ShantanuT01/DACTYL-Pretraining huggingface dataset:
Let $b$ be an algebraic number with $|b|>1$ and $\mathcal{H}$ a finite set of algebraic numbers. We study the transcendence of numbers of the form $\sum_{n=0}^\infty \frac{a_n}{b^n}$ where $a_n \in \mathcal{H}$ for all $n\in\mathbb{N}$. We assume that the sequence $(a_n)_{n=0}^\infty$ is generated by coding the orbit of a point under an irrational rotation of the unit circle. In particular, this assumption holds whenever the sequence is Sturmian. Our main result shows that all numbers of the above form are transcendental. We moreover give sufficient conditions for a finite set of such numbers to be linearly independent over~$\bar{\mathbb{Q}}$.
The firefighter problem with $k$ firefighters on an infinite graph $G$ is an iterative graph process, defined as follows: Suppose a fire breaks out at a given vertex $v\in V(G)$ on Turn 1. On each subsequent even turn, $k$ firefighters protect $k$ vertices that are not on fire, and on each subsequent odd turn, any vertex that is on fire spreads the fire to all adjacent unprotected vertices. The firefighters' goal is to eventually stop the spread of the fire. If there exists a strategy for $k$ firefighters to eventually stop the spread of the fire, then we say $G$ is $k$-containable.   We consider the firefighter problem on the hexagonal grid, which is the graph whose vertices and edges are exactly the vertices and edges of a regular hexagonal tiling of the plane. It is not known if the hexagonal grid is $1$-containable. In arXiv:1305.7076 [math.CO], it was shown that if the firefighters have one firefighter per turn and one extra firefighter on two turns, the firefighters can contain the fire. We improve on this result by showing that even with only one extra firefighter on one turn, the firefighters can still contain the fire.   In addition, we explore $k$-containability for birth sequence trees, which are infinite rooted trees that have the property that every vertex at the same level has the same degree. A birth sequence forest is an infinite forest, each component of which is a birth sequence tree. For birth sequence trees and forests, the fire always starts at the root of each tree. We provide a pseudopolynomial time algorithm to decide if all the vertices at a fixed level can be protected or not.
Recent molecular line observations with ALMA in several nearby Seyferts have revealed the existence of molecular tori, and the nature of gas flows at 10-20~pc scale. At 100~pc scale, or kpc-scale, previous NOEMA work on gravitational torques had shown that only about one third of Seyfert galaxies experienced molecular inflow and central fueling, while in most cases the gas was stalled in rings. At higher resolution, i.e. 10-20~pc scale, it is possible now to see in some cases AGN fueling due to nuclear trailing spirals, influenced by the black hole potential. This brings smoking gun evidence for nuclear fueling. In our sample galaxies, the angular resolution of up to 60~mas allows us to reach the black hole (BH) sphere of influence and the BH mass can be derived more directly than with the M-sigma relation.
Let $(B(t),\,t\ge0)$ denote the standard, one-dimensional Wiener process and $(\ell(y,t);\, y\in\mathbb{R},\, t\ge0)$ its local time at level $y$ up to time $t$. Then $\big( (B(t),\, \ell(B(t),t)),\; t\ge0 \big)$ is a random path that fills the upper half-plane, covering one unit of area per unit time.
A commonly used model for fault-tolerant computation is that of cellular automata. The essential difficulty of fault-tolerant computation is present in the special case of simply remembering a bit in the presence of faults, and that is the case we treat in this paper. We are concerned with the degree (the number of neighboring cells on which the state transition function depends) needed to achieve fault tolerance when the fault rate is high (nearly 1/2). We consider both the traditional transient fault model (where faults occur independently in time and space) and a recently introduced combined fault model which also includes manufacturing faults (which occur independently in space, but which affect cells for all time). We also consider both a purely probabilistic fault model (in which the states of cells are perturbed at exactly the fault rate) and an adversarial model (in which the occurrence of a fault gives control of the state to an omniscient adversary). We show that there are cellular automata that can tolerate a fault rate $1/2 - \xi$ (with $\xi>0$) with degree $O((1/\xi^2)\log(1/\xi))$, even with adversarial combined faults. The simplest such automata are based on infinite regular trees, but our results also apply to other structures (such as hyperbolic tessellations) that contain infinite regular trees. We also obtain a lower bound of $\Omega(1/\xi^2)$, even with purely probabilistic transient faults only.
