ShantanuT01/DACTYL-Pretraining huggingface dataset:
The scaled standard Wigner matrix (symmetric with mean zero, variance one i.i.d. entries), and its limiting eigenvalue distribution, namely the semi-circular distribution, has attracted much attention. The $2k$th moment of the limit equals the number of non-crossing pair-partitions of the set $\{1, 2, \ldots, 2k\}$. There are several extensions of this result in the literature. In this paper we consider a unifying extension which also yields additional results.   Suppose $W_n$ is an $n\times n$ symmetric matrix where the entries are independently distributed. We show that under suitable assumptions on the entries, the limiting spectral distribution exists in probability or almost surely. The moments of the limit can be described through a set of partitions which in general is larger than the set of non-crossing pair-partitions. This set gives rise to interesting enumerative combinatorial problems.   Several existing limiting spectral distribution results follow from our results. These include results on the standard Wigner matrix, the adjacency matrix of a sparse homogeneous Erd\H{o}s-R\'{e}nyi graph, heavy tailed Wigner matrix, some banded Wigner matrices, and Wigner matrices with variance profile. Some new results on these models and their extensions also follow from our main results.
We investigate the coefficients of the polynomial \[ S_{m,r}^n(\ell)=r^n+(m+r)^n+(2m+r)^n+\cdots+((\ell-1)m+r)^n. \] We prove that these can be given in terms of Stirling numbers of the first kind and $r$-Whitney numbers of the second kind. Moreover, we prove a necessary and sufficient condition for the integrity of these coefficients.
We show that certain space of vector valued harmonic weak Maass forms of half integral weight is isomorphic to a space of scalar valued ones whose Fourier coefficients are supported on suitable progressions. This kind of result for holomorphic modular forms was proved by Eichler and Zagier.
We propose an end-to-end network for image generation from given structured-text that consists of the visual-relation layout module and the pyramid of GANs, namely stacking-GANs. Our visual-relation layout module uses relations among entities in the structured-text in two ways: comprehensive usage and individual usage. We comprehensively use all available relations together to localize initial bounding-boxes of all the entities. We also use individual relation separately to predict from the initial bounding-boxes relation-units for all the relations in the input text. We then unify all the relation-units to produce the visual-relation layout, i.e., bounding-boxes for all the entities so that each of them uniquely corresponds to each entity while keeping its involved relations. Our visual-relation layout reflects the scene structure given in the input text. The stacking-GANs is the stack of three GANs conditioned on the visual-relation layout and the output of previous GAN, consistently capturing the scene structure. Our network realistically renders entities' details in high resolution while keeping the scene structure. Experimental results on two public datasets show outperformances of our method against state-of-the-art methods.
We propose a scheme which encodes information in both the longitudinal and spatial transverse phases of a continuous-wave optical beam. A split detector-based interferometric scheme is then introduced to optimally detect both encoded phase signals. In contrast to present-day optical storage devices, our phase coding scheme has an information storage capacity which scales with the power of the read-out optical beam. We analyse the maximum number of encoding possibilities at the shot noise limit. In addition, we show that using squeezed light, the shot noise limit can be overcome and the number of encoding possibilities increased. We discuss a possible application of our phase coding scheme for increasing the capacities of optical storage devices.
In a hierarchical universe, mergers may be an important mechanism not only in increasing the mass of galaxies but also in driving the color and morphological evolution of galaxies. We use a large sample of ~1000 simulated galaxies of stellar mass greater than 10^9.6 solar masses (for ~4800 observations at multiple redshifts) from a high-res (0.46 h^{-1} kpc) cosmological simulation to determine under what circumstances being a member of a pair influences galaxy properties at z <= 0.2. We identify gravitationally bound pairs, and find a relative fraction of blue-blue, red-red, and blue-red pairs that agrees with observations (Lin et al. 2010). Pairs tend to avoid the extreme environments of clusters and void centres. While pairs in groups can include galaxies that are both blue, both red, or one of each color, in the field it is rare for pair galaxies to both be red. We find that physically bound pairs closer than 250 h^{-1} kpc tend to have higher sSFRs than the galaxy population as a whole. However, the sSFR of a bound galaxy relative to galaxies in a comparable local density environment (determined by the distance to the fifth nearest neighbor, rho_5), differs depending on the local density. In regions of high rho_5 the bound population has a higher fraction of star-forming (bluer) galaxies, whereas there is very little difference between bound and unbound galaxies in low rho_5 regions. This effect on the star-forming fraction may be driven by the higher fraction of bound HI-rich galaxies compared to unbound galaxies, particularly at high local densities. It appears that being in a pair has an incremental, but not overwhelming, effect on the star formation rate of the paired galaxies, compared to the more pronounced trend where galaxies overall have low sSFR (are red) in clusters and higher sSFR (blue) at the centre of voids. This trend depends most strongly on rho_5.(abridged)
We propose models of two dimensional paramagnetic semiconductors where the intrinsic spin Hall effect is exactly quantized in integer units of a topological charge. The model describes a topological insulator in the bulk, and a "holographic metal" at the edge, where the number of extended edge states crossing the Fermi level is dictated by (exactly equal to) the bulk topological charge. We also demonstrate the spin Hall effect explicitly in terms of the spin accumulation caused by the adiabatic flux insertion.
