ShantanuT01/DACTYL-Pretraining huggingface dataset:
We consider the following convolution equation or equivalently stochastic difference equation $$\lam _k = \mu _k*\phi (\lam _{k-1}), k \in \Z \eqno (1) $$ for a given bi-sequence $(\mu _k)$ of probability measures on $\R ^d$ and a linear map $\phi $ on $\R ^d$. We study the solutions of equation (1) by realizing the process $(\mu _k)$ as a measure on $(\R ^d)^\Z$ and rewriting the stochastic difference equation as $\lam = \mu *\tau (\lam )$-any such measure $\lam$ on $(\R ^d)^\Z$ is known as $\tau$-decomposable measure with co-factor $\mu$-where $\tau$ is a suitable weighted shift operator on $(\R ^d)^\Z$. This enables one to study the solutions of (1) in the settings of $\tau$-decomposable measures. A solution $(\lam _k)$ of (1) will be called a fundamental solution if any solution of (1) can be written as $\lam _k*\phi ^k(\rho )$ for some probability measure $\rho $ on $\R ^d$. Motivated by the splitting/factorization theorems for operator decomposable measures, we address the question of existence of fundamental solutions when a solution exists and answer affirmatively via a one-one correspondence between fundamental solutions of (1) and strongly $\tau$-decomposable measures on $(\R ^d)^\Z$ with co-factor $\mu$. We also prove that fundamental solutions are extremal solutions and vice versa. We provide a necessary and sufficient condition in terms of a logarithmic moment condition for the existence of a (fundamental) solution when the noise process is stationary and when the noise process has independent $\ell _p$-paths.
We study a quite general family of nonlinear evolution equations of diffusive type with nonlocal effects. More precisely, we study porous medium equations with a fractional Laplacian pressure, and the problem is posed on a bounded space domain. We prove existence of weak solutions and suitable a priori bounds and regularity estimates.
We study certain consistent families $(F_\lambda)_{\lambda\ge 0}$ of Galton-Watson forests with lifetimes as edge lengths and/or immigrants as progenitors of the trees in $F_\lambda$. Specifically, consistency here refers to the property that for each $\mu\le\lambda$, the forest $F_\mu$ has the same distribution as the subforest of $F_\lambda$ spanned by the black leaves in a Bernoulli leaf colouring, where each leaf of $F_\lambda$ is coloured in black independently with probability $\mu/\lambda$. The case of exponentially distributed lifetimes and no immigration was studied by Duquesne and Winkel and related to the genealogy of Markovian continuous-state branching processes. We characterise here such families in the framework of arbitrary lifetime distributions and immigration according to a renewal process, related to Sagitov's (non-Markovian) generalisation of continuous-state branching renewal processes, and similar processes with immigration.
The couplings of the electroweak effective theory contain information on the heavy-mass scales which are no-longer present in the low-energy Lagrangian. We build a general effective Lagrangian, implementing the electroweak chiral symmetry breaking $SU(2)_L\otimes SU(2)_R\to SU(2)_{L+R}$, which couples the known particle fields to heavier states with bosonic quantum numbers $J^P=0^\pm$ and $1^\pm$. We consider colour-singlet heavy fields that are in singlet or triplet representations of the electroweak group. Integrating out these heavy scales, we analyze the pattern of low-energy couplings among the light fields which are generated by the massive states. We adopt a generic non-linear realization of the electroweak symmetry breaking with a singlet Higgs, without making any assumption about its possible doublet structure. Special attention is given to the different possible descriptions of massive spin-1 fields and the differences arising from naive implementations of these formalisms, showing their full equivalence once a proper short-distance behaviour is required.
A new quantum-theoretical derivation of the elastic and inelastic scattering probability of He atoms from a metal surface, where the energy and momentum exchange with the phonon gas can only occur through the mediation of the surface free-electron density, shows that the Debye-Waller exponent is directly proportional to the electron-phonon mass coupling constant $\lambda$. The comparison between the values of $\lambda$ extracted from existing data on the Debye-Waller factor for various metal surfaces and the $\lambda$ values known from literature indicates a substantial agreement, which opens the possibility of directly extracting the electron-phonon coupling strength in quasi-2D conducting systems from the temperature or incident energy dependence of the elastic Helium atom scattering intensities.