Let $K$ be a self-similar set satisfying the open set condition. Following Kaimanovich's elegant idea, it has been proved that on the symbolic space $X$ of $K$ a natural augmented tree structure ${\mathfrak E}$ exists; it is hyperbolic, and the hyperbolic boundary $\partial_HX$ with the Gromov metric is H\"older equivalent to $K$. In this paper we consider certain reversible random walks with return ratio $0< \lambda <1$ on $(X, {\mathfrak E})$. We show that the Martin boundary ${\mathcal M}$ can be identified with $\partial_H X$ and $K$. With this setup and a device of Silverstein, we obtain precise estimates of the Martin kernel and the Na\"{i}m kernel in terms of the Gromov product. Moreover, the Na\"{i}m kernel turns out to be a jump kernel satisfying the estimate $\Theta (\xi, \eta) \asymp |\xi-\eta|^{-(\alpha+ \beta)}$, where $\alpha$ is the Hausdorff dimension of $K$ and $\beta$ depends on $\lambda$. For suitable $\beta$, the kernel defines a regular non-local Dirichlet form on $K$. This extends the results of Kigami concerning random walks on certain trees with Cantor-type sets as boundaries.
We consider an arbitrarily charged polymer driven by a weak field through a gel according to the rules of the Rubinstein-Duke model. The probability distribution in the stationary state is related to that of the model in which only the head is charged. Thereby drift velocity, diffusion constant and orientation of any charged polymers are expressed in terms of those of the central model. Mapping the problem on a random walk of a tagged particle along a one-dimenional chain, leads to a unified scaling expression for the local orientation. It provides also an elucidation of the role of corrections to scaling.
We propose a robust classification algorithm for curves in 2D and 3D, under the special and full groups of affine transformations. To each plane or spatial curve we assign a plane signature curve. Curves, equivalent under an affine transformation, have the same signature. The signatures introduced in this paper are based on integral invariants, which behave much better on noisy images than classically known differential invariants. The comparison with other types of invariants is given in the introduction. Though the integral invariants for planar curves were known before, the affine integral invariants for spatial curves are proposed here for the first time. Using the inductive variation of the moving frame method we compute affine invariants in terms of Euclidean invariants. We present two types of signatures, the global signature and the local signature. Both signatures are independent of parameterization (curve sampling). The global signature depends on the choice of the initial point and does not allow us to compare fragments of curves, and is therefore sensitive to occlusions. The local signature, although is slightly more sensitive to noise, is independent of the choice of the initial point and is not sensitive to occlusions in an image. It helps establish local equivalence of curves. The robustness of these invariants and signatures in their application to the problem of classification of noisy spatial curves extracted from a 3D object is analyzed.
We consider a coupled PDE-ODE system describing the motion of the rigid body in a container filled with the incompressible, viscous fluid. The fluid and the rigid body are coupled via Navier slip boundary condition. We prove that the local in time strong solution is unique in the larger class of weak solutions on the interval of its existence. This is the first weak-strong uniqueness result in the area of fluid-structure interaction with a moving boundary.
It is standard in theoretical neutrino astrophysics to use a broken power law approximation, based on the Band function, to describe the average photon flux of the prompt emission of Gamma-Ray Bursts. We will show that this approximation overestimates the contribution of high energy gamma-rays (and underestimates low energy gamma-rays). As a consequence models that rely on this approximation overestimate neutrino event rate by a factor of approx 2 depending on Earth's column density in the direction of the GRB. Furthermore the characteristic energy of neutrinos that trigger a km^3 detector is typically 10^{16} eV, higher than previously predicted. We also provide a new broken power law approximation to the Band function and show that it properly represents the photon spectra.