We study the long-term evolution of non-radiative geometrically thick ($H/R\approx 0.5$) accretion flows using 3D global ideal MHD simulations and a pseudo-Newtonian gravity. We find that resolving the scale height with 42 grid points is adequate to obtain convergence with the product of quality factors $\langle \langle Q_{\theta} \rangle \rangle \langle \langle Q_{\phi} \rangle \rangle \geq 300$ and magnetic tilt angle $\theta_B \sim 13^{\circ}-14^{\circ}$. Like previous global isothermal thin disk simulations, we find stronger mean magnetic fields for the restricted azimuthal domains. Imposing periodic boundary conditions with the azimuthal extent smaller than $2\pi$ make the turbulent field at low $m$ appear as a mean field in the runs with smaller azimuthal extent. But unlike previous works, we do not find a monotonic trend in turbulence with the azimuthal extent. We conclude that the minimum azimuthal extent should be $\geq \pi/2$ to capture the flow structure, but a full $2 \pi$ extent is necessary to study the dynamo. We find an intermittent dynamo cycle, with $\alpha$-quenching playing an important role in the nonlinear saturated state. Unlike previous local studies, we find almost similar values of kinetic and magnetic $\alpha$-s, giving rise to an irregular distribution of dynamo-$\alpha$. The effects of dynamical quenching are shown explicitly for the first time in global simulations of accretion flows.
In this paper we address many of the fundamental open questions regarding the glassy behavior of the magnetic/electronic phase segregated state in rare earth perovskites. In particular, magnetic relaxation experiments support that the collective effects (memory, ageing, etc.) are due to interparticle interactions, rather than the double-exchange vs. superexchange competition. A careful study of the non-linear susceptibility in the critical region is performed, and the critical exponents contrasted with those of conventional spin-glasses and concentrated quenched ferrofluids. The phase segregated state constitutes a sort of self-generated assembly of magnetic particles in which magnetic interaction introduces collectivity among the clusters.
In this work, we present a general non-equilibrium ensemble formalism based on the subdynamic equation (SKE). The constructing procedure is to use a similarity transformation between Gibbsian ensemble formalism and the non-equilibrium ensemble formalism. The obtained density distribution is a projected one that can represent essence part of (irreversible) evolution of the density distribution, by which a generalized reduced density distribution for the quantum canonical ensembles is studied and applications in Cayley tree and spin network are discussed.
We study the geometry and mechanics (both classical and quantum) of potential wells described by squares of Chebyshev polynomials. We show that in a small neighbourhood of the locus cut out by them in the space of hyperelliptic curves, these systems exhibit low-orders/low-orders resurgence, where perturbative fluctuations about the vacuum determine perturbative fluctuations about non-perturbative saddles.
The cross section for $\Theta^+$ production from the reaction $\gamma p\to\pi^+K^-\Theta^+$, which was observed in the CLAS experiment at the Jefferson National Laboratory, is evaluated in a hadronic model that includes couplings of $\Theta^+$ to both $KN$ and $K^*N$. With their coupling constants determined from the empirical $\pi NN(1710)$ and $\rho NN(1710)$ coupling constants using the SU(3) symmetry, the cross section for this reaction has been evaluated by taking $\Theta^+$ to have spin 1/2 and isospin 0 but either positive or negative parity. We find that the cross section is 10-15 nb if $\Theta^+$ has positive parity as predicted by the chiral soliton model. The cross section is reduced by more than a factor of 10 if $\Theta^+$ has negative parity as given by lattice QCD studies. For both parities, the differential distribution peaks at small negative four momentum transfer as expected from the dominating $t$-channel kaon-exchange diagram that involves only the coupling of $\Theta^+$ to $KN$.
