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abstract: 'The formation and dynamics of cavities in liquids leads to focusing of kinetic energy and emission of longitudinal stress waves during the cavity collapse. Here we report that cavitation in elastic solids may additionally emit shear waves that could affect soft tissues in human bodies/brains. During collapse of the cavity close to an air-solid boundary, the cavity moves away from the boundary and forms a directed jet flow, which confines shear stresses in a volume between the bubble and the free boundary. Elastographic and high-speed imaging resolve this process and reveal the origin of a shear wave in this region. Additionally, the gelatin surface deforms and a conical crack evolves. We speculate that tissue fracture observed in medical therapy may be linked to the non-spherical cavitation bubble collapse.'
author:
- 'J. Rapet'
- 'Y. Tagawa'
- 'C.D. Ohl'
bibliography:
- 'Bib\_shear.bib'
title: 'Shear-wave generation from cavitation in soft solids'
---
In many medical applications energy is deposited within the tissue resulting in the formation of a cavity which expands and collapses. This phenomena is termed cavitation. The understanding of its dynamics helps to improve precision and mitigate side effects [@Vogel2005Nanosurgery; @brennen2015cavitation]. Examples of cavitation based therapy are histotripsy where pulsed finite amplitude ultrasound waves are focused into tissues and cornea surgery where laser pulses locally vaporize corneal tissue and produce clean intrastromal cuts [@juhasz1999corneal; @lubatschowski2000application]. The former leads to regions of intense cavitation where tissue is rapidly and locally turned into a paste[@roberts2006pulsed; @kim2011non]. The importance of bubble-tissue interaction stimulated research on individual bubbles oscillating in a tissue mimicking elastic solid. There the emission of stress and tension waves during bubble generation and collapse were documented [@brujan2006stress].
Tissue mimicking materials such as gelatin, polyacrylamide gels (PAA) or other hydrogels provide an elastic restoring force. Thus, they can transport not only longitudinal waves (with a wave velocity of $\approx 1500\,$m/s) but also transversal waves propagating at a considerably smaller velocity of $\approx 1-50$ m/s. As the shear wave velocity depends on the mechanical properties of the tissue, the tissue type may be characterized in diagnostic ultrasound from the shear front propagation [@gennisson2013ultrasound]. Beside the established methods based on acoustic focusing[@song2012comb], shear wave generation is an active research field with a number of new actuation mechanisms being reported. For example using an electric current in combination with a magnetic field displaces the elastic solid by the Lorentz force[@grasland2014imaging] or by locally displacing the hydrogel with bubbles from electrolysis[@montalescot2016electrolysis]. Recently, the generation of shear waves from local heating was demonstrated: when a laser beam heats up the surface of a soft elastic absorber[@grasland2016generation] two regimes are observed. In the thermoelastic regime the waves arise from the thermal expansion and in the ablative regime from impulse transfer due to vaporization of the surface.
Shear waves of larger amplitude, often induced by impacts[@cooper1989biophysics; @taylor2014investigation], cause the negative effects on soft tissues, e.g. in blast-induced traumatic brain injuries[@taylor2014investigation; @taber2006blast]. In the present work, we report on mechanism of shear wave generation from the non-spherical collapse of a single cavitation bubble in a soft solid. To induce the non-spherical collapse, the bubble is created in the solid near an air-solid boundary.
High-speed photography is used to record the bubble dynamics and the shear wave propagation. The latter is compared with a simple model to support the hypothesis of the mechanism of shear wave generation.
While pressure waves typically result in a change of index of refraction and can be visualized with shadowgraphy or Schlieren imaging, shear waves are more difficult to picture. Here, we utilize birefringence in the solid. Gelatin as our tissue phantom becomes doubly refractive under stress and is then suitable for photoelastic photography [@tomlinson2015photoelastic]. Those two refracted rays possess directions of polarization coinciding with the local principal stress directions. A circular polariscope is particularly suitable for measuring the stress distribution in the medium. It is obtained from two linear polarizers and two $\lambda/4$ waveplates as sketched in Figure \[fig:Setup\](a). The intensity of the light exiting the circular polariscope is a function of the retardation $\delta$ (Eq. \[eq:retardation\]) and can be expressed using trigonometry or Jones calculus [@ramesh2000digital] for 2-dimensional stress distributions $$\label{eq:circ-pola}
I = k^2\,\sin^2\bigg(\frac{\delta}{2}\bigg)\quad .$$ Monochromatic illumination assures that the relative retardation $\delta$ is only a function of the difference in amplitude of the two principal stresses and can be expressed as $$\label{eq:retardation}
\delta = \frac{2\pi h}{\lambda}C(\sigma_1-\sigma_2)\quad ,$$ where $h$ is the thickness of the sample, $\lambda$ the wavelength of the incoming light, $C$ the stress-optic coefficient, $\sigma_1$ and $\sigma_2$ the principal stresses. Thus the light is extinguished only where the sample is unstressed or where the principal stress difference $(\sigma_1 - \sigma_2)$ is causing a phase difference of $\delta = 2m\pi$ $ (m = 0,1,2,...)$. The second condition marks the presence of an isochromatic, i.e an area of constant principal stress difference.
The experimental setup shown in Figure \[fig:Setup\](a) allows to record the cavitation bubble dynamics and the shear wave emission through photoelastic imaging with a high speed camera. Single laser induced cavitation bubbles are created by focusing a laser pulse (Litron Lasers, Nano series, Q-switched Nd:YAG, 6 ns, wavelength $1064\,$nm ) into the gelatin with a microscope objective (Olympus $10\times$ Plan Achromat, N.A. $= 0.25$). At the focal point of the lens the bubble is created through an optical breakdown. The dynamics of the bubble (expansion, collapse and rebound) and the shear waves are recorded with a high speed camera (Shimadzu HPV-X2) equipped with a macro lens (Canon MP-E 65mm f/2.8 1-5$\times$ Macro). Single wavelength illumination is provided by a continuous green laser (Shaan’’xi Richeng Ltd, DPSS Green Dot Laser Module, wavelength $532\,$nm).
The soft solids samples are prepared from powdered gelatin (Gelatin 250 bloom, Yasin Gelatin CO.,LTD). It is mixed with deionized water at a mass ratio of 4% (gelatin to water) and dissolves in a flask on a hot plate with a magnetic stirrer. The hot mixture is poured into an optical glass cuvette (Aireka Scientific Co. Ltd, $3.5\,$ml, QO10204-4) to minimize birefringence distortion from the container and insure a flat gelatin-glass interface for clear observation and accurate laser focusing. The samples cool down to room temperature and are stored in a fridge. Before use, we assure that the samples have reached room temperature again.
Figure \[fig:Pola-side\] presents a typical result from the experimental setup Figure \[fig:Setup\](a). It consists of selected frames from a high-speed recording of a bubble expanding and collapsing in gelatin near a free (air) boundary. The time $t = 0$ denotes the time of bubble nucleation. Then the bubble expands and compresses the gelatin nearby resulting in a bright area around the bubble ($t = 5\,\mu$s). Upon reaching the maximum size, the bubble pushes and deforms the free surface ($t=35\,\mu$s) before it shrinks again. Thereby its upper wall bulges in and forms a funnel while the position of the interface at the bottom of the bubble remains approximately static ($t= 65\,\mu$s). Between the bubble and the free surface, two vertical dark lines develop on top of the brighter stressed area. These two lines mark the presence of isochromatics. The dark lines extend vertically following the bubble collapse with a velocity of $V_v = 25\,$m/s$\,\pm\,1$ m/s, which roughly also is the velocity of the upper bubble interface. Their horizontal position changes, too. They move outward with a velocity of $V_s =2\,$m/s$\,\pm\,0.2\,$m/s.
After reaching the minimum size the bubble rebounds ($t = 85\,\mu$s) and its volume oscillations cease shortly after the second collapse at $t = 119\,\mu$s (not shown), while it remains moving downwards.
At $t = 155\,\mu$s, a conical-shaped crack appears starting from the air-gelatin boundary and follows the path of the downwards translating bubble ($t = 225\,\mu$s). We speculate that the shrinking and moving bubble is the source for shear stress generation. As the speed of this motion is considerably higher than the shear wave velocity, shear stress is confined in the region bounded by the air interface and the bubble. Later only this confined stress propagates from this region as a shear wave into the surrounding medium.
This hypothesis is tested with a numerical simulation of the stress generation and propagation using finite element solver for an elastic solid (Solid Mechanics Module, COMSOL Multiphysics). For the sake of simplicity, the stress field resulting from a body force moving downward with a constant velocity is modeled. Figure \[fig:Simu\](a) depicts the 2D axisymmetric simulation domain, properties of the gelatin are taken from literature ($\rho = 1000$ kg/m$^3$, $\epsilon = 0.45$, $E = 10$ kPa, $C_p = 1500$ m/s, $C_s = 1.8$ m/s)[[@czerner2015determination; @gennisson2013ultrasound]]{}. The boundary conditions are stress free on the top, an axis of symmetry, and to the right and bottom of the domain impedance matched boundaries. At $t = 0$, a rectangular source of force (body load, $F = 1\,$N) moves downward from the free surface along the axis of symmetry with a velocity of $V_Z = 25$ m/s for a duration $\Delta t = 32\,\mu$s. To compare the simulations with the photoelastic images we split the axisymmetric model into slices perpendicular to the line of sight. In each slice of finite thicknesses, the polariscope equation is solved assuming a constant stress along the depth.
Figure \[fig:Simu\](b) presents selected frames from the numerical results. Black areas correspond to the gas/vapour domains (bubble and free surface drawn from the experimental results and added on the numerical results), dark gray (low light intensity) shows the unstressed gelatin and areas of stress are depicted with gray scales, with higher brightness indicating areas of higher stress. Overall the simulation results show good agreement considering the coarse simplifications. In particular we observe a bright region which extend vertically following the rectangular source of force with a velocity of $V_z$ and slowly expands radially. The main difference between simulation and experiment is that the isochromatics are not reproduced. We explain this that the pre-existing stress fields from the earlier bubble dynamics are not considered in the simulations, i.e. there the body force moves within an unstressed solid. The simulation supports the explanation that shear waves can be induced with a source of stress moving with the bubble upper interface.
{width="100.00000%"}
The motion of the bubble collapsing away from the free surface (Figure \[fig:Pola-side\]) is also reported for liquids, i.e. in water where a jet pierces through the bubble [@robinson2001interaction]. It suggests that the emergence of a jet penetrating through the bubble in gelatin could cause the generation of the shear waves. To obtain a better photographic evidence we repeated the experiment in absence of the polariscope optics, see Figure \[fig:Setup\](b) for the experimental setup. Now two high-speed cameras (Photron, Fastcam, Mini AX200) equipped with camera lenses (Canon MP-E 65 mm f/2.8 1-5$\times$ Macro) record simultaneously the bubble and the surface from the side and top.
Figure \[fig:2-cam\] presents selected frames from a bubble with very similar size and position as shown in Figure \[fig:Pola-side\]. Again the bubble expands and pushes the free surface resulting in an upward displacement visible on the surface with a gelatin jet pointing upward ((a):$t = 15\,\mu$s) and expanding laterally as the bubble grows ((a):$t = 44\,\mu$s, (b):$t = 37\,\mu$s). The bubble then collapses and moves away from the free surface. During collapse, a jet is penetrating through the center of the bubble with a velocity of $V_s = 19\,$m/s$\,\pm\,1\,$m/s ((b):$t = 68\,\mu$s). Then the jet impacts on the lower bubble interface while the bubble continues to translate downward ((b):$t = 86\,\mu$s and $t= 168\,\mu$s). These observations indicate that the jet formed by the bubble moving away from the air-gelatin boundary is causing the shear wave. At a later time the restoring force of the gelatin pulls back the residues of the bubble towards its position at creation, see (b):$t = 222\,\mu$s.
The crack originating from the air-gelatin boundary indicates that the upwards pointing tip of the jet on the gelatin-air interface ((a):$t = 15 \,\mu$s) retracts first towards the original level and then penetrates into the gelatin, visible as a conical crack in (a):$t = 178\,\mu$s. The opening angle of the crack remains while the crack tip propagates into the gelatin trailing the cavitation bubble (Figure \[fig:Pola-side\](b):$t = 222\,\mu$s). The oscillations of the free surface result in surface waves (Figure \[fig:2-cam\] sketch and (a)) traveling radially with a velocity of $V_s = 4.5\,$m/s$\,\pm\,0.5\,$m/s.
In conclusion, we have demonstrated that a single cavitation bubble in soft media can generate shear waves. The jetting of the shrinking and collapsing bubble is causing shear stresses to be confined between the air-gelatin interface and the bubble. Shear waves are then emitted from this region. We also observe the formation of a crack following the downward motion of the bubble. We speculate that this could be a potential source of tissue damage when cavitation bubbles collapse non-spherically. We speculate that confinement of shear stress near to free interfaces may lead to cracks and tissue damage.
*Acknowledgements* Y.T. acknowledges financial support from JSPS KAKENHI Grants No. R2801 and 17H01246. This project has received funding from the European Union Horizon 2020 Research and Innovation programme (No 813766).
|
---
abstract: |
In this model a collimated ultra-relativistic ejecta collides with an amorphous dense cloud surrounding the central engine, producing gamma-rays via synchrotron process. The ejecta is taken as a standard candle, while assuming a gaussian distribution in thickness and density of the surrounding cloud. Due to the cloud high density, the synchrotron emission would be an instantaneous phenomenon (fast cooling synchrotron radiation), so a GRB duration corresponds to the time that the ejecta takes to pass through the cloud. Fitting the model with the observed bimodal distribution of GRBs’ durations, the ejecta’s initial Lorentz factor, and its initial opening angle are obtained as $\Gamma_{0}\lesssim
10^{3}$, and $\zeta_{0} \approx 10^{-2}$, and the mean density and mean thickness of the surrounding cloud as $\overline{n} \sim
3 \times 10^{17} cm^{-3}$ and $\overline{L} \sim 2 \times 10^{13}
cm$. The clouds maybe interpreted as the extremely amorphous envelops of Thorne-Zytkow objects. In this model the two classes of long and short duration GRBs are explained in a unique frame.
author:
- 'F. Shekh-Momeni [^1] and J. Samimi [^2]'
title: |
A DENSE-CLOUD MODEL FOR GAMMA-RAY BURSTS\
TO EXPLAIN BIMODALITY
---
INTRODUCTION
============
Undoubtedly, gamma-ray bursts (GRBs) have remained to be one of the most exiting, intriguing, and enigmatic astrophysical phenomena since their mysterious discovery in the past several decades (for a recent expository review of GRBs the interested reader is referred to the excellent work by J. I. Katz [@katz]). Although no two GRBs resemble each other and each one has its own peculiarities which makes the problem of modeling GRBs very difficult, the whole of GRBs reveals several interesting features. Since the publication of the BATSE data [@1st] which included the observation of over $200$ GRBs and revealed an almost uniform distribution of the location of GRBs in the sky, combined with the deficiency of faint GRBs, the association of GRBs with the galactic plane has been ruled out. The successive publications confirmed the figure more and more [@3th; @4th]. However, since the observation of afterglows in X-ray [@Costa], optical [@van; @Paradijs] and radio spectrum [@Frail] and the advert of Robotic Optical Transient Search Experiment (ROTSIE) telescope [@Akerlof; @Gisler e.g.] which has revealed the red shifts for several GRBs, their cosmological origin is widely accepted.\
The BATSE data [@1st; @3th; @4th] has also revealed another equally important overall feature of the GRBs. The distribution of time duration in observed GRBs shows a double heap distribution, which the smaller one peaks around 0.2 sec and the larger one peaks around 20 sec (Fig.1). This two peaked distribution which apparently separated GRBs into the so called short duration and long duration ones was referred to as $\textit{bimodality}$ [@kouveliotou; @Norris] and led some investigators to believe that there are two distinct populations of GRBs.\
It has been widely believed that whatever the central engine is, the radiation reaching us originates from the space surrounding the central engine. It is also believed that during the collapse the energy release streams out in relativistic confined ejectas (not isotropically) and thus the total energy release during each event is far less than the unbelievable amount that one might obtain by assuming isotropic radiation [@kulkarni].\
The aim of this paper is to present a rather simple model, based on these general ideas, and show that there is no need for assuming two distinct populations of GRBs, and to show that once the geometrical considerations and cosmological effects are fully accounted for a genetic standard candle ejecta, crossing an amorphous dense cloud, the so called $\textit{bimodality}$ can be deduced.\
In § 2 the model and its general formulation is introduced. In § 3 the computational results and the fit of model parameters with BATSE data will be presented as well. § 4 is devoted to a discussion on the results. Some needed calculations and discussions are presented in appendices.
THE MODEL FORMULATION
=====================
GRBs are modeled as a central engine with an instantaneous ultra-relativistic jet of material surrounded by an amorphous dense cloud. The central engine and its jet are taken as a standard candle with a total release energy of $E$, and an initial Lorentz factor $\Gamma_{0}$ and initial opening angle $\zeta_{0}$ of the jet for all GRBs, whereas the clouds are considered to have a distribution both in thickness as well as in density. For the sake of illustration and brevity we take both of these distributions to be Gaussian.\
We want to calculate the distribution of logarithm of time duration of observed GRBs according to the above model. We should however explore the ejecta evolution since the observed gamma ray emission originates in the shock front of ejecta+shocked medium.
The Ejecta Evoution
-------------------
The equations describing the ejecta evolution are presented here, based on the notations of Paczynski & Rhoads [@Paczynski; @and; @Rhoads]. We consider the cloud to be at a distance $r_{0}$ from the source, which may be negligible in comparison to the cloud thickness $L$. The ratio of swept-up mass to ejecta mass $M_{0}$ has the form below: $$\label{}
f=\frac{1}{M_{0}}\int^{r}_{r_{0}} \rho \Omega_{m}(r^{\prime}) r
^{\prime 2} dr^{\prime},$$ where, $r$ is the ejecta distance from the source. The cloud density $\rho$ is taken to be independent of $r$. Furthermore, $\Omega_{m}(r)=2\pi[1-\cos\zeta(r)]$, in which $\zeta(r)$ represents the opening angle of the ejecta at radius $r$. So, equation(1) can be written as: $$\label{}
\frac{df}{dr}=2\pi n \frac{m_{p}\Gamma_{0}c^{2}}{
E}[1-\cos\zeta(r)]r^{2},$$ where $E$ and $\Gamma_{0}$ are the initial kinetic Energy and initial Lorentz factor of the ejecta ($E=\Gamma_{0} M_{0}
c^{2}$), and $n$ denotes the number density of the cloud ($\rho=m_{p}\:n$). Paczynski & Rhoads [@Paczynski; @and; @Rhoads] derived the relation between $f$ and $\Gamma$ from conservation of energy and momentum. Here, we use their relation in a form suitable for our computations: $$\label{}
\frac{df}{d\Gamma}=-\frac{\sqrt{\Gamma_{0}^{2}-1}}{\sqrt[3]{\Gamma^{2}-1}}~.$$ The ejecta’s opening angle $\zeta(r)$ increases with increasing $r$ as a result of lateral spreading of the cloud of ejecta+swept-up matter in the comoving frame at the sound speed $c_{s}$, which has been derived by Rhoads [@Rhoads] to be as below: $$\label{}
d\zeta(r)=\frac{c_{s}\:dt_{co}}{r},$$ where $t_{co}$ denotes the time from the event, measured in the ejecta comoving frame. Substituting $dt_{co} = dt/\Gamma$, and $dt = dr / \beta c$, we have : $$\label{}
\frac{d\zeta(r)}{dr}=\frac{c_{s}/c}{\beta\:\Gamma \: r }.$$ Now, eliminating $f$ between equations(2) and (3) yields: $$\label{}
\frac{d\Gamma}{dr}=-2\pi n
\frac{m_{p}\Gamma_{0}c^{2}}{E}[1-\cos\zeta(r)]
\frac{\sqrt[3]{\Gamma^{2}-1}}{\sqrt{\Gamma_{0}^{2}-1}}r^{2}~.$$ Let’s rewrite equations(5) and (6) in a non-dimensional form as below: $$\label{}
\left\{\begin{array}{rcl}
\frac{d\zeta}{d\eta}&=&\frac{c_{s}/c}{\eta\Gamma(\eta)(1-\frac{1}{\Gamma^{2}})^{-1/2}}\\
\frac{d\Gamma}{d\eta}&=&-\frac{\Gamma_{0}\sqrt[3]{\Gamma^{2}-1}}{\sqrt{\Gamma_{0}^{2}-1}}
[1-\cos\zeta(\eta)]\eta^{2}
\end{array}\right. \nonumber\\,$$ where, the non-dimensional parameter $\eta$ is defined as: $$\label{}
\eta\equiv \frac{r}{l(n/E)},$$ in which: $$\label{}
l(n/E)\equiv\left(\frac{2\pi m_{p}c^{2}n}{E}\right)^{-1/3}=4.75
\times 10^{11}\left(\frac{n_{18}}{E_{48}}\right)^{-1/3}\:\:cm ~.$$ These coupled first order differential equations can be solved numerically by introducing the initial conditions: $$\label{}
\left\{\begin{array}{rcl}
\Gamma(\eta_{0})=\Gamma_{0}\\
\zeta(\eta_{0})=\zeta_{0}
\end{array}\right. \nonumber\\,$$ where $\eta_{0}\equiv r_{0}/l(n/E)$.\
Noting that $\beta=dr/cdt=(1-1/\Gamma^{2})^{1/2}$, we have: $$\label{}
\frac{d\tau}{d\eta}=\left(1-\frac{1}{\Gamma^{2}(\eta)}\right)^{-1/2}~
~ ~ ; ~ ~ ~ \tau(\eta_{0})=0.$$ In equation(11) we used equation(8) and a non-dimensional time parameter $\tau$ defined as: $$\label{}
\tau\equiv\frac{c\:t}{l(n/E)}$$ The numerical results of equation(11) are used in appendix C, where we consider the effect of burster geometry on the observable time duration.
Formulation of Time Duration Distribution
-----------------------------------------
We begin with introducing the probability density for a collimated burst to occur in a direction through the cloud with a thickness $L$ and a number density $n$. We assume the cloud thickness to have a gaussian distribution in various directions from the central engine. By assuming a gaussian distribution for the cloud density as well we have: $$\label{}
\frac{d^{3}p}{dn\:dL\:d\Omega}=\frac{1}{4\pi}\frac{1}{\sqrt{2\pi}\sigma_{_{n}}}
\:\exp\left[\frac{-(n-\overline{n})^{2}}{2\sigma_{_{n}}^{2}}\right]\:\frac{1}{\sqrt{2\pi}\sigma_{_{L}}}
\:\exp\left[\frac{-(L-\overline{L})^{2}}{2\sigma_{_{L}}^{2}}\right]~.$$ The quantities $\overline{L}$ and $\sigma_{_{L}}$ denote the mean thickness of the cloud and its dispersion, respectively. Likewise, $\overline{n}$ and $\sigma_{_{n}}$ are defined in a similar manner. Denoting the angle between the ejecta symmetry axis and the line of sight by $\theta$ (Fig.2), and considering the independence of the probability density from azimuth angle in equation(13), we can write: $$\label{}
\frac{d^{3}p}{dn\:dL\:d\theta}=\frac{2\pi\:\sin\theta}{4\pi}\frac{1}{\sqrt{2\pi}\sigma_{_{n}}}
\:\exp\left[\frac{-(n-\overline{n})^{2}}{2\sigma_{_{n}}^{2}}\right]\:\frac{1}{\sqrt{2\pi}\sigma_{_{L}}}
\:\exp\left[\frac{-(L-\overline{L})^{2}}{2\sigma_{_{L}}^{2}}\right]~.$$ The synchrotron emission is a fast process for our model (see appendix A), so $T_{rec}$ (the time duration of a GRB as measured by an observer cosmologicaly near to the source and located on the line of sight), is attributed to the time that the shock front takes to cross the dense cloud. As explained in appendix C, the cloud thickness $L$ can be expressed as a function of $\theta$, $n$, and $T_{rec}$ (Eqn.\[C16\]): $$\label{}
L=L(T_{rec},n,\theta)~.$$ So we write: $$\label{}
\frac{d^{2}}{dn\:d\theta}\:\left(\frac{dp}{dL}\right)=\frac{d^{2}}{dn\:d\theta}
\left(\frac{dp}{dT_{rec}}\right)_{_{n,\theta}}\:\left(\frac{dT_{rec}}{dL}\right)_{_{n,\theta}}~,$$ or: $$\label{}
\frac{d^{3}p}{dn\:d\theta\:d\log
T_{rec}}=\frac{d^{3}p}{dn\:d\theta\:dL}\left(\frac{dL}{d\log
T_{rec}}\right)_{_{n,\theta}}~,$$ notifying that by this substitution(Eqn.\[15\]), the bursts that do not manage to cross through the cloud with a thickness $L$ (and stop in it) are practically omitted (see appendix C). Using equation(14) in equation(17), it is seen that: $$\begin{aligned}
\label{}
\frac{d^{3}p}{dn\:d\theta\:d\log
T_{rec}}&=&\frac{\sin\theta}{4\pi}\frac{1}{\sigma_{_{n}}\:\sigma_{_{L}}}\left(\frac{dL}{d\log
T_{rec}}\right)_{_{n,\theta}}\nonumber\\
&\times&\:\exp\left[\frac{-(n-\overline{n})^{2}}{2\sigma_{_{n}}^{2}}\right]
\exp\left[\frac{-(L(T_{rec},n,\theta)-\overline{L})^{2}}{2\sigma_{_{L}}^{2}}\right].\end{aligned}$$ Integrating over $\theta$ and $n$ yields: $$\begin{aligned}
\label{}
\frac{dp}{d\log T_{rec}}=\frac{1}{4 \pi\sigma_{_{n}}\sigma_{_{L}}}
\int_{\theta=0}^{\frac{\pi}{2}}\int_{n=0}^{\infty}
&&\hspace{-.7cm}\exp\left[\frac{-(n-\overline{n})^{2}}{2\sigma_{_{n}}^{2}}\right]
\exp\left[\frac{-(L(T_{rec},n,\theta)-\overline{L})^{2}}{2\sigma_{_{L}}^{2}}\right]\nonumber\\
&\times&\left(\frac{dL}{d\log
T_{rec}}\right)_{_{n,\theta}}\sin\theta\:d\theta\:dn~.\end{aligned}$$ The effect of red shift is not considered yet. Equation(19) only gives the probability density for an observed burst to have a specified logarithm of time duration, as measured by an observer near to it. We now investigate the relation between $dp / d\log
T_{rec}$ and $dp / d\log T_{\oplus}$ where $T_{\oplus}$ stands for the time duration of a GRB measured at Earth. To obtain the later, the former must be integrated over red shift z, using a weight function $F_{_{GRB}}(z)$, so that $F_{_{GRB}}(z)\:dz$ represents the probability for occurring a GRB in a red shift between $z$ and $z+dz$. To show this, let’s consider an observer located on the line from us to an occurred GRB which is (cosmologicaly) near to it. The probability density for the GRB to occur in a red shift z (with respect to us), and to have a specific $\log T_{rec}$ (measured by the observer near to the GRB), is clearly as below: $$\label{}
\frac{d^{2}p}{dz\:d\log T_{rec}}=F_{_{GRB}}(z)\frac{dp}{d\log
T_{rec}}~.$$ To obtain $d^{2}p / dz\:d\log T_{\oplus}$, which is the probability density for observing a GRB occurred at a red shift z and observed to have a specific $T_{\oplus}$, we write: $$\label{}
\frac{d^{2}p}{dz\:d\log T_{\oplus}}=\frac{d}{d z}\left(\frac{d
p}{d \log T_{\oplus}}\right)_{z}=\frac{d}{d z}\left[\left(\frac{d
p}{d \log T_{rec}}\right)_{z}\left(\frac{d \log T_{rec}}{d \log
T_{\oplus}}\right)_{z}\right]~,$$ noting that $T_{\oplus}=(1+z)T_{rec}$, the second term in the the bracket equals one. Now, using equation(20) we have : $$\label{}
\frac{d^{2}p}{dz\:d\log
T_{\oplus}}=F_{_{GRB}}(z)\:\left(\left.\frac{dp}{d\log
T_{rec}}\right|_{T_{rec}=T_{\oplus}/(1+z)}\right)~,$$ After integrating equation(22) over z, the final form of the observable probability density will be as below: $$\label{}
\frac{dp}{d\log
T_{\oplus}}=\int_{z=0}^{\infty}F_{_{GRB}}(z)\:\left(\left.\frac{dp}{d\log
T_{rec}}\right|_{T_{rec}=T_{\oplus}/(1+z)}\right)\:dz\:~.$$ The explicit form of $F_{_{GRB}}(z)$ is needed. This form is obtained in appendix D (Eqn.\[D4\] and Eqn.\[D5\]). The second term in the integrand is given by equation(19), in which the implicit form of $L(T_{rec},n,\theta)$ appears. Appendix C is devoted to the procedure of obtaining this function. For evaluation of $dp/d\log T_{\oplus}$ (Eqn.\[23\]), we need the values of eight parameters. Four of them are the cloud parameters $\overline{L}$, $\sigma_{L}$, $\overline{n}$, and $\sigma_{n}$, and the fifth one is the index $q$ corresponding to the GRB occurrence rate (see Eqn.\[D5\]). The later three ones are the ejecta parameters, $E$, $\Gamma_{0}$, and $\zeta_{0}$, which are the initial kinetic energy, the initial Lorentz factor, and the initial opening angle of the ejecta, respectively.
NUMERICAL COMPUTATIONS AND RESULTS
==================================
The $\textit{Mathmatica 4}$ software is used in the numerical computations. In the procedure we begin with solving the coupled differential equations(7) and (11) which govern the ejecta evolution, and in which $\Gamma_{0}$ and $\zeta_{0}$ are the only free parameters. Furthermore, we take $\eta_{0}=0$. For fixed values of these parameters, the functions $\Gamma(\eta)$, $\tau(\eta)$ and $\zeta_{rad}(\eta)$ (see Eqn.\[C3\]) can be uniquely obtained . Obviously $\Gamma$ decreases with increasing $\eta$ (Fig.3), and reaches to $\Gamma=1$ when $\eta$ approaches a certain value. This in fact takes an infinite time and results in an infinite non-dimensional time duration $\tau_{rec}$ (see Eqn.\[C4\] and Eqn.\[C5\]). So, to circumvent the difficulty, we consider an effective lower limit $\Gamma_{min}$ for the Lorentz factor of shocked matter, which yields an effective upper limit for non-dimensional time $\tau$. We adopted $\Gamma_{min}=2$ as a lower cut-off. Then, following the procedure explained in appendix C, for a specific cloud thickness $L$, and correspondingly a specific $\eta_{_{L}}$ (Eqn.\[C13\]), the non-dimensional time duration $\tau_{rec}(\theta,\eta_{_{L}})$ of a GRB can be calculated for every $\theta$ and $\eta_{_{L}}$ (Fig.4). Let’s denote the radius corresponding to $\Gamma_{min}$ by $\eta_{m}$ so that $\Gamma_{min}\equiv\Gamma(\eta_{m})$, and recall that in the model, for $\eta_{m}<\eta_{_{L}}$, the emitted photons would be completely scattered by the electrons of not-swept part of the cloud (that they have to pass through, before entering free space; see appendix C), and so, the whole phenomenon may be called a “$\textit{failed GRB}\:$”. But, if $\eta_{m}>\eta_{_{L}}$, the shocked matter succeeds to go out of the cloud and, as explained in § 4.3, due to a suppression process that intensively decreases the cross-section of Compton scattering,(the major part of)the emitted photons finally succeed to get released from the shocked medium and enter free space, provided that $\zeta(\eta_{_{L}})>[\sqrt{5 / 3}\:\Gamma(\eta_{_{L}})]^{-1}$ (see appendix B). It is really for this reason that $\tau_{rec}$ happens to be a function of $\eta_{_{L}}$, and practically independent of $\eta_{m}$. Then, solving $\tau_{rec}(\theta,\eta_{_{L}})$ for $\eta_{_{L}}$ numerically, we can obtain the function $\eta_{_{L}}(\tau_{rec},\theta)$ which is the equivalent non-dimensional form of expression (15). We rewrite equation(19) in a non-dimensional form suitable for numerical computations: $$\begin{aligned}
\label{}
\frac{dp}{d \log \tau_{rec}}=\frac{1}{2 \varrho\: \varsigma }
\int_{\theta=0}^{\frac{\pi}{2}}\int_{\nu=0}^{\infty}
&&\hspace{-.7cm}\exp\left[\frac{-(\nu-1)^{2}}{2\varrho^{2}}\right]
\exp\left[\frac{-\left(\frac{l(\overline{n}\nu/E)}{\overline{L}}\eta_{_{L}}(\tau_{rec},\theta)-1\right)^{2}}{2\varsigma^{2}}\right]\nonumber\\
&\times&\frac{l(\overline{n}\nu/E)}{\overline{L}}\left(\frac{d\eta_{_{L}}}{d\log
\tau_{rec}}\right)_{_{\nu,\theta}}\sin\theta\:d\theta\:d\nu,\end{aligned}$$ in which, equation(C13) and the definitions below:\
$\nu\equiv n / \overline{n}$ ,\
$\varrho\equiv \sigma_{n} / \overline{n}$ ,\
and\
$\varsigma\equiv \sigma_{L} / \overline{L}$\
are used. So, after choosing the quantities $\overline{L}$, $\sigma_{_{L}}$, $\overline{n}$, $\sigma_{n}$, and of course $E$, we can evaluate the integral appearing in equation(24). Then, after choosing a value for $q$, the integral of equation(23) can be performed to obtain the observable quantity $dp/d\textit{log} T_{\oplus}$.\
As seen in equation(24), the initial kinetic energy $E$ and the cloud mean density $\overline{n}$ appear only in the form $E/\overline{n}$, and therefore they can not be found separately in a fitting process, and all that can be obtained is only their ratio. But, the observed fluence of GRBs reveals that the released energy in GRBs is of order of $E_{iso}\sim10^{52} ergs$ for a isotropic burst, which reduces to $(\Omega / 4\pi)E_{iso}$ if the bursts were confined to a cone with a solid angle $\Omega$. We used this amount of isotropic energy to relate $E$ to $\zeta_{0}$ ($E\equiv E_{iso}\zeta_{0}^{2}/2 $), and reduce the free parameters of the model to seven as $\Gamma_{0},\zeta_{0},\overline{L},\sigma_{_{L}},\overline{n}$, $\sigma_{n}$ and $q$, which hereafter are called $\Pi$ parameters. These parameters must be chosen so that the results make the best fitting to the observed distribution of GRBs. To achieve the task, one must search in the seven dimensional $\Pi$ space, and find the point in which the statistical quantity $\chi^{2}$ takes the smallest value $\chi^{2}_{min}$. We used the BATSE 4th catalogue [@4th] of 1234 GRBs and adopted the bins $\Delta \log T _{\oplus}=0.2$ in a range $-1.9\leq\log
T_{\oplus}<2.9$, and used the $\textit{gradient search}$ technique to move toward the best point in which $\chi^{2}$ gets minimized. The obtained fitted values are: $$\begin{aligned}
\label{}
\Gamma_{0}&=&0.97\times 10^{3} \:\:\:\:\:\:\:\:\:\:\:\zeta_{0} = 0.01 \nonumber\\
\overline{L}&=&1.7 \times 10^{13} cm \:\:\:\:\:\:\:\: \overline{n} = 2.9 \times 10^{17} cm^{-3}\nonumber\\
\sigma_{_{L}}&=&0.21\overline{L}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \sigma_{n} = 0.71 \overline{n}\nonumber\\
q&=&-0.70\end{aligned}$$ with a corresponding value $\chi^{2}_{min}=1.4$ (per degree of freedom). As can be seen in Fig.5, the deviation from a more perfect coincidence to data seems for the structure in the observed distribution located around $\log T_{\oplus}\sim 0.7$. The structure has been noted before [@Yu]. We put aside the the data of the noted structure, which are ones with durations between $0.3<\log(T_{\oplus})<0.9$, and again repeated the numerical searching process in the parametric space. The new obtained values of the fitted parameters are:\
$$\begin{aligned}
\label{}
\Gamma_{0}&=&0.96 \times 10^{3} \:\:\:\:\:\:\:\:\:\:\:\zeta_{0} = 0.01 \nonumber\\
\overline{L}&=&1.8 \times 10^{13} cm \:\:\:\:\:\:\:\: \overline{n} = 2.9 \times 10^{17} cm^{-3}\nonumber\\
\sigma_{_{L}}&=&0.21\overline{L}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \sigma_{n} = 0.71 \overline{n}\nonumber\\
q&=&-0.70\end{aligned}$$ which slightly differ from what previously obtained (Eqn.\[25\]). But this time, $\chi^{2}_{min}$ reduces to $1.1$ (Fig.6). So the mentioned structure may be interpreted as a result of an independent phenomenon or effect which was not considered in our model.
DISCUSSION
==========
The GRB Source
--------------
Though the original shapes of equations (13) and (14) are naturally normalized, the resulting final equations (19) and (23) are not, because:\
1)the bursts that their symmetry axes make an angle $\theta>\zeta_{rad}(\eta_{_{L}})$ can not be detected (see appendix C),\
2)the bursts for which $\eta_{m}<\eta_{_{L}}$ were not considered in the numerical computations (because they do not manage to cross out the cloud).\
The total probability of observing the occurred bursts, $\int_{-3}^{3} (dp / d\log T_{\oplus}) ~ d\log T_{\oplus}$, is obtained to be $1.47 \times 10^{-6}$, using our best fit parameters(Eqn.\[26\]). This small probability must be interpreted to be due to the above two reasons. The first reason describes the suppression of observed GRBs by the term $ (\Omega / 4\pi)\sim
\zeta_{0}^{2} / 2 \sim 10^{-4}$, while the second is responsible for the remaining factor of $10^{-2}$. So, only about one percent of the bursts manage to produce a real GRB, and only about $10^{-4}$ of these GRBs occur in our line of sight. Of course, the lateral spreading of the ejecta+swept mass may modify these two factors, increasing the first and decreasing the second.\
With the values for the fitted parameters in equation(25) or in equation(26), the mean mass of the clouds $\overline{M}\equiv\frac{4}{3}\pi \overline{n}
m_{p}\overline{L}^{3}$ is about $1.2 \times10^{34}\:gr \approx 6
M_{\odot}$, which is of the order of the envelop mass in massive stars. Such amorphous clouds seem strange in stars, but in close neutron star-supergiant binaries where the neutron star orbits around the core and accrete the envelope, the spherical symmetry is likely removed, as pointed out by Podsiadlowski et al. [@podsiadlowski]. Terman, Taam, & Hernquist [@terman] show that the system would emerge to form a red supergiant with a massive Thorne-Zytkow Object (TZO) [@thorne]. Podsiadlowski et al. [@podsiadlowski] also estimated a TZO birth-rate of $\geq 10^{-4} yr^{-1}$ in the galaxy. Considering equation(D4) and the obtained total probability of observing a “real” GRB, which is $1.47\times 10^{-6}$ in our model (see above), it is seen that the GRB observation rate ($2$ events per day) implies a total (real+failed) GRB rate of the order of $\sim 10^{-3}Mpc^{-3}
yr^{-1}$. This is not too far from what Podsiadlowski et al. [@podsiadlowski] theoretically estimated for TZO birth-rate ($\geq 10^{-4} galaxy^{-1} yr^{-1}$). Qin et al. [@qin] introduced AICNS (Accretion-Induced Collapse of Neutron Stars) scenario as GRB engines. Katz [@katz3] introduces a dense cloud model to explain the observed $Gev$ gamma-rays in a number of GRBs [@dingus; @jones e.g.], and suggests the amorphous envelops of TZOs for these clouds.\
The initial opening angle of the ejecta in our model ($\zeta_{0}\simeq 10^{-2}rad\approx 0.6 ^{\circ}$) is obtained during the fitting process (§ 3). Such a small opening angle for an ejecta or a jet might be explained by attributing it to the collimating process of an ultra-relativistic ejecta with $\Gamma_{0}>1/\zeta_{0}$ in a sufficiently high magnetic field [@begelman]. The initial Lorentz factor of the ejecta, as obtained in our model ($\Gamma_{0}\simeq 10^{3}$, see Eqn.\[25\]) provides the necessary condition of $\Gamma_{0}>1/\zeta_{0}$. Aside from these theoretical justifications for the idea of a highly collimated ejecta at the source, there are some pieces of evidences supporting this idea [@lamb; @waxman; @granot2].
The Bimodality
--------------
At the mean time the observed bimodality must be interpreted as a result of the second reason expressed in § 4.1. Though in our model one may expect only one heap in the $log T_{\oplus}$ distribution associated with the directions in the clouds having both the most probable $L$ and $n$ (which are equal or near to $\overline{L}$ and $\overline{n}$), but as the numerical computation shows, such directions in the clouds are too thick and too dense to be crossed out by the ejecta, and therefore a real GRB would not be produced. So, we are left with four other regions with high probabilities:\
1) $n\sim\overline{n}$ and $L<\overline{L}$ (short GRBs)\
2) $n\sim\overline{n}$ and $L>\overline{L}$ (no GRBs)\
3) $n>\overline{n}$ and $L\sim\overline{L}$ (no GRBs)\
4) $n<\overline{n}$ and $L\sim\overline{L}$ (long GRBs)\
Now, we show that the directions trough the clouds associated with region(1) produce the short duration GRBs, and ones associated to regions (2) and (3) produce no GRB, while the others in the forth region produce the long duration ones.\
As to the equation(9) the quantity $l(n/E)$ (which is of the order of the sedov length) is $\sim 7\times 10^{11}cm \ll
\overline{L}$ for $n\sim \overline{n}$ (see Eqn.\[25\]). As seen in Fig.7, the time duration of GRBs associated to such directions is of the order of the time duration of short GRBs (Case (1) above). Fig.8 shows that in our model the calculated time duration of GRBs for $n\sim 10^{-6}\overline{n}$ is of the order of the duration of long GRBs. Using equation(9), we see that in this case ( $n\sim 10^{-6}\overline{n}$ ), $l(n/E)$ is about $5\times
10^{13}cm \sim \overline{L}$. So in our model the long duration GRBs are due to the passing of ejecta through the directions where $n \ll \overline{n}$ and $L \sim\overline{L}$ (case (4) above). Furthermore, Since $l(n/E) \ll \overline{L}$ when $n\sim\overline{n}$, we see that in cases (2) and (3) above, the ejecta would stop in the dense cloud and the produced photons can not scape from the optically thick cloud. In Fig.9, $T_{rec}(L;n,\theta)$ is plotted for a number of densities, when $\eta_{_{L}}=L / l(n/E)$ has the highest permitted value $\eta_{m}$.\
These general features of our calculations result from the general features of our model, and therefore we speculate that any distributions for the clouds thickness and density which are picked around a mean value could explain the general features of the duration distribution. Our chose of gaussian distributions for thickness and density was only for the few parameters needed to describe them.\
The Opacity
-----------
In “The Dense Cloud Model” at this stage we have simply omitted the bursts which ejectas can not go out of the cloud and stop in it (because the emitted photons would be scattered by the dense cloud), and we have claimed that the produced photons by all bursts that succeed to cross out the dense cloud can finally enter the free space.\
At the first glance the model might appear to have a serious problem in opacity, namely, in this model the best fit mean density and mean thickness of the clouds are found to be of the orders of $n\sim 10^{17}cm^{-3}$ and $L\sim 10^{13}cm$. So, as to the relation $\tau_{op}=\sigma_{_{T}}\:n\:L$, ($\sigma_{_{T}}=6.65\times10^{-25}cm^{2}$) one would expect an optical depth of the order of $10^{6}$. But there are two factors that remedy the situation:\
(i) As the numerical calculation shows (see § 4.2), the most probable directions characterized by $n\sim\overline{n}$ and $L\sim\overline{L}$ are dynamically too thick to be crossed out by the ejecta and the radiation produced in this case would be completely scattered by the dense cloud. On the other hand, as discussed in § 4.2, the long duration GRBs are due to the crossing of ejecta through directions where $n\sim 10
^{-6}\overline{n}$ and $L\sim\overline{L}$. since the density is reduced by the factor $10^{-6}$, the optical depth drops to the order of $1$. As explained in § 4.2, the short duration GRBs are due to the directions in the cloud where $n\sim\overline{n}$ and $L\sim 10^{-2}\overline{L}$. In this case the optical depth of the cloud reduces to $\sim10^{4}$, which is yet too high. But,\
(ii) most of photons emitted off the shock front have the chance to be overtaken by the moving shock (appendix B). Moreover, it has been shown [@Momeni] that in high temperatures $kT\sim10^{6}\:m_{e}c^{2}$ the cross section of Compton scattering for instance for $Mev$ photons effectively drops to $\sim 10^{-6}\:\sigma_{_{T}}$, so such a high temperature plasma is much more transparent than what may seem at first. The same is true for the case of a power law distribution of electrons (see Eqn.\[A4\]). Most of the electrons in such a distribution have energies of the order of $\gamma_{e,min}m_{e}c^{2}\sim 2 \times
10^{5}m_{e}c^{2}$, in which we have used equation(A5), taking $p=2.5$, $\xi_{e}=1/3$ and $\Gamma=10^{3}$. For this case the Compton cross section suppresses by the factor of $\sim10^{-3}-10^{-4}$ for $100ev-1Kev$ photons (the energy of the emitted photons in the shocked medium comoving frame is less than their observed energy by the factor of $\Gamma$), and therefore, the optical depth drops to the order of $1$. So the photons overtaken by the shocked medium may remain in it without being scattered until the shocked medium crosses up the dense cloud. Briefly, due to this second factor, the optical depth in short duration case diminishes to $\tau_{op}\sim1$, and the optical depth in long duration case diminishes to $\tau_{op}\ll1$.\
The variability seen in the light curves of long duration GRBs maybe attributed to the heterogeny of cloud’s density in the ejecta’s trajectory. This maybe the case in long duration GRBs for which $\tau_{op}\ll1$, but in short duration GRBs for which $\tau_{op}\sim1$, we expect that such a heterogeny would not be appeared.\
The lateral expansion of the shocked medium can reduce its density and therefore its optical depth. This effect of course may cool the shocked medium and cause an increase in effective Compton cross section in it. The consideration of dynamical feedback makes our modeling much more complicated and it has been neglected here. In “The Dense Cloud Model” at this stage we have simply omitted the bursts which ejectas can not go out of the cloud and stop in it, but have claimed that the produced photons by all bursts that succeed to cross out the dense cloud can finally enter free space.\
A complete study needs to include flux computations. An exact comparison with observed time duration data would be possible only when the theoretical time duration appearing in this work were exactly evaluated in the same manner as the observable duration $T_{90}$ is defined (as the time interval over which $5$ percent to $95$ percent of the burst counts accumulate). Moreover, the BATSE’s triggering mechanism made it less sensitive to short GRBs than to long ones and therefore short GRBs were detected to smaller distances [@mao; @cohen; @katz2], so a smaller number of them have been observed. Lee & Petrosian [@petrosian] studied this effect and corrected the number of short GRBs. In a more exact work this correction must be considered too.
We acknowledge the anonymous referee for valuable comments. F. S. acknowledges Mr. Mehdi Haghighi for his guides on computational methods. This research has been partly supported by Grant No. NRCI 1853 of National Research Council of Islamic Republic of Iran.
Scyncrotron Cooling Time in Dense Cloud Model
=============================================
In this part we use the review paper of Piran (1999) to estimate the synchrotron cooling time in our model. The synchrotron cooling time in the comoving frame is: $$\label{}
t_{syn,co}=\frac{3m_{e}c}{4\sigma_{T}U_{_{B}}\:\gamma_{e}}~ ,$$ where $\sigma_{T}$ is the Thompson cross section and $\gamma_{e}$ is the Lorentz factor of the emitting electron, while $U_{_{B}}$ stands for the energy density of magnetic field which in GRB literature is assumed to be proportional to $u$, the comoving internal energy density of the shocked matter: $$\label{}
U_{_{B}}=\xi_{_{B}}u ~ ,$$ so that $\xi_{_{B}}$ represents the share of magnetic field in $u$, which is given by: $$\label{}
u=4\Gamma^{2}n m_{p} c^{2}~,$$ in which, $n$ is the number density of surrounding medium (ISM, or a dense cloud as assumed in our model), and $\Gamma$ denotes the Lorentz factor of the shocked matter.\
The electrons in shocked media are assumed to develop a power law distribution of Lorentz factors: $$\label{}
N(\gamma_{e})\propto\gamma_{e}^{-p}\:\:\:\:\:\:\:\:\:\:\:\:\:for
:\:\:\:\:\:\:\:\:\:\:\:\gamma_{e}>\gamma_{e,min}~.$$ The convergence of total energy of the electrons requires the power index $p$ to be greater than $2$, while the assumed lower limit $\gamma_{e,min}$ is to prevent the divergence of electron number density and is obtained to be: $$\label{}
\gamma_{e,min}=\frac{m_{p}}{m_{e}}\:
\frac{p-2}{p-1}\:\xi_{e}\:\Gamma ~ ,$$ where $\xi_{e}\equiv u_{e} / u$ represents the share of the electrons in the internal energy of shocked matter.\
Furthermore, the time interval between emission of two photons from the same point in the comoving frame, $\delta t_{co}$, and the time interval $\delta t_{\oplus}$, which represents the time interval of their successive arrival to a cosmologically distant observer (at the earth), are related as below: $$\label{}
\delta t_{\oplus}=(1+z)\frac{\delta t_{co}}{2\Gamma}~.$$ Equations(A1-A6) are adopted from Piran (1999). Now, by substituting equations(A2),(A3), & (A5) in equation(A1), and then using equation(A6), the synchrotron cooling time measured by a terrestrial observer turns out to be: $$\begin{aligned}
\label{}
t_{syn,\oplus}<\frac{3}{32}(1+z)\xi_{e}^{-1}\xi_{B}^{-1}
\left(\frac{m_{e}}{m_{p}}\right)^{2}\left(\frac{p-1}{p-2}\right)(\sigma_{{^T}}n c)^{-1}\Gamma^{-4}sec\nonumber\\
\sim
10^{-12}(1+z)\left(\frac{\xi_{e}}{0.1}\right)^{-1}\left(\frac{\xi_{B}}{0.1}\right)^{-1}
\left(\frac{n}{10^{18}}\right)^{-1}\Gamma^{-4}sec ~,\end{aligned}$$ which is clearly much less than all observed time durations of GRBs. So, the synchrotron emission in a dense cloud model must be considered as an instantaneous process, and therefore the shock front must be regarded as the emitting surface. This allows us to attribute time duration of a GRB merely to the time that the shock front takes to cross the dense cloud.\
The relation between Lorentz factors of the shocked matter and the emitting surface
===================================================================================
Here, we want to find the relation between $\Gamma$ and $\Gamma^{\prime}$ which are respectively the Lorentz factors of the shocked matter and of the shock front (which is the emitting surface in our model; see appendix A). In Fig.10, $\beta$ and $\beta^{\prime}$ correspond to shocked matter and shock front speeds, both measured in the (central) source frame. In the shocked matter frame (Fig.11) the dense cloud which is seen to have a density $\Gamma\:n$, moves toward the shocked medium with a speed $\beta$, while being compressed to a density equal to $4\Gamma n$. Consequently, as illustrated in Fig.11, the shocked medium expands towards right with a speed $\beta^{\prime}_{co}$ which is in fact the speed of the shock front in the shocked matter frame (compare Fig.10 and Fig.11). Considering the conservation of nucleon number, it is seen that: $$\label{}
\beta^{\prime}_{co}=\frac{\beta}{4} ~ .$$ Considering the relativistic summation of velocities, we have: $$\label{}
\beta^{\prime}=\frac{\beta^{\prime}_{co}+\beta}{1+\beta
\beta^{\prime}_{co}} ~ .$$ Noting the relations $\Gamma=(1-\beta^{2})^{-1/2}$ and $\Gamma^{\prime}=(1-{\beta^{\prime}}^{2})^{-1/2}$, the substitution of equation(B1) in equation(B2) finally yields: $$\label{}
\Gamma^{\prime}\simeq\sqrt{\frac{5}{3}}\:\:\Gamma ~ ,$$ The distance from the origin to the shock front, denoted by $r^{\prime}$, can be obtained by integrating $\beta^{\prime}$ over $t$: $$\label{}
r^{\prime}=\int_{0}^{t}\beta^{\prime}\:c dt\:+r_{0}~.$$ The results are shown in Fig.12. As seen, the difference between $r$ and $r^{\prime}$ never exceeds one percent. This result justifies the use of symbol $r$ (instead of $r'$ )for the location of the shock front through out our formulation.\
Now, let’s consider a photon that is radiated from the shock front (the emitting surface) moving with the Lorentz factor $\Gamma'$ (Eqn.\[B3\]), in a direction which makes an angle $\epsilon$ with the velocity vector of the shocked matter (Fig.10). if : $$\label{}
\epsilon > \Gamma'^{-1},$$ the photon would be overtaken by the shock front. The majority of the emitted photons fulfill this condition when $\Gamma'\sim
10^{3}$ and $\zeta_{0}\sim 10^{-2}$ (see Eqn.\[25\]).
Geometrical Considerations
==========================
Here the effect of burster geometry on the observed time duration is investigated. At first, as shown in Fig.13, we consider a radiating segment on the shock front. The symmetry axis is denoted by $z^{\prime}$. Noting the definition of $\epsilon$ in the figure, it is seen that in addition to a radial component $dr/dt$, the velocity vector of the segment must have a lateral component $v_{lat}$ which is equal to: $$\label{}
v_{lat}=\frac{\epsilon}{\Gamma(\eta) \zeta(\eta)}~c_{s}~,$$ as measured in the source frame. In equation(C1) the non-dimensional radius $\eta$, as defined in equation(8), is used instead of $r$, . Equation(C1) is obtained simply by using equation(5) and assuming that the lateral speed of the segment in a frame moving (only radially and) instantaneously along with the segment, is the fraction $\epsilon / \zeta(\eta)$ of sound speed $c_{s}$ (which is the lateral speed of the emitting surface at its edges), and noting that the lateral speed in the source frame is less than its corresponding value in the (radially instantaneous) comoving frame by the factor $1/\Gamma$. It is well known that the radiation emitted by the segment is almost confined to an angle $1/\Gamma$. In Fig.13 the axis of the radiation cone emitted by the segment is denoted by $z^{\prime\prime}$, which is parallel to the velocity vector of the segment and, as seen in the figure, makes an angle $\epsilon+\delta$ with $z'$ axis, where $\delta=\tan^{-1}[\epsilon c_{s} / (\Gamma \beta c \zeta)]$. So the radiation angle $\zeta_{rad}(\eta,\epsilon)$ (as depicted in the figure) can be written as below: $$\label{}
\zeta_{rad}(\eta,\epsilon)=\epsilon+\tan^{-1}\left(\frac{\epsilon\:
c_{s}/c}{\Gamma(\eta)\:\beta(\eta)\: \zeta(\eta)
}\right)+1/\Gamma(\eta)~,$$ and the total radiation angle, defined as the radiation angle at the edge of the emitting surface, can be written as: $$\begin{aligned}
\label{}
\zeta_{rad}(\eta)\equiv\zeta_{rad}(\eta,\epsilon=\zeta(\eta))=\zeta(\eta)+\tan^{-1}\left(\frac{
c_{s}/c}{\Gamma(\eta)\:\beta(\eta)\: }\right)+1/\Gamma(\eta)~.\end{aligned}$$ This angle represents the cone of space illuminated by the emitting surface. Using the results of equation(7), the total radiation angle can be evaluated for every “$\eta$”. Having defined the total radiation angle, we explore the effect of burster geometry on its observed time duration $T_{\oplus}$. As shown in Fig.2, the problem is studied in a spherical coordinate system in which the central engine is taken as the origin, and the line-of-sight as the polar axis $z$. Consider a photon emitted off a point on the emitting surface (the shock front) with radial coordinate $r$ and polar coordinate $\Theta$; and at an instance t, measured in the source frame (the point is not shown in the figure). The relation between $t$ and the photon arrival time $t_{rec}$ to a (cosmologically) near observer is obtained by Granot, Piran, & Sari [@granot]. Making suitable for our model, it is adjusted to the form below: $$\label{}
t_{rec}=t-\frac{r\:\cos\Theta-r_{0}}{c}~.$$ In this equation the instance $t=0$ is defined as the time that the ejecta collides with the dense cloud at $r=r_{0}$; while $t_{rec}=0$ is the time that the (cosmologically) near observer receives the photon emitted at $t=0$ from the point with coordinates $r=r_{0}$ and $\Theta=0$. Defining: $$\label{}
\tau_{rec}\equiv \frac{c\:t_{rec}}{l(n/E)}~,$$ and noting equation(8) and equation(12), we rewrite equation(C4) in the form below: $$\label{}
\tau_{rec}=\tau-\eta\:\cos\Theta+\eta_{0}~.$$ Multiplying equation(C4) by the cosmological time dilation term $(1+z)$, results in the arrival time as maybe observed at the earth: $$\label{}
t_{\oplus}=(1+z)t_{rec}
=(1+z)\left(t-\frac{r\:\cos\Theta-r_{0}}{c}\right)~,$$ or equivalently: $$\label{}
\tau_{\oplus}=(1+z)\tau_{rec}
=(1+z)(\tau-\eta\:\cos\Theta+\eta_{0})~,$$ in which we used the non-dimensional time duration $\tau_{\oplus}$ defined as below: $$\label{}
\tau_{\oplus}\equiv \frac{c\:t_{\oplus}}{l(n/E)}~.$$ Now, as shown in Fig.2, we consider a situation where the ejecta’s symmetry axes makes an angle $\theta$ with the line of sight. The necessary condition that at least some photons of the emitting surface are detected by the near observer is: $$\label{}
\zeta_{rad}(\eta)>\theta ~.$$ Here we denote the inverse of function $\zeta_{rad}(\eta)$ by $\eta_{rad}(\zeta_{rad})$, which gives the radius corresponding to $\zeta_{rad}$. As seen in Fig.2, for $\theta$’s larger than $\zeta_{rad}(\eta_{0})$, the first photons reaching the detectors are those emitted at $\eta=\eta_{rad}(\theta)$. So, we use $\tau(\eta)$ (which is obtainable by solving equation(11)) to write: $$\label{}
\tau_{1}(\theta)=\cases{0&\mbox if $\:\:\:\:\:\theta <
\zeta_{rad}(\eta_{0})$ \cr
\tau(\eta_{rad}(\theta)) &\mbox if$\:\:\:\:\:\:\theta > \zeta_{rad}(\eta_{0})$},$$ where $\tau_{1}(\theta)$ represents the starting time (in the source frame) that the emitted photons can reach the (cosmologicaly near) observer. Now, using the numerical results of equation(11) we can obtain the function $\eta(\tau)$. Then, considering equation(C6), the non-dimensional time $\tau_{rec,1}(\theta)$ corresponding to $\tau_{1}(\theta)$ would be as below: $$\label{}
\tau_{rec,1}(\theta)=\cases{0&\mbox :$\:\:\theta < \zeta_{0}$ \cr
-\eta_{0}\cos(\theta-\zeta_{0})+\eta_{0} &\mbox :$\:\:\zeta_{0} < \theta < \zeta_{rad}(\eta_{0})$ \cr
\tau(\eta_{rad}(\theta))-\eta_{rad}(\theta)\cos[\theta-\zeta(\eta_{rad}(\theta))]+\eta_{0}&\mbox :$\:\:\theta > \zeta_{rad}(\eta_{0})$
}~.$$ Now, we are to find the time $\tau_{rec,2}(\theta)$ after which no photons can be detected by the near observer. In Fig.2, the photons emitted from the edge point $A$ can reach us at all times greater than $\tau_{1}(\theta)$. Let’s remind that in our model the emission process terminates at the time that the shocked matter goes out of the cloud (appendix A). As is seen in equation(C6), the closer to the point $B$ is a point on the emitting surface, the later its emitted photons would reach the near observer, of course, provided that the observer line-of-sight remains in the radiation cone of the emitting point.\
Defining: $$\label{}
\eta_{L}\equiv\frac{r_{0}+L}{l(n/E)}~,$$ and recalling equation(C2), we can solve the equation: $$\label{}
\zeta_{rad}(\eta_{_{L}},\epsilon)=\theta~,$$ to find the function $\epsilon=\epsilon(\eta_{_{L}},\theta)$, which specifies the furthest point (on the shock front at the radius $\eta_{_{L}}$) which its radiation reaches us, of course, if it remains smaller than $\zeta(\eta_{_{L}})$.\
Now, using equation(C6), we can find the instance that the last photons reach the near observer: $$\label{}
\tau_{rec,2}(\theta,\eta_{_{L}},\eta_{0})=\tau(\eta_{_{L}})-
\eta_{_{L}}\:\cos(\theta+min[\epsilon(\eta_{_{L}},\theta),\zeta(\eta_{_{L}})])+\eta_{0}~.$$ Finally, the non-dimensional time duration of a GRB, $\tau_{rec}(\theta,\eta_{_{L}})$, will be equal to $\tau_{rec,2}(\theta,\eta_{_{L}})-\tau_{rec,1}(\theta)$ and, as to equation(C12) and equation(C15), besides being a function of parameters of the ejecta and the cloud, it is also a function of the inclination angle $\theta$. So, the time duration of a GRB as measured by an observer cosmologivally near to it would be a function of $L$, $n$, and $\theta$: $$\label{}
T_{rec}=T_{rec}(L;n,\theta)~,$$ where: $$\label{}
T_{rec}=\tau_{rec}(\theta,\eta_{_{L}})
\frac{l\left(n/E\right)}{c},$$ The expression(C16) needs some explanations. In our model the ejecta parameters $\zeta_{0}$ , $E$ and $\Gamma_{0}$ and the radius $r_{0}$ are assumed to be the same in all GRBs. So, these parameters do not appear in equation(C16) explicitly , though it is implicitly a function of them too. At the mean time, the cloud’s density $n$ and its thickness $L$ are assumed to be different in different directions, and therefore they do appear explicitly in the expression (C16). Furthermore, if a cloud thickness $L$ is much more than its associated $Sedov$ length $l_{sedov}\equiv(E/n\:m_{p}\:c^{2})^{1/3}$, the ejecta may not cross it up, and finally stops in it. In such a situation the time duration of a GRB would not be a function of $L$, while it still remains dependent on $n$ and $\theta$. This is why in expression (C16), the quantity $L$ is distinguished from $n$ and $\theta$ by a semicolon. The rearrangement of the expression(C16) in the form $L=L(T_{rec},)\theta,n$ would be meaningful only when we are dealt with the situation where the ejecta succeed to cross up the cloud (see Eqn.\[15\]).\
The explicit form of $F_{_{GRB}}(z)$
====================================
Here the relation between $F_{_{GRB}}(z)$ appearing in equation(23) and the GRB occurring rate $f_{_{GRB}}(z)$ (in units of $Mpc^{-3}\:yr^{-1}$) is derived, so that by adopting a cosmological model for the occurring rate, the integral in equation(23) can be evaluated.\
The number of GRBs that their effects could reach us in a time interval $\delta t_{0}$ (which is very much less than the present comoving time $t_{0}$), and from a spherical volume element $\delta V_{z}$ (disregarding various effects, such as the geometrical ones discussed in appendix C, or those related to detectors threshold which affect the number of detected GRBs) is as below: $$\begin{aligned}
\label{}
\delta^{2}N_{_{GRB}}=f_{_{GRB}}(z)\:\delta V_{z}\:\:\frac{\delta t_{0}}{1+z}\nonumber\\
=f_{_{GRB}}(z).\:4\pi R^{^{3}}(z)\:r^{^{2}}(z)\:\delta r(z)\:
\frac{\delta t_{0}}{1+z}~,\end{aligned}$$ where $r(z)$ denotes the non-dimensional radial parameter of the source that its effects reach us with a red shift z, and $R(z)$, the scale factor at this red shift. In Einstein-de Sitter model we have: $$\label{}
R_{0}\:r(z)=\frac{2c}{H_{0}}\{1-(1+z)^{-1/2}\}~,$$ where $H_{0}$ is the Hubble constant. Furthermore, in FRW metrics: $$\label{}
R(z)=R_{0}(1+z)^{-1}~,$$ where $R_{0}\equiv R(z=0)$. Now, using equations(D2) and (D3) in equation(D1) we obtain: $$\label{}
F_{_{GRB}}(z) \propto \frac{\delta^{2}N_{_{GRB}}}{\delta t_{0}
\delta
z}=2\pi\:\left(\frac{2c}{H_{0}}\right)^{^{3}}(1+z)^{^{-11/2}}\:\{1-(1+z)^{-1/2}\}^{^{2}}\:f_{_{GRB}}(z)~.$$ What is remained is the explicit form of $f_{_{GRB}}(z)$. The high variability seen in GRB light curves has convinced the investigators to relate GRBs to stellar objects , and consequently their rate $f_{_{GRB}}(z)$ to the star formation rate $f_{_{SF}}(z)$. The simplest model is of course a proportional model $f_{_{GRB}}(z)\propto f_{_{SF}}(z)$. the proportional model may be correct if GRBs are attributed to the evolution of massive stars whose lifetime is negligible in comparison with the cosmological time scale, but in NS-NS mergers model the proportionality may not be valid (because of the delay time from the star formation to NS-NS mergers). Wijers et al. [@wijers] claimed that there is a good consistency between the proportional model and the observed GRB brightness distribution, while Petrosian & Lloyd [@petrosian2] concluded that none of the NS-NS and the proportional model are in agreement with the observed $f_{_{SF}}(z)$. Totani [@totani] ascribed this discrepancy to the uncertainties in SFR observations. Anyway we simply assume the GRB rate to be as below: $$\label{}
f_{_{GRB}}(z)=f_{_{GRB}}(0)(1+z)^{^{3+q}}~,$$ and treat $q$ as a free parameter that its best value should be obtained during the fitting procedure. In Fig.14, $F_{_{GRB}}(z)$ is plotted for a number of $q$’s. Clearly the case $q=0$ corresponds to a universe where the changes in the rates of astrophysical phenomena are only due to its expansion (non-evolutionary universe). It can be seen that $F_{_{GRB}}(z)$ takes its maximum at $z\sim1$, which is not very sensitive to the magnitude of $q$.