Galaxy formation models and simulations rely on various feedback mechanisms to reproduce the observed baryonic scaling relations and galaxy morphologies. Although dwarf galaxy and giant elliptical properties can be explained using feedback from supernova and active galactic nuclei, Milky Way-sized galaxies still represent a challenge to current theories of galaxy formation. In this paper, we explore the possible role of feedback from stellar radiation in regulating the main properties of disk galaxies such as our own Milky Way. We have performed a suite of cosmological simulations of the same $\sim10^{12} {\rm M}_{\odot}$ halo selected based on its rather typical mass accretion history. We have implemented radiative feedback from young stars using a crude model of radiative transfer for ultraviolet (UV) and infrared (IR) radiation. However, the model is realistic enough such that the dust opacity plays a direct role in regulating the efficiency of our feedback mechanism. We have explored various models for the dust opacity, assuming different constant dust temperatures, as well as a varying dust temperature model. We find that while strong radiative feedback appears as a viable mechanism to regulate the stellar mass fraction in massive galaxies, it also prevents the formation of discs with reasonable morphologies. In models with strong stellar radiation feedback, stellar discs are systematically too thick while the gas disc morphology is completely destroyed due to the efficient mixing between the feedback-affected gas and its surroundings. At the resolution of our simulation suite, we find it impossible to preserve spiral disc morphology while at the same time expelling enough baryons to satisfy the abundance matching constraints.
For a polynomial mapping from S^{n-k} to the Stiefel manifold \widetilde{V}_k(\R^{n}), where n-k is even, there is presented an effective method of expressing the corresponding element of the homotopy group \pi_{n-k}\widetilde{V}_k(\R^{n})\simeq\Z in terms of signatures of quadratic forms. There is also given a method of computing the intersection number for a polynomial immersion from S^m to \R^{2m}.
We show that the particle-hole transformation in the Hubbard model has a crucial role in relating Shastry's R-matrix to the AdS/CFT S-matrix. In addition, we construct an achiral boundary for the open Hubbard chain which possesses twisted Yangian symmetry.
Wang and White (2007) have discussed some problems with N-body simulation methods. These problems are a special case of a more general problem which has been largely unacknowledged for approximately 25 years, and affects results of all dark matter simulations on small scales (the definition of 'small' varying with time). Extensive numerical experiments with multiple types of N-body codes have demonstrated that spurious fluctuations due to particle discreteness grow rapidly even in the presence of substantial small-scale power from the intended model spectrum, and modify the results on scales smaller than the mean comoving interparticle separation. This implies that the spatial resolution of such simulations is typically limited not by the force softening length, often referred to as the 'resolution', and not by the particle density in halos. Instead it is approximately N^-1/3, where N is the mean particle density, and of course depends on and improves very slowly with increased number of particles. This calls into question many results on smaller scales over more than two decades.
We make detailed comparisons between the measured $Q^2$ dependency of nuclear transparency in quasifree electron scattering and theoretical predictions based on Glauber theory. We also describe a theory-free method to extract $T$ from data.
Although nearly one-third of barred galaxies host an inner, secondary bar, the formation and evolution of double barred galaxies remain unclear. We show here an example model of a galaxy, dominated by a live dark matter halo, in which double bars form naturally, without requiring gas, and we follow its evolution for a Hubble time. The inner bar in our model galaxy rotates almost as slowly as the outer bar, and it can reach up to half of its length. The route to the formation of a double bar may be different from that of a single strong bar. Massive dark matter halo or dynamically hot stellar disc may play an important role in the formation of double bars and their subsequent evolution.
