ShantanuT01/DACTYL-Pretraining huggingface dataset:
An edge-colored graph $G$, where adjacent edges may be colored the same, is rainbow connected if any two vertices of $G$ are connected by a path whose edges have distinct colors. The rainbow connection number $rc(G)$ of a connected graph $G$ is the smallest number of colors that are needed in order to make $G$ rainbow connected. Caro et al. showed an upper bound $rc(G)\leq n-\delta$ for a connected graph $G$ of order $n$ with minimum degree $\delta$ in "On rainbow connection, Electron. J. Combin. 15(2008), R57". Recently, Shiermeyer gave it a generalization that $rc(G)\leq n- \frac{\sigma_2} 2$ in "Bounds for the rainbow connection number of graphs, Discuss. Math Graph Theory 31(2011), 387--395", where $\sigma_2$ is the minimum degree-sum. The proofs of both results are almost the same, both fix the minimum degree $\delta$ and then use induction on $n$. This short note points out that this proof technique does not work rigorously. Fortunately, Caro et al's result is still true but under our improved proof. However, we do not know if Shiermeyer's result still hold.
We introduce a minimal extended evolving model for small-world networks which is controlled by a parameter. In this model the network growth is determined by the attachment of new nodes to already existing nodes that are geographically close. We analyze several topological properties for our model both analytically and by numerical simulations. The resulting network shows some important characteristics of real-life networks such as the small-world effect and a high clustering.
We present Gemini Multi-Object Spectrograph (GMOS) Integral Field Unit (IFU), Hubble Space Telescope (HST) and Very Large Array (VLA) observations of the OH Megamaser (OHM) galaxy IRAS 11506-3851. The HST images reveal an isolated spiral galaxy and the combination with the GMOS-IFU flux distributions and VLA data allow us to identify a partial ring of star-forming regions surrounding the nucleus with a radius of ~ 500 pc. While this ring shows starburst excitation and low velocity dispersion, the region internal to the ring shows higher excitation and velocity dispersion values, with values increasing towards its borders at ~ 240 pc from the nucleus, resembling a projected bubble. The enhanced excitation and velocity dispersion of this bubble surrounds a 8.5 GHz radio emission structure, supporting its origin in a faint AGN that is mostly shocking the surrounding gas via a plasma ejection seen in radio at the present stage. This is the fourth of the 5 OHM galaxies we have studied so far (from our sample of 15 OHM) for which GMOS-IFU data indicate the presence of a previously unknown faint AGN at the nucleus, consistent with the hypothesis that OHM galaxies harbor recently triggered AGN.
We give a new definition of an obstruction theory for infinitesimal deformation theory and relate it to earlier definitions of Artin, Fantechi-Manetti, Li-Tian, and Behrend-Fantechi.
Interpreting the short-timescale variability of the accreting, young, low-mass stars known as Classical T Tauri stars remains an open task. Month-long, continuous light curves from the Transiting Exoplanet Survey Satellite (\textit{TESS}) have become available for hundreds of T Tauri stars. With this vast data set, identifying connections between the variability observed by \TESS and short-timescale accretion variability is valuable for characterizing the accretion process. To this end, we obtained short-cadence \TESS observations of 14 T Tauri stars in the Taurus star-formation region along with simultaneous ground-based, UBVRI-band photometry to be used as accretion diagnostics. In addition, we combine our dataset with previously published simultaneous NUV-NIR \textit{Hubble Space Telescope} spectra for one member of the sample. We find evidence that much of the short-timescale variability observed in the \TESS light curves can be attributed to changes in the accretion rate, but note significant scatter between separate nights and objects. We identify hints of time lags within our dataset that increase at shorter wavelengths which we suggest may be evidence of longitudinal density stratification of the accretion column. Our results highlight that contemporaneous, multi-wavelength observations remain critical for providing context for the observed variability of these stars.
We study the electronic properties of a planar semiconductor-superconductor heterostructure, in which a thin ferromagnetic insulator layer lies in between and acts as a spin filtering barrier. We find that in such a system one can simultaneously enhance the strengths of all the three important induced physical quantities, i.e., Rashba spin-orbit coupling, exchange coupling, and superconducting pairing potential, for the hybrid mode by external gating. Our results show specific advantage of this stacked device geometry compared to conventional devices. We further discuss how to optimize geometrical parameters for the heterostructure and complement our numerical simulations with analytic calculations.
A method is presented for the implementation of edge local complementation in graph states, based on the application of two Hadamard operations and a single controlled-phase (CZ) gate. As an application, we demonstrate an efficient scheme to construct a one-dimensional logical cluster state based on the five-qubit quantum error-correcting code, using a sequence of edge local complementations. A single physical CZ operation, together with local operations, is sufficient to create a logical CZ operation between two logical qubits. The same construction can be used to generate any encoded graph state. This approach in concatenation may allow one to create a hierarchical quantum network for quantum information tasks.
