+\mathbb{N}^n \subseteq S$, whose existence is guaranteed by the properties of the good semigroups. Finally, in Section \[section5\], we address the inverse problem: given an Arf semigroup $S\subseteq \mathbb{N}^n$, find a set of vectors $G \subseteq \mathbb{N}^n$ , called set of generators of $S$, such that $\textrm{Arf}(G)=S,$ in order to find a possible generalization of the concept of characters in the numerical case. In Theorem \[T5\], we find the properties that such a $G$ has to satisfy and we focus on the problem of finding a minimal one. From this point of view we are able to give a lower and an upper bound for the minimal cardinality for a set of generators of a given Arf semigroup (Corollary \[prop2\]). With an example we also show that, given an Arf semigroup $S$, it is possible to find minimal sets of generators with distinct cardinalities. The procedures presented here have been implemented in GAP ([@gap]). Arf semigroups with a given collection of multiplicity branches {#section2} =============================================================== In this section we determine all the local Arf good semigroups having the same collection of multiplicity branches.\ First of all we need to fix some notations and recall the most important definitions. In the following, given a vector $ v$ in $\mathbb{N}^n$, we will always denote by $v[i]$ its $i$-th component. A good semigroup $S$ of $\mathbb{N}^n$ is a submonoid of $\left( \mathbb{N}^n,+\right)$ such that: (cf. [@BDF]) - for all $a,b \in S$, $ \min(a,b) \in S$; - if $a,b\in S$ and $a[i]=b[i]$ for some $i \in \left\{ 1, \ldots,n\right\}$, then there exists $c \in S$ such that $c[i] > a[i]=b[i]$, $c[j]\geq \min(a[j],b[j])$ for $j \in \left\{ 1, \ldots,n\right\} \setminus \left\{ i\right\}$ and $c[j]=\min(a[j],b[j])$ if $a[j]\neq b[j]$; - there exists $\delta \in S$ such that $\delta+\mathbb{N}^n \subseteq S$ (where we are considering the usual partial ordering in $\mathbb{N}^n$: $a \leq b$ if $a[i] \leq b[i]$ for each $i=1,\ldots,n$).\ In this paper we will always deal with local good semigroups. A good semigroup $S$ is local if the zero vector is the only vector of $S$ with some component equal to zero. However, it can be shown that every good semigroup is the direct product of local semigroups (cf. [@BDF Theorem 2.5]). An Arf semigroup of $\mathbb{N}^n$, is a good semigroup such that $ S(\alpha)-\alpha $ is a semigroup for each $\alpha \in S$ where $ S(\alpha)=\left\{\beta \in S; \beta \geq \alpha \right\}$. The multiplicity tree $T$ of a local Arf semigroup $S\subseteq \mathbb{N}^n$ is a tree where the nodes are vector $ \textbf{n}_i^j \in \mathbb{N}^n$ (where with $\textbf{n}_i^j$ we mean that this node is in the $i$-th branch on the $j$-th level. The root of the tree is $\textbf{n}_{1}^1=\textbf{n}_i^1 $ for all $i$ because we are in the local case and at level one all the branches must be glued) and we have $$S=\left\{\textbf{0}\right\} \bigcup_{T'} \left\{ \sum_{\textbf{n}_i^j \in T' } {\textbf{n}_i^j}\right\},$$ where $T'$ ranges over all finite subtree of $T$ rooted in $\textbf{n}_1^1$. Furthermore a tree $T$ is a multiplicity tree of an Arf semigroup if and only if its nodes satisfy the following properties (cf. [@BDF Theorem 5.11]): - there exists $L \in \mathbb{N}$ such that for $m \geq L$, $\textbf{n}_i^m=(0,\ldots,0,1,0\ldots,0)$ (the nonzero coordinate is in the $i$-th position) for any $i=1,\ldots,n$; - $\textbf{n}_i^j[h]=0$ if and only if $\textbf{n}_i^j$ is not in the $h$-th branch of the tree; - each $\textbf{n}_i^j$ can be obtained as a sum of nodes in a finite subtree $T'$ of $T$ rooted in $\textbf{n}_i^j$. Notice that from these properties it follows that we must have multiplicity sequences along each branch.\ \ Suppose now that $E$ is an ordered collection of $n$ multiplicity sequences (that will be the multiplicity branches of a multiplicity tree). Since any multiplicity sequence is a sequence of integers that stabilizes to 1, we can describe them by the vectors $$M(i)=[m_{i,1},\ldots, m_{i,k_i}],$$ with the convention that $m_{i,j}=1$ for all $j \geq k_i-1$ and $m_{i,k_i-2}\neq 1$; it will be clear later why do not truncate the sequence to the last non-one entry.\ If $M=\max(k_1,\ldots,k_n)$ we write for all $i=1,\ldots,n$ $$M(i)=[m_{i,1},\ldots, m_{i,M}],$$ in order to have vectors of the same length. Each $M(i)$ represents a multiplicity sequence of an Arf numerical semigroup, so it must satisfy the following property:$$\forall j\geq1 \textrm{ there exists } s_{i,j} \in \mathbb{N}, \textrm{ such that } s_{i,j}\geq j+1 \textrm{ and } m_{i,j}=\sum_{k=j+1}^{ s_{i,j}} m_{i,k}.$$ Denote by $\tau(E)$ the set of all multiplicity trees having the $n$ branches in $E$ and by $\sigma(E)$ the set of the corresponding Arf semigroups. We want to find an unambiguous way to describe distinct trees of $\tau(E)$. We define, for all $i=1,\ldots,n$, the following vectors $$S(i)=[ s_{i,1},\ldots s_{i,M}].$$ Because we have $m_ {i,j}=1$ for all $j\geq M-1$, it follows that $ s_{i,j}=j+1$ for all $j\geq M-1$. \[ex1\] Let $M(1)$ be the following multiplicity sequence: $$M(1)=[14,7,5,1,1].$$ Then $S(1)$ is: $$S(1)=[5,5,8,5,6].$$ Notice that, with this notation, from the vectors $S(i)$ we can easily reconstruct the sequences $M(i)$. It suffices to set $m_{i,M}=1$ and then to compute the values of $m_{i,j}$ using the information contained in the integers $s_{i,j}$. We will use the vectors $S(i)$ to determine the level, in a tree of $\tau(E)$, where two branches have to split up.\ For each pair of integers $i,j$ such that $i<j$ and $i,j=1,\ldots,n$ we consider the set $D(i,j)=\left\{ k : s_{i,k} \neq s_{j,k} \right\}$. If $D(i,j) \neq \emptyset $ we consider the integer $$k_E(i,j)=\min\left\{ \min( s_{i,k}, s_{j,k}), k \in D(i,j) \right\},$$ while if $D(i,j)=\emptyset$, and then the $i$-th and $j$-th branches have the same multiplcity sequence, we set $k_E(i,j)=+\infty$. We have the following proposition. \[prop\] Consider a collection of multiplicity sequences $E$ and let $T \in \tau(E)$. Then $k_E(i,j)+1$ is the lowest level where the $i$-th and the $j$-th branches are prevented from being glued in $T$ (if $k_E(i,j)$ is infinite there are no limitations on the level where the branches have to split up). **Proof** The case $k_E(i,j) = +\infty$ is trivial, because we have the same sequence along two consecutive branches and therefore no discrepancies that force the two branches to split up at a certain level. Thus suppose $k_E(i,j) \neq +\infty$ and, by contradiction, that the $i$-th and the $j$-th branches are glued at level $k_E(i,j)+1$. From the definition of $k_E(i,j)$, there exists $\overline{k} \in D(i,j)$ such that $k_E(i,j)=\min( s_{i,\overline{k}},s_{j,\overline{k}})$. Without loss of generality suppose that $\min( s_{i,\overline{k}},s_{j,\overline{k}})=s_{i,\overline{k}} \neq s_{j,\overline{k}}$ (since $\overline{k} \in D(i,j) $). So in the tree we have the following nodes, $$(\ldots,m_{i,\overline{k}},\ldots,m_{j,\overline{k}},\ldots), \ldots, (\ldots,m_{i,k_E(i,j)},\ldots,m_{j,k_E(i,j)},\ldots),$$$$, (\ldots,m_{i,k_E(i,j)+1},\ldots,m_{j,k_E(i,j)+1},\ldots).$$ We have that $k_E(i,j)=s_{i,\overline{k}} $ so $$m_{i,\overline{k}}=\sum_{t=\overline{k}+1}^{k_E(i,j)} {m_{i,t}},$$ while $k_E(i,j)+1= s_{i,\overline{k}}+1 \leq s_{j,\overline{k}} $ so $$m_{j,\overline{k}}\geq \sum_{t=\overline{k}+1}^{k_E(i,j)+1} {m_{j,t}}.$$ These facts easily imply that the first node cannot be expressed as a sum of the nodes of a subtree rooted in it, so we have a contradiction. Two branches are forced to split up only when we have this kind of problem, so the minimality of $k_E(i,j)$ guarantees that they can be glued at level $k_E(i,j)$ (and obviously at lower levels). \[ex2\] Suppose that we have $$M(1)=[14,7,5,1,1] \textrm{ and } M(2)=[7,3,1,1,1].$$ So we have the vectors $S(1)$ and $S(2)$: $$S(1)=[5,5,8,5,6] \textrm{ and } S(2)=[6,5,4,5,6].$$ We have $D(1,2)=\left\{ 1,3 \right\}$, then $k(1,2)=\min \left\{ \min(5,6), \min(4,8) \right\}=\min\left\{5,4\right\}=4$. Then the branches have to be separated at the fifth level. Notice that the first tree in the previous picture fulfills the properties of the multiplcity trees of an Arf semigroup. The second one cannot be the multiplicity tree of an Arf semigroup because the third node $(5,1)$ cannot be expressed as a sum of nodes in a subtree rooted in it. Now we prove a general lemma that will be useful in the following. \[Lemma1\] Consider $v_1,v_2$ and $v_3$ in $\mathbb{N}^n$. If $i,j \in \left\{ 1,2,3 \right\}$ with $i \neq j$ we define: - $\textrm{MIN}(v_i,v_j)=+\infty$ if $v_i=v_j$; - $\textrm{MIN}(v_i,v_j)=\min\left\{ \min(v_i[k],v_j[k]), k\in\left\{1,\ldots,n\right\}: v_i[k]\neq v_j[k]\right\}.$ Then there exists a permutation $\delta \in S^3$ such that $$\textrm{MIN}(v_{\delta(1)},v_{\delta(2)})=\textrm{MIN}(v_{\delta(2)},v_{\delta(3)})\leq\textrm{MIN}(v_{\delta(1)},v_{\delta(3)}).$$ **Proof** Suppose by contradiction that the thesis is not true. Then, renaming the indices if necessary, we have $$\textrm{MIN}(v_1,v_2)<\textrm{MIN}(v_1,v_3)\leq\textrm{MIN}(v_2,v_3).$$ From the definition of $\textrm{MIN}(v_1,v_2)=l_{1,2}$ it follows that there exists a $k \in \left\{1,\ldots,n \right\}$ such that $v_1[k] \neq v_2[k]$ and $ \min(v_1[k],v_2[k])=l_{1,2}$. We have two cases: - If $v_1[k]=l_{1,2}$ $\Rightarrow v_2[k] > l_{1,2}$. Then we must have $v_3[k]=l_{1,2}$, in fact otherwise we would have $\textrm{MIN}(v_1,v_3)\leq l_{1,2}<\textrm{MIN}(v_1,v_3)$. But then$$l_{1,2}<\textrm{MIN}(v_2,v_3) \leq \min(v_2[k],v_3[k])=l_{1,2},$$ and we have a contradiction. - If $v_2[k]=l_{1,2}$ $\Rightarrow v_1[k] > l_{1,2}$. Then we must have $v_3[k]=l_{1,2}$, in fact otherwise we would have $\textrm{MIN}(v_2,v_3)\leq l_{1,2}<\textrm{MIN}(v_2,v_3)$. But then$$l_{1,2}<\textrm{MIN}(v_1,v_3) \leq \min(v_1[k],v_3[k])=l_{1,2},$$ and we have a contradiction. \[rem\] If we have three multiplicity sequences $M(1)$, $M(2)$ and $M(3)$ then, if $E=\left\{M(1),M(2),M(3) \right\}$ then there exist a permutation $\delta \in S^3$ such that $$k_E(\delta(1),\delta(2))= k_E(\delta(2),\delta(3)) \leq k_E(\delta(1),\delta(3)).$$ In fact the integers $k_E(i,j)$ are of the same type of the integers $\textrm{MIN}(v_i,v_j)$ of the previous lemma with $v_i=S(i)$. We give now a way to describe a tree of $\tau(E)$. If $T \in \tau(E)$, it can be represented by an upper triangular matrix $ n \times n$ $$M(T)_{E}=\left( \begin{matrix} 0 & p_{1,2} & p_{1,3} & \ldots & p_{1,n} \\ 0 & 0 & p_{2,3} & \ldots & p_{2,n} \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots& p_{n-1,n} \\ 0 & 0 & 0 & \ldots & 0 \\ \end{matrix}\right),$$ where $p_{i,j}$ is the highest level such that the $i$-th and the $j$-th branches are glued in $T$. \[remark\] If $M(T)_{E}$ is the matrix of a $T \in \tau(E)$, it is clear that everytime we consider three indices $i<j<k$ we must have: $$p_{i,j} \geq \min(p_{i,k},p_{j,k}), p_{j,k} \geq \min(p_{i,j},p_{i,k}) \textrm{ and } p_{i,k} \geq \min(p_{i,j},p_{j,k}),$$ when we are using the obvious fact that the relation of being glued has the transitive property. From the previous inequalities it follows that the set $\left\{ p_{i,j}, p_{j,k}, p_{i,k}\right\}=\left\{ x,x,y \right\}$, with $x\leq y$ (independently of the order). From Proposition \[prop\] we have that $p_{i,j} \in \left\{ 1,\ldots, k_E(i,j) \right\}$ for all $i,j=1,\ldots,n$ with $i<j$. In the following, with an abuse of notation, we will identify a tree with its representation. We call a tree $T$ of $\tau(E)$ untwisted if two nonconsecutive branches are glued at level $l$ if and only if all the consecutive branches between them are glued at a level greater or equal to $l$. We will call twisted a tree that it is not untwisted. From the definition it follows that the matrix of an untwisted tree $T \in \tau(E)$ is such that: $$p_{i,j}=\min\left\{ p_{i,i+1},p_{i+1,i+2},\ldots,p_{j-1,j}\right\} \textrm{ for all } i<j.$$ So an untwisted tree can be completely described by the second diagonal of its matrix. Thus in the following we will indicate an untwisted tree by a vector $T_E=(d_1,\ldots,d_{n-1})$ where $d_i=p_{i,i+1}$. It is easy to see that a twisted tree can be converted to an untwisted one by accordingly permuting its branches. So in the following we can focus, when it is possible, only on the properties of the untwisted trees, that are easier to study, obtaining the twisted one by permutation. Let us consider the following tree of $\tau(E)$ with $$E=\left\{ M(1)=[5,4,1,1],M(2)=[2,2,1,1],M(3)=[6,4,1,1]\right\}$$ This tree is twisted because the first and the third branches are glued at level two while the first and the second are not. If we consider the permutation $(2,3)$ on the branches we obtain the tree that is untwisted, even if it belongs to a different set $\tau(E')$ where $$E'=\left\{ M(1)=[5,4,1,1],M(2)=[6,4,1,1],M(3)=[2,2,1,1]\right\},$$ and can be represented by the vector $T_E'=(2,1)$. Denote by $S(T)$ the semigroup determined by the tree $T$. In [@BDF2 Lemma 5.1] it is shown that if $T^1$ and $T^2$ are untwisted trees of $\tau(E)$, then $S(T^1) \subseteq S(T^2) $ if and only if $T^2_{E}$ is coordinatewise less than or equal to $T^1_{E}$. The previous result can be easily extended to the twisted trees. Then, in the general case we have that $S(T^1) \subseteq S(T^2) $, where $S(T^1)$ and $S(T^2)$ belong to $\sigma(E)$, if and only if each entry of $M(T^2)_{E}$ is less than or equal to the corresponding entry of $M(T^1)_{E}$. If $k_E(i,j) \neq +\infty$ for all $i<j$, we can consider $T^{\textrm{MIN}}$ such that $$M(T^{\textrm{MIN}})_{E}=\left( \begin{matrix} 0 & k_E(1,2) & k_E(1,3) & \ldots & k_E(1,n) \\ 0 & 0 & k_E(2,3) & \ldots & k_E(2,n) \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots& k_E(n-1,n) \\ 0 & 0 & 0 & \ldots & 0 \\ \end{matrix}\right),$$ that is well defined for Remark \[rem\]. Then $S(T^{\textrm{MIN}})$ is the smallest Arf semigroup belonging to $\sigma(E)$. If in the collection $E$ there are two branches with the same multiplicity sequence then $ |\sigma(E)|=+ \infty$. \[ex3\] We can count the number of untwisted trees of $\tau(E)$ by using their representation. If we call $\tau^*(E)$ the set of all the untwisted trees of $\tau(E)$, these trees are completely determined by the elements in the second diagonal of their matrix, that are bounded by $k_E(j,j+1)$. Hence the number of untwisted trees is: $$|\tau^*(E)|= \prod_{j=1}^{n-1}{k_E(j,j+1)}.$$ Suppose that $E= \left\{ M(1),M(2),M(3) \right\}$, where $$M(1)=[5,4,1,1]\textrm{, } M(2)=[6,4,1,1]\textrm{, } M(3)=[2,2,1,1].$$ We have: $$S(1)=[3,6,4,5]\textrm{, } S(2)=[4,6,4,5]\textrm{, } S(3)=[2,4,4,5].$$ Then $D(1,2)=\left\{1\right\}, D(2,3)=\left\{1,2\right\}$ and $k_E(1,2)=\min(3,4)=3$ and $k_E(2,3)=\min\left\{ \min(2,4),\min(4,6)\right\}=2$. There are $k_E(1,2) \cdot k_E(2,3)=6$ trees in $\tau^*(E)$. They are: 0.3in at(current bounding box.south)[$T_{\textrm{MIN}}=T_E=(3,2)$]{}; 0.3in at(current bounding box.south)[$T_E=(3,1)$]{}; 0.3in at(current bounding box.south)[$T_E=(2,2)$]{}; 0.3in at(current bounding box.south)[$T_E=(2,1)$]{}; 0.3in at(current bounding box.south)[$T_E=(1,2)$]{}; 0.3in at(current bounding box.south)[$T_E=(1,1)$]{}; Because we are able to determine completely $\tau^*(E)$ for each $E$ collection of multiplicity sequences we have a way to determine $\tau(E)$. If $\delta \in S_n$ is a permutation of the symmetric group $S_n$ we can consider $\delta^{-1}(\tau^*(\delta(E))) \subseteq \tau(E).$ It is trivial to see that $$\bigcup_{\delta \in S_n}{\delta^{-1}(\tau^*(\delta(E)))}=\tau(E).$$ If we apply this strategy to find $\tau(E)$ with the $E$ of the previous example we find that in $\tau(E)$ there is only one twisted tree $T$ with $$M(T)_E=\left( \begin{matrix} 0 &1 &2 \\ 0& 0 &1 \\ 0 &0 &0 \\ \end{matrix}\right).$$ 0.3in When a set of vectors determines an Arf semigroup {#section3} ================================================== In this section we want to understand when a set $G \subseteq \mathbb{N}^n$ determines uniquely an Arf semigroup of $\mathbb{N}^n$. First of all we need to fix some notations. Given $ G\subseteq \mathbb{N}^n$ we denote by $S(G)$ the following set $$S(G)=\left\{ S: S \subseteq \mathbb{N}^n \textrm{ is an Arf semigroup and } G \subseteq S \right\}.$$ If the set $ S(G)$ has a minimum (with the partial order given by the inclusion), we will denote such a minimum by $\textrm{Arf}(G)$. Hence we have to understand when $\textrm{Arf}(G)$ is well defined and, in this case, how to find it. If $i \in \left\{ 1,\ldots,n \right\}$, and $S \in S(G)$ we denote by $S_i$ the projection on the $i$-th coordinate. We know that $S_i$ is an Arf numerical semigroup and it contains the set $G[i]=\left\{ g[i]: g \in G \right\}$ where with $g[i]$ we indicate the $i$-th coordinate of the vector $g$. We recall also that, if we have a set of integers $I$ such that $\gcd(I)=1$, it is possible to compute the smallest Arf semigroup containing $I$, that is the Arf closure of the numerical semigroup generated by the elements of $I$. This computation can be made by using the modified Jacobian algorithm of Du Val (cf. [@Duval]). We have the following theorem: \[T45\] Suppose that we have $G \subseteq \mathbb{N}^n$. Then $\textrm{Arf}(G)$ is well defined if and only if the following conditions hold: - $\gcd\left\{ g[i], g \in G \right\}=1$ for $i=1,\ldots,n;$ - For all $i,j \in \left\{1,\ldots,n \right\}$ such that $i < j$ there exists $g \in G$ such that $g[i] \neq g[j]$. **Proof** ($\Rightarrow $) Suppose that $\textrm{Arf}(G)$ is well defined and suppose by contradiction that the two conditions of the theorem are not simultaneuously fulfilled. We have two cases. - **Case 1**: The first condition is not fulfilled. Then there exists an $i $ such that $\gcd(G[i])=d \neq 1$. When we apply the Jacobian algorithm to the elements of $G[i]$ we will produce a sequence of the following type: $$[m_{i,1}, \ldots, m_{i,k},\ldots]$$ where there exists a $k$ such that $m_{i,j}=d$ for all $j \geq k$ (it happens because the Jacobian algorithm performs an Euclidean algorithm on $G[i]$). Denote by $\overline{k}$ the minimum $k$ such that the Arf semigroup associated to the sequence $$[m_{i,1}, \ldots, m_{i,\overline{k}}=d,1,1],$$ contains $G[i]$ (such minimum exists for the properties of the algorithm of Du Val). Then for all $z \geq \overline{k}$ we can consider the multiplicity sequence $$M(z)=[m_{i,1}, \ldots, m_{i,\overline{k}}=d,\ldots, m_{i,z}=d,1,1]$$ and if $S(z)$ is the Arf numerical semigroup associated to $M(z)$ then $ G[i] \subseteq S(z)$. Now it is trivial to show that $S(z_1) \subseteq S(z_2)$ if $z_1 \geq z_2$. Then we have an infinite decreasing chain of Arf semigroup containing the set $G[i]$. This means that the projection on the $i$-th branch can be smaller and smaller, therefore we cannot find a minimum in the set $S(G)$. Thus we have found a contradiction in this case. An example illustrating **Case 1** is the following. If we consider $G=\left\{ [2,3],[4,4] \right\} $, we have no information on the multiplicity sequence along the first branch and so we can obtain the following infinite decreasing chain of Arf semigroups containing $G$: $\supseteq$ $\supseteq$ $\supseteq$ $\supseteq \nobreak\dots$ - **Case 2**: The first condition is fulfilled. So in this case the second condition is not fulfilled. The fact that $\gcd\left\{ g[i], g \in G \right\}=1$ for $i=1,\ldots,n$ implies that we can compute the smallest Arf numerical semigroup $S(i)$ containing $G[i]$ for all $i=1, \ldots,n$. Therefore if we denote by $M_i$ the multiplicity sequence of $S(i)$ we clearly have that $\textrm{Arf}(G) \in \sigma(E), $ where $E=\left\{ M_i, i=1,\ldots,n \right\}$. Suppose that it is defined by the matrix $$M(T)_{E}=\left( \begin{matrix} 0 & p_{1,2} & p_{1,3} & \ldots & p_{1,n} \\ 0 & 0 & p_{2,3} & \ldots & p_{2,n} \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots& p_{n-1,n} \\ 0 & 0 & 0 & \ldots & 0 \\ \end{matrix}\right).