Tree-like tableaux are objects in bijection with alternative or permutation tableaux. They have been the subject of a fruitful combinatorial study for the past few years. In the present work, we define and study a new subclass of tree-like tableaux enumerated by Baxter numbers. We exhibit simple bijective links between these objects and three other combinatorial classes: (packed or mosaic) floorplans, twisted Baxter permutations and triples of non-intersecting lattice paths. From several (and unrelated) works, these last objects are already known to be enumerated by Baxter numbers, and our main contribution is to provide a unifying approach to bijections between Baxter objects, where Baxter tree-like tableaux play the key role. We moreover get new enumerative results about alternating twisted Baxter permutations. Finally, we define a new subfamily of floorplans, which we call alternating floorplans, and we enumerate these combinatorial objects.
Let $\xi(t\,,x)$ denote space-time white noise and consider a reaction-diffusion equation of the form \[   \dot{u}(t\,,x)=\tfrac12 u"(t\,,x) + b(u(t\,,x)) + \sigma(u(t\,,x)) \xi(t\,,x), \] on $\mathbb{R}_+\times[0\,,1]$, with homogeneous Dirichlet boundary conditions and suitable initial data, in the case that there exists $\varepsilon>0$ such that $\vert b(z)\vert \ge|z|(\log|z|)^{1+\varepsilon}$ for all sufficiently-large values of $|z|$. When $\sigma\equiv 0$, it is well known that such PDEs frequently have non-trivial stationary solutions. By contrast, Bonder and Groisman (2009) have recently shown that there is finite-time blowup when $\sigma$ is a non-zero constant. In this paper, we prove that the Bonder--Groisman condition is unimproveable by showing that the reaction-diffusion equation with noise is "typically" well posed when $\vert b(z) \vert =O(|z|\log_+|z|)$ as $|z|\to\infty$. We interpret the word "typically" in two essentially-different ways without altering the conclusions of our assertions.
We study global minimizers of a continuum Landau-De Gennes energy functional for nematic liquid crystals, in three-dimensional domains, subject to uniaxial boundary conditions. We analyze the physically relevant limit of small elastic constant and show that global minimizers converge strongly, in $W^{1,2}$, to a global minimizer predicted by the Oseen-Frank theory for uniaxial nematic liquid crystals with constant order parameter. Moreover, the convergence is uniform in the interior of the domain, away from the singularities of the limiting Oseen-Frank global minimizer. We obtain results on the rate of convergence of the eigenvalues and the regularity of the eigenvectors of the Landau-De Gennes global minimizer.   We also study the interplay between biaxiality and uniaxiality in Landau-De Gennes global energy minimizers and obtain estimates for various related quantities such as the biaxiality parameter and the size of admissible strongly biaxial regions.
We report the discovery of three stars that, along with the prototype LP40-365, form a distinct class of chemically peculiar runaway stars that are the survivors of thermonuclear explosions. Spectroscopy of the four confirmed LP 40-365 stars finds ONe-dominated atmospheres enriched with remarkably similar amounts of nuclear ashes of partial O- and Si-burning. Kinematic evidence is consistent with ejection from a binary supernova progenitor; at least two stars have rest-frame velocities indicating they are unbound to the Galaxy. With masses and radii ranging between 0.20-0.28 Msun and 0.16-0.60 Rsun, respectively, we speculate these inflated white dwarfs are the partly burnt remnants of either peculiar Type Iax or electron-capture supernovae. Adopting supernova rates from the literature, we estimate that ~20 LP40-365 stars brighter than 19 mag should be detectable within 2 kpc from the Sun at the end of the Gaia mission. We suggest that as they cool, these stars will evolve in their spectroscopic appearance, and eventually become peculiar O-rich white dwarfs. Finally, we stress that the discovery of new LP40-365 stars will be useful to further constrain their evolution, supplying key boundary conditions to the modelling of explosion mechanisms, supernova rates, and nucleosynthetic yields of peculiar thermonuclear explosions.