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![Distribution of log of GRBs’ time Duration (normalized), obtained from data of BATSE 4th catalog.[]{data-label="fit"}](fig1.eps){width="8cm"}
![Geometry of Radiation. In the figure, $\eta_{rad}(\theta)$ denotes the radius where the radiation angle $\zeta_{rad}$ becomes equal to $\theta$ (see appendix C). As shown in the figure (for $\theta$’s larger than $\zeta_{rad}(\eta_{0})$) the first photons reaching us are those emitted from the edge of the shock front at this radius. If $\epsilon(\eta_{_{L}},\theta)<\zeta(\eta_{_{L}})$, then the last photons reaching us are those emitted from $\epsilon=\epsilon(\eta_{_{L}},\theta)$ on the shock front (point S in the figure). The point S would be the furthest visible point on the shock front, since the line of sight fells out of the radiation cone of the points farther than it (points between S and B). If $\epsilon(\eta_{_{L}},\theta)>\zeta(\eta_{_{L}})$, then the point B would be the last visible point (see Eqn.\[C15\]).[]{data-label="fit"}](fig2.eps){width="12cm"}
![The Lorentz factor of shocked medium $\Gamma(\eta)$ versus the non-dimensional radius $\eta$, with $\Gamma_{0}= 1000$ and $\zeta_{0}=0.01$ (see Eqn.\[7\]). []{data-label="fit"}](fig3.eps){width="8cm"}
![The evaluated non-dimensional time duration $\tau_{rec}(\theta,\eta_{_{L}})$ of GRBs, versus $\theta$ and $\eta_{_{L}}$, with $\Gamma_{0}= 1000$ and $\zeta_{0}=0.01$, (see appendix C).[]{data-label="fit"}](fig4.eps){width="12cm"}
![Results of best fitting. The solid curve shows the observed time duration distribution and the dashed one is calculated using the best of values for the model parameters. $\chi^{2}_{min}=1.4 pdf$ (see § 3).[]{data-label="fit"}](fig5.eps){width="8cm"}
![Results of best fitting. Same as Fig.5 with the data corresponding to the structure with $0.3 < \log T _{\oplus} < 0.9$ were cut. $\chi^{2}=1.1 pdf$ (see § 3).[]{data-label="fit"}](fig6.eps){width="8cm"}
![The time duration of GRBs, evaluated in our model with $n=\overline{n}$. It is plotted versus the inclination angle $\theta$ for a number of possible thicknesses $\eta_{_{L}}$. As seen, the order of time durations are of the order of ones in short duration GRBs (see § 4.2). []{data-label="fit"}](fig7.eps){width="8cm"}
![The time duration of GRBs, evaluated in our model with $n=10^{-6}\overline{n}$. It is plotted versus the inclination angle $\theta$ for a number of possible thicknesses $\eta_{_{L}}$. As seen, the order of time durations are of the order of ones in long duration GRBs (see § 4.2). []{data-label="fit"}](fig8.eps){width="8cm"}
![$\log T_{rec}(L;n,\theta)$ is plotted versus the inclination angle $\theta$ for a number of densities, with $\eta_{_{L}}\equiv L / l(n/E)$ equal to the highest permitted value $\eta_{m}$. The number near each curve is $\log (n /
\overline{n})$ ($\overline{n}=2.9 \times 10^{17} cm^{-3} $) (see § 3 and § 4.2 ). []{data-label="fit"}](fig9.eps){width="8cm"}
{width="8cm"}
(100,100)(0,0) (110,270)[$\beta^{\prime}$]{} (24,270)[$\beta$]{}
\[fit\]
{width="8cm"}
(100,100)(0,0) (90,275)[$\beta$]{} (30,275)[$\beta^{\prime}_{co}$]{}
\[fit\]
![ $(r^{\prime} - r)/r$ versus $\eta \equiv r/l(n/E)$, with $ n = 2.9 \times 10^{17} cm^{-3}$ (see appendix B). []{data-label="fit"}](fig12.eps){width="8cm"}
![Geometry of Radiation. The lateral speed of the emitting segment, $v_{lat}$, causes the radiation cone axis $z''$ to make an angle $\epsilon+\delta$ with $z'$ axis, where $\delta = \tan^{-1}[\epsilon
c_{s} / (\Gamma \beta c \zeta)]$ (see appendix C). []{data-label="fit"}](fig13.eps){width="12cm"}
![The quantity $F_{_{GRB}}(z)$ (the probability density of observing a GRB at a redshift $z$) is plotted for a number of $q$’s (the value of $q$ is written near to the peak of its corresponding curve). As seen, all curves have a maximum at $z \sim 1$. This explains why the most of GRBs have redshifts $z
\sim 1$ (see Eqn.\[D5\]).[]{data-label="fit"}](fig14.eps){width="12cm"}
[^1]: e-mail: fmomeni@mehr.sharif.edu
[^2]: e-mail: samimi@sharif.edu
|
---
author:
- |
Tomonori Kouya\
Shizuoka Institute of Science and Technology\
2200-2 Toyosawa, Fukuroi, Shizuoka 437-8555 Japan[^1][^2]
title: 'Practical Implementation of High-Order Multiple Precision Fully Implicit Runge-Kutta Methods with Step Size Control Using Embedded Formula'
---
> **Abstract**
>
> We propose a practical implementation of high-order fully implicit Runge-Kutta(IRK) methods in a multiple precision floating-point environment. Although implementations based on IRK methods in an IEEE754 double precision environment have been reported as RADAU5 developed by Hairer and SPARK3 developed by Jay, they support only 3-stage IRK families. More stages and higher-order IRK formulas must be adopted in order to decrease truncation errors, which become relatively larger than round-off errors in a multiple precision environment. We show that SPARK3 type reduction based on the so-called W-transformation is more effective than the RADAU5 type one for reduction in computational time of inner iteration of a high-order IRK process, and that the mixed precision iterative refinement method is very efficient in a multiple precision floating-point environment. Finally, we show that our implementation based on high-order IRK methods with embedded formulas can derive precise numerical solutions of some ordinary differential equations.
Introduction
============
Multiple precision floating-point (MP) arithmetic is an effective approach to solving ill-conditioned problems that cannot be solved precisely with IEEE754 double precison floating-point (DP) arithmetic. We have been developing BNCpack, a DP and MP numerical computation library based on MPFR[@mpfr], an arbitrary precision and IEEE754 standard compatible floating-point arithmetic library, for the natural number arithmetic kernel in GNU MP (GMP)[@gmp] that is well-tuned for various CPU architectures. In this paper, we propose the implementation of a practical ordinary differential equation (ODE) solver based on BNCpack and high-order implicit Runge-Kutta (IRK) methods; its availability and efficiency are verified via numerical experiments.
DP ODE solvers based on IRK methods have been developed as RADAU5 (Radau IIA formula) and SPARK3 (selectable in Radau, Gauss, and Lobatto formulas) by Haier and Jay, respectively. Both IRK implementations support only 3-stage formulas, which is not sufficient to obtain precise numerical solutions in an MP environment. The use of MP floating-point arithmetic decreases round-off errors in an IRK process, thereby increasing truncation errors. More stages and higher-order IRK formulas are neccessary in an MP environment. Consequently, higher-dimensional nonlinear equations must be solved in high-order IRK processes. Such so-called inner iteration includes linear equations of the same dimension, which must be solved efficiently. In this process, RADAU5 reduces these coeffient matrices to complex diagonal matrices, and SPARK3 reduces them to real nonsymmetric tridiagonal matrices. In an MP environment, we must also accelerate these processes. In addition, we must be able to control the step sizes in an IRK process by using local error estimation at each discrete point in the given integration interval. For this reason, embedded formulas incidental to high-order IRK ones are neccessary. Althought Hairer proposed 4-stage embedded formula incidental to 3-stage Radau IIA formula, he did not describe explicitly the existence of embedded formulas for other IRK ones.
In this paper, we first state mathematical definitions and provide a framework for IRK algorithms. In section 3, we explain the method of linear equations to be solved in the inner iteration of an IRK process. We show that the RADAU5 type reduction is not effective for high-order IRK methods, and that the mixed precision iterative refinement method can achive drastic acceleration, as shown via benchmark tests of a linear ODE. In section 4, we describe the derivation of embedded formulas for any IRK ones, and we discuss the A-stabilities. In section 5, we describe numerical experiments conducted for some test problems in order to demonstrate the high performance of our implementations. Finally, we conclude this paper and discuss the scope for future studies in section 6.
Algorithm of Implicit Runge-Kutta Method
========================================
We define the initial value proble (IVP) of the ODE to be solved as $$\left\{\begin{array}{l}
\displaystyle\frac{d\mathbf{y}}{dx} = \mathbf{f}(x, \mathbf{y}) \in\mathbb{R}^n \\
\\
\mathbf{y}(x_0) = \mathbf{y}_0.%\in\mathbb{R}^n
\end{array}\right. \label{eqn:ode}$$ The integration interval is given as $[x_0, \alpha]$.
For this IVP of the ODE, we divide the integration interval into $x_0$, $x_1 := x_0 + h_0 $, ..., $x_{k+1} := x_k + h_k$, .... In order to obtain numerical solutions $\mathbf{y}_k \approx \mathbf{y}(x_k)$ in each step by using an $m$-stage IRK method, we must solve the following nonlinear system of equations (\*). This process of solving with various iterative methods called inner iteration in IRK methods. $$\begin{split}
& (*) \left\{\begin{array}{rcl}
\mathbf{k}_1 &=& \mathbf{f}(x_k + c_1 h_k, \mathbf{y}_k + h_k\cdot \sum^{m}_{j=1} a_{1j}\mathbf{k}_j)\\
\mathbf{k}_2 &=& \mathbf{f}(x_k + c_2 h_k, \mathbf{y}_k + h_k\cdot \sum^{m}_{j=1} a_{2j}\mathbf{k}_j)\\
&\vdots& \\
\mathbf{k}_m &=& \mathbf{f}(x_k + c_m h_k, \mathbf{y}_k + h_k\cdot \sum^{m}_{j=1} a_{mj}\mathbf{k}_j)\\
\end{array}\right. \\
\mathbf{y}_{k+1} &:= \mathbf{y}_k + h_k\cdot \sum^{m}_{j=1} b_j \mathbf{k}_j
\end{split} \label{eqn:eqn_irk}$$ where the constant coefficients in the IRK formula, $c_1$, ..., $c_m$, $a_{11}$, ..., $a_{mm}$, $b_1$, ..., $b_m$, can be expressed as follows: $$\begin{array}{c|cccc}
c_1 & a_{11} & a_{12} & \cdots & a_{1m} \\
c_2 & a_{21} & a_{21} & \cdots & a_{2m} \\
\vdots & \vdots & \vdots & & \vdots \\
c_m & a_{m1} & a_{m2} & \cdots & a_{m,m}\\ \hline
\ & b_1 & b_2 & \cdots & b_m \\
\end{array} = \begin{array}{c|c}
\mathbf{c} & A \\ \hline
& \mathbf{b}^T
\end{array}.$$
All computations are the same in each discretization point $x_k$; hence, we consider only the computation $\mathbf{y}_0 \rightarrow \mathbf{y}_1 \approx \mathbf{y}(x_0 + h_0) = \mathbf{y}(x_0 + h)$.
#### Quasi-Newton Method
If Newton method is adpoted as the numerical argorithm in inner iteration, the algorithm is as follows. The initial guesses are $\mathbf{k}^{(0)}_1$, ..., $\mathbf{k}^{(0)}_m$, and the approximations of unknowns $\mathbf{k}_1$, $\mathbf{k}_2$, ..., $\mathbf{k}_m$ are calculated by iterating the computations as $$\left[\begin{array}{c}
\mathbf{k}^{(l+1)}_1 \\
\mathbf{k}^{(l+1)}_2 \\
\vdots \\
\mathbf{k}^{(l+1)}_m
\end{array}\right] := \left[\begin{array}{c}
\mathbf{k}^{(l)}_1 \\
\mathbf{k}^{(l)}_2 \\
\vdots \\
\mathbf{k}^{(l)}_m
\end{array}\right] - J^{-1}(\mathbf{k}^{(l)}_1, ..., \mathbf{k}^{(l)}_m) \left[\begin{array}{c}
\mathbf{k}^{(l)}_1 - \mathbf{f}(x_0 + c_1 h, \mathbf{y}_0 + h \sum^{m}_{j=1} a_{1j} \mathbf{k}^{(l)}_j)\\
\mathbf{k}^{(l)}_2 - \mathbf{f}(x_0 + c_2 h, \mathbf{y}_0 + h \sum^{m}_{j=1} a_{2j} \mathbf{k}^{(l)}_j)\\
\vdots \\
\mathbf{k}^{(l)}_m - \mathbf{f}(x_0 + c_m h, \mathbf{y}_0 + h \sum^{m}_{j=1} a_{mj} \mathbf{k}^{(l)}_j)
\end{array}\right]$$ where $J(\mathbf{k}^{(l)}_1, \mathbf{k}^{(l)}_2, ..., \mathbf{k}^{(l)}_m) \in\mathbb{R}^{mn\times mn}$ is given by $$J(\mathbf{k}^{(l)}_1, \mathbf{k}^{(l)}_2, ..., \mathbf{k}^{(l)}_m) = \left[\begin{array}{c|c|c|c}
I_n - J_{11} & -J_{12} & \cdots & -J_{1m} \\ \hline
-J_{21} & I_n - J_{22} & \cdots & -J_{2m} \\ \hline
\vdots & \vdots & & \vdots \\ \hline
-J_{m1} & -J_{m2} & \cdots & I_n - J_{mm} \\
\end{array}\right] ,$$ $I_n$ is an $n$-dimensional identity matrix, and $J_{pq}$ is given by $$J_{pq} = h a_{pq} \frac{\partial}{\partial \mathbf{y}} \mathbf{f}(x_0 + c_p h, \mathbf{y}_0 + h \sum^{m}_{j=1} a_{pj} \mathbf{k}^{(l)}_j)\in \mathbb{R}^{n\times n} .$$
In order to compute this part, we solve the following $mn$-dimensional system of linear equations with the coefficient matrix $J(\mathbf{k}^{(l)}_1, \mathbf{k}^{(l)}_2, ..., \mathbf{k}^{(l)}_m)$ for unknowns $[\overline{\mathbf{z}_1}\ \overline{\mathbf{z}_2}\ ...\ \overline{\mathbf{z}_n}]^T$. $$\begin{split}
& J(\mathbf{k}^{(l)}_1, \mathbf{k}^{(l)}_2, ..., \mathbf{k}^{(l)}_m) \left[\begin{array}{c}
\overline{\mathbf{z}_1} \\
\overline{\mathbf{z}_2} \\
\vdots \\
\overline{\mathbf{z}_m}
\end{array}\right] = \left[\begin{array}{c}
\mathbf{k}^{(l)}_1 - \mathbf{f}(x_0 + c_1 h, \mathbf{y}_0 + h \sum^{m}_{j=1} a_{1j} \mathbf{k}^{(l)}_j) \\
\mathbf{k}^{(l)}_2 - \mathbf{f}(x_0 + c_2 h, \mathbf{y}_0 + h \sum^{m}_{j=1} a_{2j} \mathbf{k}^{(l)}_j) \\
\vdots \\
\mathbf{k}^{(l)}_m - \mathbf{f}(x_0 + c_m h, \mathbf{y}_0 + h \sum^{m}_{j=1} a_{mj} \mathbf{k}^{(l)}_j)
\end{array} \right] \\
%& \Rightarrow J(\mathbf{K}_l) \mathbf{Z} = (\mathbf{K}_l - \overline{\mathbf{F}}(x_i, \mathbf{K}_l)
\end{split}$$ In this process, we need to have $O(m^2n^2)$ memory in order to store $J(\mathbf{k}^{(l)}_1, \mathbf{k}^{(l)}_2, ..., \mathbf{k}^{(l)}_m)$.
Actually, to accelerate this process, the following fixed coefficient matrix $J$ is used.
$$J(\mathbf{k}^{(l)}_1, \mathbf{k}^{(l)}_2, ..., \mathbf{k}^{(l)}_m) = J(\mathbf{k}^{(0)}_1, \mathbf{k}^{(0)}_2, ..., \mathbf{k}^{(0)}_m)$$
We call this method “Quasi-Newton Method."
#### Simplified Newton Method
For inner iteration in IRK methods, a more simplified Newton method is used conventionally[@hairer]. RADAU5 and SPARK3 select the simplified Newton method.
In this process, $J_{pq}$ used in the Quasi-Newton method is fixed as $$J_{pq} := h a_{pq} \frac{\partial}{\partial \mathbf{y}} \mathbf{f}(x_0, \mathbf{y}_0) = h a_{pq} J ,$$ and then, we use $Y_i = \mathbf{y}_0 + h \sum^{m}_{j=1} a_{ij} \mathbf{f}(x_0 + c_i h, Y_j)$, which is an alternative to $\mathbf{k}^{(l)}_i$. Thus, we can express the system of linear equations to be solved as $$(I_m\otimes I_n - h A\otimes J) \mathbf{Z} = -\mathbf{F}(\mathbf{Y}) \in \mathbb{R}^{mn}, \label{eqn:simplified_newton_linear_eq}$$ where $$\mathbf{F}(\mathbf{Y}) = \left[\begin{array}{c}
Y_1 - \mathbf{y}_0 - h\sum^m_{j=1} a_{1j}\mathbf{f}(x_0 + c_1 h, Y_1) \\
\vdots\\
Y_m - \mathbf{y}_0 - h\sum^m_{j=1} a_{mj}\mathbf{f}(x_0 + c_m h, Y_m)
\end{array}\right].$$
Acceleration of Inner Iteration in IRK process
==============================================
In this section, we treat the simplified Newton method as inner iteration. In this case, the system of linear equations to be solved at each step of inner iteration is expressed in (\[eqn:simplified\_newton\_linear\_eq\]). We can decrease the number of computations by applying reductions based on similarity transformations to IRK matrix $A$. RADAU5 employs complex diagonalization, and SPARK3 employs real unsymmetric tridiagonalization, which is called W-transformation. The latter transformation is better than the former because it does not treat complex arithmetic, and it can avoid the ill-conditioned transformations. Therefore, we employ SPARK3 type reduction for the implementation of high-order IRK methods. In the rest of this section, we compare RADAU5 type and SPARK3 type reductions, and we show that the acceleration due to the mixed precision iterative refinement method can drastically reduce the computational time of an IRK process.
Comparison between RADAU5 type and SPARK3 type Reductions
---------------------------------------------------------
RADAU5 type reduction[@hairer] is based on the fact that IRK matrix $A$ can be transformed into a complex diagonal matrix as $$\Lambda = \mbox{diag}(\lambda_1, \cdots, \lambda_m) = S A S^{-1} .$$ In these cases of IRK Radau, Gauss, and Lobatto formula families, eigenvalues $\lambda_i$ are generally complex numbers.
If RADAU5 type reduction is employed in the simplified Newton method, the system of linear equations to be solved has the coefficient matrix obtained by similarity transformation with $S\otimes I_n$ and $S^{-1}\otimes I_n$, as $$\begin{split}
&(S\otimes I_n)(I_m\otimes I_n - hA \otimes J)(S^{-1}\otimes I_n) = I_m\otimes I_n - h\Lambda \otimes J \\
& = \left[\begin{array}{ccc}
I_n - h\lambda_1 J & & \\
& \ddots & \\
& & I_n - h\lambda_m J
\end{array}\right].
\end{split}$$
RADAU5 type reduction has two advantages: 1. The inner iteration is completely parallelizable, and 2. the order of the required memory is $O(2mn)$. However, $\Lambda, S$, and $S^{-1}$ are complex matrices, and $\kappa_2(S) = \|S\|_2 \|S^{-1}\|_2\rightarrow$ $\infty$ $(m\rightarrow \infty)$; hence, round-off errors increase in inner iteration, especially for higher order IRK formulas, as shown in \[fig:complex\_diagonal\].
![$\kappa_2(S)$ and absolute values of eigenvalues of $A$ for IRK Gauss formulas[]{data-label="fig:complex_diagonal"}](complex_diagonal.eps){width=".5\textwidth"}
On the other hand, SPARK3 type reduction is called block triangulation through W-transformation[@hairer], given by $$\begin{split}
X &= W^T BA W = \left[\begin{array}{ccccc}
1/2 & -\zeta_1 & & & \\
\zeta_1 & 0 & \ddots & & \\
& \ddots & \ddots & -\zeta_{m-2} & \\
& & \zeta_{m-2} & 0 & -\zeta_{m-1} \\
& & & \zeta_{m-1} & 0
\end{array}\right], \\
\mbox{where }& \\
w_{ij} = \tilde{P}_{j-1}(c_i) &= \sqrt{2(j-1)+1} \sum^{j-1}_{k=1} (-1)^{j+k-1} \left(\begin{array}{c}
j-1 \\
k
\end{array}\right) \left(\begin{array}{c}
j+k-1 \\
k
\end{array}\right) c_i^k \\
\mbox{$\tilde{P}_{j-1}(x)$}&: \mbox{$j-1$-th shifted Legendre polynomial} \\
\zeta_i &= \left(2\sqrt{4i^2-1}\right)^{-1},\ B = \mbox{diag}(\mathbf{b}),\ D = W^T B W = \mbox{diag}(1\ 1\ \cdots\ 1).
\end{split}$$
By using W-transformation, we can obtain the reduced coefficient matrix in the simplified Newton method [@jay99] as $$\begin{split}
& (W^TB\otimes I_n)(I_m\otimes I_n - h A\otimes J)(W\otimes I_n) \\
&= D\otimes I_n - hX\otimes J = \left[\begin{array}{ccccc}
E_1 & F_1 & & & \\
G_1 & E_2 & F_2 & & \\
& \ddots & \ddots & \ddots & \\
& & G_{m-2} & E_{m-1} & F_{m-1} \\
& & & G_{m-1} & E_m
\end{array}\right]
\end{split},$$ where $$\begin{split}
E_1 &= I_n - \frac{1}{2}hJ,\ E_2 = \cdots = E_s = I_n \\
F_i & = h\zeta_iJ,\ G_i = -h\zeta_iJ\ (i = 1, 2, ..., m-1)
\end{split}.$$
In the implementation of SPARK3, the left preconditioner matrix $P$, $$P = \left[\begin{array}{ccccc}
\tilde{E}_1 & F_1 & & & \\
G_1 & \tilde{E}_2 & F_2 & & \\
& \ddots & \ddots & \ddots & \\
& & G_{m-2} & \tilde{E}_{m-1} & F_{m-1} \\
& & & G_{m-1} & \tilde{E}_m
\end{array}\right] \approx D\otimes I_n - hX\otimes J$$ is prepared, and then, the precondtioned system of linear equation $$P^{-1} (D\otimes I_n - hX\otimes J) (W\otimes I_n)^{-1} \mathbf{Z}= P^{-1}(W^TB\otimes I_n)(-\mathbf{F}(\mathbf{Y}))$$ is set to be solved for $\mathbf{Z}$. Jay maintained that the left precondition can decrease the number of iterations and accelerate Richardson iteration and GMRES (Generalized Minimal RESidual) methods[@jay99]. However, our numerical experiments show that such preconditioning increases the computational time in MP environment; hence, our current implementation employs such preconditions in DP environment, and does not employ in MP environment.
In comparison with RADAU5 type reduction, SPARK3 type reduction is better because all computations constitute real number arithmetic, and the order of the memory required for the coefficient matrix is $O(3mn)$. In addition, the similarity transformation matrix $W$ can remain well-conditioned when the number of stages $m$ is large, and hence, the effect of round-off errors occuring in the similarity transformation is small. \[table:cond\_w\] shows that the condition number of $\kappa_\infty(W)=\|W\|_\infty\|W^{-1}\|_\infty$ is much smaller than $\kappa_\infty(S)$ used in RADAU5 type reduction.
$m$ 3 5 10 15 20 50
-------------------- -------- -------- ------------------- ---------------------- ---------------------- ----------------------
$\kappa_\infty(S)$ $22.0$ $388$ $3.28\times 10^5$ $2.81\times 10^{8} $ $2.11\times 10^{11}$ $4.25\times 10^{28}$
$\kappa_\infty(W)$ $3.24$ $6.27$ $16.4$ $29.3$ $44.5$ $172$
: Condition numbers of two kinds of similarity transformation matrices[]{data-label="table:cond_w"}
Acceleration by using Mixed Precision Iterative Refinement Method
-----------------------------------------------------------------
The mixed precision iterative refinement method was originally proposed by Moler in 1967[@moler_iterative_ref], and then Buttari et al. showed that their revised algorithm exhibits high performance in many current computing environments. If $S$-digit floating-point arithmetic can be executed more efficiently than $L$ $(>>S)$-digit arithmetic, the system of linear equations $$\mathbf{f}(\mathbf{x}) = C\mathbf{x} - \mathbf{d}$$ can be solved by Newton method and an appropriate linear solver, shown by the following algorithm: $$\begin{aligned}
\mbox{Solve}\ & \ C^{[S]}\mathbf{x}_0^{[S]} = \mathbf{d}^{[S]}\ \mbox{for}\ \mathbf{x}_0^{[S]}. \label{eqn:simple_iterative_ref_l1} \\
\mathbf{x}_0^{[L]} &:= \mathbf{x}_0^{[S]} \nonumber \\
\mbox{For}\ & \ k=0, 1, 2, ... \nonumber \\
& \mathbf{r}_k^{[L]} := \mathbf{d}^{[L]} - C^{[L]}\mathbf{x}_k^{[L]} \label{eqn:simple_iterative_ref1} \\
& \mathbf{r}_k'^{[L]} := \mathbf{r}_k^{[L]} / \|\mathbf{r}_k^{[L]}\| \nonumber \\
& \mathbf{r}_k'^{[S]} := \mathbf{r}_k'^{[L]} \nonumber \\
& \mbox{Solve}\ C^{[S]}\mathbf{z}_{k}^{[S]} = \mathbf{r}_k'^{[S]}\ \mbox{for}\ \mathbf{z}_k^{[S]}. \label{eqn:simple_iterative_ref_l2} \\
& \mathbf{z}_k^{[L]} := \mathbf{z}_k^{[S]} \nonumber \\
& \mathbf{x}_{k+1}^{[L]} := \mathbf{x}_k^{[L]} + \|\mathbf{r}_k^{[L]}\| \mathbf{z}_k^{[L]} \label{eqn:simple_iterative_ref3} \\
& \mbox{Check convergence of $\mathbf{x}_{k+1}$.} \nonumber\end{aligned}$$ where $[S]$ and $[L]$ denote the values expressed and computed in $S$- and $L$-digit floating-point arithmetic, respectively. The above algorithm is the $S$-$L$ mixed precision iterative refinement method for a system of linear equations. The part of (\[eqn:simple\_iterative\_ref\_l1\]) and (\[eqn:simple\_iterative\_ref\_l2\]) theoretically give the solution, and hence, we do not need the above iteration. However, we cannot obtain the true solutions owing to the use of finite precision floating-point arithmetic in these parts. Thus, the residuals $\mathbf{r}_k$ are not all zero. The above algorithm executes some iterations so that the residuals $\mathbf{r}_k$ tend to zero. The number of digits required to compute the residuals increases with the precision of the obtained approximation $\mathbf{x}_k$.
Buttari et. al. also proved that the sufficient condition for convergence is satisfied if the precision of the computation (\[eqn:simple\_iterative\_ref\_l2\]) is less than that of the residual (\[eqn:simple\_iterative\_ref1\]) when the condition number $\kappa(A)=\|A\|\|A^{-1}\|$ is smaller than the precision in the computing environment. In addition, they showed that the combination of (\[eqn:simple\_iterative\_ref1\]) and (\[eqn:simple\_iterative\_ref3\]) computed DP arithmetic, and (\[eqn:simple\_iterative\_ref\_l1\]) and (\[eqn:simple\_iterative\_ref\_l2\]) computed in SP(IEEE754 single-precision) arithmetic can accelerate the solution of relatively less ill-conditioned systems of linear equations via benthmark[@mixed_prec_iterative_ref][@lawn175].
Their mixed precision iterative refinement method is just fitted to solve the system of linear equations (\[eqn:simplified\_newton\_linear\_eq\]) in inner iteration of IRK methods. This tends to be well-conditioned if the step size $h_k$ becomes smaller, and hence, the sufficient condition of convergence is satisfied in almost cases in an IRK process. In particular, for MP arithmetic, the application to the DP-MP type mixed precision iterative refinement method in which (\[eqn:simple\_iterative\_ref\_l2\]) is computed by DP arithmetic can drastically accelerate the entire computational time of an IRK process. We support direct methods and Krylov subspace methods in our current implementation of mixed precision iterative refinement method.
In the rest of this section, we show the acceleration of an IRK method with DP-MP type mixed precision iterative refinement method via benchmark tests applied to a linear ODE.
Our test problem is constructed by using a real normal matrix $R$ comprising uniform random numbers, and by its inverse matrix $R^{-1}$, as follows:
$$\begin{split}
& \left\{\begin{array}{l}
\displaystyle\frac{d\mathbf{y}}{dx} = -(R\ \mbox{diag}(n, n-1, ..., 1)\ R^{-1})\ \mathbf{y}\ \in\mathbb{R}^{128} \\
\\
\mathbf{y}(0) = [1\ ...\ 1]^T \\
\end{array}\right. \\
& \mbox{Integration Interval:} [0, 20].
\end{split}$$ The precision of MP arithmetic is fixed at $50$ decimal digits (167 bits). We compute $\mathbf{y}_1$ by using the $m$-stage $2m$ order Gauss formulas ($m=3,4, ..., 12$ and $h=1/2$), and we compare the following 4 algorithms:
1. Quasi-Newton method with the DP-MP type mixed precision iterative refinement method based on the direct method and without reduction: “Iter.Ref-DM"
2. Simplified Newton method with the simple direct method and with SPARK3 type reduction: “W-Trans"
3. Simplified Newton method with the MP (25 decimal digits)-MP iterative refinement method with SPARK3 type reduction: “W-Iter.Ref-MM"
4. Simplified Newton method with the DP-MP method with SPARK3 type reduction: “W-Iter.Ref-DM"
All computations are executed on an Intel Core i7 920 + CentOS 5.4 x86\_64 machine with gcc 4.1.2 + BNCpack 0.8 + MPFR 3.1.0/GMP 5.0.2. For convenient comparison, the maximum relative error in approximation $\mathbf{y}_1$ is expressed as a line graph in \[fig:irk\_jay\_bench\]. All 4 algorithms can obtain the same accuracy of $\mathbf{y}_1$.
![Performance of IRK methods: in case of 128-dimensional and 50 decimal digits computation[]{data-label="fig:irk_jay_bench"}](irk_jay_bench_m.eps){width=".8\textwidth"}
Although the relative errors over 9 stages undergo less reduction because of the effect of round-off errors, all numerical results from 3- to 12-stages IRK Gauss formulas are precise.
The computational time increases with number of stages employed in the IRK formulas. The Quasi-Newton method without reduction in inner iteration is competitive with less than four stages, but is extremely slow with over 9 stages. On the other hand, the computational times of the three algorithms with SPARK3 type reduction is proportional to the number of stages. Moreover, the application of the DP-MP type iterative refinement method can drastically accelerate IRK processes, especially for formulas with over 5 stages formulas. The maximum obtained speedup ratio is 4.8.
Derivation of Embedded Formula and Step Size Selection
======================================================
In order to implement actual ODE solvers, we need a mechanism of step size selection, based on the local error estimation at each discretized point. In the case of Runge-Kutta (RK) methods, embedded formulas that can reduce the amount of computation and the number of integrated function calls are widely used. For explicit RK (ERK) methods, many ODE solvers have been developed on the basis of the embedded formulas proposed by Fehlberg and Dormand-Prince. On the other hand, the embedded formula incident to 3-stage Radau IIA formula, which is proposed by Hairer[@hairer2] is only one for IRK methods, and it is constructed by a combination of the original formula and a lower order one. Hairer suggested the existence of the same type embedded formulas for other IRK formulas. In this section, we show how to automatically derive embedded formulas for IRK Gauss formulas which have wider A-stability regions.
Hairer’s Embedded Formula
-------------------------
Hairer’s embedded formula for the 3-stage 5th order Radau IIA formula is used in RADAU5, and it can be expressed as follows. The original Radau IIA is $$\begin{array}{c|ccc}
\frac{4-\sqrt{6}}{10} & \frac{88-7\sqrt{6}}{360} & \frac{296-169\sqrt{6}}{1800} & \frac{-2+3\sqrt{6}}{225} \\
\frac{4+\sqrt{6}}{10} & \frac{296+169\sqrt{6}}{1800} & \frac{88+7\sqrt{6}}{360} & \frac{-2-3\sqrt{6}}{225} \\
1 & \frac{16-\sqrt{6}}{36} & \frac{16+\sqrt{6}}{36} & \frac{1}{9} \\ \hline
& \frac{16-\sqrt{6}}{36} & \frac{16+\sqrt{6}}{36} & \frac{1}{9} \\
\end{array}$$ and its embedded formula is the 4-stage formula $$\begin{array}{c|cc}
c_0 = 0 & 0 & \mathbf{0}^T \\
\mathbf{c} & \mathbf{0} & A \\ \hline
& \gamma_0 & \hat{\mathbf{b}}^T
\end{array} \label{eqn:irk_embed}$$ where $\gamma_0$ is any non-zero constant, as recommended by Hairer for the real eigenvalue of IRK matrix $A$ in order to reduce the number of computations in local error estimation. Moreover, $\hat{\mathbf{b}}$ is expressed as $$\hat{\mathbf{b}} = \left[\begin{array}{c}
\hat{b}_1 \\
\hat{b}_2 \\
\hat{b}_3
\end{array}\right] = \left[\begin{array}{c}
b_1 - \frac{2+3\sqrt{6}}{6}\gamma_0 \\
b_2 - \frac{2-3\sqrt{6}}{6}\gamma_0 \\
b_3 - \frac{\gamma_0}{3}
\end{array}\right].$$ This $\hat{\mathbf{b}}$ satisfies the simplifying assumption $B(3)$ [@hairer] $$\mbox{$B(3)$:\ } c_0^{q-1} \gamma_0 + \sum^3_{i=1} \hat{b}_i c_{i}^{q-1} = 1/q\ (q=1, 2, 3).$$ The other coefficients $\mathbf{c}$ and $A$ are the same as the original formula, and hence, the simplifying $C(3)$, given by $$\mbox{$C(3)$:\ } \sum^3_{j=1} a_{ij} c_j^{q-1} = c_i^q / q\ (i=1, 2, 3, q=1, 2, 3),$$ is automatically satisfied. Therefore, the given embedded formula (\[eqn:irk\_embed\]) is of 3rd order at least.
Derivation of Embedded Formulas for any IRK ones
------------------------------------------------
Hairer’s embedded formulation can be generally extended to other IRK formulas. In order to satisfy the simplifying assumption $B(m)$, $\hat{\mathbf{b}}$ is the solution of the following system of linear equations with a Vandermonde matrix of coefficients, [$$\left[\begin{array}{cccc}
1 & 1 & \cdots & 1 \\
c_1 & c_2 & \cdots & c_m \\
\vdots & \vdots & & \vdots \\
c_1^{m-1} & c_2^{m-1} & \cdots & c_m^{m-1}
\end{array}\right] \left[\begin{array}{c}
\hat{b}_1 \\
\hat{b}_2 \\
\vdots \\
\hat{b}_m
\end{array}\right] = \left[\begin{array}{c}
1 - \gamma_0 \\
1/2 \\
\vdots \\
1/m
\end{array}\right] .
\label{eqn:irk_embed_intro}$$]{} By solving the above equation, we can obtain the approximation $\hat{\mathbf{y}}_{k+1}$ derived by the $m$-th order embedded formula, $$\hat{\mathbf{y}}_{k+1} := \mathbf{y}_k + h_k\left(\gamma_0 \mathbf{f}(x_k, \mathbf{y}_k) + \sum^{m}_{j=1} \hat{b}_j Y_j\right). \label{eqn:embed_formula}$$
When we fix $\gamma_0 = 1/8$ for the reason described in the next section, we compute the relative errors of the approximation $\hat{y}_k$ in (\[eqn:mxy\]) at $x = 10$ by using the embedded formula derived from the 3-stage 6th order Gauss IRK formula. $$\begin{array}{l}
\displaystyle\frac{dy}{dx} = -xy \\
y(0) = 1 \\
\mbox{Integration Interval:} [0, 10]
\end{array} \label{eqn:mxy}$$ As a result, we can confirm that the order of the embedded formula is 3, as shown in \[fig:mxy\]. For comparison, the original 6-th order $y_k$ is also plotted in the same figure.
![Relative error of the 3rd order embedded formula derived from 3-stage 6th order IRK formula[]{data-label="fig:mxy"}](mxy.eps){width=".65\textwidth"}
In our implementation of the MP ODE solver based on high-order IRK Gauss formulas, we currently use the following value $\|\mathbf{err}_k\|$ for local error estimation[@hairer2].
$$\|\mbox{\bf err}_k\| = \sqrt{\frac{1}{n}\sum^{n}_{j=1}\left(\frac{\left|\hat{y}^{(k+1)}_j - y^{(k+1)}_j\right|}{ATOL + RTOL\cdot \max\left(\left|y^{(k+1)}_j\right|, \left|y^{(k)}_j\right|\right)}\right)^2}$$
The next step size $h_{k+1}$ at $x_{k+1}$ is set as $$h_{k+1} := 0.9 \|\mathbf{err}_k\|^{1/(m+1)} h_k.$$
A-stability Regions of Embedded Formulas
----------------------------------------
One advantage of IRK formulas can be A-stable; however, the embedded formulas derived from the original IRK formulas are not A-stable. Actually, the stability region of Hairer’s embedded formula is narrower than that of the original Radau IIA formula, as shown in \[fig:irk\_radau2\].
![Stability regions: 3-stage 5th order Radau IIA formula (left) and the corresponding Hairer’s embedded formula (right).[]{data-label="fig:irk_radau2"}](irk_radau35_xy15.eps "fig:"){width=".4\textwidth"} ![Stability regions: 3-stage 5th order Radau IIA formula (left) and the corresponding Hairer’s embedded formula (right).[]{data-label="fig:irk_radau2"}](irk_radau35_embed_xy15.eps "fig:"){width=".4\textwidth"}
We must select the parameter $\gamma_0 \not= 0$ at which the embedded formula can have a wider region. As a result, we currently consider that $\gamma_0 = 1/8$ is better because it is not too small and it can be expressed in powers of 2. In addition, its region is wider than that of Hairer’s embedded formula. The stability region is shown in \[fig:irk\_gauss3\].
![Stability regions: 3-stage 6th order Gauss formula (left) and our embedded formula(left, $\gamma_0 = 1/8$)[]{data-label="fig:irk_gauss3"}](irk_gauss36_xy15.eps "fig:"){width=".4\textwidth"} ![Stability regions: 3-stage 6th order Gauss formula (left) and our embedded formula(left, $\gamma_0 = 1/8$)[]{data-label="fig:irk_gauss3"}](irk_gauss36_embed_gamma0125_xy15.eps "fig:"){width=".4\textwidth"}
Numerical Experiments
=====================
As described in previous sections, the following three techiques are applied to our MP ODE solver based on high-order IRK formulas:
1. Simplified Newton Method with SPARK3 type reduction in inner iteration
2. DP-MP type mixed precision iterative refinement method to accelarate inner iteration
3. Step size selection based on embedded formulas automatically derived
In addition, users can select DP and MP Krylov subspace methods supporting a banded Jacobi matrix in the same way as SPARK3.
To evaluate the peformance of our implementation, we present the results of numerical experiments. All computations are executed on a Intel Core i7 920 + CentOS 5.4 x86\_64 machine (gcc 4.1.2 + BNCpack 0.8 + MPFR 3.1.0/GMP 5.0.2).
Non Stiff Problem
-----------------
The Lorenz problem (\[eqn:lorenz\]) is a well-known problem in complex systems, and it is not stiff; however, the accuracy of approximation is worse in a longer integration interval. Thus, we must use MP arithmetic in propotion to the length of the integration interval.
$$\begin{array}{l}
\left\{\begin{array}{ccl}
\displaystyle\frac{d\mathbf{y}}{dx} &=& \left[\begin{array}{c}
\sigma (- y_1 + y_2) \\
-y_1 y_3 + ry_1 - y_2 \\
y_1 y_2 - by_3
\end{array}\right] \\
\mathbf{y}(0) &=& [ 0\ 1\ 0 ]^T
\end{array}\right. \\
\mbox{Integration Interval:} [0, 50]
\end{array}, \label{eqn:lorenz}$$
where $\sigma = 10$, $r = 470/19$, and $b = 8/3$. In the case of the above integration interval, we lose around 13 decimal digits of the approximation of $\mathbf{y}(50)$. Hence, we select 70 decimal digits (233 bits), 10-stage 20th order and 15-stage 30th order Gauss formulas. The numerical results are shown in \[table:lorenz\].
--------------- ---------------------- ---------------------- ---------------------- ----------------------
10 stages 15 stages 10 stages 15 stages
\# steps 41137 5112 2709021 91169
Comp.Time (s) 192.0 64.3 12911.1 1112.2
Max.Rel.Error $3.9\times 10^{-19}$ $4.4\times 10^{-19}$ $3.8\times 10^{-39}$ $4.9\times 10^{-39}$
Min.Rel.Error $7.3\times 10^{-21}$ $8.3\times 10^{-21}$ $7.1\times 10^{-41}$ $9.1\times 10^{-41}$
--------------- ---------------------- ---------------------- ---------------------- ----------------------
: Lorenz Problem[]{data-label="table:lorenz"}
The numerical results using both 10-stage and 15-stage formulas indicate that we can obtain the appropriate accuracy of corresponding $RTOL$s.
Stiff Problem
-------------
Next, we solve the van del Pol equation (\[eqn:vdpol\]), which is provided in Testset[@testset_ode]. $$\begin{array}{l}
\left\{\begin{array}{ccl}
\displaystyle\frac{d\mathbf{y}}{dx} &=& \left[\begin{array}{c}
y_2 \\
( (1 - y_1^2)y_2 - y_1 ) / 10^{-6}
\end{array}\right] \\
\mathbf{y}(0) &=& \left[\begin{array}{c}
2 \\
0
\end{array}\right]
\end{array}\right. \\
\mbox{Integration Interval:} [0, 2]
\end{array} \label{eqn:vdpol}$$ By using a 15-stage 30th order Gauss formula computed in 50 decimal digits (167 bits) MP arithmetic, we set $ATOL=0$ and $RTOL=10^{-40}$ or $10^{-30}$. In this case, \[fig:local\_error\_est\] shows the history of $\|\mathbf{err}_k\|$ and step size.
![History of local error estimation and step size: van der Pol equaiton[]{data-label="fig:local_error_est"}](van_der_pol_eps_dec50_new.eps){width=".9\textwidth"}
As a result, we can obtain the appropriate approximations corresponding to each $RTOL$s listed in \[table:van\_del\_pol\].
$RTOL=10^{-30}$ $RTOL=10^{-40}$
--------------- ---------------------- ----------------------
\# steps 4325 6202
Comp.Time (s) 127.8 208.9
Max.Rel.Error $1.2\times 10^{-29}$ $1.0\times 10^{-39}$
Min.Rel.Error $2.2\times 10^{-36}$ $8.6\times 10^{-47}$
: Van del Pol equation: The number of steps, computational time and relative errors.[]{data-label="table:van_del_pol"}
1-Dimensional Brusselator Problem
---------------------------------
As a large-scale problem, we solve the 1-dimensional Brusselator problem[@spark3] by using our MP ODE solver. $$\left\{\begin{array}{l}
\frac{\partial u}{\partial t} = 1 + u^2 v - 4 + 0.02\cdot \frac{\partial^2 u}{\partial x^2} \\
\frac{\partial v}{\partial t} = 3u - u^2v + 0.02\cdot \frac{\partial^2 v}{\partial x^2}
\end{array}\right. \label{eqn:bruss1d_org}$$
The above original partial differential equation (\[eqn:bruss1d\_org\]) can be discretetized as a large-scale ODE as follows: $$\begin{array}{l}
\left\{\begin{array}{l}
\frac{d u_i}{d t} = 1 + u_i^2 v_i - 4 + 0.02\cdot \frac{u_{i+1} - 2u_i + u_{i-1}}{(\Delta x)^2} \\
\frac{d v_i}{d t} = 3u_i - u_i^2v_i + 0.02\cdot \frac{v_{i+1} - 2v_i + v_{i-1}}{(\Delta x)^2} \\
u_0(t) = u_{N+1}(t) = 1,\ v_0(t) = v_{N+1}(t) = 3,\\
u_i(0) = 1 + \sin(2\pi i\Delta x),\ v_i(0) = 3
\end{array}\right. \ (i =1, 2, ..., N) \\
\mbox{Integration Interval:} [0, 10]
\end{array}
\label{eqn:bruss1d}$$
We solve the above ODE for the following situation:
$N=500, n=2N=1000$, $\Delta x = 1/(N+1) = 1/501$
$RTOL = ATOL = 10^{-30}$
DP-MP($L=50$) type mixed precision BiCGSTAB method with band matrix-vector multiplication.
In many large-scale problems, $J$ can be sometimes expressed as a sparse matrix. In particular, for an MP enviroment, we must treat $J$ as a sparse matrix in order to overcome the limitation of main memory. In this case, the dense matrix of $J$ need about 36 MB for 50 decimal digits, and hence $3\times 36 \times 10 = 1.08$ GB for a 10-stage IRK method. On the other hand, the band matrix of $J$ need be about 0.18 MB, and hence, $3\times 0.18 \times 10 = 5.4$ MB. This problem is sutable for DP-MP Krylov subspace methods in inner iteration. In this case, we use the left preconditioned DP-MP BiCGSTAB and normal DP-MP BiCGSTAB methods.
\[table:bruss1d\] shows the result of numerical experiments for (\[eqn:bruss1d\]). The left preconditioned DP BiCGSTAB methods at (\[eqn:simple\_iterative\_ref\_l1\]) and (\[eqn:simple\_iterative\_ref\_l2\]) in the DP-MP type iterative refinement method is needed in order to be converged successfully.
--------------- ---------------------- ---------------------- ---------------------- ----------------------
10 stages 20 stages 10 stages 20 stages
\# steps 2966 341 15056 7587
Comp.Time (s) 11205 3642 77717 64573
Max.Rel.Error $2.2\times 10^{-25}$ $1.3\times 10^{-20}$ $3.9\times 10^{-25}$ $3.8\times 10^{-25}$
Min.Rel.Error $1.3\times 10^{-29}$ $6.1\times 10^{-23}$ $2.1\times 10^{-27}$ $2.1\times 10^{-27}$
--------------- ---------------------- ---------------------- ---------------------- ----------------------
: 1-dimensional Brusselator Problem[]{data-label="table:bruss1d"}
By comparing the numerical values provided in SPARK3, we confirmed that all elements of the approximations have over true 14 decimal digits. Moreover, we can find that they have about 20 - 29 true decimal digits by comparing the results obtained by using 100 decimal digits MP computation. In the above-mentioned problems, faster computations with the same accuracy of approximations can be achieved with higher-order formulas.
However, there are some problems to be solved in order to speed up our MP ODE solver as shown \[fig:bruss1d\_local\_error\_est\].
![History of step size and $\|\mathbf{err}_k\|$ : 10 stages, 20th order : Left preconditioned DP-MP BiCGSTAB(Left), DP-MP BiCGSTAB(Right)[]{data-label="fig:bruss1d_local_error_est"}](bruss1d_irk10_precond.eps "fig:"){width=".45\textwidth"} ![History of step size and $\|\mathbf{err}_k\|$ : 10 stages, 20th order : Left preconditioned DP-MP BiCGSTAB(Left), DP-MP BiCGSTAB(Right)[]{data-label="fig:bruss1d_local_error_est"}](bruss1d_irk10_normal.eps "fig:"){width=".45\textwidth"}
The two sets of graphs of $\|\mathbf{err}_k\|$ and step size show that the right ones are widely wiggled because of non-convergence of the unpreconditioned DP BiCGSTAB method used in the iterative refinement method. If the DP BiCGSTAB method is not convergent, the approximation computation is rejected and a new one is recomputed after the step size becomes smaller at the discretized point. These bottlenecks can be overcome by using predconditionings or other robust DP iterative methods for a system of linear equations in inner iteration.
Conlusion and Future Works
==========================
We showed that our MP ODE solver based on high-order IRK formulas can obtain accurate approximations in some problems. However, some problems remain such as the unsupported DP robust linear solver for sparse Jacobi matrix.
Our final objective is to provide practical and high-performance DP and MP ODE solvers based on high-order IRK methods.
In order to achieve this objective, we plan to tackle the following issues in the future:
1. Parallelization of inner iteration in IRK method for multi-core CPU and GPU. For this purpose, we plan to use a well-tuned linear computation library based on LAPACK and BLAS.
2. We plan to accumulate many numerical experiments, especially for ill-conditioned problems requiring MP arithmetic, and we also plan to provide some selection for solvers of system of linear equations in inner iteration in order to optimize the computational time and user-required accuracy.
Acknowledgments {#acknowledgments .unnumbered}
===============
I thank Hideko Nagasaka and Masatsugu Tanaka for encouraging me when writing my doctoral thesis that is one of origins of this paper.
[10]{}
The [GNU]{} [M]{}ultiple [P]{}recision arithmetic library, <http://gmplib.org/>.
A.Buttari, J.Dogarra, Julie Langou, Julien Langou, P.Luszczek, and J.Karzak, Mixed precision iterative refinement techniques for the solution of dense linear system. , Vol. 21, No. 4, pp. 457–466, 2007.
S.P.N[ø]{}rsett E.Hairer and G.Wanner, . Springer-Verlarg, 1996.
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[^1]: 2010 Mathematics Subject Classification: 65L06, 65F10, 65G50
[^2]: Keywords and phrases: Implicit Runge-Kutta Method, Multiple Precision Floating-point Arithmetic, Iterative Refinement Method
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abstract: 'We study the asymptotic dynamics of dark energy as a mixture of presssureless matter and an interacting vacuum component. We find that the only dynamics compatible with current observational data favors an asymptotically vanishing matter-vacuum energy interaction in a model where dark energy is simulated by a generalized Chaplygin gas cosmology.'
address: |
Department of Mathematics, College of Engineering,\
American University of the Middle East\
P.O.Box: 220 Dasman, 15423 Kuwait
author:
- Georgia Kittou
date: today
title: 'Is interacting vacuum viable? '
---
Dark energy,Dark matter,Interacting vacuum ,Chaplygin gas model,Asymptotic analysis
Introduction {#sec1}
============
In recent years most of the cosmological studies have been focused on variations of General Relativity and modifications of the Standard Cosmological model [@mod]. This is done in order to provide a more reliable framework to explain the present physical evidence of the universe. It is observed today that only $5\%$ of the matter content is of baryonic form and additional evidence coming from the high redshift surveys of type I supernovae [@obs1; @obs2] indicate that we currently live in a universe that undergoes an accelerated expansion.
In the present literature many cosmological models involve the presence of an exotic type of matter component that lies beyond the framework of standard cosmology, cf. [@c1; @c2; @c3; @c4], in an attempt to explain the present acceleration of the observed universe. Additionally, the case of coupling and energy transfer between dark energy and dark matter leads to research efforts that try to alleviate the so-called coincidence problem [@co1; @co2]. One unified dark energy model that has attracted the interest for research is the Generalized Chaplygin Gas model (GCG in short). This model has a dual character since at early times it satisfies the properties of a matter-dominated universe whereas at late times it approaches the limiting behaviour of dark-energy dominated universe [@Cha].
In previous works [@p1; @p2; @phd], we have studied the asymptotic dynamics near finite-time singularities of flat and curved universes filled with two interacting fluids using an interaction term that was first introduced by Barrow and Clifton, cf. [@ba1; @ba2]. In the present work, we consider the case of energy exchange between dark matter (as pressureless dust) and dark energy (as vacuum) with the local energy transfer being associated with the energy density of the vacuum (that is $\rho_V$) so that $Q_{\mu}=-\nabla_{\mu}\rho_V$ [@w1].
In the limit of zero energy exchange ($Q_{\mu}=0$), or equivalently if the vacuum energy is covariantly conserved ($\nabla_{\mu}\rho_V=0$), then the vacuum energy must be homogeneous in spacetime and equal to a cosmological constant [@Cha2]. Under these conditions, we address the question of the viability and stability of the interacting vacuum model on approach to the finite-time singularity by studying the asymptotic properties of solutions of the scale factor, the total energy density and total pressure of the universe.
The cosmological model is expected to be stable, and therefore acceptable, if asymptotically it reproduces the dominant features of dark matter and dark energy at both early and late times respectively. We show that asymptotically at early times the energy exchange is vanishing and the energy density of the vacuum is approximately zero, in contrast to what occurs in the standard cosmological model. Hence, at early times and in the absence interaction, our model is indistinguishable from the CDM model [@Bento].
The asymptotic analysis of the solutions is carried out using the method of asymptotic splittings, cf. [@met1; @met2]. The analysis provides a complete description of all possible dominant features that the solution possesses as it is driven to a blow-up.
The plan of this paper is as follows. In the next section, we write down all possible asymptotic decompositions of the basic differential equations of our problem describing the GCG model. Sections \[subsection1\]-\[subsection3\] present a detailed study of the various asymptotic solutions. In the last section we discuss our results and point out some interesting open problems in this field.
Decomposed Dark energy models {#sec2}
=============================
We study the case of the generalised Chaplygin gas model in flat FRW universe as a mean to explain the accelerated expansion of the universe [@2]. In the GCG approach the exotic cosmological fluid is defined by the barotropic equation of state $$P_{cgc}=A\rho_{cgc}^{-\alpha},$$
where $A$ is a positive constant and $0<\alpha\leq1$. This leads to a cosmological solution for the density $$\label{gcg}
\rho_{cgc}=\left(A+Ba^{-3(\alpha+1)}\right)^{1/(\alpha+1)},$$
where $a$ is the scale factor of the universe and $B$ is a positive integration of constant for a well defined $\rho_{cgc}$ at all times. From Eq. (\[gcg\]), one can conclude that at early times the asymptotic solution for the energy density reproduces the CDM model as described by $$\rho_{cgc}\sim a^{-3}\quad a\rightarrow 0,$$
in the limit of vanishing constant $\alpha\rightarrow0$ [@Sand; @Park]. At late times the solution (\[gcg\]) implies that the fluid behaves as a cosmological constant $$\rho_{cgc}\sim A^{1/(\alpha+1)}\quad a\rightarrow\infty.$$
This interpolation of the model between two different fluids at different stages of the evolution of the universe suggests that the GCG model can be interpreted as a mixture of two cosmological fluids with energy exchange.
Now, any unphysical oscillations or exponential blow-up in the matter spectrum produced by such a unified model [@Sand] can be avoided, if one excludes coupling with phantom fields [@Bento]. Therefore, the unique coupling between dark matter (pressureless dust) and dark energy (cosmological constant) makes the GCG model a well-behaved model both at early times (approach the successful CDM model) and at late times (approach de Sitter Universe).
It is interesting to mention here that the interaction between the fluid components allows energy to be transferred from dark matter to dark energy, since $\alpha$ is a positive constant. As we will show below, this energy transfer is vanishingly small at early times making the model indistinguishable from a CDM model in the past. Whereas when interaction starts off the contribution of the cosmological constant is significant and the model approaches de Sitter universe [@Bento].