$$ Now if we consider an element $ h \in G[i]$ we have that $h \in S(i)$ and therefore there exists an index $\textrm{pos}_E(i,h)$ such that $$h= \sum_{k=1}^{\textrm{pos}_E(i,h)} {m_{i,k}}.$$ If $g\in G$ we can define $\textrm{pos}_E(g)=[\textrm{pos}_E(1,g[1]),\ldots, \textrm{pos}_E(n,g[n])]$. Notice that, if we consider $i,j \in \left\{ 1,\ldots,n \right\}$, with $i<j$ and $g \in G$ such that $\textrm{pos}_E(i,g[i]) \neq \textrm{pos}_E(j,g[j])$, we can easily deduce that in a multiplicity tree of an Arf semigroup of $\sigma(E)$ containing $G$ the $i$-th and $j$-th branches cannot be glued at a level greater than $\min(\textrm{pos}_E(i,g[i]), \textrm{pos}_{E}(j,g[j]))$ . Then $p_{i,j}$ is at most $\min(\textrm{pos}_E(i,g[i]), \textrm{pos}_{E}(j,g[j]))$, and we also have to recall that $p_{i,j}$ is at most $k_E(i,j)$. So denote by $$U_E(G)=\left\{(i,j) \in \left\{ 1,\ldots, n \right\}^2 : i<j; \textrm{pos}_E(i,g[i])=\textrm{pos}_{E}(j,g[j]) \textrm{ for all } g \in G \right\}.$$ For each $ (i,j) \notin U_E(G)$ we define $$\textrm{MIN}_E(i,j,G)=\min\left(k_E(i,j),\min\left\{\min(\textrm{pos}_E(i,g[i]), \textrm{pos}_E(j,g[j])): g \in G, \right. \right.$$ $$\left. \left. \textrm{pos}_E(i,g[i]) \neq \textrm{pos}_{E}(j,g[j]) \right\} \right) .$$ Notice that we need $(i,j) \notin U_E(G)$ to have the previous integers well defined. So from the previous remark it follows that an Arf semigroup $S(T^1)$ of $\sigma(E)$ containing $G$ with $$M(T^1)_{E}=\left( \begin{matrix} 0 & a_{1,2} & a_{1,3} & \ldots & a_{1,n} \\ 0 & 0 & a_{2,3} & \ldots & a_{2,n} \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots& a_{n-1,n} \\ 0 & 0 & 0 & \ldots & 0 \\ \end{matrix}\right)$$ is such that $a_{i,j}$ is at most $k_E(i,j)$ for $(i,j) \in U_E(G)$ and $a_{i,j}$ is at most $\textrm{MIN}_E(i,j,G)$ for $(i,j) \notin U_E(G)$. So for the Arf closure we want to choose the biggest possible values, therefore we have: $$p_{i,j}=k_E(i,j) \textrm{ for } (i,j) \in U_E(G) \textrm{ and } p_{i,j}=\textrm{MIN}_E(i,j,G) \textrm{ for } (i,j) \notin U_E(G).$$ We need to prove that this integers are compatible with the transitive property of a matrix of an Arf semigroup tree. Therefore we consider a triad of integers $i<j<k$ and we want to show that $p_{i,j},p_{j,k}$ and $p_{i,k}$ are in a $ \left\{x,x,y\right\}$ configuration. We have the following cases: 1. $(i,j),(j,k),(k,i) \in U_E(G)$. Then $p_{i,j}=k_E(i,j),$$p_{i,k}=k_E(i,k)$ and $p_{j,k}=k_E(j,k)$ and for the Remark \[rem\] they satisfy our condition; 2. $(i,j),(j,k),(k,i) \notin U_E(G)$. We consider the vectors $$v_l=[\textrm{pos}_E(l,g_1[l]),\ldots,\textrm{pos}_E(l,g_m[l])],$$ where $l\in \left\{i,j,k\right\}$ and $G=\left\{g_1,\ldots,g_m\right\}$. Then, using the notations of Lemma \[Lemma1\], we have that $$p_{i,j}=\min(k_E(i,j),\textrm{MIN}(v_i,v_j)),p_{i,k}=\min(k_E(i,k),\textrm{MIN}(v_i,v_k)) \textrm{ and }$$$$p_{j,k}=\min(k_E(j,k),\textrm{MIN}(v_j,v_k)).$$ Then suppose by contradiction that they are not compatible. Without loss of generality we can assume that $$p_{i,j}<p_{i,k} \leq p_{j,k}.$$ We have two cases - $p_{i,j}=k_E(i,j).$ Then we would have $$k_E(i,j)=p_{i,j}<p_{j,k} \leq k_E(j,k) \textrm{ and } k_E(i,j)=p_{i,j}<p_{i,k} \leq k_E(i,k),$$ and this is absurd for the Remark \[rem\]; - $p_{i,j}=\textrm{MIN}(v_i,v_j).$ Then we would have $$\textrm{MIN}(v_i,v_j)=p_{i,j}<p_{j,k} \leq\textrm{MIN}(v_j,v_k)\textrm{ and } \textrm{MIN}(v_i,v_j)=p_{i,j}<p_{i,k} \leq\textrm{MIN}(v_i,v_k),$$ and this is absurd against Lemma \[Lemma1\] applied to the vectors $v_i,v_j$ and $v_k$. 3. $(i,j) \in U_E(G)$ and $(j,k),(k,i) \notin U_E(G)$ (and the similar configurations). In this case we have that $v_i=v_j$. Then $$p_{i,j}=k_E(i,j), p_{i,k}=\min(k_E(i,k),x), \textrm{ and } p_{j,k}=\min(k_E(j,k),x),$$ where $x=\textrm{MIN}(v_i,v_k)= \textrm{MIN}(v_j,v_k)$. We have two cases: - $k_E(i,j)=k_E(j,k) \leq k_E(i,k) $ (or equivalently $k_E(i,j)=k_E(i,k) \leq k_E(j,k) $). If $x<k_E(j,k)\leq k_E(i,k)$ then we have $ p_{j,k}=p_{i,k}=x<k_E(i,j)$ and it is fine. If $x \geq k_E(j,k)$ then $p_{j,k}=k_E(j,k)=p_{i,j} \leq p_{i,k}$ that is compatible too. - $k_E(i,k)=k(j,k) < k_E(i,j)$. In this case we have $p_{i,k}=p_{j,k}<k_E(i,j)=p_{i,j}$ and it is fine. So we actually have a well defined tree. Anyway, because the second condition is not fulfilled, then there exists a pair $ (i,j)\in \left\{1,\ldots,n \right\}^2$ such that for all $g \in G$ we have $g[i] = g[j]$. So $(i,j) \in U_E(G)$, and, since in this case the two sequences are the same, we obtain $p_{i,j}=k_E(i,j)=+\infty$. Thus we have found a contradiction because $\textrm{Arf}(G)$ is not well defined. An example illustrating **Case 2** is the following. If we consider $G=\left\{ [3,3,2],[2,2,3] \right\}$, we will have the same multiplicity sequences in the first two branches, with no clues about the splitting point so we can obtain the following infinite decreasing chain in $S(G)$: 0.3in $\supseteq$ $\supseteq$ $\supseteq \nobreak\dots$ ($\Leftarrow$) The previous proof gives us a way to compute $\textrm{Arf}(G)$. We have to compute, using the modified Jacobian algorithm of Du Val, the Arf closure of each $G[i]$, finding the collection $E$ (the first condition guarantees that it is possible to do that). Then we can find the matrix describing the semigroup using the set $U_E(G)$ and the integers $\textrm{MIN}_E(i,j,G)$ with the procedure present in the first part (we cannot have $p_{i,j}=+\infty$ for the second condition). \[ex6\] Suppose that we have $ G=\left\{\textrm{G}(1)=[5,6,5], \textrm{G}(2)=[9,11,4], \textrm{G}(3)=[9,10,2] \right\}, $ that satisfies the conditions of the theorem. Then we have to apply the modified Jacobian algorithm to the sets $$G[1]=\left\{5,9\right\}, G[2]=\left\{6,10,11 \right\} \textrm{ and } G[3]=\left\{ 2,4,5\right\}.$$ We will find the following multiplicity sequences: $$M_1=[5,4,1,1], M_2=[6,4,1,1] \textrm{ and } M_3=[2,2,1,1].$$ We have $k_E(1,2)=3$, $k_E(2,3)=2$ and $k_E(1,3)=2$. So we have $ \textrm{pos}_E(\textrm{G}(1))=[1,1,3]$, $ \textrm{pos}_E(\textrm{G}(2))=[2,3,2]$ and $ \textrm{pos}_E(\textrm{G}(3))=[2,2,1].$ In this case $U_E(G)=\emptyset$. We have $ \textrm{MIN}_E(1,2,G)=\min(2,k_E(1,2))=2$, $ \textrm{MIN}_E(2,3,G)=\min(1,k_E(2,3))=1$ and $ \textrm{MIN}_E(1,3,G)=\min(1,k_E(1,3))=1$. So the Arf closure is described by the matrix $$M(T)_E=\left( \begin{matrix} 0 &2 &1 \\ 0& 0 &1 \\ 0 &0 &0 \\ \end{matrix}\right).$$ with $$E=\left\{M_1=[5,4,1,1], M_2=[6,4,1,1] \textrm{ and } M_3=[2,2,1,1] \right\}.$$ Notice that in this case we find that the Arf closure is an untwisted tree of $\tau(E)$ represented by the vector $T_E=(2,1)$. Suppose that we have $ G=\left\{\textrm{G}(1)=[8,6,10], \textrm{G}(2)=[5,10,5], \textrm{G}(3)=[10,13,8] \right\}, $ that satisfies the conditions of the theorem. Then we have to apply the modified Jacobian algorithm to the sets $$G[1]=\left\{5,8,10\right\}, G[2]=\left\{6,10,13 \right\} \textrm{ and } G[3]=\left\{ 5,8, 10\right\}.$$ We will find the following multiplicity sequences: $$M_1=[5,3,2,1,1], M_2=[6,4,2,1,1] \textrm{ and } M_3=[5,3,2,1,1].$$ We have $k_E(1,2)=4$, $k_E(2,3)=4$ and $k_E(1,3)=+ \infty $. So we have $ \textrm{pos}_E(\textrm{G}(1))=[2,1,3]$, $ \textrm{pos}_E(\textrm{G}(2))=[1,2,1]$ and $ \textrm{pos}_E(\textrm{G}(3))=[3,4,2].$ In this case $U_E(G)=\emptyset$. We have $ \textrm{MIN}_E(1,2,G)=\min(1,k_E(1,2))=1$, $ \textrm{MIN}_E(2,3,G)=\min(1,k_E(2,3))=1$ and $ \textrm{MIN}_E(1,3,G)=\min(2,k_E(1,3))=2$. So the Arf closure is described by the matrix $$M(T)_E=\left( \begin{matrix} 0 &1 &2\\ 0& 0 &1 \\ 0 &0 &0 \\ \end{matrix}\right).$$ with $$E=\left\{M_1=[5,3,2,1,1], M_2=[6,4,2,1,1] \textrm{ and } M_3=[5,3,2,1,1] \right\}.$$ Notice that in this case we find that the Arf closure is a twisted tree. 0.3in Arf closure of a good semigroup of $\mathbb{N}^n$ {#section4} ================================================= Denote by $S$ a good semigroup of $\mathbb{N}^n$. In this section we describe a way to find the smallest Arf semigroup of $\mathbb{N}^n$ containing $S$, that is the Arf closure of $S$ (the existence of the Arf closure is proved in [@DPMT]). We denote this semigroup by $\textrm{Arf}(S)$. If $S$ is a good semigroup of $\mathbb{N}^n$, we denote by $S_i$ the projection on the $i$-th coordinate. The properties of the good semigroups guarantee that $S_i$ is a numerical semigroup. Thus it is clear that an Arf semigroup $T$ containing $S$ is such that $\textrm{Arf}(S_i) \subseteq T_i$ for all $i=1,\ldots,n$, where $\textrm{Arf}(S_i)$ is the Arf closure of the numerical semigroup $S_i$ (we can compute it using the algorithm of Du Val on a minimal set of generators of $S_i$). Therefore, in order to have the smallest Arf semigroup containing $S$, we must have $\textrm{Arf}(S) \in \sigma(E)$ where $E=\left\{ M_1,\ldots, M_n \right\}$ and $M_i$ is the multiplicity sequence associated to the Arf numerical semigroup $\textrm{Arf}(S_i)$.\ Now we need to find the matrix $$M(T)_{E}=\left( \begin{matrix} 0 & p_{1,2} & p_{1,3} & \ldots & p_{1,n} \\ 0 & 0 & p_{2,3} & \ldots & p_{2,n} \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots& p_{n-1,n} \\ 0 & 0 & 0 & \ldots & 0 \\ \end{matrix}\right).$$that describes the tree of $\textrm{Arf}(S)$. We recall that from the properties of good semigroups, it follows that there exists a minimal vector $\delta\in \mathbb{N}^n$ such that $\delta+\mathbb{N}^n \subseteq S$ (we will call this vector the conductor of $S$). Suppose that $\delta=(c[1],\ldots,c[n])$. We denote by $$\textrm{Small}(S)=\left\{ \textbf{s}: \textbf{0}<\textbf{s}\leq \delta \right\} \cap S,$$ the finite set of the small elements of $S$ (the elements of $S$ that are coordinatewise smaller than the conductor). In [@DPMT] it is shown that $\textrm{Small}(S)$ describes completely the semigroup $S$ (in this paper we are not including the zero vector in $\textrm{Small}(S)$ to enlight the notations of the following procedures). We can recover the collection $E$ from $\textrm{Small}(S)$. In fact, the multiplicity sequence $M_i$ can be determined applying the Du Val algorithm to the set $ \left\{ s[i], s\in \textrm{Small}(S)\right\} \cup \left\{ c[i]+1 \right\} \subseteq S_i$. In order to apply the Du Val algorithm we may have to add $c[i]+1$ because we can have $\gcd( \left\{ s[i], s\in \textrm{Small}(S)\right\}) \neq 1$. Because $c[i]$ and $c[i]+1$ belong to $S_i$, we know that $\textrm{Arf}(S_i)$ has conductor smaller than $c[i]$ and this implies that we only have to consider the elements that are smaller than $c[i]+1$. Now, we notice that $p_{i,j} \leq \min(\textrm{pos}_E(i,c[i]),\textrm{pos}_{E}(j,c[j]))$ for all $i,j\in \left\{1,\ldots,n\right\}$, with $i<j$, where we are using the notations of the previous section. In fact, if $\textrm{pos}_E(i,c[i]) \neq \textrm{pos}_E(j,c[j])$, we have already noticed that in an Arf semigroup containing $\delta$ the $i$-th and the $j$-th branches cannot be glued at a level greater than $\min(\textrm{pos}_E(i,c[i]),\textrm{pos}_E(j,c[j]))$, then $p_{i,j} \leq \min(\textrm{pos}_E(i,c[i]),\textrm{pos}_{E}(j,c[j]))$ . If $\textrm{pos}_E(i,c[i]) =\textrm{pos}_{E}(j,c[j])$ then we have $\delta_1=(c[1],\ldots,c[i]+1,c[i+1],\ldots,c[n]) \in S$, and $\textrm{pos}_E(i,c[i]+1)=\textrm{pos}_E(i,c[i])+1>\textrm{pos}_{E}(j,c[j])$. Therefore in an Arf semigroup containing $\delta_1$ the $i$-th and the $j$-th branches cannot be glued at a level greater than $$\min(\textrm{pos}_E(i,c[i])+1,\textrm{pos}_{E}(j,c[j]))=\textrm{pos}_{E}(j,c[j])=$$ $$=\min(\textrm{pos}_E(i,c[i]),\textrm{pos}_{E}(j,c[j]),$$ hence we have again $p_{i,j} \leq \min(\textrm{pos}_E(i,c[i]),\textrm{pos}_{E}(j,c[j]))$.\ Furthermore, we always have to take in account that $p_{i,j} \leq k_E(i,j)$ for all $i,j \in \left\{1,\ldots,n\right\}$.\ Now let us consider the following subset of $\left\{1,\ldots,n\right\}^2$, $$U_E(\textrm{Small}(S))=\left\{ (i,j): \textrm{pos}_E(i,s[i]) =\textrm{pos}_{E}(j,s[j]) \textrm{ for all } s \in \textrm{Small}(S) \right\}.$$ If $(i,j) \in \left\{1,\ldots,n\right\}^2 \setminus U_E(\textrm{Small}(S))$, and $i<j$ we can consider the following integers $$\textrm{MIN}_E(i,j,\textrm{Small}(S))=\min\left(k_E(i,j),\min\left\{ \min(\textrm{pos}_{E}(i,s[i]),\textrm{pos}_{E}(j,s[j])): s \in \textrm{Small}(S), \right. \right.$$ $$\left. \left. \textrm{pos}_E(i,s[i]) \neq \textrm{pos}_{E}(j,s[j]) \right\}\right).$$ Notice that we need only to consider the vectors of $\textrm{Small}(S)$ because if $ s\in S $ then $ s_1=\min(s,\delta) \in \textrm{Small}(S)$ and we clearly have $$\min(\textrm{pos}_{E}(i,s[i]),\textrm{pos}_{E}(j,s[j])) \geq \min(\textrm{pos}_E(i,s_1[i]),\textrm{pos}_{E}(j,s_1[j])),$$ therefore $s_1 \in \textrm{Small}(S)$ gives us more precise information on the ramification level than $s$ (it can happen that $\textrm{pos}_E(i,s_1[i])=\textrm{pos}_{E}(j,s_1[j])$ and $\textrm{pos}_{E}(i,s[i]) \neq \textrm{pos}_{E}(j,s[j]) $, but only when $ \min(\textrm{pos}_{E}(i,s[i]),\textrm{pos}_{E}(j,s[j])) \geq \min(\textrm{pos}_E(i,c[i]),\textrm{pos}_{E}(j,c[j]))$). Thus, if $T^1$ is an Arf semigroup of $\sigma(E)$ containing $S$, represented by $$M(T^1)_{E}=\left( \begin{matrix} 0 & a_{1,2} & a_{1,3} & \ldots & a_{1,n} \\ 0 & 0 & a_{2,3} & \ldots & a_{2,n} \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots& a_{n-1,n} \\ 0 & 0 & 0 & \ldots & 0 \\ \end{matrix}\right)$$ we have: - $a_{i,j} \leq \textrm{MIN}_E(i,j,\textrm{Small}(S))$ for $(i,j) \in \left\{1,\ldots,n\right\}^2 \setminus U_E(\textrm{Small}(S));$ - $a_{i,j} \leq \min(k_E(i,j),\textrm{pos}_{E}(i,c[i]))$, for $i \in U_E(\textrm{Small}(S))$ (we have $\textrm{pos}_E(i,c[i]) =\textrm{pos}_{E}(j,c[j])$). Then we can finally deduce that the $p_{i,j}$ that defines the matrix of $\textrm{Arf}(S)$ are such that - $p_{i,j} =\textrm{MIN}_E(i,j,\textrm{Small}(S))$, for $(i,j) \in \left\{1,\ldots,n\right\}^2 \setminus U_E(\textrm{Small}(S));$ - $p_{i,j} = \min(k_E(i,j),\textrm{pos}_{E}(i,c[i]))$, for $i \in U_E(\textrm{Small}(S))$ (we have $\textrm{pos}_E(i,c[i]) =\textrm{pos}_{E}(j,c[j])$), and it is easy to see that the $p_{i,j}$ fulfill the condition of Remark \[remark\]. In other words we showed that $\textrm{Arf}(S)$ can be computed by computing $\textrm{Arf}(G)$ where: $$G=\textrm{Small}(S) \bigcup \left\{(c[1]+1,\ldots,c[i],c[i+1],\ldots,c[n]),\ldots,(c[1],\ldots,c[i]+1,c[i+1],\ldots, c[n]),\ldots,\right.$$ $$\left.(c[1],\ldots,c[i],c[i+1],\ldots,c[n]+1) \right\}.$$ Let us consider the good semigroup $S$ with the following set of small elements, $$\textrm{Small}(S)= \left\{ [ 5, 6, 5 ], [ 5, 10, 5 ], [ 5, 12, 5 ], [ 8, 6, 8 ], [ 8, 10, 8 ], [ 8, 12, 8 ], [ 8, 6, 10 ], [ 8, 10, 10 ], \right.$$ $$\left. [ 8, 12, 10 ], [ 10, 6, 8 ], [ 10, 10, 8 ], [ 10, 12, 8 ], [ 10, 6, 10 ], [ 10, 10, 10 ], [ 10, 12, 10 ] \right\}.$$ The conductor is $\delta=[10,12,10]$. First of all we need to recover from $\textrm{Small}(S)$ the collection of multiplicity sequences $E$. We have to apply the Du Val algorithm to the following sets: $$\left\{ 5, 8, 10, 11\right\}, \left\{ 6, 10, 12, 13\right\} \textrm{ and } \left\{ 5, 8, 10, 11\right\},$$ therefore we find that $E=\left\{ [5,3,2,1,1], [6,4,2,1,1], [5,3,2,1,1] \right\}.$ We have $$\textrm{pos}(\textrm{Small}(S))=\left\{\textrm{pos}_E(s): s \in \textrm{Small}(S) \right\}=\left\{ [ 1, 1, 1 ], [ 1,2, 1 ], [ 1, 3, 1 ], [ 2, 1, 2 ], [ 2, 2, 2 ], \right.$$ $$\left.[ 2, 3, 2 ], [ 2, 1,3 ], [ 2, 2, 3 ], [ 2, 3, 3 ], [ 3, 1, 2 ], [ 3, 2, 2 ], [ 3, 3, 2 ], [3, 1,3 ], [3, 2,3 ], [ 3,3, 3 ] \right\}.$$ It is easy to check that $U_E(\textrm{Small}(S))=\emptyset.$ Thus we have - $p_{1,2}=\textrm{MIN}_E(1,2,\textrm{Small}(S))=\min(k_E(1,2)=4,1)=1,$ because we have the element $[1,2,1] \in \textrm{pos}(\textrm{Small}(S))$ corresponding to $s=[5,10,5] \in \textrm{Small}(S)$ such that $1=\textrm{pos}_E(1,s[1]) \neq \textrm{pos}_{E}(2,s[2])=2 $ and $\min(\textrm{pos}_{E}(1,s[1]),\textrm{pos}_{E}(2,s[2]))=1 $ . - $p_{2,3}=\textrm{MIN}_E(2,3,\textrm{Small}(S))=\min(k_E(2,3)=4,1)=1,$ because we have the element $[1,2,1] \in \textrm{pos}(\textrm{Small}(S))$ corresponding to $s=[5,10,5] \in \textrm{Small}(S)$ such that $2=\textrm{pos}_E(2,s[2]) \neq \textrm{pos}_{E}(3,s[3])=1 $ and $\min(\textrm{pos}_{E}(2,s[2]),\textrm{pos}_{E}(3,s[3]))=1 $ . - $p_{1,3}=\textrm{MIN}_E(1,3,\textrm{Small}(S))=\min(k_E(1,3)=+\infty,2)=2,$ because we have the element $[2,2,3] \in \textrm{pos}(\textrm{Small}(S))$ corresponding to $s=[8,10,10] \in \textrm{Small}(S)$ such that $2=\textrm{pos}_E(1,s[1]) \neq \textrm{pos}_{E}(3,s[3])=3 $ and $\min(\textrm{pos}_{E}(1,s[1]),\textrm{pos}_{E}(3,s[3]))=2 $, and we cannot find a smaller discrepancy. So the Arf closure of $S$ is described by the matrix $$M(T)_E=\left( \begin{matrix} 0 &1 &2\\ 0& 0 &1 \\ 0 &0 &0 \\ \end{matrix}\right).$$ with $$E=\left\{M_1=[5,3,2,1,1], M_2=[6,4,2,1,1] \textrm{ and } M_3=[5,3,2,1,1] \right\}.$$ The following procedure, implemented in GAP, has as argument the set of small elements of a good semigroup and give as a result the Arf Closure of the given good semigroup. The Arf clousure is described by a list $[E,M(T)_E]$. gap> S:=[[5,6,5],[5,10,5],[5,12,5],[8,6,8],[8,10,8],[8,12,8], [8,6,10],[8,10,10],[8,12,10],[10,6,8],[10,10,8],[10,12,8], [10,6,10],[10,10,10], [10,12,10]]; [ [ 5, 6, 5 ], [ 5, 10, 5 ], [ 5, 12, 5 ], [ 8, 6, 8 ], [ 8, 10, 8 ], [ 8, 12, 8 ], [ 8, 6, 10 ], [ 8, 10, 10 ], [ 8, 12, 10 ], [ 10, 6, 8 ], [ 10, 10, 8 ], [ 10, 12, 8 ], [ 10, 6, 10 ], [ 10, 10, 10 ], [ 10, 12, 10 ] ] gap> ArfClosureOfGoodsemigroup(S); [ [ [ 5, 3, 2 ], [ 6, 4, 2 ], [ 5, 3, 2 ] ], [ [ 0, 1, 2 ], [ 0, 0, 1 ], [ 0, 0, 0 ] ] ] Bounds on the minimal number of vectors determining a given Arf semigroup {#section5} ========================================================================== Suppose that $E$ is a collection of $n$ multiplicity sequences. Let $T\in \tau(E)$ and given a semigroup $S(T)$ in $\sigma(E)$, we want to study the properties that a set of vectors $G(T) \subseteq \mathbb{N}^n$ has to satisfy to have $S(T)=\textrm{Arf}(G(T))$, with the notations given in the previous section. We call such a $G(T)$ a set of generators for $S(T)$. In particular we want to find bounds on the cardinality of a minimal set of generators for a $S(T) \in \sigma(E)$. Since we want to find a $G(T)$ such that $\textrm{Arf}(G(T))$ is well defined, it has to satisfy the following properties: - For all $i=1,\ldots,n$ $$\gcd(v[i]; v \in G(T))=1,$$ where $v[i]$ is the $i-$th coordinate of the vector $v \in G(T)$. - For all $i,j\in \left\{1,\ldots,n\right\}$, with $i<j$ there exists $v \in G(T)$ such that $v[i] \neq v[j]$. We recall that, given a Arf numerical semigroup $S$, there is a uniquely determined smallest semigroup $N$ such that the Arf closure of $N$ is $S$. The minimal system of generators for such $N$ is called the Arf system of generators for $S$, or the set of characters of $S$. Now we want that $\textrm{Arf}(G(T))$ is an element of $\sigma(E)$. This implies that, when we apply the algorithm of Du Val to $G(T)[i]$, we have to find the $i$-th multiplicity sequence of $E$. This means that, if we call $S_i$ the Arf numerical semigroup corresponding to the projection on the $i$-th coordinate, we must have $G(T)[i] \subseteq S_i$ and furthermore $G(T)[i]$ has to contain a minimal system of generators for $S_i$. In fact, in [@Arf] it is proved that if we have $G=\left\{ g_1,\ldots,g_m \right\} \subseteq \mathbb{N}$ with $\gcd(G)=1$ then $G$ must contain the set of characters of the Arf closure of the semigroup $N=\langle G \rangle$.