We demonstrate the existence of distribution functions that can be used to represent spherical massless cored stellar systems embedded in cuspy dark matter halos with constant mildly tangential velocity anisotropy. In particular, we derive analytically the functional form of the distribution function for a Plummer stellar sphere in a Hernquist dark halo, for \beta_0 = -0.5 and for different degrees of embedding. This particular example satisfies the condition that the central logarithmic slope of the light profile \gamma_0 > 2 \beta_0. Our models have velocity dispersion profiles similar to those observed in nearby dwarf spheroidal galaxies. Hence they can be used to generate initial conditions for a variety of problems, including N-body simulations that may represent dwarf galaxies in the Local Group.
In this article, we study the masses and pole residues of the pseudoscalar-diquark-pseudoscalar-antidiquark type and vector-diquark-vector-antidiquark type scalar hidden-charm $cu\bar{c}\bar{d}$ ($cu\bar{c}\bar{s}$) tetraquark states with QCD sum rules by taking into account the contributions of the vacuum condensates up to dimension-10 in the operator product expansion. The predicted masses can be confronted with the experimental data in the future. Possible decays of those tetraquark states are also discussed.
Packing problems, even of objects with regular geometries, are in general non-trivial. For few special shapes, the features of crystalline as well as random, irregular two-dimensional (2D) packings are known. The packing of 2D crosses does not yet belong to the category of solved problems. We demonstrate in experiments with crosses of different aspect ratios (arm width to length) which packing fractions are actually achieved by random packing, and we compare them to densest regular packing structures. We determine local correlations of the orientations and positions after ensembles of randomly placed crosses were compacted in the plane until they jam. Short-range orientational order is found over 2 to 3 cross lengths. Similarly, correlations in the spatial distributions of neighbors extend over 2 to 3 crosses. Apparently, there is no simple relation between the geometries of the crosses and peaks in the spatial correlation functions. Some features of the orientational correlations, however, are intuitively evident.
Balls and bins models are classical probabilistic models where balls are added to bins at random according to a certain rule. The balls and bins model with feedback is a non-linear generalisation of the P\'olya urn, where the probability of a new ball choosing a bin with $m$ balls is proportional to $m^{\alpha}$, with $\alpha$ being the feedback parameter. It is known that if the feedback is positive (i.e. $\alpha>1$) then the model is monopolistic: there is a finite time after which one of the bins will receive all incoming balls. We consider a time-dependent version of this model, where $\sigma_n$ independent balls are added at time $n$ instead of just one. We show that if $\alpha>1$ then one of the bins gets all but a negligible number of balls, and identify a phase transition in the growth of $(\sigma_n)$ between the monopolistic and non-monopolistic behaviour. We also describe the critical regime, where the probability of monopoly is strictly between zero and one. Finally, we show that in the feedback-less case $\alpha=1$ no dominance occurs, that is, each bin gets a non-negligible proportion of balls eventually. This is in sharp contrast with a similar model where new balls added at time $n$ are all placed in the same bin rather than independently.
Ruthenium-based perovskite systems are attractive because their Structural, electronic and magnetic properties can be systematically engineered. SrRuO$_3$/SrTiO$_3$ superlattice, with its period consisting of one unit cell each, is very sensitive to strain change. Our first-principles simulations reveal that in the high tensile strain region, it transits from a ferromagnetic (FM) metal to an antiferromagnetic (AFM) insulator with clear tilted octahedra, while in the low strain region, it is a ferromagnetic metal without octahedra tilting. Detailed analyses of three spin-down Ru-t$_{2g}$ orbitals just below the Fermi level reveal that the splitting of these orbitals underlies these dramatic phase transitions, with the rotational force constant of RuO$_6$ octahedron high up to 16 meV/Deg$^2$, 4 times larger than that of TiO$_6$. Differently from nearly all the previous studies, these transitions can be probed optically through the diagonal and off-diagonal dielectric tensor elements. For one percent change in strain, our experimental spin moment change is -0.14$\pm$0.06 $\mu_B$, quantitatively consistent with our theoretical value of -0.1 $\mu_B$.