Ferromagnetic (FM) semimetals Fe$_n$GeTe$_2$(n=3, 4, 5), exhibit several symmetry-protected band-crossing points or lines near the Fermi energy (E$_F$) and these topological properties of energy bands lead to interesting transport properties. We study these materials employing the first-principle calculations and the tight-binding Hamiltonian constructed by fitting the parameters of the first principles calculation. In the presence of spin-orbit coupling (SOC) for n=3,5 a large Berry curvature (BC) concentrated on the nodal lines is observed. The consequence of the correlation of the topological nodal line and magnetic moments on anomalous Hall conductivity (AHC) $\sigma_{xy}$ and anomalous Nernst conductivity (ANC) $\alpha_{xy}$ have been investigated. We find $\sigma_{xy}=150$ S/cm for n=3, 295 S/cm for n=4, and 90 S/cm for n=5 at 0 K, while the ANC is observed as $\alpha_{xy}=0.55$ A/Km for n=3, 0.10 A/Km for n=5, and 0.80 A/Km for n=4, at the E$_F$ at room temperature. Our calculated AHC values at 0 K, i.e., 150 S/cm for Fe$_3$GeTe$_2$ and 90 S/cm Fe$_5$GeTe$_2$, are consistent with the experimentally reported values. Also the experimentally reported value of ANC for Fe$_5$GeTe$_2$ is close to our calculated value at room temperature, i.e., 0.10 A/Km.
We look at algebraic embeddings of the Cremona group in $n$ variables $Cr_n(C)$ to the group of birational transformations $Bir(M)$ of an algebraic variety $M$. First we study geometrical properties of an example of an embedding of $Cr_2(C)$ into $Cr_5(C)$ that is due to Gizatullin. In a second part, we give a full classification of all algebraic embeddings of $Cr_2(C)$ into $Bir(M)$, where $dim(M)=3$, and generalize this result partially to algebraic embeddings of $Cr_n(C)$ into $Bir(M)$, where $dim(M)=n+1$, for arbitrary $n\geq 2$. In particular, this yields a classification of all algebraic $PGL_{n+1}(C)$-actions on smooth projective varieties of dimension $n+1$ that can be extended to rational actions of $Cr_n(C)$.
In this article, we study the potential theory of normal tempered stable process which is obtained by time-changing the Brownian motion with a tempered stable subordinator. Precisely, we study the asymptotic behavior of potential density and Levy density associated with tempered stable subordinator and the Green function and the Levy density associated with the normal tempered stable process. We also provide the corresponding results for normal inverse Gaussian process which is a well studied process in literature.
We unify two paradigmatic mesoscopic mechanisms for the emergence of nonextensive statistics, namely the multiplicative noise mechanism leading to a {\it linear} Fokker-Planck (FP) equation with {\it inhomogenous} diffusion coefficient, and the non-Markovian process leading to the {\it nonlinear} FP equation with {\it homogeneous} diffusion coefficient. More precisely, we consider the equation $\frac{\partial p(x,t)}{\partial t}=-\frac{\partial}{\partial x}[F(x) p(x,t)] + 1/2D \frac{\partial^2}{\partial x^2} [\phi(x,p)p(x,t)]$, where $D \in {\cal R}$ and $F(x)=-\partial V(x) /\partial x$, $V(x)$ being the potential under which diffusion occurs. Our aim is to find whether $\phi(x,p)$ exists such that the inhomogeneous linear and the homogeneous nonlinear FP equations become unified in such a way that the (ubiquitously observed) $q$-exponentials remain as stationary solutions. It turns out that such solutions indeed exist for a wide class of systems, namely when $\phi(x,p)=[A+BV(x)]^\theta [p(x,t)]^{\eta}$, where $A$, $B$, $\theta$ and $\eta$ are (real) constants.   Our main result can be sumarized as follows: For $\theta \neq 1$ and arbitrary confining potential $V(x)$, $p(x,\infty) \propto \lbrace 1-\beta(1-q)V(x)\rbrace ^{1/(1-q)} \equiv e_q^{-\beta V(x)}$, where $q= 1+ \eta/(\theta-1)$. The present approach unifies into a single mechanism, essentially {\it long memory}, results currently discussed and applied in the literature.