\ So we need to recall a way to compute the characters of an Arf numerical semigroup. We suppose that $E=\left\{ M(1),\ldots, M(n) \right\} $. Given $$M(i)=[m_{i,1},\ldots, m_{i,M}],$$ we define the restricion number $r(m_{i,j})$ of $m_{i,j}$ as the number of sums $\displaystyle m_{i,q}=\sum_{h=1}^k {m_{i,q+h}}$ where $m_{i,j}$ appears as a summand. With this notation we have that the characters of the multiplicity sequence $M(i)$ are the elements of the set (cf. [@BDF2 Lemma 3.1]) $$\textrm{Char}_E(i)=\left\{\sum_{k=1}^j{m_{i,k}} :r(m_{i,j})<r(m_{i,j+1}) \right\}.$$ Notice that, from our assumptions on $M$, it follows that the last two entries in each $M(i)$ are $1$, and it is easy to see how it guarantees that we cannot find characters in correspondence of indices greater than $M$. We define $\textrm{PChar}_E(i)=\left\{ j: r(m_{i,j})<r(m_{i,j+1}) \right\}$. Given the collection $E$, we denote by $$V_E(j_1,j_2,\ldots,j_n)=\left[ \sum_{k=1}^{j_1}{m_{1,k}}, \sum_{k=1}^{j_2}{m_{2,k}},\ldots, \sum_{k=1}^{j_n}{m_{n,k}}\right].$$ Now, the elements of $G(T)$ must be of the type $V_E(j_1,j_2,\ldots,j_n)$ (in fact we noticed that when we project on the $k$-th coordinate we must find an element of the corresponding numerical semigroup that has the previous representation for some $j_k$). From the previous remarks and notations we have the following property: $$G(T)=\left\{ \textrm{Gen}(1)=V_E(j_{1,1},\ldots,j_{1,n}), \ldots, \textrm{Gen}(t)=V_E(j_{t,1},\ldots,j_{t,n})\right\}$$ are generators of a semigroup of $\sigma(E)$ if and only if $$\textrm{PChar}_E(i) \subseteq \left\{ j_{1,i},\ldots,j_{t,i} \right\} \textrm{ for all }i=1,\ldots,n.$$ In particular we have found a lower bound for the cardinality of a minimal set of generators for a $S(T) \in \sigma(E)$. In fact $G(T)$ has at least $C_E=\max\left\{ |\textrm{PChar}_E(i)|, i=1,\ldots,n\right\}$ elements. Now we want to determine the generators of a given semigroup $S(T) \in \sigma({E})$. We have the following theorem. \[T5\] Let $S(T) \in \sigma(E)$ with $$M(T)_{E}=\left( \begin{matrix} 0 & p_{1,2} & p_{1,3} & \ldots & p_{1,n} \\ 0 & 0 & p_{2,3} & \ldots & p_{2,n} \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots& p_{n-1,n} \\ 0 & 0 & 0 & \ldots & 0 \\ \end{matrix}\right).$$ Denote by $P=\left\{ (q,r): p_{q,r}=k_E(q,r) \right\}$. Then $G(T)=\left\{\textrm{Gen}(1),\ldots,\textrm{Gen}(t) \right\} \subseteq \mathbb{N}^n$ is such that $\textrm{Arf}(G(T))=S(T)$ if and only if the following conditons hold - $\textrm{Gen}(1)=V_E(j_{1,1},\ldots,j_{1,n}), \ldots, \textrm{Gen}(t)=V_E(j_{t,1},\ldots,j_{t,n})$ for some values of the indices $j_{1,1},\ldots,j_{t,n}$; - $\textrm{PChar}_E(i) \subseteq \left\{ j_{1,i},\ldots,j_{t,i} \right\}$ for all $i=1,\ldots,n$. Furthermore, if we consider the following integer $$\textrm{MIN}_{G(T)}(q,r)=\min\left( k_E(q,r),\min\left\{ \min(j_{p,q},j_{p,r}) : j_{p,q} \neq j_{p,r}, p=1,\ldots,t \right\} \right),$$ for the $(q,r)\in \left\{1,\ldots,n \right\}^2$ with $q<r$ and where it is well defined, we have: - for $(q,r) \in P$ we have either $ j_{p,q}=j_{p,r} $ for all $p=1,\ldots,t$, or $ \textrm{MIN}_{G(T)}(q,r)$ is well defined and it equals $k_E(q,r)$; - $ \textrm{MIN}_{G(T)}(q,r)$ is well defined and it equals $p_{q,r}$, for all $(q,r) \notin P$. **Proof** ($\Leftarrow$) Suppose that we have $G(T)=\left\{\textrm{Gen}(1),\ldots,\textrm{Gen}(t) \right\} \subseteq \mathbb{N}^n$ satisfying the conditions of the theorem. The first two conditions ensure that if we apply the algorithm defined in the previous section on $G(T)$ it will produce an element of $\sigma(E)$. Now it is easy, using the notations of Theorem \[T45\], to show that $j_{p,q}=\textrm{pos}_E(q,\textrm{Gen}(p)[q])$ and from this it follows that, when $\textrm{MIN}_{G(T)}(q,r)$ is well defined, it is equal to $\textrm{MIN}_E(q,r,G(T))$. Furthermore we have $U_E(G(T)) \subseteq P$. In fact we have $$U_E(G(T))=\left\{(q,r)\in \left\{1,\ldots,n\right\}^2: \textrm{pos}_E(q,\textrm{Gen}(p)[q])=\textrm{pos}_E(r,\textrm{Gen}(p)[r]) \right.$$ $$\left. \textrm{ for all } p=1,\ldots,t \right\}=\left\{(q,r)\in \left\{1,\ldots,n\right\}^2: j_{p,q}=j_{p,r} \textrm{ for all } p=1,\ldots,t \right\},$$ therefore if $(q,r) \in U_E(G(T))$ then $(q,r) \in P$, since $G(T)$ satisfy the fourth condition in the statement of the theorem (we cannot have $(q,r) \notin P$ because in this case $\textrm{MIN}_{G(T)}(q,r)=\textrm{MIN}_E(q,r,G(T))$ has to be well defined). So it will follows that, if $S(T^1)$ is $\textrm{Arf}(G(T))$ then $$M(T^1)_{E}=\left( \begin{matrix} 0 & a_{1,2} & a_{1,3} & \ldots & a_{1,n} \\ 0 & 0 & a_{2,3} & \ldots & a_{2,n} \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots& a_{n-1,n} \\ 0 & 0 & 0 & \ldots & 0 \\ \end{matrix}\right)$$ where - $a_{i,j}=\textrm{MIN}_E(i,j,G(T))$ if $(i,j) \notin U_E(G(T))$; - $a_{i,j}=k_E(i,j)$ if $(i,j) \in U_E(G(T))$. Therefore if $(i,j) \notin P$ then $(i,j) \notin U_E(G(T))$ and we have $a_{i,j}=\min(\textrm{MIN}_E(i,j,G(T)))=\textrm{MIN}_{G(T)}(i,j)=p_{i,j}$. If $(i,j) \in P$ then - if $(i,j) \in U_E(G(T))$ then $a_{i,j}=k_E(i,j)$; - if $(i,j) \notin U_E(G(T))$ then $a_{i,j}=\textrm{MIN}_E(i,j,G(T))=\textrm{MIN}_{G(T)}(i,j)=k_E(i,j)$, for the properties of the set $G(T)$ ($(i,j)\in P$). So we showed that $\textrm{Arf}(G(T))=S(T)$. Thus the proof of this implication is complete. ($\Rightarrow$) It follows immediately by contradiction, using the first part of the proof. \[ex7\] Suppose that we have $E= \left\{ M(1),M(2),M(3) \right\}$, where $$M(1)=[5,4,1,1], M(2)=[6,4,1,1], M(3)=[2,2,1,1].$$ We have, $k_E(1,2)=3, k_E(2,3)=2$ and $k_E(1,3)=2$. We can define: $$R(i)=[r(m_{i,1}),r(m_{i,2}),\ldots,r(m_{i,N}) ].$$ Notice that $r(m_{i,1})=0, r(m_{i,2})=1$. The values of $\textrm{PChar}(i)$ are the indices where this sequence has an increase (it can be easily shown that when the sequence has an increase we have $r(m_{i,j})=r(m_{i,j+1})-1$ cf. [@BDF2 Lemma 3.2]). Furthermore $R(1)=[0,1,2,2,2,2]$, $R(2)=[0,1,2,3,2,2]$ and $R(3)=[0,1,1,2]$. So $\textrm{PChar}_E(1)=\left\{1,2 \right\}, \textrm{PChar}_E(2)=\left\{1,2,3\right\}$ and $\textrm{PChar}_E(3)=\left\{1,3\right\}$. Suppose that we want to find generators for the untwisted tree $T^1$ such that $ T^1_{E}=(2,1) $. We need at least three vectors because $C_E=3$. Consider the vectors $\textrm{Gen}(1)=V_E(1,1,3), \textrm{Gen}(2)=V_E(2,3,2)$ and $\textrm{Gen}(3)=V_E(2,2,1)$. The second condition, that guarantees that we have a tree belonging to $\tau(E)$, is satisfied. Furthermore $\textrm{MIN}_{G(T)}(1,2)=\min(k_E(1,2),2)=2$ , $\textrm{MIN}_{G(T)}(2,3)=\min(k_E(2,3),1)=1$, and $\textrm{MIN}_{G(T)}(1,3)=\min(k_E(1,3),1)=1=\min(d_1,d_2)$ where $G(T)=\left\{ \textrm{Gen}(1),\textrm{Gen}(2),\textrm{Gen}(3)\right\}$. Thus we have $\textrm{Arf}(G(T))=S(T^1)$. They are the vectors $ \textrm{Gen}(1)=[5,6,5], \textrm{Gen}(2)=[9,11,4], \textrm{Gen}(3)=[9,10,2] $ which appeared in the Example \[ex6\]. Now, we want to find an upper bound for the cardinality of a minimal set $G(T)$ such that $\textrm{Arf}(G(T)) \in \sigma(E)$. \[remark24\] Suppose that $T^1$ is a twisted tree of $\tau(E)$, where $E$ is a collection of $n$ multiplicity sequences. Then there exists a permutation $\delta\in S^n$ such that $\delta(T^1)$ is an untwisted tree of $\tau(\delta(E))$. Then if $G$ is a set of generators for $\delta(T^1)$ then it is clear that we have $$\delta^{-1}(G)=\left\{ \delta^{-1}(g); g \in G \right\},$$ is a set of generators for the twisted tree $T^1$. From the previous remark it follows that we can focus only on the untwisted trees of $\tau(E)$ to find an upper bound for the cardinality of $G(T)$. Our problem is clearly linked to the following question. Let us consider a vector $\textbf{d}=[d_1,\ldots,d_n] \in \mathbb{N}^n$. For all the $G \subseteq \mathbb{N}^{n+1}$ we denote by $\textrm{MIN}(G,i,j)$ the integers (if they are well defined) $$\textrm{MIN}(G,i,j)= \min\left\{ \min(g[i],g[j]): g\in G \textrm { with } g[i]\neq g[j] \right\},$$ for all the $i < j$ and $i,j \in \left\{1,\ldots,n+1\right\}.$ We define a solution for the vector $\textbf{d}$ as a set $G \subseteq \mathbb{N}^{n+1}$ such that: $$\textrm{MIN}(G,i,j)=\min\left\{ d_i,\ldots,d_{j-1}\right\} \textrm{ for all } i<j.$$ Consider $n\in \mathbb{N}$ with $n \geq 1$ . Find the smallest $t \in \mathbb{N}$, such that for all $[d_1,\ldots,d_{n}] \in \mathbb{N}^{n}$ there exists a solution with $t$ vectors. We denote such a number $t$ by $\textrm{NS}(n)$. \[thm\] Consider $n\in \mathbb{N}$ with $n \geq 1$ . Then $\textrm{NS}(n)=\left \lceil{ \log_2{(n+1)}}\right \rceil $, where $\left \lceil{d}\right \rceil=\min\left\{m\in \mathbb{N}: m\geq d \right\}$. **Proof** First of all we show that given an arbitrary vector $\textbf{d}$ of $\mathbb{N}^n$ we are able to find a solution of $\textbf{d}$ consisting of $N=\left \lceil{ \log_2{(n+1)}}\right \rceil$ vectors. We will do it by induction on $n$. The base of induction is trivial. In fact if $n=1$ then for each vector $[d_1]$ we find the solution $G=\left\{ [d_1,d_1+1]\right\}$ that has cardinality $\left \lceil{ \log_2{(1+1)}}\right \rceil=1$. Thus we suppose that the theorem is true for all the $m<n$ and we prove it for $n$. Let $\textbf{d}$ an arbitrary vector of $\mathbb{N}^n$. We fix some notations. Given a vector $\textbf{d}$, we will denote by $\textrm{sol}(\textbf{d})$ a solution with $\left \lceil{ \log_2{(n+1)}}\right \rceil$ vectors. We denote by $\textrm{Inf}(\textbf{d})=\min\left\{ d_i: i=1,\ldots,n \right\}$ and by $\textrm{Pinf}(\textbf{d})=\left\{i \in \left\{1,\ldots,n\right\}: d_i=\textrm{Inf}(\textbf{d}) \right\}$. We have $1\leq |\textrm{Pinf}(\textbf{d})|=k(\textbf{d}) \leq n$. Suppose that $\textrm{Pinf}(\textbf{d})=\left\{ i_1 < i_2 < \dots < i_{k(\textbf{d})}\right\}$. Then we can split the vector $\textbf{d}$ in the following $k(\textbf{d})+1$ subvectors: $$\begin{cases}\textbf{d}_1=\textbf{d}(1 \ldots i_1-1), \\ \textbf{d}_{j}=\textbf{d}(i_{j-1}+1 \ldots i_{j}-1) \textrm{ for } j=2,\ldots,k(\textbf{d}),\\ \textbf{d}_{k(\textbf{d})+1}=\textbf{d}(i_{k(\textbf{d})}+1\ldots n), \end{cases}$$ where with $\textbf{d}(a \ldots b)$ we mean - $ \emptyset$ if $b<a$; - The subvector of $\textbf{d}$ with components between $a$ and $b$ if $a\leq b$. Then the subvectors $\textbf{d}_j$ are either empty or with all the components greater than $\textrm{Inf}(\textbf{d})$. We briefly illustrate with an example the construction of the subvectors $\textbf{d}_j$. Suppose that $\textbf{d}=[2,3,2,2,5,4,5]$. Then $\textrm{Inf}(\textbf{d})=2$, $\textrm{Pinf}(\textbf{d})=\left\{ 1,3,4\right\}$ and then we have the four subvectors: - $ \textbf{d}_1=\textbf{d}(1\ldots 0)=\emptyset,$ - $ \textbf{d}_2=\textbf{d}(2\ldots 2)=[3]$, - $ \textbf{d}_3=\textbf{d}(4\ldots 3)=\emptyset $, - $ \textbf{d}_4=\textbf{d}(5\ldots 7)=[5,4,5]$. Then we can consider the list of $k(\textbf{d})+1$ subvectors: $$p(\textbf{d})=[\textbf{d}_1,\ldots,\textbf{d}_{k(\textbf{d})+1}],$$ and, because all the $\textbf{d}_i$ have length strictly less than $n$ we can find a solution for each of them with $N=\left \lceil{ \log_2{(n+1)}}\right \rceil$ or less vectors. For the $\textbf{d}_i=\emptyset$ we will set $\textrm{sol}(\emptyset)=\left\{ [x]\right\} $, where $x$ is an arbitrary integer that is strictly greater than all the entries of $\textbf{d}$. It is quite easy to check that the following equality holds: $$\label{form}n=k(\textbf{d})+\sum_{i=1}^{k(\textbf{d})+1}{\textrm{Length}(\textbf{d}_i}).$$ We associate to the list of vectors $p(\textbf{d})$ another list of vector $c(\textbf{d})$ such that $$c(\textbf{d})=[\textbf{c}_1,\ldots,\textbf{c}_{k(\textbf{d})+1}],$$ where $\textrm{Length}(\textbf{c}_j)=\textrm{Length}(\textbf{d}_j)+1$ and the entries of $\textbf{c}_j$ are all equal to $\textrm{Inf}(\textbf{d})$ for all $j=1,\ldots, k(\textbf{d})+1. $ Now we consider the set $ I(N)=\left\{0,1\right\}^N$. For each $\textbf{t} \in I(N)$ we denote by $O(\textbf{t})$ the number of one that appear in $\textbf{t}$. Because we have $N=\left \lceil{ \log_2{(n+1)}}\right \rceil$ , it follows $$k(\textbf{d})+1 \leq n+1 \leq 2^N=|I(N)|,$$ therefore it is always possible to associate to each subvectors of the list $p(\textbf{d})$ distinct elements of $I(N)$. We actually want to show that it is possible to associate to all the subvectors $\textbf{d}_i$ distinct vectors of $\textbf{t} \in I(N)$ such that $O(\textbf{t})\geq |\textrm{sol}(\textbf{d}_i)|$ (for $\textbf{d}_i=\emptyset$ we can also associate the zero vector). We already know for the inductive step that all the $\textbf{d}_i$ have solutions with at most $N$ vectors. Suppose therefore that $m \leq N$. It is easy to see that $$|\left\{ \textbf{t} \in I(N) : O(t)\geq m \right\}|= \sum_{k=m}^N{\binom{N}{k}}.$$ Then we suppose by contradiction that in $p(\textbf{d})$ we have $\sum_{k=m}^N{\binom{N}{k}}+1$ subvectors with solution with cardinality $m$. From the inductive step it follows that all these subvectors have at least length $2^{m-1}$, and from the formula \[form\] it follows: $$n \geq \sum_{k=m}^N{\binom{N}{k}}+\left(\sum_{k=m}^N{\binom{N}{k}}+1\right)2^{m-1} \Rightarrow n+1 \geq \left(\sum_{k=m}^N{\binom{N}{k}}+1\right)(1+2^{m-1}).$$ But we also have that: $$\sum_{k=m}^N{\binom{N}{k}}+1 \geq 2^{N-m+1},$$ in fact $\sum_{k=m}^N{\binom{N}{k}}$ is the number of ways to select a subset of $\left\{ 1,\ldots,N\right\}$ of at least $m$ elements while there are $2^{N-m+1}-1$ ways to select a subset which contains at least $m$ elements and contains $\left\{ 1,2,\ldots,m-1\right\}$. Therefore we can continue the inequality: $$n +1 \geq 2^{N-m+1} (1+2^{m-1})=2^N+2^{N-m+1}>2^N.$$ But $N=\left \lceil{ \log_2{(n+1)}}\right \rceil$ and therefore $n+1 \leq 2^N$ and we find a contradiction. Then in $\left\{ \textbf{t} \in I(N) : O(t)\geq m \right\} $ we have enough vectors to cover all the subvectors with solution with cardinality $m$. We still also have to exclude the following possibility. Suppose that we have $x$ subvectors with solutions of cardinality $m_1$ and $y$ subvectors with solutions of cardinality $m_2 >m_1$. If $|\left\{ \textbf{t} \in I(N) : O(t)\geq m_1 \right\} |-x<y$ then it would not be possible to associate to all the subvectors of the second type an element $\textbf{t}$ of $I(N)$ with $O(\textbf{t}) \geq m_2.$ Indeed if this happen we would have: $$n \geq x+y-1+x\cdot2^{m_1-1}+y\cdot 2^{m_2-1} >x+y-1+(x+y)2^{m_1-1} \Rightarrow$$ $$\Rightarrow n+1 \geq (x+y)(1+2^{m_1-1}) \geq \left(\sum_{k=m_1}^N{\binom{N}{k}}+1\right)(1+2^{m_1-1}),$$ and we already have seen that this is not possible. Therefore we showed that we can consider a matrix $A$ with $N$ rows and $ k(\textbf{d})+1$ distinct columns with only zeroes and ones as entries and such that the $i$-th column of $A$ is a vector $\textbf{t}$ of $I(N)$ such that $O(\textbf{t}) \geq | \textrm{sol}(\textbf{d}_i)|$ for each $1\leq i \leq k(\textbf{d})+1$. Now we can complete the construction of a solution for $\textbf{d }$. We consider a matrix $B$ with $N$ rows and $ k(\textbf{d})+1$ columns. We fill the matrix $B$ following these rules: - If $A[i,j]=0$ then in $B[i,j]$ we put the vector $\textbf{c}_j$; - If $A[i,j]=1$ then in $B[i,j]$ we put an element of $ \textrm{sol}(\textbf{d}_j)$; - All the elements of $\textrm{sol}(\textbf{d}_j)$ have to appear in the $j$-th column for all $j=1,\ldots, k(\textbf{d})+1$. Then if we glue all the vectors appearing in the $i$-th row of $B$ for each $i=1,\ldots,N$ we obtain a solution $G$ for the vector $\textbf{d}$. In fact if we consider $i_1,j_1$ such that $i_1<j_1$ we have two possibilities: - $i_1$ and $j_1$ both correspond to elements in the $j$-th column of $B$. Then because in this column we have either vectors of a solution for $\textbf{d}_j$ or costant vectors, it follows that they fulfill our conditions. - $i_1$ and $j_1$ correspond to elements in distinct columns. This implies that we must have $\textrm{MIN}(G,i_1,j_1)=\textrm{Inf}(\textbf{d})$. In fact, for construction, between two distinct subvectors we have an element equal to $\textrm{Inf}(\textbf{d})$ in $\textbf{d}$ forcing $\textrm{MIN}(G,i_1,j_1)=\textrm{Inf}(\textbf{d})$. Now suppose that $i_1$ and $j_1$ correspond respectively to elements in the $i$-th and $j$-th columns of $B$. Because we suppose $i \neq j$ we have that the $i$-th column and the $j$-th column of the matrix $A$ are distinct so there exists a $k$ such that $A[k,i]=0$ and $A[k,j]=1$ (or viceversa). This implies that in $B$ we have a row where in the $i$-th column there is the constant vector equal to $\textrm{Inf}(\textbf{d})$ while in the $j$-th column we have a vector corresponding to a solution of a subvectors of $\textbf{d}$ (that has all the components greater than $\textrm{Inf}(\textbf{d})$ by construction). This easily implies that $\textrm{MIN}(G,i_1,j_1)=\textrm{Inf}(\textbf{d})$. Suppose that $\textbf{d}=[2,3,2,2,5,4,5]$. We have $n=7$, then we want to show that there exists a solution with three vectors. We have already seen that in this case we have: $$p(\textbf{d})=[\emptyset,[3],\emptyset,[5,4,5]].$$ We need to compute a solution for each entry of $p(\textbf{d})$. We have: - $\textrm{sol}(\emptyset)=\left\{ [6]\right\}$ ( $6$ is greater than all the entries of $\textbf{d}$); - $\textrm{sol}([3])=\left\{ [3,4]\right\}$; - Let us compute a solution for $\textbf{f}=[5,4,5]$ with the same techniques. Because $\textrm{Length}(\textbf{f})=3$ we expect to find a solution with at most two vectors. We have: $$p(\textbf{f})=[[5],[5]],$$ and we have $\textrm{sol}([5])=\left\{ [5,6] \right\}$. Then in $I(2)$ we want to find two distinct vectors with at least an entry equal to one. We can choose $[1,1]$ and $[0,1].$ Therefore we have: $$A=\left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right) \textrm{ and } B=\left( \begin{matrix} [5,6] & [4,4] \\ [5,6] & [5,6] \\ \end{matrix} \right).$$ Then $\textrm{sol}([5,4,5])=\left\{ [5,6,4,4],[5,6,5,6]\right\}$. Now we want to find in $I(3)$ four vectors $\textbf{t}_i$ for $i=1,\ldots,4$. We have free choice for the $\textbf{t}_1$ and $\textbf{t}_3$ , while we need $O(\textbf{t}_2) \geq 1$ and $O(\textbf{t}_4) \geq 2$. For instance we choose $\textbf{t}_1=[0,0,0],\textbf{t}_2=[1,0,0],\textbf{t}_3=[1,1,0],\textbf{t}_4=[1,0,1]$. Then we have: $$A=\left( \begin{matrix} 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 0\\ 0 & 0 &0 &1 \\ \end{matrix} \right) \textrm{ and } B=\left( \begin{matrix} [2] & [3,4] & [6] & [5,6,4,4] \\ [2] & [2,2] & [6] & [2,2,2,2]\\ [2] & [2,2] & [2] & [5,6,5,6]\\ \end{matrix} \right).$$ Then a solution for $\textbf{d}$ is the set $$G=\left\{[2, 3, 4, 6, 5, 6, 4, 4], [2, 2, 2, 6, 2, 2, 2, 2], [2, 2, 2, 2, 5, 6, 5, 6]\right\}.$$ So we proved that $\textrm{NS}(n) \leq \left \lceil{ \log_2{(n+1)}}\right \rceil $. To prove that the equality holds we notice that for each $n$ a constant vector needs exactly $\left \lceil{ \log_2{(n+1)}}\right \rceil $ vectors in its solutions. Now we can return to the problem of determining an upper bound for the cardinality of $G(T)$. We need another lemma: \[Lemma2\] Let $E=\left\{M(1),M(2) \right\}$ be a collection of two multiplicity sequences. Then, with the previous notations we have: $$k_E(1,2) \leq \min\left\{j: j\in (\textrm{PChar}_E(1)\cup \textrm{PChar}_E(2))\setminus (\textrm{PChar}_E(1)\cap \textrm{PChar}_E(2)) \right\}.