Gravitational microlensing events of high magnification have been shown to be promising targets for detecting extrasolar planets. However, only a few events of high magnification have been found using conventional survey techniques. Here we demonstrate that high magnification events can be readily found in microlensing surveys using a strategy that combines high frequency sampling of target fields with online difference imaging analysis. We present 10 microlensing events with peak magnifications greater than 40 that were detected in real-time towards the Galactic Bulge during 2001 by MOA. We show that Earth mass planets can be detected in future events such as these through intensive follow-up observations around the event peaks. We report this result with urgency as a similar number of such events are expected in 2002.
It this paper we revisit the fast stylization method introduced in Ulyanov et. al. (2016). We show how a small change in the stylization architecture results in a significant qualitative improvement in the generated images. The change is limited to swapping batch normalization with instance normalization, and to apply the latter both at training and testing times. The resulting method can be used to train high-performance architectures for real-time image generation. The code will is made available on github at https://github.com/DmitryUlyanov/texture_nets. Full paper can be found at arXiv:1701.02096.
The use of variational method in functional integral approach is discussed for fermion and boson systems with Coulomb interaction. The formal general expression of thermodynamic potential is obtained by Feynman path integral technique and representation of Coulomb interaction with functional integrals. Introduced additional complex field show to transform the problem to calculation of functional integrals containing third order vertices. The thermodynamic potential can be found from variational principle with respect to field cumulants. The calculation of the equation of state and critical properties is demonstrated for symmetrical plasma by variation of finite number of parameters in the propagator.
The aim of this paper is to study certain multiple series which can be regarded as multiple analogues of Eisenstein series. As a prior research, the second-named author considered double analogues of Eisenstein series and expressed them as polynomials in terms of ordinary Eisenstein series. This fact was derived from the analytic observation of infinite series involving hyperbolic functions which were based on the study of Cauchy, and also Ramanujan. In this paper, we prove an explicit relation formula among these series. This gives an alternative proof of this fact by using the technique of partial fraction decompositions of multiple series which was introduced by Gangl, Kaneko and Zagier. By the same method, we further show a certain multiple analogue of this fact and give some examples of explicit formulas. Finally we give several remarks about the relation between our present result and the previous work for infinite series involving hyperbolic functions.
We work out the interplay between lowest-order perturbative computations in the 't Hooft coupling, $g^2=g^2_{YM} N$, operator mixing, renormalization-group (RG) improved ultraviolet (UV) asymptotics of leading-order (LO) nonplanar/planar contributions to $2$-point correlators, and nonperturbative large-$N$ expansion of perturbatively massless QCD-like theories. As concrete examples, we compute to the lowest perturbative order in $SU(N)$ YM theory the ratios, $r_i$, of LO-nonplanar to planar contributions to the $2$-point correlators in the orthogonal basis in the coordinate representation of the gauge-invariant dimension-$8$ scalar operators and all the twist-$2$ operators. We demonstrate that -- if $\frac{\gamma_0}{\beta_0}$ has no LO-nonplanar contribution, with $\gamma_0$ and $\beta_0$ the one-loop coefficients of the anomalous-dimension matrix and beta function respectively -- $r_i$ actually coincides with the corresponding ratio in the large-$N$ expansion of the RG-improved UV asymptotics of the $2$-point correlators, provided that a certain canonical nonresonant diagonal renormalization scheme exists for the corresponding operators. Contrary to the aforementioned scalar operators, for the first $10^3$ twist-$2$ operators we actually verify the above conditions, and we get the universal value $r_i=-\frac{1}{N^2}$. Hence, nonperturbatively such $r_i$ must coincide with the UV asymptotics of the ratio of the glueball self-energy loop to the glueball tree contribution to the $2$-point correlators above. As a consequence, the universality of $r_i$ reflects the universality of the effective coupling in the nonperturbative large-$N$ YM theory for the twist-$2$ operators in the coordinate representation.
We have observed 79 supernova remnants (SNRs) with the Urumqi 25m telescope at 6cm during the Sino-German 6cm polarization survey of the Galactic plane. We measured flux densities of SNRs at 6cm, some of which are the first ever measured or the measurements at the highest frequency, so that we can determine or improve spectra of SNRs. Our observations have ruled out spectral breaks or spectral flattening that were suggested for a few SNRs, and confirmed the spectral break of S147. By combining our 6cm maps with 11cm and 21cm maps from the Effelsberg 100 m telescope, we calculated the spectral index maps of several large SNRs. For many remnants we obtained for the first time polarization images, which show the intrinsic magnetic field structures at 6cm. We disapproved three objects as being SNRs, OA184, G192.8-1.1 and G16.8-1.1, which show a thermal spectrum and no polarization. We have discovered two large supernova remnants, G178.2-4.2 and G25.1-2.3., in the survey maps.