We deal with linear parabolic (in sense of Petrovskii) systems of order 2b with discontinuous principal coefficients. A'priori estimates in Sobolev and Sobolev--Morrey spaces are proved for the strong solutions by means of potential analysis and boundedness of certain singular integral operators with kernels of mixed homogeneity. As a byproduct, precise characterization of the Morrey, BMO and H\"older regularity is given for the solutions and their derivatives up to order 2b-1. Some counterexamples of recent publications announcing estimates similar to the obtained here are also given.
Measurements of galaxy cluster kinematics are important in understanding the dynamical state and evolution of clusters of galaxies, as well as constraining cosmological models. While it is well established that clusters exhibit non-spherical geometries, evident in the distribution of galaxies on the sky, azimuthal variations of galaxy kinematics within clusters have yet to be observed. Here we measure the azimuthal dependence of the line-of-sight velocity dispersion profile in a stacked sample of 1743 galaxy clusters from the Sloan Digital Sky Survey (SDSS). The clusters are drawn from the SDSS DR8 redMaPPer catalog. We find that the line-of-sight velocity dispersion of galaxies lying along the major axis of the central galaxy is larger than those that lie along the minor axis. This is the first observational detection of anisotropic kinematics of galaxies in clusters. We show that the result is consistent with predictions from numerical simulations. Furthermore we find that the degree of projected anisotropy is strongly dependent on the line-of-sight orientation of the galaxy cluster, opening new possibilities for assessing systematics in optical cluster finding.
We extend the Cohen-Lenstra heuristics to the setting of ray class groups of imaginary quadratic number fields, viewed as exact sequences of Galois modules.   By asymptotically estimating the mixed moments governing the distribution of a cohomology map, we prove these conjectures in the case of $4$-ranks.
Ultrasonication is widely used to exfoliate two dimensional (2D) van der Waals layered materials such as graphene. Its fundamental mechanism, inertial cavitation, is poorly understood and often ignored in ultrasonication strategies resulting in low exfoliation rates, low material yields and wide flake size distributions, making the graphene dispersions produced by ultrasonication less economically viable. Here we report that few-layer graphene yields of up to 18% in three hours without introduction of basal plane defects can be achieved by optimising inertial cavitation during ultrasonication. We demonstrate that the yield and the graphene flake dimensions exhibit a power law relationship with inertial cavitation dose. Furthermore, inertial cavitation is shown to preferentially exfoliate larger graphene flakes which causes the exfoliation rate to decrease as a function of sonication time. This study demonstrates that measurement and control of inertial cavitation is critical in optimising the high yield sonication-assisted aqueous liquid phase exfoliation of size-selected nanomaterials. Future development of this method should lead to the development of high volume flow cell production of 2D van der Waals layered nanomaterials.
In this article, we introduce Skellam process of order k and its running average. We also discuss the time-changed Skellam process of order k. In particular we discuss space-fractional Skellam process and tempered space-fractional Skellam process via time changes in Poisson process by independent stable subordinator and tempered stable subordinator, respectively. We derive the marginal probabilities, Levy measures, governing difference-differential equations of the introduced processes. Our results generalize Skellam process and running average of Poisson process in several directions.
Most existing virtual try-on applications require clean clothes images. Instead, we present a novel virtual Try-On network, M2E-Try On Net, which transfers the clothes from a model image to a person image without the need of any clean product images. To obtain a realistic image of person wearing the desired model clothes, we aim to solve the following challenges: 1) non-rigid nature of clothes - we need to align poses between the model and the user; 2) richness in textures of fashion items - preserving the fine details and characteristics of the clothes is critical for photo-realistic transfer; 3) variation of identity appearances - it is required to fit the desired model clothes to the person identity seamlessly. To tackle these challenges, we introduce three key components, including the pose alignment network (PAN), the texture refinement network (TRN) and the fitting network (FTN). Since it is unlikely to gather image pairs of input person image and desired output image (i.e. person wearing the desired clothes), our framework is trained in a self-supervised manner to gradually transfer the poses and textures of the model's clothes to the desired appearance. In the experiments, we verify on the Deep Fashion dataset and MVC dataset that our method can generate photo-realistic images for the person to try-on the model clothes. Furthermore, we explore the model capability for different fashion items, including both upper and lower garments.