$$ **Proof** Let us choose an arbitrary element $t \in (\textrm{PChar}_E(1)\cup \textrm{PChar}_E(2))\setminus (\textrm{PChar}_E(1)\cap \textrm{PChar}_E(2))$. We want to show that $k_E(1,2) \leq t$. Suppose by contradiction that $t<k_E(1,2)$. Without loss of generality we suppose that $t \in \textrm{PChar}_E(1)$. It follows that $t \notin \textrm{PChar}_E(2)$ and we have: $$r(m_{1,t}) <r(m_{1,t+1}) \textrm{ and } r(m_{2,t}) \geq r(m_{2,t+1}).$$ Notice that if an entry of $M(1)$ has $m_{1,t+1}$ as a summand and it is not $m_{1,t}$, it is forced to have $m_{1,t}$ as a summand too. So from $r(m_{1,t}) <r(m_{1,t+1}) $ we deduce that in $M(1)$ there are no entries involving only $m_{1,t}$. Similarly from $r(m_{2,t}) \geq r(m_{2,t+1})$ we deduce that in $M(2)$ we must have at least one entry $m_{2,s}$ which involves $m_{2,t}$ as a summand but not $m_{2,t+1}$. Namely $$m_{2,s}= \sum_{k=s+1}^t{m_{2,k}}. \label{equation}$$ Now, we have assumed that $t<k_E(1,2)$ hence $t+1\leq k_E(1,2)$. This imply that the untwisted tree $T$ such that $T_E=(t+1)$ is well defined. In $T$ we have the following nodes: $$(m_{1,s},m_{2,s}), \ldots, (m_{1,t},m_{2,t}), (m_{1,t+1},m_{2,t+1}).$$ Then from (\[equation\]) and from the fact that the two branches are still glued at level $t+1$ it must follow that $$\label{eq} m_{1,s}= \sum_{k=s+1}^t{m_{1,k}}$$ and we have still noticed how it contradicts the assumption $ r(m_{1,t}) <r(m_{1,t+1}) $.\ \ Now we can prove the following result: \[prop2\] Let $E$ be a collection of $n$ multiplicity sequences. Then, if $S(T) \in \sigma(E)$, there exists $G(T) \subseteq \mathbb{N}^n$ with $\textrm{Arf}(G(T))=S(T)$ and $|G(T)|=C_E+\left \lceil{ \log_2{(n})}\right \rceil $. **Proof** For the Remark \[remark24\] it suffices to prove the theorem only for the untwisted trees. Therefore we suppose that $T_E=(d_1,\ldots,d_{n-1})$. First of all we have to satisfy the condition on the characters to ensure that $\textrm{Arf}(G(T)) \in \sigma(E)$. From the Lemma (\[Lemma2\]) it follows that we can use $C_E$ vectors to satisfy all the conditions. To see it, let us fix some notations. Denote by $ \tau(i)=| \textrm{PChar}_E(i)|$ for all $i=1,\ldots,n$. Therefore $C_E=\max\left\{\tau(i), i=1,\ldots,n\right\}$. Suppose that $$\textrm{PChar}_E(i)=\left\{a_{i,1}<\dots<a_{i,\tau(i)}\right\},$$ and we define $$L=\max\left(\bigcup_{i=1}^{n}{ \textrm{PChar}_E(i)}\right)+1.$$ For all $i=1,\ldots,n$ we consider the vector $J(i)=[a_{i,1},\ldots,a_{i,\tau(i)},L,\ldots,L] \in \mathbb{N}^{C_E}.$ Thus we can use the following set of vectors to satisfy the condition on the characters, $$G=\textrm{Gen}(1)=V_E(j_{1,1},\ldots,j_{1,n}), \ldots, \textrm{Gen}(C_E)=V_E(j_{C_E,1},\ldots,j_{C_E,n}),$$ where $j_{p,q}=J(q)[p]$ for all $p=1,\ldots,C_E$ and $q=1,\ldots,n$. Now it is clear that we have $\textrm{PChar}_E(i) \subseteq \left\{ j_{1,i},\ldots,j_{C_E,i} \right\}$ for all $i=1,\ldots,n$. We also need to show that this choice does not affect the condition on $(d_1,\ldots,d_{n-1})$. We define $P=\left\{ (q,r) \in \left\{1,\ldots,n\right\}^2: j_{p,q}=j_{p,r} \textrm{ for all } p=1,\ldots,C_E \right\}.$ Thus for each $(q,r) \in P$ the previous vectors are compatible with the conditions on the element $p_{q,r}$ of $M(T)_E$. For each $ (q,r) \notin P $, we consider $$p(q,r)=\min\left\{ p: j_{p,q} \neq j_{p,r} \right\}.$$ Now, because the entries of the vectors $J(q)$ are in an increasing order, it is clear that we have $$\textrm{MIN}_{G}(q,r)=\min \left(k_E(q,r), \min \left\{ \min(j_{p,q},j_{p,r}) : j_{p,q} \neq j_{p,r} \right\}\right)=$$$$=\min \left( k_E(q,r), \min(j_{p(q,r),q},j_{p(q,r),r})\right),\textrm{for all }(q,r)\notin P.$$ Furthermore, for the particular choice of the vectors $\textrm{Gen}(i)$ and of $L$, it is clear that from $ j_{p(q,r),q} \neq j_{p(q,r),r}$, it follows that $$\min(j_{p(q,r),q},j_{p(q,r)r}) \in (\textrm{PChar}_E(q)\cup \textrm{PChar}_E(r))\setminus (\textrm{PChar}_E(q)\cap \textrm{PChar}_E(r)),$$ and from the Lemma (\[Lemma2\]), we finally have $$\min(j_{p(q,r),q},j_{p(q,r),r}) \geq k_E(q,r) \textrm{ for all } (q,r) \notin P,$$ so the vectors $\textrm{Gen}(i)$ are compatible with our tree. Now from the Theorem (\[thm\]) it follows that we can use $\left \lceil{ \log_2{(n})}\right \rceil $ vectors to have a solution for the vector $[d_1,\ldots,d_{n-1}]$. Adding the vectors corresponding to this solution to the previous $C_E$ we obtain a set $G(T)$ such that $\textrm{Arf}(G(T))=S(T)$. Notice that the first $C_E$ vectors may satisfy some conditions on the $d_i$, therefore it is possible to find $G(T)$ with smaller cardinality than the previous upper bound. Let us consider the Arf semigroup of the Example \[ex7\]. It was $T=T_E=(2,1)$, where $$E= \left\{M(1)=[5,4,1,1], M(2)=[6,4,1,1], M(3)=[2,2,1,1] \right\},$$ with $$\textrm{PChar}_E(1)=\left\{1,2 \right\}, \textrm{PChar}_E(2)=\left\{1,2,3\right\} \textrm{ and } \textrm{PChar}_E(3)=\left\{1,3\right\}.$$ We found $G=\left\{ V_E(1,1,3),V_E(2,3,2),V_E(2,2,1)\right\}$ as a set such that $\textrm{Arf}(G)=S(T)$, and it is also minimal because we have $|G|=C_E$ and we clearly cannot take off any vector from it. Using the strategy of the previous corollary we would find the vectors: $$\textrm{Gen}(1)=V_E(1,1,1), \textrm{Gen}(2)=V_E(2,2,3) \textrm{ and } \textrm{Gen}(3)=V_E(4,3,4),$$ that satisfy the conditions on the characters ($L=4$). We have to add vectors that correspond to a solution for the vector $[2,1]$. For istance it suffices to consider $ [3,2,1]$ and therefore we will add the vector $\textrm{Gen}(4)=V_E(3,2,1)$. Notice how the set $G'=\left\{ V_E(1,1,1),V_E(2,2,3),V_E(4,3,4),V_E(3,2,1)\right\}$, with $|G'|>|G|$, is still minimal because we cannot remove any vector from it without disrupting the condition on the tree. Therefore we can have minimal sets of generators with distinct cardinalities. Let us consider $$E=\left\{ M_1=[4,4,1,1],M_2=[6,4,1,1],M_3=[2,2,1,1],M_4=[3,2,1,1] \right\}.$$ We want to find a set of generators for the twisted tree $T$ of $\tau(E)$ such that: $$M(T)_E=\left( \begin{matrix} 0 &2& 1& 2 \\ 0& 0& 1& 3 \\ 0& 0 &0 &1 \\ 0& 0 &0 &0 \\\end{matrix}\right).$$ First of all we notice that it is well defined because it satisfies the conditions given by the Remark \[rem\] and we have $$k(1,2)=2, k(1,3)=4,k(1,4)=2, k(2,3)=2, k(2,4)=3 \textrm{ and } k(3,4)=2.$$ We consider the permutation $\delta=(3,4)$ of $S^4$. Then $\delta(T)$ is an untwisted tree of $\tau(\delta(E))$ and it is described by the vector $T_{\delta(E)}=(2,3,1)$. We have: - $ \textrm{PChar}_{\delta(E)}(1)=\left\{1,3 \right\};$ - $ \textrm{PChar}_{\delta(E)}(2)=\left\{1,2,3 \right\};$ - $ \textrm{PChar}_{\delta(E)}(3)=\left\{1,2 \right\};$ - $ \textrm{PChar}_{\delta(E)}(4)=\left\{1,3 \right\}.$ Then with the vectors $V_{\delta(E)}(1,1,1,1),V_{\delta(E)}(3,2,2,3),V_{\delta(E)}(4,3,4,4)$, we satisfy the condition on the characters. We need to add the vectors corresponding to a solution for $[2,3,1]$. It suffices to add $V_{\delta(E)}(2,4,3,1)$. Then $$G(T)=\left\{ [4,6,3,2],[9,10,5,5],[10,11,7,6],[8,12,6,2]\right\},$$ is a set of generators for $\delta(T)$. Because $\delta^{-1}=(3,4)$, we have that $$\delta^{-1}( G(T))=\left\{ [4,6,2,3],[9,10,5,5],[10,11,6,7],[8,12,2,6]\right\}$$ is a set of generators for the twisted tree $T$. The author would like to thank Marco D’Anna for his helpful comments and suggestions. Special thanks to Pedro García-Sánchez for his careful reading of an earlier version of the paper and for many helpful hints regarding the implementation in GAP of the presented procedures. [english]{} [Dillo 100]{} C. Arf [*Une interpretation algebrique de la suite des ordres de multiplicite d’une branche algebrique*]{} Proc. London Math. Soc. (2), 50:256-287, 1948. V. Barucci, M. D’Anna, R. Fröberg [*Analitically unramifed one-dimensional semilocal rings and their value semigroups*]{}, in J. Pure Appl. Algebra, [*147*]{} (2000), 215-254. V. Barucci, M. D’Anna, R. Fröberg [*Arf characters of an algebroid curve*]{}, in JP Journal of Algebra, Number Theory and Applications, vol.3(2)(2003), p. 219-243. A. Campillo, [*Algebroid curves in positive characteristic*]{}, Lecture Notes in Math. 813, Springer-Verlag, Heidelberg, (1980). M.  D’Anna [*The canonical module of a one-dimensional reduced local ring*]{}, Comm. Algebra , [*25*]{} (1997), 2939-2965. M.  D’Anna, P. A. Garcia Sanchez, V. Micale, L. Tozzo [*Good semigroups of $\mathbb{N}^n$,*]{} preprint, arXiv:1606.03993 (2017) P.  Du Val [*The Jacobian algorithm and the multiplicity sequence of an algebraic branch*]{}, Rev. Fac. Sci. Univ. Istanbul. Ser. A., [*7*]{} (1942), 107-112. The GAP Group, [*GAP-Groups, Algorithms and Programming*]{}, Version 4.7.5 , [*25*]{} (2014). GIUSEPPE ZITO-Dipartimento di Matematica e Informatica-Università di Catania-Viale Andrea Doria, 6, I-95125 Catania- Italy. E-mail address: giuseppezito@hotmail.it

--- abstract: 'We consider a non-projective class of inhomogeneous random graph models with interpretable parameters and a number of interesting asymptotic properties. Using the results of [@Bollobas2007], we show that i) the class of models is sparse and ii) depending on the choice of the parameters, the model is either scale-free, with power-law exponent greater than 2, or with an asymptotic degree distribution which is power-law with exponential cut-off. We propose an extension of the model that can accommodate an overlapping community structure. Scalable posterior inference can be performed due to the specific choice of the link probability. We present experiments on five different real-world networks with up to 100,000 nodes and edges, showing that the model can provide a good fit to the degree distribution and recovers well the latent community structure.' author: - 'Juho Lee[^1]' - 'Lancelot F. James' - Seungjin Choi - François Caron bibliography: - 'paper.bib' title: '**A Bayesian model for sparse graphs with flexible degree distribution and overlapping community structure**' --- Introduction {#sec:introduction} ============ Simple graphs are composed of a set of vertices with undirected connections between them. The graph may represent a set of friendship relationships between individuals, a physical infrastructure network, or a protein-protein interaction network. Defining flexible and realistic statistical graph models is of great importance in order to perform link prediction or for uncovering interpretable latent structure, and has been the subject of a large body of work in recent years, see e.g. [@Newman2009; @Kolaczyk2009; @Goldenberg2010]. Our objective is to develop a class of models with interpretable parameters and realistic asymptotic properties. Of particular interest for this paper are the notions of sparsity and scale-freeness. A sequence of graphs is said to be sparse if the number of edges scales subquadratically with the number of nodes. The degree of a node is the number of connections of that node. The sequence of graphs is said to be scale-free if the proportion of nodes of degree $k$ is approximately $k^{-\eta}$ when the number $n$ of nodes is large, where the exponent $\eta$ is greater than 1. That is, for large $n$, the degree distribution behaves like a power-law. These notions of sparsity and scale-freeness have received a lot of attention in the network literature in the past years [@Barabasi1999; @Newman2009; @Orbanz2015; @Barabasi2016; @Caron2017]; some authors argued that they are desirable properties of random graph models, and that many networks exhibit this scale-free behavior, usually with an exponent $\eta>2$. Other authors have recently challenged the scale-free assumption, showing that a power-law distribution with exponential cut-off provides a good fit to many real-world networks [@Newman2009; @Broido2018], see the appendix for more discussion about testing for network scale-freeness. Besides these global asymptotic properties, we are also interested in capturing some latent structure in graphs. Individuals may belong to some latent communities, and their level of affiliation to the community defines the probability that two nodes connect. We propose a class of sparse graph models with overlapping community structure and well-specified asymptotic degree distributions. The graph can either be scale-free with exponent $\eta>2$, or non-scale-free, with asymptotic degree distribution being a power-law distribution with exponential cut-off. The construction builds on inhomogeneous random graphs, a class of models exhibiting degree heterogeneity. This class of models has been studied extensively in the applied probability literature [@Aldous1997; @Chung2002; @Bollobas2007; @vanderHofstad2016], but has been left unexplored for the statistical analysis of real-world networks. In \[sec:sparsedef\] we provide a formal description of sparsity and scale-freeness for sequences of graphs. In \[sec:rank1\] we describe the rank-1 inhomogeneous random graphs, and present their sparsity property and asymptotic degree distribution. The model is then extended in \[sec:rankc\] in order to accommodate a latent community structure. Posterior inference is discussed in \[sec:inference\]. In \[sec:discussion\] we discuss the relative merits and drawbacks of our approach compared to other random graph models. \[sec:experiments\] provides an illustration of the approach on several real-world networks, showing that the model can provide a good fit to the empirical degree distribution and recover the latent community structure. **Notations.** Throughout the article, $X_n\pto X$ denotes convergence in probability, and $a_n\sim b_n$ indicates $\lim_{n\rightarrow\infty} a_n/b_n\rightarrow 1$. Sparse and scale-free networks {#sec:sparsedef} ============================== We first provide a formal definition of sparsity and scale-freeness, as there is no general agreement on the definition of a scale-free network and these notions are core to the results of this paper. Let $(G_n)_{n\geq 1}$ be a sequence of simple random graphs of size $n$ where $G_n=(V_n,E_n)$, $V_n=\{1,\ldots,n\}$ is the set of vertices and $E_n$ the set of edges. Denote $|E_n|$ the number of edges. The graph is said to be sparse if $\mathbb E (|E_n|)/n^2\rightarrow 0$. Let $N_k^{(n)}$ be the number of nodes of degree $k$ in $G_n$. We now formally give the definition of a scale-free network informally introduced in Section \[sec:introduction\]. \[def:scalefree\] A random graph sequence $(G_n)_{n\geq 1}$ is said to be scale-free with exponent $\eta$ iff there exists a slowly varying function $\ell$ and $\eta>1$ such that, for each $k=1,2,\ldots$ $$\begin{aligned} \frac{N^{(n)}_k}{n}\pto \pi_k$$ as $n$ tends to infinity, where $$\pi_k\sim\ell(k) k^{-\eta}\text{ as }k\rightarrow\infty.\label{eq:regvardegree}$$ Background definitions and properties of slowly and regularly varying functions are given in the appendix. Intuitively, slowly varying functions are functions that vary more slowly than any power of $x$. The term scale-free comes from the fact that the asymptotic degree distribution satisfies some (asymptotic) scale-invariance. For any integer $m\geq 1$, $$\lim_{k\rightarrow\infty}\frac{\pi_{mk}}{\pi_k}= m^{-\eta}.$$ The most classical case is when $\ell(k)=C$ is constant. In this case, the asymptotic degree distribution behaves as a pure power-law for $k$ large. More generally, the scale-invariance property defined above will be satisfied for any slowly varying function $\ell$, which can be e.g. logarithm, or iterated logarithm. \[def:scalefree\] is slightly more restrictive than the definition of a scale-free graph sequence in [@vanderHofstad2016 Definition 1.4], which is implied from \[def:scalefree\] by properties of regularly varying functions (see appendix). Rank-1 inhomogeneous random graphs {#sec:rank1} ================================== Definition ---------- Let $(G_n)_{n\geq 1}$ be a sequence of simple random graphs of size $n$ defined as follows. The probability that two nodes $i$ and $j$ are connected in the graph $G_n$ is given by $$p_{ij}^{(n)}=1-\exp\left ( -\frac{w_iw_j}{s^{(n)}}\right )\label{eq:norros-reittu}$$ where $s{^{(n)}}=\sum_{i=1}^n w_i$ and the positive weights $(w_1,w_2,\ldots)$ are independently and identically distributed (iid) from some distribution $F$ with $\mathbb E(w_1)<\infty$. The model  is known as the Norros-Reittu (NR) inhomogeneous random graph model [@Norros2006]. This model has been the subject of a lot of interest in the applied probability and graph theory literature [@Bollobas2007; @Bhamidi2012; @vanderHofstad2013; @vanderHofstad2016; @Broutin2018]. The parameter $w_i>0$ accounts for degree heterogeneity in the graph and can be interpreted as a sociability parameter of node $i$. The larger this parameter, the more likely node $i$ is to connect to other nodes. Sparsity and scale-free properties ---------------------------------- The random graph sequence defined by Equation satisfies a number of remarkable asymptotic properties. The first result, which follows from [@Bollobas2007] (see details in the appendix), shows that the resulting graphs are sparse. \[thm:rank1\_sparsity\] Let $|E_n|$ denote the number of edges in the graph $G_n$. Then $$\begin{aligned} \frac{\mathbb E(|E_n|)}{n}\rightarrow \frac{\mathbb E (w_1)}{2}~~\text{ and }~~ \frac{|E_n|}{n}\pto \frac{\mathbb E (w_1)}{2}.\end{aligned}$$ The following result is a corollary of Theorem 3.13, remark 2.4 and the discussion in Section 16.4 in [@Bollobas2007]. It states that the asymptotic degree distribution is a mixture of Poisson distributions, with mixing distribution $F$. \[thm:rank1\_degree\] Let $N^{(n)}_k$ be the number of vertices of degree $k$ in the graph $G_n$ of size $n$ and link probability $p_{ij}^{(n)}$ given by Equation . Then, for each $k=1,2,\ldots$, $N^{(n)}_k/n\pto \pi_k$ as $n$ tends to infinity, where $$\begin{aligned} \pi_k:= \int_0^\infty \frac{x^k}{k!}e^{-x}dF(x).\end{aligned}$$ Our analysis on the asymptotic degree distribution is based on the following theorem for the asymptotic behavior of mixed Poisson distributions. \[thm:mixed\_poisson\_asymp\] [@Willmot1990] Suppose that $$f(x) \sim \ell(x) x^\eta e^{-\zeta x}, \quad x \to \infty,$$ where $\ell(x)$ is a locally bounded function on $(0,\infty)$ which varies slowy at infinity, $\zeta \geq 0$, and $-\infty < \eta < \infty$ (with $\eta < -1$ when $\zeta=0$). For $\lambda > 0$, define the probabilities of the mixed Poisson distribution as $$\pi_k = \int_0^\infty \frac{(\lambda x)^k e^{-\lambda x}}{k!} f(x) dx; \quad k=0, 1, 2, \dots.$$ Then, $$\pi_k \sim \frac{\ell(k)}{(\lambda + \zeta)^{\eta+1}} \bigg( \frac{\lambda}{\lambda+\zeta}\bigg)^k k^\eta, \quad k \to \infty.$$ The following result is a corollary of \[thm:rank1\_degree\] and \[thm:mixed\_poisson\_asymp\]. It states that if the random variables $w_i$ are regularly varying (see definition in the appendix), then the sequence of random graphs is scale-free. \[thm:scalefreemixed\] Let $N^{(n)}_k$ be the number of vertices of degree $k$ in the graph $G_n$ of size $n$ and link probability $p_{ij}^{(n)}$ given by Equation . Assume that the distribution F is absolutely continuous with pdf $f$ verifying $f(w)\sim \ell(w)w^{-\eta}$ as $w$ tends to infinity, for some locally bounded slowly varying function $\ell$ and $\eta> 1$. Then, for each $k=1,2,\ldots$, $N^{(n)}_k/n\pto \pi_k$ as $n$ tends to infinity, where $$\begin{aligned} \pi_k\sim \ell(k) k^{-\eta}, \quad k \to \infty.