The Einstein equations[^1] for a flat Friedman universe filled with pressureless dust ($\rho_m$) and vacuum ($\rho_V$), scale factor $a(t)$ and Hubble expansion rate $H=\dot{a}/a$ reduce to the Friedman equation $$\label{f1}
3H^2=\rho_m+\rho_V=\rho_{gcg}.$$
The total energy momentum tensor of the pair is the algebraic some of the individual energy-momentum tensor given by $$T_{total}^{\mu\nu}= T_{m}^{\mu\nu}+ T_{V}^{\mu\nu}.$$
Since the two fluids are not separately conserved it occurs that $$\nabla_{\nu}T^{\mu\nu}_m=-u^{\mu}\quad \nabla_{\nu}T^{\mu\nu}_V=u^{\mu},$$
where $u^{\mu}$ is the total 4-velocity so that $\nabla_{\nu}T^{\mu\nu}_{total}=0$.
It is shown in [@phd] that by taking the covariant derivative of each energy density momentum separately one obtains $$-u^0=\nabla_{\nu}T^{0\nu}_m=-a^3\dot{p_m}+\frac{d}{dt}[a^3(p_m+\rho_m)]$$ and similarly for the second one $$u^0=\nabla_{\nu}T^{0\nu}_V=-a^3\dot{p_V}+\frac{d}{dt}[a^3(p_V+\rho_V)].$$
The forms of the continuity equations then read $$\begin{aligned}
\dot{\rho_m}+3H\rho_m&=&-\frac{u^0}{a^3}\\
\dot{\rho_V}&=&\frac{u^0}{a^3}.\end{aligned}$$
If we set $Q=u^0/a^3$ as the interaction term, one can show after some calculations that the interaction term reads $$\label{interaction function}
Q=\frac{u^0 H}{\dot{a}a^2}.$$
Therefore, the interaction function (\[interaction function\]) is generally dependent on the expansion rate $H$, the scale factor $a(t)$, its time derivative $dot{a(t)}$, as well as on the energy densities and pressures of the fluid components. We note here that if the expansion of the universe ceases, that is for $H=0$, the interaction between the fluid components will also vanish due to the fact that interaction is coupled to the $3$-geometry of the slice with a mean curvature described from the Hubble parameter $H$.
We assume a fluid interaction of the form [@w1] $$Q=3\alpha H\left(\frac{\rho_m\rho_V}{\rho_m+\rho_V}\right),$$
and the final forms of the continuity equations for matter and vacuum are given by $$\begin{aligned}
\dot{\rho_m}+3H\rho_m&=&-Q\label{f2}\label{c1}\\
\dot{\rho_V}&=&Q\label{f3}\label{c2},\end{aligned}$$
respectively. Equations (\[f1\]),(\[f2\]) and (\[f3\]) describe a $3$-dimensional system with unknowns $(a, \rho_m, \rho_V)$ satisfying the constraint given by Eq. ($\ref{f1}$). After some manipulations it is proved that the above set of equations leads to the following master differential equation $$\label{master}
\ddot{H}+3(\alpha+1)H\dot{H}+2\alpha\frac{\dot{H}^2}{H}=0.$$
Equation (\[master\]) is a nonlinear differential equation of second order for the Hubble parameter $H$. If we assume a power-law type solution for the scale factor $a=t^p$ that is $H=p/t$, where $p\in\mathcal{Q}$, then one can provide all possible exact solutions for the Hubble parameter as well for the scale factor based on the GCG parameter $\alpha$ at both early and late times.
Indeed if we substitute the form $H=p/t$ in Eq. (\[master\]) we get the following equation for the exponent $p$ $$p^2[2-3p(\alpha+1)+2\alpha]=0.$$
We find two possible solutions to the equation above; The first one describes the case of no interaction with $p=0$ while the non-trivial solution satisfies the form $$p=\frac{2+2\alpha}{3(\alpha+1)},$$
and the exact solution for the scale factor reads $$\label{scalef}
a(t)=t^{(2+2\alpha)/[3(\alpha+1)]}.$$
However, in this work we are interested in an asymptotic analysis of solutions of (\[master\]) near finite-time singularities. To do so, it will be very useful for our calculations to rewrite the master equation (\[master\]) in a suitable dynamical system form. In this respect, we rename $H=x$ and find the $2$-dimensional system $$\begin{aligned}
\dot{x}&=&y\nonumber\\
\dot{y}&=&-3(\alpha+1)xy-2\alpha\frac{y^2}{x}\label{ds}.\end{aligned}$$
Equivalently, we have the vector field $$f(x,y)=[y,-3(\alpha+1)xy-2\alpha\frac{y^2}{x}]\label{vf}.$$
The vector field can split [@met1] in three different ways namely $$\begin{aligned}
\label{dc1}f_{I}(x,y)&=&[y,-3(\alpha+1)xy]+(0,-2\alpha\frac{y^2}{x}),\\
\label{dc2}f_{II}(x,y)&=&(y,-2\alpha\frac{y^2}{x})+[0,-3(\alpha+1)xy],\\
\label{dc3}f_{III}(x,y)&=&[y,3(\alpha+1)-2\alpha\frac{y^2}{x}].
\end{aligned}$$
In the following sections we apply the method of asymptotic splittings, analytically expounded in [@met1; @met2], to describe the asymptotic properties of the solutions of the dynamical system (\[ds\]) in the vicinity of its finite-time singularities.
Early times asymptotics {#subsection1}
=======================
In this section, we give necessary conditions in terms of the parameter $\alpha$ for the existence of generalised Fuchsian series type solutions [@met2] towards the finite-time singularity of the first decomposition $f_{I}(x,y)=[y,-3(\alpha+1)xy]+(0,-2\alpha\frac{y^2}{x})$.
To do so, we look for possible dominant balances by substituting the forms $x(t)=\theta t^p,\quad y(t)=\xi t^q$ in the dominant part of the decomposition described by the system below $$\begin{aligned}
\label{ds1}
\dot{x}&=&y\nonumber\\
\dot{y}&=&-3(\alpha+1)xy.\end{aligned}$$ We assume here that $\mathbf{\Xi}=(\theta, \xi)\in \mathbb{C}$ and $\mathbf{p}=(p, q)\in \mathbb{Q}$. This leads to the unique balance $$\mathcal{B}_{I}=[\mathbf{\Xi},\mathbf{ p}]=\left[\left(\frac{2}{3(\alpha+1)},-\frac{2}{3(\alpha+1)}\right),(-1,-2)\right],$$
for $0<\alpha\leq1$. The subdominant part of the splitting (\[dc1\]) satisfies $$\frac{f^{(sub)}_{I}(\mathbf{\Xi},t^\mathbf{p})}{t^{\mathbf{p}-1}}=\left(0,-\frac{4\alpha}{3(\alpha+1)}\right),$$
and is asymptotically subdominant [@p1] in the sense that $$\label{conditionofsubdomiannce}
\lim_{t\rightarrow 0}\frac{f^{(sub)}_{I}(\mathbf{\Xi},t^\mathbf{p})}{t^{\mathbf{p}-1}}\rightarrow 0,$$
only if $\alpha\rightarrow 0$. We therefore conclude that in the neighbourhood of the finite-time singularity the asymptotic solution is meaningful only in the limit of vanishing $\alpha$, that is in the absence of interaction between the two fluids.
Next we calculate the Kovalevskaya matrix given by, $$\mathcal{K}_{I}=\mathcal{D}f_{I}(\mathbf{\Xi})-diag(\textbf{p}),$$
where $\mathcal{D}f$ is the Jacobian matrix of the decomposition. For this case the Kovalevskaya matrix reads $$\mathcal{K}_{I}=\left[ {\begin{array}{cc}
1 & 1 \\
2 & 0 \\
\end{array} } \right].$$
As discussed in [@met1] the number of non-negative $\mathcal{K}$- exponents equals the number of arbitrary constants expected to appear in the series solution, while the $-1$ exponent corresponds to the arbitrary constant relevant to the position of the singularity (for notational convenience taken to be $t=0$). Therefore, if the balance is to correspond to a general solution, two arbitrary constants are expected to appear in the series expansion (since the original system (\[ds\]) is two dimensional). Here we find $$spec(\mathcal{K}_{I})=(-1,2),$$
with a corresponding eigenvector $$v_{2}^T=(1,-1).$$
Hence, it is expected that the balance $\mathcal{B}_{I}$ will correspond to a general solution. Substituting the series expansions $$\label{forms}
x=\sum_{j=0}^{\infty}c_{j1}t^{j-1}, \quad y=\sum_{j=0}^{\infty}c_{j2}t^{j-2},$$
in the system (\[ds\]) we arrive after manipulations at the following asymptotic solution around the singularity $$\label{sol1}
x(t)=\frac{2}{3}t^{-1}+c_{21}t+\cdots,\quad as\quad t\rightarrow0,\quad \alpha\rightarrow0.$$
The $y$-expansion is derived from the above by differentiation. As a final test for the validity of this solution, a compatibility condition has to be satisfied for every positive $\mathcal{K}$-exponent [@p1]. For the positive eigenvalue $2$ and an associated vector $ v_{2}^T=(1,-1)$ it reads $$c_{21}=c_{22},$$
and this is indeed true based on previous recursive calculations.
It follows from Eq. (\[sol1\]) that all solutions are dominated by the $x=H\sim \frac{2}{3}t^{-1}$ solution which in terms of the scale factor reads $$\label{sf1}
a(t)\sim t^{2/3}\quad as\quad t\rightarrow0,\quad \alpha\rightarrow0.$$
The dominant term of the series expansion (\[sf1\]) is the same as the exact solution for the scale factor described by Eq. (\[scalef\]) in the limit $\alpha\rightarrow0$. It also follows from Eq. (\[sf1\]) that in the vicinity of the finite-time singularity and in the absence of interaction the total energy density of the model satisfies the form $$\rho_{tot}\sim a^{-3}\quad t\rightarrow0,\quad\alpha\rightarrow0$$
Now, the geometric character of the singularity is completely described in terms of the asymptotic behaviour of the total energy density and pressure of the model, the asymptotic behaviour of the scale factor and the Hubble parameter [@p2]. For the solutions above it occurs that $$\rho_{tot}\rightarrow \infty\quad P_{tot}\rightarrow0,\quad a\rightarrow\infty,\quad H\rightarrow\infty,$$
as $t\rightarrow0$ and $\alpha\rightarrow0$. The asymptotic conditions above describe the case of a Big Bang type of singularity. Consequently, the singularity is necessarily placed at early times. It is discussed in cf. [@Bento] that the contribution from the cosmological constant is negligibly small at early times, hence we conclude that our decomposition describes a model that is indistinguishable from a CDM dominated universe in the past.
It is discussed in [@Bento] that the energy density perturbations at early times regarding the dark matter component (and the baryon perturbations) are linear and small in scale ($\delta_m<<1$) and in the absence of interaction one can easily recover the standard energy perturbations in the CDM model.
\[subsection2\]Quasi de Sitter Universe
=======================================
Let us now move on to the asymptotic analysis of the decomposition with dominant part given by the vector field $$\label{an}
f_{II}(x,y)=(y,-2\alpha y^2/x) .$$
Now by substituting in the asymptotic system $(\dot{x},\dot{y})=[y,-2\alpha (y^2/x)]$ the forms $x(t)=\theta t^p$ and $y(t)=\xi t^q$ we find the following dominant balance $$\label{bal2}
\mathcal{B}_{II}=\left[\left(\theta, \frac{\theta}{2\alpha+1}\right),\left(\frac{1}{2\alpha+1},\frac{1}{2\alpha+1}\right)\right],$$
where $\theta$ is an arbitrary constant. The candidate subdominant part of the vector field, namely $f_{II}^{(sub)}(x,y)=[0,-3(\alpha+1)xy]$ is vanishing asymptotically without any restrictions on the values of the parameter $\alpha$ nor the constant $\theta$. Hence the decomposition is acceptable. To continue with, the Kovalevskaya matrix is given by $$\mathcal{K}_{II}=\left[ {\begin{array}{cc}
-1/(2\alpha+1) & 1 \\
(2\alpha)/(2\alpha+1)^2 & -(2\alpha)/(2\alpha+1) \\
\end{array} } \right],$$
with corresponding eigenvalues $$spec(\mathcal{K}_{II})={(-1,0)},$$
and an eigenvector $$v_{2}^T=\left(1,\frac{1}{2\alpha+1}\right).$$
We note here that the second $\mathcal{K}$-exponent is zero. Hence the arbitrary constants at the $j=0$ level of expansion (cf. [@p1; @p2; @met2; @phd] for this terminology) are the coefficients given by the dominant balance (\[bal2\]), that is $(c_{01},c_{02})=\left(\theta,\frac{\theta}{2\alpha+1}\right)$. Therefore, the asymptotic solution is general since two arbitrary constants appear in the asymptotic solution as described below $$\label{sol2}
x(t)=\theta t^{1/(2\alpha+1)},\quad t\rightarrow0,$$
for $0<\alpha\leq 1$. Since we are interested in expanding universes $(H>0)$, it follows that the arbitrary constant $\theta$ attains only positive values. By integrating the solution above one obtains asymptotically the general solution for the scale factor described by the expression $$\label{sols}
a(t)=a_0\exp{\left(\theta C t^{1/C}\right)}\quad{as}\quad t\rightarrow0,$$
where $C=(2\alpha+1)/(2\alpha+2)$.
A comment about the asymptotic behaviour of the scale factor is in order. The specific form of Eq. (\[sols\]) describes an exponential evolution of the universe, with slower rate of expansion than the de Sitter universe, valid for a time interval. Clearly, as interaction kicks off the transfer of energy from dark matter to dark energy (described by Eqs. (\[c1\])-(\[c2\])) results in an important growth of the energy density of the vacuum. However the presence of dark matter decelerates the rate of expansion.
As shown in the asymptotic solution (\[sols\]) the $\alpha$ parameter determines the asymptotic states of the universe. For $0<\alpha\leq 1$ the universe enters (for a time interval) a quasi de Sitter space where the total energy density, total pressure, scale factor and Hubble parameter are asymptotically equal to $$\label{forms}
\rho_{tot}\rightarrow0,\quad |P_{tot}|\rightarrow \infty,\quad a\rightarrow a_0,\quad H\rightarrow 0,$$
respectively as $t\rightarrow0$, while higher derivatives of $H$ diverge. This is a new type of singularity, a combination of Type $IV$ [@sing; @sd2; @sd3; @sd4] and Type $II$ (sudden) singularity placed at late times.
It is interesting to note here that the present decomposition describes an intermediate phase in the evolution of our interacting model. In the limiting case $\alpha\rightarrow 0$ (limit of no interaction) it is expected that the universe asymptotically (as $t\rightarrow0$) will approach the CDM model. This is indeed true since for $\alpha\rightarrow0$ the asymptotic analysis is identical to the one performed for the first decomposition in section \[subsection1\]. Consequently, the particular decomposition successfully reproduces the CDM model (as $t\rightarrow0$ and $\alpha\rightarrow0$).
In addition, it is also expected at late times that the dominance of dark energy will drive the evolution of the universe towards de Sitter space. This is indeed feasible in the limit $\alpha\rightarrow\infty$, that is the case where energy is being transferred from dark matter to dark energy without bound. This results in the following asymptotic forms for the total energy density, total pressure, scale factor and the Hubble parameter [^2] $$\label{sols1}
\rho_{tot}\rightarrow\rho_0,\quad |P_{tot}|\rightarrow\infty,\quad a\sim a_0\exp{(\theta t)},\quad x=H\sim \theta,$$
as $t\rightarrow0$, $\alpha\rightarrow\infty$. The forms above describe a dark energy dominated universe with a sudden type singularity placed at late times. It is discuss in [@Bento] that at late times the energy density perturbations of dark matter and baryons deviate from the linear behaviour explaining the large energy transfer from dark matter to dark energy.
We note here that the exact solution described by Eq. (\[scalef\]) fails to reproduce this specific behaviour of the scale factor at late times since it describes only possible power-law type solutions.
\[subsection3\]Interacting Vacuum
=================================
We now focus on the asymptotic analysis of all-terms-dominant case, that is the decomposition (\[f3\]), or equivalently described by the asymptotic system $$\begin{aligned}
\label{sys}
\dot{x}&=&y\nonumber\\
\dot{y}&=&-3(\alpha+1)xy-2\alpha\frac{y^2}{x}.\end{aligned}$$
The subdominant vector field is the zero field in this case and there is one distinct balance given by $$\label{b3}
\mathcal{B}_{III}=\left[\left(\frac{2}{3},-\frac{2}{3}\right),(-1,-2)\right].$$
The Kovalevskaya matrix is given by $$\mathcal{K}_{III}=\left[ {\begin{array}{cc}
1 & 1 \\
4\alpha+2 & 2\alpha \\
\end{array} } \right],$$
with corresponding eigenvalues $$spec(\mathcal{K}_{III})={[-1,2(\alpha+1)]}.$$ Even though the parameter $\alpha$ is present in the second $\mathcal{K}$-exponent, the form of the dominant the balance (\[b3\]) indicates that on approach to the finite-time singularity the dominant part of asymptotic solution $x\sim(2/3) t^{-1}$ is independent from the choice of the parameter $\alpha$.
For the whole series expansion though, the choice of the parameter $\alpha$ will determine the level of expansion at which the second arbitrary constant is expected to appear. For purposes of illustration we choose $\alpha=1/2$ so that $ spec(\mathcal{K}_{III})={(-1,3)}$. Then the associated eigenvector reads $$v_{2}^T=(1,-1).$$
The candidate asymptotic solution is expected to be general if two arbitrary constants (the position of the singularity and one constant at the $j=3$ level of expansion) appear in the series expansion. After substituting the forms (\[forms\]) into the asymptotic system (\[sys\]), and for $\alpha=1/2$, we find the following asymptotic solution $$\label{sol3}
x(t)=\frac{2}{3}t^{-1}+c_{31}t^{-2}+\cdots,\quad t\rightarrow0.$$ The compatibility condition at the $j=3$ level reads $$2c_{31}=c_{32}$$
and it is indeed satisfied after recursive calculations. Hence the asymptotic solution found above is general. In particular, the dominant behaviour of the solution (\[sol3\]) on approach to the finite-time singularity is identical to the one of the decomposition $f_{I}(x,y)$.
We conclude here that the decomposition describes asymptotically the model in the very early universe before interaction becomes significantly large. Having said that, the model described here has the same asymptotic features as in the case where the interaction is switched off asymptotically and the universe is matter dominated. Hence, at early times the contribution of dark energy is negligible.
Discussion {#sec3}
==========
In this paper we analysed the stability of the singular flat space solutions that arise in the content of a unified dark energy model (GCG model) on approach to the finite-time singularity. We have shown that spacetime evolves from a phase that is initially dominated by dark matter to a phase that is asymptotically de-Sitter under some restrictions. The transition period in our model, between dark matter and dark energy domination corresponds to a quasi-inflationary regime that posses a new type of singularity asymptotically.
We conclude that the current observational data are supportive towards an asymptotically vanishing interaction in a model where dark energy is simulated by a generalised Chaplygin gas cosmology. In particular, it is shown that for such unified model the interaction is asymptotically vanishing at early times and the contribution of dark energy (as cosmological constant) is negligible. Hence, the model is indistinguishable from CDM universe. Such a model attains a pole-like [@phd] type of singularity and it is proved in previous works [@p1; @p2] that such a dominant behaviour is an attractor of all possible asymptotic solutions on approach to the finite-time singularity.
An interesting era of expansion arises in an intermediate phase of expansion when the vector field decomposition admits a quasi de-Sitter solution on approach to the finite-time singularity for $0<\alpha\leq 1$. In particular, the decomposition reproduces the successful CDM model at early times (in the limit as $\alpha\rightarrow0$) and approaches de-Sitter Universe at late times respectively. This intermediate phase of evolution is in alignment with the predictions of the GCG model for both early and late times.
To conclude with, it would be interesting to apply the central projection technique of Poincarè to the dominant part of each of the asymptotic solutions on approach to the finite-time singularity to discuss the asymptotic stability of the model at infinity. This is examined in [@wip].
Acknowledgements
================
We thank Prof. David Wands and Prof. Elias C. Vagenas for discussions and useful comments.
References
==========
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[^1]: Here we consider the case where the baryons have a similar behaviour to that of a pressureless dust, i.e dark matter and we exclude the possibility of an energy exchange between baryons and dark energy (see [@Bento] for more information.
[^2]: If we assume here for purposes of notation that the constant $\theta$ is positive and plays the role of the cosmological constant it can be proved that our solution (\[sols\]) can also describe a de-Sitter Universe at a finite time at late epoch $t_f\not=0$.
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---
abstract: 'It has been shown by Ibragimov and Zaporozhets \[to appear in Prokhorov Festschrift; arXiv:1102.3517\] that the complex roots of a random polynomial $G_n(z)=\sum_{k=0}^n \xi_k z^k$ with i.i.d. coefficients $\xi_0,\ldots,\xi_n$ concentrate a.s. near the unit circle as $n\to\infty$ if and only if ${\mathbb E}\log_{+} |\xi_0|<\infty$. We study the transition from concentration to deconcentration of roots by considering coefficients with tails behaving like $L(\log |t|)(\log |t|)^{-\alpha}$ as $t\to\infty$, where $\alpha\geq 0$ and $L$ is a slowly varying function. Under this assumption, the structure of complex and real roots of $G_n$ is described in terms of the least concave majorant of the Poisson point process on $[0,1]\times (0,\infty)$ with intensity $\alpha v^{-(\alpha+1)} dudv$.'
author:
- Zakhar Kabluchko
- Dmitry Zaporozhets
bibliography:
- 'lognuli\_bib.bib'
title: Roots of random polynomials whose coefficients have logarithmic tails
---
Introduction and statement of results {#sec:intro}
=====================================
Introduction {#subsec:intro}
------------
Let $\xi_0,\xi_1,\ldots$ be i.i.d. non-degenerate random variables with values in ${\mathbb{C}}$. Let $z_{1n},\ldots, z_{nn}$ be the complex roots (counted with multiplicities) of the random polynomial $$G_n(z)=\sum_{k=0}^n \xi_k z^k.$$ For $0\leq a\leq b$ denote by $R_n(a,b)$ the number of roots of $G_n$ in the ring $\{z\in{\mathbb{C}}: a\leq |z|\leq b\}$. Improving on a result of @sparo_sur, @iz_log showed that $$\frac 1n R_n(1-{\varepsilon},1+{\varepsilon})
{\overset{a.s.}{\underset{n\to\infty}\longrightarrow}}1$$ for every ${\varepsilon}\in(0,1)$, if and only if, $$\label{eq:log_moment}
{\mathbb E}\log_+ |\xi_0|<\infty.$$ Here, $\log _+ x=\max (\log x, 0)$. Without any assumptions on the distribution of $\xi_0$, @iz_log also proved that for every $\alpha,\beta$ such that $0\leq\alpha<\beta\leq 2\pi$, $$\frac1n\sum_{k=1}^n{\mathbbm{1}}_{\{\alpha\leq\arg z_{kn}\leq\beta\}}
{\overset{a.s.}{\underset{n\to\infty}\longrightarrow}}\frac{\beta-\alpha}{2\pi}.$$ Thus, under a very mild moment condition, the complex roots of $G_n$ concentrate near the unit circle uniformly by the argument as $n\to \infty$.
Imposing additional conditions on the distribution of $\xi_0$ it is possible to obtain more precise information about the asymptotic concentration of the roots near the unit circle. In the case when $\xi_0$ belongs to the domain of attraction of an $\alpha$-stable law, $\alpha\in(0,2]$, @ibr_zeit showed that for every $t>0$, $$\label{eq:ibr_zeit}
\lim_{n\to\infty}\frac1n {\mathbb E}R_n\left(1-\frac {t}{n},1+\frac {t}{n}\right)=
\frac{1+e^{-\alpha t}}{1-e^{-\alpha t}}-\frac2{\alpha t}.$$ This is a generalization of the result of @shepp_vanderbei who considered real-valued standard Gaussian coefficients.
On the other hand, if ${\mathbb E}\log_+ |\xi_0|=\infty$ and thus there is no concentration near the unit circle, it is also possible to describe the asymptotic behavior of the roots when the tail of $|\xi_0|$ is extremely heavy. @goetze_zaporozhets proved that if the distribution of $\log_+\log_+|\xi_0|$ has a slowly varying tail, then the complex roots of $G_n$ concentrate in probability on two circles centered at the origin whose radii tend to zero and infinity, respectively. See also [@z], [@zn] for more results in the case of extremely heavy tails.
Up to now, the behavior of the roots has been unknown when the tail of $\xi_0$ is somewhere between the two cases described above. The aim of this paper is to consider a class of distributions which in some sense continuously links the above cases. We will consider coefficients with logarithmic power-law tails. More precisely, we make the following assumption: for some $\alpha\geq 0$, $$\label{eq:tail1}
\bar F(t):={\mathbb{P}}[\log |\xi_0|>t] \text{ is regularly varying at } +\infty \text{ with index } -\alpha.$$ This class of distributions includes distributions with both finite ($\alpha>1$) and infinite ($\alpha<1$) logarithmic moments. We will obtain a precise information on how the concentration of the roots near the unit circle becomes destroyed as $\alpha$ approaches $1$ from above and how the roots behave when there is no concentration ($\alpha<1$).
The case $\alpha=+\infty$ corresponds formally to the light or power-law tails studied in [@shepp_vanderbei], [@ibr_zeit]. The roots are concentrated near the unit circle and, apart from this, no global organization is apparent. We will prove that as $\alpha$ becomes finite, the distribution of roots becomes highly organized; see Figure \[fig:roots\]. The roots “freeze” on a random set of circles centered at the origin. Both the radii of the circles and the distribution of the roots among the circles are random, however the distribution of the roots on each circle is uniform by argument. As long as $\alpha$ stays above $1$, the logarithmic moment is finite and the circles approach the unit circle at rate $n^{1/\alpha-1}$ (ignoring a slowly varying term), in full agreement with the result of [@iz_log]. Note also that for $\alpha$ close to $+\infty$ this rate is close to the rate $1/n$ appearing in . As $\alpha$ becomes equal to $1$, we have a transition from finite to infinite logarithmic moment. We will show that if $\bar F(t)\sim c/t$ as $t\to+\infty$, then the empirical measure formed by the roots of $G_n$ converges weakly (without normalization) to a random probability measure concentrated on an infinite number of circles with random radii. For the first time, the roots are not concentrated near the unit circle. As $\alpha$ becomes smaller than $1$, the circles divide into two groups approaching $0$ and $\infty$ at the rates $\pm n^{1/\alpha-1}$, on the logarithmic scale. The number of circles, which was infinite for $\alpha\geq 1$, becomes finite for $\alpha<1$ and decreases to $2$ as $\alpha\to 0$. At $\alpha=0$ the roots freeze on just $2$ circles located very close to $0$ and $\infty$, in accordance with [@goetze_zaporozhets] whose results we will strengthen. At $\alpha=0$, the empirical measure formed by the roots becomes almost deterministic: the only parameter which remains random after taking the limit $n\to\infty$ is the distribution of the roots between $0$ and $\infty$, which is uniform on $[0,1]$.
![Roots of a random polynomial of degree $n=2000$ whose (real) coefficients are a) standard normal, b) such that ${\mathbb{P}}[\log \xi_0>t]=1/t^{2}$ for $t\geq 1$.[]{data-label="fig:roots"}](roots2.png "fig:"){height="49.00000%" width="49.00000%"} ![Roots of a random polynomial of degree $n=2000$ whose (real) coefficients are a) standard normal, b) such that ${\mathbb{P}}[\log \xi_0>t]=1/t^{2}$ for $t\geq 1$.[]{data-label="fig:roots"}](roots1.png "fig:"){height="49.00000%" width="49.00000%"}
An interesting phenomenon we will encounter is the appearance of the long-range dependence between the roots under condition . Consider a random polynomial $G_n$ of high degree and suppose that we know that it has a root at some point $z_0\in{\mathbb{C}}$. In the case of coefficients from the domain of attraction of a stable law this information has almost no influence on the other roots of $G_n$ except for the roots located in an infinitesimal neighborhood of $z_0$. However, for coefficients with logarithmic power-law tails, the knowledge about the existence of a root at $z_0$ implies that there exists (with high probability) a circle of roots containing $z_0$. Moreover, the radii of the other circles of roots are influenced by the existence of the root at $z_0$. We observe a long-range dependence between the roots: the conditional distribution of roots given that there is a root at $z_0$ differs, even on the global scale, from the unconditional distribution of roots.
If the random variables $\xi_i$ are real-valued, we will also analyze the real roots of $G_n$. For a particular family of distributions satisfying with $\alpha>1$, @shepp_farahmand have shown that the expected number of real roots of $G_n$ is asymptotically $C(\alpha)\log n$ with $C(\alpha)=\frac{2\alpha-2}{2\alpha-1}$. As $\alpha$ decreases from $+\infty$ to $1$ the function $C(\alpha)$ decreases from $1$ to $0$. We will complement this result by showing that for $\alpha\in (0,1)$ the number of real roots of $G_{n}$ has two subsequential distributional limits as $n\to\infty$ along the subsequence of even/odd integers. This means that for $\alpha\in (0,1)$ the polynomial $G_n$ has roughly speaking $O(1)$ real roots. Finally, we will prove that for $\alpha=0$ the number of real roots of $G_n$ can take asymptotically only the values $0,\ldots,4$ and compute the probabilities of these values.
Complex roots {#subsec:complex_roots}
-------------
Given a complex number $z=|z|e^{i \arg z}$ and $a\in{\mathbb{R}}$ we write $$z^{\langle a\rangle}=|z|^{a}e^{i\arg z}.$$ The next theorem describes the structure of complex roots of $G_n$. Let $\delta(z)$ be the unit point mass at $z$. Denote by $\bar {\mathbb{C}}={\mathbb{C}}\cup\{\infty\}$ the Riemann sphere. We need normalizing sequences $a_n,b_n$ such that $$\label{eq:def_an}
\bar F(a_n)\sim \frac 1n \;\; \text{ as } n\to\infty, \qquad b_n=\frac n{a_n}.$$
\[theo:complex\] If the tail condition is satisfied with some $\alpha>0$, then we have the following weak convergence of random probability measures on $\bar {\mathbb{C}}$: $$\frac 1 n\sum_{k=1}^n \delta({z_{kn}^{\langle b_n \rangle}}){\overset{w}{\underset{n\to\infty}\longrightarrow}}\Pi_{\alpha}.$$ The limiting random probability measure $\Pi_{\alpha}$ is a.s. a convex combination of at most countably many uniform measures concentrated on circles centered at the origin.
For $\alpha\geq 1$ the logarithmic moment condition is satisfied, which by [@iz_log] means that the roots should concentrate near the unit circle. In the next corollary we compute the rate of convergence of the roots to the unit circle.
Let $\alpha\geq 1$. As $n\to\infty$, the random probability measure $$\frac 1 n\sum_{k=1}^n \delta(b_n(|z_{kn}|-1))$$ converges weakly to some random, a.s. purely atomic probability measure on ${\mathbb{R}}$.
In the case $\bar F(t)\sim c/t$ as $t\to+\infty$, where $c>0$, the logarithmic moment condition just fails. We have no concentration of the roots near the unit circle for the first time. In this case, Theorem \[theo:complex\] simplifies as follows.
Suppose that $\bar F(t)\sim c/t$ as $t\to+\infty$. Then, the empirical measure $\frac 1n \sum_{k=1}^n \delta(z_{kn})$ converges weakly to some nontrivial limiting random probability measure on ${\mathbb{C}}$.
We proceed to the description of the random probability measure $\Pi_{\alpha}$. Let $\rho=\sum_{k=1}^{\infty}\delta(U_k,V_k)$ be a Poisson point process on $[0,1]\times (0,\infty)$ with intensity measure $\alpha v^{-(\alpha+1)} du dv$. Equivalently, $U_k$, $k\in{\mathbb{N}}$, are i.i.d. random variables with a uniform distribution on $[0,1]$ and, independently, $V_k=W_k^{-1/\alpha}$, where $W_1,W_2,\ldots$ are the arrival times of a homogeneous Poisson point process on $(0,\infty)$ with intensity $1$. Of major importance for the sequel is the *least concave majorant* (called simply *majorant*) of $\rho$, see Figure \[fig:majorant\], which is a function ${\mathfrak{C}}_{\rho}:[0,1]\to [0,\infty)$ defined by $${\mathfrak{C}}_{\rho}(t):=\inf_{f} f(t), \qquad t\in[0,1],$$ where the infimum is taken over the set of all concave functions $f:[0,1]\to [0,\infty)$ satisfying $f(U_k)\geq V_k$ for all $k\in {\mathbb{N}}$. From a constructive viewpoint, the least concave majorant ${\mathfrak{C}}_{\rho}$ may be defined as follows. Let $(X_0,Y_0)$ be the a.s. unique atom of $\rho$ having a maximal second coordinate $Y_0$ among all atoms of $\rho$. Consider a horizontal line passing through $(X_0,Y_0)$. Rotate this line around $(X_0,Y_0)$ in a *clock-wise* direction until it hits some atom of $\rho$, denoted by $(X_1,Y_1)$, other than $(X_0,Y_0)$. Continue to rotate the line in the clock-wise direction, this time around $(X_1,Y_1)$, until it hits some atom of $\rho$, denoted by $(X_2,Y_2)$, other than $(X_1,Y_1)$. Continue to rotate the line around $(X_2,Y_2)$, and so on. The procedure is terminated if at some time the line hits the point $(1,0)$. (As we will see later, this happens a.s. if and only if $\alpha\in (0,1)$). Otherwise, the procedure is repeated indefinitely. Analogously, we can start with a horizontal line passing through $(X_0,Y_0)$ and rotate it in an *anti-clockwise* direction obtaining a sequence of points $(X_{-1},Y_{-1}), (X_{-2},Y_{-2}),\ldots$. The sequence may eventually terminate at $(0,0)$. (We will see that this happens a.s. if and only if $\alpha\in (0,1)$). Now, join any point $(X_k,Y_k)$ to the next point $(X_{k+1},Y_{k+1})$ by a line segment. The polygonal path constructed in this way is the graph of the majorant ${\mathfrak{C}}_{\rho}$. The points $(X_k,Y_k)$ are called the *vertices* of the majorant, the intervals $[X_k,X_{k+1}]$ are called the *linearity intervals* of the majorant. The least concave majorant ${\mathfrak{C}}_{\rho}$ is thus a piecewise linear function with at most countably many linearity intervals. We write ${\mathfrak{C}}_{\rho}$ in the form $$\label{eq:CCC_rho}
{\mathfrak{C}}_{\rho}(t)=S_k-R_k t, \qquad t\in [X_k,X_{k+1}].$$ The limiting random probability measure $\Pi_{\alpha}$ in Theorem \[theo:complex\] can be constructed as follows. For $r>0$ let $\Lambda_r$ be the length measure (normalized to have total mass $1$) on the circle $\{z\in{\mathbb{C}}: |z|=r\}$. Then, $$\Pi_{\alpha}=\sum_{k} (X_{k+1}-X_k) \Lambda_{\exp(R_k)},$$ where the (finite or infinite) sum is taken over all linearity intervals $[X_k,X_{k+1}]$ of the majorant ${\mathfrak{C}}_{\rho}$. Thus, Theorem \[theo:complex\] states that the roots of $G_n$ asymptotically concentrate on random circles which correspond to the linearity intervals of the majorant. The radii of these random circles are $\exp({R_k})$, where the $R_k$’s are the negatives of the slopes of the majorant. The proportion of roots on any circle is the length of the corresponding linearity interval.
![The least concave majorant of the Poisson point process $\rho$ on $[0,1]\times(0,\infty)$ with intensity $\alpha v^{-(\alpha+1)} du dv$, where $\alpha=2$.[]{data-label="fig:majorant"}](maj1.png){height="49.00000%" width="98.00000%"}
Our next results describes the distribution of the complex roots of $G_n$ in the case $\alpha=0$. We assume that $$\label{eq:tail_alpha_0}
\bar F(t):= {\mathbb{P}}[\log |\xi_0|>t] \text{ is slowly varying at } +\infty.$$ We will show that under with probability close to $1$ the complex roots of $G_n$ are located on just $2$ circles, one of them with a radius close to $0$ and the other one with a radius close to $\infty$. A weaker result was obtained by @goetze_zaporozhets under a more restrictive assumption on the tails. Let $\tau_n$ be the index of the maximal (in the sense of absolute value) coefficient of $G_n$, that is $\tau_n\in\{0,\ldots,n\}$ is such that $|\xi_{\tau_n}|=\max_{k=0,\ldots,n}|\xi_k|$. Denote by $w_{1n},\ldots,w_{\tau_n n}$ the roots of the equation $\xi_{\tau_n} z^{\tau_n}+\xi_0=0$ and by $w_{(\tau_n+1)n},\ldots,w_{nn}$ the roots of the equation $\xi_n z^{n-\tau_n}+\xi_{\tau_n}=0$.
\[theo:complex\_alpha0\] Suppose that is satisfied and $\xi_0\neq 0$ a.s. Fix some $A>0$. Then, the probability that the following three statements hold simultaneously goes to $1$ as $n\to\infty$:
1. $\tau_n$ is uniquely defined;
2. it is possible to renumber the roots $z_{1n},\ldots,z_{nn}$ of $G_n$ such that $$|z_{kn}-w_{kn}|<e^{-n^A} |w_{kn}|, \qquad 1\leq k\leq n;$$
3. we have $|w_{kn}|<e^{-n^A}$ for $1\leq k\leq \tau_n$ and $|w_{kn}|>e^{n^A}$ for $\tau_n<k\leq n$.
\[cor:complex\_alpha0\] Under , the empirical measure $\frac 1n \sum_{k=1}^n \delta(z_{kn})$ converges weakly, as a random probability measure on the Riemann sphere $\bar {\mathbb{C}}$, to $U\delta(0)+(1-U)\delta(\infty)$, where $U$ is a random variable with a uniform distribution on $[0,1]$.
Properties of the majorant {#subsec:prop_maj}
--------------------------
In this section we study some of the properties of the least concave majorant ${\mathfrak{C}}_{\rho}$. Note that random convex hulls similar to ${\mathfrak{C}}_{\rho}$ appeared in the literature; see [@majumdar] and the references therein. The next proposition will be used frequently.
\[prop:number\_vertices\_majorant\] Let $L_{\alpha}$ be the number of linearity intervals of the majorant ${\mathfrak{C}}_{\rho}$. If $\alpha\in (0,1)$, then $L_{\alpha}<\infty$ a.s. If $\alpha\geq 1$, then $L_{\alpha}=\infty$ a.s. Moreover, in this case any neighborhood of $0$ (as well as any neighborhood of $1$) contains infinitely many linearity intervals of ${\mathfrak{C}}_{\rho}$ a.s. and we have $\lim_{k\to-\infty}R_k=-\infty$ and $\lim_{k\to +\infty}R_k=+\infty$ a.s.
Take any ${\varepsilon}>0$ and consider the set $D_{{\varepsilon}}$ of all pairs $(x,y)\in [0,1]\times (0,\infty)$ such that $y>{\varepsilon}x$. Integrating the intensity of $\rho$ over $D_{{\varepsilon}}$ we see that $\rho(D_{{\varepsilon}})=\infty$ a.s. if and only if $\alpha\geq 1$. If $\alpha\in (0,1)$, we have only finitely many points above any line $y>{\varepsilon}x$ and hence, the majorant ${\mathfrak{C}}_{\rho}$ has a well-defined first segment starting at $(0,0)$. On the other hand, if $\alpha\geq 1$, then no such first segment exists and consequently, we have infinitely many linearity intervals of $\rho$ in any neighborhood of $0$. By symmetry, the same is true for the point $1$.
The distribution of $L_{\alpha}$ in the case $\alpha\in (0,1)$ seems difficult to characterize. In the next theorem we compute the expectation of $L_{\alpha}$ in terms of the modular constant $C(\beta)$ introduced by @barnes in his theory of the double Gamma function. Let $\psi(z)=\Gamma'(z)/\Gamma(z)$ be the logarithmic derivative of the Gamma function. @barnes showed that the following limit exists for $\beta>0$: $$\label{eq:barnes_modular_const}
C(\beta):=\lim_{n\to\infty}\left\{\sum_{m=1}^{n} \psi(m\beta)-\left(n+\frac 12-\frac 1{2\beta}\right) \log (n\beta)+n\right\}.
$$ The role of the constant $C(\beta)$ in the theory of the double Gamma function is similar to the role of the Euler–Mascheroni constant $\gamma=\lim_{n\to\infty}(\sum_{k=1}^n\frac 1k-\log n)$ in the theory of the usual Gamma function.
\[theo:intensity\] For $\alpha\in (0,1)$, $\alpha\neq 1/2$, we have $$\label{eq:theo_expect_number_vertices}
{\mathbb E}L_{\alpha}
=
2+\frac{2-2\alpha}{2\alpha-1}\left(1-2C(1-\alpha)+\frac{\log (1-\alpha)-\alpha\gamma}{1-\alpha}\right).$$ For $\alpha=1/2$ the result should be interpreted by continuity.
We will provide a representation of ${\mathbb E}L_{\alpha}$ as a definite integral in Eqn. below. Using this representation it is possible to compute the value of ${\mathbb E}L_{\alpha}$ in closed form for any rational $\alpha$. Here are some examples:
[|l|c|c|c|c|c|c|c|]{} $\alpha$ & $0$ & $1/4$ & $1/3$ & $1/2$ & $2/3$ & $3/4$ & $1$\
${\mathbb E}L_{\alpha}$ & 2 & $\left(\frac 32-\frac{4}{3\sqrt 3}\right)\pi$ & $\frac{4\pi}{3\sqrt 3}$ &$\frac 32 +\frac{\pi^2}{8}$
------------------------------------------------------------------------
------------------------------------------------------------------------
& $2+\frac{2\pi}{3\sqrt 3}$ & $2+\frac{\pi}{2}$ & $+\infty$\
The values at $\alpha=0$ and $\alpha=1$ should be understood as one-sided limits. As a corollary, we have $L_\alpha\to 2$ in distribution as $\alpha\downarrow 0$. Another way to see this is the following theorem.
\[theo:probab\_two\_seg\] For $\alpha\in (0,1)$ we have ${\mathbb{P}}[L_{\alpha}=2]=1-\alpha$.
Real roots {#subsec:real_roots}
----------
Suppose now that the coefficients of the polynomial $G_n(z)=\sum_{k=0}^n \xi_k z^k$ are i.i.d. *real-valued* random variables. Denote by $z_{11},\ldots,z_{N_nn}$ the real roots of $G_n$, the number of real roots being $N_n$. For a special family of distributions satisfying with $\alpha>1$ @shepp_farahmand showed that ${\mathbb E}N_n\sim \frac{2\alpha-2}{2\alpha-1}\log n$ as $n\to\infty$. In the next theorem we describe the positions of the real roots of $G_n$ in the limit $n\to\infty$ for every $\alpha>0$. Recall the notation $z^{\langle a\rangle}=|z|^a {\mathop{\mathrm{sgn}}\nolimits}(z)$, where $z,a\in{\mathbb{R}}$. Define a point process $\Upsilon_n$ on ${\mathbb{R}}$ by $$\Upsilon_n=\sum_{k=1}^{N_n} \delta(z^{\langle b_n \rangle}_{kn}).$$ In addition to we assume that the following limit exists $$\label{eq:tail_real}
c:=\lim_{t\to+\infty}\frac{{\mathbb{P}}[\xi_0>t]}{{\mathbb{P}}[|\xi_0|>t]}\in[0,1].$$
\[theo:real\] Suppose that and hold with some $\alpha>0$. Write $p={\mathbb{P}}[\xi_0>0]$ and suppose that $\xi_0\neq 0$ a.s.
1. For $\alpha\geq 1$ the point process $\Upsilon_{n}$ converges weakly to some point process $\Upsilon_{\alpha,c}$ on ${\mathbb{R}}{\backslash}\{0\}$.
2. For $\alpha\in (0,1)$ the point process $\Upsilon_{2n}$ (respectively, $\Upsilon_{2n+1}$) converges weakly to some point process $\Upsilon_{\alpha,c,p}^{+}$ (respectively, $\Upsilon_{\alpha,c,p}^{-}$) on $[-\infty,+\infty]$ and on ${\mathbb{R}}$.
The somewhat technical description of the point processes $\Upsilon_{\alpha,c}$, $\Upsilon_{\alpha,c,p}^{\pm}$ is postponed to Section \[sec:proof\_real\_def\_proc\]. Recall that by Theorem \[theo:complex\] the complex roots of $G_n$ are located asymptotically on a set of random circles. Each circle crosses the real line at $2$ points. We will show that any of these points may or may not be a real root of $G_n$ with some probabilities. For $\alpha\in (0,1)$ the point processes $\Upsilon_{\alpha,c,p}^{\pm}$ have a.s. finitely many atoms, whereas for $\alpha\geq 1$ the atoms of the point process $\Upsilon_{\alpha,c}$ accumulate a.s. at $\pm 0$ and $\pm\infty$. (Of course, this is related to Proposition \[prop:number\_vertices\_majorant\]). Since the map assigning to a finite counting measure on $[-\infty,\infty]$ its total mass is continuous (locally constant) in the weak topology, we obtain the following statement on the number of real roots of $G_n$.
\[cor:real\_number\] Suppose that and hold with $\alpha\in (0,1)$ and let $\xi_0\neq 0$ a.s. Then, the sequence $\{N_{2n}\}_{n\in{\mathbb{N}}}$ (respectively, $\{N_{2n+1}\}_{n\in{\mathbb{N}}}$) converges in distribution to a random variable $N^{+}_{\alpha,c,p}$ (respectively, $N^{-}_{\alpha,c,p}$).
The expectations ${\mathbb E}N^{+}_{\alpha,c,p}$ and ${\mathbb E}N^{-}_{\alpha,c,p}$ can be computed using the representation of the point processes $\Upsilon_{\alpha,c,p}^{\pm}$ given in Section \[sec:proof\_real\_def\_proc\]: $${\mathbb E}N^{+}_{\alpha,c,p}=
{\mathbb E}N^{-}_{\alpha,c,p}=
\left(2c(1-c)+\frac 12\right)({\mathbb E}L_{\alpha}-2)+2(p+c-2pc)+1.$$ For instance, if the distribution of $\xi_0$ is symmetric with respect to the origin, then both expectation are equal to ${\mathbb E}L_{\alpha}$. We conjecture that the convergence in Corollary \[cor:real\_number\] holds in the $L^1$-sense.
The behavior of $N_n$ in the case $\alpha=1$ remains open. For $\alpha=1$ the result of [@shepp_farahmand] turns formally into ${\mathbb E}N_n=o(\log n)$, whereas the fact that $\Upsilon_{1,c}$ has infinitely many atoms a.s. suggests that ${\mathbb E}N_n$ should be infinite. It is natural to conjecture that for $\alpha=1$, we should have ${\mathbb E}N_n\sim K\log\log n$ for some $K>0$.
Finally, we investigate the number of real roots of $G_n$ in the case $\alpha=0$.
\[theo:real\_alpha0\] Suppose that and hold, $\xi_0\neq 0$ a.s., and write $p={\mathbb{P}}[\xi_0>0]$. Then, the sequence $\{N_{2n}\}_{n\in{\mathbb{N}}}$ converges weakly to a random variable $N^{+}_{0,c,p}$ and the sequence $\{N_{2n+1}\}_{n\in{\mathbb{N}}}$ converges weakly to a random variable $N^{-}_{0,c,p}$ such that $$\begin{aligned}
{\mathbb{P}}[N^{+}_{0,c,p}=m]&=
\begin{cases}
\frac 12 (cp^2+(1-c)(1-p)^2), &m=0,\\
\frac 12+ p(1-p), &m=2,\\
\frac 12 (c(1-p)^2+(1-c)p^2), &m=4;
\end{cases}\label{eq:theo_real_alpha0_eq1}\\
{\mathbb{P}}[N^{-}_{0,c,p}=m]&=
\begin{cases}
1-p-c+2p c, &m=1,\\
p+c-2p c, &m=3.
\end{cases}\label{eq:theo_real_alpha0_eq2}\end{aligned}$$
If the distribution of $\xi_0$ is symmetric with respect to the origin, we obtain the following results: $N^{+}_{0,1/2,1/2}$ takes the values $0,2,4$ with probabilities $ 1/8, 3/4, 1/8$ and $N^{-}_{0,1/2,1/2}$ takes the values $1,3$ with probabilities $1/2, 1/2$.
For fixed $p\in[0,1]$ both $\min_{c\in[0,1]} {\mathbb E}N^{+}_{0,c,p}$ and $\min_{c\in[0,1]} {\mathbb E}N^{-}_{0,c,p}$ are equal to $1+2\min(p,1-p)$. The same number appeared in [@zn] as the minimal expected number of real roots of a random polynomial.
Emergence of the majorant
-------------------------
The least concave majorant which we encountered above is reminiscent of the Newton polygons appearing when solving polynomial equations with non-archimedian (for example, $p$-adic) coefficients; see [@koblitz_book Chapter IV]. Of course, our random polynomial $G_n$ has complex (archimedian) coefficients. However, non-archimedian effects will appear in the following way. Consider the sum $c_1e^{nx_1}+\ldots+c_d e^{nx_d}$, where $x_i>0$ and $c_i\in{\mathbb{C}}$. If $n$ is large, then the most easy way such sum may become zero is if two terms, say $c_ke^{nx_k}$ and $c_le^{nx_l}$, cancel each other and the other terms are much smaller than these two. We will show that under similar considerations apply to the polynomial $G_n(z)=\sum_{j=0}^n \xi_jz^j$ with high probability: $z\in{\mathbb{C}}$ is a root of $G_n$ essentially only if two of the terms, $\xi_kz^k$ and $\xi_lz^l$, cancel each other and all other terms are of smaller order. Geometrically, this means that the points $(k,\log |\xi_k|)$ and $(l,\log|\xi_l|)$ are neighboring vertices of the least concave majorant of the set $\{(j,\log |\xi_j|): j=0,\ldots,n\}$. The non-zero roots of $\xi_kz^k+\xi_lz^l=0$ form a regular polygon inscribed into the circle whose radius is the exponential of minus the slope of the line joining the points $(k,\log |\xi_k|)$ and $(l,\log|\xi_l|)$. Taking the union of such circles over all segments of the majorant we obtain essentially all the roots of $G_n$. To complete the argument, we need to find the limiting form of the majorant as $n\to\infty$. This is done using the following proposition which is known in the extreme-value theory; see [@resnick_book Cor. 4.19(ii)].
\[prop:resnick\] Let $\xi_0,\xi_1,\ldots$ be i.i.d. random variables satisfying . Then, the following convergence holds weakly on the space of locally finite counting measures on $[0,1]\times (0,\infty]$: $$\rho_n:=\sum_{k=0}^n \delta\left(\frac kn, \frac{\log |\xi_k|}{a_n}\right)
{\overset{w}{\underset{n\to\infty}\longrightarrow}}\sum_{i=1}^{\infty}\delta(U_i,V_i)
=:
\rho.$$ Here, $\rho$ is a Poisson point process on $[0,1]\times (0,\infty)$ with intensity $\alpha v^{-(\alpha+1)}dudv$. We agree that the points for which $\log |\xi_k|\leq 0$ are not counted in $\rho_n$.
The paper of @shepp_farahmand seems to be the only work where random polynomials with coefficients satisfying have been considered. The method used there (characteristic functions) is very different from our approach based on majorants. Whether the results of [@shepp_farahmand] can be recovered (or strengthened) using our approach remains open.
The main lemma {#sec:main_lemma}
==============
The next lemma is the key step in the proof. Let $g(z)=\sum_{j=0}^n a_jz^j$ be a (deterministic) polynomial with complex coefficients. Suppose that the points $(k,\log |a_k|)$ and $(l,\log |a_l|)$, where $0\leq k<l\leq n$, are neighboring vertices on the least concave majorant of the set $\{(j,\log |a_j|):j=0,\ldots,n\}$. That is to say, for some $s,r\in{\mathbb{R}}$, we have $$\label{eq:modulus_xi_kl}
\log |a_{k}|=s-kr,
\;\;
\log |a_{l}|=s-lr,
\;\;
h:=\min_{\substack{0\leq j\leq n\\j\neq k,l}}(s-jr-\log |a_j|)>0.$$ Here, we have assumed that no three points of the majorant are on the same line. Note that $h$ measures the gap between the line passing through the points $(k,\log |a_k|)$, $(l,\log |a_l|)$ and the points lying below this line.
\[lem:main\] If $\delta>0$ is such that $ne^{\delta n-h}<1-e^{-\delta}$, then in the ring $e^{r-\delta}<|z|<e^{r+\delta}$ there are exactly $l-k$ roots of $g$. Moreover, if $\zeta$ is such that $2ne^{2\delta n-h}<\zeta<\frac{\pi}{l-k}$, then the set $$\label{eq:set_sector_ring}
\left \{
z\in{\mathbb{C}}:
e^{r-\delta}<|z|<e^{r+\delta},\;\;
\left|\arg z-\frac{\varphi+ 2\pi m}{l-k}\right|\leq \zeta
\right\},$$ where $\varphi=\arg (-a_k/a_l)$, contains exactly one root of $g$ for every $m=1,\ldots,l-k$.
Here, we agree to understand the distance between the arguments of complex numbers as the geodesic distance on the unit circle. Also, let the index $j$ be always restricted to $0\leq j\leq n$.
We will prove a stronger version of the lemma. Namely, we will show that the statement holds for the family of polynomials $$g_t(z)=a_kz^k+a_lz^l+t\sum_{j\ne k,l}a_jz^j,\quad 0\leqslant t\leqslant1.$$ Note that in particular, $g_0(z)=a_kz^k+a_lz^l$ and $g_1(z)=g(z)$. Let $z\in{\mathbb{C}}$ be such that $|z|=e^{r-\delta}$. It follows from that $$t\left|\sum_{j\neq k, l} a_j z^j\right|
\leq
\sum_{j\neq k, l} e^{s-jr-h} e^{j(r-\delta)}
<
n e^{s-h}.$$On the other hand, again by , $$|a_{k}z^{k}+a_{l}z^{l}|
\geqslant
|a_{k}z^{k}|-|a_{l}z^{l}|
=
e^{s}e^{-\delta k}(1-e^{-\delta (l-k)})
>
e^{s-\delta n}(1-e^{-\delta}).$$ Since $ne^{\delta n-h}<1-e^{-\delta}$ holds, everywhere on the circle $|z|=e^{r-\delta}$ we have $$\label{eq:cond_rouche}
|a_{k}z^{k}+a_{l}z^{l}|
>
t\left|\sum_{j\neq k, l} a_j z^j\right|.$$ Hence, by Rouché’s theorem, the polynomial $g_t$ has exactly $k$ roots in the circle $|z|\leq e^{r-\delta}$.
Let now $z\in{\mathbb{C}}$ be such that $|z|=e^{r+\delta}$. Then, $$\label{eq:est_rouche1}
t\left|\sum_{j\neq k, l} a_j z^j\right|
\leq
\sum_{j\neq k,l} e^{s-jr-h} e^{j(r+\delta)}
<
ne^{s-h+\delta n}.$$ On the other hand, $$|a_{k}z^{k}+a_{l}z^{l}|
\geq
|a_{l}z^{l}|-|a_{k}z^{k}|
=
e^{s}e^{\delta l}(1-e^{\delta (k-l)})
>
e^{s}(1-e^{-\delta}).$$ Therefore, inequality also holds everywhere on the circle $|z|=e^{r+\delta}$. It follows from Rouché’s theorem that the polynomial $g_t$ has exactly $l$ roots in the circle $|z|\leq e^{r+\delta}$. Hence, the polynomial $g_t$ has exactly $l-k$ roots in the ring $e^{r-\delta}\leq |z|\leq e^{r+\delta}$.
Let us now show that these $l-k$ roots are located approximately at the same positions as the non-zero roots of the equation $a_lz^{l}+a_kz^k=0$. Let $z_0$ be some root of $g_t$ satisfying $e^{r-\delta}\leq |z_0|\leq e^{r+\delta}$. Then, repeating the argument of we obtain that $$\label{eq:est_arg1}
\left|a_lz_0^{l}+a_kz_0^k\right|
=
\left|\sum_{j\neq k,l}a_jz_0^j\right|
<
n e^{s-h+\delta n}.
$$ Recall that $\varphi=\arg (-a_k/a_l)$. The arguments of the non-zero roots of the equation $a_lz^{l}+a_kz^k=0$ are given by $\frac{\varphi+ 2\pi m}{l-k}$, where $m=1,\ldots,l-k$, and their moduli are equal to $e^{r}$. Let $$\varsigma=\min_{m=1,\ldots,l-k}\left|\arg z_0-\frac{\varphi+ 2\pi m}{l-k}\right|.$$ Note that $\varsigma\in [0,\frac{\pi}{l-k}]$ by definition. Then, $$|\arg (a_lz_0^{l})-\arg (-a_kz_0^k)|=|\arg z_0^{l-k}-\varphi|=(l-k) \varsigma.$$ By the inequality $|z_1-z_2|\geq 2|z_1|\sin (|\arg z_1-\arg z_2|/2)$ valid for $|z_1|\leq |z_2|$ and the inequality $\sin x\geq \frac 2 {\pi} x$ valid for $x\in[0,\frac {\pi} 2]$ we obtain $$\label{eq:est_arg2}
\left|a_lz_0^{l}+ a_kz_0^k\right|
\geq
2e^{s-\delta l}\sin \left(\frac {(l-k)\varsigma}{2}\right)
\geq
\frac 1 {2}e^{s-\delta n} \varsigma.$$ It follows from and that $\varsigma < 2ne^{2\delta n-h}$ and hence $\varsigma<\zeta$. Therefore, every root $z_0$ of $g_t$ such that $e^{r-\delta}\leq |z_0|\leq e^{r+\delta}$ is contained in a set of the form for some $m=1,\ldots,l-k$. To complete the proof, it remains to show that every set contains exactly one root of $g_t$. Since $\zeta<\frac{\pi}{l-k}$, all these sets are disjoint. By the above, $g_t$ does not vanish on their boundaries. It follows from this and the argument principle that the number of roots of $g_t$ in any set is continuous as a function of $t\in[0,1]$ and hence, constant. Obviously, every set contains exactly one root of $g_0$ and hence, exactly one root of $g_t$.
Least concave majorants and weak convergence {#sec:majorant_conv}
============================================
Proposition \[prop:resnick\] states the convergence of the point process $\rho_n$ formed by the logarithms of the coefficients of the random polynomial $G_n$ to the limiting Poisson process $\rho$. We will need to deduce from this the weak convergence of certain functionals of $\rho_n$ to the same functionals of $\rho$. This will be done using the following well-known continuous mapping theorem; see [@resnick_book p. 152] or [@billingsley_book p. 30].
\[prop:cont\_mapping\] Let $F: M_1\to M_2$ be a map between two metric spaces $(M_1,d_1)$ and $(M_2,d_2)$. Let $X_n$ be a sequence of $M_1$-valued random variables converging weakly to some $M_1$-valued random variable $X$. If $F$ satisfies $${\mathbb{P}}[ F \text{ is discontinuous at } X]=0,$$ then $F(X_n)$ converges weakly to $F(X)$ on $(M_2,d_2)$.