The von Neumann entanglement entropy is used to estimate the critical point $h_c/J \simeq 0.143(3)$ of the mixed ferro-antiferromagnetic three-state quantum Potts model $H = \sum_i [ J ( X_i X_{i+1}^{\,2} + X_i^{\,2} X_{i+1} ) - h\, R_i ]$, where $X_i$ and $R_i$ are standard three-state Potts spin operators and $J>0$ is the antiferromagnetic coupling parameter. This critical point value gives improved estimates for two Kosterlitz-Thouless transition points in the antiferromagnetic ($\beta < 0$) region of the $\Delta$--$\beta$ phase diagram of the three-state quantum chiral clock model, where $\Delta$ and $\beta$ are, respectively, the chirality and coupling parameters in the clock model. These are the transition points $\beta_c \simeq - 0.143(3)$ at $\Delta = \frac12$ between incommensurate and commensurate phases and $\beta_c \simeq - 7.0(1)$ at $\Delta = 0$ between disordered and incommensurate phases. The von Neumann entropy is also used to calculate the central charge $c$ of the underlying conformal field theory in the massless phase $h \le h_c$. The estimate $c \simeq 1$ in this phase is consistent with the known exact value at the particular point $h/J = -1$ corresponding to the purely antiferromagnetic three-state quantum Potts model. The algebraic decay of the Potts spin-spin correlation in the massless phase is used to estimate the continuously varying critical exponent $\eta$.
The organic metal \theta$-(BETS)$_4$HgBr$_4$(C$_6$H$_5$Cl) is known to undergo a phase transition as the temperature is lowered down to about 240 K. X-ray data obtained at 200 K indicate a corresponding modification of the crystal structure, the symmetry of which is lowered from quadratic to monoclinic. In addition, two different types of cation layers are observed in the unit cell. The Fermi surface (FS), which can be regarded as a network of compensated electron and hole orbits according to band structure calculations at room temperature, turns to a set of two alternating linear chains of orbits at low temperature. The field and temperature dependence of the Shubnikov-de Haas oscillations spectrum have been studied up to 54 T. Eight frequencies are observed which, in any case, points to a FS much more complex than predicted by band structure calculations at room temperature, even though some of the observed Fourier components might be ascribed to magnetic breakdown or frequency mixing. The obtained spectrum could result from either an interaction between the FS's linked to each of the two cation layers or to an eventual additional phase transition in the temperature range below 200 K.
We present the one-loop scalar field effective potential for the $N=2$ supersymmetric nonrelativistic self-interacting matter fields coupled to an Abelian Chern-Simons gauge field and for its generalization when bosonic matter fields are coupled to non-Abelian Chern-Simons field. In both models, Gauss's law linearly relates the magnetic field to the matter field densities; hence, we also include radiative effects from the background gauge field. We compute the scalar field effective potentials in two gauge families, a gauge reminiscent of the $R_\xi$-gauge in the limit $\xi\rightarrow 0$ and in the Coulomb family gauges. We regularize the theory with operator regularization and a cutoff to demonstrate that the results are independent of the regularization scheme.
Due to the confined focal length of optical sensors, focusing all objects in a scene with a single sensor is a difficult task. To handle such a situation, image fusion methods are used in multi-focus environment. Discrete Cosine Transform (DCT) is a widely used image compression transform, image fusion in DCT domain is an efficient method. This paper presents a low complexity approach for multi-focus image fusion and its VLSI implementation using DCT. The proposed method is evaluated using reference/non-reference fusion measure criteria and the obtained results asserts it's effectiveness. The maximum synthesized frequency on FPGA is found to be 221 MHz and consumes 42% of FPGA resources. The proposed method consumes very less power and can process 4K resolution images at the rate of 60 frames per second which makes the hardware suitable for handheld portable devices such as camera module and wireless image sensors.
We consider polygonal tilings of certain regions and use these to give intuitive definitions of tiling-based perimeter and area. We apply these definitions to rhombic tilings of Elnitsky polygons, computing sharp bounds and average values for perimeter tiles in convex centrally symmetric 2n-gons. These bounds and values have implications for the combinatorics of reduced decompositions of permutations. We also classify the permutations whose polygons gave minimal perimeter, defined in two different ways. We conclude by looking at some of these questions in the context of domino tilings, giving a recursive formula and generating function for one family, and describing a family of minimal-perimeter regions.