\end{aligned}$$ Particular examples ------------------- We now consider two special cases. The first case yields scale-free graphs with asymptotic power-law degree distributions with exponent $\eta>2$. The second yields non-scale-free graphs, where the asymptotic degree distribution is power-law with exponential cut-off. ### Scale-free graph with power-law degree distribution For $i=1,2,\ldots$, let $w_i \iidsim \mathrm{invgamma}(\alpha, \beta)$ where $\mathrm{invgamma}(\alpha, \beta)$ denotes the inverse gamma distribution with parameters $\alpha>1$ and $\beta>0$, whose probability density function (pdf) is given by $$f(w)=\frac{\beta^\alpha}{\Gamma(\alpha)} w^{-\alpha-1} e^{-\beta/w}.$$ Here, the constraint $\alpha > 1$ is required for the condition $\bbE[w_1]< \infty$. By \[thm:rank1\_degree\], the asymptotic degree distribution is a mixed Poisson-inverse-gamma distribution with probability mass function $$\pi_k = \frac{2\beta^{\frac{k+\alpha}{2}}}{k!\Gamma(\alpha)}K_{k-\alpha}(2\sqrt{\beta}) ,$$ where $K$ is the modified Bessel function of the second kind. Using \[thm:scalefreemixed\], we obtain $$\pi_k \sim \frac{\beta^\alpha }{\Gamma(\alpha)} k^{-\alpha-1} \quad \textrm{ as }k \to \infty.$$ The resulting asymptotic degree distribution is a power-law and the graph is scale-free with arbitrary index $\alpha + 1 > 2$. The two hyperparameters of the inverse gamma prior play an important role to decide the asymptotic properties of graphs. The shape parameter $\alpha$ tunes the index of power-law, and is also related to the sparsity of graphs. The scale parameter $\beta$ is also related to the sparsity of graphs. \[fig:rank1\_degree\_sparsity\] shows the empirical degree distributions and number of edges of graphs generated from inverse gamma NR model. {width="0.48\linewidth"} {width="0.48\linewidth"} {width="0.48\linewidth"} {width="0.48\linewidth"} ### Non scale-free graph with power-law degree distribution with exponential cut-off Now we consider another model with generalized inverse Gaussian (GIG) prior. Let $w_i \iidsim \mathrm{GIG}(\nu, a, b)$ where the density of the GIG distribution with parameter $\nu$, $a>0$ and $b>0$ is given by $$f(w) = \frac{(a/b)^{\nu/2}}{2K_\nu(\sqrt{ab})} w^{\nu-1} \exp\bigg\{ -\frac{1}{2}\bigg(aw + \frac{b}{w}\bigg)\bigg\}.$$ Note that by taking $a\rightarrow 0$, one obtains the pdf of an inverse gamma distribution as a limiting case. By \[thm:rank1\_degree\], the asymptotic degree distribution is $$\pi_k = \frac{(a/b)^{\nu/2}}{k!\{(a+2)/b\}^{(k+\nu)/2}} \frac{K_{k+\nu}(\sqrt{(a+2)b})}{K_\nu(\sqrt{ab})}.$$ This distribution is sometimes called the Sichel distribution, after Herbert Sichel [@Sichel1974]. Note that $f(w)\sim (a/b)^{\nu/2}/2/K_\nu(\sqrt{ab}) w^{\nu-1} \exp(-aw/2)$ as $w\rightarrow\infty$ hence, by \[thm:mixed\_poisson\_asymp\], $$\pi_k \sim \frac{(a/b)^{\nu/2} k^{\nu-1} e^{-\log(1+a/2)k} }{2(1 + a/2)^\nu K_\nu(\sqrt{ab})} \quad \textrm{ as } k \to \infty.$$ In this case, the asymptotic degree distribution is not of the form of Equation , and the graph sequence is therefore not scale-free. However, the asymptotic degree distribution has the form $k^{\nu-1}e^{-\tau k}$ of a power-law distribution with exponential cut-off. This class of probability distributions has been shown to provide a good fit to the degree distributions of a wide range of real-world networks [@Clauset2009]. As for the inverse gamma NR model, the hyperparameters $(\nu, a, b)$ tunes the asymptotic properties. $\nu$ determines the power-law index of degree distribution, $a$ is related to the exponential cutoff and sparsity, and $b$ is related to the sparsity. \[fig:rank1\_degree\_sparsity\] shows the empirical degree distributions and the number of edges of graphs generated from GIG NR model. Extension to Latent Overlapping Communities {#sec:rankc} =========================================== Definition ---------- The inhomogeneous random graphs considered so far only account for degree heterogeneity. However, the connections in real-world networks are often due to some latent interactions between the vertices. Recently, several models that combine a degree correction together with a latent structure to define edge probabilities were proposed [@Zhou2015; @Todeschini2016; @Herlau2016; @Lee2017]. In this section, we propose an extension of the NR model that includes some latent overlapping structure, and study the sparsity, scale-freeness properties and asymptotic degree distribution of this model. Let the edge probability between the vertex $i$ and $j$ be given by $$\label{eq:rankc_link} p^{(n)}_{ij} = 1 - \exp\bigg( - \frac{w_iw_j}{s{^{(n)}}}\sum_{q=1}^c \frac{v_{iq} v_{jq}}{r_q{^{(n)}}/n}\bigg).$$ where $(w_i)_{i=1,2,\ldots}$ are iid random variables with distribution $F$ with $\mathbb E(w_1)<\infty$ and $(v_{i1},\ldots,v_{ic})_{i=1,2,\ldots}$ are i.i.d. with $\mathbb E(v_{1q})<\infty$ for all $q$ and $r_q{^{(n)}}= \sum_{i=1}^n v_{iq}$. We call this model with $c$ communities the *rank-$c$ model*. As in the rank-1 model, the parameter $w_i$ can be interpreted as an overall sociability parameter of node $i$, or degree-correction. The parameter $v_{iq}$ can be interpreted as the level of affiliation of individual of $i$ to community $q$. Similar models, in a different asymptotic framework have been used in  [@Yang2013; @Zhou2015; @Todeschini2016]. \[thm:rankc\_sparsity\_and\_degree\] Let $|E_n|$ denote the number of edges in the graph $G_n$ defined with link probability . Then, $$\frac{\bbE(|E_n|)}{n} &\rightarrow \frac{\bbE(w_1) \sum_{q=1}^c \bbE(v_{1q})}{2}\\ \frac{|E_n|}{n} &\pto \frac{\bbE (w_1) \sum_{q=1}^c\bbE (v_{1q})}{2}.$$ Recall that $N^{(n)}_k$ is the number of vertices of degree $k$ in the graph $G_n$ of size $n$. Then, for each $k=1,2,\ldots$, $N^{(n)}_k/n \pto \pi_k$ as $n$ tends to infinity, where $$\begin{aligned} \pi_k= \int_0^\infty\int_0^\infty \frac{(uw)^k}{k!}e^{-uw}dF(w)dH(u)\end{aligned}$$ where $H$ is the distribution of the random variable $U=\sum_{q=1}^c v_{1q}$. If additionally $F$ is absolutely continuous with pdf $f$ verifying $ f(w)\sim \ell(w)w^{-\eta}$ as $w\rightarrow\infty$ for some locally bounded slowly varying function $\ell$ and $\eta> 1$ and $\mathbb E(U^{\eta-1+\epsilon})<\infty$ for some $\epsilon>0$, then $$\pi_k\sim \mathbb E(U^\eta)\ell(k)k^{-\eta}~~\text{as }k\rightarrow\infty.$$ The proof of \[thm:rankc\_sparsity\_and\_degree\] is given in the appendix. In this paper, we consider in particular $$(v_{i1}, \dots, v_{iq}) \sim \mathrm{Dir}(\gamma),$$ where $\mathrm{Dir}(\gamma)$ denotes the standard Dirichlet distribution with parameter $\gamma=(\gamma_1,\ldots,\gamma_c)$, where $\gamma_q>0$ for $q=1,\ldots,c$. Posterior inference {#sec:inference} =================== Posterior inference for the rank-1 NR ------------------------------------- Let $Y = \{y_{ij}\}_{1\leq i < j \leq n}$ be an (upper triangular part of) adjacency matrix of a graph $G_n$ and $w=(w_1,\ldots,w_n)$. The joint density is written as $$p(Y, w) = \prod_{i=1}^n f(w_i) \prod_{i<j} \Big(1-e^{-\frac{w_iw_j}{s{^{(n)}}}}\Big)^{y_{ij}} e^{(y_{ij}-1)\frac{w_iw_j}{s{^{(n)}}}}$$ Following [@Caron2017] and [@Zhou2015], we introduce a set of auxiliary truncated Poisson random variables $m_{ij}$ for the pairs with $y_{ij}=1$. $$p(m_{ij}|w) = \frac{(\frac{w_iw_j}{s{^{(n)}}})^{m_{ij}}\exp(-\frac{w_iw_j}{s{^{(n)}}})\indicator{m_{ij}>0} }{m_{ij}!(1-\exp(-\frac{w_iw_j}{s{^{(n)}}})}.$$ The log joint density is then given as $$\label{eq:rank1_joint} \log p(Y, M, w) = \sum_{(i,j)\in E_n} \bigg(m_{ij} \log \frac{w_iw_j}{s{^{(n)}}} - \log m_{ij}!\bigg) + \frac{1}{2} \bigg( \sum_{i=1}^n \frac{w_i^2}{s{^{(n)}}} - s{^{(n)}}\bigg) + \sum_{i=1}^n \log f(w_i).$$ Note that the terms for the pairs without edges $(y_{ij}=0)$ are collapsed into a single summation, and hence the overall computations of the log joint density and its gradient take $O(n + |E_n|)$ time. This is a huge advantage of the link function of NR model, while other link functions for rank-1 inhomogeneous random graphs [@Britton2006; @Chung2002; @Chung2002a; @Chung2003] suffer from $O(n^2)$ computing times. For the posterior inference, we use a Markov chain Monte Carlo (MCMC) algorithm. At each step, given the gradient of the log joint density, we update $w$ via Hamiltonian Monte Carlo (HMC, [@Duane1987; @Neal2011]). Then we resample the auxiliary variables $m$ from truncated Poisson, and update hyperparameters for $f(w)$ using a Metropolis-Hastings step. Details can be found in the appendix. Posterior inference for the rank-$c$ NR --------------------------------------- The posterior inference for the rank-$c$ model is similar to that of the rank-1 model. Following [@Todeschini2016], for tractable inference, we introduce a set of multivariate truncated Poisson random variables $M = ((m_{ijq})_{q=1}^c)_{(i,j)\in E_n}$, $$p(M|w, V) = \prod_{(i,j)\in E_n} \prod_{q=1}^c \frac{\lambda_{ijq}^{m_{ijq}} e^{-\lambda_{ijq}} \indicator{\sum_{q'=1}^c m_{ijq'}>0}}{1 - \exp(-\sum_{q'=1}^c \lambda_{ijq'})}.$$ where $\lambda_{ijq} = \frac{w_iw_j}{s{^{(n)}}} \frac{v_{iq}v_{jq}}{r_q{^{(n)}}/n}$ and $V=(v_{iq})_{i=1,\ldots,n,q=1,\ldots,c}$. The log joint density is $$\log p(Y, M, w, V) &= \prod_{(i,j)\in E_n} \sum_{q=1}^c (m_{ijq} \log \lambda_{ijq} - \log m_{ijq}!) - \sum_{i<j} \sum_{q=1}^c \lambda_{ijq} + \sum_{i=1}^n \log f(w_i) \nonumber\\ & + \sum_{i=1}^n \log g(v_{i1}, \dots, v_{ic};\gamma),$$ where $g(\cdot;\gamma)$ is the density for Dirichlet distribution with parameters $\gamma$. As for the rank-1 model, we can efficiently compute this log joint density and its gradient w.r.t. $w$ and $V$ with $O(cn+c|E_n|)$ time. At each step of MCMC, we first sample $w$ and $V$ via HMC, resample $M$ from multivariate truncated Poisson, and update hyperparameters via Metropolis-Hastings. The detailed procedure can be found in the appendix. Discussion {#sec:discussion} ========== The models described in this paper can capture sparsity, scale-freeness with exponent $\eta>2$ and latent community structure. One drawback of the construction is that the model lacks projectivity, due to normalisation by $s_n$ in the link probability . While this is an undesirable feature of the approach, we stress that there does not exist any projective class of random graphs that can capture all those properties, as we explain below. A popular class of models is the graphon-based or vertex-exchangeable graphs, which include as special cases stochastic blockmodels, latent factor models and their extensions, see [@Orbanz2015] for a review. While these models have been successfully applied in a wide range of application, they produce dense graphs with probability one, as stressed by [@Orbanz2015]. Alternative models have been proposed, either based on exchangeable point processes [@Caron2017; @Veitch2015; @Borgs2016], or on the notion of edge-exchangeability [@Crane2015; @Crane2017; @Cai2016]. @Caron2017a showed that using exchangeable point processes, one can obtain scale-free graphs with exponent $\eta\in(1,2]$, but not above. While no results exist for the scale-freeness of edge-exchangeable random graphs in the sense of \[def:scalefree\] (see [@Janson2017 Problem 9.8]), it is likely that a similar range is achieved for this class of models. Another family of models are non-exchangeable models based on preferential attachment [@Barabasi1999]. The generated graphs are scale-free with exponent $\eta>2$. However, the generative process makes it difficult to consider more general constructions that take into account community structure. Additionally, the non-exchangeability implies that the ordering of nodes must be known or need to be inferred for inference, which limits its applicability. By contrast, our model is finitely exchangeable for each $n$, and so the ordering of the nodes needs not to be known in order to make inference. As a consequence, no other projective class of model can give scale-free networks with exponent $\eta>2$, interpretable parameters capturing community structure, and scalable inference, as described in this paper. While the model has a number of attractive properties, it also has some limitations. The mean number of triangles in inhomogeneous random graphs converges to a constant as $n$ tends to infinity [@vanderHofstad2018]. Although the latent community structure introduced may mitigate this effect for reasonable $n$, this property appears undesirable for real-world network. Experiments {#sec:experiments} =========== Experiments with the rank-1 models ---------------------------------- In this section, we test our inverse-gamma NR model (IG-NR) and generalized inverse Gaussian NR model (GIG-NR) on synthetic and real world graphs. For all experiments, we ran three MCMC chains for 10,000 iterations for our algorithms, and collected every 10th samples after 5,000 burn-in samples. The prior distributions for the hyperparameters of the different models are given in the appendix. The code for our experiments is available at <https://github.com/OxCSML-BayesNP/BNRG>. {width="0.24\linewidth"} {width="0.24\linewidth"} {width="0.24\linewidth"}{width="0.24\linewidth"} {width="0.24\linewidth"} {width="0.24\linewidth"} {width="0.24\linewidth"} #### Experiments with synthetic graphs. We first fitted the basic models with Inverse-gamma prior (IG) and generalized inverse Gaussian prior (GIG) on synthetic graphs generated from IG-NR model and GIG-NR model. For IG, we generated a graph with $n=5,000$ nodes with parameters $\alpha=1.5$ and $\beta=3.0$. For GIG, we generated a graph with 5,000 nodes with parameters $\nu=0.5, a=0.1, b=3.0$. As summarized in \[fig:synth\], the posterior distribution recovers the hyperparameter values used to generated the graphs, and the posterior predictive distribution provides a good fit to the empirical degree distribution. #### Experiments with real-world graphs. Now we evaluate our models on three real-world networks:\    $\bullet$ `cond-mat`[^2]: co-authorship network based on arXiv preprints for condensed matter, 16,264 nodes and 47,594 edges.\ $\bullet$ `Enron`[^3]: Enron collaboration e-mail network, 36,692 nodes and 183,831 edges.\ $\bullet$ `internet`[^4]: Network of internet routers, 124,651 nodes and 193,620 edges.\ To evaluate the goodness-of-fit in terms of degree distributions, as suggested in @Clauset2009, we sample graphs from the posterior predictive distribution based on the posterior samples, and computed the reweighted Kolmogorov-Sminorov (KS) statistic: $$D = \max_{x\geq x_{\min}} \frac{|S(x) - P(x)|}{\sqrt{P(x)(1-P(x))}},$$ where $S(x)$ is the CDF of observed degrees, $P(x)$ is the CDF of degrees of graphs sampled from the predictive distribution, and $x_{\min}$ is the minimum $x$ values among the observed degree and predictive degree. We compare our model to the random graph model with generalized gamma process prior (GGP, [@Caron2017]), whose asymptotic degree distribution is a power-law with exponent in $(1, 2)$. We ran MCMC for the GGP model with 40,000 iterations and three chains. Posterior predictive degree distribution are reported in \[fig:rank1\_real\]. Credible intervals of the hyperparameters and KS statistics for the different models are given in \[tab:rank1\_real\]. Both IG and GIG provide a good to the degree distribution, with an exponent greater than 2, while the GGP model fails to capture the shape of the degree distribution. [@ c c M[2.5cm]{} c M[2.5cm]{} c M[2.5cm]{} @]{}& & &\ &$D$ & hyperparams &$D$ & hyperparams &$D$ & hyperparams\ IG &**0.07**$\pm$0.01 & $\alpha\in(2.55, 2.72)$ $\beta\in(9.20,9.95)$ &0.13$\pm$0.05 & $\alpha\in(1.29, 1.34)$ $\beta\in(3.23, 3.41)$ &**0.19**$\pm$0.00& $\alpha\in(3.20, 3.28)$ $\beta\in (6.51,6.72)$\ GIG & **0.07**$\pm$0.01 & $\nu\in(-2.61,-2.37)$$a\in(0.01,0.02)$$b\in(17.41,19.14)$ & **0.12**$\pm$0.01 & $\nu\in(-1.33,-1.28)$$a\in(0.00,0.00)$$b\in(6.42, 6.75)$ &**0.19**$\pm$0.00 & $\nu\in(-3.25,-3.18)$$a\in(0.00,0.00)$$b\in(12.93,13.30)$\ GGP &0.15$\pm$0.06 & $\sigma \in(-0.93, -0.80)$ $\tau\in(75.81,85.52)$ & 0.18$\pm$0.02 & $\sigma \in(0.19, 0.22)$ $\tau\in(11.53,12.98)$ & 0.40$\pm$0.10 & $\sigma \in(-0.18, -0.04)$ $\tau\in(92.05,196.17)$\ \[tab:rank1\_real\] Experiments with latent overlapping communities ----------------------------------------------- Finally, we tested our models with latent overlapping communities on two real-world graphs with ground-truth communities.\ $\bullet$ `polblogs`[^5]: the network of Americal political blogs. 1,224 nodes and 16.715 edges, two true communities (left or right).\ $\bullet$ `DBLP`[^6]: Co-authorship network of DBLP computer science bibliography. The original network has 317,080 nodes. Based on the ground-truth communities extracted in @Yang2012, we took three largest communities and subsampled 1,990 nodes among them. The subsampled graph contains 4,413 edges. We compared our two models IG-NR and GIG-NR models to the random graph model based on compound generalized gamma process (CGGP, [@Todeschini2016]), and mixed membership stochastic blockmodel (MMSB, [@Airoldi2009]). CGGP can capture the latent overlapping communities and has asymptotic power-law degree distsribution of exponent in $(1,2)$. MMSB can capture the latent communities, but does not include a degree correction term. For all three models, we set the number of communities to be equal to two for `polblogs`, and three for `DBLP`. The CGGP was ran for 200,000 iterations after 10,000 initial iterations where $w$ was initialized by running the model without communities (GGP). Each iteration of the sampler for MMSB scales quadratically with the number of nodes, and the sampler was therefore ran for a smaller number of iterations (5,000) for fair comparison. We found that longer iterations did not lead to improved performances. All methods were ran with three MCMC chains. For CGGP and MMSB methods, point estimates of the parameters measuring the level of affiliation of each individual were obtained using the Bayesian estimator described in @Todeschini2016. For IG-NR and GIG-IR, we simply took the maximum a posteriori estimate of $V$. To compare to the ground truth communities, nodes are then assigned to the community where they have the strongest affiliation. The learned communities are shown in the appendix. Posterior predictive of the degree distributions for the different models are given in \[fig:rankc\_real\], and the KS statistic in \[tab:KS2\]. Both GIG-NR and CGGP exhibit a good fit to the `polblogs` dataset, where there does not seem to be evidence for a power-law exponent greater than 2. For the DBLP, both IG-NR and GIG-NR provide a good fit, while CGGP fails to capture adequately the degree distribution. The classification accuracy is also reported in \[tab:KS2\]. The classification accuracy is similar for IG-NR, GIG-NR and CGGP on `polblogs`. IG-NR and GIG-NR outperform other methods on the `DBLP` network. MMSB failed to capture both degree distributions and community structures, due to the large degree heterogeneity, a limitation already reported in previous articles [@Karrer2011; @Gopalan2013]. ------ ------------------- ----------- ------------------- ----------- $D$ Acc (%) $D$ Acc (%) IG 0.71$\pm$0.50 **94.28** **0.08**$\pm$0.03 72.46 GIG 0.14 $\pm$ 0.03 93.79 0.09$\pm$0.03 **76.58** CGGP **0.12**$\pm$0.03 94.12 0.33$\pm$0.02 57.49 MMSB 3.74$\pm$1.18 52.12 0.37$\pm$0.07 39.94 ------ ------------------- ----------- ------------------- ----------- : Average reweighted KS statistics and clustering accuracies. \[tab:KS2\] [^1]: Corresponding author, `juho.lee@stats.ox.ac.uk` [^2]: <https://toreopsahl.com/datasets/#newman2001> [^3]: <https://snap.stanford.edu/data/email-Enron.html> [^4]: <https://www.cise.ufl.edu/research/sparse/matrices/Pajek/internet.html> [^5]: <http://www.cise.ufl.edu/research/sparse/matrices/Newman/polblogs> [^6]: <https://snap.stanford.edu/data/com-DBLP.html>