In order to apply Proposition \[prop:cont\_mapping\] we need to prove the a.s. continuity of the functionals under consideration. This is the aim of the present section. First we introduce some notation. Let ${\mathfrak{M}}$ be the set of locally finite counting measures $\mu$ on $[0,1]\times (0,\infty]$ such that $\mu([0,1]\times\{\infty\})=0$. We endow ${\mathfrak{M}}$ with the topology of vague convergence. Every $\mu\in{\mathfrak{M}}$ can be written in the form $\mu=\sum_{i}\delta(u_i, v_i)$, where $i$ ranges in some at most countable index set and $u_i\in[0,1]$, $v_i\in(0,\infty)$. The number of atoms of $\mu$ in a set of the form $[0,1]\times [{\varepsilon},\infty)$ is finite for every ${\varepsilon}>0$, but the atoms of $\mu$ may (and often will) have accumulation points in the set $[0,1]\times\{0\}$. The least concave majorant of $\mu\in{\mathfrak{M}}$ is a function ${\mathfrak{C}}_{\mu}:[0,1]\to [0,\infty)$ defined by ${\mathfrak{C}}_{\mu}(t)=\inf_f f(t)$, where the infimum is taken over all concave functions $f:[0,1]\to [0,\infty)$ such that $f(u_i)\geq v_i$ for all $i$. We write the piecewise linear function ${\mathfrak{C}}_{\mu}$ in the form $$\label{eq:def_LCM_mu}
{\mathfrak{C}}_{\mu}(t)=s_k-r_kt, \qquad t\in[x_{k}, x_{k+1}],$$ where $k$ ranges over a finite or infinite discrete subinterval of ${\mathbb{Z}}$. We set $y_k={\mathfrak{C}}_{\mu}(x_k)$. The intervals $[x_k,x_{k+1}]$ (called the linearity intervals of the majorant) are always supposed to be chosen in such a way that the points $(x_k,y_k)$ and $(x_{k+1},y_{k+1})$ are atoms of $\mu$ and there are no further atoms of $\mu$ on the segment joining these two points. Fix some small $\kappa\in (0,1/2)$. Given a counting measure $\mu\in{\mathfrak{M}}$ we define the indices $q'=q'_{\kappa}(\mu)$ and $q''=q''_{\kappa}(\mu)$ by the conditions $x_{q'}\leq \kappa < x_{q'+1}$ and $x_{q''-1}<1-\kappa\leq x_{q''}$. Let ${\mathfrak{M}}_1$ be the set of all counting measures $\mu\in{\mathfrak{M}}$ with the following properties:
1. both $0$ and $1$ are accumulation points for the linearity intervals of ${\mathfrak{C}}_{\mu}$;
2. $\mu(L)\leq 2$ for every line $L\subset {\mathbb{R}}^2$;
3. no atom of $\mu$ has first coordinate $\kappa$ or $1-\kappa$.
Note that every $\mu\in{\mathfrak{M}}_1$ has only simple atoms. Denote by ${\mathfrak{N}}$ the space of finite measures on ${\mathbb{R}}$ endowed with the weak topology. Let $V_k(\mu)$ be the subset of $[0,1]\times [0,\infty)$ consisting of $[0,1]\times\{0\}$ together with all atoms of $\mu$ except for $(x_k,y_k)$ and $(x_{k+1}, y_{k+1})$.
\[lem:Psi\_eta\_cont\] The following mappings are continuous on ${\mathfrak{M}}_1$:
1. $\Psi_{1}:{\mathfrak{M}}\to{\mathfrak{N}}$ defined by $\Psi_{1}(\mu)=\sum_{k=q'}^{q''-1} (x_{k+1}-x_k) \delta(r_k)$;
2. $H_1:{\mathfrak{M}}\to {\mathbb{R}}$ defined by $H_1(\mu)=\min\{s_k-r_ku-v\}$, where the minimum is over $q'\leq k< q''$ and $(u,v)\in V_k(\mu)$;
3. $L_1:{\mathfrak{M}}\to {\mathbb{R}}$ defined by $L_1(\mu)=\min_{q'\leq k<q''} (x_{k+1}-x_k)$.
Let $\{\mu_n\}_{n\in{\mathbb{N}}} \subset {\mathfrak{M}}$ be a sequence converging to $\mu\in{\mathfrak{M}}_1$ in the vague topology on $[0,1]\times (0,\infty]$. Let ${\varepsilon}>0$ be such that $
2{\varepsilon}<\min_{q'\leq k<q''}\{s_{k}, s_{k}-r_{k}\}
$. Note that the minimum is strictly positive by the definition of ${\mathfrak{M}}_1$. Denote by $(u_l,v_l)$, where $1\leq l\leq m$, all atoms of $\mu$ (excluding those which are vertices of ${\mathfrak{C}}_{\mu}$) with the property that $v_l>{\varepsilon}$. Since $\mu_n\to\mu$ vaguely, we can find (see [@resnick_book Prop. 3.13]) atoms of $\mu_n$ denoted by $(x_{kn}, y_{kn})$ (where $q'\leq k\leq q''$) and $(u_{ln},v_{ln})$ (where $1\leq l\leq m$) such that $$\begin{aligned}
&\lim_{n\to\infty} (x_{kn},y_{kn})=(x_k,y_k),&\qquad &q'\leq k\leq q'',\label{eq:lem_cont_1a} \\
&\lim_{n\to\infty} (u_{ln},v_{ln})=(u_l,v_l),&\qquad &1\leq l\leq m. \label{eq:lem_cont_1b}\end{aligned}$$ Moreover, since the vague convergence was required to hold on $[0,1]\times (0,\infty]$, there are no other atoms of $\mu_n$ having a second coordinate exceeding $2{\varepsilon}$ provided that $n$ is sufficiently large. It follows that as $n\to\infty$, $$\label{eq:lem_cont_2}
r_{kn}:=-\frac{y_{(k+1)n}-y_{kn}}{x_{(k+1)n}-x_{kn}} \to r_k,
\;\;
s_{kn}:=y_{kn}+r_{kn} x_{kn}\to s_k,
\;\;
q'\leq k<q''.$$ In particular, for sufficiently large $n$, all $q'\leq k<q''$ and all $1\leq l\leq m$, $$s_{kn}-r_{kn} u_{ln}>v_{ln},
\qquad
\inf_{u\in[0,1]} (s_{kn}-r_{kn}u)>2{\varepsilon}.$$ It follows that for sufficiently large $n$ the segment joining the points $(x_{kn}, y_{kn})$ and $(x_{(k+1)n}, y_{(k+1)n})$ belongs to the majorant of $\mu_n$ for every $q'\leq k<q''$. Also, $x_{q'n}<\kappa<x_{(q'+1)n}$ and $x_{(q''-1)n}<1-\kappa< x_{q''n}$. By using , , and letting ${\varepsilon}\downarrow 0$ we obtain that $H_1(\mu_n)\to H_1(\mu)$ and $L_1(\mu_n)\to L_1(\mu)$ as $n\to\infty$. This proves the continuity of $H_1$ and $L_1$ on ${\mathfrak{M}}_1$. To prove the continuity of $\Psi_{1}$ note that for every continuous, bounded function $f:{\mathbb{R}}\to{\mathbb{R}}$, $$\int_{{\mathbb{R}}}fd\Psi_{1}(\mu_n)
=
\sum_{k=q'}^{q''-1} (x_{(k+1)n}-x_{kn}) f(r_{kn})
{\overset{}{\underset{n\to\infty}\longrightarrow}}\sum_{k=q'}^{q''-1} (x_{k+1}-x_{k}) f(r_{k})
=
\int_{{\mathbb{R}}}fd\Psi_{1}(\mu).$$ Thus, $\Psi_{1}(\mu_n) \to \Psi_{1}(\mu)$ weakly, which proves the continuity of $\Psi_{1}$.
The next lemma will be needed to prove our main results for $\alpha\in (0,1)$. Let ${\mathfrak{M}}_0$ be the set of all non-zero counting measures $\mu\in{\mathfrak{M}}$ with the following properties:
1. the number of linearity intervals of ${\mathfrak{C}}_{\mu}$ is finite and ${\mathfrak{C}}_{\mu}(0)={\mathfrak{C}}_{\mu}(1)=0$;
2. $\bar \mu(L)\leq 2$ for every line $L\subset {\mathbb{R}}^2$, where $\bar \mu=\mu+\delta(0,0)+\delta(1,0)$;
3. no atom of $\mu$ has first coordinate $\kappa$ or $1-\kappa$.
\[lem:Psi\_cont\] The following mappings are continuous on ${\mathfrak{M}}_0$:
1. $\Psi_0:{\mathfrak{M}}\to{\mathfrak{N}}$ defined by $\Psi_0(\mu)=\sum_{k} (x_{k+1}-x_k) \delta(r_k)$, where the sum is over all linearity intervals $[x_k,x_{k+1}]$ of the majorant ${\mathfrak{C}}_{\mu}$;
2. $H_0:{\mathfrak{M}}\to [0,\infty]$ defined by $H_0(\mu)= \min\{s_k-r_ku-v\}$, where the minimum is over $q'< k<q''-1$ and $(u,v)\in V_k(\mu)$;
3. $L_0:{\mathfrak{M}}\to [0,\infty]$ defined by $L_0(\mu)=\min_{q'< k<q''-1} (x_{k+1}-x_k)$.
In fact, $\Psi_0$ is continuous on the whole of ${\mathfrak{M}}$, but we will not need this. The minimum over an empty set is $+\infty$.
Let $\{\mu_n\}_{n\in{\mathbb{N}}} \subset {\mathfrak{M}}$ be a sequence converging vaguely to $\mu\in{\mathfrak{M}}_0$. The majorant ${\mathfrak{C}}_{\mu}$ is a piecewise linear function whose graph is a broken line connecting the points denoted by $(x_k,y_k)$, where $p'\leq k\leq p''$ and $(x_{p'},y_{p'})=(0,0)$, $(x_{p''},y_{p''})=(1,0)$. For $p'<k<p''$, the point $(x_k,y_k)$ is an atom of $\mu$. Denote by $(u_l,v_l)$, where $1\leq l\leq m$, all atoms of $\mu$ (excluding those which are vertices of the majorant) with the property that $v_l>{\varepsilon}$, where ${\varepsilon}>0$ is a number such that $
2{\varepsilon}<\min_{p'<k<p''-1}\{s_{k}, s_{k}-r_{k}\}
$. Note that the minimum is taken over the set of linearity intervals of the majorant excluding the first and the last interval. If the majorant consists of just two segments, then the minimum is $+\infty$. The vague convergence $\mu_n\to\mu$ implies (see [@resnick_book Prop. 3.13]) that we can find atoms of $\mu_n$ denoted by $(x_{kn}, y_{kn})$ (where $p'<k<p''$) and $(u_{ln},v_{ln})$ (where $1\leq l\leq m$) such that $$\begin{aligned}
&\lim_{n\to\infty} (x_{kn},y_{kn})=(x_k,y_k),& \qquad &p'<k<p'',\label{eq:lem_cont_10a} \\
&\lim_{n\to\infty} (u_{ln},v_{ln})=(u_l,v_l),& \qquad &1\leq l\leq m. \label{eq:lem_cont_10b}\end{aligned}$$ Moreover, if $n$ is sufficiently large, then there are no other atoms of $\mu_n$ having a second coordinate exceeding $2{\varepsilon}$. It follows that as $n\to\infty$, $$\label{eq:lem_cont_20}
r_{kn}:=-\frac{y_{(k+1)n}-y_{kn}}{x_{(k+1)n}-x_{kn}} \to r_{k},
\;\; s_{kn}:=y_{kn}+r_{kn} x_{kn}\to s_{k},
\;\; p'<k<p''-1.$$ Note that by concavity $s_k-r_ku_l>v_l$ for all $p'<k<p''-1$ and $1\leq l\leq m$. Thus, for sufficiently large $n$, $$s_{kn}-r_{kn} u_{ln}>v_{ln},
\qquad
\inf_{u\in[0,1]} (s_{kn}-r_{kn}u)>2{\varepsilon}.$$ This means that for sufficiently large $n$ the segment joining the points $(x_{kn}, y_{kn})$ and $(x_{(k+1)n}, y_{(k+1)n})$ belongs to the majorant of $\mu_n$ for every $p'< k<p''-1$. Also, $x_{q'n}<\kappa<x_{(q'+1)n}$ and $x_{(q''-1)n}<1-\kappa< x_{q''n}$.
From , , we obtain that $H_0(\mu_n)\to H_0(\mu)$ and $L_0(\mu_n)\to L_0(\mu)$ as $n\to\infty$. This proves the continuity of $H_0$ and $L_0$ on ${\mathfrak{M}}_0$. To prove the continuity of $\Psi_0$ we need to show that for every continuous, bounded function $f:{\mathbb{R}}\to [0,\infty)$, $$\label{eq:lem_need10}
\lim_{n\to\infty}\int_{{\mathbb{R}}}f d\Psi_0(\mu_n)= \int_{{\mathbb{R}}}fd\Psi_0(\mu).$$ By and we have $$\label{eq:lem_cont_30}
\lim_{n\to\infty} \sum_{p'<k<p''-1} (x_{(k+1)n}-x_{kn}) f(r_{kn})=
\sum_{p'<k<p''-1} (x_{k+1}-x_{k}) f(r_{k}).$$
However, we have to be more careful about approximating the first and the last segments of ${\mathfrak{C}}_{\mu}$. Denote by $(x_{kn},y_{kn})$, where $k\leq p'+1$, the vertices of the majorant of $\mu_n$ (counted from left to right) with the property $x_{kn}\leq x_{(p'+1)n}$. Note that the number of such vertices is, in general, arbitrary and may be infinite. Since the first segment of the majorant of $\mu$ joins $(0,0)$ and $(x_{p'+1},y_{p'+1})$, all points $(u_{ln},v_{ln})$, where $1\leq l\leq m$, are located below the line joining $(0,0)$ and $(x_{(p'+1)n},y_{(p'+1)n})$ for large $n$. Therefore, for large $n$ there are no atoms of $\mu_n$ above the line joining $(0,2{\varepsilon})$ and $(x_{(p'+1)n},y_{(p'+1)n})$. Hence, $$r_{p'n}:=-\frac{y_{(p'+1)n}-y_{p'n}}{x_{(p'+1)n}-x_{p'n}}\in \left[ -\frac{y_{(p'+1)n}-2{\varepsilon}}{x_{(p'+1)n}}, -\frac{y_{(p'+1)n}}{x_{(p'+1)n}}\right],
\;\;
x_{p'n}<2{\varepsilon}\frac{y_{(p'+1)n}}{x_{(p'+1)n}}.
$$ It follows that $r_{p'n}\to r_{p'}$ as $n\to\infty$. The contribution of linearity intervals to the left of $x_{p'n}$ can be estimated as follows: for large $n$, $$\sum_{k< p'} (x_{(k+1)n}-x_{kn}) f(r_{kn})\leq x_{p'n}\|f\|_{\infty}\leq 4{\varepsilon}\frac{y_{p'+1}}{x_{p'+1}}\|f\|_{\infty}.$$ Since ${\varepsilon}>0$ can be made as small as we like, we have $$\label{eq:lem_cont_4a}
\lim_{n\to\infty} \sum_{k\leq p'} (x_{(k+1)n}-x_{kn}) f(r_{kn})=x_{p'+1}f(r_{p'}).$$ Similar arguments can be applied to the part of the majorant of $\mu_n$ located to the right of $(x_{(p''-1)n}, y_{(p''-1)n})$: with straightforward notation, $$\label{eq:lem_cont_4b}
\lim_{n\to\infty} \sum_{k\geq p''-1} (x_{(k+1)n}-x_{kn}) f(r_{kn})=(1-x_{p''-1})f(r_{p''-1}).$$ Bringing , , together we obtain .
In our proofs we will often consider some “good” random event $E_n(\kappa)$ under which we will be able to localize the roots of $G_n$. The next lemma will be useful.
\[lem:good\_event\_conv\_distr\] Let $\{S_n\}_{n\in{\mathbb{N}}}$ and $S$ be random variables defined on a common probability space. Suppose that for each $\kappa>0$ we have random events $\{E_n(\kappa)\}_{n\in{\mathbb{N}}}$ and random variables $\{S_n(\kappa)\}_{n\in{\mathbb{N}}}$, $S(\kappa)$ such that the following conditions hold:
1. for every $\kappa>0$, $S_n(\kappa)\to S(\kappa)$ in distribution as $n\to\infty$;
2. $S(\kappa)\to S$ in distribution as $\kappa\downarrow 0$;
3. $\lim_{\kappa\downarrow 0}\liminf_{n\to\infty}{\mathbb{P}}[E_n(\kappa)]=1$;
4. $|S_n(\kappa)-S_n|<m_n(\kappa)$ on $E_n(\kappa)$, where $\lim_{\kappa\downarrow 0}\limsup_{n\to\infty} m_n(\kappa)=0$.
Then, $S_n \to S$ in distribution as $n\to\infty$.
Let $f:{\mathbb{R}}\to{\mathbb{R}}$ be a continuous function with compact support. Write $C=\|f\|_{\infty}$. Take some ${\varepsilon}>0$. We can choose $\kappa=\kappa({\varepsilon})>0$ such that $$\label{eq:lemma_12}
|{\mathbb E}f(S(\kappa))- {\mathbb E}f(S)| <{\varepsilon},
\;\;\;
\limsup_{n\to\infty}{\mathbb{P}}[E_n^c(\kappa)]<{\varepsilon},
\;\;\;
\limsup_{n\to\infty} m_n(\kappa)<{\varepsilon}.$$ Here, $E_n^c(\kappa)$ denotes the complement of $E_n(\kappa)$. After having fixed $\kappa$ we choose $n_0=n_0({\varepsilon})$ such that for all $n>n_0$, $$\label{eq:lemma_14}
|{\mathbb E}f(S_n(\kappa))- {\mathbb E}f(S(\kappa))|< {\varepsilon},
\;\;\;
{\mathbb{P}}[E_n^c(\kappa)]<2{\varepsilon},
\;\;\;
m_n(\kappa)<2{\varepsilon}.$$ Denoting by $\omega_f(\delta)=\sup_{|z_1-z_2|\leq \delta} |f(z_1)-f(z_2)|$ the continuity modulus of $f$, we have $$\label{eq:lemma_16}
|{\mathbb E}f(S_n)-{\mathbb E}f(S_n(\kappa))|\leq \omega_f(m_n(\kappa))+2C{\mathbb{P}}[E_n^c(\kappa)]\leq \omega_f(2{\varepsilon})+4C{\varepsilon}.$$ Taking ${\varepsilon}\downarrow 0$ in , , , we obtain $\lim_{n\to\infty}{\mathbb E}f(S_n)={\mathbb E}f(S)$.
Proof of Theorem \[theo:complex\]
=================================
Notation {#subsec:proof_main_not}
--------
Let $\xi_0,\xi_1,\ldots$ be i.i.d. random variables satisfying . Consider the least concave majorant ${\mathfrak{C}}_n$ of the set $\{(k,\log |\xi_k|):k=0,\ldots,n\}$, where we agree to exclude points with $\log |\xi_k|\leq 0$ from consideration. By definition, ${\mathfrak{C}}_n(t)=\inf_f f(t)$ for all $t\in[0,n]$, where the infimum is taken over all concave functions $f:[0,n]\to [0,\infty)$ satisfying $f(k)\geq \log |\xi_k|$ for all $k=0,\ldots,n$. For simplicity, we will call ${\mathfrak{C}}_n$ the majorant of the polynomial $G_n$. Denote the vertices of ${\mathfrak{C}}_n$ (from left to right) by $(k_{in}, \log_+|\xi_{k_{in}}|)$, where $0\leq i\leq d_n$ and $k_{0n}=0$, $k_{d_nn}=n$. On the interval $[k_{in},k_{(i+1)n}]$ the majorant is a linear function which we write in the form $$\label{eq:def_LCM_n}
{\mathfrak{C}}_n(t)=S_{in}-R_{in}t, \qquad t\in [k_{in},k_{(i+1)n}],\qquad 0\leq i<d_n.$$
Further, denote by $\rho$ a Poisson point process on $[0,1]\times (0,\infty)$ with intensity $\alpha v^{-(\alpha+1)}dudv$. The majorant of $\rho$ is denoted by ${\mathfrak{C}}_{\rho}$. As in Section \[subsec:complex\_roots\], we denote the vertices of ${\mathfrak{C}}_{\rho}$, counted from left to right, by $(X_k,Y_k)$. In the case $\alpha\geq 1$ the index $k$ ranges (with probability $1$) in ${\mathbb{Z}}$ by Proposition \[prop:number\_vertices\_majorant\]. In the case $\alpha\in (0,1)$ the index $k$ ranges in $p'\leq k\leq p''$, where $p',p''$ are a.s. finite random variables and $(X_{p'},Y_{p'})=(0,0)$, $(X_{p''},Y_{p''})=(1,0)$. On each interval $[X_k,X_{k+1}]$ the majorant ${\mathfrak{C}}_{\rho}$ is a linear function written in the form $$\label{eq:def_LCM_Poi}
{\mathfrak{C}}_{\rho}(t)=S_{k}-R_{k}t,\qquad t\in[X_{k},X_{k+1}].$$
We will be mostly interested in the “main” parts of the majorants ${\mathfrak{C}}_n$ and ${\mathfrak{C}}_{\rho}$. To make this precise, we take some small $\kappa\in (0,1/2)$ and let $0\leq q_n'< q_n''\leq d_n$ and $q'< q''$ be indices (depending on $\kappa$) defined by the conditions $$\begin{aligned}
&k_{q_n'n}\leq \kappa n<k_{(q_n'+1)n},
&\qquad
&k_{(q_n''-1)n}<(1-\kappa)n\leq k_{q_n''n},\label{eq:q_n}\\
&X_{q'}\leq \kappa<X_{q'+1},
&\qquad
&X_{q''-1}<1-\kappa\leq X_{q''}.\label{eq:q}\end{aligned}$$
In our proof of Theorem \[theo:complex\] it will be convenient to consider the logarithms of the roots of $G_n$ rather than the roots themselves. We will prove the following weak convergence of random probability measures on the space $E=[-\infty,\infty]\times [0,2\pi]$: $$\label{eq:main_restated}
\frac 1n\sum_{j=1}^n \delta(b_n \log |z_{jn}|, \arg z_{jn})
{\overset{w}{\underset{n\to\infty}\longrightarrow}}\sum_{k}(X_{k+1}-X_{k}) \lambda_{R_k},$$ where $\lambda_r$ is the Lebesgue measure on $\{r\}\times [0,2\pi]$ normalized to have total mass $1$. The sum on the right-hand side is over all linearity intervals $[X_k,X_{k+1}]$ of the majorant ${\mathfrak{C}}_{\rho}$. To see that implies the statement of Theorem \[theo:complex\] note that the map $F:E\to \bar {\mathbb{C}}$ given by $F(r,\varphi)=e^{r+ i \varphi}$ is continuous and hence, it induces a weakly continuous map between the corresponding spaces of probability measures; see [@resnick_book Prop. 3.18]. By Proposition \[prop:cont\_mapping\] we can apply $F$ to the both sides of which yields Theorem \[theo:complex\]. So, let $f:E\to [0,\infty)$ be a continuous function. To prove Theorem \[theo:complex\] it suffices to show that $$\label{eq:proof_main}
S_n:=\frac 1 n\sum_{j=1}^n f\left(b_n\log |z_{jn}|, \arg z_{jn}\right)
{\overset{d}{\underset{n\to\infty}\longrightarrow}}\sum_{k}(X_{k+1}-X_{k})\bar f(R_k)
=:S,$$ where $\bar f:[-\infty, \infty]\to{\mathbb{R}}$ is defined by $\bar f(r)=\int_E fd\lambda_r=\frac 1 {2\pi}\int_{0}^{2\pi} f\left(r,\varphi\right)d\varphi$.
We will need to consider the cases $\alpha\geq 1$ and $\alpha\in (0,1)$ separately. The main difference is that in the former case the linearity intervals of the majorant ${\mathfrak{C}}_{\rho}$ cluster at $0$ and $1$, whereas in the latter case we have a well-defined first and a well-defined last linearity interval of ${\mathfrak{C}}_{\rho}$. These intervals cannot be ignored and have to be considered separately. This makes the case $\alpha\in (0,1)$ somewhat more difficult.
Proof in the case $\alpha\geq 1$
--------------------------------
The next lemma shows that with probability approaching $1$ the majorant of $G_n$ has some “good” properties. In particular, there is a gap between the majorant and the points lying below the majorant. Let $W_{in}\subset [0,n]\times [0,\infty)$ be the set consisting of $[0,n]\times\{0\}$ together with the points $(k, \log_+ |\xi_k|)$ for all $0\leq k\leq n$ such that $k\neq k_{in},k_{(i+1)n}$.
\[lem:En\_geq1\] Fix sufficiently small ${\varepsilon}>0$ and consider a random event $E_n:=E_n^1\cap E_n^2$, where $$\begin{aligned}
E_n^1&=\left\{\min_{q_n'\leq i<q_n''} \min_{(u,v)\in W_{in}} (S_{in}-R_{in}u-v)>n^{\frac 1 {\alpha}-{\varepsilon}}\right\},\label{eq:En1_geq1}\\
E_n^2&=\left\{\min_{q'_n\leq i<q''_n} (k_{(i+1)n}-k_{in})>\sqrt n\right\}. \label{eq:En2_geq1}\end{aligned}$$ Then, $\lim_{n\to\infty}{\mathbb{P}}[E_n]=1$.
By Proposition \[prop:resnick\] the point process $\rho_n=\sum_{k=0}^n\delta(\frac kn, \frac{\log |\xi_k|}{a_n})$ converges to $\rho$ weakly on ${\mathfrak{M}}$, where the points with $\log |\xi_k|\leq 0$ are ignored. Recall the definition of the functionals $H_1$ and $L_1$ in Lemma \[lem:Psi\_eta\_cont\]. By scaling, $$\begin{aligned}
H_1(\rho_n)&=\frac 1{a_n}\min_{q_n'\leq i<q_n''} \min_{(u,v)\in W_{in}} (S_{in}-R_{in}u-v),\\
L_1(\rho_n)&=\frac 1n \min_{q'_n\leq i<q''_n} (k_{(i+1)n}-k_{in}).\end{aligned}$$ It follows that $${\mathbb{P}}[E_n^1]={\mathbb{P}}[H_1(\rho_n)>a_n^{-1}n^{\frac 1 {\alpha}-{\varepsilon}}],
\;\;\;
{\mathbb{P}}[E_n^2]={\mathbb{P}}[L_1(\rho_n)>n^{-1/2}].$$ By Lemma \[lem:Psi\_eta\_cont\] and Proposition \[prop:cont\_mapping\] (which is applicable since ${\mathbb{P}}[\rho\in {\mathfrak{M}}_1]=1$ for $\alpha\geq 1$), we have $H_1(\rho_n)\to H_1(\rho)$ and $L_1(\rho_n)\to L_1(\rho)$ in distribution as $n\to\infty$. Note that $H_1(\rho)>0$ and $L_1(\rho)>0$ a.s. Also, $a_n>n^{\frac 1 {\alpha}-\frac{{\varepsilon}}{2}}$ for large $n$ by and . It follows that $\lim_{n\to\infty}{\mathbb{P}}[E_n]=1$.
In the next lemma we localize most complex roots of $G_n$ under the event $E_n$.
\[lem:local\_zero\_geq1\] On the random event $E_n$ the following holds: for every $q_n'\leq i<q_n''$ and $1\leq m\leq k_{(i+1)n}-k_{in}$ there is exactly one root of $G_n$ in the set $$Z_{i,m}(n):=
\left\{z\in {\mathbb{C}}:\;\; |\log |z|-R_{in}|< \delta_n ,\;
\left|\arg z-\frac{\varphi_{in}+ 2\pi m}{k_{(i+1)n}-k_{in}}\right|< \delta_n\right\},$$ where $\delta_n=\exp(-n^{\frac 1 {\alpha}-2{\varepsilon}})$ and $\varphi_{in}=\arg (-\xi_{k_{in}}/\xi_{k_{(i+1)n}})$. The above sets are disjoint and there are no other roots in the ring $R_{q_n'n}-\delta_n\leq \log |z|< R_{(q_n''-1)n}+\delta_n$.
First note that on $E_n$ it is impossible that $q_n'=0$ and $\log |\xi_0|\leq 0$. Similarly, on $E_n$ it is impossible that $q_n''=d_n$ and $\log |\xi_n|\leq 0$. It follows from that on the event $E_n$ the conditions of Lemma \[lem:main\] are fulfilled for the polynomial $G_n$ with $k=k_{in}$, $l=k_{(i+1)n}$, $\delta=\zeta=\delta_n$ for every $q_n'\leq i<q_n''$. Hence, every set $Z_{i,m}(n)$ contains exactly one root of $G_n$. Also, it follows from the proof of Lemma \[lem:main\] that there are exactly $k_{q_n'n}$ roots of $G_n$ in the disk $\log |z|<R_{q_n'n}-\delta_n$ and exactly $k_{q_n''n}$ roots in the disc $\log |z|<R_{(q_n''-1)n}+\delta_n$. Hence, there are exactly $k_{q_n''n}-k_{q_n'n}$ roots in the ring $R_{q_n'n}-\delta_n\leq \log |z|< R_{(q_n''-1)n}+\delta_n$, which coincides with the number of different sets $Z_{i,m}(n)$. It remains to show that the sets $Z_{i,m}(n)$ are disjoint on $E_n$. To this end, it suffices to show that on $E_n$ it holds that $R_{(i+1)n}-R_{in}>3\delta_n$ for every $q_n'\leq i<q_n''-1$. We have $$(k_{(i+2)n}-k_{(i+1)n})(R_{(i+1)n}-R_{in})= S_{in}-R_{in}k_{(i+2)n}-\log|\xi_{k_{(i+2)n}}|>n^{\frac 1 {\alpha}-{\varepsilon}}$$ on $E_n$. Since $k_{(i+2)n}-k_{(i+1)n}\leq n$, this implies the required.
Our aim is to show that $S_n\to S$ in distribution as $n\to\infty$; see . Define random variables $S_n(\kappa)$ and $S(\kappa)$ which approximate $S_n$ and $S$ by $$\begin{aligned}
S_n(\kappa)=\frac 1n \sum_{q_n'\leq i<q_n''}(k_{(i+1)n}-k_{in}) \bar f\left(b_nR_{in}\right),
\;\;
S(\kappa)=\sum_{q'\leq i<q''} (X_{i+1}-X_i) \bar f(R_{i}).\end{aligned}$$ Let $\omega_f(\delta)=\sup_{|z_1-z_2|\leq \delta}|f(z_1)-f(z_2)|$, where $\delta>0$, be the continuity modulus of the function $f$.
On the random event $E_n$ it holds that $$|S_n-S_n(\kappa)|\leq \omega_f( 10/\sqrt n)+2\kappa \|f\|_{\infty}.$$
We always assume that the event $E_n$ occurs. Take some $q'_n\leq i<q''_n$. By Lemma \[lem:local\_zero\_geq1\], the polynomial $G_n$ has a unique root, denoted by $z_{i,m}(n)$, in the set $Z_{i,m}(n)$, where $1\leq m\leq \Delta_{in}$ and $\Delta_{in}=k_{(i+1)n}-k_{in}$. Denote by ${\mathcal{Z}}_{in}$ the finite set $\{z_{i,m}(n): 1\leq m\leq \Delta_{in}\}$. By we have $\Delta_{in}>\sqrt n$. By the definition of $Z_{i,m}(n)$ in Lemma \[lem:local\_zero\_geq1\], $$\left|f(b_n\log |z_{i,m}(n)|,\arg z_{i,m}(n))- \frac{\Delta_{in}}{2\pi}\int_{\frac{\varphi_{in}+2\pi m-\pi}{\Delta_{in}}}^{\frac{\varphi_{in}+2\pi m+\pi}{\Delta_{in}}}f(b_nR_{in},\varphi)d\varphi\right|
<\omega_f(10/\sqrt n).$$ Taking the sum over $1\leq m\leq \Delta_{in}$, we obtain $$\label{eq:wspom_est1_geq1}
\frac 1n\left|\sum_{z\in {\mathcal{Z}}_{in}} f\left(b_n\log |z|, \arg z\right)-
\Delta_{in}\bar f\left(b_nR_{in}\right)\right|
\leq \frac{\Delta_{in}}n \omega_f(10/{\sqrt n}).$$ Let ${\mathcal{Z}}_n^*$ be the set of roots (counted with multiplicities) of the polynomial $G_n$ not belonging to $\cup_{q_n'\leq i< q_n''} {\mathcal{Z}}_{in}$. The number of roots in ${\mathcal{Z}}_{n}^*$ is $n-k_{q_n''n}+k_{q_n'n}$, which is at most $2\kappa n$ by . Hence, $$\label{eq:wspom_est2_geq1}
\frac 1 n\sum_{z\in {\mathcal{Z}}_n^*} f\left(b_n\log |z|, \arg z\right)
\leq 2\kappa \|f\|_{\infty}.$$ Taking the sum of over all $q'_n\leq i<q''_n$ and applying we obtain the required inequality.
\[lem:conv\_major\_point\_proc\_geq1\] We have $S_n(\kappa)\to S(\kappa)$ in distribution as $n\to\infty$.
By Proposition \[prop:resnick\] the point process $\rho_n=\sum_{k=0}^n\delta(\frac kn, \frac{\log |\xi_k|}{a_n})$ converges to $\rho$ weakly on ${\mathfrak{M}}$. By Lemma \[lem:Psi\_eta\_cont\] and Proposition \[prop:cont\_mapping\] (which is applicable since ${\mathbb{P}}[\rho\in{\mathfrak{M}}_1]=1$ for $\alpha\geq 1$) we obtain that $\Psi_{1}(\rho_n)$ converges weakly (as a random finite measure on ${\mathbb{R}}$) to $\Psi_{1}(\rho)$. This implies that $
\int_{{\mathbb{R}}}\bar f d \Psi_{1}(\rho_n)
$ converges in distribution to $
\int_{{\mathbb{R}}}\bar f d \Psi_{1}(\rho),
$ which is exactly what is stated in the lemma.
The proof of Theorem \[theo:complex\] in the case $\alpha\geq 1$ can be completed as follows. Recall that $\lim_{n\to\infty}{\mathbb{P}}[E_n]=1$ by Lemma \[lem:En\_geq1\]. Trivially, $S(\kappa)\to S$ as $\kappa\downarrow 0$ a.s. and hence, in distribution. By Lemma \[lem:good\_event\_conv\_distr\] (whose conditions have been verified above) we obtain that $S_n\to S$ in distribution as $n\to\infty$. This proves .
Proof in the case $\alpha\in (0,1)$
-----------------------------------
This case is somewhat more difficult since we have to analyze the first and the last segment of the majorant of $G_n$ separately. In our proof we will assume that $\xi_0\neq 0$ a.s. This assumption will be removed afterwards. Let $0<\tau_n\leq n$, $0\leq \theta_n< n$ be indices (for concreteness, we choose the smallest possible values) such that $$\frac{\log |\xi_{\tau_n}|}{\tau_n}=\max_{k=1,\ldots,n} \frac{\log|\xi_k|}{k},
\qquad
\frac{\log |\xi_{\theta_n}|}{n-\theta_n}=\max_{k=0,\ldots,n-1} \frac{\log|\xi_k|}{n-k}.$$ Recall that $W_{in}\subset [0,n]\times [0, +\infty)$ denotes the set consisting of $[0,n]\times\{0\}$ together with the points $(k, \log_+ |\xi_k|)$ for all $0\leq k\leq n$ such that $k\neq k_{in},k_{(i+1)n}$.
\[lem:E\_n\_01\] For sufficiently small ${\varepsilon}>0$ and $\kappa\in (0,1/2)$ consider a random event $E_n:=\cap_{i=1}^6 E_n^i$, where $$\begin{aligned}
E_n^1&=\left\{\min_{0<i<d_n-1} \min_{(u,v)\in W_{in}} (S_{in}-R_{in}u-v)>n^{\frac 1 {\alpha}-{\varepsilon}}\right\},\label{eq:En1_01}\\
E_n^2&=\left\{\min_{0\leq i<d_n} (k_{(i+1)n}-k_{in})>\sqrt n\right\}, \label{eq:En2_01}\\
E_n^3&=\left\{\min_{j\neq 0,\tau_n} \left(\frac{\log |\xi_{\tau_n}|}{\tau_n}-\frac{\log_+|\xi_j|}{j}\right)>n^{\frac 1 {\alpha}-1-{\varepsilon}}\right\},\label{eq:En3_01}\\
E_n^4&=\left\{\min_{j\neq n,\theta_n} \left(\frac{\log |\xi_{\theta_n}|}{n-\theta_n}-\frac{\log_+|\xi_j|}{n-j}\right)>n^{\frac 1 {\alpha}-1-{\varepsilon}}\right\},\label{eq:En4_01}\\
E_n^5&=\{\tau_n>\kappa n, \theta_n<(1-\kappa)n\},\label{eq:En5_01}\\
E_n^6&=
\{
|\log |\xi_0||<n^{{\varepsilon}}, |\log |\xi_n||<n^{{\varepsilon}}
\}. \label{eq:En6_01}\end{aligned}$$ Then, $\lim_{\kappa\downarrow 0}\liminf_{n\to\infty} {\mathbb{P}}[E_n]=1$ for every ${\varepsilon}>0$.
Note that $E_n^1$ states that all segments of the majorant except for the first and the last one are well-separated from the points below the majorant. For the first and the last segment the well-separation property is stated in random events $E_n^3$ and $E_n^4$.
\[rem:E\_n\_01\] We will see that on $E_n^3\cap E_n^6$ the segment joining the points $(0,\log_+|\xi_0|)$ and $(\tau_n, \log|\xi_{\tau_n}|)$ is the first segment of the majorant of $G_n$. Similarly, on $E_n^4\cap E_n^6$ the segment joining $(\theta_n, \log |\xi_{\theta_n}|)$ and $(n, \log_+|\xi_n|)$ is the last segment of the majorant of $G_n$. It follows that $q_n'=0$ and $q_n''=d_n$ on the event $\cap_{i=3}^6 E_n^i$.
We start by considering $E_n^3$. Let $\tilde \rho$ be a Poisson point process on $(0,\infty)$ with intensity $\frac {\alpha} {1-\alpha} v^{-(\alpha+1)}dv$. We will show that the following weak convergence of point processes on $(0,\infty]$ holds: $$\label{eq:wspom_conv_poiss}
\tilde \rho_n:=\sum_{k=1}^n\delta\left(\frac {b_n \log |\xi_k|}{k}\right){\overset{w}{\underset{n\to\infty}\longrightarrow}}\tilde \rho.$$ Again, we agree that the terms with $\log |\xi_k|\leq 0$ are ignored. Recall from that $\bar F(a_n)\sim 1/n$ as $n\to\infty$. Take some $t>0$. By and a well-known uniform convergence theorem for regularly varying functions we have, uniformly in $\kappa n\leq k\leq n$, $$\label{eq:wspom1_01}
{\mathbb{P}}\left[\frac {b_n\log |\xi_k|}{k}>t\right]
= \bar F\left(\frac{kta_n}{n}\right)
\sim n^{\alpha-1}k^{-\alpha}t^{-\alpha},\qquad n\to\infty.$$ To estimate the terms with $1\leq k\leq \kappa n$ recall the following Potter bound: for every small $\delta>0$ we have $\bar F(x)/\bar F(y)\leq 2(x/y)^{-\alpha-\delta}$ as long as $x<y$ are sufficiently large; see [@bingham_book Them. 1.5.6]. We have $$\label{eq:wspom2_01}
\sum_{k=1}^{[\kappa n]}{\mathbb{P}}\left[\frac {b_n\log |\xi_k|}{k}>t\right]
= \sum_{k=1}^{[\kappa n]} \bar F\left(\frac{kta_n}{n}\right)
\leq 2\bar F(\kappa ta_n) \sum_{k=1}^{[\kappa n]} \left(\frac{\kappa n}{k}\right)^{\alpha+\delta}
<C \kappa^{1-\alpha}t^{-\alpha}.$$ From and with $\kappa\downarrow 0$ we get $$\label{eq:wspom3_01}
\lim_{n\to\infty}\sum_{k=1}^n {\mathbb{P}}\left[\frac {b_n\log |\xi_k|}{k}>t\right]=\frac{1}{(1-\alpha)}t^{-\alpha}.$$ By a standard argument this implies . Since the weak convergence of point processes in implies (via Proposition \[prop:cont\_mapping\]) the weak convergence of the corresponding upper order statistics, we have $$\min_{j\neq 0,\tau_n}\left\{b_n\left(\frac{\log |\xi_{\tau_n}|}{\tau_n}-\frac{\log_+|\xi_j|}{j}\right)\right\}{\overset{d}{\underset{n\to\infty}\longrightarrow}}\tilde V_{1}-\tilde V_2,$$ where $\tilde V_1,\tilde V_2$ are the largest and the second largest points of $\tilde \rho$. Since $b_n^{-1}>n^{\frac 1{\alpha}-1-\frac{{\varepsilon}}{2}}$ for large $n$ and since $\tilde V_1>\tilde V_2$ a.s., we have $\lim_{n\to\infty}{\mathbb{P}}[E_n^3]=1$. By symmetry, $\lim_{n\to\infty}{\mathbb{P}}[E_n^4]=1$.
Let us consider $E_n^5$. By and we have, for every $t>0$ and sufficiently large $n$, $$\begin{aligned}
{\mathbb{P}}[\tau_n\leq \kappa n]
&\leq
{\mathbb{P}}\left[\max_{k=1,\ldots,n}\frac{b_n\log |\xi_k|}{k}\leq t\right]+{\mathbb{P}}\left[\max_{k=1,\ldots,[\kappa n]}\frac{b_n\log |\xi_k|}{k}>t\right]\\
&<
2\exp\left\{-\frac{1}{1-\alpha}t^{-\alpha}\right\}+C\kappa^{1-\alpha}t^{-\alpha}.\end{aligned}$$ Taking $t^{\alpha}=\kappa^{(1-\alpha)/2}$ and letting $\kappa\downarrow 0$ we obtain $\lim_{\kappa\downarrow 0}\limsup_{n\to\infty}{\mathbb{P}}[\tau_n\leq \kappa n]=0$. By symmetry, $\lim_{\kappa\downarrow 0}\liminf_{n\to\infty}{\mathbb{P}}[E_n^5]=1$. Since we assume that $\xi_0\neq 0$ a.s., we have $\lim_{n\to\infty}{\mathbb{P}}[E_n^6]=1$.
To proceed further we need to prove Remark \[rem:E\_n\_01\]. Let $s,r\in{\mathbb{R}}$ be such that $s=\log_+|\xi_0|$ and $s-\tau_n r=\log |\xi_{\tau_n}|$. On the random event $E_n^3\cap E_n^6$ we have that for every $1\leq j\leq n$, $j\neq \tau_n$, $$s-jr-\log |\xi_j|=j \left(\frac{\log |\xi_{\tau_n}|}{\tau_n}-\frac{\log|\xi_j|}{j}-s\left(\frac{1}{\tau_n}-\frac 1j\right)\right)
>
n^{\frac 1 {\alpha}-1-{\varepsilon}}-2n^{{\varepsilon}}
>0.$$ This proves the required. Let us turn our attention to $E_n^1$ and $E_n^2$. By Proposition \[prop:resnick\] the point process $\rho_n=\sum_{k=0}^n \delta(\frac k n, \frac{\log |\xi_k|}{a_n})$ converges weakly to $\rho$. Recall the definition of the functionals $H_0$ and $L_0$ in Lemma \[lem:Psi\_cont\]. By a scaling argument, $$\begin{aligned}
H_0(\rho_n)&=\frac 1{a_n}\min_{q_n'< i<q_n''-1} \min_{(u,v)\in W_{in}} (S_{in}-R_{in}u-v),\\
L_0(\rho_n)&=\frac 1n \min_{q'_n< i<q''_n-1} (k_{(i+1)n}-k_{in}).
$$ As observed in Remark \[rem:E\_n\_01\], on the event $\cap_{i=3}^6 E_n^i$ we have $q_n'=0$ and $q_n''=d_n$. Hence, $$\begin{aligned}
{\mathbb{P}}[E_n^1]&\geq {\mathbb{P}}[H_0(\rho_n)>a_n^{-1} n^{\frac 1 {\alpha}-{\varepsilon}}]-(1-{\mathbb{P}}[\cap_{i=3}^6 E_n^i]),\\
{\mathbb{P}}[E_n^2]&\geq {\mathbb{P}}[L_0(\rho_n)>n^{-1/2}]-(1-{\mathbb{P}}[\cap_{i=3}^6 E_n^i]).\end{aligned}$$ By Lemma \[lem:Psi\_cont\] and Proposition \[prop:cont\_mapping\] (which is applicable since ${\mathbb{P}}[\rho\in {\mathfrak{M}}_0]=1$ for $\alpha\in (0,1)$), we have $H_0(\rho_n)\to H_0(\rho)$ and $L_0(\rho_n)\to L_0(\rho)$ weakly on $[0,\infty]$ as $n\to\infty$. Note that $H_0(\rho)>0$ and $L_0(\rho)>0$ a.s. and $a_n>n^{\frac 1 {\alpha}-\frac{{\varepsilon}}{2}}$ for large $n$. Also, we have already shown that the probability of the event $\cap_{i=3}^6 E_n^i$ can be made arbitrary close to $1$ by choosing $\kappa$ small and $n$ large. It follows that $\lim_{n\to\infty}{\mathbb{P}}[E_n^1]=\lim_{n\to\infty}{\mathbb{P}}[E_n^2]=1$, as required.
In the next lemma we isolate all roots of $G_n$ under the event $E_n$. It will be convenient to modify the definition of the slopes of the majorant of $G_n$. Let $R_{0n}'$ be such that $\log|\xi_0|-R_{0n}'k_{1n}=\log |\xi_{k_{1n}}|$. This is well-defined since $\xi_0\neq 0$ a.s. Note that if $\log|\xi_0|<0$, then $R_{0n}'$ is not the same as $R_{0n}$. On $E_n$ we have the estimate $$\label{eq:diff_R_in_R_in_prime}
|R_{0n}-R_{0n}'|
\leq
\tau_n^{-1}|\log|\xi_0||
<n^{2{\varepsilon}-1}.$$ In a similar way, we can define $R_{(d_n-1)n}'$. For all $0<i<d_n-1$ set $R_{in}'=R_{in}$.
\[lem:isol\_roots\_01\] On the random event $E_n$ the following holds: for every $0\leq i<d_n$ and $1\leq m\leq k_{(i+1)n}-k_{in}$ there is exactly one root of $G_n$ in the set $$Z_{i,m}(n):=\left\{z\in {\mathbb{C}}:\;\; \left|\log |z|-R_{in}'\right|< \delta_{n} ,\;
\left|\arg z-\frac{\varphi_{in}+ 2\pi m}{k_{(i+1)n}-k_{in}}\right|< \delta_{n}\right\},$$ where $\varphi_{in}=\arg (-\xi_{k_{in}}/\xi_{k_{(i+1)n}})$ and $\delta_{n}=\exp (-n^{\frac{1}{\alpha}-1-3{\varepsilon}})$. The above sets are disjoint and there are no other roots of $G_n$.
Consider the case $i=0$ first. Let $s=\log |\xi_{0}|$ (well-defined since $\xi_0\neq 0$ a.s.) and $r=R_{0n}'$. Note that $\tau_n=k_{1n}$ on $E_n$ by Remark \[rem:E\_n\_01\]. In order to apply Lemma \[lem:main\] with $k=0$, $l=\tau_n$ we need to estimate $h:=\min_{j\neq 0,\tau_n} (s-jr-\log |\xi_j|)$. On the event $E_n$ we have $$\min_{j\neq 0,\tau_n}\frac{s-jr-\log |\xi_j|}{j}
=
\min_{j\neq 0,\tau_n} \left(\frac{\log |\xi_{\tau_n}|}{\tau_n}-\frac{\log|\xi_j|}{j}-s\left(\frac{1}{\tau_n}-\frac 1j\right)\right)
>n^{\frac 1 {\alpha}-1-2{\varepsilon}},$$ which implies that $h>n^{\frac 1 {\alpha}-1-2{\varepsilon}}$. To prove the lemma for $i=0$ apply Lemma \[lem:main\] with $k=0$, $l=\tau_n$ and $\delta=\zeta=\delta_n$. The case $i=d_n-1$ is similar. Let us now consider the case $0<i<d_n-1$. On the event $E_n$ the conditions of Lemma \[lem:main\] are fulfilled for the polynomial $G_n$ with $k=k_{in}$, $l=k_{(i+1)n}$ and $\delta=\zeta=\delta_n$; see . The statement follows by Lemma \[lem:main\].
It remains to prove that the sets $Z_{i,m}(n)$ are disjoint. It suffices to show that on $E_n$ it holds that $R_{(i+1)n}'-R_{in}'>3\delta_n$ for every $0\leq i<d_n$. We have $$\label{eq:est_dist_R_in}
(k_{(i+2)n}-k_{(i+1)n})(R_{(i+1)n}-R_{in})= S_{in}-R_{in}k_{(i+2)n}-\log_+|\xi_{k_{(i+2)n}}|.$$ For $i\neq 0, d_n-1$ it follows from that the right-hand side can be estimated below by $n^{\frac 1 {\alpha}-{\varepsilon}}$ on $E_n$. The required follows since $k_{(i+2)n}-k_{(i+1)n}\leq n$. Using we obtain that for $i=0$ on the event $E_n$ it holds that $$\frac{k_{2n}-k_{1n}}{k_{2n}}(R_{1n}-R_{0n})
=
\frac{\log |\xi_{\tau_n}|}{\tau_n}-\frac{\log |\xi_{k_{2n}}|}{k_{2n}}- \log_+|\xi_0| \left(\frac 1{\tau_n}-\frac{1}{k_{2n}}\right)
>
n^{\frac{1}{\alpha}-1-2{\varepsilon}},$$ where the last inequality follows from , . It follows that $R_{1n}-R_{0n}>n^{\frac{1}{\alpha}-1-2{\varepsilon}}$. Recalling we obtain $R_{1n}'-R_{0n}'>3\delta_n$. The case $i=d_n-1$ is similar.
Recall from that we need to prove that $S_n\to S$ in distribution as $n\to\infty$. Define a random variable $S_n^*$ which approximates $S_n$ by $$S_n^*=\frac 1n \sum_{0\leq i<d_n-1}(k_{(i+1)n}-k_{in}) \bar f\left(b_nR_{in}\right).$$
On the random event $E_n$ it holds that $|S_n^*-S_n|<\omega_f(n^{-{\varepsilon}})$.
Assume that the event $E_n$ occurs. Take some $0\leq i<d_n$. Write $\Delta_{in}=k_{(i+1)n}-k_{in}$. By Lemma \[lem:isol\_roots\_01\], the polynomial $G_n$ has a unique root, denoted by $z_{i,m}(n)$, in the set $Z_{i,m}(n)$ for every $1\leq m\leq \Delta_{in}$. Denote by ${\mathcal{Z}}_{in}$ the finite set $\{z_{i,m}(n): 1\leq m\leq \Delta_{in}\}$. Recall from that $\Delta_{in}>\sqrt n$. It follows from the definition of the set $Z_{i,m}(n)$ that for every $1\leq m\leq \Delta_{in}$, $$\left|f(b_n\log |z_{i,m}(n)|,\arg z_{i,m}(n))- \frac{\Delta_{in}}{2\pi}\int_{\frac{\varphi_{in}+2\pi m-\pi}{\Delta_{in}}}^{\frac{\varphi_{in}+2\pi m+\pi}{\Delta_{in}}}f(b_nR_{in},\varphi)d\varphi\right|
<\omega_f(n^{-{\varepsilon}}).$$ Note that for $i=0$ and $i=d_n-1$ we need to use to prove this estimate. Taking the sum over $1\leq m\leq \Delta_{in}$, we obtain $$\frac 1n\left|\sum_{z\in {\mathcal{Z}}_{in}} f\left(b_n\log |z|, \arg z\right)-
\Delta_{in}\bar f\left(b_nR_{in}\right)\right|
\leq \frac{\Delta_{in}}n \omega_f(n^{-{\varepsilon}}).$$ Taking the sum over $0\leq i<d_n$ we obtain the required.
\[lem:Psi\_maj\_G\_n\] We have $S_n^*\to S$ in distribution as $n\to\infty$.
By Proposition \[prop:resnick\] the point process $\rho_n=\sum_{k=0}^n\delta(\frac kn, \frac{\log |\xi_k|}{a_n})$ converges weakly to $\rho$. By Lemma \[lem:Psi\_cont\] and Proposition \[prop:cont\_mapping\] (which is applicable since ${\mathbb{P}}[\rho\in{\mathfrak{M}}_0]=1$ for $\alpha\in (0,1)$) we have that $\Psi_0(\rho_n)$ converges weakly (as a random probability measure on ${\mathbb{R}}$) to $\Psi_0(\rho)$. It follows that $\int_{{\mathbb{R}}}\bar f d \Psi_0(\rho_n)$ converges in distribution to $\int_{{\mathbb{R}}}\bar f d \Psi_0(\rho)$, which is exactly what is stated in the lemma.
The proof of Theorem \[theo:complex\] in the case $\alpha\in (0,1)$ can be completed as follows. By Lemma \[lem:good\_event\_conv\_distr\] with $S_n(\kappa)=S_n^*$ and $S(\kappa)=S$ we obtain $S_n\to S$ in distribution as $n\to\infty$. This proves .
The following explains how to get rid of the assumption $\xi_0\neq 0$ a.s. Let ${\mathbb{P}}[\xi_0\neq 0]$ be strictly positive. Denote the first (respectively, last) non-zero coefficient of $G_n$ by $\xi_{l_n}$ (respectively, $\xi_{n-m_n}$). For fixed $l,m\in {\mathbb{N}}_0$ consider the conditional distribution ${\mathbb{P}}_{l,m}^{n}$ of the random variables $\xi_k$, $l\leq k\leq n-m$, given that $l_n=l$, $m_n=m$. Under ${\mathbb{P}}_{l,m}^{n}$, these variables are independent and, apart from the first and the last variable, identically distributed. It is easily seen that the above proof applies to the polynomial $\sum_{k=l}^{n-m}\xi_k z^k$ under ${\mathbb{P}}_{l,m}^{n}$. Since this holds for all $l,m\in{\mathbb{N}}_0$, the proof is complete.
Proof of Theorem \[theo:complex\_alpha0\]
=========================================
Recall that $\tau_n\in\{0,\ldots,n\}$ is such that $
M_n:=\max_{k=0,\ldots,n} \log |\xi_k|=\log |\xi_{\tau_n}|.
$ Intuitively, under the slow variation condition , the maximum $M_n$ is with probability close to $1$ much larger than all the other terms $\log |\xi_k|$, $1\leq k\leq n$. The majorant of the set $\{(j,\log |\xi_j|): j=0,\ldots,n\}$ consists, with high probability, of two segments joining the endpoints $(0, \log_+ |\xi_0|)$ and $(n,\log_+ |\xi_n|)$ to the maximum $(\tau_n, \log|\xi_{\tau_n}|)$. The roots of $G_n$ group around two circles corresponding to these segments. Our aim is to make this precise. Let the index $k$ be always restricted to $0\leq k\leq n$. We may always assume that the index $\tau_n$ is defined uniquely, since this event has probability converging to $1$ as $n\to\infty$; see [@darling].
\[lem:0\_En\] For $\kappa\in (0,1/2)$, $A>0$ define a random event $E_n=\cap_{i=1}^4E_n^i$, where $$\begin{aligned}
E_n^1&= \left\{\min_{k\neq 0,\tau_n}\left(\frac {M_n}{\tau_n}-\frac{\log |\xi_k|} {k}\right) > n^{2A}\right\},\\
E_n^2&=\left\{\min_{k\neq \tau_n,n}\left(\frac {M_n}{\tau_n}-\frac{\log |\xi_k|} {n-k}\right) > n^{2A}\right\},\\
E_n^1&=\{\kappa n<\tau_n<(1-\kappa)n\},\\
E_n^4&=\{\|\log|\xi_0||<n^{A}, M_n>n^{2A+1}, |\log|\xi_n||<n^{A}\}.
$$ Then, for every $A>0$, $\lim_{\kappa\downarrow 0}\liminf_{n\to\infty}{\mathbb{P}}[E_n]=1$.
By symmetry, $\tau_n/n$ converges as $n\to\infty$ to the uniform distribution, which implies that $\lim_{\kappa\downarrow 0}\liminf_{n\to\infty}{\mathbb{P}}[E_n^3]=1$. By [@darling Thm. 3.2] the slow variation condition implies that $$\frac 1{M_n}\max_{\substack{0\leq k\leq n\\k\neq \tau_n}} \log |\xi_k| {\overset{P}{\underset{n\to\infty}\longrightarrow}}0.$$ It follows that $$\label{eq:proof0_1}
{\mathbb{P}}\left[\max_{\substack{\kappa n\leq k<n\\k\neq \tau_n} }\frac {\log |\xi_k|}{k}>\frac {M_n} {2n}\right]
\leq
{\mathbb{P}}\left[\max_{\substack{0\leq k\leq n\\k\neq \tau_n}} \log |\xi_k|>\frac {\kappa}{2} M_n\right] {\overset{}{\underset{n\to\infty}\longrightarrow}}0. $$ Put $c_n=\inf \{s: \bar F(s)\leq 1/(\sqrt{\kappa} n)\}$. Then, $\bar F(c_n)\sim 1/(\sqrt{\kappa} n)$ by [@resnick_book pp.15–16] and $\lim_{n\to\infty}c_n/n=\infty$. Recall the Potter bound for slowly varying functions: for every $\delta>0$ we have $\bar F(y)/\bar F(x)< 2(x/y)^{\delta}$ provided that $x>y$ are sufficiently large; see [@bingham_book Thm. 1.5.6]. We have $$\begin{aligned}
{\mathbb{P}}\left[\max_{1\leq k\leq \kappa n}\frac {\log |\xi_k|}{k} >\frac{M_n}{2n} \right]
&\leq
\sum_{1\leq k\leq \kappa n} \bar F\left(\frac{k}{2n} c_n\right)+{\mathbb{P}}\left[M_n<c_n\right]\label{eq:proof0_2}\\
&<
\frac{3}{\sqrt{\kappa} n}\sum_{1\leq k\leq \kappa n} \left(\frac{2n}{k}\right)^{1/4}+\left(1-\frac{1}{2\sqrt{\kappa} n}\right)^{n+1}\notag\\
&<
C(\kappa^{1/4}+e^{-1/(2\sqrt{\kappa})}).\notag\end{aligned}$$ Since $\bar F$ decays more slowly than any negative power of $n$, $$\label{eq:proof0_3}
{\mathbb{P}}\left[\frac{M_n}{2n}>n^{2A}\right]=1-(1-\bar F(2n^{2A+1}))^{n+1}>1-\left(1-\frac 1{n^{2}}\right)^{n+1}{\overset{}{\underset{n\to\infty}\longrightarrow}}1.
$$ Putting , and together and letting $\kappa\downarrow 0$ we obtain $\lim_{n\to\infty}{\mathbb{P}}[E_n^1]=1$. By symmetry, we also have $\lim_{n\to\infty}{\mathbb{P}}[E_n^2]=1$. From it also follows that $\lim_{n\to\infty}{\mathbb{P}}[E_n^4]=1$.
In the sequel, we always suppose that the event $E_n$ occurs. The roots of the equation $\xi_{\tau_n} z^{\tau_n}+\xi_0=0$, denoted by $w_{1n},\ldots,w_{\tau_n n}$, satisfy $$|w_{kn}|=(|\xi_0|/|\xi_{\tau_n}|)^{1/\tau_n}=e^{(\log |\xi_0|-M_n)/\tau_n}< e^{-n^A}, \qquad 1\leq k\leq \tau_n.$$ Similarly, the roots of the equation $\xi_{n} z^{n-\tau_n}+\xi_{\tau_n}=0$, denoted by $w_{(\tau_n+1)n},\ldots,w_{nn}$, satisfy $$|w_{kn}|=(|\xi_{\tau_n}|/|\xi_{n}|)^{1/(n-\tau_n)}=e^{(M_n-\log|\xi_n|)/(n-\tau_n)}> e^{n^A}, \qquad \tau_n<k\leq n.$$ Choose $s,r\in{\mathbb{R}}$ so that $s=\log|\xi_0|$ and $s-r\tau_n=\log |\xi_{\tau_n}|=M_n$. To apply Lemma \[lem:main\] with $k=0$, $l=\tau_n$ we need to estimate $h:=\min_{k\neq 0,\tau_n}(s-rk-\log |\xi_k|)$. We have, by definition of $E_n$, $$\min_{k\neq 0,\tau_n} \frac {s-rk-\log |\xi_k|}k
=\min_{k\neq 0,\tau_n} \left(\frac{M_n}{\tau_n}-\frac{\log |\xi_k|}{k}+s\left(\frac 1k -\frac 1 {\tau_n}\right) \right)
>n^{\frac 32 A}.$$ Hence, $h>n^{\frac 32 A}$. It follows that on the event $E_n$ the conditions of Lemma \[lem:main\] are fulfilled for $k=0$, $l=\tau_n$ and $\delta=\zeta=e^{-n^A}$. Then, for every $1\leq k\leq\tau_n$, the set $$\left\{z\in {\mathbb{C}}:\;\; \left|\log |z|-r\right|\leq e^{-n^A} ,\;
\left|\arg z-\arg w_{kn}\right|\leq e^{-n^A}\right\}$$ contains exactly one root, say $z_{kn}$, of the polynomial $G_n$. It follows that $$|z_{kn}-w_{kn}|< 10\delta e^r= 10 e^{-n^A}|w_{kn}|,\qquad 1\leq k\leq \tau_n.
$$ By symmetry, a similar inequality holds for $\tau_n<k\leq n$.