We prove that if $G$ is a $k$-partite graph on $n$ vertices in which all of the parts have order at most $n/r$ and every vertex is adjacent to at least a $1-1/r+o(1)$ proportion of the vertices in every other part, then $G$ contains the $(r-1)$-st power of a Hamiltonian cycle
We study the star formation quenching mechanism in cluster galaxies by fitting the SED of the Herschel Reference Survey, a complete volume-limited K-band-selected sample of nearby galaxies including objects in different density regions, from the core of the Virgo cluster to the general field. The SED are fitted using the CIGALE SED modelling code. The truncated activity of cluster galaxies is parametrised using a specific SFH with 2 free parameters, the quenching age QA and the quenching factor QF. These 2 parameters are crucial for the identification of the quenching mechanism which acts on long timescales if starvation while rapid and efficient if ram pressure. To be sensitive to an abrupt and recent variation of the star formation activity, we combine in a new way 20 UV to FIR photometric bands with 3 age-sensitive Balmer line absorption indices extracted from available medium-resolution integrated spectroscopy and with Halpha narrow band imaging data. The use of a truncated SFH significantly increases the quality of the fit in those objects whose atomic gas content has been removed during the interaction with the hostile cluster environment. The typical QA of the perturbed late-type galaxies is QA < 300 Myr whenever the activity of star formation is reduced by 50% < QF <= 80% and QA < 500 Myr for QF > 80%, while that of the quiescent early-types is QA ~ 1-3 Gyr. The fraction of late-types with a star formation activity reduced by QF > 80% and with an HI-deficiency parameter HI-def > 0.4 drops by a factor of ~ 5 from the inner half virial radius of the Virgo cluster, where the hot diffuse X-ray emitting gas of the cluster is located, to the outer regions. The efficient quenching of the star formation activity observed in Virgo suggests that the dominant stripping process is ram pressure. We discuss the implication of this result in the cosmological context of galaxy evolution.
A Born--Infeld theory describing a $D2$--brane coupled to a 3--form RR potential is reconsidered and a new type of static solution is obtained which is even stable.
Let $D_n(1,x)$ be the $n$th reversed Dickson polynomial. The power sums $\sum_{a\in\Bbb F_q}D_n(1,a)^i$, $i=1,2$, have been determined recently. In this paper we give an evaluation of the sum $\sum_{a\in\Bbb F_q}D_n(1,a)^3$. This result implies new necessary conditions for $D_n(1,x)$ to be a permutation polynomial over $\Bbb F_q$.
We show that the $\operatorname{Sine}_{\beta}$ point process, defined as the scaling limit of the Circular Beta Ensemble when the dimension goes to infinity, and generalizing the determinantal sine-kernel process, is rigid in the sense of Ghosh and Peres: the number of points in a given bounded Borel set $B$ is almost surely equal to a measurable function of the position of the points outside $B$.
Let $r,n$ be two natural numbers and let $H(r,n)$ denote the maximal absolute value of $r$th coefficient of divisors of $x^n-1$. In this paper, we show that $\sum_{n\leq x}H(r,n)$ is asymptotically equal to $c(r)x(\log x)^{2^r-1}$ for some constant $c(r)>0$. Furthermore, we give an explicit expression of $c(r)$ in terms of $r$.
Alloying Ti-Al-N coatings with Ta has proven to enhance their hardness, thermal stability, and oxidation resistance. However, especially for arc-evaporated Ti-Al-Ta-N coatings only limited information on the detailed influence of the elements on various properties is available. Therefore, we have developed arc-evaporated Ti1-x-yAlxTayN coatings with various Al (x = 0.50 - 0.65) and Ta (y = 0.00 - 0.15) contents. While the thermal stability of our coatings during annealing in inert He atmosphere increases with increasing Ta content, best results are obtained for specific Ta-Al ratios during oxidation. Single phase cubic Ti0.32Al0.60Ta0.08N yields a mass-gain of only ~5 % after 5 h at 950 {\deg}C in synthetic air, whereas Ti0.35Al0.65N is completely oxidized after 15 min. This is in part based on the suppressed anatase and direct rutile TiO2 formation at a defined Ta-Al content. Consequently, the anatase-to-rutile transformation, generally observed for Ti1-xAlxN, is absent. This reduces the generation of pores and cracks within the oxide scale and especially at the nitride-oxide interface, leading to the formation of a protective rutile and corundum based oxide scale. This is also reflected in the pronounced decrease in activation energy for the protective scale formation from 232 kJ/mol for Ti0.35Al0.65N down to 14.5 kJ/mol for Ti0.32Al0.60Ta0.08N. Based on our results we can conclude that especially phase transformations within the oxide scale need to be suppressed, as the connected volume changes lead to the formation of cracks and pores.