--- abstract: 'This paper deals with a BMO Theorem for $\epsilon$ distorted diffeomorphisms from $\mathbb R^D$ to $\mathbb R^D$ with applications to manifolds of speech and sound.' author: - 'Steven B. Damelin [^1]' - 'Charles Fefferman, [^2]' - 'William Glover[^3]' title: 'A BMO theorem for $\epsilon$ distorted diffeomorphisms from $\mathbb R^D$ to $\mathbb R^D$ with applications to manifolds of speech and sound' --- 1991 AMS(MOS) Classification: 58C25, 42B35, 94A08, 94C30, 41A05, 68Q25, 30E05, 26E10, 68Q17. [**Keywords and phrases**]{} Measure, Diffeomorphism, Small Distortion, Whitney Extension, Manfold, Noise, Sound, Speech, Isometry, Almost Isometry, BMO. Introduction ============ Music, Speech and Mathematics ----------------------------- From the very beginning of time, mathematicians have been intrigued by the facinating connections which exist between music, speech and mathematics. Indeed, these connections were already in some subtle form in the writings of Gauss. The aim of this paper is to study estimates in measure for diffeomorphisms from $\mathbb R^D$ to $\mathbb R^D$, $D\geq 2$ of small distortion and provide an application to music and speech manifolds. Preliminaries ============= Fix a dimension $D\geq 2$. We work in $\mathbb R^D$. We write $B(x,r)$ to denote the open ball in $\mathbb R^D$ with centre $x$ and radius $r$. We write $A$ to denote Euclidean motions on $\mathbb R^D$. A Euclidean motion may be orientation-preserving or orientation reversing. We write $c$, $C$, $C'$ etc to denote constants depending on the dimension $D$. These expressions need not denote the same constant in different occurrences. For a $D\times D$ matrix, $M=(M_{ij})$, we write $|M|$ to denote the Hilbert-Schmidt norm $$|M|=\left(\sum_{ij}|M_{ij}|^2\right)^{1/2}.$$ Note that if $M$ is real and symmetric and if $$(1-\lambda)I\leq M\leq (1+\lambda)I$$ as matrices, where $0<\lambda<1$, then $$|M-I|\leq C\lambda.$$ This follows from working in an orthonormal basis for which $M$ is diagonal. One way to understand the formulas above is to think of $\lambda$ as being close to zero. See also (2.6) below. A function $f:\mathbb R^D\to \mathbb R$ is said to be BMO (Bounded mean oscillation )if there is a constant $K\geq 0$ such that, for every ball $B\subset \mathbb R^D$, there exists a real number $H_B$ such that $$\frac{1}{{\rm vol}\, B}\int_{B}|f(x)-H_B|dx\leq K.$$ The least such $K$ is denoted by $||f||_{{\rm BMO}}$. In harmonic analysis, a function of bounded mean oscillation, also known as a BMO function, is a real-valued function whose mean oscillation is bounded (finite). The space of functions of bounded mean oscillation (BMO), is a function space that, in some precise sense, plays the same role in the theory of Hardy spaces, that the space of essentially bounded functions plays in the theory of $Lp$-spaces: it is also called a John-Nirenberg space, after Fritz John and Louis Nirenberg who introduced and studied it for the first time. See [@J; @JN]. The John-Nirenberg inequality asserts the following: Let $f\in BMO$ and let $B\subset \mathbb R^D$ be a ball. Then there exists a real number $H_B$ such that $${\rm vol}\left\{x\in B:\, |f(x)-H_B|>C\lambda ||f||_{BMO}\right\}\leq \exp(-\lambda){\rm vol}\,B,\, \lambda \geq 1.$$ As a corollary of the John-Nirenberg inequality, we have $$\left(\frac{1}{{\rm vol}\, B}\int_{B}|f(x)-H_B|^4dx\right)^{1/4}\leq C\lambda ||f||_{BMO}.$$ There is nothing special about the 4th power in the above; it will be needed later. The definition of BMO, the notion of the BMO norm, the John-Nirenburg inequality (2.3) and its corollary (2.4) carry through to the case of functions $f$ on $\mathbb R^D$ which take their values in the space of $D\times D$ matrices. Indeed, we take $H_B$ in (2.2-2.4) to be a $D\times D$ matrix for such $f$. The matrix valued norms of (2.3-2.4) follow easily from the scalar case. We will need some potential theory. If $f$ is a smooth function of compact support in $\mathbb R^D$, then we can write $\Delta^{-1}f$ to denote the convolution of $f$ with the Newtonian potential. Thus, $\Delta^{-1}f$ is smooth and $\Delta(\Delta^{-1}f)=f$ on $\mathbb R^D$. We will use the estimate: $$\left\|\frac{\partial}{\partial x_i}\Delta^{-1}\frac{\partial}{\partial x_j}f\right\|_{L^2(\mathbb R^D)} \leq C||f||_{L^2(\mathbb R^D)},\, ij=1,...,D$$ valid for any smooth function $f$ with compact support. Estimate (2.5) follows by applying the Fourier transform. We will work with a positive number $\varepsilon$. We always assume that $\varepsilon\leq {\rm min}(1,C)$. An $\varepsilon$ distorted diffeomorphism of $\mathbb R^D$ is a one to one and onto diffeomorphism $\Phi:\mathbb R^D\to \mathbb R^D$ such as $$(1-\varepsilon)I\leq (\Phi'(x))^{T}(\Phi'(x))\leq (1+\varepsilon)I$$ as matrices. Thanks to (2.1), such $\Phi$ satisfy $$\left|(\Phi'(x))^{T}(\Phi'(x))-I\right|\leq C\varepsilon.$$ We end this section, with the following inequality from [@FD]: [**Approximation Lemma**]{} Let $\Phi:\mathbb R^D\to \mathbb R^D$ be an $\varepsilon$ distorted diffeomorphism. Then, there exists an Euclidean motion $A$ such that $$\left|\Phi(x)-A(x)\right|\leq C\varepsilon$$ for all $x\in B(0, 10)$. An overdetermined system ======================== We will need to study the following elemetary overdetermined system of partial differential equations. $$\frac{\partial \Omega_i}{\partial x_j}+\frac{\partial \Omega_j}{\partial x_i}=f_{ij}, i,j=1,...,D$$ on $\mathbb R^D$. Here, $\Omega_i$ and $f_{ij}$ are $C^{\infty}$ functions on $\mathbb R^D$. A result concerning (3.1) we need is: [**PDE Theorem**]{} Let $\Omega_1$,...,$\Omega_D$ and $f_{ij}$, $i,j=1,...,D$ be smooth functions on $\mathbb R^D$. Assume that (3.1) holds and suppose that $$||f_{ij}||_{L^2(B(0,4))}\leq 1.$$ Then, there exist real numbers $\Delta_{ij}$, $i,j=1,...,D$ such that $$\Delta_{ij}+\Delta_{ji}=0,\, \forall i,j$$ and $$\left\|\frac{\partial \Omega_i}{\partial x_j}-\Delta_{ij}\right\|_{L^2(B(0,1))}\leq C.$$ [**Proof**]{} From (3.1), we see at once that $$\frac{\partial \Omega_i}{\partial x_i}=\frac{1}{2}f_{ii}$$ for each $i$. Now, by differentiating (3.1) with respect to $x_j$ and then summing on $j$, we see that $$\Delta \Omega_i +\frac{1}{2}\frac{\partial}{\partial x_i}\left(\sum_j f_{jj}\right)=\sum_j \frac{\partial f_{ij}}{\partial x_j}$$ for each $i$. Therefore, we may write $$\Delta \Omega_i=\sum_j \frac{\partial}{\partial x_j} g_{ij}$$ for smooth functions $g_{ij}$ with $$||g_{ij}||_{L^2(B(0,4)}\leq C.$$ This holds for each $i$. Let $\chi$ be a $C^{\infty}$ cutoff function on $\mathbb R^D$ equal to 1 on $B(0,2)$ vanishing outside $B(0,4)$ and satisfying $0\leq \chi\leq 1$ everywhere. Now let $$\Omega_i^{{\rm err}}=\Delta^{-1}\sum_j\frac{\partial}{\partial x_j}\left(\chi g_{ji}\right)$$ and let $$\Omega_i^*=\Omega_i-\Omega_i^{err}.$$ Then, $$\Omega_i=\Omega_i^*+ \Omega_i^{err}$$ each $i$. $$\Omega_i^*$$ is harmonic on $B(0,2)$ and $$\left||\nabla \Omega_i^{{\rm err}}\right||_{L^2(B(0,2))}\leq C$$ thanks to (2.5). By (3.1, 3.2, 3.5, 3.7), we can write $$\frac{\partial \Omega_i^*}{\partial x_j}+\frac{\partial \Omega_j^*}{\partial x_i}=f_{ij}^*, i,j=1,...,D$$ on $B(0,2)$ and with $$\left||f_{ij}^*\right||_{L^2(B(0,2)}\leq C.$$ From (3.6) and (3.8), we see that each $f_{ij}^*$ is a harmonic function on $B(0,2)$. Consequently, (3.9) implies $$sup_{B(0,1)}\left|\nabla f_{ij}^*\right|\leq C.$$ From (3.8), we have for each $i,j,k$, $$\begin{aligned} && \frac{\partial^2 \Omega_i^*}{\partial x_{j}\partial x_k}+ \frac{\partial^2 \Omega_k^*}{\partial x_{i}\partial x_j}=\frac{\partial f_{ik}^*}{\partial x_j}; \frac{\partial^2 \Omega_i^*}{\partial x_{j}\partial x_k}+ \frac{\partial^2 \Omega_j^*}{\partial x_{i}\partial x_k}=\frac{\partial f_{ij}^*}{\partial x_k} \\ && \frac{\partial^2 \Omega_j^*}{\partial x_{i}\partial x_k}+ \frac{\partial^2 \Omega_k^*}{\partial x_{i}\partial x_j}=\frac{\partial f_{jk}^*}{\partial x_i}.\end{aligned}$$ Now adding the first two equations above and subtracting the last, we obtain: $$2\frac{\partial^2 \Omega_i^*}{\partial x_{j}\partial x_k}=\frac{\partial f_{ik}^*}{\partial x_j}+\frac{\partial f_{ij}^*}{\partial x_k}-\frac{\partial f_{jk}^*}{\partial x_i}$$ on $B(0,1)$. Now from (3.10) and (3.13), we obtain the estimate $$\left|\frac{\partial^2 \Omega_i^*}{\partial x_{j}\partial x_k}\right|\leq C$$ on $B(0,1)$ for each $i,j,k$. Now for each $i,j$, let $$\Delta_{ij}^*=\frac{\partial \Omega_i^*}{\partial x_j}(0).$$ By (3.14), we have $$\left|\frac{\partial \Omega_i^*}{\partial x_j}-\Delta_{ij}^*\right|\leq C$$ on $B(0,1)$ for each $i,j$. Recalling (3.5) and (3.7), we see that (3.16) implies that $$\left\|\frac{\partial \Omega_i}{\partial x_j}-\Delta_{ij}^*\right\|_{L^2(B(0,1))}\leq C.$$ Unfortunately, the $\Delta_{ij}^*$ need not satisfy (3.3). However, (3.1), (3.2) and (3.17) imply the estimate $$\left|\Delta_{ij}^*+\Delta_{ji}^*\right|\leq C$$ for each $i,j$. Hence, there exist real numbers $\Delta_{ij}$, $(i,j=1,...,D)$ such that $$\Delta_{ij}+\Delta_{ji}=0$$ and $$\left|\Delta_{ij}^*-\Delta_{ij}\right|\leq C$$ for each $i,j$. From (3.17) and (3.19), we see that $$\left\|\frac{\partial \Omega_i}{\partial x_j}-\Delta_{ij}\right\|_{L^2(B(0,1))}\leq C$$ for each $i$ and $j$. Thus (3.18) and (3.20) are the desired conclusions of the Theorem. $\Box$ A BMO Theorem ============= In this section, we prove the following: [**BMO Theorem 1**]{} Let $\Phi:\mathbb R^D\to \mathbb R^D$ be an $\varepsilon$ diffeomorphism and let $B\subset \mathbb R^D$ be a ball. Then, there exists $T\in O(D)$ such that $$\frac{1}{{\rm vol}\, B}\int_{B}\left|\Phi'(x)-T\right|dx\leq C\varepsilon^{1/2}.$$ [**Proof**]{} Estimate (4.1) is preserved by translations and dilations. Hence we may assume that $$B=B(0,1).$$ Now we know that there exists an Euclidean motion $A:\mathbb R^D\to \mathbb R^D$ such that $$\left|\Phi(x)-A(x)\right|\leq C\varepsilon$$ for $x\in B_{(0,10)}$. Our desired conclusion (4.1) holds for $\Phi$ iff it holds for $A^{-1}o\Phi$ (with a different T). Hence, without loss of generality, we may assume that $A=I$. Thus, (4.3) becomes $$\left|\Phi(x)-x\right|\leq C\varepsilon, x\in B(0,10).$$ We set up some notation: We write the diffeomorphism $\Phi$ in coordinates by setting: $$\Phi(x_1,...,x_D)=(y_1,...,y_D)$$ where for each $i$, $1\leq i\leq D$, $$y_i=\psi_i(x_1,...,x_D).$$ First claim: For each $i=1,...,D,$ $$\int_{B(0,1)}\left|\frac{\partial \psi_i(x)}{\partial x_i}-1\right|\leq C\varepsilon.$$ For this, for fixed $(x_2,...,x_D)\in B'$, we apply (4.4) to the points $x^{+}=(1,...,x_D)$ and $x^{-}=(1,...,x_D)$. We have $$\left|\psi_1(x^+)-1\right|\leq C\varepsilon$$ and $$\left|\psi_1(x^{-1})+1\right|\leq C\varepsilon.$$ Consequently, $$\int_{-1}^{1}\frac{\partial \psi_1}{\partial x_1}(x_1,...,x_D)dx_1\geq 2-C\varepsilon.$$ On the other hand, since, $$\left(\psi'(x)\right)^{T}\left(\psi'(x)\right)\leq (1+\varepsilon)I,$$ we have the inequality for each $i=1,...D$, $$\left(\frac{\partial \psi_i}{\partial x_i}\right)^2\leq 1+\varepsilon.$$ Therefore, $$\left|\frac{\partial \psi_i}{\partial x_i}\right|-1\leq \sqrt{1+\varepsilon}-1\leq \varepsilon.$$ Set $$I^{+}=\left\{x_1\in [-1,1]:\, \frac{\partial \psi_1}{\partial x_1}(x_1,...,x_D)-1\leq 0\right\},$$ $$I^{-1}=\left\{x_1\in [-1,1]:\, \frac{\partial \psi_1}{\partial x_1}(x_1,...,x_D)-1\geq 0\right\},$$ $$\Delta^{+}=\int_{I^+}\left(\frac{\partial \psi_1}{\partial x_1}(x_1,...,x_D)-1\right)dx_1$$ and $$\Delta^{-}=\int_{I^-}\left(\frac{\partial \psi_1}{\partial x_1}(x_1,...,x_D)-1\right)dx_1.$$ The inequality (4.8) implies that $-\Delta^{-1}\leq C\varepsilon+\Delta^{+}$. The inequality (4.9) implies that $$\frac{\partial \psi_1}{\partial x_1}-1\leq C\varepsilon.$$ Integrating the last inequality over $I^+$, we obtain $\Delta^+\leq C\varepsilon$. Consequently, $$\int_{-1}^{1}\left|\frac{\partial \psi_1}{\partial x_1}(x_1,...,x_D)-1\right|dx_1=\Delta^+ -\Delta^-\leq C\varepsilon.$$ Integrating this last equation over $(x_2,..., x_D)\in B'$ and noting that $B(0,1)\subset [-1,1]\times B'$, we conclude that $$\int_{B(0,1)}\left|\frac{\partial \psi_1}{\partial x_1}(x_1,...,x_D)-1\right|dx\leq C\varepsilon.$$ Similarly, for each $i=1,...,D$, we obtain (4.7). Second claim: For each $i,j=1,...,D, i\neq j$, $$\int_{B(0,1)}\left|\frac{\partial \psi_i(x)}{\partial x_j}\right|dx\leq C\sqrt{\varepsilon}.$$ Since $$(1-\varepsilon)I\leq (\Phi'(x))^{T}(\Phi'(x))\leq (1+\varepsilon)I,$$ we have $$\sum_{i,j=1}^D \left(\frac{\partial \psi_i}{\partial x_j}\right)^2\leq (1+C\varepsilon)D.$$ Therefore, $$\sum_{i\neq j}\left(\frac{\partial \psi_i}{\partial x_j}\right)^2 \leq C\varepsilon+\sum_{i=1}^D\left(1-\frac{\partial \psi_i}{\partial x_i}\right) \left(1+\frac{\partial \psi_i}{\partial x_i}\right).$$ Using (4.9) for $i$, we have $\left|\frac{\partial \psi_i}{\partial x_i}\right|+1\leq C$. Therefore, $$\sum_{i\neq j}\left(\frac{\partial \psi_i}{\partial x_j}\right)^2 \leq C\varepsilon+C\left\|\frac{\partial \psi_i}{\partial x_i}-1\right|.$$ Now integrating the last inequality over the unit ball and using (4.7), we find that $$\int_{B(0,1)}\sum_{i\neq j}\left(\frac{\partial \psi_i}{\partial x_j}\right)^2dx\leq C\varepsilon+\int_{B(0,1)} \left\|\frac{\partial \psi_i}{\partial x_i}-1\right|dx\leq C\varepsilon.$$ Consequently, by the Cauchy-Schwartz inequality, we have $$\int_{B(0,1)}\sum_{i\neq j}\left|\frac{\partial \psi_i}{\partial x_j}\right|dx\leq C\sqrt{\varepsilon}.$$ Third claim: $$\int_{B(0,1)}\left|\frac{\partial \psi_i}{\partial x_i}\right|dx\leq C\sqrt{\varepsilon}.$$ Since, $$\int_{B(0,1)} \left(\frac{\partial \psi_i}{\partial x_i}-1\right)^2dx\leq \int_{B(0,1)} \left|\frac{\partial \psi_i}{\partial x_i}-1\right|\left|\frac{\partial \psi_i}{\partial x_i}+1\right|dx,$$ using (4.7) and $\left|\frac{\partial \psi_i}{\partial x_i}\right|\leq 1+C\varepsilon$, we obtain $$\int_{B(0,1)}\left(\frac{\partial \psi_i}{\partial x_i}\right)^2dx\leq C\varepsilon.$$ Thus, an application of Cauchy Schwartz, yields (4.15). Final claim: By the Hilbert Schmidt definition, we have $$\begin{aligned} && \int_{B(0,1)}|\Psi'(x)-I|dx=\int_{B(0,1)}\left(\sum_{i,j=1}^D\left(\frac{\partial \psi_i}{\partial x_j}- \delta_{ij}\right)^2\right)^{1/2} \\ && \leq \int_{B(0,1)}\sum_{i,j=1}^D\left|\frac{\partial \psi_i}{\partial x_j}-\delta_{ij}\right|dx.\end{aligned}$$ The estimate (4.11) combined with (4.15) yields: $$\int_{B(0,1)}\left|\Phi'(x)-I\right|dx \leq C\varepsilon^{1/2}.$$ Thus we have proved (4.1) with $T=I$. The proof of the BMO Theorem 1 is complete. $\Box$ [**Corollary**]{} Let $\Phi:\mathbb R^D\to \mathbb R^D$ be an $\varepsilon$-distorted diffeomorphism. For each, ball $B\subset \mathbb R^D$, there exists $T_B\in O(D)$, such that $$\left(\frac{1}{{\rm vol}\, B}\int_{B}\left|\Phi'(x)-T\right|^4 dx\right)^{1/4}\leq C\varepsilon^{1/2}.$$ The proof follows from the first BMO Theorem just proved and the John Nirenberg inequality. (See (2.4). $\Box$. A Refined BMO Theorem ===================== We prove: [**BMO Theorem 2**]{} Let $\Phi:\mathbb R^D\to \mathbb R^D$ be an $\varepsilon$ diffeomorphism and let $B\in \mathbb R^D$ be a ball. Then, there exists $T\in O(D)$ such that $$\frac{1}{{\rm vol}\, B}\int_{B}\left|\Phi'(x)-T\right|dx\leq C\varepsilon.$$ [**Proof**]{} We may assume without loss of generality that $$B=B(0,1).$$ We know that there exists $T_B^*\in O(D)$ such that $$\left(\int_{B}|\Phi'(x)-T^*_{B}|^4 dx\right)^{1/4}\leq C\varepsilon^{1/2}.$$ Our desired conclusion holds for $\Phi$ iff it holds for $(T_{B}^{*})^{-1}o\Phi$. Hence without loss of generality, we may assume that $T_B^{*}=I$. Thus we have $$\left(\int_{B}|\Phi'(x)-I)|^4 dx\right)^{1/4}\leq C\varepsilon^{1/2}.$$ Let $$\Omega(x)=\left(\Omega_1(x),\Omega_2(x),....,\Omega_{D}(x)\right)=\Phi(x)-x,\, x\in \mathbb R^{D}.$$ Thus (5.3) asserts that $$\left(\int_{B(0,1)}\left|\nabla\Omega(x)\right|^4dx\right)^{1/4}\leq C\varepsilon^{1/2}.$$ We know that $$\left|(\Phi'(x))^{T}\Phi'(x)-I\right|\leq C\varepsilon,\, x\in \mathbb R^{D}.$$ In coordinates, $\Phi'(x)$ is the matrix $\left(\delta_{ij}+\frac{\partial \Omega_i(x)}{\partial x_j}\right)$, hence $\Phi'(x)^{T}\Phi'(x)$ is the matrix whose $ij$th entry is $$\delta_{ij}+\frac{\partial \Omega_j(x)}{\partial x_i}+\frac{\partial \Omega_i(x)}{\partial x_j} +\sum_{l}\frac{\partial \Omega_l(x)}{\partial x_i}\frac{\partial \Omega_l(x)}{\partial x_j}.$$ Thus (5.6) says that $$\left|\frac{\partial \Omega_j}{\partial x_i}+\frac{\partial \Omega_i}{\partial x_j} +\sum_{l}\frac{\partial \Omega_l}{\partial x_i}\frac{\partial \Omega_l}{\partial x_j}\right|\leq C \varepsilon$$ on $\mathbb R^{D},\, i,j=1,...,D.$ Thus, we have from (5.5), (5.7) and the Cauchy Schwartz inequality the estimate $$\left\|\frac{\partial \Omega_i}{\partial x_j}+\frac{\partial \Omega_j}{\partial x_i}\right\|_{L^2(B(0,10))}\leq C\varepsilon.$$ By the PDE Theorem, there exists, for each $i,j$, an antisymmetric matrix $S=(S)_{ij}$, such that $$\left\|\frac{\partial \Omega_i}{\partial x_j}-S\right\|_{L^2(B(0,1))}\leq C\varepsilon.$$ Recalling (5.4), this is equivalent to $$\left\|\Phi'-(I+S)\right\|_{L^2(B(0,1))}\leq C\varepsilon.$$ Note that (5.5) and (5.8) show that $$|S|\leq C\varepsilon^{1/2}$$ and thus, $$\left|\exp(S)-(I+S)\right|\leq C\varepsilon.$$ Hence, (5.9) implies via Cauchy Schwartz. $$\int_{B(0,1)}\left|\Phi'(x)-\exp(S)(x)\right|dx \leq C\varepsilon^{1/2}.$$ This implies the result because $S$ is antisymmetric which means that $\exp(S)\in O(D)$. $\Box$. A BMO Theorem for Diffeomorphisms of Small Distortion ===================================================== In this section, we prove the following theorem. [**Theorem**]{} Let $\Phi:\mathbb R^D\to \mathbb R^D$ be an $\varepsilon$ distorted diffeomorphism. Let $B\subset \mathbb R^D$ be a ball. Then, there exists $T_B\in O(D)$ such that for every $\lambda\geq 1$, $${\rm vol}\left\{x\in B:\, |\Phi'(x)-T_B|>C\lambda \varepsilon\right\}\leq \exp(-\lambda){\rm vol}\,(B).