Main purpose of this paper is to study the local propagation of singularities of viscosity solution to contact type evolutionary Hamilton-Jacobi equation $$ D_tu(t,x)+H(t,x,D_xu(t,x),u(t,x))=0. $$ An important issue of this topic is the existence, uniqueness and regularity of the strict singular characteristic. We apply the recent existence and regularity results on the Herglotz' type variational problem to the aforementioned Hamilton-Jacobi equation with smooth initial data. We obtain some new results on the local structure of the cut set of the viscosity solution near non-conjugate singular points. Especially, we obtain an existence result of smooth strict singular characteristic from and to non-conjugate singular initial point based on the structure of the superdifferential of the solution, which is even new in the classical time-dependent case. We also get a global propagation result for the $C^1$ singular support in the contact case.
A three-dimensional weak topological insulator (WTI), being equivalent to stacked layers of two-dimensional quantum spin-Hall insulators, accommodates massless Dirac electrons on its side surface. A notable feature of WTIs is that surface states typically consist of two Dirac cones in the reciprocal space. We study the Landau quantization of Dirac electrons of WTIs in a perpendicular magnetic field. It is shown that when the magnetic length $l_B$ is much larger than the interlayer distance $a$, surface electrons are quantized into Landau levels according to the ordinary quantization rule for Dirac electrons. It is also shown that, with decreasing $l_B$ toward $a$, each Landau level and its spin state become modulated in a nontrivial manner. We demonstrate that this is attributed to the mixing of two Dirac cones induced by the discreteness of the layered structure.
The problem of (pathwise) large deviations for conditionally continuous Gaussian processes is investigated. The theory of large deviations for Gaussian processes is extended to the wider class of random processes -- the conditionally Gaussian processes. The estimates of level crossing probability for such processes are given as an application.
In a quantum system with a smoothly and slowly varying Hamiltonian, which approaches a constant operator at times $t\to \pm \infty$, the transition probabilities between adiabatic states are exponentially small. They are characterized by an exponent that depends on a phase integral along a path around a set of branch points connecting the energy level surfaces in complex time. Only certain sequences of branch points contribute. We propose that these sequences are determined by a topological rule involving the Stokes lines attached to the branch points. Our hypothesis is supported by theoretical arguments and results of numerical experiments.
Brownian motion occurs in a variety of fluids, from rare gases to liquids. The Langevin equation, describing friction and agitation forces in statistical balance, is one of the most successful ways to treat the phenomenon. In rare gases, it is appropriate to model both friction and agitation in terms of independent molecular impacts with the particle. But in relatively dense fluids, such as water and air at standard temperature and pressure, the mean free path between collisions of fluid molecules is much smaller than the size of the Brownian particle, and the friction is normally treated as a mesoscopic viscous effect described by Stokes' Law which treats the fluid as continuous. The appropriateness of using Stokes' Law will be discussed in terms of recent experimental research in the ballistic or "coasting" phase of motion occurring at a very short time scale. Given the mesoscopic nature of the friction force for relatively dense fluids, we should expect the agitation force to also be mesoscopic. But it is often unrealistically modeled as uncorrelated individual impacts. It has been suggested occasionally that mesoscopic pressure fluctuations are appropriate for denser fluids. The purpose of this paper is to model friction as a result of mesoscopic pressure fluctuations. First, the simple random walk will be used to approximate the time and space scales below which ballistic motion begins and diffusive motion ends. Following that, pressure fluctuations and the associated time scale will be introduced to explain Brownian motion. As successful as the pressure fluctuation model is for many fluids, it will be shown to fail for fluids like glycerin that have viscosities a thousand times and more that of water.
The measurement of shape parameters of sources in astronomical images is usually performed by assuming that the underlying noise is uncorrelated. Spatial noise correlation is however present in practice due to various observational effects and can affect source shape parameters. This effect is particularly important for measurements of weak gravitational lensing, for which the sought image distortions are typically of the order of only 1%. We compute the effect of correlated noise on two-dimensional gaussian fits in full generality. The noise properties are naturally quantified by the noise autocorrelation function (ACF), which is easily measured in practice. We compute the resulting bias on the mean, variance and covariance of the source parameters, and the induced correlation between the shapes of neighboring sources. We show that these biases are of second order in the inverse signal-to-noise ratio of the source, and could thus be overlooked if bright stars are used to monitor systematic distortions. Radio interferometric surveys are particularly prone to this effect because of the long-range pixel correlations produced by the Fourier inversion involved in their image construction. As a concrete application, we consider the search for weak lensing by large-scale structure with the FIRST radio survey. We measure the noise ACF for a FIRST coadded field, and compute the resulting ellipticity correlation function induced by the noise. In comparison with the weak-lensing signal expected in CDM models, the noise correlation effect is important on small angular scales, but is negligible for source separations greater than about 1 arcmin. We also discuss how noise correlation can affect weak-lensing studies with optical surveys.