The Walsh transform $\widehat{Q}$ of a quadratic function $Q:F_{p^n}\rightarrow F_p$ satisfies $|\widehat{Q}(b)| \in \{0,p^{\frac{n+s}{2}}\}$ for all $b\in F_{p^n}$, where $0\le s\le n-1$ is an integer depending on $Q$. In this article, we study the following three classes of quadratic functions of wide interest. The class $\mathcal{C}_1$ is defined for arbitrary $n$ as $\mathcal{C}_1 = \{Q(x) = Tr(\sum_{i=1}^{\lfloor (n-1)/2\rfloor}a_ix^{2^i+1})\;:\; a_i \in F_2\}$, and the larger class $\mathcal{C}_2$ is defined for even $n$ as $\mathcal{C}_2 = \{Q(x) = Tr(\sum_{i=1}^{(n/2)-1}a_ix^{2^i+1}) + {\rm Tr_{n/2}}(a_{n/2}x^{2^{n/2}+1}) \;:\; a_i \in F_2\}$. For an odd prime $p$, the subclass $\mathcal{D}$ of all $p$-ary quadratic functions is defined as $\mathcal{D} = \{Q(x) = Tr(\sum_{i=0}^{\lfloor n/2\rfloor}a_ix^{p^i+1})\;:\; a_i \in F_p\}$. We determine the distribution of the parameter $s$ for $\mathcal{C}_1, \mathcal{C}_2$ and $\mathcal{D}$. As a consequence we obtain the distribution of the nonlinearity for the rotation symmetric quadratic Boolean functions, and in the case $p > 2$, our results yield the distribution of the co-dimensions for the rotation symmetric quadratic $p$-ary functions, which have been attracting considerable attention recently. We also present the complete weight distribution of the subcodes of the second order Reed-Muller codes corresponding to $\mathcal{C}_1$ and $\mathcal{C}_2$.
This paper studies vehicle attribute recognition by appearance. In the literature, image-based target recognition has been extensively investigated in many use cases, such as facial recognition, but less so in the field of vehicle attribute recognition. We survey a number of algorithms that identify vehicle properties ranging from coarse-grained level (vehicle type) to fine-grained level (vehicle make and model). Moreover, we discuss two alternative approaches for these tasks, including straightforward classification and a more flexible metric learning method. Furthermore, we design a simulated real-world scenario for vehicle attribute recognition and present an experimental comparison of the two approaches.
A complete Lagrangian and Hamiltonian description of the theory of self-gravitating light-like matter shells is given in terms of gauge-independent geometric quantities. For this purpose the notion of an extrinsic curvature for a null-like hypersurface is discussed and the corresponding Gauss-Codazzi equations are proved. These equations imply Bianchi identities for spacetimes with null-like, singular curvature. Energy-momentum tensor-density of a light-like matter shell is unambiguously defined in terms of an invariant matter Lagrangian density. Noether identity and Belinfante-Rosenfeld theorem for such a tensor-density are proved. Finally, the Hamiltonian dynamics of the interacting system: ``gravity + matter'' is derived from the total Lagrangian, the latter being an invariant scalar density.
The most amazing (at least to me) result in Enumerative Combinatorics is Dominique Gouyou-Beauchamps and Xavier Viennot's theorem that states that the number of so-called directed animals with compact source (that are equivalent, via Viennot's beautiful concept of heaps, to towers of dominoes, that I take the liberty of renaming xaviers) with n+1 points equals 3^n. This amazing result received an even more amazing proof by Jean B\'etrema and Jean-Guy Penaud. Both theorem and proof deserve to be better known! Hence this article, that is also accompanied by a comprehensive Maple package http://www.math.rutgers.edu/~zeilberg/tokhniot/BORDELAISE that implements everything (and much more)
The paper has been withdrawn by the authors for modification.