$$ Moreover, the result (6.1) is sharp in the sense of small volume if one takes a slow twist defined as follows: For $x\in \mathbb R^D$, let $S_x$ be the block-diagonal matrix $$\left( \begin{array}{llllll} D_1(x) & 0 & 0 & 0 & 0 & 0 \\ 0 & D_2(x) & 0 & 0 & 0 & 0 \\ 0 & 0 & . & 0 & 0 & 0 \\ 0 & 0 & 0 & . & 0 & 0 \\ 0 & 0 & 0 & 0 & . & 0 \\ 0 & 0 & 0 & 0 & 0 & D_r(x) \end{array} \right)$$ where, for each $i$, either $D_i(x)$ is the $1\times 1$ identity matrix or else $$D_i(x)=\left( \begin{array}{ll} \cos f_i(|x|) & \sin f_i(|x|) \\ -\sin f_i(|x|) & \cos f_i(|x|) \end{array} \right)$$ for a function $f_i$ of one variable. Now define for each $x\in \mathbb R^D$, $\Phi(x)=\Theta^{T}S_x(\Theta x)$ where $\Theta$ is any fixed matrix in $SO(D)$. One checks that $\Phi$ is $\varepsilon$-distorted, provided for each $i$, $t|f_i'(t)|<c\varepsilon$ for all $t\in [0,\infty)$. [**Proof**]{} The theorem follows from the BMO Theorem 2 and the John Nirenburg inequality. The sharpness can be easily checked. $\Box$. On the Approximate and Exact Allignment of Data in Euclidean Space, Speech and Music Manifolds ============================================================================================== Approximate and Exact Allignment of Data ---------------------------------------- The following is a classical question in Euclidean Geometry, see for example [@WW]: Suppose we are that given two sets of distinct data points in Euclidean $D\geq 2$ space, say from two manifolds. We do not know what the manifolds are apriori but we do know that the pairwise distances between the points are equal. Does there exist an isometry (distance preserving map) that maps the one set of points 1-1 onto the other. This is a fundamental question in data analysis for most often, we are only given sampled function points from two usually unknown manifolds and we seek to know what can be said about the manifolds themselves. A typical example might be a face recognition problem where all we have is multiple finite images of people’s faces from various views. An added complication in the above question is that in general, we are not given exact distances between function value points. We have noise and so we need to demand that instead of the pairwise distances being equal, they should be “ close” in some reasonable metric. It is well known, see [@WW] that any isometry of a subset of a Euclidean space into the space can be extended to an isometry of that Euclidean space onto itself. Some results on almost isometries in Euclidean spaces can be found for example in [@J], [@ATV] and [@FD]. Some results on almost isometries in Euclidean spaces can be found for example in [@J] and [@ATV]. In [@FD], the following two theorems are established which tell us alot about how to handle manifold identification when the point set function values given are not exactly equal but are close. [**Theorem**]{} Given $\varepsilon>0$ and $k\geq 1$, there exists $\delta>0$ such that the following holds. Let $y_1,...y_k$ and $z_1,...,z_k$ be points in $\mathbb R^D$. Suppose $$(1+\delta)^{-1}\leq \frac{|z_i-z_j|}{|y_i-y_j|}\leq 1+\delta,\, i\neq j.$$ Then, there exists a Euclidean motion $\Phi_0:x\to Tx+x_0$ such that $$|z_i-\Phi_0(y_i)|\leq \varepsilon {\rm diam}\left\{y_1,...,y_k\right\}$$ for each $i$. If $k\leq D$, then we can take $\Phi_0$ to be a proper Euclidean motion on $\mathbb R^{D}$. [**Theorem**]{} Let $\varepsilon>0$, $D\geq 1$ and $1\leq k\leq D$. Then there exists $\delta>0$ such that the following holds: Let $E:=y_1,...y_k$ and $E':=z_1,...z_k$ be distinct points in $\mathbb R^D$. Suppose that $$(1+\delta)^{-1}\leq \frac{|z_i-z_j|}{|y_i-y_j|}\leq (1+\delta),\, 1\leq i,j\leq k,\, i\neq j.$$ Then there exists a diffeomorphism, 1-1 and onto map $\Psi:\mathbb R^D\to \mathbb R^D$ with $$(1+\varepsilon)^{-1}\leq \frac{|\Psi(x)-\Psi(y)|}{|x-y|}\leq (1+\varepsilon),\, x,y\in \mathbb R^D,\, x\neq y$$ satisfying $$\Psi(y_i)=z_i,\, 1\leq i\leq k.$$ Given the two theorems above, we now need to ask ourselves. Can we take, in any particular data application, a smooth $\varepsilon$ distortion and approximate it by an element of $O(D)$. Clearly this is very important. We understand that the results of this paper tell us that at least in measure, the derivative of a smooth $\varepsilon$ distortion may be well approximated by an element of $O(D)$. Speech and Music manifolds -------------------------- Recently, see for example, [@DM] and the references cited therein there has been much interest in geometrically motivated dimensionality reduction algorithms. The reason for this is that these algorithms exploit low dimensional manifold structure in certain natural datasets to reduce dimensionality while preserving categorical content. In [@JN1], the authors motivated the existence of low dimensional music and speech manifold structure to the existence of certain rigid motions approximating smooth distortions of voice and speech sounds maps. The theorems proved in this paper and in [@FD] provide a fascinating insight into these very interesting questions. Acknowledgement --------------- The second author thanks Charlie Fefferman for wonderfully inspiring collaborations. The second author also thanks Princeton for financial support. The second author acknowledges inspiring collaborations with William Glover on his “passion” for music (piano in particular) and for introducing him to the beautiful world of beats, movements, scales, measures and time signatures. [99]{} P. Alestalo, D. A. Trotsenko and J. Vaisala, [*The linear extension property of bi-Liptchitz mappings*]{}, Siberian Math, J. 44 (2003), (6), pp 959-968. S. B. Damelin and W. Miller, [*The Mathematics of Signal Processing*]{}, Cambridge University Press (2012). C. Fefferman and S. B. Damelin, [*Extensions, interpolation and matching in $\mathbb R^D$*]{}, preprint, arxiv 1411.2451. H. Helmhotz, [*On the sensations of tone as a physiologial basis for the theory of music*]{}, New York, Dover Publications. G. Holton, [*The story of sound*]{}, New York, Harcourt. F. John, [*Rotation and Strain*]{}, Commun. Pure Appl. Math, [**14**]{}(1961), pp 391-413. F. John and L. Nirenberg, [*On functions of bounded mean oscillation*]{}, Commun. Pure Appl. Math, [**14**]{}(3)(1961), pp 415-426. A. Jansen and P. Niyogi, [*Intrinsic Fourier analysis on the manifold of speech sounds*]{}, Proceedings of the International Conference on Acoustics, Speech, and Signal Processing, Toulose, France, 2006. C. Miller, [*The science of musical sounds*]{}, New York, The McMillian Company 1916. G. Reuben, [*What is sound?*]{}, Chicago, Benefic Press. J. H. Wells, L. R. Williams, [*Embeddings and extensions in analysis*]{}, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84, Springer-Verlag, New York-Heidelberg, 1975. [^1]: Mathematical Reviews, 416 Fourth Street, Ann Arbor, MI 48103, USA email: damelin@umich.edu [^2]: Department of Mathematics; Fine Hall, Washington Road, Princeton NJ 08544-1000 USA, email: cf@math.princeton.edu [^3]: Department of Music, Albany State University, Albany, GA. email: will25655@yahoo.com

--- abstract: 'Recently, Lagos et al. (Nature Nanotechnology $\textbf{4}$, 149 (2009)) reported the discovery of the smallest possible silver square cross-section nanotube. A natural question is whether similar carbon nanotubes can exist. In this work we report *ab initio* results for the structural, stability and electronic properties for such hypothetical structures. Our results show that stable (or at least metastable) structures are possible with metallic properties. They also show that these structures can be obtained by a direct interconversion from SWNT(2,2). Large finite cubane-like oligomers, topologically related to these new tubes were also investigated.' author: - P A S Autreto - S B Legoas - M Z S Flores - D S Galvao title: 'Carbon Nanotube with Square Cross-section: An *Ab Initio* Investigation' --- Introduction ============ The study of the mechanical properties of nanoscale systems presents new theoretical and experimental challenges [@ray; @book]. The arrangements of atoms at nano and macro scales can be quite different and affect electronic and mechanical properties. Of particular interest are the structures that do not exist at macroscale but can be formed (at least as metastable states) at nanoscale, specially when significant stress/strain is present. Examples of these cases are atomic suspended chains [@onishi; @ruit; @alloy] and helical nanowires [@tosatti]. More recently [@agtube], it was discovered the smallest metal (silver) nanotube from high resolution transmission electron microscopy (HRTEM) experiments. These tubes are spontaneously formed during the elongation of silver nanocontacts. *Ab initio* theoretical modeling [@agtube] suggested that the formation of these hollow structures requires a combination of minimum size and high gradient stress. This might explain why these structures had not been discovered before in spite of many years of investigations. Even from theoretical point of view, where low stress regimes and small structures have been the usual approach, no study predicted their existence. The unexpected discovery of this new family of nanotubes suggests that such other ‘exotic’ nanostructures may exist. One question that naturally arises is whether carbon-based similar nanotubes (i.e., carbon nanotubes with square cross-section - CNTSC) could exist (figure \[figure1\]). From the topological point of view CNTSC tubes would require carbon atoms arranged in multiple square-like configurations. Molecular motifs satisfying these conditions, the so-called cubanes, do exist and they are stable at room temperature (figure \[figure2\]) [@cub1]. Cubane (C$_8$H$_8$) is a hydrocarbon molecule consisting of 8 carbon atoms in an almost perfect cubic arrangement (thus its name). Hydrogen atoms are bonded to each carbon atom (figure \[figure2\]), completing its four-bond valency. The 90 degrees angles among carbon atoms create a very high strained structure. Because of this unusual carbon hybridization, cubane was considered a ‘platonic’ hydrocarbon and believed to be almost impossible to be synthesized [@cub1]. However, in 1964 Eaton and Cole [@cub2] achieved its breakthrough synthesis. Since then the cubane chemistry evolved quickly [@cub1; @cub3]. Solid cubane [@cub4] proved to be remarkably stable and polymers containing up to substituted 40 cubanes units have been already synthesized [@cub1]. However, up to now no tubular structure has been reported [@cub5; @cub6]. ![(color online). Structural nanotube models. Frontal and lateral views: (a) Silver tube [@agtube], (b) SWNT(2,2), and (c) CNTSC. See text for discussions.[]{data-label="figure1"}](figure1.eps) In this work we have theoretically investigated structural, stability and electronic properties of CNTSC tubes. We have considered infinite (considering periodic boundary conditions) and finite (oligomers up to ten square units, figure \[figure2\]) structures. Methodology =========== We have carried out *ab initio* total energy calculations in the framework of density functional theory (DFT), as implemented in the DMol$^3$ code [@dmol3]. Exchange and correlation terms were treated within the generalized gradient (GGA) functional by Perdew, Burke, and Ernzerhof [@pbe]. Core electrons were treated in a non-relativistic all electron implementation of the potential. A double numerical quality basis set with polarization function (DNP) was considered, with a real space cutoff of 3.7 Å. The tolerances of energy, gradient, and displacement convergence were 0.00027 eV, 0.054 eV/Å and 0.005 Å, respectively. ![(color online). Cubane and its ‘polymerized’ units. The label refers to the number of square cross-sections in the structure. We considered structures from cubane up to n=10. Stick models, C and H atoms are in grey and white colors, respectively.[]{data-label="figure2"}](figure2.eps) Initially we optimized the CNTSC unit cell with fixed $\textit{a}$ and $\textit{b}$ parameters set to 20 Å in order to ensure isolated (non-interacting) structures. The axial $\textit{c}$ lattice parameter was varied, and the total energy per unit cell calculated. All internal atomic coordinates were free to vary. Total energy versus unit cell volume was fitted using the Murnaghan procedure [@murnaghan] to obtain the equilibrium *c* lattice parameter. For comparison purposes we have also considered graphite, diamond and carbon nanotube SWNT(2,2). SWNT(2,2) was chosen because, although ultra-small carbon nanotubes (USCNTs) have been theoretically investigated [@cox; @scipioni; @yuan; @kuzmin], it remains the smallest CNT experimentally observed with an estimated diameter of 3 Å [@22prl1; @22prl2]. Graphite and diamond were also included in our study, because they are the two most stable carbon form and to provide a benchmark for the relative stability between the different tubes and these structures. Results and Discussions ======================= The results are presented in table \[table1\]. As expected, graphite is the structure with the lowest energy (most stable), followed by diamond, SWNT(2,2) and CNTSC, respectively. Although the CNTSC energy per atom is relatively high (in part due to the strained C-C bonds, as in cubanes) its relative energy difference to SWNT(2,2) (0.0384 Ha) is similar to the difference between SWNT(2,2) and cubic diamond (0.0395 Ha). (a) [(b)]{} (c) (d) ---------------------- ----------- ----------- ----------- ----------- Energy/atom (Ha) $-38.085$ $-38.081$ $-38.041$ $-38.003$ Lattice parameters: *a* (Å) $2.46$ $3.57$ $20.0$ $20.0$ *c* (Å) $6.80$ $3.57$ $2.53$ $1.62$ C-C bond-length (Å): $1.423$ $1.537$ $1.447$ $1.580$ $1.464$ $1.616$ : \[table1\]DMol$^3$ results for crystalline carbon allotropic structures: (a) Graphite, (b) Cubic diamond, (c) SWNT(2,2), and (d) CNTSC. ![Binding energy per unit cell as a function of axial *c* lattice parameter for SWNT(2,2) (circles) and CNTSC (squares). It is also shown its interconversion curve (triangles). See text for discussions.[]{data-label="figure3"}](figure3.eps) In figure \[figure3\] we present the binding energy per unit cell. As a direct comparison it is not possible because the number of atoms in the minimum unit cell is different for SWNT(2,2) and CNTSC (eight and four, respectively), we used a double CNTSC unit cell. As can be seen from figure \[figure3\], the results suggest that stable (or at least metastable, as a well defined minimum is present) CNTSC structures are possible. Our results also suggest that a direct interconversion from SWNT(2,2) to CNTSC is unlikely to occur via axial(longitudinal) stretching. The extrapolation of the stretched SWNT(2,2) curve (figure \[figure3\], circle data points) could be misleading suggesting that it would be possible an interception with the stretched CNTSC curve (figure \[figure3\], square data points). However, this did not occur, the SWNT(2,2) can not preserve its tubular topology when its c-value is beyond 3.2 Å. We then investigated whether if an assisted interconversion would be possible, in our case we considered a continuously decrease of the tube radii value (in order to mimic an applied external (radially) pressure) while keeping the tube free to expand/contract longitudinally (see figure \[figure4\] and video1 [@EPAPS]). ![(color online). Snapshots from the axial compression process, showing the interconversion of SWNT(2,2) to CNTSC. (a) Initial SWNT(2,2), (b) and (c) intermediates, and (c) final CNTSC structures.[]{data-label="figure4"}](figure4.eps) We have performed these calculations starting from an optimized SWNT(2,2) unit cell and then continuously decreasing its radii value and re-equilibrating the system and measuring the new c-values (figure \[figure3\], triangle data points). Our results show that under these conditions there is a pathway that permits a direct interconversion from SWNT(2,2) to CNTSC. In figure \[figure4\] we present a sequence of snapshots from the simulations representing the interconversion process. The strain energy injected into the system by the radial compression (figure \[figure4\](a)) produces a $\textit{c}$-lattice expansion, leading to a structural transition (figure \[figure4\](d)). The compression process produces new C-C bonds followed by carbon rehybridizations. The processes is better visualized in the supplementary materials (video1) [@EPAPS]. ![Band structure and total density of states (in electrons/Ha) results for the (a) SWNT(2,2) and (b) carbon square-cross-section CNTSC. Energy is relative to Fermi level (dashed horizontal lines). Primitive unit cells have eight and four carbon atoms for SWNT(2,2) and CNTSC, respectively.[]{data-label="figure5"}](figure5.eps){width="6cm"} We then proceeded with the CNTSC electronic structure analysis. In figure \[figure5\] we displayed the band structure and the density of states (DOS) for SWNT(2,2) and CNTSC tubes. Both structures present a finite DOS at the Fermi energy, characteristic of metallic regimes. Although CNTSC exhibits non-usual carbon hybridization it follows the general trends that small diameter carbon nanotubes are metallic [@metallic; @kamal]. It is possible that synthetic methods used to produce CNTs (such as laser ablation, chemical vapor deposition and arc discharge [@book]) could also produce CNTSC, specially inside CNTs of different chiralities, as in the case of SWNT(2,2) [@22prl1; @22prl2]. Another possibility could be a polymeric synthetic approach, like the topochemical ones to produce carbon nanotube of specific types that have been recently discussed in the literature [@raytopo; @jacstopo]. Considering that cubane molecules and their polymers exist and are stable, we decided to investigate the local stability and endothermicity of cubane-like tubular oligomers that are topologically related to CNTSC. We carried out DMol$^3$ calculations for the molecular structures displayed in figure \[figure2\]. The terminal C atoms were passivated with H atoms. $\textit{n}$ $e_t(n)$ -------------- -------------- [2]{} -38.647 [3]{} [-38.429]{} [4]{} [-38.323]{} [5]{} [-38.259]{} [6]{} [-38.216]{} [7]{} [-38.186]{} [8]{} [-38.163]{} [9]{} [-38.145]{} [10]{} [-38.131]{} [$\vdots$]{} [$\vdots$]{} [CNTSC]{} [-38.003]{} : \[table2\]Total energy per carbon atom (in Ha) for the structures shown in figure \[figure2\]. The corresponding value for the infinite structure is also presented. In table \[table2\] we present the results for the total energy per carbon atom as a function of the number of square cross-sections. Our results show that although the oligomer formation would require an endothermal (energetically assisted) process, the structures are stable and the energy per carbon atom converges assyntotically to the corresponding value of the infinite tube (see also supplementary materials [@EPAPS]). In summary, based on a recent discovery of the smallest possible silver nanotube with a square cross-section [@agtube], we have investigated whether a similar carbon-based structure could exist. We have used *ab initio* methodology to investigate the structural, stability and electronic properties of carbon nanotubes with square cross-section (CNTSC). Our results show that stable (or at least metastable) CNTSC (finite and infinite) structures can exist. They also show that it is possible to convert SWNT(2,2) to CNTSC under radial contraction. CNTSCs should share most of the general features exhibited by ’standard’ nanotubes. Although the CNTSCs have not yet been observed, we believe our results had proven their feasibility.We hope the present work will estimulate further works for these new family of carbon nanotubes. This work was supported in part by the Brazilian Agencies CNPq, CAPES and FAPESP. DSG thanks Prof. D. Ugarte for helpful discussions. [29]{} bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{} , , , , , , , ****, (). , ** , CRC Press, FL, USA, . , , , ****, (). , , , ****, (). , , , , , , ****, (). , , , ****, (). , , , , , , ****, (). , ****, (). , , ****, (). , ****, (). , , , , , ****, (). , , , , , ****, (). , ****, (). , ****, (). , , , ****, (). , ****, (). , ****, (). , , , ****, (). , ****, (). , ****, (). , , , ****, (). , , , , , , ****, (). See supplementary material at \[URL will be inserted by AIP\] for windows powerpoint file and movies in wmv format, of simulations discussed in the paper. , , , , ****, (). , ****, (). , , , , ****, (). , , , ****, ().