We consider the inverse problem of determining different type of information about a diffusion process, described by ordinary or fractional diffusion equations stated on a bounded domain, like the density of the medium or the velocity field associated with the moving quantities from a single boundary measurement. This properties will be associated with some general class of time independent coefficients that we recover from a single Neumann boundary measurement, on some parts of the boundary, of the solution of our diffusion equation with a suitable boundary input, located on some parts of the boundary.
Scale-free percolation is a stochastic model for complex networks. In this spatial random graph model, vertices $x,y\in\mathbb{Z}^d$ are linked by an edge with probability depending on i.i.d.\ vertex weights and the Euclidean distance $|x-y|$. Depending on the various parameters involved, we get a rich phase diagram. We study graph distances and compare it to the Euclidean distance of the vertices. Our main attention is on a regime where graph distances are (poly-)logarithmic in the Euclidean distance. We obtain improved bounds on the logarithmic exponents. In the light tail regime, the correct exponent is identified.
Here we examine the noncommutative counterpart of QED, which is called as noncommutative QED. The theory is obtained by examining the consistent minimal coupling to noncommutative U(1) gauge field. The *-product admits the coupling of the matter with only three varieties of charges, i.e., 0, +1 and -1. Ultraviolet divergence can be absorbed into the rescaling of the fields and the parameters at least at one loop level. To examine the infrared aspect of the theory the anomalous magnetic dipole moment is calculated. The dependence on the direction of photon momentum reflects the Lorentz symmetry violation of the system. The explicit calculation of the finite part of the photon vacuum polarization shows the singularity ln({q C^TC q}) (C^{\mu\nu} is a noncommutative parameter.) in the infrared side which also exists in noncommutative Yang-Mills theory. It is associated with the ultraviolet behavior of the theory. We also consider the extension to chiral gauge theory in the present context, but the requirement of anomaly cancellation allows only noncommutative QED.
In this work we study the possibility that primordial magnetic fields observed in galaxies could be produced by a dark matter halo made of charged ultra-light bosons. In the model, we assume that ultra-light bosons arise as excitations of a complex scalar field described by the Klein-Gordon equation with local $U(1)$ symmetry which introduces electromagnetic fields that minimally couple to the complex scalar current. We use classical solutions of the Klein-Gordon-Maxwell system to describe the density profile of dark matter and magnetic fields in the galaxies. We consider two cases assuming spherical and dipolar spatial symmetries respectively. For the particular case of the LSB spherical galaxy F563, we test the sensitivity of the predicted rotation curves in the charged scalar field dark matter (cSFDM) model to variations of the electromagnetic coupling and, by using the Fisher matrix error estimator, we set a constraint over that coupling by requiring that theoretical rotation curves lay inside the $1\sigma$ confidence region of observational data . We find that cSFDM haloes are able to generate magnetic fields of the order of $\mu G$ and reproduce the observed rotation curves of F563-V2 at the same time if the ultra-light boson has a charge lower than $\sim 10^{-13}e$ for the monopole-like density profile and lower than $10^{-14}e$ for the dipole-like one.
Multimode spinless p-wave superconducting wires with a width W much smaller than the superconducting coherence length \xi are known to have multiple low-energy subgap states localized near the wire's ends. Here we compare the typical energies of such endstates for various terminations of the wire: A superconducting wire coupled to a normal-metal stub, a weakly disordered superconductor wire and a wire with smooth confinement. Depending on the termination, we find that the energies of the subgap states can be higher or lower than for the case of a rectangular wire with hard-wall boundaries.
Using the Rowland idea, we find two infinite sets of generators of primes. We also pose some conjectures concerning twin primes.