Proof of Theorem \[theo:real\] and Theorem \[theo:real\_alpha0\] {#sec:proof_real}
=================================================================
Limiting point processes {#sec:proof_real_def_proc}
------------------------
First of all, we describe the limiting point processes $\Upsilon_{\alpha,c}$ and $\Upsilon_{\alpha,c,p}^{\pm}$. Let $\rho$ be a Poisson point process on $[0,1]\times (0,\infty)$ with intensity $\alpha v^{-(\alpha+1)}dudv$ and majorant ${\mathfrak{C}}_{\rho}$ as in Section \[subsec:complex\_roots\]. Recall that the vertices of the majorant ${\mathfrak{C}}_{\rho}$ are denoted by $(X_k,Y_k)$. For $\alpha\geq 1$ the index $k$ ranges in ${\mathbb{Z}}$, whereas for $\alpha\in (0,1)$ we have $p'\leq k\leq p''$ and $(X_{p'},Y_{p'})=(0,0)$, $(X_{p''},Y_{p''})=(1,0)$. Let $\sigma_{k},\pi_{k}$ be independent $\{-1,1\}$-valued random variables (attached to the *vertices* $(X_k,Y_k)$ of ${\mathfrak{C}}_{\rho}$ except for the boundary vertices $(0,0)$ and $(1,0)$ in the case $\alpha\in (0,1)$) such that $${\mathbb{P}}[\sigma_{k}=1]=c,\qquad {\mathbb{P}}[\pi_{k}=1]=1/2. $$ In the case $\alpha\in (0,1)$ we have to add the boundary conditions
1. $\pi_{p'}=1$;
2. $\pi_{p''}=1$ in the definition of $\Upsilon_{\alpha,c,p}^{+}$ and $\pi_{p''}=-1$ in the definition of $\Upsilon_{\alpha,c,p}^{-}$;
3. ${\mathbb{P}}[\sigma_{p'}=1]={\mathbb{P}}[\sigma_{p''}=1]=p$.
Define random variables ${\varepsilon}_k^+$ and ${\varepsilon}_k^-$ attached to the *linearity intervals* $[X_k,X_{k+1}]$ of the majorant ${\mathfrak{C}}_{\rho}$ by $$\begin{aligned}
{\varepsilon}_k^+={\mathbbm{1}}_{\{\sigma_{k}\neq \sigma_{k+1}\}},
\qquad
{\varepsilon}_k^-=
{\mathbbm{1}}_{\{\sigma_{k}\pi_k\neq \sigma_{k+1}\pi_{k+1}\}}.\end{aligned}$$ With this notation, the limiting point processes $\Upsilon_{\alpha,c}$ and $\Upsilon_{\alpha,c,p}^{\pm}$ are defined by $$\label{eq:Pi_real}
\Upsilon_{\alpha,c(,p)}^{(\pm)}
=
\sum_{k}{\varepsilon}_k^+\delta(e^{R_k})+
\sum_{k}{\varepsilon}_k^-\delta(-e^{R_k}),$$ where the sum is over all linearity intervals of the majorant ${\mathfrak{C}}_{\rho}$ and $R_k$ is the negative of the slope of the $k$-th segment of ${\mathfrak{C}}_{\rho}$ as in . We proceed to the proof of Theorem \[theo:real\].
Proof in the case $\alpha\geq 1$
--------------------------------
We will show that the following weak convergence of point processes on $E={\mathbb{R}}\times \{-1,1\}$ holds true: $$\label{eq:main_real_restated}
\sum_{j=1}^{N_n} \delta(b_n\log |z_{jn}|, {\mathop{\mathrm{sgn}}\nolimits}z_{jn}){\overset{w}{\underset{n\to\infty}\longrightarrow}}\sum_{k} {\varepsilon}_k^+ \delta(R_k,1)+\sum_{k}{\varepsilon}_k^-\delta(R_k,-1),$$ where the sum on the right-hand side is over all linearity intervals of the majorant ${\mathfrak{C}}_{\rho}$. To see that implies Theorem \[theo:real\] for $\alpha\geq 1$ note that the mapping $F:E\to {\mathbb{R}}{\backslash}\{0\}$ given by $F(r,\sigma)=\sigma e^r$ is continuous and proper (preimages of compact sets are compact). By [@resnick_book Prop. 3.18] it induces a vaguely continuous mapping between the spaces of locally finite counting measures on $E$ and ${\mathbb{R}}{\backslash}\{0\}$. By Proposition \[prop:cont\_mapping\] we may apply this mapping to the both sides of , which implies the statement of Theorem \[theo:real\] for $\alpha\geq 1$. Denote by ${\mathcal{Z}}_n^+$ (respectively, ${\mathcal{Z}}_n^-$) the set of positive (respectively, negative) real roots of $G_n$, counted with multiplicities. Let $f^+,f^-:{\mathbb{R}}\to [0,\infty)$ be two continuous functions supported on an interval $[-A,A]$. Define random variables $S_n$ and $S$ by $$\begin{aligned}
S_n&=\sum_{z\in {\mathcal{Z}}_n^+} f^+(b_n\log z)+\sum_{z\in {\mathcal{Z}}_n^-} f^-(b_n\log |z|),\label{eq:Sn_real}\\
S&=\sum_{k} {\varepsilon}_k^+ f^+(R_k)+\sum_{k} {\varepsilon}_k^- f^-(R_k),\label{eq:S_real}\end{aligned}$$ where the sum in is over all linearity intervals of ${\mathfrak{C}}_{\rho}$. To prove it suffices to show that $S_n\to S$ in distribution as $n\to\infty$. In fact, we may even suppose additionally that $f^+$ and $f^-$ are Lipschitz, that is $|f^{\pm}(z_1)-f^{\pm}(z_2)|<L|z_1-z_2|$ for some $L>0$ and all $z_1,z_2\in{\mathbb{R}}$. The first step is to localize the real roots of $G_n$ under some “good” event. We use the same notation as in Section \[subsec:proof\_main\_not\]. Take $\kappa\in (0,1/2)$ and recall that the random indices $q_n'$ and $q_n''$ have been defined in . Define a random event $E_n$ as in Lemma \[lem:En\_geq1\]. Additionally, we will need another “good” event $F_n$. The next lemma states that it has probability close to $1$.
\[lem:Fn\] Consider a random event $
F_n=\{ b_n R_{q_n'n}<-2A\}\cap \{b_n R_{(q_n''-1)n}>2A\}.
$ Then, $\lim_{\kappa\downarrow 0} \liminf_{n\to\infty} {\mathbb{P}}[F_n]=1$.
Recall from Section \[sec:majorant\_conv\] that ${\mathfrak{M}}$ is the space of locally finite counting measures on $[0,1]\times (0,\infty]$ which do not charge the set $[0,1]\times \{\infty\}$. Given $\mu\in {\mathfrak{M}}$ we denote by $[x_{q'},x_{q'+1}]$ the unique linearity interval of the majorant ${\mathfrak{C}}_{\mu}$ such that $x_{q'}\leq \kappa< x_{q'+1}$. Denote by $r_{q'}$ the negative of the slope of the corresponding segment of ${\mathfrak{C}}_{\mu}$. Define a map $T_{\kappa}:{\mathfrak{M}}\to {\mathbb{R}}$ by $T_{\kappa}(\mu)=r_{q'}$. Then, the same argument as in Lemma \[lem:Psi\_eta\_cont\] shows that $T_{\kappa}$ continuous on ${\mathfrak{M}}_1$; see . Applying Proposition \[prop:resnick\] together with Proposition \[prop:cont\_mapping\] and noting that $T_{\kappa}(\rho_n)=b_n R_{q_n'n}$ we obtain that for every $\kappa>0$, $b_n R_{q_n'n}\to T_{\kappa}(\rho)$ in distribution as $n\to\infty$. By Proposition \[prop:number\_vertices\_majorant\] we have $T_{\kappa}(\rho)\to -\infty$ a.s. as $\kappa\downarrow 0$. It follows easily that $\lim_{\kappa\downarrow 0} \liminf_{n\to\infty} {\mathbb{P}}[b_n R_{q_n'n}<-2A]=1$. The statement of the lemma follows by symmetry.
In the next lemma we will localize, under the event $E_n\cap F_n$, those real roots of $G_n$ which are contained in $[-A,A]$. Recall that the vertices of the majorant of $G_n$ are denoted (from left to right) by $(k_{in}, \log_+|\xi_{k_{in}}|)$, where $0\leq i\leq d_n$ and $k_{0n}=0$, $k_{d_nn}=n$. We already know that any linearity interval $[k_{in},k_{(i+1)n}]$ of the majorant corresponds to a “circle” of *complex* roots of $G_n$ located approximately at the same positions as the non-zero roots of the polynomial $\xi_{k_{in}}z^{k_{in}}+\xi_{k_{(i+1)n}}z^{k_{(i+1)n}}$. In order to localize the *real* roots of $G_n$ we have to keep track of two things: the signs of the coefficients $\xi_{k_{in}}, \xi_{k_{{(i+1)n}}}$ and the parities of the indices $k_{in},k_{(i+1)n}$. Write $$\begin{aligned}
{\varepsilon}_{in}^+&={\mathbbm{1}}\{{\mathop{\mathrm{sgn}}\nolimits}(\xi_{k_{in}})\neq {\mathop{\mathrm{sgn}}\nolimits}(\xi_{k_{(i+1)n}})\},
\label{eq:eps_in_plus_real}\\
{\varepsilon}_{in}^-&={\mathbbm{1}}\{(-1)^{k_{in}}{\mathop{\mathrm{sgn}}\nolimits}(\xi_{k_{in}})\neq (-1)^{k_{(i+1)n}}{\mathop{\mathrm{sgn}}\nolimits}(\xi_{k_{(i+1)n}})\}.\label{eq:eps_in_minus_real}\end{aligned}$$ The next lemma shows that ${\varepsilon}_{in}^+$ (respectively, ${\varepsilon}_{in}^-$) is the indicator of the presence of a real root of $G_{n}$ near $e^{R_{in}}$ (respectively, $-e^{R_{in}}$).
\[lem:local\_zero\_geq1\_real\] On the random event $E_n$ the following holds: For every $q_n'\leq i<q_n''$ such that ${\varepsilon}_{in}^+=1$ (respectively, ${\varepsilon}_{in}^-=1$) there is exactly one positive (respectively, negative) real root of $G_n$ satisfying $|\log |z|-R_{in}|\leq \exp(-n^{\frac{1}{\alpha}-2{\varepsilon}})$. Moreover, if additionally $F_n$ occurs, then all real roots of $G_n$ satisfying $b_n\log |z|\in [-A,A]$ are among the described above.
We will use the notation of Lemma \[lem:local\_zero\_geq1\]. Recall that on the event $E_n$ for every $q_n'\leq i< q_n''$ and every $1\leq m\leq k_{(i+1)n}-k_{in}$ there is a unique complex root of $G_n$, denoted by $z_{i,m}(n)$, in the set $Z_{i,m}(n)$. Let ${\varepsilon}_{in}^+=1$ for some $q_n'\leq i<q_n''$. Then, $\varphi_{in}=0$ in Lemma \[lem:local\_zero\_geq1\]. Setting $m=k_{(i+1)n}-k_{in}$ we have that $z:=z_{i,m}(n)$ satisfies $|\log |z|-R_{in}|<\delta_n$ and $|\arg z|<\delta_n$. Since the coefficients of $G_n$ are real, the root $z$ must in fact be real (and positive). Indeed, otherwise, we would have a pair complex conjugate roots (rather than a single root) in the set $Z_{i,m}(n)$. Similarly, if ${\varepsilon}_{in}^-=1$ for some $q_n'\leq i<q_n''$, then we have a real negative root of the form $z_{i,m}(n)$ for a suitable $m$. By Lemma \[lem:local\_zero\_geq1\] all real roots in the set $R_{q_n'n}-\delta_n\leq \log |z|\leq R_{(q_n''-1)n}+\delta_n$ are of the above form. To complete the proof note that this set contains the set $-A\leq b_n\log |z|\leq A$ on the event $F_n$.
The random variables $S_n$ and $S$ will be approximated by the random variables $S_n(\kappa)$ and $S(\kappa)$ defined by $$\begin{aligned}
S_n(\kappa)
&=\sum_{q_n'<i<q_n''-1} ({\varepsilon}_{in}^+ f^+(b_nR_{in})+ {\varepsilon}_{in}^- f^-(b_nR_{in})),
\label{eq:Sn_kappa_real}\\
S(\kappa)
&=
\sum_{q'<i<q''-1} ({\varepsilon}_i^+ f^+(R_i)+{\varepsilon}_i^- f^-(R_i)). \label{eq:S_kappa_real}\end{aligned}$$
On the random event $E_n\cap F_n$ we have $|S_n-S_n(\kappa)|<1/n$.
Recall that $f^+$ and $f^-$ are functions supported on $[-A,A]$ with Lipschitz constant at most $L$. By Lemma \[lem:local\_zero\_geq1\_real\] and the definition of $F_n$ we have, on $E_n\cap F_n$, $$\left|\sum_{z\in {\mathcal{Z}}_n^+} f^+(b_n\log z)-\sum_{i=0}^{d_n-1} {\varepsilon}_{in}^+ f^+(b_nR_{in})\right|\leq Ld_n b_n \exp(-n^{\frac{1}{\alpha}-2{\varepsilon}}) \leq \frac 1{2n}.$$ A similar inequality holds for the negative roots and the statement follows.
The next proposition determines the limiting structure of the coefficients of $G_n$ together with attached signs and parities. Let $\tilde{{\mathfrak{M}}}$ be the space of locally finite counting measures on $[0,1]\times (0,\infty]\times \{-1,1\}^2$ which do not charge the set $[0,1]\times \{\infty\}\times \{-1,1\}^2$. We endow $\tilde {\mathfrak{M}}$ with the topology of vague convergence. Every element $\tilde \mu\in\tilde {\mathfrak{M}}$ can be written in the form $\tilde \mu=\sum_{i}\delta(u_i,v_i, \varsigma_i,\varpi_i)$, where $\mu=\sum_{i}\delta(u_i,v_i)\in{\mathfrak{M}}$ is the projection of $\tilde \mu$ on ${\mathfrak{M}}$ and $(\varsigma_i, \varpi_i)\in\{-1,1\}^2$ is considered as a mark attached to the point $(u_i,v_i)$. In the marks $(\varsigma_i, \varpi_i)$ we will record the signs of the coefficients of $G_n$ and the parities of the corresponding indices.
\[prop:resnick\_real\] Let $\xi_0,\xi_1,\ldots$ be i.i.d. random variables satisfying and . Then, the following convergence holds weakly on the space $\tilde {\mathfrak{M}}$: $$\label{eq:tilde_rhon_tilde_rho}
\tilde \rho_n:=\sum_{k=0}^{n} \delta\left(\frac kn, \frac{\log |\xi_k|}{a_n}, {\mathop{\mathrm{sgn}}\nolimits}\xi_k, (-1)^{k}\right)
{\overset{w}{\underset{n\to\infty}\longrightarrow}}\sum_{i=1}^{\infty}\delta(U_i,V_i, \varsigma_i,\varpi_i)
=:
\tilde\rho.$$ Here, $\rho=\sum_{i=1}^{\infty} \delta(U_i,V_i)$ is a Poisson point process on $[0,1]\times (0,\infty)$ with intensity $\alpha v^{-(\alpha+1)}dudv$ and independently, $\varsigma_i, \varpi_i$ are $\{-1,1\}$-valued random variables with ${\mathbb{P}}[\varsigma_{i}=1]=c$ and ${\mathbb{P}}[\varpi_{i}=1]=1/2$. Terms with $\log|\xi_k|\leq 0$ are ignored.
Write $\xi_k^+=\xi_k {\mathbbm{1}}_{\xi_k>0}$ and $\xi_k^-=|\xi_k| {\mathbbm{1}}_{\xi_k\leq 0}$. Note that by , and , $${\mathbb{P}}\left[\frac{\log \xi_k^+}{a_n}> t\right] \sim \frac c {nt^{\alpha}},
\qquad
{\mathbb{P}}\left[\frac{\log \xi_k^-}{a_n}> t\right] \sim \frac {1-c} {nt^{\alpha}},
\qquad
n\to\infty.$$ Fix some $(\varsigma,\varpi)\in\{-1,1\}^2$. We will consider only coefficients $\xi_k$ with sign $\varsigma$ and parity $\varpi$. By Proposition \[prop:resnick\] the point process $$\tilde \rho_n(\varsigma, \varpi):=\sum_{k=0}^{n} \delta\left(\frac {k}n, \frac{\log |\xi_{k}|}{a_{n}}\right){\mathbbm{1}}\{{\mathop{\mathrm{sgn}}\nolimits}(\xi_k)=\varsigma, (-1)^k=\varpi\}$$ converges weakly to the Poisson point process with intensity $(\alpha/2) c v^{-(\alpha+1)}dudv$ if $\varsigma=1$ and $(\alpha/2) (1-c) v^{-(\alpha+1)}dudv$ if $\varsigma=-1$. Taking the union over all $4$ choices of $(\varsigma,\varpi)$, we obtain the statement.
In order to pass from the convergence of the coefficients to the convergence of the point process of real roots we need a continuity argument. Consider $\tilde \mu\in\tilde {\mathfrak{M}}$ with a projection $\mu\in{\mathfrak{M}}$. We denote the vertices of the majorant of $\mu$ counted from left to right by $(x_k, y_k)$. Denote by $r_k$ the negative of the slope of the majorant of $\mu$ on the interval $[x_k,x_{k+1}]$. Let $\kappa\in (0,1/2)$ be fixed and define indices $q'$ and $q''$ by the conditions $x_{q'}\leq \kappa<x_{q'+1}$ and $x_{q''-1}<1-\kappa\leq x_{q''}$. For $q'< k< q''$ we denote by $(\sigma_k, \pi_k)\in\{-1,1\}^2$ the mark attached to the vertex $(x_k,y_k)$. Let $\tilde {\mathfrak{M}}_1$ be the set of all $\tilde \mu\in\tilde {\mathfrak{M}}$ such that $\mu\in{\mathfrak{M}}_1$, where ${\mathfrak{M}}_1\subset {\mathfrak{M}}$ is defined as in Section \[sec:majorant\_conv\]. Let ${\mathfrak{P}}$ be the space of locally finite counting measures on ${\mathbb{R}}$ endowed with the topology of vague convergence. Define a map $\Phi_{1}: \tilde {\mathfrak{M}}\to {\mathfrak{P}}\times {\mathfrak{P}}$ by $$\Phi_{1}(\tilde \mu)
=
\left(
\sum_{q'<k<q''-1} {\mathbbm{1}}_{\sigma_k\neq \sigma_{k+1}}\delta(r_k),
\sum_{q'<k<q''-1} {\mathbbm{1}}_{\sigma_k\pi_k\neq \sigma_{k+1}\pi_{k+1}}\delta(r_k)
\right).$$
\[lem:Psi\_cont\_real\_geq1\] The map $\Phi_{1}$ is continuous on $\tilde {\mathfrak{M}}_1$.
Let $\{\tilde \mu_n\}_{n\in{\mathbb{N}}}\subset \tilde {\mathfrak{M}}$ be a sequence converging vaguely to $\tilde \mu\in \tilde {\mathfrak{M}}_1$. This implies the vague convergence of the corresponding projections: $\mu_n\to\mu\in{\mathfrak{M}}_1$. Arguing as in the proof of Lemma \[lem:Psi\_eta\_cont\] (and using the same notation) we arrive at the following conclusions. There exist points $(x_{kn},y_{kn})$, $q'\leq k\leq q''$, which are vertices of the majorant of $\mu_n$, such that $(x_{kn},y_{kn})\to (x_k,y_k)$ as $n\to\infty$. Further, $x_{q'n}< \kappa<x_{(q'+1)n}$ and $x_{(q''-1)n}<1-\kappa<x_{q''n}$ for sufficiently large $n$. Also, with the same notation as in , $r_{kn}\to r_k$ as $n\to\infty$. Finally, $\tilde \mu_n\to\tilde \mu$ implies that for sufficiently large $n$ the mark $(\sigma_{kn}, \pi_{kn})$ attached to $(x_{kn},y_{kn})$ is the same as the mark $(\sigma_k,\pi_k)$ attached to $(x_k,y_k)$, for all $q'\leq k\leq q''$. This implies that $\Phi_{1}(\tilde\mu_n)\to \Phi_{1}(\tilde\mu)$ as $n\to\infty$, whence the continuity.
We have $S_n(\kappa)\to S(\kappa)$ in distribution as $n\to\infty$.
By Proposition \[prop:resnick\_real\] we have $\tilde \rho_n\to \tilde \rho$ weakly on $\tilde {\mathfrak{M}}$. Define a map $I:{\mathfrak{P}}\times{\mathfrak{P}}\to {\mathbb{R}}$ by $I(\nu^+,\nu^-)=\int_{{\mathbb{R}}} f^+ d\nu^++ \int_{{\mathbb{R}}}f^- d\nu^-$. Clearly, $I$ is continuous on ${\mathfrak{M}}_1$. By Lemma \[lem:Psi\_cont\_real\_geq1\] the map $I\circ \Phi_{1}:\tilde {\mathfrak{M}}\to{\mathbb{R}}$ is continuous. By Proposition \[prop:cont\_mapping\] (which is applicable since ${\mathbb{P}}[\tilde \rho\in\tilde {\mathfrak{M}}_1]=1$ for $\alpha\geq 1$) we have that $I(\Phi_{1}(\tilde \rho_n))\to I(\Phi_{1}(\tilde \rho))$ in distribution. This is exactly what is stated in the lemma.
The proof of Theorem \[theo:real\] in the case $\alpha\geq 1$ can be completed as follows. Trivially, we have $S(\kappa)\to S$ a.s. as $\kappa\downarrow 0$. All the other assumptions of Lemma \[lem:good\_event\_conv\_distr\] have been verified above. Applying Lemma \[lem:good\_event\_conv\_distr\] we obtain $S_n\to S$ in distribution as $n\to\infty$.
Proof in the case $\alpha\in (0,1)$
-----------------------------------
We will show that the weak convergence of point processes in holds, this time on the space $E=[-\infty,+\infty]\times \{-1,1\}$ with the restriction that $n$ stays either even or odd and ${\varepsilon}_k^+, {\varepsilon}_k^-$ on the right-hand side of is defined accordingly to this choice (see the boundary conditions in Section \[sec:proof\_real\_def\_proc\]). Let $f^+,f^-: [-\infty,\infty] \to [0,\infty)$ be two continuous functions such that $|f^{\pm}(z_1)-f^{\pm}(z_2)|<L|z_1-z_2|$ for all $z_1,z_2\in{\mathbb{R}}$. With the same notation as in and it suffices to prove that $S_n\to S$ in distribution as $n\to\infty$. The next lemma localizes all real roots of $G_n$ under a “good” event.
\[lem:local\_zero\_01\_real\] On the random event $E_n$ defined as in Lemma \[lem:E\_n\_01\] the following holds: For every $0\leq i<d_n$ such that ${\varepsilon}_{in}^+=1$ (respectively, ${\varepsilon}_{in}^-=1$) there is exactly one positive (respectively, negative) real root $z$ of $G_n$ satisfying $|\log |z|-R_{in}'|\leq \exp(-n^{\frac 1 {\alpha}-1-3{\varepsilon}})$. Moreover, there are no other real roots of $G_n$.
Follows from Lemma \[lem:isol\_roots\_01\]; see the proof of Lemma \[lem:local\_zero\_geq1\_real\].
Take $\kappa\in (0,1/2)$ and define random variables $S_n(\kappa)$ and $S(\kappa)$ as in and , but with summation over $q_n'\leq k<q_n''$ and $q'\leq k<q''$.
On the random event $E_n$ we have $|S_n-S_n(\kappa)|<1/\sqrt{n}$.
By Remark \[rem:E\_n\_01\] we have $q_n'=0$ and $q_n''=d_n$ on $E_n$. The rest follows from Lemma \[lem:local\_zero\_01\_real\], the Lipschitz property of $f^+$ and $f^-$ and .
Again, we need a continuity argument to transform the convergence of the coefficients in Proposition \[prop:resnick\_real\] into the convergence of real roots. This time, we have to take care of the first and the last coefficients of the random polynomial $G_n$. Write ${\mathfrak{K}}=\tilde{{\mathfrak{M}}}\times \{-1,1\}^2$. Every element of ${\mathfrak{K}}$ can be written in the form $(\tilde \mu, \sigma',\sigma'')$, where $\tilde \mu\in\tilde {\mathfrak{M}}$ and $(\sigma',\sigma'')\in\{-1,1\}^2$. In $\sigma'$ and $\sigma''$ we will record the signs of the first and the last coefficients of $G_n$. As above, the vertices of the majorant of $\mu$ counted from left to right are denoted by $(x_k, y_k)$ and the indices $q'$ and $q''$ are defined by the conditions $x_{q'}\leq \kappa<x_{q'+1}$ and $x_{q''-1}<1-\kappa\leq x_{q''}$. For $q'< k< q''$ (note the strict inequalities) we denote by $(\sigma_k, \pi_k)\in\{-1,1\}^2$ the mark attached to the vertex $(x_k,y_k)$. We will need the following boundary conditions: Define $(\sigma_{q'},\pi_{q'})=(\sigma',1)$ and put $(\sigma_{q''},\pi_{q''})=(\sigma'',1)$ (if we are proving the convergence of $\Upsilon_{2n}$) or $(\sigma_{q''},\pi_{q''})=(\sigma'',-1)$ (if we are proving the convergence of $\Upsilon_{2n+1}$). Let ${\mathfrak{K}}_0$ be the set of all $(\tilde \mu, \sigma',\sigma'')\in {\mathfrak{K}}$ such that the projection $\mu$ of $\tilde \mu$ satisfies $\mu\in{\mathfrak{M}}_0$. Here, ${\mathfrak{M}}_0\subset {\mathfrak{M}}$ is defined as in Section \[sec:majorant\_conv\]. Let ${\mathfrak{Q}}$ be the space of finite counting measures on $[-\infty,\infty]$ endowed with the topology of weak convergence. Define a map $\Phi_{0}: {\mathfrak{K}}\to {\mathfrak{Q}}\times {\mathfrak{Q}}$ by $$\Phi_{0}(\tilde \mu, \sigma',\sigma'')
=
\left(
\sum_{k=q'}^{q''-1} {\mathbbm{1}}_{\sigma_k\neq \sigma_{k+1}}\delta(r_k),
\sum_{k=q'}^{q''-1} {\mathbbm{1}}_{\sigma_k\pi_k\neq \sigma_{k+1}\pi_{k+1}}\delta(r_k)
\right).$$
\[lem:Psi\_cont\_real\_01\] The map $\Phi_{0}$ is continuous on ${\mathfrak{K}}_0$.
Let $\{(\tilde \mu_n, \sigma_n',\sigma_n'')\}_{n\in{\mathbb{N}}}\subset {\mathfrak{K}}$ be a sequence converging vaguely to $(\tilde \mu, \sigma',\sigma'')\in {\mathfrak{K}}_0$. This implies that for sufficiently large $n$, $\sigma_n'=\sigma'$ and $\sigma_n''=\sigma''$. Also, $\tilde \mu_n\to\tilde \mu$ vaguely. Consequently, we have the vague convergence of the corresponding projections: $\mu_n\to\mu$. As in the proof of Lemma \[lem:Psi\_cont\] we obtain the following results. There exist points $(x_{kn},y_{kn})$, $q'<k<q''$, which are vertices of the majorant of $\mu_n$, such that $(x_{kn},y_{kn})\to (x_k,y_k)$ as $n\to\infty$. Also, $x_{q'n}< \kappa<x_{(q'+1)n}$ and $x_{(q''-1)n}<1-\kappa<x_{q''n}$ for sufficiently large $n$. Furthermore, with the same notation as in , $r_{kn}\to r_k$ as $n\to\infty$. It follows from $\tilde \mu_n\to\tilde \mu$ that for sufficiently large $n$ the mark $(\sigma_{kn}, \pi_{kn})$ attached to $(x_{kn},y_{kn})$ is the same as the mark $(\sigma_k,\pi_k)$ attached to $(x_k,y_k)$ for all $q'< k< q''$. The same statement holds for $k=q'$ and $k=q''$ by the boundary conditions. This implies that $\Phi_{0}(\tilde\mu_n, \sigma_n', \sigma_n'')\to \Phi_{0}(\tilde\mu, \sigma', \sigma'')$ as $n\to\infty$.
We have $S_n(\kappa)\to S(\kappa)$ in distribution as $n\to\infty$.
By Proposition \[prop:resnick\_real\] we have $\tilde \rho_n\to \tilde \rho$ weakly on $\tilde {\mathfrak{M}}$. The sum in can be taken from $1$ to $n-1$. Consequently, $(\tilde \rho_n, {\mathop{\mathrm{sgn}}\nolimits}\xi_0,{\mathop{\mathrm{sgn}}\nolimits}\xi_n)$ converges weakly, as a random element in ${\mathfrak{K}}$, to $(\tilde \rho, \sigma',\sigma'')$, where $\sigma'$ and $\sigma''$ are independent (and independent of $\tilde \rho$) $\{-1,1\}$-valued random variables with the same distribution as ${\mathop{\mathrm{sgn}}\nolimits}\xi_0$. By Lemma \[lem:Psi\_cont\_real\_01\] and Proposition \[prop:cont\_mapping\] (which is applicable since ${\mathbb{P}}[(\tilde \rho, \sigma',\sigma'')\in{\mathfrak{K}}_0]=1$ for $\alpha\in (0,1)$) we have that $\Phi_{0}(\tilde \rho_n, {\mathop{\mathrm{sgn}}\nolimits}\xi_0,{\mathop{\mathrm{sgn}}\nolimits}\xi_n)$ converges, as a random element in ${\mathfrak{Q}}\times {\mathfrak{Q}}$, to $\Phi_{0}(\tilde \rho, \sigma',\sigma'')$ as $n\to\infty$. Taking the integrals of $f^+$ and $f^-$ over the components of $\Phi_{0}(\tilde \rho_n, {\mathop{\mathrm{sgn}}\nolimits}\xi_0,{\mathop{\mathrm{sgn}}\nolimits}\xi_n)$ and $\Phi_{0}(\tilde \rho, \sigma',\sigma'')$ we arrive at the statement of the lemma.
The proof of Theorem \[theo:real\] in the case $\alpha\in (0,1)$ can be completed as follows. Trivially, we have $S(\kappa)\to S$ a.s. as $\kappa\downarrow 0$. All the other assumptions of Lemma \[lem:good\_event\_conv\_distr\] have been verified above. Applying Lemma \[lem:good\_event\_conv\_distr\] we obtain $S_n\to S$ in distribution as $n\to\infty$. The proof is complete.
Proof of Theorem \[theo:real\_alpha0\]
--------------------------------------
It follows from the proof of Theorem \[theo:complex\_alpha0\] that on the event $E_n$ defined as in Lemma \[lem:0\_En\] the number of real roots of $G_n$ is the same as the number of real solution of the equation $$\label{eq:equiv_poly_for_alpha0}
\left(\xi_{\tau_n} z^{\tau_n}+\xi_0\right)
\left(\xi_n z^{n-\tau_n}+\xi_{\tau_n}\right)=0.$$ The number of real solutions of depends on whether the numbers $0,\tau_n,n$ are even or odd and on whether the coefficients $\xi_0, \xi_{\tau_n},\xi_n$ are positive or negative. It is not difficult to show that $(-1)^{\tau_n}$ and ${\mathop{\mathrm{sgn}}\nolimits}\xi_{\tau_n}$ become asymptotically independent and that ${\mathbb{P}}[(-1)^{\tau_n}=1]\to 1/2$ and ${\mathbb{P}}[{\mathop{\mathrm{sgn}}\nolimits}\xi_{\tau_n}=1]\to c$ as $n\to\infty$. Considering all possible cases leads to and .
Proof of Theorem \[theo:intensity\] and Theorem \[theo:probab\_two\_seg\]
=========================================================================
Proof of Theorem \[theo:intensity\]
-----------------------------------
Let $\rho$ be a Poisson point process with intensity $\nu(dudv)=\alpha v^{-(\alpha+1)}dudv$ on $E=[0,1]\times (0,\infty)$, where $\alpha\in (0,1)$. We are going to compute the expectation of $L_{\alpha}$, the number of segments of the least concave majorant of $\rho$. Denote by $\rho^2_{\ne}$ the set of all ordered pairs of distinct atoms of the point process $\rho$. For $P_1,P_2\in E$ consider an indicator function $f_{\rho}(P_1,P_2)$ taking value $1$ if and only if there are no points of the Poisson process $\rho$ lying above the line passing through $P_1$ and $P_2$. Counting the first and the last segments of the majorant of $\rho$ separately, we have ${\mathbb E}L_{\alpha}=2+I_{\alpha}/2$, where $$I_{\alpha}={\mathbb E}\left[\sum_{(P_1,P_2)\in\rho^2_{\ne}}f_{\rho}(P_1,P_2)\right].$$ In the sequel we compute $I_{\alpha}$. Applying the Slyvnyack–Mecke formula (see, e.g., [@schneider_weil_book Cor. 3.2.3]), we obtain $$I_{\alpha}=\int_{E^2}{\mathbb E}[f_{\rho}(P_1,P_2)]\nu(dP_1)\nu(dP_2).$$ Denoting $P_1=(x_1,y_1),P_2=(x_2,y_2)$, we have $$I_{\alpha}=\alpha^2\int_0^\infty\int_0^\infty\int_0^1\int_0^1{\mathbb E}[f_{\rho}(P_1,P_2)]y_1^{-\alpha-1}y_2^{-\alpha-1}\,dx_1dx_2dy_1dy_2.$$ The probability of the event that there are no points of $\rho$ lying above the line $P_1P_2$ is non-zero only if the line $P_1P_2$ intersects both vertical sides of the boundary of $E$. Therefore, $$I_{\alpha}=2\alpha^2\int_X\int_{Y} {\mathbb E}[f_{\rho}(P_1,P_2)]y_1^{-\alpha-1}y_2^{-\alpha-1}\,dy_1dy_2 dx_1dx_2,$$ where $X=\{(x_1,x_2): 0<x_1<x_2<1\}$ and $Y=Y_{x_1,x_2}$ is a set defined by $$Y=\left\{(y_1,y_2)\in (0,\infty)^2\,:\,y_1x_2-y_2x_1>0,\quad y_2-y_1+y_1x_2-y_2x_1>0\right\}.$$ Let us replace the variables $y_1,y_2$ by $$r=-\frac{y_2-y_1}{x_2-x_1},\quad u=1+\frac{y_2-y_1}{y_1x_2-y_2x_1}.$$ Then, $(y_1,y_2)\in Y$ if and only if $(r,u)\in(-\infty,0)\times(1,\infty)$ or $(r,u)\in (0,\infty)\times(0,1)$. The inverse transformation is given by $$y_1=r\left(\frac{1}{1-u}-x_1\right), \qquad y_2=r\left(\frac{1}{1-u}-x_2\right).$$ The Jacobian determinant of the transformation $(r,u)\mapsto (y_1,y_2)$ is equal to $
r(x_2-x_1)/(1-u)^2
$. Write $\tilde f_{\rho}(u,r)=f_{\rho}((x_1,y_1(u,r)),(x_2,y_2(u,r)))$. By symmetry, we can consider only the case $r>0$, $u\in (0,1)$. Indeed, considering the case $r>0$ means that we restrict ourselves to segments of the majorant with positive slope. By a change of variables formula, $$\begin{gathered}
I_{\alpha}=4\alpha^2\int_0^\infty\int_0^1\int_X {\mathbb E}[\tilde f_{\rho}(u,r)]
\\\times r^{-2\alpha-1}\left(\frac{1}{1-u}-x_1\right)^{-\alpha-1}\left(\frac{1}{1-u}-x_2\right)^{-\alpha-1}\frac{x_2-x_1}{(1-u)^2}\,dx_1dx_2dudr.\end{gathered}$$ Further, by definition of the Poisson process, $$\begin{aligned}
{\mathbb E}[\tilde f_{\rho}(u,r)]
&=
\exp\left(-\int_{\{(x,y)\in E\,:\,y\geq-rx+\frac{r}{1-u}\}} \alpha y^{-(\alpha+1)}dydx\right) \label{0024}\\
&=
\exp\left(-\int_{0}^1\left(-rx+\frac{r}{1-u}\right)^{-\alpha}dx\right)
\notag \\
&=
\exp\left(-\frac{r^{-\alpha}}{(1-\alpha)}\frac{1-u^{1-\alpha}}{(1-u)^{1-\alpha}}\right).
\notag\end{aligned}$$ The integral $J:=\int_X(c-x_1)^\beta(c-x_2)^\beta(x_2-x_1)dx_1dx_2$, where $c>1$, can be evaluated by writing $(x_2-x_1)=(c-x_1)-(c-x_2)$. We obtain $$\label{0317}
J=
\begin{cases}
\frac{c^{2\beta+3}-(c-1)^{2\beta+3}-(2\beta+3)c^{\beta+1}(c-1)^{\beta+1}}{(\beta+1)(\beta+2)(2\beta+3)}, & \text { if } \beta\ne -1,-3/2,-2,\\
-4\ln\left(\frac{c}{c-1}\right)+\frac 4{\sqrt {c(c-1)}}, &\text{ if }\beta=-3/2.
\end{cases}$$ In the case $\alpha\neq 1/2$, we apply and to obtain $$\begin{gathered}
I_{\alpha}=\frac{4\alpha}{(1-\alpha)(2\alpha-1)}\int_0^\infty\int_0^1r^{-2\alpha-1}\exp\left(-\frac{r^{-\alpha}}{(1-\alpha)}\frac{1-u^{1-\alpha}}{(1-u)^{1-\alpha}}\right)
\\\times (1-u)^{2\alpha-3}\left[1-u^{1-2\alpha}-(1-2\alpha)u^{-\alpha}(1-u)\right]\,dudr.\end{gathered}$$ In the case $\alpha=1/2$ we get, combining with , $$I_{\alpha}=4\int_0^\infty\int_0^1r^{-2}\exp\left(-2r^{-1/2}\frac{1-u^{1/2}}{(1-u)^{1/2}}\right)(1-u)^{-2}\left[u^{-1/2}(1-u)+\ln u\right]\,dudr.$$ Applying in both cases the formula $\int_0^\infty r^{-2\alpha-1}e^{-cr^{-\alpha}}\,dr=(c^2\alpha)^{-1}$ we arrive at $$\label{eq:exp_L_alpha_int}
{\mathbb E}L_{\alpha}=
\begin{cases}
2+\frac{2(1-\alpha)}{(2\alpha-1)}\int_0^1
\frac{1-u^{1-2\alpha}-(1-2\alpha)u^{-\alpha}(1-u)}{(1-u)(1-u^{1-\alpha})^{2}}\, du
&\text{ if } \alpha\neq 1/2,\\
2+\int_0^1\frac{u^{-1/2}(1-u)+\ln u}{(1-u)(1-u^{1/2})^2}\,du,
&\text { if } \alpha=1/2.
\end{cases}$$
The second line is just the limit of the first line as $\alpha\to 1/2$, so that ${\mathbb E}L_{\alpha}$ depends on $\alpha$ continuously. If $\alpha=p/q\neq 1/2$ is rational, then the substitution $v=u^{1/q}$ reduces the integral in to an integral of a rational function which can be computed in closed form; see the table in Section \[subsec:prop\_maj\]. Numerical computation suggests that ${\mathbb E}L_{\alpha}$ is increasing in $\alpha\in (0,1)$.
In the rest of the proof we compute the integral on the right-hand side of in terms of the Barnes modular constant. Let $$K_{\alpha}=\int_0^1
\frac{1-u^{1-2\alpha}-(1-2\alpha)u^{-\alpha}(1-u)}{(1-u)(1-u^{1-\alpha})^{2}}\, du.$$ Write $\beta=1-\alpha$. Recall that $\psi(z)=\Gamma'(z)/\Gamma(z)$ is the logarithmic derivative of the Gamma function. Using the geometric series $\frac 1 {1-u}=\sum_{n=0}^{\infty}u^n$ and the formula $\psi(z)=-\gamma-\frac 1 z+\sum_{n=1}^{\infty} (\frac 1 n-\frac 1 {z+n})$ (see [@bateman § 1.7]) we obtain that for every $m>0$, $$\begin{aligned}
\int_{0}^1 u^{m\beta}\frac{1-u^{1-2\alpha}}{1-u}\,du
&=
\int_{0}^1 \sum_{n=0}^{\infty} u^{n+m\beta}(1-u^{1-2\alpha})\,du\\
&=
\sum_{n=1}^{\infty}\left(\frac 1{n+m\beta}-\frac{1}{n+(m+2)\beta}\right)
-\frac {1}{(m+2)\beta}\\
&=
\psi((m+2)\beta)-\psi(m\beta)-\frac 1 {m\beta}.\end{aligned}$$ For $m=0$ the value of the integral is $\psi(2\beta)+\gamma$, where $\gamma=-\psi(1)$ is the Euler–Mascheroni constant; see [@bateman § 1.7.2]. Using the expansion $\frac 1{(1-u)^2}=\sum_{m=0}^{\infty}(m+1)u^m$ we obtain that $K_{\alpha}=\lim_{N\to\infty}S_N$, where $$\begin{aligned}
S_N
&=
\sum_{m=1}^{N}(m+1)\left(\psi((m+2)\beta)-\psi(m\beta)-\frac{1}{m\beta} \right)-(N+1)\frac{1-2\alpha}{1-\alpha}+\psi(2\beta)+\gamma\\
&=
-2\sum_{m=1}^{N}\psi(m\beta)+(N+1)\psi((N+2)\beta)+N\psi ((N+1)\beta)-2N\\
&-\sum_{m=1}^N\frac 1{m\beta}+\gamma-\frac{1-2\alpha}{1-\alpha}.\end{aligned}$$ The second equality follows by an elementary transformation of the telescopic sum. Using the asymptotic expansion $\psi(z)=\log z-\frac 1 {2z}+o(\frac 1z)$ as $z\to\infty$, we obtain $$S_N=-2\sum_{m=1}^{N}\psi(m\beta)+(2N+1)\log (\beta N)-2N-\frac 1{\beta}\log N+1-\frac{\alpha \gamma}{1-\alpha}+o(1).$$ Comparing this with yields $$K_\alpha=1-2C(1-\alpha)+\frac{\log (1-\alpha)}{1-\alpha}-\frac{\alpha\gamma}{1-\alpha}.$$ The proof of Theorem \[theo:intensity\] is completed by inserting this into .
Proof of Theorem \[theo:probab\_two\_seg\]
------------------------------------------
We prove that ${\mathbb{P}}[L_{\alpha}=2]=1-\alpha$. For a point $P\in E=[0,1]\times (0,\infty)$ let $g_{\rho}(P)$ be the indicator of the following event: there are no atoms of $\rho$ above the lines joining $P$ to the points $(0,0)$ and $(1,0)$. Then, $${\mathbb{P}}[L_{\alpha}=2]={\mathbb E}\left[\sum_{P\in {\mathop{\mathrm{supp}}\nolimits}\rho} g_{\rho}(P)\right].$$ By the Slivnyak–Mecke formula [@schneider_weil_book Cor. 3.2.3], $$\label{eq:N_alpha_2_sliv}
{\mathbb{P}}[L_{\alpha}=2]=\int_E {\mathbb E}[ g_{\rho}(P)] \nu(dP)= \alpha\int_{0}^1\int_{0}^{\infty} {\mathbb E}[g_{\rho}(x,y)] y^{-(\alpha+1)} dydx.$$ The intensity of the Poisson process $\rho$ integrated over the set $\{(u,v)\in E\,:\, u\in [0,x], v>yu/x\}$ is $$\int_{0}^x \int_{yu/x}^{\infty} \alpha v^{-(\alpha+1)}\,dudv=\int_0^x \left(\frac{yu}{x}\right)^{-\alpha} \,du=\frac{1}{1-\alpha} xy^{-\alpha}.$$ By symmetry, the intensity of $\rho$ integrated over the set $\{(u,v)\in E\,:\, u\in [x,1],v>y(u-1)/(x-1)\}$ is $\frac{1}{1-\alpha} (1-x)y^{-\alpha}$. It follows that $${\mathbb E}[g_{\rho}(x,y)]=\exp\left(-\frac{1}{(1-\alpha)y^{\alpha} }\right).$$ Inserting this into we obtain ${\mathbb{P}}[L_{\alpha}=2]=1-\alpha$.
|
---
abstract: 'In this paper, we consider the application of Reed-Solomon (RS) channel coding for joint error correction and cooperative network coding on non-binary phase shift keying (PSK) modulated signals. The relay first decodes the RS channel coded messages received each in a time slot from all sources before applying network coding (NC) by the use of bit-level exclusive OR (XOR) operation. The network coded resulting message is then channel encoded before its transmission to the next relay or to the destination according to the network configuration. This scenario shows superior performance in comparison with the case where the relay does not perform channel coding/decoding. For different orders of PSK modulation and different wireless configurations, simulation results demonstrate the improvements resulting from the use of RS channel codes in terms of symbol error rate (SER) versus signal-to-noise ratio (SNR).'
author:
-
title: 'Joint Channel Coding and Cooperative Network Coding on PSK Constellations in Wireless Networks\'
---
Cooperative network coding, wireless networks, Reed-Solomon channel codes, $M$−PSK modulation, symbol error rate.
Introduction
============
The growing interest for reliable wireless networks compels researchers to explore new techniques that exploit network information theory and telecommunications principles, such as the widely investigated network coding and cooperation. This strategy, since its proposal [@b1], attracts a great attention because of its proven efficiency and is already the subject of many research papers including those considering the use of non-binary modulations and joint channel-network coding and decoding. In contrast to traditional communication networks where the nodes can only forward messages individually from different sources, network coding technique allows nodes to process the incoming independent information flows. Namely, a simple linear combination of incoming messages forms a unique message called network coded (NC) message to be forwarded to one or several destinations. This action provides an optimization in network throughput, resources usage and security. With the addition of cooperative communication concept, and thanks to the broadcast nature of wireless networks, a gain in diversity can be achieved as well.
Exploiting network coding and cooperation principle, the authors in [@b2] investigated the use of low density lattice codes (LDLC) in a multiple access relay channel (MARC) network formed by two source nodes communicating with one destination via one relay. In two time slots, the two sources broadcast their LDLC coded messages to both the relay and the destination then the relay decodes them and forwards, in the third time slot, their XOR combination to the destination where a joint iterative decoding is performed. With a simulation, using $4-$PAM (Pulse Amplitude Modulation) constellation, they showed that the modulo-addition LDLC outperforms the superposition LDLC and their proposed method provides more diversity and coding gains. In [@b3], channel coded physical layer network coding in a two-way relay scenario was investigated. The authors adopted non-binary $M-$PSK modulation where $M\in\{2,3,4,5\}$. The channel coding schemes used were concatenated RS and convolutional code for $M-$PSK where $M \in \{3,4,5\}$ and LDPC code for BPSK (Binary PSK $i.e.~M=2$). They confirmed by simulation how non-binary constellations $(i.e.$ finite fields $GF(M))$ outperform the binary case in terms of frame error rate (FER) versus signal-to-noise ratio (SNR).
![Wireless network topologies considered: (a) X-structure, (b) Extended X-structure, (c) Butterfly network and (d) Extended butterfly network.[]{data-label="fig"}](Imag3.png)
The system model used in [@b4] contains two source nodes communicating with one sink via one relay and over direct links for cooperation. The authors combined linear network coding with bit interleaved coded modulation (BICM) scheme using LDPC codes for the time-division 2 users MARC with orthogonal quasi-static fading channels. After simulations with $2^q-ary$ PAM modulation for $q\in\{1,2,3\}$ and $GF(2^q)$ LDPC channel codes, they showed that non-binary LDPC codes outperform the binary case. The authors in [@b5] proposed a practical scheme called non-binary joint network-channel decoding (NB-JNCD). In their simulation, they used block fading Rayleigh channels $i.e.$ a constant fading coefficient for each transmitted packet of 1000 LDPC coded symbols (coding rate of $0.8$) over $GF(2^4)$ with $16-$QAM (Quadrature AM) modulation.
The rest of the paper is organized as follows. Section II describes the system model and the equations handling the different scenarios in the wireless network configurations considered. In Section III, we first present the assumptions and parameters of our simulations and then we expose the numerical results in graphs showing the evolution of SER versus SNR for all scenarios and all network topologies. Finally, we conclude our work in Section IV.
System Model
============
In this paper, we aim to highlight the advantages of combining NC and cooperation (NCC) with RS channel codes on some widely used wireless topologies for higher orders of PSK constellations. In [@b6], the authors exposed in details many sophisticated channel coding schemes such as RS codes, LDPC codes and Turbo codes. We opt for Reed-Solomon codes because they are well-suited for non-binary symbols transmission over wireless channels. RS codes are non-binary cyclic codes with symbols made up of $q-$bit sequences, where $q$ is any positive integer having a value greater than 2 which makes them suitable for coding symbols in a finite field $GF(M)=GF(2^q)$, $i.e.$, taking values in $\{0,1,...,M-1\}$. The RS coded values of elements in $GF(M)$ are themselves elements of the same field, $i.e.$, if for instance $M=8$ then we have: $\forall s \in GF(8), \Gamma(s) \in GF(8)$ where $\Gamma$ represents the RS coding operator. In the studied networks, the MAC phase, where $N$ source nodes transmit to the relay, lasts $N$ time slots (straightforward network coding). The broadcast phase, where the relay broadcasts only one network coded symbol instead of $N$, is the key to reducing the amount of information transmitted through the network and hence to throughput increase. Furthermore, the use of high order modulation provides high bandwidth efficiency. In this work, we modulate symbols in $GF(8)$, $GF(16)$ and $GF(32)$ using $8-ary$, $16-ary$ and $32-ary$ PSK constellations respectively. All channels are independent (correlation coefficient $\rho=0$) and subject to AWGN and fast Rayleigh attenuation. Perfect channel state information (CSI) is assumed available at each receiver node from its direct transmitter node. All source nodes have channel coding/decoding capabilities, all relays in charge of NC procedure can also perform channel coding/decoding and finally all destination nodes are equipped with channel and network decoders. We first show the advantages of NCC scheme in comparison with the direct path with and without channel coding in terms of throughput, transmit power and diversity gains, then demonstrate how RS codes can reduce considerably the SER and finally we compare the performance of the two following schemes combining RS channel coding and NCC:
1. **Scheme 1**: The relay receiving the RS coded symbols from the sources performs the network coding on them without channel decoding. The destinations apply network decoding before RS decoding. In other words, the relays in charge of network coding are not equipped with channel coders and decoders.
2. **Scheme 2**: The relay receiving the RS coded symbols from the sources first decodes each one of them before applying network coding and then re-encodes the NC symbol resulting. The destinations apply RS channel decoding before network decoding . In other words, the relays in charge of network coding can also execute channel coding/decoding.
Our simulation results show clearly how scheme 2 outperforms scheme 1. The network coding operation is performed by the use of a bit-level XOR operator denoted $\oplus$ on the symbols. For example, in $GF(16)$ with 4 bits symbols, $5 \oplus 12 = 9$ and $3 \oplus 5 = 6$. The XOR operation is applied bit per bit in the binary representation of symbols.
RS codes are non-binary cyclic codes with symbols made up of $q-$bit sequences, where $q$ is any positive integer having a value greater than 2. RS($n, k$) codes on $q-$bit symbols exist for all $n$ and $k$ for which $0 < k < n < 2^q + 2$ where $k$ is the number of data symbols being encoded, and $n$ is the total number of code symbols in the encoded block. For the most conventional RS($n, k$) code, $(n, k) = (2^q - 1, 2^q - 1 - 2t)$ where $t$ is the symbol-error correcting capability of the code, and $n - k = 2t$ is the number of parity symbols. An extended RS code can be made up with $n = 2^q$ or $n = 2^q + 1$, but not any further [@b7].
Hence, the choice of RS code parameters $n$ and $k$ is not random. Similarly to an optimization problem with constraints, we need to maximize the difference $n-k=2t$, $i.e.$ maximize the number $t$ of corrected symbols while respecting some conditions on $n$ and $k$. These constraints are: $n$ and $k$ must satisfy $0<k<n<2^q+2$ and for our simulations, $k$ must be divisor of the number of symbols in the test sequence.
Our contribution in this paper, in contrast to the previous works dealing with the application of NCC in combination with channel codes to a small network, is the generalization to a wide panoply of larger wireless network topologies using one or two relays and a higher number of source-destination pairs. We also confirm the efficiency of RS channel codes application on finite fields elements $GF(M)$ and the superiority of scheme 2 over scheme 1 in providing best network reliability and performance in terms of symbol error probability.
The wireless network topologies studied in this paper are divided in two categories. The first one uses one relay and the second one uses two in series. In the first one, we simulated the $X-$structure configuration and proposed an extended version containing four sources and four corresponding destinations. With two relays in series, we considered the butterfly network topology and proposed an extended version with four source nodes communicating with four respective destination nodes. All these topologies are illustrated for one source communicating with its destination while the remaining nodes are cooperating and hence increasing the spatial diversity and the overall throughput of the wireless network. We assume that the radio range of each source $S_i$ ($i=1, ..., N$) can reach the relay $R$ and the destinations $D_j$ ($j=1, ..., N, j \neq i$).
X-structure network
-------------------
The first wireless network configuration studied is the well known $X-$topology illustrated in Fig. 1a. In this network, two sources are transmitting messages towards two respective destinations where each source is cooperating to ensure the reliable reception of the other source’s message. On the other hand, the relay handles the network coding procedure and broadcasts the result (combination of both messages) to both destinations. This is achieved in 3 time slots while in the same network but without NCC these transmissions would take 4 time slots which lead to a throughput gain of 4/3. The diversity order of the NCC scheme is 2 which can be proved by the following system outage probability calculation. For this network, system outage occurs when the data from source 1 and source 2 cannot both be correctly recovered at their respective destinations [@b8]. Let $P_S$ be the system outage probability and let $p_1$, $p_2$ and $p_R$ be the error rates of uplink channels of $S_1$, $S_2$ and the relay R respectively. The expression of $P_S$ is given by
$$\begin{aligned}
P_S&=p_1p_R(1-p_2)+p_2p_R(1-p_1)+p_1p_2(1-p_R)\\
& +p_1p_2p_R\\
& =p_1p_2+p_1p_R+p_2p_R-2p_1p_2p_R.\end{aligned}$$
If we assume similar uplink error rates for all channels transmissions, $i.e.$ $p_1=p_2=p_R=p<<1$, then we have $$\label{}
P_S=3p^2-2p^3 \sim p^2 \sim O\left(\frac{1}{SNR}\right)^2$$
Let us first define some useful applications:
- Let $\mathbb{F}_M$ be the finite field $GF(M)$ to which belong the symbols to transmit. The application $\Gamma : \mathbb{F}_M^k \rightarrow \mathbb{F}_M^n$ that maps $k$ symbols $s_i$ from $\mathbb{F}_M$ into $n$ symbols $x_i$ from $\mathbb{F}_M$ represents our channel encoder RS($n,k$).
- The application $\Phi : \mathbb{F}_M \rightarrow \mathbb{C}$ that maps a symbol $s$ or $x$ from $\mathbb{F}_M$ into a complex value representing a constellation is our $M-$PSK modulation.
- The application $\psi : \mathbb{F}_M^N \rightarrow \mathbb{F}_M$ that maps $N$ symbols from $\mathbb{F}_M$ into one is our network coding operation performed in practice by a bit-level XOR between the $N$ symbols, $i.e$, $\psi(x_1,x_2,...,x_N) = x_1\oplus x_2\oplus ... \oplus x_N=\bigoplus_{i=1}^{N} x_i.$
- The application $\xi : \mathbb{C} \rightarrow \mathbb{C}$ that maps a complex value into another one represents the equalization process at each receiver which consists of multiplying the received signal by the conjugate of the channel fading coefficient.
In the absence of NC, we consider two scenarios. In the first scenario (without RS coding), the message from $S_1$ to $D_1$ is processed as follows: The symbol $s_1$ is mapped into the $M-$PSK constellation using the application $\Phi$ (PSK modulation) before its transmission to the relay $R$ through a complex AWGN channel with complex fast Rayleigh attenuation. The received symbol at $R$ is $y_{S_1R}=\sqrt{P_1}\Phi(s_1)h_{S_1R}+n_{S_1R}$ where $P_1$ is the transmit power of source 1, $n_{S_1R}$ is the additive white Gaussian noise of the link ($S_1 \rightarrow R$) and $h_{S_1R}$ is the fast fading Rayleigh coefficient (the value of $h_{S_1R}$ varies with each symbol). After reception, the relay first equalizes $y_{S_1R}$ (as we already assumed perfect CSI available) using the equalization operator $\xi$ then simply amplifies and forwards (AF relay) the result. The amplification factor used at the relay is $\beta_{S_1R}$ given by [@b9]: $$\label{}
\beta_{S_1R}=\sqrt\frac{P_R}{P_1|h_{S_1R}|^2+\sigma_{S_1R}^2}$$ where $\sigma_{S_1R}^2$ is the variance of noise $n_{S_1R}$ on the link ($S_1 \rightarrow R$). That is, the forwarded symbol from relay $R$ to destination $D_1$ is $y_{RD_1}=\sqrt{P_R}\beta_{S_1R}\xi(y_{S_1R})h_{RD_1}+n_{RD_1}$ where $P_R$ is the relay transmit power, $h_{RD_1}$ and $n_{RD_1}$ are the Rayleigh coefficient and noise component of the link ($R~\rightarrow~D_1$). At destination node $D_1$, the symbol $s_1$ is recovered using the operations $\xi$ and $\Phi$ as follows: $\tilde{s}_1=\Phi^{-1}(\xi(y_{RD_1}))$.
In the second scenario (with RS coding), the symbol received at relay R from $S_1$ is $$\label{}
y_{S_1R}=\sqrt{P_1}\Phi(\Gamma(s_1))h_{S_1R}+n_{S_1R}.$$ The symbol received at destination $D_1$ has the same form as in the first scenario (assuming the relay unable to perform channel coding/decoding) but, of course, with a different value of $y_{S_1R}$, $i.e.$, $$\label{}
y_{RD_1}=\sqrt{P_R}\beta_{S_1R}\xi(y_{S_1R})h_{RD_1}+n_{RD_1}.$$ The destination (now equipped with RS decoder) recovers the transmitted symbol $s_1$ by evaluating $$\label{}
\tilde{s}_1=\Gamma^{-1}(\Phi^{-1}(\xi(y_{RD_1}))).$$ The scenarios above (without NCC) serve as reference to show the improvements in network reliability provided by the application of NCC procedure described below.
When NCC is applied in the *X*-topology (without channel coding application $\Gamma$), both relay $R$ and destination $D_2$ receive the message broadcast from $S_1$ in time slot 1 over the links ($S_1 \rightarrow R$) and ($S_1 \rightarrow D_2$). At $R$, the received signal is $y_{S_1R}=\sqrt{P_1}\Phi(s_1)h_{S_1R}+n_{S_1R}$ and at $D_2$ $y_{S_1D_2}=\sqrt{P_1}\Phi(s_1)h_{S_1D_2}+n_{S_1D_2}$. In time slot 2, $S_2$ sends $s_2$ through both channels ($S_2 \rightarrow R$) and ($S_2 \rightarrow D_1$) and the received signals are, respectively, $y_{S_2R}=\sqrt{P_2}\Phi(s_2)h_{S_2R}+n_{S_2R}$ and $y_{S_2D_1}=\sqrt{P_2}\Phi(s_2)h_{S_2D_1}+n_{S_2D_1}$. The relay demodulates the received symbols using $\Phi^{-1}$ then encodes them using $\psi$, remodulates the result with $\Phi$ and broadcasts the coded symbol to both $D_1$ and $D_2$. Each destination node uses its received symbols to recover its intended message. At $D_1$ for instance, $s_1$ is recovered by $$\label{}
\tilde{s}_1=\psi^{-1}(\Phi^{-1}(\xi(y_{RD_1})), \Phi^{-1}(\xi(y_{S_2D_1})))$$ where $$\begin{aligned}
y_{RD_1}& =\sqrt{P_R}\Phi(\psi(\Phi^{-1}(\xi(y_{S_1R})),\Phi^{-1}(\xi(y_{S_2R}))))\nonumber\\
& .h_{RD_1}+n_{RD_1}\end{aligned}$$ and $y_{S_2D_1}=\sqrt{P_2}\Phi(s_2)h_{S_2D_1}+n_{S_2D_1}$.
From now on, we use only $\psi$ for both encoding and decoding operations since we have $\psi^{-1} = \psi$ (the attractive property of XOR).