As is well known, the 0 - 0 component of the Schwarzschild space can be obtained by the requirement that the geodesic of slowly moving particles match the Newtonian equation. Given this result, we show here that the remaining components can be obtained by requiring that the inside of a Newtonian ball of dust matched at a free falling radius with the external space determines that space to be Schwarzschild, if no pathologies exist. Also we are able to determine that the constant of integration that appears in the Newtonian Cosmology, coincides with the spatial curvature of the FLRW metric. These results are of interest at least in two respects, one from the point of view of its pedagogical value of teaching General Relativity without in fact using Einstein's equation and second, the fact that some results attributed to General Relativity can be obtained without using General Relativity indicates that these results are more general than the particular dynamics specified by General Relativity.
Cluster concepts have been extremely useful in elucidating many problems in physics. Percolation theory provides a generic framework to study the behavior of the cluster distribution. In most cases the theory predicts a geometrical transition at the percolation threshold, characterized in the percolative phase by the presence of a spanning cluster, which becomes infinite in the thermodynamic limit. Standard percolation usually deals with the problem when the constitutive elements of the clusters are randomly distributed. However correlations cannot always be neglected. In this case correlated percolation is the appropriate theory to study such systems. The origin of correlated percolation could be dated back to 1937 when Mayer [1] proposed a theory to describe the condensation from a gas to a liquid in terms of mathematical clusters (for a review of cluster theory in simple fluids see [2]). The location for the divergence of the size of these clusters was interpreted as the condensation transition from a gas to a liquid. One of the major drawback of the theory was that the cluster number for some values of thermodynamic parameters could become negative. As a consequence the clusters did not have any physical interpretation [3]. This theory was followed by Frenkel's phenomenological model [4], in which the fluid was considered as made of non interacting physical clusters with a given free energy. This model was later improved by Fisher [3], who proposed a different free energy for the clusters, now called droplets, and consequently a different scaling form for the droplet size distribution. This distribution, which depends on two geometrical parameters, has the nice feature that the mean droplet size exhibits a divergence at the liquid-gas critical point.
In this review we discuss a recent proposal to perform partial Bell-state (parity) measurements on two-electron spin states for electrons confined to quantum dots. The realization of this proposal would allow for a physical implementation of measurement-based quantum computing. In addition, we consider the primary sources of energy relaxation and decoherence which provide the ultimate limit to all proposals for quantum information processing using electron spins in quantum dots. We give an account of the Hamiltonians used for the most important interactions (spin-orbit and hyperfine) and survey some of the recent work done to understand dynamics, control, and decoherence under the action of these Hamiltonians. We conclude the review with a table of important decay times found in experiment, and relate these time scales to the potential viability of measurement-based quantum computing.
This paper presents estimates for the distribution of the exit time from balls and short time asymptotics for measure metric Dirichlet spaces. The estimates cover the classical Gaussian case, the sub-diffusive case which can be observed on particular fractals and further less regular cases as well. The proof is based on a new chaining argument and it is free of volume growth assumptions.
The main finding of this work is that the standard image classification pipeline, which consists of dictionary learning, feature encoding, spatial pyramid pooling and linear classification, outperforms all state-of-the-art face recognition methods on the tested benchmark datasets (we have tested on AR, Extended Yale B, the challenging FERET, and LFW-a datasets). This surprising and prominent result suggests that those advances in generic image classification can be directly applied to improve face recognition systems. In other words, face recognition may not need to be viewed as a separate object classification problem.   While recently a large body of residual based face recognition methods focus on developing complex dictionary learning algorithms, in this work we show that a dictionary of randomly extracted patches (even from non-face images) can achieve very promising results using the image classification pipeline. That means, the choice of dictionary learning methods may not be important. Instead, we find that learning multiple dictionaries using different low-level image features often improve the final classification accuracy. Our proposed face recognition approach offers the best reported results on the widely-used face recognition benchmark datasets. In particular, on the challenging FERET and LFW-a datasets, we improve the best reported accuracies in the literature by about 20% and 30% respectively.
Superconductivity, superfluidity, condensation, cluster formation, etc. are phenomena that might occur in many-particle systems. These are due to residual interactions between the particles. To explain these phenomena consistently in a microscopic approach, at some point, one needs to solve few-body equations that are modified because of the Pauli principle and the interactions of the many particles around.