We propose a new formulation of gauge theories as a quantum theory which has the gauge theory action $S$ as its dynamical variable. This system is described by a simple actional $I(S)$ (that is, an action for the action $S$) whose equation of motion gives the Batalin-Vilkovisky (BV) master equation for $S$. Upon quantization we find that our new formulation is reduced to something like a topological field theory having a BRST exact gauge-fixed actional. Therefore the present formulation can reproduce ordinary gauge theories since the path-integral over $S$ is dominated by the classical configuration which satisfies the BV master equation. This ``theory of theories'' formulation is intended to be applied to closed string field theory.
The connection between the quantum-vacuum geometric phases (which originates from the vacuum zero-point electromagnetic fluctuation) and the non-normal product procedure is considered in the present Letter. In order to investigate this physically interesting geometric phases at quantum-vacuum level, we suggest an experimentally feasible scheme to test it by means of a noncoplanarly curved fiber made of gyrotropic media. A remarkable feature of the present experimental realization is that one can easily extract the nonvanishing and nontrivial quantum-vacuum geometric phases of left- and/or right- handed circularly polarized light from the vanishing and trivial total quantum-vacuum geometric phases.
Context. This work is part of a long-term spectroscopic study of a sample of 30 multiple stars with early-type components. In this second paper we present the results of six multiple systems in which new stellar components have been detected.   Aims. The main aim is to increase the knowledge of stellar properties and dynamical structure of early-type multiple stellar systems.   Methods. Using spectroscopic observations taken over a time baseline of more than 5 years we measured RVs by cross-correlations and applied a spectral disentangling method to double-lined systems. Besides the discovery of objects with double-lined spectra, the existence of new spectroscopic subsystems have been inferred from the radial velocity variations of single-lined components and through the variation of the barycentric velocity of double-lined subsystems. Orbital elements have been calculated when possible.   Results. Seven new stellar components and two members that we expect to confirm with new observations have been discovered in the six studied multiples. We present orbital parameters for two double-lined binaries and preliminary orbits for three single-lined spectroscopic binaries. Five of the six analysed systems are quadruples, while the remaining has five components distributed in four hierarchical levels. These multiplicity orders are in fact lower limits, since these systems lack high-resolution visual observations and additional hierarchical level might exist in that separation range.   Conclusions. The six analysed systems have greater multiplicity degree and a more complex hierarchical structure than previously known, which suggests that high-order multiple systems are significantly more frequent that it is currently estimated. The long term spectroscopic monitoring of multiple systems has shown to be useful for the detection of companions in intermediate hierarchical levels.
In this paper, we study the {\it parabolic Anderson model} starting from the Dirac delta initial data: \[ \left(\frac{\partial}{\partial t} -\frac{\nu}{2}\frac{\partial^2}{\partial x^2} \right) u(t,x) = \lambda u(t,x) \dot{W}(t,x), \qquad u(0,x)=\delta_0(x), \quad x\in\mathbb{R}, \] where $\dot{W}$ denotes the space-time white noise. By evaluating the threefold contour integral in the third moment formula by Borodin and Corwin [2], we obtain some explicit formulas for $\mathbb{E}[u(t,x)^3]$. One application of these formulas is given to show the exact phase transition for the intermittency front of order three.
Three terminal tunnelling experiments on quantum dots in the Coulomb blockade regime allow a quantitative determination of the coupling strength of individual quantum states to the leads. Exploiting this insight we have observed independent fluctuations of the coupling strengths as a function of electron number and magnetic field due to changes in the shape of the wave function in the dot. Such a detailed understanding and control of the dot-lead coupling can be extended to more complex systems such as coupled dots, and is essential for building functional quantum electronic systems.
The paper studies the connectivity properties of facet graphs of simplicial complexes of combinatorial interest. In particular, it is shown that the facet graphs of $d$-cycles, $d$-hypertrees and $d$-hypercuts are, respectively, $(d+1)$, $d$, and $(n-d-1)$-vertex-connected. It is also shown that the facet graph of a $d$-cycle cannot be split into more than $s$ connected components by removing at most $s$ vertices. In addition, the paper discusses various related issues, as well as an extension to cell-complexes.