When we add RS coding/decoding to the scenario above and using scheme 1, (6) becomes $$\label{}
\tilde{s}_1=\Gamma^{-1}(\psi(\Phi^{-1}(\xi(y_{RD_1})), \Phi^{-1}(\xi(y_{S_2D_1}))))$$ where $$\label{}
y_{S_2D_1}=\sqrt{P_2}\Phi(\Gamma(s_2))h_{S_2D_1}+n_{S_2D_1}$$ and $$\begin{aligned}
y_{RD_1} & = \sqrt{P_R}\Phi(\psi(\Phi^{-1}(\xi(y_{S_1R})),\Phi^{-1}(\xi(y_{S_2R}))))\nonumber\\
& .h_{RD_1}+n_{RD_1}.\end{aligned}$$
Now, using scheme 2 instead of scheme 1 for the combination of NCC and RS codes, (8) becomes $$\label{}
\tilde{s}_1=\psi(\Gamma^{-1}(\Phi^{-1}(\xi(y_{RD_1}))),\Gamma^{-1}(\Phi^{-1}(\xi(y_{S_2D_1}))))$$ where $y_{S_2D_1}$ is described in (9) and $$\begin{aligned}
y_{RD_1} & =\sqrt{P_R}\Phi(\Gamma(\psi(\Gamma^{-1}(\Phi^{-1}(\xi(y_{S_1R}))),\nonumber\\
& \Gamma^{-1}(\Phi^{-1}(\xi(y_{S_2R}))))))h_{RD_1}+n_{RD_1} \end{aligned}$$ In the next subsection, we propose an extended version of the *X*-structure dealing with four source-destination pairs instead of two.
Extended $X-$structure network
------------------------------
Fig. 1b shows the proposed extended wireless $X-$network where 4 $(S-D)$ pairs are communicating via one relay. Here, each source $S_i$ broadcasts in the MAC phase the message $s_i$ to the relay $R$ and all destinations $D_j$ ($j\neq i$), $i,j\in \{1,2,3,4\}$. In Fig. 2, we have $i=1$ and the relations obtained in the previous section are extended as follows: Without NCC, the same relations hold for both scenarios (with and without RS channel coding). When NCC is applied without RS code, 4 time slots are needed to broadcast the 4 signals $s_i$ to the relay and the destinations $D_j$ ($j\neq i$). The received signals at $R$ have the same form $$y_{S_iR}=\sqrt{P_i}\Phi(s_i)h_{S_iR}+n_{S_iR}, (i=1,\cdots,4)$$ The received signal at $D_1$ is the extended version of (6), $i.e.$ $$\begin{aligned}
\tilde{s}_1 & =\psi(\Phi^{-1}(\xi(y_{RD_1})), \Phi^{-1}(\xi(y_{S_2D_1})),\nonumber\\
& \Phi^{-1}(\xi(y_{S_3D_1})),\Phi^{-1}(\xi(y_{S_4D_1})))\end{aligned}$$ where $$\begin{aligned}
y_{RD_1}& =\sqrt{P_R}\Phi(\psi(\Phi^{-1}(\xi(y_{S_1R})),\Phi^{-1}(\xi(y_{S_2R})),\nonumber\\
& \Phi^{-1}(\xi(y_{S_3R})),\Phi^{-1}(\xi(y_{S_4R}))))h_{RD_1}+n_{RD_1}\end{aligned}$$ that we can write as $$y_{RD_1} =\sqrt{P_R}\Phi(\bigoplus_{i=1}^{4}\Phi^{-1}(\xi(y_{S_iR})))h_{RD_1}+n_{RD_1}.$$ Likewise, (14) can be expressed as $$\tilde{s}_1 =\Phi^{-1}(\xi(y_{RD_1}))\oplus\bigoplus_{i=2}^{4}\Phi^{-1}(\xi(y_{S_iD_1})).$$ For all scenarios, the results are the same but with 4 $(S-D)$ pairs instead of 2. For instance, with NCC and RS codes in scheme 2, (11) becomes $$\begin{aligned}
\tilde{s}_1&=\psi(\Gamma^{-1}(\Phi^{-1}(\xi(y_{RD_1}))),\Gamma^{-1}(\Phi^{-1}(\xi(y_{S_2D_1}))),\nonumber\\
& \Gamma^{-1}(\Phi^{-1}(\xi(y_{S_3D_1}))),\Gamma^{-1}(\Phi^{-1}(\xi(y_{S_4D_1}))))\end{aligned}$$
Butterfly network
-----------------
In the butterfly wireless network of Fig. 1c, there are two source-destination ($S-D$) pairs communicating via 2 relays in series. The first relay $R_1$ is the one in charge of network coding and channel coding/decoding while the second $R_2$ simply forwards (AF protocol) the received signal from $R_1$. Then, destinations $D_1$ and $D_2$ receive the broadcast message from $R_2$. In the conventional scenario (without NCC), a transmission between both $(S-D)$ pairs requires 6 time slots while with NCC, we need only 4 which leads to a throughput gain of $3/2$.
In the direct path scenario with no NCC and no RS codes, the symbol received at $R_1$ is $y_{S_1R_1}=\sqrt{P_1}\Phi(s_1)h_{S_1R_1}+n_{S_1R_1}$. Relay $R_2$ receives $y_{R_1R_2}=\sqrt{P_{R_1}}\beta_{S_1R_1}\xi(y_{S_1R_1})h_{R_1R_2}+n_{R_1R_2}$ from $R_1$ where $$\beta_{S_1R_1}=\sqrt\frac{P_{R_1}}{P_1|h_{S_1R_1}|^2+\sigma_{S_1R_1}^2}$$ and finally, destination $D_1$ receives $y_{R_2D_1}=\sqrt{P_{R_2}}\beta_{R_1R_2}\xi(y_{R_1R_2})h_{R_2D_1}+n_{R_2D_1}$ from relay $R_2$ where $$\beta_{R_1R_2}=\sqrt\frac{P_{R_2}}{P_{R_1}|h_{R_1R_2}|^2+\sigma_{R_1R_2}^2}$$ and retrieves its intended symbol $\tilde{s}_1=\Phi^{-1}(\xi(y_{R_2D_1}))$.
Considering the application of channel coding at both sources and channel decoding at their respective destinations for the previous scenario, we have $y_{S_1R_1}=\sqrt{P_1}\Phi(\Gamma(s_1))h_{S_1R_1}+n_{S_1R_1}$ and $\tilde{s}_1=\Gamma^{-1}(\Phi^{-1}(\xi(y_{R_2D_1})))$. When NCC is performed without RS channel coding, the relay $R_2$ receives from $R_1$ the signal $$y_{R_1R_2}=\sqrt{P_{R_1}}\Phi(\bigoplus_{i=1}^{2}\Phi^{-1}(\xi(y_{S_iR_1})))h_{R_1R_2}+n_{R_1R_2}.$$ The retrieved symbol at $D_1$ is $$\tilde{s}_1 =\Phi^{-1}(\xi(y_{R_2D_1}))\oplus\Phi^{-1}(\xi(y_{S_2D_1})).$$ Combining RS codes with NCC respecting scheme 1 leads to the following relations: $y_{R_1R_2}$ is given by (19) with the difference that received symbols at $R_1$ are now RS coded and have the form $$y_{S_iR_1}=\sqrt{P_i}\Phi(\Gamma(s_i))h_{S_iR_1}+n_{S_iR_1}.$$ In this case, destination $D_1$ recovers the symbol $s_1$ with the operation $$\begin{aligned}
\tilde{s}_1&=\Gamma^{-1}(\psi(\Phi^{-1}(\xi(y_{R_2D_1})), \Phi^{-1}(\xi(y_{S_2D_1}))))\nonumber\\
&=\Gamma^{-1}(\Phi^{-1}(\xi(y_{R_2D_1}))\oplus\Phi^{-1}(\xi(y_{S_2D_1}))).\end{aligned}$$ Now, combining RS codes with NCC respecting scheme 2, we have $$\begin{aligned}
y_{R_1R_2} & =\sqrt{P_{R_1}}\Phi(\Gamma(\psi(\Gamma^{-1}(\Phi^{-1}(\xi(y_{S_1R_1}))),\nonumber\\
& \Gamma^{-1}(\Phi^{-1}(\xi(y_{S_2R_1}))))))h_{R_1R_2}+n_{R_1R_2}. \end{aligned}$$ After equalization of $y_{R_1R_2}$, the relay $R_2$ amplifies it using $\beta_{R_1R_2}$ and broadcasts it to $D_1$ and $D_2$. Hence, $D_1$ decodes the symbol $s_1$ as follows: $$\label{}
\tilde{s}_1=\psi(\Gamma^{-1}(\Phi^{-1}(\xi(y_{R_2D_1}))),\Gamma^{-1}(\Phi^{-1}(\xi(y_{S_2D_1})))).$$
Extended Butterfly network
--------------------------
In this subsection, we propose to extend the number of $(S-D)$ pairs to 4 as illustrated in Fig. 1d. The results found for the butterfly network hold for its extended version with the addition of sources $S_3$ and $S_4$ and destinations $D_3$ and $D_4$. In the scenario of scheme 1 for example, (21) becomes $$\begin{aligned}
\tilde{s}_1&=\Gamma^{-1}(\psi(\Phi^{-1}(\xi(y_{R_2D_1})), \Phi^{-1}(\xi(y_{S_2D_1})),\nonumber\\
&\Phi^{-1}(\xi(y_{S_3D_1})), \Phi^{-1}(\xi(y_{S_4D_1})))).\end{aligned}$$ that can be expressed as $$\tilde{s}_1=\Gamma^{-1}(\Phi^{-1}(\xi(y_{R_2D_1})) \oplus \bigoplus_{i=2}^{4}\Phi^{-1}(\xi(y_{S_iD_1}))).$$ Application of NCC on the extended butterfly network allows a throughput gain of $2$ with the diversity order 4. These values become $3N/(N+2)$ and $N$ for the general case of $N$ $(S-D)$ pairs.
In the next section, we get the results that confirm the efficiency of combining NCC with RS channel codes by conducting simulations of all the scenarios above.
Simulation Results
==================
The scenarios detailed in the wireless topologies presented in the previous section are simulated using MATLAB. We generate a set of 14 SNR values from $0 dB$ to $26 dB$ with a step of $2 dB$. For each $M-$PSK constellation, we generate (at each source node) a sequence of 1000 random integer values in the corresponding finite field $GF(M)$. Depending on the coding rate $k/n$ of our RS coder, these values are coded into a set of $\frac{1000n}{k}$ values in the same finite field. We assume equal transmit power at all nodes. All channels are independent and each one is defined by its complex AWGN components $n \sim CN(0, \sigma_n^2)$ and complex Rayleigh fading coefficients $h \sim CN(0, \sigma_h^2)$. Each simulation is the average of 1000 iterations to ensure accurate reliable results.
To derive the values of SER, we compare the received sequence of symbols $\tilde{s_1}$ evaluated using the corresponding scenario equation to the original one $s_1$ generated at source node $S_1$.
![SER vs. SNR obtained for $X$-structure with 16-PSK and RS(15,5).[]{data-label="fig"}](Figure2.png)
Due to the large number of scenarios in our work, we gathered the main relevant results in one SER versus SNR graph for each topology in addition to one graph ($X-$structure with $16-$PSK modulation and RS(15,5)) to illustrate the advantage of NCC application with and without Reed-Solomon channel coding. In Fig. 2, we can notice how the application of cooperative network coding, at the cost of a small degradation of SER values, produces an important improvement of the overall system throughput and diversity order [@b10] in addition to transmit power reduction. Table I shows the values of throughput gain and diversity orders attained by exploiting NCC technique on the general cases of the networks studied in this paper. Also, RS channel coding (we used scheme 2) allows considerable reductions of symbol error probability. At $SNR=14.35dB$, the SER with RS code is equal to $10^{-3}$ while its value is $10^{-0.9}$ without RS code. For all the four wireless network configurations, the graphs on Figs. 3 and 4 show clearly how scheme 2 outperforms scheme 1 in terms of symbol error probability for the three PSK constellations adopted. For instance, in the extended butterfly network using $32-$PSK and RS(31,10), the SER decreases from $10^{-1}$ in scheme 1 to $10^{-3}$ in scheme 2 at $SNR=16.65dB$. Furthermore, with a small extra computation at the relay in scheme 2 to perform $RS(n,k)$ codes of the received messages and re-encoding of the resulting network coded message, we can save a large amount of system transmit energy [@b11; @b12; @b13], since, in scheme 1, the number of NC symbols to transmit is $\frac{n}{k}$ times greater than the number in scheme 2.
\[\]
----------------- --------------- ------------------ --------------- ------------------
****
**X-net.** **Extended-X** **Butter.** **Ext. Butter.**
Throughput Gain $\frac{4}{3}$ $\frac{2N}{N+1}$ $\frac{3}{2}$ $\frac{3N}{N+2}$
Diversity Order $2$ $N$ $2$ $N$
----------------- --------------- ------------------ --------------- ------------------
: Throughput gain and Diversity Order.
![Results of scheme 1 and scheme 2 on X-structure and extended X-structure networks with RS(7,2), RS(15,5) and RS(31,10) for 8-PSK, 16-PSK and 32-PSK respectively.[]{data-label="fig"}](Figure3.png)
![Results of scheme 1 and scheme 2 on butterfly and extended butterfly networks with RS(7,2), RS(15,5) and RS(31,10) for 8-PSK, 16-PSK and 32-PSK respectively.[]{data-label="fig"}](Figure4.png)
Conclusion
==========
In this paper, we put the light on the advantage of using Reed-Solomon codes jointly with cooperative network coding in wireless network architectures for non-binary PSK constellations. Besides the optimizations achieved in terms of system throughput, diversity order and network transmit power by NCC strategy, we showed how RS channel codes can improve considerably the symbol error rate. RS codes give high performance with non-binary constellations since they deal with Galois fields $GF(M)$ elements. We also proved that scheme 2, where the relay operates RS decoding before network coding and then RS re-encoding, decreases the symbol error probability in comparison with scheme 1 where the relay executes network coding directly on channel coded symbols. As a future work, we will investigate the application of random linear network coding (RLNC) at the NC relay.
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address:
- 'Department of Mathematics, University of Haifa, 3498838 Haifa, Israel'
- 'Department of Mathematics, Bilkent University, 06800 Ankara, Turkey'
author:
- Toufik Mansour
- gökhan Yildirim
title: Enumerations of bargraphs with respect to corner statistics
---
abstract {#abstract .unnumbered}
========
We study the enumeration of bargraphs with respect to some corner statistics. We find generating functions for the number of bargraphs that tracks the corner statistics of interest, the number of cells, and the number of columns. The bargraph representation of set partitions is also considered and some explicit formulas are obtained for the number of some specific types of corners in such representations.
Introduction
============
Combinatorial analysis of certain geometric cluster models such as polygons, polycubes, polyominos is an important research endeavor for understanding many statistical physics models [@G; @Fer; @JR]. A finite connected union of unit squares on two dimensional integer lattice is called a *polyomino*, and a *bargraph* is a column-convex polyomino in the first quadrant of the lattice such that its lower boundary lies on the $x$-axis. A *bargraph* can also be considered as a self-avoiding path in the integer lattice $\mathbb{L}=\mathbb{Z}_{\geq 0}\times \mathbb{Z}_{\geq 0} $ with steps $u=(0,1)$, $h=(1,0)$ and $d=(0,-1)$ that starts at the origin, ends on the $x$-axis and never touches the $x$-axis except at the endpoints. The steps $u,h$ and $d$ are called *up, horizontal* and *down* steps respectively. Emumerations of bargraphs with respect to some statistics have been an active area of research recently [@PB; @Fer; @Man]. Bosquet-Melóu and Rechnitzer [@BMR] obtained the site-perimeter generating function for bargraphs, and also showed that the width and site-perimeter generating function for bargraphs is not D-finite. Blecher et al. investigated the generating functions for bargraphs with respect to some statistics such as the number of levels [@Bl1], descents [@Bl2], peaks [@Bl3], and walls [@Bl4]. Deutsch and Elizalde [@DE] used a bijection between bargraphs and cornerless Motzkin paths, and determined more than twenty generating functions for bargraphs according to the number of up steps, the number of horizontal steps, and the statistics of interest such as the number of double rises and double falls, the length of the first descent, the least column height. Bargraphs are also used in statistical physics to model vesicles or polymers [@SP; @AP; @PB].
We shall study the enumerations of bargraphs and set partitions with respect to some corner statistics. Let’s first introduce some definitions. A unit square in the lattice $\mathbb{L}$ is called a *cell*. A bargraph is usually identified with a sequence of numbers $\pi=\pi_1\pi_2\cdots \pi_m$ where $m$ is the number of horizontal steps of the bargraph and $\pi_j$ is the number of *cells* beneath the $j^{th}$ horizontal step which is also called the *height* of the $j^{th}$ *column*. A vertex on a bargraph is called a *corner* if it is at the intersection of two different types of steps. A corner is called an *$(a,b)$-corner* if it is formed by maximum $a$ number of one type of consecutive steps followed by maximum $b$ number of another type of consecutive steps. A corner is called of *type A* if it is formed by down steps followed by horizontal steps $(\llcorner)$. Similarly, a corner is of *type B* if it is formed by horizontal steps followed by down steps $(\urcorner)$, see Figure \[barFig\]. We shall use $\BB_n$ and $\BB_{n,k}$ to denote the set of all bargraphs with $n$ cells, and the set of all bargraphs with $n$ cells and $k$ columns respectively.
Bargraphs are also related to the set partitions. Recall that a of set $[n]:=\{1,2,\cdots,n\}$ is any collection of nonempty, pairwise disjoint subsets whose union is $[n]$. Each subset in a partition is called a *block* of the partition. A partition $p$ of $[n]$ with $k$ blocks is said to be in the *standard form* if it is written as $p=A_1/A_2/\cdots/A_k$ where $\min(A_1)<\min(A_2)<\cdots<\min(A_k)$. There is also a unique *canonical sequential representation* of a partition $p$ as a word of length $n$ over the alphabet $[k]$ denoted by $\pi=\pi_1\pi_2\cdots \pi_n$ where $\pi_i=j$ if $i\in A_{\pi_j}$ which can be considered a bargraph representation. For instance, the partition $\pi=\{1,3,6\}/\{2,5\}/\{4,7\}/\{8\}$ has the canonical sequential representation $\pi=12132134$. Mansour [@Man] studied the generating functions for the number of set partitions of $[n]$ represented as bargraphs according to the number of interior vertices. For some other enumeration results, see also [@Bl5; @MShaSha]. Henceforth, we shall represent set partitions as bargraphs corresponding to their canonical sequential representations.
(30,20) (0,0)[(0,1)[6.8]{}]{} (0,6.8)[(1,0)[3.4]{}]{} (3.4,6.8)[(0,1)[6.8]{}]{} (3.4,13.6)[(1,0)[10.2]{}]{} (13.6, 13.6)[(0,-1)[10.2]{}]{} (13.6,3.4)[(1,0)[6.8]{}]{} (20.4,3.4)[(0,1)[6.8]{}]{} (20.4,10.2)[(1,0)[3.4]{}]{} (23.8,10.2)[(0,-1)[3.4]{}]{} (23.8,6.8)[(1,0)[6.8]{}]{} (30.6,6.8)[(0,-1)[6.8]{}]{} (13.8,13.9)[$ a$]{} (12.6,2.0)[$b$]{} (23.9,10.5)[$ c$]{} (22.8,4.8)[$d$]{} (0,0)(3.4,0)[11]{} [(0,1)[17]{}]{} (0,0)(0,3.4)[6]{} [(1,0)[34]{}]{}
The rest of the paper is organized as follows. In section \[secA\], we find the generating function for the number of bargraphs according to the number of cells, the number of columns, and the number of $(a,b)$-corners of type A for any given positive integers $a,b$. As a corollary, we determine the total number of $(a,b)$-corners of type A, and the total number of type A corners over all bargraphs having $n$ cells. In section \[secRB\] and section \[secSPA\], we extend these results to the restricted bargraphs in which the height of each column is restricted to be atmost $N$ for any given positive integer $N$, and to the set partitions respectively. We obtain similar results for corners of type B in section \[secB\]. One of the main results of the paper, Theorem \[thmcAsp\], shows that the total number of corners of type A over the set partitions of $[n+1]$ with $k$ blocks is given by $$\frac{n}{2}S_{n+1,k}-\frac{1}{4}S_{n+2,k}-\frac{n}{2}S_{n,k}+\frac{1}{4}S_{n+1,k}+S_{n,k-2},$$ where $S_{n,k}$ is the Stirling number of second kind. Similarly, Theorem \[thmcBsp\], shows that the total number of corners of type B over the set partitions of $[n+1]$ with $k$ blocks is given by $$\frac{n}{2}S_{n+1,k}-\frac{1}{4}S_{n+2,k}-\frac{n}{2}S_{n,k}+\frac{5}{4}S_{n+1,k}+S_{n,k-2}.$$
Counting Corners of type A {#secA}
==========================
Let $H:=H(x,y,\qq)$ be the generating function for the number of bargraphs $\pi$ according to the number of cells in $\pi$, the number of columns of $\pi$, and the number of $(a,b)$-corners of type A in $\pi$ corresponding to the variables $x,y$ and $\qq=(q_{a,b})_{a,b\geq 1}$ respectively. That is, $$H=\sum_{n\geq 0}\sum_{\pi\in \BB_n}x^ny^{\text{col}(\pi)}\prod_{a,b\geq
1}q_{a,b}^{\Lambda_{(a,b)}(\pi)}$$ where $\Lambda_{(a,b)}(\pi)$ is the number of $(a,b)$-corners of type A in $\pi$, and $\text{col}(\pi)$ denotes the number of columns of $\pi$.
From the definitions, we have $$\label{eqcA1}
H=1+\sum_{a\geq 1}H_a,$$ where $1$ counts the empty bargraph, and $H_a:=H_a(x,y,\qq)$ is the generating function for the number of bargraphs $\pi=a\pi'$ in which the hight of the first column is $a$. Since each bargraph $\pi=a\pi'$ can be decomposed as either $a$, $aj\pi''$ with $j\geq a$ or $ab\pi''$ with $1\leq b\leq a-1$, we have $$\label{eqcA2}
H_a=x^ay+x^ay\sum_{j\geq a}H_j+\sum_{b=1}^{a-1}H_{ab}$$ Note that each bargraph $\pi=ab\pi''$, $1\leq b\leq a-1$, can be written as either $ab^m$ (where we define $b^m$ to be the word $bb\cdots b$), $ab^mj\pi'$ with $j\geq b+1$, or $ab^mj\pi'$ with $j\leq b-1$. Thus, for all $1\leq b\leq a-1$, we have $$\begin{aligned}
H_{ab}&=\sum_{m\geq 1}x^{a+bm}y^{m+1}q_{a-b,m}+\sum_{m\geq 1}\left(x^{a+bm}y^{m+1}q_{a-b,m}\sum_{j\geq b+1}H_{j}\right)\\
&+\sum_{m\geq1}\left(x^{a+b(m-1)}y^mq_{a-b,m}\sum_{c=1}^{b-1}H_{bc}\right),\end{aligned}$$ which is equivalent to $$\begin{aligned}
\label{eqcA3}
H_{ab}&=\sum_{m\geq 1}x^{a+bm}y^{m+1}q_{a-b,m}\\
&+\sum_{m\geq1}\left(x^{a+b(m-1)}y^{m}q_{a-b,m}\left(x^by\sum_{j\geq
b+1}H_{j}+\sum_{c=1}^{b-1}H_{bc}\right)\right).\notag\end{aligned}$$ Thus, by , we have that $H_a-x^ay-x^ayH_a=x^ay\sum_{j\geq a+1}H_j +
\sum_{b=1}^{a-1}H_{ab}$, which, by , leads to $$H_{ab}=\alpha_{ab}(1-x^by)H_b\mbox{ with }\alpha_{ab}=\sum_{m\geq
1} x^{a+b(m-1)}y^{m}q_{a-b,m}.$$ Therefore, by we can write $$\label{eqcA4}
H_a=x^ayH+\sum_{b=1}^{a-1}\beta_{ab}H_b$$ with $\beta_{ab}=\alpha_{ab}(1-x^by)-x^ay$.
\[lemcA1\] For all $a\geq1$, $$\begin{aligned}
H_a=H\left(x^ay+\sum_{j=1}^a\left(x^jy\sum_{s\geq0}L_a(j,s)\right)\right),\end{aligned}$$ where $L_a(j,s)=\sum_{j=i_{s+1}<i_s<\cdots<i_1<i_0=a}\prod_{\ell=0}^s\beta_{i_\ell
i_{\ell+1}}$.
The proof is given by induction on $a$. For $a=1$, this gives $H_1=xyH$ as expected (by removing the leftmost column of the bargraph $1\pi'$). Assume that the claim holds for $1,2,\cdots,a$, and let us prove it for $a+1$. By , we have $$H_{a+1}=x^{a+1}yH+\sum_{b=1}^{a}\beta_{(a+1)b}H_b.$$ Thus, by induction assumption, we obtain $$\begin{aligned}
H_{a+1}&=x^{a+1}yH+\sum_{b=1}^{a}\beta_{(a+1)b}H\left(x^by+\sum_{j=1}^bx^jy\left(\sum_{s\geq0}L_b(j,s)\right)\right)\\
&=H\left(x^{a+1}y+\sum_{b=1}^{a}\beta_{(a+1)b}x^by+\sum_{b=1}^{a}\sum_{j=1}^bx^jy\beta_{(a+1)b}\left(\sum_{s\geq0}L_b(j,s)\right)\right)\\
&=H\left(x^{a+1}y+\sum_{b=1}^{a}\beta_{(a+1)b}x^by\right)\\
&\qquad+H\left(\sum_{j=1}^{a}\sum_{b=j}^ax^jy\left(\sum_{s\geq0}\sum_{j=i_{s+1}<i_s<\cdots<i_1<i_0=b<i_{-1}=a+1}\prod_{\ell=-1}^s\beta_{i_\ell i_{\ell+1}}\right)\right)\\
&=H\left(x^{a+1}y+\sum_{b=1}^{a}x^by\sum_{i_1=b<i_0=a+1}\beta_{i_0i_1}\right)\\
&\qquad+H\sum_{j=1}^ax^jy\left(\sum_{s\geq0}\sum_{j=i_{s+1}<i_s<\cdots<i_1<i_0<i_{-1}=a+1}\prod_{\ell=-1}^s\beta_{i_\ell i_{\ell+1}}\right)\\
&=H\left(x^{a+1}y+\sum_{j=1}^{a}x^jy\left(\sum_{s\geq0}L_{a+1}(j,s)\right)\right),\end{aligned}$$ which completes the proof.
By and Lemma \[lemcA1\], we can state our first main result.
\[mth1\] The generating function $H(x,y,\qq)$ is given by $$H(x,y,\qq)=\frac{1}{1-\frac{xy}{1-x}-\sum_{j\geq 1}\left(x^jy
\sum_{s\geq0}L(j,s)\right)}$$ where $L(j,s)=\sum_{j=i_{s+1}<i_s<\cdots<i_1<i_0}\prod_{\ell=0}^s\beta_{i_\ell
i_{\ell+1}}$.
For instance, if $q_{a,b}=1$ for all $a,b\geq 1$, then $\alpha_{ab}=\sum_{m\geq 1}x^{a+b(m-1)}y^m=\frac{x^ay}{1-x^by}$, which yields $\beta_{ab}=\alpha_{ab}(1-x^by)-x^ay=0$. Thus, in this case, Theorem \[mth1\] shows that $H(x,y,1,1,\ldots)=\frac{1-x}{1-x-xy}$, as expected.
Counting all corners of type A
------------------------------
Let $q_{a,b}=q$ for all $a,b\geq 1$. By definitions, we have $\alpha_{ab}=q\frac{x^ay}{1-x^by}$ and $\beta_{ab}=(q-1)x^ay$. Therefore, $$\begin{aligned}
L(j,s)&=\sum_{j=i_{s+1}<i_s<\cdots<i_1<i_0}\prod_{\ell=0}^s(q-1)x^{i_{\ell}y}\\
&=\sum_{j=i_{s+1}<i_s<\cdots<i_1<i_0}(q-1)^{s+1}y^{s+1}x^{i_0+i_1+\cdots+i_s}\\
&=(q-1)^{s+1}y^{s+1}\frac{x^{(s+1)j+\binom{s+2}{2}}}{(1-x)(1-x^2)\cdots(1-x^{s+1})}.\end{aligned}$$ Thus the generating function $F=H(x,y,q,q,\cdots)$ is given by $$\begin{aligned}
F&=\frac{1}{1-\frac{xy}{1-x}-\displaystyle \sum_{j\geq 1}x^jy\sum_{s\geq 0}\frac{(q-1)^{s+1}y^{s+1}x^{(s+1)j+\binom{s+2}{2}}}{(1-x)(1-x^2)\cdots(1-x^{s+1})}}\\
&=\frac{1}{1-\frac{xy}{1-x}-\displaystyle \sum_{s\geq 0}\frac{(q-1)^{s+1}y^{s+2}x^{(s+2)+\binom{s+2}{2}}}{(1-x)(1-x^2)\cdots(1-x^{s+2})}}\\
&=\frac{1}{1-\frac{xy}{1-x}-\displaystyle \sum_{s\geq 1}\frac{(q-1)^sy^{s+1}x^{\binom{s+2}{2}}}{(1-x)(1-x^2)\cdots(1-x^{s+1})}}.\end{aligned}$$
Let $\cora(\pi)$ be the number of corners of type A in $\pi$. We define $g_{n,k}=\sum_{\pi \in \BB_{n,k}}\cora(\pi)$ and $g_n=\sum_{k\geq 1}g_{n,k}$. Let $G(x,y)=\sum_{n,k\geq 1}g_{n,k}x^ny^k$ be the generating function for the total number of type A corners over all bargraphs according to the number of cells and columns. Then, it follows that $$\begin{aligned}
G(x,y)=\frac{\partial F}{\partial
q}\Bigr|_{\substack{q=1}}=\frac{y^2x^3}{(1-x-xy)^2(1+x)}.\end{aligned}$$ Note that $G(x,1)=\frac{x^3}{(1-2x)(1+x)}$ is the generating function for the total number of type A corners over all bargraphs according to the number of cells. Hence, $$g_n=\left(\frac{n+1}{12}-\frac{2}{9}\right)2^n-\frac{1}{9}(-1)^n.$$
Counting $(v,w)$-corners of type A
----------------------------------
Fix $v,w\geq1$. Define $q_{v,w}=q$ and $q_{a,b}=1$ for all $(a,b)\neq(v,w)$. Then we have $$\begin{aligned}
\alpha_{ab}&=\sum_{m\geq1} x^{a+b(m-1)}y^{m}q_{a-b,m}=\sum_{m\geq1} x^{a+b(m-1)}y^{m}+x^{a+b(w-1)}y^w(q_{a-b,w}-1)\\
&=\frac{x^ay}{1-x^by}+x^{a+b(w-1)}y^w(q-1)\delta_{a-b=v}\end{aligned}$$ where $\delta_{\chi}=1$ if $\chi$ holds, and $\delta_{\chi}=0$ otherwise. And hence $$\begin{aligned}
\label{eqcA5}
\beta_{ab}&=\alpha_{ab}(1-x^by)-x^ay
=x^{a+b(w-1)}y^w(q-1)\delta_{a-b=v}(1-x^by).\end{aligned}$$ Recall that $L(j,s)=\sum_{j=i_{s+1}<i_s<\cdots<i_1<i_0}\prod_{\ell=0}^s\beta_{i_\ell
i_{\ell+1}}$. By using , we have $$\begin{aligned}
L(j,s)&=\sum_{j=i_{s+1}<i_s<\cdots<i_1<i_0}\prod_{\ell=0}^s\left(x^{i_{\ell}+i_{\ell+1}(w-1)}y^w(q-1)\delta_{i_{\ell}-i_{\ell+1}=v}(1-x^{i_{\ell+1}}y)\right)\\
&=\sum_{j=i_{s+1}<i_s<\cdots<i_1<i_0}(q-1)^{s+1}y^{w(s+1)}x^{\sum_{\ell=0}^si_{\ell}+(w-1)i_{\ell+1}}\prod_{\ell=0}^s\left(\delta_{i_{\ell}-i_{\ell+1}=v}(1-x^{i_{\ell+1}}y)\right)\\
&=(q-1)^{s+1}y^{w(s+1)}x^{wj(s+1)+v\binom{s+2}{2}+(w-1)v\binom{s+1}{2}}\prod_{\ell=0}^s(1-x^{j+(s-l)v}y).\end{aligned}$$ From Theorem \[mth1\], we obtain that the generating function $F=H(x,y,\qq)$ is given by $$F=\frac{1}{1-\frac{xy}{1-x}-\displaystyle \sum_{j\geq
1}x^jy\sum_{s\geq
0}(q-1)^{s+1}y^{w(s+1)}x^{wj(s+1)+v\binom{s+2}{2}+(w-1)v\binom{s+1}{2}}\prod_{\ell=0}^s(1-x^{j+\ell
v}y)}.$$ Recall that $\Lambda_{(v,w)}(\pi)$ denotes the number of $(v,w)$-corners of type A in $\pi$. We define $t_{n,k}=\sum_{\pi \in \BB_{n,k}}\Lambda_{(v,w)}(\pi)$ and $t_n=\sum_{k\geq 1}t_{n,k}$. Let $T(x,y)=\sum_{n,k\geq 1}t_{n,k}x^ny^k$ be the generating function for the total number of $(v,w)$-corners of type A over all bargraphs according to the number of cells and columns. Then, it follows that $$\begin{aligned}
T(x,y)=\frac{\partial F}{\partial
q}\Bigr|_{\substack{q=1}}=\frac{x^{v+w+1}y^{w+1}}{(1-\frac{xy}{1-x})^2}\frac{1-xy-x^{w+2}(1-y)}{(1-x^{w+1})(1-x^{w+2})},\end{aligned}$$ which leads to $T(x,1)=\frac{x^{v+w+1}}{(1-2x)^2}\frac{(1-x)^3}{(1-x^{w+1})(1-x^{w+2})}$; the generating function for the total number of $(v,w)$-corners of type A over all bargraphs according to the number of cells. As consequence, we have the following result.
The total number of $(v,w)$-corners of type A over all bargraphs having $n$ cells is given by $$t_n=\frac{n}{(2^{w+1}-1)(2^{w+2}-1)}2^{w-v+n-1}.$$
Restricted bargraphs {#secRB}
--------------------
Theorem \[mth1\] can be refined as follows. Fix $N\geq1$. Let $H^{(N)}:=H^{(N)}(x,y,\qq)$ be the generating function for the number of bargraphs $\pi$ such that the height of each column is at most $N$ according to the number of cells in $\pi$, the number of columns of $\pi$, and the number of $(a,b)$-corners of type A in $\pi$ corresponding to the variables $x,y$ and $\qq=(q_{a,b})_{a,b\geq 1}$ respectively. Then by using similar arguments as in the proof of , we obtain $$\label{eqcNA1}
H_a^{(N)}=x^ayH^{(N)}+\sum_{b=1}^{a-1}\beta_{ab}H_b^{(N)},$$ where $H^{(N)}_a:=H^{(N)}_a(x,y,\qq)$ is the generating function for the number of bargraphs $\pi=a\pi'$ such that the height of each column is at most $N$. Clearly, $H^{(N)}=1+\sum_{a=1}^NH_a^{(N)}$. By the proof of Theorem \[mth1\], we can state its extension as follows.
\[mthN1\] The generating function $H^{(N)}(x,y,\qq)$ is given by $$H^{(N)}(x,y,\qq)=\frac{1}{1-y\sum_{j=1}^Nx^j-\sum_{j=1}^N\left(x^jy
\sum_{s\geq0}L(j,s)\right)}$$ where $L(j,s)=\sum_{j=i_{s+1}<i_s<\cdots<i_1<i_0\leq
N}\prod_{\ell=0}^s\beta_{i_\ell i_{\ell+1}}$. Moreover, for all $a=1,2,\ldots,N$, we have $$\begin{aligned}
H_a^{(N)}=H^{(N)}\left(x^ay+\sum_{j=1}^a\left(x^jy\sum_{s\geq0}L_a(j,s)\right)\right),\end{aligned}$$ where $L_a(j,s)=\sum_{j=i_{s+1}<i_s<\cdots<i_1<i_0=a}\prod_{\ell=0}^s\beta_{i_\ell
i_{\ell+1}}$.
For instance, Theorem \[mthN1\] for $N=1,2$ gives $H^{(1)}(x,y,\qq)=\frac{xy}{1-xy}$ and $$H^{(2)}(x,y,\qq)=\frac{1}{1-(x+x^2)y-x^2(1-xy)\sum_{m\geq1}x^my^mq_{1,m}+x^3y^2}.$$
Counting corners of type A in set partitions {#secSPA}
--------------------------------------------
Recall that we represent any set partition as a bargraph corresponding to its canonical sequential representation. Let $P_k(x,y,\qq)$ be the generating function for the number of set partitions $\pi$ of $[n]$ with exactly $k$ blocks according to the number of cells in $\pi$, the number of columns of $\pi$ (which is $n$), and the number of $(a,b)$-corners of type A in $\pi$ corresponding to the variables $x,y$ and $\qq=(q_{a,b})_{a,b\geq 1}$ respectively.
Note that each set partition with exactly $k$ blocks can be decomposed as $1\pi^{(1)}\cdots k\pi^{(k)}$ such that $\pi^{(j)}$ is a word over alphabet $[j]$. Thus, by Theorem \[mthN1\], we have the following result.
\[mthPk1\] The generating function $P_k(x,y,\qq)$ is given by $$\begin{aligned}
P_k(x,y,\qq)=\prod_{N=1}^kH^{(N)}_N(x,y,\qq)=\prod_{N=1}^k\frac{x^Ny+\sum_{j=1}^N\left(x^jy\sum_{s\geq0}L_N(j,s)\right)}
{1-y\sum_{j=1}^Nx^j-\sum_{j=1}^N\left(x^jy\sum_{s\geq0}L(j,s)\right)}\end{aligned}$$ where $\displaystyle L(j,s)=\sum_{j=i_{s+1}<i_s<\cdots<i_0\leq
N} \prod_{\ell=0}^s\beta_{i_\ell i_{\ell+1}}$ and $\displaystyle L_N(j,s)=\sum_{j=i_{s+1}<i_s<\cdots<i_0=N} \prod_{\ell=0}^s\beta_{i_\ell i_{\ell+1}}$.
Now, we consider counting all corners of type A in set partitions. Let $q_{a,b}=q$ for all $a,b\geq 1$, and $Q_k(x,y)=\frac{\partial}{\partial q}P_k(x,y,\qq)\Bigr|_{\substack{q=1}}$. Note that for any $s\geq 0$ and $1\leq j\leq N-1$, $$L_N(j,s)=(q-1)^{s+1}y^{s+1}\sum_{j=i_{s+1}<i_s<\cdots<i_0=N}x^{\sum_{\ell=0}^sx_{\ell}}.$$ We have similar expression for $L(j,s)$. By Theorem \[mthPk1\], we have the generating function $Q_k(x,y)$ given by $$\begin{aligned}
Q_k(x,y)=\prod_{N=1}^k\frac{x^Ny}{1-y\sum_{j=1}^Nx^j}
\sum_{N=1}^k\frac{\sum_{j=1}^{N-1}\left(x^jy(1-y\sum_{j=1}^Nx^j)+x^jy^2(x^{j+1}+\cdots+x^N)\right)} {1-y\sum_{j=1}^Nx^j}.\end{aligned}$$ Let $\phi(t)=\frac{t^k}{(1-t)(1-2t)\cdots(1-kt)}$. Then we have $\phi^{\prime}(t)=\frac{t^{k-1}}{(1-t)\cdots(1-kt)}\sum_{j=1}^k\frac{1}{1-jt}$.
Note that $$\begin{aligned}
Q_k(1,t)&=\phi(t)\sum_{N=2}^k\left((N-1)t+\frac{t^2\sum_{j=1}^{N-1}(N-j)}{1-Nt}\right)\\
&=\phi(t) t\sum_{N=2}^k\left((N-1)+\frac{t\binom{N}{2}}{1-Nt}\right)\\
&=\phi(t)t\left(\binom{k}{2}+\frac{1}{2}\sum_{N=2}^k\frac{tN(N-1)}{1-Nt}\right)\\
&=\phi(t)t\left(\frac{1}{2}\binom{k}{2}+\frac{1}{2}\sum_{N=1}^k\frac{N-1}{1-Nt}\right)\\
&=\frac{1}{2}\binom{k}{2}\phi(t)t+\frac{1}{2}\phi(t)\sum_{N=1}^k\left(-1-t\frac{1}{1-Nt}+\frac{1}{1-Nt}\right)\\
&=\frac{1}{2}\binom{k}{2}t\phi(t)-\frac{1}{2}k\phi(t)+\frac{1}{2}t\phi^{\prime}(t)-\frac{1}{2}t^2\phi^{\prime}(t).\end{aligned}$$ Let $q_{n,k}$ be the coefficient of $t^n$ in $Q_k(1,t)$. Define $\tilde{Q}_k(t)=\sum_{n\geq k}q_{n,k}\frac{t^n}{n!}$ to be the exponential generating function for $q_{n,k}$. Recall that the ordinary and exponential generating functions for Stirling numbers of the second kind $S_{n,k}$ is given by $\phi(t)$ and $\frac{(e^t-1)^k}{k!}$, respectively.
Thus, $$\begin{aligned}
\tilde{Q}_k(t)&=\frac{1}{2}\binom{k}{2}\int_0^t\frac{(e^r-1)^k}{k!}dr
-\frac{k(e^t-1)^k}{2k!}+\frac{kt(e^t-1)^{k-1}e^t}{2k!}
-\int_0^t\frac{rk(e^r-1)^{k-1}e^r}{2k!}dr.\end{aligned}$$ Hence, the exponential generating function $\tilde{Q}(t,y)=\sum_{k\geq0}\tilde{Q}_k(t)y^k$ for the total number of corners over set partitions of $[n]$ with $k$ blocks is given by $$\begin{aligned}
\tilde{Q}(t,y)&=\frac{y^2}{4}\int_0^t(e^r-1)^2e^{y(e^r-1)}dr+\frac{yt}{2}e^{t+y(e^t-1)}
-\frac{y}{2}(e^t-1)e^{y(e^t-1)}-\frac{y}{2}\int_0^tre^{r+y(e^r-1)}dr.\end{aligned}$$ In particular, we have $$\begin{aligned}
\frac{\partial}{\partial t}\tilde{Q}(t,y)&=\frac{y^2}{4}(2te^{2t+y(e^t-1)}-e^{2t+y(e^t-1)}+e^{y(e^t-1)})\\
&=\frac{2t-1}{4}\frac{\partial^2}{\partial t^2}e^{y(e^t-1)}-\frac{2t-1}{4}\frac{\partial}{\partial t}e^{y(e^t-1)}+\frac{y^2}{4}e^{y(e^t-1)}.\end{aligned}$$ Hence, we can state the following result.
\[thmcAsp\] The total number of corners of type A over set partitions of $[n+1]$ with $k$ blocks is given by $$\frac{n}{2}S_{n+1,k}-\frac{1}{4}S_{n+2,k}-\frac{n}{2}S_{n,k}+\frac{1}{4}S_{n+1,k}+S_{n,k-2}.$$ Moreover, the total number of corners of type A over set partitions of $[n+1]$ is given by $$\frac{2n+1}{4}B_{n+1}-\frac{1}{4}B_{n+2}-\frac{n-2}{2}B_n,$$ where $B_n$ is the $n$th Bell number.
Counting Corners of type B {#secB}
==========================
Let $J:=J(x,y,\pp)$ be the generating function for the number of bargraphs $\pi$ according to the number of cells in $\pi$, the number of columns of $\pi$, and the number of $(a,b)$-corners of type B in $\pi$ corresponding to the variables $x,y$ and $\pp=(p_{a,b})_{a,b\geq 1}$ respectively, that is, $$J=\sum_{n\geq 0}\sum_{\pi\in \BB_n}x^ny^{\text{col}(\pi)}\prod_{a,b\geq
1}p_{a,b}^{\Lambda_{(a,b)}(\pi)},$$ where $\Lambda_{(a,b)}(\pi)$ denotes the number of $(a,b)$-corners of type B in $\pi$. From the definitions, we have $$\label{eqcB1}
J=1+\sum_{a\geq 1}J_a,$$ where $J_a$ is the generating function for the number of bargraphs $\pi=a\pi'$ in which the hight of the first column is $a$. Since each bargraph $\pi=a\pi'$ can be decomposed as either $\pi=a^m$, $\pi=a^mb\pi''$ with $b\geq a+1$, or $\pi=a^mb\pi''$ with $1\leq b\leq a-1$, we have $$\label{eqcB2}
J_a=\sum_{m\geq 1}x^{am}y^mp_{m,a} + \sum_{m\geq 1}x^{am}y^m(J_{a+1}+J_{a+2}+\cdots) + \sum_{m\geq 1}\sum_{b=1}^{a-1}x^{am}y^mp_{m,a-b}J_b.$$ Define $\gamma_a:=\sum_{m\geq 1}x^{am}y^mp_{m,a}$ and since $J-1-\sum_{b=1}^aJ_b=\sum_{b\geq a+1}J_b$ (see ), we obtain $$\begin{aligned}
J_a&=\gamma_a+\frac{x^ay}{1-x^ay}\left(J-1-\sum_{b=1}^aJ_b\right)+\sum_{m\geq 1}\left(x^{am}y^m\sum_{b=1}^{a-1}p_{m,a-b}J_b\right)\\
&=\gamma_a+\frac{x^ay}{1-x^ay}(J-1)-\frac{x^ay}{1-x^ay}J_a+\sum_{m\geq
1}\left(x^{am}y^m\sum_{b=1}^{a-1}(p_{m,a-b}-1)J_b\right),\end{aligned}$$ which, by solving for $J_a$, gives $$\begin{aligned}
J_a&=x^ay(J-1)+(1-x^ay)\gamma_a+(1-x^ay)\sum_{m\geq
1}\left(x^{am}y^m\sum_{b=1}^{a-1}(p_{m,a-b}-1)J_b\right).\end{aligned}$$ If we define $$\theta_a:=x^ay(J-1)+(1-x^ay)\gamma_a \hbox{ and }\mu_{a,b}:=(1-x^ay)\sum_{m\geq 1}\left(x^{am}y^m(p_{m,a-b}-1)\right),$$ then we obtain $$\label{eqcB3}
J_a=\theta_a+\sum_{b=1}^{a-1}\mu_{a,b}J_b.$$ By similar techniques as in the proof of Lemma \[lemcA1\], we can state the following.
\[lemcB1\] For all $a\geq1$, $$\begin{aligned}
J_a=\theta_a+\sum_{j=1}^{a-1}\Gamma_{a,j}\theta_j,\end{aligned}$$ where $\Gamma_{a,j}=\sum_{s\geq0}\sum_{j=i_{s+1}<i_s<\cdots<i_0=a}\prod_{\ell=0}^s\mu_{i_\ell
i_{\ell+1}}$.
\[mth2\]The generating function $J(x,y,\pp)$ is given by $$J(x,y,\pp)=1+\frac{\sum_{m\geq 1}\sum_{j\geq 1}(1+\Gamma_j)(1-x^jy)x^{jm}y^mp_{m,j}}{1-\frac{xy}{1-x}-\sum_{j\geq 1}x^jy
\Gamma_j}$$ where $\Gamma_j=\sum_{s\geq 0}\sum_{j=i_{s+1}<i_s<\cdots<i_0}\prod_{\ell=0}^s\mu_{i_{\ell}i_{\ell+1}}$.
For instance, if $p_{a,b}=1$ for all $a,b\geq 1$, then $\mu_{a,b}=0$ which implies that $\Gamma_a=0$. Thus Theorem \[mth2\] shows that $J(x,y,1,1,\cdots)=\frac{1-x}{1-x-xy}$.
Counting all corners of type B
------------------------------
Let $p_{a,b}=p$ for all $a,b\geq 1$. By definitions, we have $\mu_{a,b}=(p-1)x^ay$ which yields $$\begin{aligned}
\label{eqcB4}
\Gamma_j&=\sum_{s\geq 0}\left((p-1)^{s+1}\sum_{j=i_{s+1}<i_s<\cdots<i_0}x^{i_0+\cdots+i_s}\right)\notag\\
&=\sum_{s\geq 0}\frac{(p-1)^{s+1}y^{s+1}x^{(s+1)j+\binom{s+2}{2}}}{(1-x)(1-x^2)\cdots(1-x^{s+1})}.\end{aligned}$$ From Theorem \[mth2\] and , the generating function $F=J(x,y,p,p,\cdots)$ is given by $$\begin{aligned}
F&=1+\frac{p\frac{xy}{1-x}+p\sum_{j\geq 1}\Gamma_jx^jy}{1-\frac{xy}{1-x}-\sum_{j\geq 1}\Gamma_jx^jy}\\
&=1+\frac{p\frac{xy}{1-x}+p\sum_{s\geq 0}\frac{(p-1)^{s+1}y^{s+2}x^{\binom{s+3}{2}}}{(1-x)(1-x^2)\cdots(1-x^{s+2})}}{1-\frac{xy}{1-x}-\sum_{s\geq 0}\frac{(p-1)^{s+1}y^{s+2}x^{\binom{s+3}{2}}}{(1-x)(1-x^2)\cdots(1-x^{s+2})}}.\end{aligned}$$ Let $\corb(\pi)$ be the number of corners of type B in $\pi$. We define $h_{n,k}=\sum_{\pi \in \BB_{n,k}}\corb(\pi)$ and $h_n=\sum_{k\geq 1}h_{n,k}$. Let $H(x,y)=\sum_{n,k\geq 1}h_{n,k}x^ny^k$ be the generating function for the total number of type B corners over all bargraphs according to the number of cells and columns. Then, it follows that $$H(x,y)=\frac{\partial F}{\partial
p}\Bigr|_{\substack{p=1}}=\frac{xy(1-x-xy+x^2y^2)}{(1-x-xy)^2}.$$ Note that $H(x,1)=\frac{x(x-1)^2}{(1-2x)^2}$ is the generating function for the total number of type B corners over all bargraphs according to the number of cells.
Counting $(v,w)$-corners of type B
----------------------------------
Fix $v,w\geq1$. Define $p_{v,w}=p$ and $p_{a,b}=1$ for all $(a,b)\neq(v,w)$. Then we have $\mu_{a,b}=(1-x^ay)x^{av}y^v(p-1)\delta_{a-b=w}$ which yields $$\begin{aligned}
\label{eqcB5}
\Gamma_j&=\sum_{s\geq 0}\sum_{j=i_{s+1}<i_s<\cdots<i_0}\prod_{\ell=0}^s\left((1-x^{i_{\ell}}y)x^{i_{\ell}v}y^v(p-1)\delta_{i_{\ell+1}-i_{\ell}=w}\right)\notag\\
&=\sum_{s\geq 0} (p-1)^{s+1}y^{v(s+1)}x^{vj(s+1)+vw\binom{s+1}{2}}\prod_{\ell=0}^s(1-x^{j+(\ell+1) w}y).\end{aligned}$$ From Theorem \[mth2\] and , the generating function $F=J(x,y,\pp)$ is given by $$\begin{aligned}
F=1+\frac{\frac{yx}{1-x}+(1+\Gamma_w)(1-x^wy)x^{wv}y^v(p-1)+y\displaystyle\sum_{j\geq 1}x^j\Gamma_j}{1-\frac{xy}{1-x}-y\displaystyle\sum_{j\geq 1}x^j\Gamma_j}.\end{aligned}$$
Recall that $\Lambda_{(v,w)}(\pi)$ denotes the number of $(v,w)$-corners of type B in $\pi$. We define $t_{n,k}=\sum_{\pi \in \BB_{n,k}}\Lambda_{(v,w)}(\pi)$ and $t_n=\sum_{k\geq 1}t_{n,k}$. Let $T(x,y)=\sum_{n,k\geq 1}t_{n,k}x^ny^k$ be the generating function for the total number of $(v,w)$-corners of type B over all bargraphs according to the number of cells and columns. Then, it follows that $$\begin{aligned}
T(x,y)=\frac{\partial F}{\partial
p}\Bigr|_{\substack{p=1}}=\frac{(1-x^wy)x^{vw}y^w}{\left(1-\frac{xy}{1-x}\right)^2}+\frac{y^{v+1}x^{2v+3}(1-x^wy)+(yx)^{v+1}(1-x^{w+1}y)}{(1-x^{v+1})(1-x^{v+2})\left(1-\frac{xy}{1-x}\right)^2},\end{aligned}$$ which leads to $T(x,1)=\frac{(1-x^w)x^{vw}}{\left(1-\frac{x}{1-x}\right)^2}+\frac{x^{2v+3}(1-x^w)+x^{v+1}(1-x^{w+1})}{(1-x^{v+1})(1-x^{v+2})\left(1-\frac{x}{1-x}\right)^2}$; the generating function for the total number of $(v,w)$-corners of type B over all bargraphs according to the number of cells.
Restricted bargraphs {#restricted-bargraphs}
--------------------
Theorem \[mth2\] can be refined as follows. Fix $N\geq1$. Let $J^{(N)}:=J^{(N)}(x,y,\pp)$ be the generating function for the number of bargraphs $\pi$ such that the height of each column is at most $N$ according to the number of cells in $\pi$, the number of columns of $\pi$, and the number of $(a,b)$-corners of type B in $\pi$ corresponding to the variables $x,y$ and $\pp=(p_{a,b})_{a,b\geq 1}$ respectively. Then by using similar arguments as in the proof of and , we obtain that $J^{(N)}=1+\sum_{a=1}^NJ_a^{(N)}$ and $J_a^{(N)}=\theta_a+\sum_{b=1}^{a-1}\mu_{a,b}J_b^{(N)}$, for all $a=1,2,\ldots,N$, where $J^{(N)}_a:=J^{(N)}_a(x,y,\pp)$ is the generating function for the number of bargraphs $\pi=a\pi'$ such that the height of each column is at most $N$. By the proof of Theorem \[mth2\], we can state its extension as follows.
\[mthN2\] The generating function $J^{(N)}=J^{(N)}(x,y,\pp)$ is given by $$J^{(N)}=1+\frac{\sum_{j=1}^N(1+\Gamma_j)(1-x^jy)\gamma_j}{1-y\sum_{j=1}^Nx^j-\sum_{j=1}^Nx^jy
\Gamma_j}$$ where $\Gamma_j=\sum_{s\geq 0}\sum_{j=i_{s+1}<i_s<\cdots<i_0\leq N}\prod_{\ell=0}^s\mu_{i_{\ell}i_{\ell+1}}$. Moreover, for all $a=1,2,\ldots,N$, we have $$\begin{aligned}
J_a^{(N)}=\left(x^ay+\sum_{j=1}^{a-1}x^jy\Gamma_{a,j}\right)(J^{(N)}-1)
+(1-x^ay)\gamma_a+\sum_{j=1}^{a-1}\Gamma_{a,j}(1-x^jy)\gamma_j,\end{aligned}$$ where $\Gamma_{N,j}=\sum_{s\geq0}\sum_{j=i_{s+1}<i_s<\cdots<i_0=N}\prod_{\ell=0}^s\mu_{i_\ell
i_{\ell+1}}$.
For instance, Theorem \[mthN2\] for $N=1$ gives $$J^{(1)}(x,y,\pp)=1+\frac{(1-xy)\gamma_1}{1-xy}=1+\sum_{m\geq1}x^my^mp_{m,1}.$$
Counting corners of type B in set partitions
--------------------------------------------
Recall that we represent any set partition as a bargraph corresponding to its canonical sequential representation. Let $P_k(x,y,\pp)$ be the generating function for the number of set partitions $\pi$ of $[n]$ with exactly $k$ blocks according to the number of cells in $\pi$, the number of columns of $\pi$ (which is $n$), and the number of $(a,b)$-corners of type B in $\pi$ corresponding to the variables $x,y$ and $\pp=(p_{a,b})_{a,b\geq 1}$ respectively.
Note that each set partition with exactly $k$ blocks can be decomposed as $1\pi^{(1)}\cdots k\pi^{(k)}$ such that $\pi^{(j)}$ is a word over alphabet $[j]$. Thus, by Theorem \[mthN2\], we have the following result.
\[mthPk1\] Let $p_{a,b}=p$ for all $a,b\geq 1$. Then the generating function $P_k(x,y,\pp)$ is given by $$\begin{aligned}
P_k(x,y,\pp)=p^{1-k}\prod_{N=1}^kJ^{(N)}_N(x,y,\pp).\end{aligned}$$ where $J_N^{(N)}$ is given in statement Theorem \[mthN2\].
Now, we consider counting all corners of type B in set partitions. Let $p_{a,b}=p$ for all $a,b\geq 1$, and $Q_k(x,y)=\frac{\partial}{\partial p}P_k(x,y,\pp)\Bigr|_{\substack{p=1}}$. By Theorem \[mthN2\], we have that $J^{(N)}(x,y,{\bf 1})=\frac{1}{1-y\sum_{j=1}^Nx^j}$ and $$\frac{\partial }{\partial p}J^{(N)}(x,y,\pp)\mid_{p=1}=
\frac{y\sum_{j=1}^Nx^j-\left(y\sum_{j=1}^Nx^j\right)^2+y^2\sum_{j=1}^Nx^j\frac{x^{j+1}-x^{N+1}}{1-x}}
{\left(1-y\sum_{j=1}^Nx^j\right)^2}.$$ Moreover, Theorem \[mthN2\] gives that $J_N^{(N)}(x,y,{\bf 1})=\frac{x^Ny}{1-y\sum_{j=1}^Nx^j}$ and $$\frac{\partial }{\partial p}J_N^{(N)}(x,y,\pp)\mid_{p=1}=x^Ny\left(\frac{\partial }{\partial p}J^{(N)}(x,y,\pp)\mid_{p=1}
+\frac{1-x^Ny}{1-\sum_{j=1}^Nx^jy}\right).$$ Hence, by Theorem \[mthPk1\] we have $$Q_k(x,y)=\prod_{N=1}^k\frac{x^Ny}{1-y\sum_{j=1}^Nx^j}\left(\sum_{N=1}^k\frac{\frac{\partial }{\partial p}J_N^{(N)}(x,y,\pp)\mid_{p=1}}{\frac{x^Ny}{1-y\sum_{j=1}^Nx^j}}-k+1\right).$$ In particular, the generating function for the total number of corners of type B over all set partitions of $[n]$ with $k$ blocks is given by $$Q_k(1,t)=\frac{t^k}{(1-t)(1-2t)\cdots(1-kt)}
\left(\sum_{N=1}^k\frac{\frac{\partial}{\partial p}J_N^{(N)}(1,t,\pp)\mid_{p=1}}{\frac{t}{1-Nt}}-k+1\right),$$ which, by $\frac{\partial}{\partial p}J_N^{(N)}(1,t,\pp)\mid_{p=1}=t\left(\frac{Nt}{1-Nt}+\frac{t^2N(N-1)}{2(1-Nt)^2}+\frac{1-t}{1-Nt}\right)$, is equivalent to $$Q_k(1,t)=\frac{t^k}{(1-t)(1-2t)\cdots(1-kt)}
\left(1-k+\sum_{N=1}^k\left(1+(N-1)t+\frac{t^2N(N-1)}{2(1-Nt)}\right)\right).$$ Hence, $$Q_k(1,t)=\frac{t^k}{(1-t)(1-2t)\cdots(1-kt)}
\left(1+\frac{t}{2}\binom{k}{2}+\frac{t}{2}\sum_{N=1}^k\frac{N-1}{1-Nt}\right).$$ Define $\tilde{Q}_k(t)$ to be the corresponding exponential generating function to $Q_k(1,t)$, that is $\tilde{Q}_k(t)=\sum_{n\geq0}q_{n,k}\frac{t^n}{n!}$ where $q_{n,k}$ is the coefficient of $t^n$ in $Q_k(1,t)$. Similar to Section \[secSPA\], we have $$\tilde{Q}_k(t)=\frac{(e^t-1)^k}{k!}+\binom{k}{2}\int_0^t\frac{(e^r-1)^k}{2k!}dr
+\frac{kt(e^t-1)^{k-1}e^t}{2k!}-\frac{k(e^t-1)^k}{2k!} -\int_0^t\frac{rk(e^r-1)^{k-1}e^r}{2k!}dt.$$ Define $\tilde{Q}(t,y)=\sum_{k\geq0}\tilde{Q}_k(t)y^k$, thus by multiplying by $y^k$ and summing over $k\geq1$, we obtain $$\begin{aligned}
\tilde{Q}(t,y)&=e^{y(e^t-1)}-1+\frac{y^2}{4}\int_0^t(e^r-1)^2e^{y(e^r-1)}dr\\
&+\frac{ty}{2}e^{t+y(e^t-1)}-\frac{y}{2}(e^t-1)e^{y(e^t-1)}-\frac{y}{2}\int_0^tre^{r+y(e^r-1)}dr.\end{aligned}$$ In particular, $$\frac{\partial}{\partial t}\tilde{Q}(t,y)=
\frac{y}{4}\left(y(2t-1)e^{2t+y(e^t-1)}+ye^{y(e^t-1)}+4e^{t+y(e^t-1)}\right),$$ which is equivalent to $$\frac{\partial }{\partial t}\tilde{Q}(t,y)=
\frac{2t-1}{4}\frac{\partial^2}{\partial t^2}e^{y(e^t-1)}-\frac{2t-5}{4}\frac{\partial}{\partial t}e^{y(e^t-1)}+\frac{y^2}{4}e^{y(e^t-1)}.$$ Since $e^{y(e^t-1)}=\sum_{n\geq0}\sum_{k=0}^nS_{n,k}\frac{x^ny^k}{n!}$, $S_{n,k}$ is the Stirling number of the second kind, we obtain the following result.
\[thmcBsp\] The total number of corners of type B over set partitions of $[n+1]$ with $k$ blocks is given by $$\frac{n}{2}S_{n+1,k}-\frac{1}{4}S_{n+2,k}-\frac{n}{2}S_{n,k}+\frac{5}{4}S_{n+1,k}+S_{n,k-2}.$$ Moreover, the total number of corners of type B over set partitions of $[n+1]$ is given by $$\frac{2n+5}{4}B_{n+1}-\frac{1}{4}B_{n+2}-\frac{n-2}{2}B_n$$ where $B_n$ is the $n$th Bell number.