--- abstract: 'Criticisms of so called ‘subjective probability’ come on the one hand from those who maintain that probability in physics has only a frequentistic interpretation, and, on the other, from those who tend to ‘objectivise’ Bayesian theory, arguing, e.g., that subjective probabilities are indeed based ‘only on private introspection’. Some of the common misconceptions on subjective probability will be commented upon in support of the thesis that coherence is the most crucial, universal and ‘objective’ way to assess our confidence on events of any kind.' author: - 'G. D’AGOSTINI' title: | ROLE AND MEANING OF SUBJECTIVE PROBABILITY\ SOME COMMENTS ON COMMON MISCONCEPTIONS --- Introduction ============ The role of scientists is, generally speaking, to understand Nature, in order to forecast as yet unobserved (‘future’) events, independently of whether or not these events can be influenced. In laboratory experiments and all technological applications, observations depend on our intentional manipulation of the external world. However, other scientific activities, like astrophysics, are only observational. Nevertheless, to claim that cosmology, climatology or geophysics are not Science, because “experiments cannot be repeated” - as pedantic interpreters of Galileo’s scientific method do - is, in my opinion, short-sighted (for a recent defence of this strict Galilean point of view, advocating ‘consequently’ frequentistic methods, see Ref. [@Giunti]). The link between past observations and future observations is provided by theory (or model). It is accepted that quantitative (and, often, also qualitative) forecasting of future observations is invariably uncertain, from the moment that we define sufficiently precisely the details of the future events. The uncertainty may arise because we are not certain about the parameters of the theory (or of the theory itself), and/or about the initial state and boundary conditions of the phenomenon we want to describe. But it may also be due to the stochastic nature of the theory itself, which would produce uncertain predictions even if all parameters and boundary conditions [*were*]{} precisely known. Nevertheless, the constant state of uncertainty does not prevent us from doing science. As Feynman wrote, “it is scientific only to say what is more likely and what is less likely”.[@Feynman] This observation holds not only for observations, but also for the values of physical quantities (i.e. parameters of the theory which have effect on the real observations). And indeed, physicists find probabilistic statements about, for example, top quark mass or gravitational constant very natural,[@Maxent98] and several equivalent expressions are currently used, such as “to be more or less [*confident*]{}”, “to consider something more or less [*probable*]{}, or more or less [*likely*]{}”, “to [*believe*]{} more or less something”. However, the subjective definition of probability, the only one consistent with the above expressions, is usually rejected because of educational bias according to which “the only scientific definition of probability is the frequentistic one,” “quantum mechanics only allows the frequency based definition of probability,” “probability is an objective property of the physical world,” etc. In this paper I will comment on these and other objections against the so called ‘subjective Bayesian’ point of view. Indeed, some criticisms come from ‘objective Bayesians’, who have been, traditionally, in a clear majority during this workshop series. I don’t expect to solve these debates in this short contribution, especially considering that many aspects of the debate are of a psychological and sociological nature. Neither will I be able to analyse in detail every objection or to cite all the counter-arguments. I prefer, therefore, to focus here only on a few points, referring to other papers [@YR; @anxiety; @ajp] and references therein for points already discussed elsewhere. Subjective Probability and Role of Coherence ============================================ The main aim of subjective probability is to recover the intuitive concept of probability as degree of belief. Probability is then related to uncertainty and not (only) to the outcomes of repeated experiments. Since uncertainty is related to knowledge, probability is only meaningful as long as there are human beings interested in knowing (or forecasting) something, no matter if “the events considered are in some sense [*determined*]{}, or known by other people.”[@deFinetti] Since - fortunately! - we do not share identical states of information, we are in different conditions of uncertainty. Probability is therefore only and always conditional probability, and depends on the different subjects interested in it (and hence the name [*subjective*]{}). This point of view about probability is not related to a single evaluation rule. In particular, symmetry arguments and past frequencies, as well as their combination properly weighted by means of Bayes’ theorem, can be used. Since beliefs can be expressed in terms of betting odds, as is well known and done in practice, betting odds can be seen as the most general way of making relative beliefs explicit, independently of the kind of events one is dealing with, or of the method used to define the odds. For example, everybody understands Laplace’s statement concerning Saturn’s mass, that “it is a bet of 10000 to 1 that the error of this result is not 1/100th of its value.”[@Sivia] I wish all experimental results to be provided in these terms, instead of the misleading [@CLWdag] “such and such percent CL’s.” What matters is that the bet must be reversible and that no bet can be arranged in such a way that one wins or loses with certainty. The second condition is a general condition concerning bets. The first condition forces the subject to assess the odds consistently with his/her beliefs and also to accept the second condition: once he/she has fixed the odds, he/she must be ready to bet in either direction. Coherence has two important roles: the first is, so to speak, moral, and forces people to be honest; the second is formal, allowing the basic rules of probability to be derived as theorems, including the formula relating conditional probability to probability of the conditionand and their joint probability (note that, consistent with the use of probability in practice and with the fact that in a theory where only conditional probabilities matter, it makes no sense to have a formula that [*defines*]{} conditional probability, see e.g. Section 8.3 of Ref. [@YR] for further comments and examples). Once coherence is included in the subjective Bayesian theory, it becomes evident that ‘subjective’ cannot be confused with ‘arbitrary’, since all ingredients for assessing probability must be taken into account, including the knowledge that somebody else might assess different odds for the same events. Indeed, the coherent subjectivist is far more responsible (and more “objective”, in the sense that ordinary parlance gives to this word) than those who blindly use standard ‘objective’ methods (see examples in Ref. [@YR]). Another source of objections is the confusion between ‘belief’ and ‘imagination’, for which I refer to Ref. [@anxiety]. Subjective Probability, Objective Probability, Physical Probability =================================================================== To those who insist on objective probabilities I like to pose practical questions, such as how they would evaluate probability in specific cases, instead of letting them pursue mathematical games. Then it becomes clear that, at most, probability evaluations can be intersubjective, if we all share the same education and the same real or conventional state of information. The probability that a molecule of $N_2$ at a certain temperature has a velocity in a certain range [*seems*]{} objective: take the Maxwell velocity p.d.f., make an integral and get a number, say $p=0.23184\ldots$ This mathematical game gets immediately complicated if one thinks about a real vessel, containing real gas, and the molecule velocity measured in a real experiment. The precise ‘objective’ number obtained from the above integral might no longer correspond to our confidence that the velocity is really in that interval. The idealized “physical probability” $p$ can easily be a misleading “metaphysical” concept which does not correspond to the confidence of real situations. In most cases, in fact, $p$ is a number that one gets from a model, or a free parameter of a model. Calling $E$ the [*real*]{} event and $P(E)$ the probability we attribute to it, the idealized situation corresponds to the following conditional probability: $$P(E\,|\,\mbox{Model}\rightarrow p) = p\,.$$ But, indeed, our confidence on $E$ relies on our confidence on the model: $$P(E\,|\,I) = \sum_{\mbox{Models}}P(E\,|\,I,\mbox{Model}\rightarrow p)\cdot P(\mbox{Model}\rightarrow p\,|\,I) \,, \label{eq:sumModels}$$ where $I$ stands for a background state of information which is usually implicit in all probability assessments. Describing our uncertainty on the [*parameter*]{} $p$ by a p.d.f. $f(p)$ (continuity is assumed for simplicity), the above formula can be turned into $$P(E\,|\,I)=\int_0^1\!P(E\,|\,p,I)\cdot f(p\,|\,I)\,\mbox{d}p\,. \label{eq:intp}$$ The results of Eqs. (\[eq:sumModels\]) and (\[eq:intp\]) really express the meaning of probability, describing our beliefs, and upon which (virtual) bets can be set (‘virtual’ because it is well known that real bets are delicate decision problems of which beliefs are only one of the ingredients). For those who still insist that probability is a property of the world, I like to give the following example, readapted from Ref. [@Scozzafava]. Six externally indistinguishable boxes each contain five balls, but with differing numbers of black and white balls (see Ref. [@ajp] for details and for a short introduction of Bayesian inference based on this example). One box is chosen at random. What will be its white ball content? If we extract a ball, what is the probability that it will be white? Then a ball is extracted and turns out to be white. The ball is reintroduced into the box, and the above two questions are asked again. As a simple application of Bayesian inference, the probability of extracting a white ball in the second extraction becomes $P(E_2=W)=73\%$, while it [*was*]{} $P(E_1=W)=50\%$ for the first extraction. One does not need to be a Bayesian to solve this simple text book example, and everybody will agree on the two values of probability (we have got “objective” results, so to say). But it is easy to realize that these probabilities do not represent a ‘physical property of the box’, but rather a ‘state of our mind’, which changes as the extractions proceed. In particular, ‘measuring’, or ‘verifying’, that $P(E_2=W)=73\%$ using the relative frequency makes no sense. We could imagine a large number of extractions. It is easy to understand, given our prior knowledge of the box contents, that the relative frequencies “will tend” (in a probabilistic sense) to $\approx 20\%$, $\approx 40\%$, $\approx 60\%$, $\approx 80\%$, or $\approx 100\%$, but ‘never’ 73% [^1]. This certainly appears to be a paradox to those who agree that $P(E_2=W)=73\%$ is the ‘correct’ probability, but still maintain that probability as degree of belief is a useless concept. In this simple case the six a priori probabilities $p_i=i/5$, with $i=0,1,\ldots,5$, can be seen as the possible “physical probabilities”, but the “real” probability which determines our confidence on the outcome is given by a discretized version of Eq. (\[eq:intp\]), with $f(p_i)$ changing from one extraction to the next. I imagine that at this point some readers might react by saying that the above example proves that only the frequentistic definition of probability is sensible, because the relative frequency will tend for $n\rightarrow \infty$ to the ‘physical probability’, identified in this case by the white ball ratio in the box. But this reaction is quite naïve. First, a definition valid for $n\rightarrow\infty$ is of little use for practical applications (“In the long run we are all dead”[@Keynes]). Second, it is easy to show [@ajp] that, for the cases in which the ‘physical probability’ can be checked and the number of extraction is finite, though large, the convergence behaviour of the frequency based evaluation is far poorer than the Bayesian solution and can also be in paradoxical contradiction with the available status of information. Moreover, only the Bayesian theory answers consistently, in unambiguous probabilistic terms, the legitimate question “what is the box content?”, since the very concept of probability of hypotheses is banished in the frequentistic approach. Similarly, only in the Bayesian approach does it make sense to express in a logical, consistent way the confidence on the different causes of observed events and on the possible values of physics quantities (which are unobserved entities). To state that “in high energy physics, where experiments are repeatable (at least in principle) the definition of probability normally used”[@PDG] is to ignore the fact that the purpose of experiments is not to predict which electronic signal will come out next from the detector (following the analogy of the six box example), but rather to narrow the range in which we have high confidence that the physics quantities lie.[^2] Observed Frequencies, Expected Frequencies, Frequentistic Approach and Quantum Mechanics ======================================================================================== Many scientists think they are frequentists because they are used to assessing their beliefs in terms of expected frequencies, without being aware of the implications for a sane person of sticking strictly to frequentistic ideas. Certainly, past frequencies can be a part of the information upon which probabilities can be assessed [@ajp; @CLWdag]. Similarly, probability theory teaches us how to predict future frequencies from the assessment of beliefs, under well defined conditions. But identifying probability with frequency is like confusing a table with the English word ‘table’. This confusion leads some authors, because they lack other arguments to save the manifestly sinking boat of the frequentistic collection of [*adhoc-eries*]{}, to argue[^3] that “probability in quantum mechanics is frequentistic probability, and is defined as long-term frequency. Bayesians will have to explain how they handle that problem, and they are warned in advance.”[@James] Probability deals with the belief that an event may happen, given a particular state of information. It does not matter if the fundamental laws are ‘intrinsically probabilistic’, or if it is just a limitation of our present ignorance. The impact on our minds remains the same. If we think of two possible events resulting from a quantum mechanics experiment, and say (after computing no matter how complicated calculations to also take into account detector effects) that $P(E_1)\gg P(E_2)$, this simply means that we feel more [*confident*]{} in $E_1$ than in $E_2$, or that we will be [surprised]{} if $E_2$ happens instead of $E_1$. If we have the opportunity to repeat the experiment we believe that events of the ‘class’ $E_1$ will happen more frequently than events of class $E_2$. This is what we find carefully reviewing the relevant literature and discussing with theorists: probability is ‘probability’, although it might be expressed in terms of expected frequencies, as discussed above. Take for example Hawking’s [*A Brief History of Time*]{},[@Hawking] (which a statistician[@Berry] said should be called ‘a brief history of beliefs’, so frequently do the words belief, believe and synonyms appear in it). For example: “In general, quantum mechanics does not predict a single definite result for an observation. Instead, it predicts a number of different possible outcomes and tells us how likely each of these is” [@Hawking]. Looking further to the past, it is worth noting the concept of “degree of truth” introduced by von Weizsäcker, as reported by Heisenberg [@Heisenberg]. It is difficult to find any difference between this concept and the usual degree of belief, especially because both Heisenberg and von Weizsäcker were fully aware that “nature is earlier than man, but man is earlier than natural science” [@Heisenberg], in the sense that science is done by our brains, mediated by our senses. It is true that, reading some text books on quantum mechanics one gets the idea that “probability is frequentistic probability”[@James], but one should remember the remarks at the beginning of this section, and the fact that many authors have used, uncritically, the dominant ideas on probability in the past decades. But some authors also try to account for probability of [*single*]{} events, instead of ‘repeated events’, and have to admit that this is possible if probability is meant as degree of belief. In conclusion, invoking intriguing fundamental aspects of quantum mechanics in the discussion of inferential frameworks shows little awareness of the real issues involved in the two classes of problems. First, the debate about the interpretations of quantum mechanics is far from being settled [@tHooft]. Second, as far as ‘natural science’ is concerned, it doesn’t really matter if nature is deeply deterministic or probabilistic, as eloquently said by Hume: “Though there is no such thing as Chance in the world; our ignorance of the real cause of any event has the same influence on the understanding, and begets a like species of belief or opinion.”[@Hume] Who is Afraid of Subjective Bayesian Theory? ============================================ “It is curious that, even when different workers are in substantially complete agreement on what calculations should be done, they may have radically different views as to what we are actually doing and why we are doing it.” [@Jaynes] It is indeed surprising that strong criticisms of subjective probability come from people who essentially agree that probability represents “our degree of confidence” [@Jeffreys] and that Bayes’ theorem is the proper inferential tool. I would like to comment here on criticisms (and invitations to convert…) which I have received from objective Bayesian friends and colleagues, and which can be traced back essentially to the same source. [@Jaynes] The main issue in the debate is the choice of the [*prior*]{} to enter in the Bayesian inference. I prefer subjective priors because they seem to me to correspond more closely to the spirit of the Bayesian theory and the results of the methods based on them are more reliable and never paradoxical [@anxiety]. Nevertheless, I agree, in principle, that a “concept of a ‘minimal informative’ prior specification – appropriately defined!”[@BS] can be useful in particular applications. The problem is that those who are not fully aware of the intentions and limits of the so called [*reference priors*]{} tend to perceive the Bayesian approach as dogmatic. Let us analyse, then, some of the criticisms. - “Subjective Bayesians have settled into a position intermediate between orthodox statistics and the theory expounded here.”[@Jaynes] I think exactly the opposite. Now, it is obvious that frequentistic methods are conceptually a mess, a collection of arbitrary prescriptions. But those who stick too strictly to the theory expounded in Ref. [@Jaynes] tend to give up the real (unavoidably subjective!) knowledge of the problem in favour of mathematical convenience or blindly following the stance taken by the leading figures in their school of thought. This is exactly what happens with practitioners using blessed ‘objective’ frequentistic ‘procedures’ (for example, see Ref. [@anxiety] for a discussion on the misuse of Jeffreys’ priors). - “While perceiving that probabilities cannot represent only frequencies, they \[subjective Bayesians\] still regard sampling probabilities as representing frequencies of ‘random variables’”.[@Jaynes] The name‘random variable’ is avoided by the most authoritative subjective Bayesians [@deFinetti] and the terms ‘uncertain (aleatoric) numbers’ and ‘aleatoric vectors’ (form multi-dimensional cases) are currently used. Even the idea of ‘repeated events’ is rejected [@deFinetti], as every event is unique, though one might think of classes of analogous events to which we can attribute the same [*conditional*]{} probability, but these events are usually stochastically dependent (like the outcomes black and white in the six box example of Ref. [@ajp]). In this way the ideas of uncertainty and probability are completely disconnected from that of [*randomness*]{} à la von Mises. [@vonMises] Nevertheless, I admit that there are authors, including myself [@YR], who mix the terms ‘uncertain numbers’ and ‘random variables’, to make life easier for those who are not accustomed to the concept of uncertain numbers, since the formal properties (like p.d.f., expected value, variance, etc) are the same for the two objects. - “Subjective Bayesians face an awkward ambiguity at the beginning of a problem, when one assigns prior probabilities. If these represent merely prior opinions, then they are basically arbitrary and undefined”. [@Jaynes] Here the confusion between subjective and arbitrary, discussed above, is obvious. - “It seems that only private introspection could assign them and different people will make different assignments”. No knowledge, no science, and therefore no probability, is conceivable if there is no brain to analyse the [*external*]{} world. Fortunately there are no two identical brains (yet), and therefore no two identical states of knowledge are conceivable, though intersubjectivity can be achieved in many cases. - “Our goal is that inferences are to be completely ‘objective’ in the sense that two persons with the prior information must assign the same prior probability.” [@Jaynes] This is a very naïve idealistic statement of little practical relevance. - “The natural starting point in translating a number of pieces of prior information is the state of complete ignorance.” [@Jaynes] When should we define the state of complete ignorance? At conception or at birth? How much is learned and how much was already coded in the DNA? Conclusions =========== To conclude, I think that none of the above criticisms is really justified. As for criticisms put forward by frequentists or by self-designed frequentistic practitioners (who are more Bayesian than they think they are [@Maxent98]) there is little more to comment in the context of this workshop. I am much more interested in making some final comments addressed to fellow Bayesians who do not share some of the ideas expounded here. I think that users and promoters of Bayesian methods of the different schools should make an effort to smooth the tones of the debate, because the points we have in common are without doubt many more, and more relevant, than those on which there is disagreement. Working on similar problems and exchanging ideas will certainly help us to understand each other. There is no denying that Maximum Entropy methods are very useful in solving many complicated practical problems, as this successful series of workshops has demonstrated. But I don’t see any real contradiction with coherence: I am ready to take seriously the result of any method, if the person responsible for the result is honest and is ready to make any combination of reversible bets based on the declared probabilities. [99]{} C. Giunti, Proc. Workshop on Confidence Limits, CERN, Geneva, Switzerland, January 2000. CERN Report 2000–005, May 2000, pp. 63–72, e-print arXiv: hep-ex/0002042. R. Feynman, [*The character of physical law*]{}, MIT Press, 1967. G. D’Agostini, Proc. of the XVIII International Workshop on Maximum Entropy and Bayesian Methods, Garching (Germany), July 1998. (Kluwer Academic Publishers, Dordrecht, 1999), pp. 157–170, e-print arXiv: physics/9811046. G. D’Agostini, CERN Report 99–03, July 1999, electronic version available at the author’s URL. G. D’Agostini, Rev. R. Acad. Cienc. Exact. Fis. Nat., Vol. 93, Nr. 3, 1999, pp. 311–319, e-print arXiv: physics/9906048. G. D’Agostini, Am. J. Phys. [**67**]{} (1999) 1260-1268, e-print arXiv: physics/9908014. B. de Finetti, [*Theory of Probability*]{} (J. Wiley & Sons, 1974). \[bib:deFinetti\] D.S. Sivia, [*Data analysis – a Bayesian tutorial*]{}, Oxford University Press, 1997. G. D’Agostini, Proc. Workshop on Confidence Limits, CERN, Geneva, Switzerland, January 2000. CERN Report 2000–005, May 2000, pp. 3–23. R. Scozzafava, Pure Math. and Appl., Series C 2, 1991, pp. 223-235. J.M. Keynes, [*A tract on Monetary Reform*]{} (Macmillan, London, 1923). Particle Data Group, D.E. Groom et al., [*Eur. Phys. J.*]{} [**C15**]{} (2000) 1 (Section 28). DIN Deutsches Institut für Normung, [*“Grundbegriffe der Messtechnick – Behandlung von Unsicheratine bei der Auswertung von Messungen”*]{}, (DIN 1319 Teile 1–4), Beuth Verlag GmBH, Berlin, Germany, 1985.\ ISO International Organization for Standardization, [*International vocabulary of basic and genaral terms in metrology*]{},” Geneva, Switzerland, 1993. F. James, Proc. Workshop on Confidence Limits, CERN, Geneva, Switzerland, January 2000. CERN Report 2000–005, May 2000, pp. 1–2. S.W. Hawking, [*A Brief history of time*]{}, Bantam Doubleday Dell, 1988. D.A. Berry, Talk at the Annual Meeting of American Statistical Association, Chicago, August 1996. W. Heisenberg, [*Physics and philosophy. The revolution in modern science*]{}, 1958 (Harper & Row Publishers, New York, 1962). See e.g. G. ’t Hooft, [*In search of the ultimate building blocks*]{}, 1992 (Cambridge University Press, 1996). D. Hume, [*Enquiry concerning human understanding*]{}”, 1748. E.T. Jaynes, [*Probability Theory: The logic of Science*]{}, Chapter 12 on [*Ignorance priors and transformation groups*]{}, (book available at http://bayes.wustl.edu/etj/prob.html and http://omega.albany.edu:8008/JaynesBook, but Chapter 12 is missing since beginning of 2000). H. Jeffreys, [*Theory of probability*]{}, 1939 (Clarendon Press, Oxford, 1998). J.M. Bernardo and A.F.M. Smith, [*Bayesian Theory*]{}, John Wiley & Sons, 1994. R. von Mises, [*Probability, Statistics and Truth*]{}, 1928, (George Allen & Unwin, 1957), second edition. [^1]: Does this violate Bernoulli’s theorem? I leave the solution to this apparent paradox as amusing problem to the reader. [^2]: To be more rigorous, simple laboratory experiments can be performed in conditions of repeatability[@DIN-ISO], but thinking of repeating very complex particle physics experiments run for a decade make no sense (even in principle!). Perhaps the remark “in principle” in the above quotation from Ref. [@PDG] is to justify Monte Carlo simulation of the experiments. But one has to be aware that a Monte Carlo program is nothing but a collection of our best beliefs about the behaviour of the studied reaction, background reactions and apparatus. [^3]: A similar desperate attempt is try to throw a bad shadow over Bayesian theory, saying that in this theory “frequency and probability are completely disconnected”[@James], using as argument an ambiguous sentence picked up from the large Bayesian literature, and severed from its context.