[**Acknowledgment:**]{} G. Yildirim would like to thank the Department of Mathematics at the University of Haifa for their warm hospitality during the writing of this paper.
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|
---
author:
- 'I. Zhelyazkov [^1]'
date: 'Received: date / Revised version: date'
title: 'MHD waves and instabilities in flowing solar flux-tube plasmas in the framework of Hall magnetohydrodynamics'
---
[leer.eps]{} gsave 72 31 moveto 72 342 lineto 601 342 lineto 601 31 lineto 72 31 lineto showpage grestore
Introduction {#sec:intro}
============
Various waves and oscillations which occur in structured solar atmosphere were intensively studied over the past three decades and an exclusive review of their theory the reader can find in [@valery07] and references therein. Next step in studying the wave phenomena in solar and stellar atmospheres was the consideration of steady flows there. Satellite measurements of plasma characteristics of, for instance, the solar wind and coronal plumes flows, such as the magnetic field, the thermal and flow velocity and density of plasma or plasma compositions, are important to understand the various plasma wave modes which may arise. However, wave analysis requires further information and special tools to identify which set of modes is contributing to observed wave features. In practice, one may use filters to perform the so-called *pattern recognition* to detect the various kind of modes that may propagate in plasma and to determine their contribution to the wave energy [@vocks99]. Another important issue is the question for the waves’ stability. The magnetosonic waves in structured atmospheres with steady flows have been examined by Nakariakov and Roberts [@valery95], Nakariakov et al. [@valery96], Andries and Goossens [@andries01], and Terra-Homem et al. [@terra03]. Andries and Goossens studied also the conditions at which resonant flow and Kelvin–Helmholtz instabilities take place, and Terra-Homem et al. have investigated the effect of a steady flow on the linear and nonlinear wave propagation in various solar structures.
It is worth mentioning that all the aforementioned studies were performed in the framework of standard magnetohydrodynamics. It was Lighthill [@lighthill60] who pointed out almost $50$ years ago that for an adequate description of wave phenomena in fusion and astrophysical plasmas one has to include the Hall term, $m_{\rm i} (\mathbf{j} \times \mathbf{B})/e\rho$, in the generalized Ohm’s law. That approach is termed Hall magnetohydrodynamics (Hall MHD). In this way, it is possible to describe waves with frequencies up to $\omega \approx \omega_{\rm ci}$. Because the model still neglects the electron mass, it is limited to frequencies well below the lower hybrid frequency: $\omega \ll \omega_{\rm lh}$. Generally speaking, the theory of Hall MHD is relevant to plasma dynamics occurring on length scales shorter than an ion inertial length, $L < l_{\rm Hall} = c/\omega_{\rm pi}$ (where $c$ is the speed of light and $\omega_{\rm pi}$ is the ion plasma frequency), and time scales of the order or shorter than the ion cyclotron period ($t < {\omega_{\rm ci}}^{-1}$) [@huba95]. Thus the Hall MHD should affect the dispersion properties of MHD waves in spatially bounded magnetized plasmas.
The first paper devoted to the propagation of fast MHD and ion-cyclotron surface waves at a plasma–vacuum interface in the limit of cold plasma was that by Cramer and Donnelly [@cramer83]. Later on, Cramer [@cramer91] generalizes that model obtaining the dispersion of nonlinear surface Alfvén waves. It is worth noticing that even in unbounded magnetoplasma at purely parallel propagation the Alfvén waves become dispersive when the Hall current is taken into account in the basic MHD equations [@almaguer92]. In Ref. [@ivan96] the Hall MHD has been applied in studying the parallel propagation of fast wave in an ideal static plasma slab. There were derived four boundary conditions (see Sec. III in [@ivan96]) necessary for obtaining the dispersion relations of sausage and kink modes in spatially bounded magnetized plasmas. That approach has been applied in studying the oblique propagation in the same geometry both for low-$\beta$ and finite-$\beta$ plasmas [@ivan00; @ivan03].
The first study on surface-wave parallel propagation in a flowing ideal MHD flux tube surrounded by a static plasma environment (both embedded in a constant magnetic field $\mathbf{B}_0$) in the framework of the Hall MHD was performed by Miteva et al. [@rossi03]. It has been shown that while in a static plasma slab the hydromagnetic surface waves (sausage and kink modes) are Alfvén ones (their phase velocities are close to the Alfvén speed in the layer), in slabs with steady flows they become super-Alfvénic waves. Moreover, as it is logical to expect, there exist two type of waves: forward and backward propagating ones, bearing in mind that the flow velocity defines the positive (forward) direction. An in-deep examination of wave modes in flowing solar-wind-flux-tube magnetized plasmas for finite-$\beta$ and zero-$\beta$ ionized media has been performed in [@rossi04; @sikka04] (see also the references in those papers).
Hall MHD is relevant not only to linear MHD waves, but also to nonlinear ones [@michael02a; @michael02b; @ballai03; @mahajan05]. Dispersive effects caused by Hall currents perpendicular to the ambient magnetic field can influence the generation and propagation of shock waves [@ballai07]. Recently Clack and Ballai [@clack08] studied the nonlinear wave propagation of resonant slow magnetoacoustic waves in plasmas with strongly anisotropic viscosity and thermal conductivity alongside of dispersive effects due to Hall currents. They show that the nonlinear governing equation describing the dynamics of nonlinear resonant slow waves is supplemented by a term which describes nonlinear dispersion and is of the same order of magnitude as nonlinearity and dissipation. In the case of stationary nonlinear waves the Hall-MHD equations can be rewritten in the so-called Sakai–Sonnerup set of equations that describe the plasma state and provide oscillatory and solitary types of solutions [@rossi08]. The overall parameter study on the polarization characteristics, together with the magnetic field components and density variations in different ranges of solutions performed in that paper might be further on applied to the theoretical treatment of particle interaction with such waves, e.g., at shocks in space plasmas, possibly leading to particle acceleration. MHD parametric instabilities of parallel propagating incoherent Alfvén waves are influenced by the Hall effect and that is especially important for the left-hand polarized Alfvén waves [@nariyuki07]. Ruderman and Caillol [@michael08] claim that the left-hand polarized Alfvén waves in a Hall plasma are actually subject to three different types of instabilities, namely modulational, decay, and beat instability, while the right-hand polarized waves are subject only to decay instability. A new trend in the Hall MHD is its application to partially ionized plasmas [@pandey08] – the Hall effect may play an important role in the dynamics of weakly ionized systems such as the Earth’s ionosphere and protoplanetary discs. The exact nonlinear cylindrical solution for incompressible Hall-MHD waves, including dissipation, essentially from electron–neutral collisions, obtained in a uniformly rotating, weakly ionized plasma, derived by Krishan and Varghese [@krishan08], demonstrates the dispersive nature of the waves, introduced by the Hall effect, at large axial and radial wave numbers. Such short-wavelength and arbitrarily large-amplitude waves could contribute toward the heating of the solar atmosphere. One of the most important mechanisms for solar atmosphere’s heating and especially the solar wind is the wave turbulence, which is also affected by the Hall currents [@krishan04; @galtier06; @galtier07; @galtier09; @shaikh09]. The strongest competitor of the wave turbulence heating mechanism, notably the magnetic reconnection, is similarly heavily influenced by the Hall effect [@ma01; @morales05; @arber06; @cassak07; @craig08]. One even speaks for Hall reconnection being akin to Petschek reconnection model. It is worth mentioning that the Hall effect can influence the magnetic field dynamics in dense molecular clouds [@wardle99], magnetorotational instability [@sano02], star formation [@wardle04], as well as compact objects [@elena08]. As seen, the Hall MHD has impact on many important astrophysical phenomena and objects.
Here, we investigate the influence of flow velocities on the dispersion characteristics and stability of hydromagnetic surface waves (sausage and kink modes) travelling along an infinitely conducting, magnetized jet moving past also (with a different speed) infinitely conducting, magnetized plasmas. If in the solar corona plasma $\beta$ (the ratio of gas to magnetic pressure) is much less than unity, in the solar wind flux tubes it is $\beta \approx 1$. Since we are going to study the wave propagation in flowing solar wind plasma, we can assume that we have a ‘high-$\beta$’ magnetized plasma and treat it as an incompressible fluid. For simplicity, we consider a planar jet of width $2x_0$ (embedded together with environments in a constant magnetic field $\mathbf{B}_0$ directed along the $z$ axis), allowing for different plasma densities within and outside the jet, $\rho_{\rm o}$ and $\rho_{\rm e}$, respectively. The most natural discontinuity which occurs at the surfaces bounding the layer is the tangential one because it is the discontinuity that ensures a static pressure balance [@cravens97]. For typical values of the ambient constant magnetic field $B_0 = 5 \times 10^{-9}$ T and the electron number density inside the jet $n_{\rm o} = 2.43 \times 10^6$ m$^{-3}$ at 1 AU [@cravens97], the ion cyclotron frequency $\omega_{\rm ci}/2\pi = 76$ mHz, and the Hall scale length ($= v_{\rm Ao}/\omega_{\rm ci}$, which is equivalent to $c/\omega_{\rm pi}$) is $l_{\rm Hall} \approx 150$ km. This scale length is small, but not negligible compared to layer’s width of a few hundred kilometers. Here, we introduce a scale parameter $\varepsilon = l_{\rm Hall}/x_0$ called the *Hall parameter*. In the limit of $\varepsilon \to 0$, the Hall MHD system reduces to the conventional MHD system. Our choice for that parameter is $\varepsilon = 0.4$. The flow speeds of the jet and its environment are generally rather irregular. For investigating the stability of the travelling MHD waves it is convenient to consider the wave propagation in a frame of reference attached to the flowing environment. Thus we can define the relative flow velocity $\mathbf{U}^{\rm rel} = \mathbf{U}_{\rm o} - \mathbf{U}_{\rm e}$ ($U_{\rm o}$ and $U_{\rm e}$ being the steady flow speeds inside and outside the flux tube, respectively) as an entry parameter whose value determines the stability/instability status of the jet. As usual, we normalize that relative flow velocity with respect to the Alfvén speed in the jet and call it Alfvénic Mach number $M_{\rm A}$, omitting for simplicity the superscript “rel”. Another important entry parameter of the problem is $\eta = \rho_{\rm e}/\rho_{\rm o}$. It turns out that the waves’ dispersion characteristics and their stability critically depend on the magnitude of $\eta$. Our preliminary numerical studies, based on a criterion for rising the Kelvin–Helmholtz instability (namely that the modulus of the steady flow velocity must be larger than the sum of Alfvén speeds inside and outside the flux tube) proposed by Andries and Goossens [@andries01], showed that in order to expect instability onset at some reasonable values of the relative Alfvénic Mach number $M_{\rm A}$, one should assume fairly large values of the parameter $\eta$. In this study we take $\eta = 10$ which means that the Alfvén speed inside the jet is roughly three times larger than that in the environment – actually an “edge” choice, but still acceptable. Thus the waves’ dispersion characteristics (the dependence of the wave phase velocity $v_{\rm ph} = \omega/k$ on the wave number $k$) and their stability states are determined by the three parameters $\eta$, $\varepsilon$, and $M_{\rm A}$, two of which are fixed ($\eta$ and $\varepsilon$) and the third one, $M_{\rm A}$, is running.
The basic equations and wave dispersion relations will be exposed in Sec. 2 of the paper, and the numerical results and discussions – in Sec. 3. Section 4 summarizes the new findings and comments on future improvements of this study.
Basic equations and dispersion relations {#sec:basiceqns}
========================================
The jet’s interfaces are the surfaces $x = \pm x_0$, the uniform magnetic field $\mathbf{B}_0$ and the steady flow velocities $\mathbf{U}_{\rm o,e}$ point in the $z$ direction. The wave vector $\mathbf{k}$ lies also along the $z$ axis and its direction is the same as that of $\mathbf{B}_0$ and $\mathbf{U}_{\rm o,e}$. As we have already mentioned we will study the waves propagation in a reference frame affixed to the environment. Thus the steady flow velocity in the slab is $\mathbf{U} \equiv \mathbf{U}^{\rm rel} = \mathbf{U}_{\rm o} - \mathbf{U}_{\rm e}$ and zero outside. The basic equations for the incompressible Hall-MHD waves are the linearized equations governing the evolution of the perturbed fluid velocity $\delta \mathbf{v}$ and wave magnetic field $\delta \mathbf{B}$: $$\rho \frac{\partial}{\partial t} \delta \mathbf{v} + \rho(\mathbf{U} \cdot \nabla) \delta \mathbf{v} + \nabla(\frac{1}{\mu_0}\mathbf{B}_0 \cdot \delta \mathbf{B}) - \frac{1}{\mu_0} (\mathbf{B}_0 \cdot \nabla) \delta \mathbf{B} = 0,
\label{eq:movm}$$ $$\frac{\partial}{\partial t} \delta \mathbf{B} + (\mathbf{U} \cdot \nabla) \delta \mathbf{B} - (\mathbf{B}_0 \cdot \nabla) \delta \mathbf{v} + \mathbf{B}_0 \nabla \cdot \delta \mathbf{v} + \frac{v_{\rm A}^2}{\omega_{\rm ci}}\hat{z} \cdot \nabla \nabla \times \delta \mathbf{B} = 0,
\label{eq:magfield}$$ with the constrains $$\nabla \cdot \delta \mathbf{v} = 0,
\label{eq:cont}$$ $$\nabla \cdot \delta \mathbf{B} = 0,
\label{eq:magind}$$ where $v_{\rm A} = B_0/(\mu_0 \rho)^{1/2}$ is the Alfvén speed and $\mu_0$ is the permeability of free space. After Fourier transforming all perturbed quantities $\propto\!\!g(x)\exp(-\mathrm{i}\omega t + \mathrm{i}kz)$, we derive two coupled second order differential equations for $\delta v_x$ and $\delta v_y$, notably $$\left( {\displaystyle \frac{\mathrm{d}^2}{\mathrm{d} x^2} - k^2} \right)\delta v_x - \mathrm{i} \frac{a - 1}{\epsilon} k^2 \delta v_y = 0,
\label{eq:short_1}$$ and $$\left( {\displaystyle \frac{\mathrm{d}^2}{\mathrm{d} x^2} - k^2} \right)\delta v_x + \mathrm{i}{\displaystyle \frac{\epsilon}{a - 1} \left( {\displaystyle \frac{\mathrm{d}^2}{\mathrm{d} x^2} - k^2} \right)} \delta v_y = 0,
\label{eq:short_2}$$ where $$\epsilon = \frac{\omega - \mathbf{k} \cdot \mathbf{U}}{\omega_{\rm ci}}, \quad \mbox{and} \quad a = \left( \frac{\omega - \mathbf{k} \cdot \mathbf{U}}{kv_{\rm A}} \right)^2$$ with $\omega_{\rm ci} = eB_0/m_{\rm i}$ being the ion angular cyclotron frequency ($m_{\rm i}$ is ion/proton mass and $e$ is the elementary electric charge). Note that we have different $\epsilon$s and $a$s inside and outside the jet. We seek the solutions to coupled equations (\[eq:short\_1\]) and (\[eq:short\_2\]) in the form $$\delta v_x(x) = f \left[ \exp(-\kappa x) \mp \exp(\kappa x)
\right],$$ $$\delta v_y(x) = \mathrm{i} h \left[ \exp(-\kappa x) \mp \exp(\kappa x)
\right],$$ anticipating surface waves with attenuation coefficient $\kappa$, and obtain the following set of equations $$\begin{aligned}
\left( \kappa^2 - k^2 \right)f + \frac{a - 1}{\epsilon}
k^2h = 0, \\
\left( \kappa^2 - k^2 \right)f - \frac{\epsilon}{a - 1}
\left( \kappa^2 - k^2 \right)h = 0.\end{aligned}$$ This set of equations yields the following expressions for $\kappa$: $$\begin{aligned}
\kappa_1 &=& k, \nonumber
\\
\kappa_2 = k\left[ 1 - (a - 1)^2/\epsilon^2 \right]^{1/2} &\equiv& m. \nonumber
\label{eq:kappa}\end{aligned}$$ That means that there are in fact two pairs of attenuation coefficients: ($k,m_{\rm o}$) inside the flux tube and ($k,m_{\rm e}$) outside the jet, respectively.
As is known, on a bounded MHD plasma wave guide (cylinder or layer) two types of waves may exist. Recall that for a slab geometry the general solutions to the equations governing $\delta v_x$ and $\delta v_y$ are sought in the form of superpositions of $\cosh \kappa_{\rm o}x$ and $\sinh \kappa_{\rm o}x$. Those solutions contain waves whose shape is defined by the $\cosh$ function (they are called *kink* waves) and another type of waves associated with the $\sinh$ function (which are termed *sausage* waves). The transverse structure of both waves inside the slab is determined by the two attenuation coefficients $\kappa_{\rm o1,2}$ (i.e., $k$ and $m_{\rm o}$). Thus the solutions for $\delta
v_x$ and $\delta v_y$ inside the slab ($|x| < x_0$) assuming a sausage wave form, accordingly, are $$\delta v_x(x) = f_1 {\displaystyle \frac{\sinh\,k x} {\sinh\,k x_0}}
+ f_2 {\displaystyle \frac{\sinh\,m_{\rm o} x}
{\sinh\,m_{\rm o} x_0}},$$ and $$\delta v_y(x) = \mathrm{i} f_1 G_{\rm o1} \displaystyle
{\frac{\sinh\,k x} {\sinh\,k x_0}}
+ \mathrm{i} f_2 G_{\rm o2} \displaystyle {\frac{\sinh\,m_{\rm o} x}
{\sinh\,m_{\rm o} x_0}},$$ where $$G_{\rm o1,2} = \displaystyle{
-\frac{\epsilon_{\rm o}}{a_{\rm o} - 1} \frac{\kappa_{\rm o1,2}^2 - k^2}{k^2}}.$$ These solutions have been obtained by imposing the compatibility condition for the set of equations for $f$ and $h$ as unknown quantities. We also notice that $G_{\rm o1} = 0$ and $G_{\rm o2} = (a_{\rm o} - 1)/\epsilon_{\rm o}$. For a kink surface-wave form the expressions for perturbed fluid velocity components have the same description – it is only necessary to replace $\sinh$ with $\cosh$. The solutions outside the layer (identical for both modes) are $$\delta v_x(x) = \begin{cases}
\alpha_1 \exp \left[ -k(x - x_0) \right] + \alpha_2 \exp \left[ -m_{\rm e}(x - x_0) \right] & \text{for $x > x_0$,}
\\
\beta_1 \exp \left[ k(x + x_0) \right] + \beta_2 \exp \left[ m_{\rm e}(x + x_0) \right] & \text{for $x < -x_0$,} \end{cases}$$ and $$\delta v_y(x) = \begin{cases}
\mathrm{i} \alpha_1 G_{{\rm e}1} \exp\left[ -k(x - x_0) \right] + \mathrm{i}
\alpha_2 G_{{\rm e}2} \exp \left[ -m_{\rm e}(x - x_0) \right] &
\text{for $x > x_0$,}
\\
\mathrm{i} \beta_1 G_{{\rm e}1} \exp \left[ k(x + x_0) \right] + \mathrm{i}
\beta_2 G_{{\rm e}2} \exp \left[ m_{\rm e}(x + x_0) \right] &
\text{for $x < -x_0$.} \end{cases}$$ Here, as above, $$G_{\rm e1,2} = \displaystyle{
-\frac{\epsilon_{\rm e}}{a_{\rm e} - 1} \frac{\kappa_{\rm e1,2}^2 - k^2}{k^2}},$$ and, similarly, $G_{\rm e1} = 0$ and $G_{\rm e2} = (a_{\rm e} - 1)/\epsilon_{\rm e}$.
Having derived the expressions for the perturbed fluid velocity components $\delta v_x$ and $\delta v_y$, one can calculate the perturbed total pressure, which in our case is the perturbed magnetic pressure only, and arrive at $$\begin{aligned}
\delta p_{\rm total}(x) = \frac{1}{\mu_0}\mathbf{B}_0
\cdot \delta \mathbf{B}
{}= \mathrm{i} \frac{\rho}{\omega - \mathbf{k} \cdot \mathbf{U}} v_{\rm
A}^2 \\ \\
\times\left\{ {\displaystyle
\left[ \frac{\left( \omega - \mathbf{k} \cdot \mathbf{U}
\right)^2}{k^2 v_{\rm A}^2} - 1 \right] \frac{\mathrm{d}}{\mathrm{d} x}
\delta v_x}(x) +
\mathrm{i}\:{\displaystyle \frac{\omega - \mathbf{k} \cdot \mathbf{U}}
{\omega_{\rm ci}} \frac{\mathrm{d}}{\mathrm{d} x} \delta v_y(x)}
\right\}.\end{aligned}$$
The perturbed wave electric field $\delta \mathbf{E}$ can be obtained from the generalized Ohm’s law $$\mathbf{E} = - \mathbf{v} \times \mathbf{B} +
\frac{m_{\rm i}}{e \rho}\, \mathbf{j} \times \mathbf{B},$$ which, by means of Ampère’s law (multiplied vectorially by $\mathbf{B}$) yields $$\delta \mathbf{E} = \mathbf{B}_0 \times
\left( \delta \mathbf{v}
- {\displaystyle l_{\rm Hall} \frac{v_{\rm A}}{B_0}}
\nabla \times \delta
\mathbf{B} \right) - \mathbf{U} \times \delta \mathbf{B}.$$ Its components are $$\delta E_x(x) = -{\displaystyle \frac{\omega}{\omega -
\mathbf{k} \cdot \mathbf{U}} \left( \frac{\omega - \mathbf{k}
\cdot \mathbf{U}}{k v_{\rm A}} \right)^2} B_0 \delta v_y(x),$$ and $$\delta E_y(x) = {\displaystyle \frac{\omega}{\omega -
\mathbf{k} \cdot \mathbf{U}} B_0 \left[ \delta v_x(x) -
\mathrm{i}\: \frac{\omega - \mathbf{k} \cdot \mathbf{U}}{\omega_{\rm ci}}
\delta v_y(x) \right]}.$$ The above expressions for perturbed quantities are used when implementing the boundary conditions.
It follows from the solutions to the basic equations (\[eq:short\_1\]) and (\[eq:short\_2\]) for the perturbed fluid velocity components $\delta v_x$ and $\delta v_y$ that the number of integration constants is six. However, because of the symmetry (or antisymmetry), the two $\beta_{1,2}$ amplitudes are in fact directly obtainable from the $x > x_0$ solutions – indexed $\alpha$s and $\beta$s are not independent. Thus we can derive the dispersion relations by applying only four boundary conditions at one interface, for example, at $x = x_0$. These boundary conditions, as we already mentioned in Sec. \[sec:intro\], are derived and discussed in [@ivan96]. We can borrow them except the first one, the continuity of $\delta v_x$ across the interface, as in the present case of a flowing slab this condition must be replaced by the continuity of $\delta v_x/\left(\omega - \mathbf{k} \cdot \mathbf{U} \right)$ [@chandra61]. The rest of the boundary conditions are the continuity of the perturbed pressure $\delta p_{\rm total}$, the $y$-component of perturbed wave electric field $\delta E_y$, and the $x$-component of perturbed electric displacement $\delta D_x = \varepsilon_0 \left( K_{xx}\delta E_x + K_{xy}\delta E_y \right)$ (where $\varepsilon_0$ is the permittivity of the free space) at the interface. In the last boundary condition, $K_{xx}$ and $K_{xy}$ are the low-frequency components of the plasma dielectric tensor [@swanson89] $$K_{xx} \approx \frac{c^2}{v_{\rm A}^2} \qquad {\rm and}
\qquad K_{xy} \cong \mathrm{i} \frac{\omega -
\mathbf{k} \cdot \mathbf{U}}{\omega_{\rm ci}} \frac{c^2}{v_{\rm A}^2}.$$
By imposing the boundary conditions at the interface $x = x_0$, and after some straightforward algebra, we finally arrive at the dispersion relations for parallel propagation of sausage and kink waves in a planar jet, surrounded by steady plasma media $$\begin{aligned}
\label{eq:dispeqn}
\left( \frac{\omega - \mathbf{k} \cdot \mathbf{U}}
{k v_{\rm Ao}}\right)^2 - 1 + \left[ \frac{\rho_{\rm e}}{\rho_{\rm o}}
\left( \frac{\omega}{k v_{\rm Ao}}\right)^2 - 1 \right] {\tanh \choose \coth}kx_0
\nonumber \\
\\ \nonumber
{}- \epsilon_{\rm o}^2 \left[ 1 + \tilde{\omega}^2 \frac{\rho_{\rm e}}{\rho_{\rm o}}
{\tanh \choose \coth}kx_0 \right]\frac{1 - \tilde{\omega}
\rho_{\rm e}/\rho_{\rm o}}{1 - \tilde{\omega}
\left( \rho_{\rm e}/\rho_{\rm o} \right)^2 } = 0,\end{aligned}$$ where $$\tilde{\omega} = \frac{\omega}{\omega - \mathbf{k} \cdot \mathbf{U}} \quad \mbox{and} \quad \epsilon_{\rm o} = \frac{\omega - \mathbf{k}\cdot \mathbf{U}}
{\omega_{\rm ci}}.$$ As can be seen, the wave frequency $\omega$ is Doppler-shifted inside the jet. These dispersion relations can be extracted from Eq. (12) in [@rossi04] in the limit $c_{\rm s} \to \infty$ (sound speed much larger than the Alfvén one) by letting $\mathbf{U}_{\rm e} = 0$ and $\mathbf{U}_{\rm o} = \mathbf{U}$. When the flux tube is static ($\mathbf{U} = 0$), Eq. (\[eq:dispeqn\]) coincides with Eq. (18) in Ref. [@ivan96]. Finally, when one neglects the Hall effect ($\epsilon_{\rm o} = 0$) one gets the well-known, derived by Edwin and Roberts [@pat82], dispersion relations in the limit $c_{\rm s} \to \infty$. We should also emphasize that in an incompressible jet both modes are pure surface waves. A question which immediately arises is what type of MHD waves are those described by Eq. (\[eq:dispeqn\])? From the three well-known MHD linear modes propagating in infinite compressible magnetized plasmas (Alfvén, fast and slow magnetosonic waves) in incompressible limit survive only two, notably the shear and pseudo Alfvén waves. The latter is the incompressible vestige of the slow mode of compressible MHD. The displacement vector of a shear Alfvén wave is perpendicular to the plane defined by its wave vector, $\mathbf{k}$, and a uniform background magnetic field, $\mathbf{B}_0$, whereas that of a pseudo Alfvén wave lies in this plane. The two wave modes share the dispersion relation $$\omega^2 = \frac{(\mathbf{k} \cdot \mathbf{B}_0)^2}{\mu_0 \rho} \equiv \left( k_{\parallel}v_{\rm A} \right)^2$$ and propagate with group velocity, $\mathbf{v}_{\rm A}$, either parallel or antiparallel to $\mathbf{B}_0$ depending upon the sign of $k_\parallel$. It seems that our Hall-MHD modes travelling along the jet are akin to the pseudo Alfvén waves.
The dispersion relation of each mode can be symbolically written down in the form: $$\mathcal{D}(\omega,k,\mathrm{parameters}) = 0,
\label{eq:symb}$$ where the function argument ‘parameters’ includes data specific for the jet – they will be listed shortly. As we are interested in the stability of the surface waves running at the jet interfaces, we have to assume that the wave frequency is complex, i.e., $\omega \to \omega + \mathrm{i}\gamma$, where $\gamma$ is the expected instability growth rate. Thus dispersion equations (\[eq:dispeqn\]) become complex and their solving is not a trivial problem [@acton90]. When studying dispersion characteristics of MHD waves, one usually plots the dependence of the wave phase velocity $v_{\rm ph}$ as function of the wave number $k$. For numerical solving of equations (\[eq:dispeqn\]) we normalize all quantities by defining the dimensionless wave phase velocity $V_{\rm ph} = \omega/kv_{\rm Ao}$, wave number $K = kx_0$, and the (relative) Alfvénic Mach number $M_{\rm A} = U/v_{\rm Ao}$, respectively. Accordingly $$\left[ (\omega - \mathbf{k} \cdot \mathbf{U})/(kv_{\rm Ao}) \right]^2 = \left( V_{\rm ph} - M_{\rm A} \right)^2, \quad \omega^2/(kv_{\rm Ao})^2 = V_{\rm ph}^2,$$ $$\tilde{\omega} = V_{\rm ph}/
\left( V_{\rm ph} - M_{\rm A} \right) \quad \mbox{and} \quad
\epsilon_{\rm o} = K \left( V_{\rm ph} - M_{\rm A} \right)
l_{\rm Hall}/x_0.$$ Note, that $l_{\rm Hall}/x_0 = \varepsilon$ (alongside with $\eta$ and $M_{\rm A}$) is an entry parameter which has to be specified at the start of the numerical procedure. Thus we have to solve normalized dispersion relations (\[eq:dispeqn\]), having now the form $$\begin{aligned}
\label{eq:numdisp}
\left( V_{\rm ph} - M_{\rm A} \right)^2 - 1 + \left( \eta V_{\rm ph}^2 - 1 \right){\tanh \choose \coth}K \nonumber \\
\\ \nonumber
{}- K^2 \varepsilon^2 \left[ \left( V_{\rm ph} - M_{\rm A} \right)^2 + V_{\rm ph}^2 \eta {\tanh \choose \coth}K \right]\frac{V_{\rm ph}(1 - \eta) - M_{\rm A}} {V_{\rm ph}(1 - \eta^2) - M_{\rm A}} = 0.\end{aligned}$$ Recall that we consider the normalized wave phase velocity $V_{\rm ph}$ as a complex number and we shall look for the dependencies of the real and imaginary parts of $V_{\rm ph}$ as functions of the real dimensionless wave number $K$ at given values of the three entry parameters. It can be easily seen from above equations that they are cubic ones with respect to $V_{\rm ph}$. Hence, the dimensionless dispersion relation of, for example, the kink mode can be displayed in the form: $$A\,V_{\rm ph}^3 + B\,V_{\rm ph}^2 + C\,V_{\rm ph} + D = 0,
\label{eq:kink}$$ where $$\begin{aligned}
A &=& (1 + \eta \coth K)(1 - \eta)(1 + \eta - \varepsilon^2 K^2),\\
B &=& -M_{\rm A}\left[ 2(1 - \eta)(1 + \eta - \varepsilon^2 K^2) + (1 + \eta \coth K)(1 - \varepsilon^2 K^2) \right],\\
C &=& M_{\rm A}^2 \left[ (1 - \eta)(1 + \eta - \varepsilon^2 K^2) + 1 - \varepsilon^2 K^2 \right] - (1 + \coth K)(1 - \eta^2),\\
D &=& M_{\rm A} \left[ -M_{\rm A}^2 (1 - \varepsilon^2 K^2) + 1 + \coth K \right].\end{aligned}$$ The dimensionless dispersion relation for sausage mode is similar – one has simply to replace $\coth$ by $\tanh$.
Cubic equations will be solved on using Cardano’s formulas [@kurosh65]. First let us define a variable $f$: $$f = -\frac{1}{3}\frac{B^2}{A^2} + \frac{C}{A}.$$ Next we define $g$: $$g = \frac{2}{27}\frac{B^3}{A^3} - \frac{BC}{3A^2} + \frac{D}{A}.$$ Finally we define $h$: $$h = (f/3)^3 + (g/2)^2.
\label{eq:h}$$ If $h > 0$, there is only one real root and two complex conjugate ones. When $h \leqslant 0$, all three roots are real.
When $h < 0$, the real solutions to the cubic Eq. (\[eq:kink\]) are: $$\label{eq:v1}
V_1 = 2\sqrt{-\frac{f}{3}}\cos \frac{\alpha}{3} - \frac{B}{3A},$$ $$\label{eq:v23}
V_{2,3} = -2\sqrt{-\frac{f}{3}}\cos \left( \frac{\alpha}{3} \pm \frac{\pi}{3} \right) - \frac{B}{3A},$$ where $$\alpha = \cos^{-1}\left[-\frac{g}{2\sqrt{-(f/3)^3}}\right].$$ For simplicity, we have dropped the subscript ‘ph’ to the normalized phase velocity $V$.
When $h > 0$, the real root of Eq. (\[eq:kink\]) is given by $$\label{eq:v0}
V_0 = M + N - \frac{B}{3A},$$ where $$M = \left( -g/2 + \sqrt{h} \right)^{1/3} \quad \mbox{and} \quad N = \left( -g/2 - \sqrt{h} \right)^{1/3}.$$ The two complex roots are given by $V_{\rm r} \pm \mathrm{i}V_{\rm i}$, where $$\label{eq:Vr}
V_{\rm r} = -\frac{1}{2}(M + N) - \frac{B}{3A},$$ and $$\label{eq:Vi}
V_{\rm i} = \frac{\sqrt{3}}{2}(M - N).$$ During the numerical solving of dispersion equations (\[eq:numdisp\]) we can look what sign possesses $h$, defined by Eq. (\[eq:h\]), and in the ranges of the normalized wave numbers $K$, where $h$ is positive, we can expect complex solutions, i.e., amplification or damping of the waves due to their interaction with the flow.
Numerical results and discussion {#sec:results}
================================
Before starting the numerical solving of dispersion equations (\[eq:numdisp\]) we have to specify the entry jet’s parameters \[c.f., Eq. (\[eq:symb\])\]. As was told in the Introduction section, we take the Hall parameter $\varepsilon = 0.4$. The relative Alfvénic Mach number, $M_{\rm A}$, will be a running parameter, from $0$ (for a static flux tube) to some reasonable values. These values to some extent depend upon the choice of the third parameter, $\eta$, equal to the ratio of plasma densities outside and inside the jet. When studying Kelvin–Helmholtz instabilities of MHD waves in the coronal plume–interplume region in the framework of standard magnetohydrodynamics, Andries and Goossens [@andries01] find that in the $\beta = 0$ case one can expect that the instability will occur approximately for $|U| > v_{\rm Ao} + v_{\rm Ae}$. As we have already mentioned in Sec. \[sec:intro\], we assume that this estimation is valid for our case of an incompressible plasma jet. After normalizing all velocities with respect to the Alfvén speed inside the slab, $v_{\rm Ao}$, we get $$\frac{|U|}{v_{\rm Ao}} > 1 + \frac{v_{\rm Ae}}{v_{\rm Ao}}.$$ or $$\label{eq:eta}
\left| M_{\rm A} \right| > 1 + \frac{1}{\sqrt{\eta}}.$$ As seen from above inequality, for some small $\eta$s the relative Alfvénic Mach number, $M_{\rm A}$, might become rather large in order to register the onset of Kelvin–Helmholtz instability. Such large Mach numbers imply that for magnitudes of the Alfvén speed inside a solar wind jet in the range, say, of $60$–$100$ kms$^{-1}$ the difference between Alfvén speeds in the jet and its environment should be of the order of a few hundred kilometers per second which is unlikely to occur. Moreover, for relatively large Alfvénic Mach numbers dispersion curves of Hall-MHD surface modes become rather complicated. That is why we choose $\eta = 10$, which means that one can expect a onset of instability at $\left| M_{\rm A} \right| > 1.316$ that looks as a reasonable value.
A specific feature of the surface Hall-MHD waves travelling along an incompressible static plasma layer is that there exists a limiting dimensionless wave number $K_{\rm limit}$ beyond which the wave propagation is no longer possible. That limiting wave number is given by [@ivan96] $$\label{eq:klimit}
K_{\rm limit} = (1 + \eta)^{1/2}/\varepsilon.$$ For our choice of $\eta$ and $\varepsilon$, $K_{\rm limit} = 8.292$. With approaching that wave number the wave phase velocity becomes very large. It is interesting to see whether the steady flow will change that limiting value.
Let us first start with the kink mode. For solving the corresponding dispersion relation (\[eq:numdisp\]) we generally use the roots of the cubic equation given by Eqs. (\[eq:v1\])–(\[eq:Vi\]). We begin the numerical solving with the relative $M_{\rm A} = 0$ (corresponding to a static flux tube) running the dimensionless wave number $K$ from $0.005$ to $10$. As naturally to expect, in that wave number region $h$ is negative and we get real values for the normalized phase velocity – the corresponding dispersion curve is labeled by ‘0’ in Fig. \[fig:fg1\]. As seen from that figure, the kink wave is generally a sub-Alfvénic one; at $K \approx 7.2$ its phase velocity becomes equal to the Alfvén speed and starts quickly to grow up reaching rather large values (up to $800$ even more) whence $K \to K_{\rm limit}$. What is going on as the layer possesses any flow velocity? If we take, for instance, $M_{\rm A} = 0.5$, one can construct from the roots of the cubic equation two dispersion curves, both being real solutions, however, one curve with a positive phase velocity, and another curve with a negative one. The first dispersion curve as seen from Fig. \[fig:fg2\] is slightly above the ‘$0$’th curve, while the second dispersion curve starts with the negative value of $-0.32$ which becomes large in magnitude at $K$ approaching its critical value, and what is more interesting, it goes beyond the $K_{\rm limit}$, now decreasing in magnitude. In this extended propagation range there exists a complex solution to the cubic equation with positive imaginary part, i.e., the wave becomes unstable. However, bearing in mind that it is unlikely to observe/detect such a backward wave in a solar wind tube (supposing that the wave phase and group velocities have the same direction), we have to drop all the solutions with negative phase velocities. The dispersion curves of the second type will be discussed in another, physically acceptable situation, soon.
With increasing the relative Alfvénic Mach number the dispersion curves with positive velocities initially lie below the ‘$0$’th curve, but afterwards, for values of the normalized wave number between $3$ and $5.5$, they cross the curve with label ‘0’ staying on the left side of that curve. In other words, the flow velocity slightly diminishes the value of the $K_{\rm limit}$ without allowing them to propagate beyond those limiting $K$s. Note also that the course of the dispersion curves is not monotonous – initially, for small $M_{\rm A}$s, they lie above the neutral (‘$0$’th labeled) curve in the range of small and average dimensionless wave numbers, while for $M_{\rm A} \geqslant 1$ those curves set up below neutral dispersion curve.
For any negative value of the relative Alfvénic Mach number $M_{\rm A}$ we have as before two set of solutions. The real ones are negative and represent a mirror image of dispersion curves shown in Fig. \[fig:fg1\]. We do not plot them for the same reason; although mathematically correct they are not acceptable from a physical point of view. The most interesting case are the dispersion curves shown in Figs. \[fig:fg3\] and \[fig:fg4\]. It is clearly seen that for each $M_{\rm A}$, in fact, two distinctive dispersion curves merge at $K \approx K_{\rm limit}$. The solutions to the dispersion relation for the curves lying on the right side of the merging vertical line at some $K$s become complex with positive imaginary part, i.e., there the waves are unstable. The other important observation is that now the waves’ phase velocities do not grow too much when $K \to K_{\rm limit}$. That is especially true for the dispersion curves associated with relatively large in modulus $M_{\rm A}$s – see, for example, the dispersion curve labeled by ‘$-1.45$.’ Another important feature is the circumstance that all dispersion curves lying on the left side of the merging vertical line correspond to a stable (generally with complicated shapes of the dispersion curves) waves’ propagation! We also note that parts of the dispersion curves, corresponding to a stable wave propagation, continue smoothly beyond the $K_{\rm limit}$ – see, for example, in Fig. \[fig:fg4\] the dispersion curve labeled by ‘$-1.25$’ that ends at $kx_0 = 10$ with $v_{\rm ph}/v_{\rm Ao} = 0.13$. It turns out that the Hall current makes the waves stable in the wave number propagation range between $0$ and $K_{\rm limit}$ – an instability onset is only possible at some critical wave numbers larger than $K_{\rm limit}$. The growth rates of kink waves in the instability region are plotted in Fig. \[fig:fg5\]. We would like to emphasize that all complex solutions were checked for some selected dimensionless wave numbers on using the complex versions of Newton–Raphson and Müller [@muller56] methods.
The case with sausage Hall-MHD waves is similar in many ways, although there are some specific features. First, for positive relative Alfvénic Mach numbers the real solutions in the long-wavelength limit (small $K$s) are much more complicated. That can be seen in Figs. \[fig:fg6\] and \[fig:fg7\]. The most striking issue is the existence of a bordering dispersion curve (with label ‘1.315’) which divides the rest dispersion curves into two types. The dispersion curves with $M_{\rm A} < 1.315$ begin as super-Alfvénic waves with decreasing phase velocities which passing through a minimum start to grow and at around $K \approx 5$ cross the neutral dispersion curve (that corresponding to a static slab). After that they quickly increase their magnitudes reaching very large values at $K \to K_{\rm limit}$. The second type of dispersion curves consist of two families of curves: ones at very narrow regions of small $K$s, and others starting with negligibly small negative phase velocities which later on passing through inverted “s-shaped” parts continuously increase their speeds reaching large values at $K \to K_{\rm limit}$. It is worth noticing that the second type’s dispersion curves possess multiple values at a fixed $K$ in the range between $0$ and $1.22$.
The most intriguing question is how the dispersion curves will behave as $M_{\rm A}$ is negative? The answer is illustrated in Figs. \[fig:fg8\] and \[fig:fg9\]. Actually there is no a big surprise – more or less the dispersion curves are similar to those of the kink mode. However, we should immediately notice two differences: (i) the dramatic changes in the shapes of the dispersion curves start at $M_{\rm A} = -1.3$ (vs. $-1.25$ for the kink mode), and (ii) parts of dispersion curves for $M_{\rm A} \leqslant -1.3$ with dimensionless wave number between $0$ and $8.2$ are negative. All these forward and backward sausage waves are stable. Unstable are only those waves (like for the case of kink waves) whose dispersion curves are on the right of the merging vertical line (look at Fig. \[fig:fg8\]). The growth rates of such unstable sausage Hall-MHD modes are shown in Fig. \[fig:fg10\] and each of them starts at some critical normalized wave number.
We should recall that our theory is a linear one and since at the instability onset the waves amplitudes begin rising, for a further waves’ evolution one must employ a nonlinear approach. Nevertheless, results, obtained here, can be used as a start point for a deeper investigation of the wave propagation in flowing solar flux-tube plasmas in the framework of the Hall magnetohydrodynamic.
An interesting issue which springs to mind is how the instability growth rate depends on the value of the entry parameter $\eta$, say, at a fixed dimensionless wave number. Let us do such an examination for the kink mode and let our choice for the fixed wave number be (see Fig. \[fig:fg5\]) $kx_0 = 8.5$. The wave growth rate depends not only on $K$ but also on the relative Alfvénic Mach number $M_{\rm A}$. In Fig. \[fig:fg11\] we show a family of curves depicting the dependence of the normalized wave growth rate as a function of the plasma densities ratio $\eta = \rho_{\rm e}/\rho_{\rm o}$ for various Alfvénic Mach numbers between $-1.45$ and $-0.5$. As seen, one observes two local maxima: one at $\eta = 1$ and another around $\eta \sim 10$. The first maximum is not surprising – it is well known that the Kelvin–Helmholtz instability is easily excited when the densities of the two adjacent flowing media are approximately the same. The only exception here is the curve corresponding to $M_{\rm A} = -0.5$ – a maximum of the wave growth rate for that value of the Alfvénic Mach number one can expect for values of $\eta$ bigger than $10$. The second local maxima are obviously depending on the magnitude of the relative Alfvénic Mach number $M_{\rm A}$.
Another curious question is how the propagation and stability properties of the Hall-MHD surface waves change with the value of $\eta$. If we take $\eta = 4$, the dispersion curves of kink waves for negative values of the relative Alfvénic Mach number are plotted in Fig. \[fig:fg12\]. As seen, the picture is similar to that shown in Fig. \[fig:fg4\], however, with an distinctive feature, notably the curves corresponding to $M_{\rm A} < -1$ represent stable wave propagation even on the right side of the vertical merging line (look at curves labeled by ‘$-1.25$’ and ‘$-1.5$’, respectively). This really surprising observation indicates that the Hall-MHD kink surface waves for that value of $\eta$ ($=4$) are unstable only for negative relative Alfvénic Mach numbers $M_{\rm A} \geqslant -1$ – their normalized growth rates are plotted in Fig. \[fig:fg13\]. This example shows us that we must be very cautious with stating some very general criteria for the Kelvin–Helmholtz instability onset with Hall-MHD waves propagating in flowing solar flux-tube plasmas.
Conclusion and outlook {#sec:concl}
======================
Let us now summarize the basic results obtained in this study. In investigating the wave propagation along a jet moving with respect to the environment with a constant speed $U$ we had to take into account the influence of two factors: (i) the Hall term in the generalized Ohm’s law, and (ii) the flow itself. The combining effect of these two factors can be expressed as follows:
- The Hall term generally limits the range of propagation of the wave modes not only for static tubes/layers but also for jets with positive Alfvénic Mach numbers. The limiting normalized wave number, $K_{\rm limit}$, is specified by two plasma parameters: the densities ratio of the two plasma media (outside and inside the jet), $\eta$, and the Hall parameter, $\varepsilon$ \[look at Eq. (\[eq:klimit\])\]. If this is the exact value for the waves propagating on a static flux tube, it is approximately the same for the waves travelling along a jet. One should emphasize that all dispersion curves, at relatively large $K$s, lie on the left to the dispersion curve corresponding to $M_{\rm A} = 0$ (see Figs. \[fig:fg1\] and \[fig:fg6\]). However, when $M_{\rm A}$ becomes negative the real part of the phase velocity of the eigenmodes (kink and sausage waves) is forced to go beyond that limiting wave number in a region where the wave becomes unstable (or if you prefer, overstable). The instability which occurs should be of Kelvin–Helmholtz type. It is rather surprising that the instability onset starts with relatively large growth rates gradually decreasing with increasing the modulus of the Alfvénic Mach number (look at Figs. \[fig:fg5\] and \[fig:fg10\]). We note that such growth rates in the extended range of the waves’ propagation was obtained for the same plasma-jet configuration for a different value of the parameter $\eta$ ($= 0.64$) [@ivan07]. It is worth mentioning that for negative Alfvénic Mach numbers we actually have (for each mode) two dispersion curves merging at $K_{\rm limit}$, but unstable for negative $M_{\rm A}$ are the dispersion curves situated on the right to the cusp (see Fig. \[fig:fg3\] and \[fig:fg8\]). Our conclusion that the Kelvin–Helmholtz instability onset is only possible for negative relative Alfvénic Mach numbers is in agreement with a similar inference of Andries and Goossens [@andries01].
- It seems that the critical relative Alfvénic Mach number at which the Kelvin–Helmholtz instability starts, given by Eq. (\[eq:eta\]), is unfortunately not applicable, as a rough estimation, in the Hall magnetohydrodynamics. Notwithstanding, it is still useful because yields that value of the negative Alfvénic Mach number which is associated with a dramatic change in the shape of the waves’ dispersion curves.
- A safely general conclusion is that the Hall current keeps stable the surface modes travelling in flowing solar plasmas within the dimensionless wave number range between $0$ and $K_{\rm limit}$ for each relative Alfvénic Mach number. An instability of the Kelvin–Helmholtz type is possible only at negative Alfvénic Mach numbers in a wave number range lying beyond the $K_{\rm limit}$ and it (the instability) starts at some critical normalized wave number depending on the magnitude of $M_{\rm A}$. The maximum instability growth rate is largest for small in magnitude Alfvénic Mach numbers gradually decreasing with the increase of $\left|M_{\rm A}\right|$. The instability can turn off as $\left|M_{\rm A}\right|$ reaches some value depending on the magnitude of the parameter $\eta$ (look at Figs. \[fig:fg12\] and \[fig:fg13\]).
This study can be extended in two directions. The first one is to consider finite-valued sound speeds. In that case, however, the waves’ dispersion equations become rather complicated and they can be solved (looking for complex roots) only numerically which is a highly difficult task. Any way, one can expect that the plasma compressibility will not substantially change the Kelvin–Helmholtz instability pictures. Moreover, one can state that the sausage and kink waves represented in Figs. 5 and 7 in Ref. [@rossi04] are definitely stable because with $\eta \approx 0.6$ and $\varepsilon = 0.4$ the value of $K_{\rm limit}$, beyond which one can expect the instability onset, is $3.16$, while the waves’ propagation in that paper was examined till $kx_0 = 2.5$ only. The second direction is to study a more realistic geometry, for instance, cylindrical one, and conduct the investigations with appropriate observable plasma and magnetic field parameters. This is in progress and will be reported elsewhere.
We do believe that the present results might be useful in studying wave turbulence in the solar wind as well as in solving other problems associated with wave propagation in structured/spatially bounded magnetized plasmas.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author would like to thank Michael Ruderman for useful discussions and advices, as well as the referee for his valuable critical remarks and helpful suggestions.
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![(Online colour) Dispersion curves of kink Hall-MHD waves travelling along an incompressible flowing plasma layer for positive values of the relative Alfvénic Mach number $M_{\rm A}$ – all waves are stable. For details in curves’ labeling see Fig. \[fig:fg2\]. \[fig:fg1\]](izh-epjdfg1){height=".30\textheight"}
![(Online colour) A zoom of the bottom part of Fig. \[fig:fg1\]. The dispersion curves sandwiched between curves labeled by ‘1.25’ and ‘1.45’ correspond to $M_{\rm A}$ equal to $1.3$, $1.35$, and $1.4$, respectively. \[fig:fg2\]](izh-epjdfg2){height=".30\textheight"}
![(Online colour) Dispersion curves of kink Hall-MHD waves travelling along an incompressible flowing plasma layer for negative values of the relative Alfvénic Mach number $M_{\rm A}$. All the curves lying on the left side of the vertical merging line/cusp represent stable waves’ propagation. The waves become unstable only on the family of curves labeled by ‘$-0.5$’ and ‘$-1.45$’. For details in curves’ labeling see Fig. \[fig:fg4\]. \[fig:fg3\]](izh-epjdfg3){height=".30\textheight"}
![(Online colour) A zoom of the bottom part of Fig. \[fig:fg3\]. The dispersion curves situated between the curves labeled by ‘$-1.25$’ and ‘$-1.45$’ correspond to $M_{\rm A}$ equal to $-1.3$, $-1.35$, and $-1.4$, respectively. \[fig:fg4\]](izh-epjdfg4){height=".30\textheight"}
![(Online colour) Growth rates of unstable kink Hall-MHD waves travelling along an incompressible flowing plasma layer for negative values of the relative Alfvénic Mach number $M_{\rm A}$ in the short-wavelength region (beyond the $K_{\rm limit}$). \[fig:fg5\]](izh-epjdfg5){height=".30\textheight"}
![(Online colour) Dispersion curves of sausage Hall-MHD waves travelling along an incompressible flowing plasma layer for positive values of the relative Alfvénic Mach number $M_{\rm A}$ – all waves are stable. For details in curves’ labeling see Fig. \[fig:fg7\]. \[fig:fg6\]](izh-epjdfg6){height=".30\textheight"}
![(Online colour) A zoom of the bottom part of Fig. \[fig:fg6\]. The dispersion curve corresponding to $M_{\rm A} = 1.315$ divides the rest curves into two types, notably “simple” dispersion curves (for $M_{\rm A} < 1.315$) and much complex families of dispersion curves (for $M_{\rm A} > 1.315$). \[fig:fg7\]](izh-epjdfg7){height=".30\textheight"}
![(Online colour) Dispersion curves of sausage Hall-MHD waves travelling along an incompressible flowing plasma layer for negative values of the relative Alfvénic Mach number $M_{\rm A}$. All the curves lying on the left side of the vertical merging line/cusp represent stable waves’ propagation. The waves become unstable only on the family of curves labeled by ‘$-0.5$’ and ‘$-1.45$’. For details in curves’ labeling see Fig. \[fig:fg9\]. \[fig:fg8\]](izh-epjdfg8){height=".30\textheight"}
![(Online colour) A zoom of the bottom part of Fig. \[fig:fg8\]. The dispersion curves sandwiched between the curves labeled by ‘$-1.3$’ and ‘$-1.45$’ correspond to $M_{\rm A}$ equal to $-1.35$ and $-1.4$, respectively. The curves situated at the left bottom corner of the dispersion diagram represent backward stable sausage surface waves. \[fig:fg9\]](izh-epjdfg9){height=".30\textheight"}
![(Online colour) Growth rates of unstable sausage Hall-MHD waves travelling along an incompressible flowing plasma layer for negative values of the relative Alfvénic Mach number $M_{\rm A}$ in the short-wavelength region (beyond the $K_{\rm limit}$). \[fig:fg10\]](izh-epjdfg10){height=".30\textheight"}
![(Online colour) Dependence of the normalized instability growth rate on the ratio $\rho_{\rm e}/\rho_{\rm o} = \eta$ at a fixed dimensionless wave number ($kx_0 = 8.5$) for various relative Alfvénic Mach numbers in the range between $-1.45$ and $-0.5$ The beginning of the horizontal axis starts at $\eta = 0.2$. The family of curves sandwiched between the curves labeled by ‘$-1.25$’ and ‘$-1.45$’ correspond to relative Alfvénic Mach numbers equal to $-1.3$, $-1.35$, and $-1.4$, respectively. \[fig:fg11\]](izh-epjdfg11){height=".30\textheight"}
![(Online colour) Dispersion curves of kink Hall-MHD waves travelling along an incompressible flowing plasma layer for negative values of the relative Alfvénic Mach number $M_{\rm A}$ and $\eta = 4$. All the curves lying on the left side of the vertical merging line/cusp represent stable waves’ propagation. The waves become unstable only on the family of curves labeled by ‘$-0.25$’ and ‘$-1$’. The curves with labels ‘$-1.25$’ and ‘$-1.5$’ represent stable waves’ propagation. \[fig:fg12\]](izh-epjdfg12){height=".30\textheight"}
![(Online colour) Growth rates of unstable kink Hall-MHD waves travelling along an incompressible flowing plasma layer for negative values of the relative Alfvénic Mach number $M_{\rm A}$ and $\eta = 4$ in the short-wavelength region (beyond the $K_{\rm limit} = 5.59$). \[fig:fg13\]](izh-epjdfg13){height=".30\textheight"}
[^1]: *e-mail:* izh@phys.uni-sofia.bg
|
---
author:
-
title: HVDC loss factors in the Nordic Market
---
Electricity markets, HVDC losses, HVDC transmission, loss factors, market operation, Nordics, power losses, zonal pricing.
Introduction
============
Formulation {#sec:2}
===========
Nordic test case {#sec:3}
================
Numerical simulations {#sec:4}
=====================
Conclusion {#sec:5}
==========
|
[**[Stability of multi antipeakon-peakons profile]{}**]{}
0.2cm
[**Khaled El Dika$^\sharp$ and Luc Molinet$^\clubsuit$**]{}\
[$\sharp$ L.A.G.A., Institut Galilée, Université Paris-Nord,\
93430 Villetaneuse, France.\
$\clubsuit$ L.M.P.T., UFR Sciences et Techniques, Université de Tours, Parc Grandmont, 37200 Tours, FRANCE.]{} 0.3cm khaled@math.univ-paris13.fr\
Luc.Molinet@lmpt.univ-tours.fr
0.5cm [**Abstract.**]{} [ The Camassa-Holm equation possesses well-known peaked solitary waves that can travel to both directions. The positive ones travel to the right and are called peakon whereas the negative ones travel to the left and are called antipeakons. Their orbital stability has been established by Constantin and Strauss in [@CS1]. In [@EL2] we have proven the stability of trains of peakons. Here, we continue this study by extending the stability result to the case of ordered trains of anti-peakons and peakons.]{}\
Introduction
============
The Camassa-Holm equation (C-H), $$u_t -u_{txx}=- 3 u u_x +2 u_x u_{xx} + u u_{xxx}, \quad
(t,x)\in{{ I\!\!R}}^2, \label{CH}$$ can be derived as a model for the propagation of unidirectional shalow water waves over a flat bottom by writing the Green-Naghdi equations in Lie-Poisson Hamiltonian form and then making an asymptotic expansion which keeps the Hamiltonian structure ([@CH1], [@Johnson]). Note that the Green-Naghdi equations arise as approximations to the governing equations for shallow-water medium-amplitude regime which captures more nonlinear effects than the classical shallow-water small amplitude KdV regime and thus can accommodate models for breaking waves (cf. [@AL], [@CL], [@CE]). The Camassa-Holm equation was also found independently by Dai [@dai] as a model for nonlinear waves in cylindrical hyperelastic rods and was, actually, first discovered by the method of recursive operator by Fokas and Fuchsteiner [@FF] as an example of bi-Hamiltonian equation. Let us also mention that it has also a geometric derivation as a re-expression of geodesic flow on the diffeomorphism group on the line (cf. [@K1], [@K2]) and that this framework is instrumental in showing that the Least Action Principle holds for this equation (cf. [@C], [@CK]).
(C-H) is completely integrable (see [@CH1],[@CH2], [@C1] and [@CGI]). It possesses among others the following invariants $$E(v)=\int_{{{ I\!\!R}}} v^2(x)+v^2_x(x)
\, dx \mbox{ and } F(v)=\int_{{{ I\!\!R}}} v^3(x)+v(x)v^2_x(x) \, dx\;
\label{E}$$ and can be written in Hamiltonian form as $$\partial_t E'(u) =-\partial_x F'(u) \quad .$$ Camassa and Holm [@CH1] exhibited peaked solitary waves solutions to (C-H) that are given by $$u(t,x)=\varphi_c(x-ct)=c\varphi(x-ct)=ce^{-|x-ct|},\; c\in{{ I\!\!R}}.$$ They are called peakon whenever $ c>0 $ and antipeakon whenever $c<0$. Let us point out here that the feature of the peakons that their profile is smooth, except at the crest where it is continuous but the lateral tangents differ, is similar to that of the waves of greatest height, i.e. traveling waves of largest possible amplitude which are solutions to the governing equations for water waves (cf. [@C2], [@CE2] and [@T]). Note that (C-H) has to be rewriten as $$u_t +u u_x +(1-\partial_x^2)^{-1}\partial_x (u^2+u_x^2/2)=0
\label{CH3} \; .$$ to give a meaning to these solutions. Their stability seems not to enter the general framework developed for instance in [@Benjamin], [@GSS]. However, Constantin and Strauss [@CS1] succeeded in proving their orbital stability by a direct approach. In [@EL2] we combined the general strategy initiated in [@MMT](note that due to the reasons mentioned above, the general method of [@MMT] is not directly applicable here ), a monotonicity result proved in [@EL] on the part of the energy $ E(\cdot) $ at the right of a localized solution traveling to the right and localized versions of the estimates established in [@CS1] to derive the stability of ordered trains of peakons. In this work we pursue this study by proving the stability of ordered trains of anti-peakons and peakons. The main new ingredient is a monotonicity result on the part of the functional $ E(\cdot)-\lambda F(\cdot) $, $ \lambda\ge 0 $, at the right of a localized solution traveling to the right. It is worth noticing that the sign of $ \lambda $ plays a crucial role in our analysis.
Before stating the main result let us introduce the function space where we will define the flow of the equation. For $ I $ a finite or infinite interval of $ {{ I\!\!R}}$, we denote by $ Y(I) $ the function space[^1] $$Y(I):= \Bigl\{ u\in C(I;H^1({{ I\!\!R}})) \cap L^\infty(I;W^{1,1}({{ I\!\!R}})), \;
u_x\in L^\infty(I; BV({{ I\!\!R}}))\Bigr\} \, . \label{theoweak}$$ In [@CM1], [@dan1] and [@E] (see also [@L]) the following existence and uniqueness result for this class of initial data is derived.
\[wellposedness\] Let $ u_0\in H^1({{ I\!\!R}}) $ with $ m_0:=u_0-u_{0,xx} \in {\cal M}({{ I\!\!R}})
$ then there exists $ T=T(\|m_0\|_{\cal M})>0 $ and a unique solution $ u \in Y([-T,T]) $ to (C-H) with initial data $ u_0 $. The functionals $ E(\cdot) $ and $ F(\cdot) $ are constant along the trajectory and if $ m_0 $ is such that [^2] $ \supp m_0^- \subset ]-\infty, x_0] $ and $ \supp m_0^+ \subset [x_0,+\infty[ $ for some $ x_0\in {{ I\!\!R}}$ then $ u $ exists for all positive times and belongs to $ Y([0,T]) $ for all $ T>0 $.\
Moreover, let $ \{u_{0,n}\}\subset H^1({{ I\!\!R}}) $ such that $ u_{0,n}\to u_0 $ in $ H^1({{ I\!\!R}}) $ with $
\{m_{0,n}:=u_{0,n}-\partial^2_x u_{0,n} \} $ bounded in $ {\cal M} ({{ I\!\!R}}) $, $ \supp m_{0,n}^- \subset ]-\infty, x_{0,n}] $ and $ \supp m_{0,n}^+ \subset [x_{0,n},+\infty[ $ for some sequence $ \{x_{0,n}\}\subset {{ I\!\!R}}$. Then, for all $ T> 0
$, $$u_n \longrightarrow u \mbox{ in } C([0,T]; H^1({{ I\!\!R}})) \; .$$
Let us emphasize that the global existence result when the negative part of $ m_0 $ lies completely to the left of its positive part is proven in [@E] and that the last assertion of the above theorem is not explicitly contained in this paper. However, following the same arguments as those developed in these works (see for instance Section 5 of [@L]), one can prove that there exists a subsequence $ \{u_{n_k}\} $ of solutions of [(\[CH\])]{} that converges in $ C([0,T]; H^1({{ I\!\!R}})) $ to some solution $ v$ of [(\[CH\])]{} belonging to $ Y([0,T[) $. Since $ u_{0,n_k} $ converges to $ u_0 $ in $ H^1 $, it follows that $ v(0)=u_0 $ and thus $ v=u $ by uniqueness. This ensures that the whole sequence $ \{u_n\} $ converges to $ u $ in $C([0,T]; H^1({{ I\!\!R}}))$ and concludes the proof of the last assertion.
It is worth pointing out that recently, in [@BC1] and [@BC2], Bressan and Constantin have constructed global conservative and dissipative solutions of the Camassa-Holm equation for any initial data in $ H^1({{ I\!\!R}}) $. However, even if for the conservative solutions, $ E(\cdot) $ and $ F(\cdot) $ are conserved quantities, these solutions are not known to be continuous with values in $ H^1({{ I\!\!R}}) $. Therefore even one single peakon is not known to be orbitally stable in this class of solutions. For this reason we will work in the class of solutions constructed in Theorem \[wellposedness\].
We are now ready to state our main result.
\[mult-peaks\] Let be given $ N $ non vanishing velocities $c_1<..<c_{k}<0<c_{k+1}<..< c_N $ . There exist $ \gamma_0 $, $
A>0 $, $ L_0>0 $ and $ \varepsilon_0>0 $ such that if $ u \in Y([0,T[)$, with $ 0<T\le \infty $, is a solution of (C-H) with initial data $ u_0 $ satisfying $$\|u_0-\sum_{j=1}^N \varphi_{c_j}(\cdot-z_j^0) \|_{H^1} \le \varepsilon^2 \label{ini}$$ for some $ 0<\varepsilon<\varepsilon_0$ and $ z_j^0-z_{j-1}^0\ge L$, with $ L>L_0 $, then there exist $x_1(t), ..,x_N(t) $ such that $$\sup_{[0,T[} \|u(t,\cdot)-\sum_{j=1}^N
\varphi_{c_j}(\cdot-x_j(t)) \|_{H^1} \le
A(\sqrt{\varepsilon}+L^{-{1/8}})\; .\label{ini2}$$ Moreover there exists $ C^1 $-functions $ {\tilde x_1}, ..,
{\tilde x_N} $ such that, $ \forall j\in \{1,..,N\} $, $$\label{gy}
|x_j(t)-{\tilde x_{j}}(t)|=O(1) \mbox{ and }
\frac{d}{d t} {\tilde x_j}(t) = c_j +O(\varepsilon^{1/4}) +O(L^{-\frac{1}{16}}), \forall t\in[0,T[ \; .$$
\[rem1\] We do not know how to prove the monotonicity result in Lemma \[monotonicitylem\], and thus Theorem \[mult-peaks\], for solutions that are only in $ C([0,T[;H^1({{ I\!\!R}})) $ which is the hypothesis required for the stability of a single peakon (cf. [@CS1]). Note anyway that there exists no well-posedness result in the class $ C([0,T[;H^1({{ I\!\!R}})) $ for general initial data in $ H^1({{ I\!\!R}}) $. On the other hand, according to Theorem \[wellposedness\] above, $ u\in Y([0,T[) $ as soon as $u_0 \in
H^1({{ I\!\!R}}) $ and $ (1-\partial_x^2) u_0 $ is a Radon measure with bounded variations.
Note that under the hypotheses of Theorem \[mult-peaks\], $$\sum_{j=1}^N \varphi_{c_j}(\cdot-z_j^0)$$ belongs to the class $ v\in H^1({{ I\!\!R}}) $with $m:=v-v_{xx} \in {\cal M}({{ I\!\!R}}) $, $ \supp m^- \subset ]-\infty, x_0] $ and $ \supp m^+ \subset [x_0,+\infty[ $ for some $ x_0\in {{ I\!\!R}}$. Therefore, in view of Theorem \[wellposedness\], Theorem \[mult-peaks\] leads to the orbital stability (for positive times) of such ordered sum of antipeakons and peakons with respect to $ H^1 $-perturbations that keep the initial data in this same class.
As discovered by Camassa and Holm [@CH1], (C-H) possesses also special solutions called multipeakons given by $$u(t,x)=\sum_{i=1}^N p_j(t) e^{-|x-q_j(t)|} \,,$$ where $ (p_j(t),q_j(t)) $ satisfy the differential system : $$\label{systH}
\Bigl\{
\begin{array}{l}
\dot{q}_i=\sum_{j=1}^N p_j e^{-|q_i-q_j|} \\
\dot{p}_i = \sum_{j=1}^N p_i p_j \sgn (q_i-q_j) e^{-|q_i-q_j|}\; .
\end{array}
\Bigr.$$ In [@Beals] (see also [@Beals0] and [@CH1]), the asymptotic behavior of the multipeakons is studied. In particular, the limits as $ t $ tends to $ +\infty $ and $ -\infty $ of $ p_i(t) $ and $ \dot{q_i}(t) $ are determined. Combining these asymptotics with the preceding theorem and the continuity with respect to initial data stated in Theorem \[wellposedness\] we get the following result on the stability for positive times of the variety $ {\cal N}_{N,k} $ of $ H^1({{ I\!\!R}}) $ defined for $ N\ge 1 $ and $ 0\le k \le N $ by $${\cal N}_{N,k}:= \Bigl\{ v=\sum_{i=1}^N p_j e^{-|\cdot-q_j|}, \,
(p_1,..,p_{N})\in ({{ I\!\!R}}_-^*)^k\times ({{ I\!\!R}}_+^*)^{N-k} ,\,
q_1<q_2<..<q_N \,\Bigr\} \; .$$
\[cor-mult-peaks\] Let be given $ k $ negative real numbers $ p_1^0,.., p_{k}^0 $, $
N-k $ positive real numbers $ p_{k+1}^0,..,p_N^0 $ and $ N $ real numbers $ q_1^0< ..< q_N^0 $. For any $ B> 0 $ and any $ \gamma
>0 $ there exists $ \alpha>0 $ such that if $ u_0\in H^1({{ I\!\!R}}) $ is such that $ m_0:=u_0-u_{0,xx} \in
{\mathcal M}({{ I\!\!R}}) $ with $ \supp m_0^- \subset ]-\infty, x_0] $ and $ \supp m_0^+ \subset [x_0,+\infty[ $ for some $ x_0\in {{ I\!\!R}}$, and satisfies $$\|m_0\|_{\cal M}\le B \quad \mbox{ and }\quad
\|u_0-\sum_{j=1}^N p_j^0 \exp (\cdot-q_j^0) \|_{H^1}\le \alpha
\label{ini3}$$ then $$\forall t\in{{ I\!\!R}}_+, \quad \inf_{P\in {{ I\!\!R}}_-^k\times{{ I\!\!R}}_+^{N-k},Q\in {{ I\!\!R}}^N}
\|u(t,\cdot)-\sum_{j=1}^N p_j \exp (\cdot-q_j) \|_{H^1} \le
\gamma\; . \label{ini33}$$ Moreover, there exists $ T>0 $ such that $$\forall t\ge T, \quad \inf_{Q\in {\mathcal G}} \|u(t,\cdot)-\sum_{j=1}^N \lambda_j \,\exp
(\cdot-q_j) \|_{H^1} \le \gamma \label{ini4}$$ where $ {\mathcal G}:=\{Q\in {{ I\!\!R}}^N, \, q_1<q_2<..<q_N\} $ and $
\lambda_1<..<\lambda_N $ are the eigenvalues of the matrix $ \Bigl( p_j^0 e^{-|q_i^0-q_j^0|/2}\Bigr)_{1\le i,j\le N}
$.
Again, note that for $(p_1^0,..,p_N^0)\in ({{ I\!\!R}}_-^*)^k\times({{ I\!\!R}}_+^*)^{N-k}$ and $ q_1^0<..<q_N^0 $, $$\sum_{j=1}^N p_j^0 \exp (\cdot-q_j^0)$$ belongs to the class $ v\in H^1({{ I\!\!R}}) $with $m:=v-v_{xx} \in {\cal M}({{ I\!\!R}}) $, $ \supp m^- \subset ]-\infty, x_0] $ and $ \supp m^+ \subset [x_0,+\infty[ $ for some $ x_0\in {{ I\!\!R}}$. Corollary \[cor-mult-peaks\] thus ensures that the variety $ {\cal N}_{N,k} $ is stable with respect to $ H^1$-perturbations that keeps the initial data in this same class.
This paper is organized as follows. In the next section we sketch the main points of the proof of Theorem \[mult-peaks\] whereas the complete proof is given in Section \[sec-multi\]. After having controlled the distance between the different bumps of the solution we establish the new monotonicity result and state local versions of estimates involved in the stability of a single peakon. Finally, the proof of Theorem \[mult-peaks\] is completed in Subsection \[end-proof\].
Sketch of the proof {#section1peak}
===================
Our proof as in [@MMT] combined the stability of a single peakon and a monotonicity result for functionals related to the conservation laws. Recall that the stability proof of Constantin and Strauss (cf. [@CS1]) is principally based on the following lemma of $\cite{CS1}$.
\[1peakon-lemme\] For any $u\in H^1({{ I\!\!R}})$, $ c\in {{ I\!\!R}}$ and $\xi\in{{ I\!\!R}}$, $$\label{eq1}
E(u)-E(\varphi_c)= \|u-\varphi_c(\cdot-\xi)\|^2_{H^1}+4c(u(\xi)-c).$$ For any $u\in H^1({{ I\!\!R}})$, let $M=\max_{x\in{{ I\!\!R}}}\{u(x)\}$, then $$\label{eq2}
M E(u)-F(u)\geq \frac{2}{3}M^3.$$
Indeed, with this lemma at hand, let $u\in C([0, T[ ; H^1({{ I\!\!R}}))$ be a solution of ${(\ref{CH})}$ with $\|u(0)-\varphi_c\|_{H^1}\leqslant\varepsilon^2$ and let $\xi(t)\in{{ I\!\!R}}$ be such that $u(t,\xi(t))=\max_{{{ I\!\!R}}} u(t,\cdot) $. Assuming that $ u(t)$ is sufficiently $ H^1 $-close to $ \{r\in {{ I\!\!R}}, \varphi(\cdot-r)\} $, setting $\delta=c-u(t,\xi(t))$, and using that $ E(u(t))=E(u_0)=2c^2+O(\varepsilon^2) $ and $ F(u(t))=F(u_0)=\frac{4}{3}c^3+O(\varepsilon^2) $, [(\[eq2\])]{} leads to $$\delta^2 (c-\delta/3)\le O(\varepsilon^2))\Longrightarrow \delta \lesssim \varepsilon$$ and then [(\[eq1\])]{} yields $$\label{eq33}
\|u(t)-\varphi_c(\cdot-\xi(t))\|_{H^1}\lesssim \sqrt{\varepsilon} \,.$$ This proves the stability result. At this point, a crucial remark is that, instead of using the conservation of $ E $ and $F $, we can only use that, for any fixed $ \lambda \ge 0 $, $ E(\cdot)-\lambda F(\cdot) $ is non increasing. Indeed, for $M=\max_{x\in{{ I\!\!R}}}\{u(t,x)\}=u(t,\xi(t))$ and $ \lambda=1/M $, [(\[eq2\])]{} then implies $$M E(u_0)-F(u_0)\ge \frac{2}{3} M^3$$ and, for $ \lambda=0 $, [(\[eq1\])]{} implies $$E(u_0)-E(\varphi_c)\ge \|u-\varphi_c(\cdot-\xi(t))\|^2_{H^1}+4c(u(\xi(t))-c).$$ This leads to [(\[eq33\])]{} exactly as above.\
Now, in [@EL2] it is established that [(\[eq1\])]{} and [(\[eq2\])]{} almost still hold if one replaces $ E(\cdot) $ and $ F(\cdot) $ by their localized version, $ E_j(\cdot) $ and $ F_j(\cdot) $, around the j$th $ bump. Therefore to prove our result it will somehow suffices to prove that the functionals $ E_j(\cdot) +\lambda F_j(\cdot) $ are almost decreasing.
One of the very important discovering of the works of Martel-Merle is that for one dimensional dispersive equations with a linear group that travels to the left, the part of the energy at the right of a localized solution traveling to the right is almost decreasing. In [@MMT] it is noticed that this holds also for the part of the energy at the right of each bump for solutions that are close to the sum of solitary waves traveling to the right. In this paper we will use that, for a fixed $ \lambda\ge 0$ and $ j\ge k+1 $, if we call by $ I_j =\sum_{q=j}^N ( E_q-\lambda F_q)$ the part of the functionals $ E(\cdot)-\lambda F(\cdot) $ that is at the right of the (j-1)$th
$ bumps, then $ I_j(\cdot) $ is almost decreasing in time. Since $ I_N=E_N-\lambda F_N $, we infer from above that the $ Nth $ bump of the solution stays $H^1 $-close to a translation of $\varphi_{c_N} $. Then, since $ I_{N-1} =E_{N-1}-\lambda F_{N-1}+I_N$ and $I_{N-1} $ is almost decreasing, we obtain that $ E_{N-1}-\lambda F_{N-1} $ is also almost decreasing which leads to the stability result for the $ (N-1)th $ bump. Iterating this process until $ j=k+1 $, we obtain that each bump moving to the right remains close to the orbit of the suitable peakon. Finally, since (C-H) is invariant by the change of unknown $
u(t,x)\to -u(t,-x) $, this also ensures that each bump moving to the left remains close to the orbit of the suitable antipeakon. This leads to the desired result since the total energy is conserved.
Actually we will not proceed exactly that way since by using such iterative process one loses some power of $\varepsilon$ at each step. More precisely this iterative scheme would prove Theorem \[mult-peaks\] but with $
\varepsilon^\beta $ with $ \beta =4^{1/2-\max(q, N-q)} $ instead of $\varepsilon^{1/2} $ in [(\[ini2\])]{}. To derive the desired power of $\varepsilon$ we will rather sum all the contributions of bumps that are traveling in the same direction and use Abel’s summation argument to get the stability of all these bumps in the same time.
Stability of multipeakons {#sec-multi}
=========================
For $ \alpha>0 $ and $ L>0 $ we define the following neighborhood of all the sums of N antipeakons and peakons of speed $ c_1,..,c_N $ with spatial shifts $ x_j $ that satisfied $ x_j-x_{j-1}\ge L $. $$U(\alpha,L) = \Bigl\{
u\in H^1({{ I\!\!R}}), \, \inf_{x_j-x_{j-1}> L} \|u-\sum_{j=1}^N \varphi_{c_j} (\cdot-x_j) \|_{H^1} < \alpha \Bigr\}\; .$$
By the continuity of the map $ t\mapsto u(t) $ from $ [0,T[ $ into $ H^1({{ I\!\!R}}) $, to prove the first part of Theorem \[mult-peaks\] it suffices to prove that there exist $ A>0 $, $ \varepsilon_0>0 $ and $ L_0>0 $ such that $ \forall L>L_0 $ and $ 0<\varepsilon<\varepsilon_0 $, if $u_0$ satisfies [(\[ini\])]{} and if for some $ 0<t_0< T $, $$\label{e11}
u(t)\in U\left(A(\sqrt{\varepsilon}+L^{-{1/8}}),L/2\right) \textrm{ for all }t\in[0,t_0]$$ then $$\label{e12}
u(t_0) \in U\left(\frac{A}{2}(\sqrt{\varepsilon}+L^{-{1/8}}),\frac{2L}{3}\right).$$ Therefore, in the sequel of this section we will assume [(\[e11\])]{} for some $ 0<\varepsilon<\varepsilon_0 $ and $ L>L_0 $, with $ A$, $ \varepsilon_0 $ and $ L_0 $ to be specified later, and we will prove [(\[e12\])]{}.
Control of the distance between the peakons
-------------------------------------------
In this subsection we want to prove that the different bumps of $ u $ that are individualy close to a peakon or an antipeakon get away from each others as time is increasing. This is crucial in our analysis since we do not know how to manage strong interactions. The following lemma is principally proven in [@EL2].
\[eloignement\] Let $ u_0 $ satisfying [(\[ini\])]{}. There exist $\alpha_0>0$, $L_0>0$ and $C_0>0$ such that for all $0<\alpha<\alpha_0$ and $0<L_0<L$ if $u\in U(\alpha, L/2) $ on $ [0,t_0] $ for some $
0<t_0< T $ then there exist $ C^1
$-functions $ {\tilde x_1}, .., {\tilde x_N} $ defined on $
[0,t_0] $ such that $\forall t\in [0,t_0] $, $$\frac{d}{d t} {\tilde x_i}(t) = c_i +O(\sqrt{\alpha}) +O(L^{-1}) , \; i=1,..,N \, ,
\label{vitesse}$$ $$\label{distH1}
\|u(t)-\sum_{i=1}^N \varphi_{c_i} (\cdot -{\tilde x_i}(t)) \|_{H^1} =
O(\sqrt{\alpha}) \, ,$$ $${\tilde x_i}(t)-\tilde{x}_{i-1}(t) \ge 3L/4+(c_{i}-c_{i-1}) t/2 ,
\quad i=2,..,N . \label{eloi}$$ Moreover, for $ i=1,..,N $, it holds $$|x_i(t)-{\tilde x_i}(t)| =O(1) , \label{prox}$$ where $ x_{i}(t)\in [\tilde{x}_i(t)-L/4, \tilde{x}_{i}(t)+L/4] $ is any point such that $$|u(t,x_i(t))|=\max_{[\tilde{x}_i(t)-L/4, \tilde{x}_{i}(t)+L/4]} |u(t)| . \label{maxi}$$
[*Proof.* ]{} We only sketch the proof and refer to [@EL2] for details. The strategy is to use a modulation argument to construct $ N $ $C^1$-functions $ t\mapsto {\tilde x_i}(t)$, $i=1,..,N $ on $ [0,t_0] $ satisfying the following orthogonality conditions : $$\int_{{{ I\!\!R}}} \Bigl( u(t,\cdot) -\sum_{j=1}^N \varphi_{c_j}(\cdot-{\tilde x_j}(t)) \Bigr)
\partial_x \varphi_{c_i} (\cdot -{\tilde x_i}(t)) \, dx = 0 \; . \label{mod2}$$ Moreover, setting $$\label{ZZ}
R_Z(\cdot)=\sum_{i=1}^N \varphi_{c_i}(\cdot -z_i)$$ for any $ Z=(z_1,..,z_N)\in {{ I\!\!R}}^N $, one can check that $$\|u(t)-R_{{\tilde X}(t)}\|_{H^1} \lesssim C_0 \sqrt{\alpha}\, , \quad \forall t\in
[0,t_0] \; . \label{estepsilon}$$ To prove that the speed of $ {\tilde x}_i $ stays close to $ c_i $, we set $$R_j(t)=\varphi_{c_j}(\cdot-{ \tilde x}_j(t)) \mbox{ and }
v(t)=u(t)-\sum_{i=1}^N R_j(t)=u(t,\cdot)-R_{{\tilde X}(t)} \; .$$ and differentiate [(\[mod2\])]{} with respect to time to get $$\int_{{{ I\!\!R}}} v_t \partial_x R_i =\dot{\tilde x}_i \, \langle
\partial_x^2 R_i \, ,\, v \rangle_{H^{-1}, H^1} \, ,$$ and thus $$\Bigl|\int_{{{ I\!\!R}}} v_t \partial_x R_i\Bigr|
\le |\dot{\tilde x}_i| O(\|v\|_{H^1}) \le |\dot{\tilde x}_i-c_i|
O(\|v\|_{H^1})+O(\|v\|_{H^1})\; . \label{huhu}$$ Substituting $ u $ by $ v+\sum_{j=1}^N R_j $ in [(\[CH3\])]{} and using that $ R_j $ satisfies $$\partial_t R_j +(\dot{\tilde x}_j-c_j) \partial_x R_j + R_j \partial_x R_j
+(1-\partial_x^2)^{-1}\partial_x [R_j^2+(\partial_x R_j)^2/2] = 0 \;,$$ we infer that $ v$ satisfies on $ [0,t_0] $, $$\begin{aligned}
v_t& - & \sum_{j=1}^N (\dot{\tilde x}_j-c_j) \partial_x R_j
= -\frac{1}{2} \partial_x \Bigl[(v+\sum_{j=1}^N
R_j)^2- \sum_{j=1}^N R_j^2 \Bigr] \nonumber \\
& &-(1-\partial_x^2)^{-1}\partial_x \Bigl[(v+\sum_{j=1}^N R_j)^2- \sum_{j=1}^N R_j^2
+\frac{1}{2} (v_x +\sum_{j=1}^N \partial_x R_j)^2 -\frac{1}{2}\sum_{j=1}^N
(\partial_x R_j)^2\Bigr]\; . \nonumber
\end{aligned}$$ Taking the $ L^2 $-scalar product with $ \partial_x R_i $, integrating by parts, using the decay of $ R_j $ and its first derivative, [(\[estepsilon\])]{} and [(\[huhu\])]{}, we find $$|\dot{\tilde x}_i-c_i|\Bigl(\|\partial_x R_i \|_{L^2}^2 +O(\sqrt{\alpha}) \Bigr)
\le O(\sqrt{\alpha}) +O(e^{ -L/8})\; . \label{fofo}$$ Taking $\alpha_0$ small enough and $ L_0 $ large enough we get $ |\dot{\tilde x}_i-c_i| \le (c_i-c_{i-1})/4 $ and thus, for all $ 0<\alpha<\alpha_0
$ and $ L\ge L_0>3C_0\varepsilon $, it follows from [(\[ini\])]{}, [(\[estepsilon\])]{} and [(\[fofo\])]{} that $${\tilde x}_j(t)-{\tilde x}_{j-1}(t)>
L-C_0\varepsilon+(c_j-c_{j-1}) t/2 , \quad \forall t\in[0,t_0]
\; . \label{xj-xj-1}$$ which yields [(\[eloi\])]{}.\
Finally from [(\[estepsilon\])]{} and the continuous embedding of $
H^1({{ I\!\!R}})$ into $ L^\infty({{ I\!\!R}}) $, we infer that $$u(t,x) = R_{{\tilde
X}(t)}(x)+O(\sqrt{\alpha}), \quad \forall x\in {{ I\!\!R}}\, .$$ Applying this formula with $ x=\tilde{x}_i$ and taking advantage of [(\[eloi\])]{}, we obtain $$|u(t,\tilde{x}_i)|
=|c_i|+O(\sqrt{\alpha})+O(e^{-L/4}) \ge
3|c_i|/4 \; .$$ On the other hand, for $ x\in [\tilde{x}_i(t)-L/4, \tilde{x}_{i}(t)+L/4]\backslash
]{\tilde x_i}(t)-2,{\tilde x_i}(t)+2[ $, we get $$|u(t,x)|\le |c_i|
e^{-2}+O(\sqrt{\alpha})+O(e^{-L/4}) \le |c_i|/2 \; .$$ This ensures that $x_i$ belongs to $ [{\tilde x_i}-2,{\tilde
x_i}+2] $.
Monotonicity property {#Sectmonotonie}
---------------------
Thanks to the preceding lemma, for $ \varepsilon_0> 0 $ small enough and $ L_0>0 $ large enough, one can construct $C^1$-functions $ {\tilde x_1}, .., {\tilde x_N} $ defined on $
[0,t_0] $ such that [(\[vitesse\])]{}-[(\[prox\])]{} are satisfied. In this subsection we state the almost monotonicity of functionals that are very close to the $ E(\cdot) -\lambda F(\cdot) $ at the right of the $ i $th bump, $ i=k,..,N-1 $ of $ u $. The proof follows the same lines as in Lemma 4.2 in [@EL] but is more delicate since we have also to deal with the functional $ F $. Moreover, $ F $ generates a term ( $J_4 $ in [(\[go2\])]{}) that we are not able to estimate in a suitable way but which fortunately is of the good sign.
Let $ \Psi $ be a $ C^\infty $ function such that $ 0<\Psi\le 1 $, $ \Psi'>0 $ on $ {{ I\!\!R}}$, $ |\Psi'''|\le 10 |\Psi'| \mbox{ on } [-1,1]
$, $$\label{psipsi}
\Psi(x)=\left\{ \begin{array}{ll}
e^{-|x|} & \quad x<-1\\
1-e^{-|x|}& \quad x>1
\end{array}
\right. .$$ Setting $ \Psi_K=\Psi(\cdot/K) $, we introduce for $ j\in \{q,..,N\}
$ and $ \lambda\ge 0 $, $$I_{j,\lambda}(t)=I_{j,\lambda,K}(t,u(t))= \int_{{{ I\!\!R}}}\Bigl( (u^2(t)+u_x^2(t)) -\lambda (u^3(t)+u
u_x^2(t))\Bigr) \Psi_{j,K}(t) \, dx\,,$$ where $ \Psi_{j,K}(t,x)=\Psi_K(x-y_j(t)) $ with $ y_j(t)$, $ j=k+1,..,N $, defined by $$y_{k+1}(t)=\tilde{x}_{k+1}(0)+c_{k+1} t/2 -L/4$$ $$\label{defyi}
\mbox{ and } y_i(t)=\frac{{\tilde x_{i-1}}(t)+{\tilde x_i}(t)}{2},\quad
i=k+2,..,N.$$ Finally, we set $$\sigma_0=\frac{1}{4} \min \Bigl(c_{k+1},c_{k+2}
-c_{k+1},..,c_N-c_{N-1}\Bigr) \; .$$
\[monotonicitylem\] Let $ u\in Y([0,T[) $ be a solution of (C-H) satisfying [(\[distH1\])]{} on $[0,t_0] $. There exist $ \alpha_0>0 $ and $ L_0>0 $ only depending on $ c_{k+1} $ and $c_N$ such that if $ 0<\alpha<\alpha_0 $ and $ L\ge L_0 $ then for any $ 4\le K \lesssim L^{1/2} $ and $ 0\le \lambda\le 2/c_{k+1} $, $$\label{monotonicityestim}
I_{j,\lambda,K}(t)-I_{j,\lambda,K}(0)\le O( e^{-\frac{\sigma_0 L }{8K}}) ,
\quad \forall j\in\{k+1,..,N\}, \; \quad \forall t\in [0,t_0] \; .$$
[*Proof.* ]{} Let us assume that $ u $ is smooth since the case $ u\in Y([0,T[) $ follows by modifying slightly the arguments (see Remark 3.2 of [@EM]).
$$\begin{aligned}
\frac{d}{dt}\int_{{{ I\!\!R}}} (u^2+u_x^2) g \, dx&=&\int_{{{ I\!\!R}}}(u^3+4uu_x^2)g^{'} \, dx\nonumber\\
&&\hspace*{-10mm}
-\int_{{{ I\!\!R}}}u^3g^{'''} \, dx- 2 \int_{{{ I\!\!R}}}u h g^{'} \, dx.
\label{go}\end{aligned}$$
and $$\begin{aligned}
\frac{d}{dt}\int_{{{ I\!\!R}}} (u^3+uu_x^2) g \, dx&=&\int_{{{ I\!\!R}}}(u^4/4+u^2 u_x^2)g^{'} \, dx\nonumber\\
&&\hspace*{-10mm}
+\int_{{{ I\!\!R}}}u^2 h g^{'} \, dx+\int_{{{ I\!\!R}}}( h^2-h_x^2) g^{'} \, dx.
\label{gogo}\end{aligned}$$ where $ h:=(1-\partial_x^2)^{-1} (u^2+u_x^2/2) $.
[Proof. ]{} Since [(\[go\])]{} is proven in [@EL2] we concentrate on the proof of [(\[gogo\])]{}. $$\begin{aligned}
\frac{d}{dt} \intR (u^3+u u_x^2) g & = & 3 \intR u_t u^2 g + 2 \intR u_{tx} u_x u g +\intR u_t u_x^2 g
\nonumber\\
& = & 2\intR u_t (u^2+u_x^2/2) g +\intR u_t u^2 g -\intR u_{txx} u^2 g -\intR u_{tx} u^2 g' \nonumber\\
&= & 2\intR u_t (u^2+u_x^2/2) g +\intR (u_t -u_{txx}) u^2 g -\intR u_{tx} u^2 g' \nonumber \\
= I_1+I_2+I_3 \, . \label{hu1}\end{aligned}$$ Setting $ h:=(1-\partial_x^2)^{-1} (u^2+u_x^2/2) $ and using the equation we get $$\begin{aligned}
I_1 & = & -2 \intR u u_x (u^2+u_x^2/2) g -2 \intR g h_x (1-\partial_x^2) h \nonumber \\
& = & -2 \intR u^3 u_x g -\intR u u_x^3 g -2 \intR h h_x g +2\intR h_x h_{xx} g \nonumber\\
& = & \frac{1}{2} \intR u^4 g' -\intR u u_x^3 g +\intR (h^2-h_x^2) g' \; .\end{aligned}$$ In the same way, $$\begin{aligned}
I_2 & = & -3 \intR u^3 u_x g -\frac{1}{2} \intR \partial_x(u_x^2) u^2 g +\frac{1}{2} \intR \partial^3_x (u^2) u^2 g \nonumber \\
& = & \frac{3}{4}\intR u^4 g' -\frac{1}{2} \intR \partial_x(u_x^2) u^2 g -\frac{1}{2} \intR
\partial_x^2 (u^2) \partial_x(u^2) g -\frac{1}{2}\intR \partial_x^2 (u^2) u^2 g' \nonumber\\
& = & \frac{3}{4}\intR u^4 g'+\intR u u_x^3 g +\frac{1}{2} \intR u_x^2 u^2 g' +\frac{1}{4} \intR
[\partial_x (u^2) ]^2 g' +\intR \partial_x(u^2) u u_x g' \nonumber \\
& & +\frac{1}{2}\intR \partial_x (u^2) u^2 g^{''} \nonumber\\
& = & \frac{3}{4}\intR u^4 g'+\intR u u_x^3 g +\frac{1}{2} \intR u_x^2 u^2 g' +\intR
u^2 u_x^2 g' +2 \intR u^2_x u^2 g' +\intR u^3 u_x g^{''} \nonumber\\
& = & \frac{3}{4}\intR u^4 g'-\frac{1}{4}\intR u^4 g^{'''}+\frac{7}{2} \intR u_x^2 u^2 g'
+\intR u u_x^3 g \; .\end{aligned}$$ At this stage it is worth noticing that the terms $\intR u u_x^3 g $ cancels with the one in $ I_1 $. Finally, $$\begin{aligned}
I_3 & = & \intR \partial_x ( u u_x) u^2 g' +\intR g' u^2 \partial_x^2 h \nonumber \\
& = & -2\intR u^2 u_x^2 g' -\intR u^3 u_x g^{''} -\intR u^2( u^2 +u_x^2/2) g' +\intR u^2 h g' \nonumber\\
& = & -2 \intR u^2 u_x^2 g' +\frac{1}{4} \intR u^4 g^{'''}-\intR u^4 g' -\frac{1}{2}\intR u^2 u_x^2 g' +\intR u^2 h g' \nonumber\\
& = & -\frac{5}{2} \intR u^2 u_x^2 g' +\frac{1}{4} \intR u^4 g^{'''}-\intR u^4 g' +\intR u^2 h g' \label{hu4}
\end{aligned}$$ where we used that $ \partial_x^2 (I-\partial_x^2)^{-1} = -I +(I-\partial_x^2)^{-1} $. Gathering [(\[hu1\])]{}-[(\[hu4\])]{}, [(\[gogo\])]{} follows.\
Applying [(\[go\])]{}-[(\[gogo\])]{} with $ g=\Psi_{j,K} $, $ j\ge k+1 $, one gets $$\begin{aligned}
\frac{d}{dt} I_{j,\lambda,K} & : = &
\frac{d}{dt}\int_{{{ I\!\!R}}} \Psi_{j,K}[(u^2+u_x^2)-\lambda (u^3+ u u_x^2) ] \, dx \nonumber \\
&=&-\dot{y_j}
\int_{{{ I\!\!R}}} \Psi_{j,K}' (u^2+u_x^2) \nonumber \\
& & +\int_{{{ I\!\!R}}}\Psi_{j,K}' \Bigl[[(u^3+4uu_x^2)-\lambda\Bigl(\dot{y_j} (u^3+u u_x^2)-(u^4/4 +u^2 u_x^2)\Bigr) \Bigr] \, dx\nonumber\\
&&-\int_{{{ I\!\!R}}}\Psi_{j,K}^{'''} u^3\, dx-\int_{{{ I\!\!R}}}\Psi_{j,K}' (2 u+\lambda u^2) h \, dx \nonumber \\
& &- \lambda \intR \Psi_{j,K}' (h^2-h_x^2) \, dx\nonumber \\
& = & -\dot{y_j}\int_{{{ I\!\!R}}} \Psi_{j,K}'(u^2+u_x^2) +J_1+J_2+J_3+J_4 \nonumber \\
& \le & -\frac{c_{k+1}}{2} \int_{{{ I\!\!R}}} \Psi_{j,K}'(u^2+u_x^2) +J_1+J_2+J_3+J_4 \; .
\label{go2}\end{aligned}$$ We claim that $ J_4 \le 0 $ and that for $ i\in \{1,2,3\} $, it holds $$J_i \le \frac{c_{k+1}}{8} \int_{{{ I\!\!R}}} \Psi_{j,K}' (u^2+u_x^2) + \frac{C}{K} \, e^{-\frac{1}{K}
(\sigma_0 t+L/8)}\; . \label{go4}$$ To handle with $ J_1 $ we divide $ {{ I\!\!R}}$ into two regions $ D_j $ and $ D_j^c $ with $$D_j=[{\tilde x_{j-1}}(t) +L/4, {\tilde x_j}(t) -L/4]$$ First since from [(\[eloi\])]{}, for $ x\in D_j^c $ , $$|x-y_j(t)| \ge \frac{{\tilde x_{j}}(t)-{\tilde x_{j-1}}(t)}{2}-L/4 \ge \frac{c_j-c_{j-1}}{2} \, t +L/8\, ,$$ we infer from the definition of $ \Psi $ in Section \[Sectmonotonie\] that $$\Bigl|\int_{D_j^c} \Psi_{j,K}' \Bigl[[(u^3+4uu_x^2)-\lambda\Bigl(\dot{y_j} (u^3+u u_x^2)-(u^4/4 +u^2 u_x^2)\Bigr) \Bigr] \, dx\Bigr|$$ $$\le \frac{C}{K} \, (1+2 \lambda c_N) (\|u_0\|_{H^1}^3+ \|u_0\|_{H^1}^4)
e^{-\frac{1}{K}(\sigma_0 t +L/8)} \; .$$ On the other hand, on $ D_j $ we notice, according to [(\[distH1\])]{}, that $$\begin{aligned}
\|u(t)\|_{L^\infty_{D_j}}& \le & \sum_{i=1}^N \|\varphi_{c_i}(\cdot -{\tilde x_i(t))\|_{L^\infty}(D_j)}
+ \|u-\sum_{i=1}^N\varphi_{c_i}(\cdot -{\tilde x_i(t)})\|_{L^\infty(D_j)} \nonumber \\
& \le & C \, e^{-L/8} +O(\sqrt{\alpha}) \; .\label{go3}
\end{aligned}$$
Therefore, for $ \alpha $ small enough and $ L $ large enough it holds $$J_1 \le \frac{c_{k+1}}{8} \int_{{{ I\!\!R}}} \Psi_{j,K}' (u^2+u_x^2) + \frac{C}{K} \, e^{-\frac{1}{K}(\sigma_0 t +L/8)}\; .$$ Since $ J_2 $ can be handled in exactly the same way, it remains to treat $ J_3 $. For this, we first notice as above that $$\begin{aligned}
& &\hspace*{-15mm} -\int_{D_j^c}(2u+\lambda u^2) \Psi_{j,K}'
(1-\partial_x^2)^{-1}(u^2+u_x^2/2 ) \nonumber \\
& & \le (2+\lambda\|u\|_{\infty}) \|u\|_{\infty} \sup_{x\in D_j^c}
|\Psi_{j,K}'(x-y_j(t))|\int_{{{ I\!\!R}}} e^{-|x|} \ast (u^2+u_x^2/2 ) \, dx \nonumber \\
& & \le \frac{C}{K} \|u_0\|_{H^1}^3 \, e^{-\frac{1}{K}(\sigma_0 t +L/8)}\; ,
\label{J31}\end{aligned}$$ since $$\forall f\in L^1({{ I\!\!R}}), \quad (1-\partial_x^2)^{-1} f
=\frac{1}{2} e^{-|x|} \ast f \; .
\label{tytu}$$ Now in the region $D_j $, noticing that $ \Psi_{j,K}' $ and $ u^2+u_x^2/2 $ are non-negative, we get $$\begin{aligned}
& & \hspace*{-15mm} -\int_{D_j}(2 +\lambda u ) u \Psi_{j,K}'
(1-\partial_x^2)^{-1}(u^2+u_x^2/2 ) \nonumber \\
& \le &
(2+\lambda \|u(t)\|_{L^\infty({D_j})} ) \|u(t)\|_{L^\infty({D_j})} \int_{D_j}\Psi_{j,K}'(
(1-\partial_x^2)^{-1}(2u^2+u_x^2) \nonumber \\
& \le & (2+\lambda \|u(t)\|_{L^\infty({D_j})} ) \|u(t)\|_{L^\infty({D_j})} \int_{{{ I\!\!R}}} (2u^2+u_x^2) (1-\partial_x^2)^{-1}
\Psi_{j,K}'\; .\end{aligned}$$ On the other hand, from the definition of $ \Psi $ in Section \[Sectmonotonie\] and [(\[tytu\])]{} we infer that for $ K\ge 4 $, $$(1-\partial_x^2) \Psi_{j,K}' \ge (1-\frac{10}{K^2}) \Psi_{j,K}' \Rightarrow
(1-\partial_x^2)^{-1} \Psi_{j,K}'\le (1-\frac{10}{K^2})^{-1} \Psi_{j,K}' \; .$$ Therefore, taking $ K\ge 4 $ and using [(\[go3\])]{} we deduce for $ \alpha $ small enough and $ L $ large enough that $$-\int_{D_j} (2u+\lambda u^2) \Psi_K'
(1-\partial_x^2)^{-1}(u^2+u_x^2/2)
\le \frac{ c_q}{8}
\int_{{{ I\!\!R}}} (u^2+u_x^2/2)
\Psi_K' \; . \label{J32}$$ This completes the proof of [(\[go4\])]{}. It remains to prove that $ J_4 $ is non positive. Recall that $ h=(I-\partial_x^2)^{-1} v $ with $ v:=u^2+u_x^2/2 \ge 0 $. Therefore, following [@CE1], it holds $$\begin{aligned}
h(x)& =& \frac{1}{2} e^{-|\cdot|} \ast v(\cdot) \\
& = & \frac{1}{2} e^{-x} \int_{-\infty}^x e^y v(y) \, dy +\frac{1}{2} e^{x} \int_{-\infty}^x e^{-y} v(y) \, dy
\end{aligned}$$ and $$h'(x)=-\frac{1}{2} e^{-x} \int_{-\infty}^x e^y v(y) \, dy +\frac{1}{2} e^{x} \int_{-\infty}^x e^{-y} v(y) \, dy$$ which clearly ensures that $ h^2 \ge h_x^2 $. Since $ \Psi_{j,K}'\ge 0$ and $ \lambda \ge 0 $, this leads to the non positivity of $ J_4=- \lambda \intR \Psi_{j,K}' (h^2-h_x^2) \, dx $.
Gathering [(\[go2\])]{} and [(\[go4\])]{} we infer that $$\frac{d}{dt}\int_{{{ I\!\!R}}} \Psi_{j,K}[u^2+u_x^2-\lambda(u^3+u u_x^2) ] \, dx\le -\frac{c_1}{8}
\int_{{{ I\!\!R}}}\Psi_{j,K}' (u^2+u_x^2) +
\frac{C}{K} (1+\|u_0\|_{H^1}^4) \, e^{-\frac{1}{K}(\sigma_0 t +L/8)} \; .$$ Integrating this inequality between $ 0 $ and $ t $, [(\[monotonicityestim\])]{} follows.\
Localized estimates {#Localized energy estimates}
-------------------
We define the function $ \Phi_i=\Phi_i(t,x) $, $i=k+1,..,N$, by $
\Phi_N=\Psi_{N,K}=\Psi_K(\cdot-y_N(t)) $ and for $i=k+1,..,N-1 $ $$\label{defphii}
\Phi_i=\Psi_{i,K}-\Psi_{i+1,K}=\Psi_K(\cdot-y_i(t))-\Psi_K(\cdot-y_{i+1}(t))\; ,$$ where $ \Psi_{i,K} $ and the $ y_i $’s are defined in Section \[Sectmonotonie\]. It is easy to check that the $ \Phi_i$’s are positive functions and that $\displaystyle \sum_{i=k+1}^N \Phi_{i}\equiv
\Psi_{k+1,K} $. We will take $L/K>4$ so that [(\[psipsi\])]{} ensures that $ \Phi_i $ satisfies for $i\in \{k+1,..,N\} $, $$|1-\Phi_{i}| \le 2 e^{-\frac{L}{8K}} \mbox{ on } ] y_i+L/8, y_{i+1}-L/8[
\label{de1}$$ and $$|\Phi_{i}| \le 2 e^{-\frac{L}{8K}} \mbox{ on } ]y_i-L/8,
y_{i+1}+L/8[^c \; ,\label{de2}$$ where we set $ y_{N+1}:=+\infty $.\
It is worth noticing that, somehow, $ \Phi_i(t) $ takes care of only the ith bump of $u(t)$. We will use the following localized version of $ E $ and $ F $ defined for $i\in \{k+1,..,N\}, $ by $$\label{defEi}
E_i^t(u) = \int_{{{ I\!\!R}}} \Phi_{i}(t) (u^2+u_x^2) \mbox{ and }
F_i^t(u)= \int_{{{ I\!\!R}}} \Phi_i(t) (u^3+u u_x^2) \; .$$ [**Please note that henceforth we take $K=L^{1/2}/8 $.**]{}\
The following lemma gives a localized version of [(\[eq2\])]{}. Note that the functionals $ E_i $ and $ F_i $ do not depend on time in the statement below since we fix $
y_{k+1}<..<y_{N+1}=+\infty$.
\[m-p-lemme\] Let be given $u\in H^1({{ I\!\!R}}) $ with $\|u\|_{H^1}=\|u_0\|_{H^1} $ and $ N-k $ real numbers $ y_{k+1}<..<y_N$ with $y_i-y_{i-1} \ge 2L/3 $. For $ i=k+1,..,N $, set $J_i:=]y_i-L/4,y_{i+1}+L/4[ $ with $ y_{N+1}=+\infty$, and assume that there exist $ x_i\in ]y_i+L/4,y_{i+1}-L/4[ $ such that $ u(x_i)=\displaystyle \max_{J_i} u:=M_i>0 $. Then, defining the functional $ E_i $’s and $ F_i$’s as in [(\[defphii\])]{}-[(\[defEi\])]{}, it holds $$\label{eq2m}
F_i(u)\leqslant M_i E_i(u)-\frac{2}{3}M_i^3+\|u_0\|_{H^1}^3 O(L^{-{1/2}}), \quad
i\in \{k+1,..,N\} \, .$$ and for any $ x_1<..<x_k$ with $x_k <y_{k+1}-L/4$, setting $X:=(x_{1},..,x_N)\in {{ I\!\!R}}^{N} $, it holds $$\label{eq3m}
E_i(u)-E(\varphi_{c_i})=E_i (u-R_{X})
+4c_i (M_i-c_i)+ \|u_0\|_{H^1}^2 O(L^{-1/2})
, \quad
i\in \{k+1,..,N\} ,$$ where $ R_X $ is defined in [(\[ZZ\])]{}.
[*Proof.* ]{} Let $ i\in \{k+1,..,N\}$ be fixed. Following [@CS1], we introduce the function $g$ defined by $$g(x)=\left\{
\begin{array}{l}
u(x)-u_x(x) \; \mbox{ for } \; x<{ x_i} \\
u(x)+u_x(x) \; \mbox{ for }\; x>{ x_i}
\end{array}.
\right.$$ Integrating by parts we compute $$\begin{aligned}
\label{ug2}
\int ug^2\Phi_{i}&=&\int_{-\infty}^{
x_i}(u^3+uu_x^2-2u^2u_x)\Phi_i
+\int_{ x_i}^{+\infty}(u^3+uu_x^2+2u^2u_x)\Phi_i\nonumber\\
&=&F_i(u)-\frac{4}{3}u({x_i})^3\Phi_i({
x_i})+\frac{2}{3}\int_{-\infty}^{ x_i}u^3\Phi_i^{'}
-\frac{2}{3}\int_{ x_i}^{+\infty}u^3\Phi_i^{'}\, .\end{aligned}$$ Recall that we take $ K=\sqrt{L}/8 $ and thus $ |\Phi'|\le C/K = O(L^{-1/2}) $. Moreover, since $ x_i\in ] y_i+L/4, y_{i+1}-L/4[$, it follows from [(\[de1\])]{} that $ \Phi_i({ x
_i})=1+O(e^{-L^{1/2}}) $ and thus $$\int u g^2 \Phi_i = F_i(u)-\frac{4}{3} M_i^3+\|u\|_{H^1}^3 O(L^{-1/2}) \; .$$ On the other hand, with [(\[de2\])]{} at hand, $$\begin{aligned}
\label{hg2}
\int u g^2\Phi_i& \le & M_i \int_{J_i} g^2 \Phi_i + \int_{J_i^c} | u| g^2 \Phi_i \nonumber \\
& \le & M_i \int_{-\infty}^{+\infty} g^2 \Phi_i + \|u\|_{L^\infty({{ I\!\!R}})} \int_{J_i^c} g^2 \Phi_i \nonumber \\
&\leq &M_i \Bigl( E_i(u)-2\int_{-\infty}^{x_i}uu_x\Phi_i+2\int_{
x_i}^{+\infty}
uu_x\Phi_i \Bigr)+ \|u\|_{H^1}^3 \sup_{x\in J_i^c} |\Phi_i(x)|\nonumber\\
&\leq &M_iE_i(u)-2M_i^3+\|u\|_{H^1}^3 O(L^{-1/2}) \; .\end{aligned}$$ This proves [(\[eq2m\])]{}. To prove [(\[eq3m\])]{}, we use the relation between $ \varphi $ an its derivative and integrate by parts, to get $$\begin{aligned}
E_i (u-R_X) & = & E_i (u)+E_i (R_X) -2 \int \Phi_i
\Bigl( u\, \varphi_{c_i}
(\cdot-x_i) + u_x \, \partial_x \varphi_{c_i}(\cdot-x_i) \Bigr) \\
& =& E_i(u)+E_i (R_X) -2 \int \Phi_i u \, \varphi_{c_i}(\cdot-x_i) \\
& & +2
\int_{x_i}^{+\infty} \Phi_i u_x \, \varphi_{c_i}(\cdot-x_i) -2
\int^{x_i}_{-\infty}\Phi_i u_x \, \varphi_{c_i}(\cdot-x_i) \\
& =& E_i(u)+E_i (R_X) -2 \int \Phi_i u \, \varphi_{c_i}(\cdot-x_i)
+2 \int \Phi_i' u \, \varphi_{c_i}(\cdot-x_i)\\
& & +2
\int_{z_i}^{+\infty} \Phi_i u_x \, \varphi_{c_i}(\cdot-x_i) -2
\int^{z_i}_{-\infty}\Phi_i u_x \, \varphi_{c_i}(\cdot-x_i) \\
& = & E_i(u)+E_i (R_X) -4 c_i u(x_i) \Phi_i(x_i)
+2 \int \Phi_i' u \, \varphi_{c_i}(\cdot-x_i)\\
& & -2
\int_{x_i}^{+\infty} \Phi_i' u \, \varphi_{c_i}(\cdot-x_i) +2
\int^{x_i}_{-\infty}\Phi_i' u \, \varphi_{c_i}(\cdot-x_i) \; .
\end{aligned}$$ From [(\[de1\])]{}-[(\[de2\])]{}, it is easy to check that $
E_i(R_X)=E(\varphi_{c_i}) +O(e^{-\sqrt{L}/8}) $. Since $ C/K = O(L^{-1/2})$ and, in view of [(\[de1\])]{}, $ \Phi_i({ x_i})=1+O(e^{-L^{1/2}}) $, it follows that $$E_i(u)+E_i(\varphi_{c_i})=E_i(u-R_X)+4c_i M_i +\|u\|_{H^1}^2 O(L^{-1/2}) \; .$$ This yields the result by using that $ E(\varphi_{c_i})=2c_i^2 $.
End of the proof of Theorem $\ref{mult-peaks}$ {#end-proof}
----------------------------------------------
\[last\] There exists constants $ C,C'>0 $ independent of $ A $ such that
$$\label{oo}
I_{k+1,0}\Bigl( t_0,u(t_0)-R_{X(t_0)}\Bigr) =\sum_{i=k+1}^N
E_i^{t_0}\Bigl( u(t_0)-R_{X(t_0)}\Bigr) \le C ({\varepsilon}+L^{-{1/4}})$$
and $$\label{oo2}
I_{k+1,0}(t_0)=\sum_{i=k+1}^N
E_i^{t_0}(u(t_0))=\sum_{i=k+1}^N E(\varphi_{c_i}) + O ({\varepsilon}+L^{-{1/4}})\, .$$ with $ |O(x)|\le C' x , \, \forall x\in {{ I\!\!R}}_+^* $.
[*Proof.* ]{} First it is worth noticing that according to Lemma \[eloignement\], $u(t_0)$, $(y_{k+1}(t_0),..,y_{N+1}) $, constructed in [(\[defyi\])]{}, and $ X(t_0)=(x_1(t_0),..,x_N(t_0)) $, constructed in [(\[maxi\])]{}, satisfy the hypotheses of Lemma \[m-p-lemme\]. Indeed, by construction for $ i\in\{k+1,..,N\} $, $ x_i\in [\tilde{x}_i(t_0)-L/4,\tilde{x}_i(t_0)+L/4, ]\subset
]y_i(t_0)+L/4, y_{i+1}(t_0)-L/4[ $ and it is easy to check that $|u(t_0)|\le O(e^{-\sqrt{L}})+O(\alpha)<3 c_i/4 \le |u(x_i)| $ on $ ]y_i(t_0)-L/4,y_{i+1}(t_0)+L/4[\backslash [\tilde{x}_i(t_0)-L/4,\tilde{x}_i(t_0)+L/4 ] $ so that $$0<u(t_0,x_i(t_0))=\max_{]y_i(t_0)-L/4,y_{i+1}(t_0)+L/4[} u(t_0) \; .$$ Therefore, setting $M_i=u(t_0,x_i(t_0)) $, $ \delta_i=c_i-M_i$ and taking the sum over $i=k+1,..,N $ of [(\[eq2m\])]{} one gets : $$\sum_{i=k+1}^{N}\Bigl( M_i E_i^{t_0}(u(t_0)) -
F_i^{t_0}(u(t_0))\Bigr) \ge
-\frac{2}{3}\sum_{i=k+1}^{N}M_i^3+O(L^{-{1/2}})$$ Note that by [(\[distH1\])]{} and the continuous embedding of $ H^1({{ I\!\!R}}) $ into $
L^\infty({{ I\!\!R}}) $, $ M_i=c_i+O(\sqrt{\alpha})+O(e^{-L/8})$, and thus $$0<M_{k+1}<\cdot \cdot<M_N \mbox{ and } \delta_i<c_i/2 , \, \forall i\in \{k+1,..,N\} \; .\label{zq}$$ We set $
\Delta_0^{t_0} F_i(u)=F_i^{t_0}(u(t_0))-F^0(u(0)) $, $ \Delta_0^{t_0} E(u)=E^{t_0}(u(t_0))-E^0(u(0)) $, $\Delta_0^{t_0} I_{i,\lambda}(u)=I_{i,\lambda}(t_0,u(t_0))-I_{i,\lambda}(0,u(0)) $. Using the Abel transformation and the monotonicity estimate [(\[monotonicityestim\])]{} (note that $ 0\le 1/M_i\le 2 /c_{k+1} $ for $ i\in \{k+1,..,N\} $), we get $$\sum_{i=k+1}^{N}M_i \Bigl(\Delta_0^{t_0} E(u) -\frac{1}{M_i}
\Delta_0^{t_0} F(u)\Bigr) =\sum_{i=k+1}^{N}(M_i-M_{i-1})\Delta_0^t
I_{i,1/M_i} \leqslant O(e^{- \sigma_0\sqrt{L}})$$ and thus $$\label{e3}
\sum_{i=k+1}^{N}\Bigl( M_i E_i^{0}(u_0) -
F_i^{0}(u_0))\Bigr) \ge
-\frac{2}{3}\sum_{i=k+1}^{N}M_i^3+O(L^{-{1/2}})\; .$$ By [(\[ini\])]{}, the exponential decay of the $ \varphi_{c_i} $’s and the $ \Phi_i
$’s, and the definition of $ E_i $ and $ F_i $, it is easy to check that $$\label{zs}
|E_i^0(u_0)-E(\varphi_{c_i})|+ |F_i^0(u_0)-F (\varphi_{c_i})|\le
O(\varepsilon^2)+O(e^{-\sqrt{L}}), \; \forall i\in\{1,..,N\}\, .$$ Injecting this in $(\ref{e3})$, taking advantage of [(\[zq\])]{} and using that $ E(\varphi_{c_i})=2c_i^2 $ and $ F(\varphi_{c_i})=4c_i^3/3 $, we obtain $$\begin{aligned}
\sum_{i=k+1}^{N}(c_i\delta_i^2-\frac{1}{3}\delta_i^3)
&=&\sum_{i=k+1}^{N}\delta_i^2(c_i-\frac{1}{3}\delta_i)\leqslant
O({\varepsilon}^2+L^{-{1/2}}) \nonumber \\
& \Longrightarrow & \label{e4}
\sum_{i=k+1}^{N} \delta_i^2=
O({\varepsilon}^2+L^{-{1/2}}).\label{zqq}\end{aligned}$$ On the other hand, summing [(\[eq3m\])]{} for $ i=k+1, .., N $ one gets $$\label{lo}
I_{k+1,0}(t_0) - \sum_{i=k+1}^N E(\varphi_{c_i}) = \sum_{i=k+1}^N E_i^{t_0}\Bigl(u(t_0)-R_{X(t_0)}\Bigr)
+4 \sum_{i=k+1}^N c_i \delta_i +O(L^{-1/2}) \, .$$ Using [(\[zq\])]{} and the almost monotonicity of $ t\mapsto I_{k+1,0}(t) $, we infer that $$\sum_{i=k+1}^N E_i^{t_0}\Bigl(u(t_0)-R_{X(t_0)}\Bigr)\le I_{k+1,0}(0)- \sum_{i=k+1}^N E(\varphi_{c_i})
+O(\varepsilon +L^{-1/4})$$ and [(\[zs\])]{}-[(\[e4\])]{} then yield [(\[oo\])]{}. Finally, with [(\[oo\])]{} at hand, [(\[oo2\])]{} follows directly from [(\[zqq\])]{}-[(\[lo\])]{}.\
Now, it is crucial to note that (C-H) is invariant by the change of unknown $ u(t,x)\mapsto -u(t,-x) $. Therefore setting, for any $ v\in H^1({{ I\!\!R}}) $, $${\tilde I}_{k,0}(t,v) :=\int_{{{ I\!\!R}}} \Psi(y_{k}(t)-x)[v^2(x)+ v_x^2(x)]\, dx \, ,$$ with $$y_k(t)=\tilde{x}_k(0)+c_k t/2+L/4\, ,$$ we infer from Proposition \[last\] that $$\label{ooo}
{\tilde I}_{k,0}\Bigl(t_0,u(t_0)-R_{X(t_0)}\Bigr) \le C ({\varepsilon}+L^{-{1/4}})\;$$ and $$\label{ooo2}
{\tilde I}_{k,0}(t_0,u(t_0)) = \sum_{i=1}^k E(\varphi_{c_i}) +O({\varepsilon}+L^{-{1/4}})\; .$$ Hence, $$\begin{aligned}
\tilde{I}_{k,0}(t_0,u(t_0))+I_{k+1,0}(t_0,u(t_0)) & =&
\sum_{i=1}^N E(\varphi_{c_i})+O({\varepsilon}+L^{-{1/4}})\\
& = & E(u_0)+O({\varepsilon}+L^{-{1/4}})\; .\end{aligned}$$ Since $ E(u(t_0))=E(u_0) $ we deduce that $$\int_{{{ I\!\!R}}}\Bigl[ 1-\Psi(y_{k}(t_0)-x)-\Psi(x-y_{k+1}(t_0))\Bigr] [u^2(t_0,x)+ u_x^2(t_0,x)]\, dx
=O({\varepsilon}+L^{-{1/4}})\; .$$ Therefore, since $ | 1-\Psi(y_{k}(t_0)-x)-\Psi(x-y_{k+1}(t_0))|\le O(e^{-\sqrt{L}}) $ for $ x\in {{ I\!\!R}}\backslash ]y_k-L/4, y_{k+1}+L/4[ $ and by the exponentional decay of $ \varphi $, [(\[vitesse\])]{} and [(\[eloi\])]{}, $$\int_{y_k-L/4}^{y_{k+1}+L/4} |R_{X(t_0)}|^2+|\partial_x R_{X(t_0)})|^2 \le
O(e^{-\sqrt{L}/4}) \; ,$$ it follows that $$\int_{{{ I\!\!R}}}\Bigl[ 1-\Psi(y_{k}(t_0)-\cdot)-\Psi(\cdot-y_{k+1}(t_0))\Bigr] [(u(t_0)-R_{X(t_0})^2
+ (u_x(t_0)-\partial_x R_{X(t_0)})^2]
=O({\varepsilon}+L^{-{1/4}}) \; . \label{oooo}$$ Combining [(\[oo\])]{}, [(\[ooo\])]{} and [(\[oooo\])]{} we infer that $$E(u(t_0)-R_{X(t_0)}) =O({\varepsilon}+L^{-{1/4}})\;$$ which concludes the proof of [(\[e12\])]{} since, according to Proposition \[last\], $ |O(x)|\le C |x| $ for some constant $ C>0 $ independent of $ A $. This proves [(\[ini2\])]{} whereas [(\[gy\])]{} Êfollows from [(\[vitesse\])]{} and [(\[prox\])]{}.\
[**Acknowledgements**]{} L.M. would like to thank the Oberwolfach Mathematical center where this work was initiated.
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[^1]: $ W^{1,1}({{ I\!\!R}}) $ is the space of $ L^1({{ I\!\!R}}) $ functions with derivatives in $ L^1({{ I\!\!R}}) $ and $ BV({{ I\!\!R}}) $ is the space of function with bounded variation
[^2]: ${\mathcal
M}({{ I\!\!R}})$ is the space of Radon measures on $ {{ I\!\!R}}$ with bounded total variation. For $ m_0\in {\mathcal M}({{ I\!\!R}})$ we denote respectively by $ m_0^- $ and $ m_0^+ $ its positive and negative part.
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