Integer lattice
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On probability measures arising from lattice points on circles Open
A circle, centered at the origin and with radius chosen so that it has non-empty intersection with the integer lattice , gives rise to a probability measure on the unit circle in a natural way. Such measures, and their weak limits, are …
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Invariant measures of stochastic delay lattice systems Open
This paper is concerned with the existence and uniqueness of invariant measures for infinite-dimensional stochastic delay lattice systems defined on the entire integer set. For Lipschitz drift and diffusion terms, we prove the existence of…
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Lattice position optimization for LATTICE therapy Open
Background LATTICE radiation therapy delivers 3D heterogenous dose of high peak‐to‐valley dose ratio (PVDR) to the tumor target, with peak dose at lattice vertices inside the target and valley dose for the rest of the target. Although the …
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Integer points on spheres and their orthogonal lattices Open
ISSN:0020-9910
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Lattice Point Visibility on Generalized Lines of Sight Open
For a fixed $b\in\mathbb{N}=\{1,2,3,\ldots\}$ we say that a point $(r,s)$ in the integer lattice $\mathbb{Z} \times \mathbb{Z}$ is $b$-visible from the origin if it lies on the graph of a power function $f(x)=ax^b$ with $a\in\mathbb{Q}$ an…
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Fast Maximization of Non-Submodular, Monotonic Functions on the Integer Lattice Open
The optimization of submodular functions on the integer lattice has received much attention recently, but the objective functions of many applications are non-submodular. We provide two approximation algorithms for maximizing a non-submodu…
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Discrete scattering by a pair of parallel defects Open
Scattering of a time-harmonic anti-plane shear wave due to either a pair of crack tips or a pair of rigid constraint tips on square lattice is considered. The two problems correspond to the so-called zero-offset case of scattering due to a…
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EXCEEDINGLY LARGE DEVIATIONS OF THE TOTALLY ASYMMETRIC EXCLUSION PROCESS Open
Consider the Totally Asymmetric Simple Exclusion Process (TASEP) on the integer lattice $ \mathbb{Z} $. We study the functional Large Deviations of the integrated current $ \mathsf{h} (t,x) $ under the hyperbolic scaling of space and time …
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Current Trends in Random Walks on Random Lattices Open
In a classical random walk model, a walker moves through a deterministic d-dimensional integer lattice in one step at a time, without drifting in any direction. In a more advanced setting, a walker randomly moves over a randomly configured…
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Spatial statistics for lattice points on the sphere I: Individual\n results Open
We study the spatial distribution of point sets on the sphere obtained from\nthe representation of a large integer as a sum of three integer squares. We\nexamine several statistics of these point sets, such as the electrostatic\npotential,…
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The Apollonian structure of integer superharmonic matrices Open
We prove that the set of quadratic growths attainable by integer-valued superharmonic functions on the lattice $\mathbb{Z}^2$ has the structure of an Apollonian circle packing. This completely characterizes the PDE which determines the con…
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Lattice multipolygons Open
We discuss generalizations of some results on lattice polygons to certain piecewise linear loops which may have a self-intersection but have vertices in the lattice $\\mathbb{Z}^{2}$ . We first prove a formula on the rotation number of a u…
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Three-dimensional lattice polytopes with two interior lattice points Open
We classify the three-dimensional lattice polytopes with two interior lattice points. Up to unimodular equivalence there are 22,673,449 such polytopes. This classification allows us to verify, for this case only, a conjectural upper bound …
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Large deviation principles of stochastic reaction-diffusion lattice systems Open
This paper is concerned with the large deviation principle of the stochastic reaction-diffusion lattice systems defined on the $ N $-dimensional integer set, where the nonlinear drift term is locally Lipschitz continuous with polynomial gr…
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Variance of Lattice Point Counting in Thin Annuli Open
We give asymptotic estimates of the variance of the number of integer points in translated thin annuli in any dimension.
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Fluctuations of the Propagation Front of a Catalytic Branching Walk Open
We consider a supercritical catalytic branching random walk (CBRW) on a\nmultidimensional lattice Z^d (d is positive integer). The main subject of study\nis the behavior of particles cloud in space and time. For CBRW on an integer\nline, C…
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The weak order on integer posets Open
We explore lattice structures on integer binary relations (i.e. binary relations on the set $\{1, 2, \dots, n\}$ for a fixed integer $n$) and on integer posets (i.e. partial orders on the set $\{1, 2, \dots, n\}$ for a fixed integer $n$). …
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Mixing time and eigenvalues of the abelian sandpile Markov chain Open
The abelian sandpile model defines a Markov chain whose states are integer-valued functions on the vertices of a simple connected graph . By viewing this chain as a (nonreversible) random walk on an abelian group, we give a formula for its…
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Spanning Lattice Polytopes and the Uniform Position Principle Open
A lattice polytope $P$ is called IDP if any lattice point in its $k$th dilate is a sum of $k$ lattice points in $P$. In 1991 Stanley proved a strong inequality in Ehrhart theory for IDP lattice polytopes. We show that his conclusion holds …
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Discrete Optimization: The Case of Generalized BCC Lattice Open
Recently, operations research, especially linear integer-programming, is used in various grids to find optimal paths and, based on that, digital distance. The 4 and higher-dimensional body-centered-cubic grids is the nD (n≥4) equivalent of…
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The Maximum of an Asymmetric Simple Random Walk with Reflection Open
Consider the extreme value of a Bernoulli random walk on the one-dimensional integer lattice, with reflection at 0, over a finite discrete time interval. Only the asymmetric (biased) case is discussed. Asymptotic mean/variance results are …
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Spatial statistics for lattice points on the sphere I: Individual results Open
We study the spatial distribution of point sets on the sphere obtained from the representation of a large integer as a sum of three integer squares. We examine several statistics of these point sets, such as the electrostatic potential, Ri…
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The near-critical two-point function for weakly self-avoiding walk in high dimensions Open
We use the lace expansion to study the long-distance decay of the two-point function of weakly self-avoiding walk on the integer lattice $\mathbb{Z}^d$ in dimensions $d>4$, in the vicinity of the critical point, and prove an upper bound $|…
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On the Directional Movement of a Collective of Automata without a Compass on a One-dimensional Integer Lattice Open
ИНФОРМАТИКА УДК
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On the ergodic theory of maps associated with the nearest integer complex continued fractions over imaginary quadratic fields Open
We consider the nearest integer complex continued fraction map associated to the Euclidean field $ \mathbb Q(\sqrt{-d}) $ for each $ d = 1, 2, 3, 7, 11 $. For each map, we see that there is an absolutely continuous ergodic invariant probab…
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Moments of moments of characteristic polynomials of random unitary matrices and lattice point counts Open
In this note, we give a combinatorial and noncomputational proof of the asymptotics of the integer moments of the moments of the characteristic polynomials of Haar distributed unitary matrices as the size of the matrix goes to infinity. Th…
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Notions of Maximality for Integral Lattice-Free Polyhedra: The Case of Dimension Three Open
Lattice-free sets and their applications for cutting-plane methods in mixed-integer optimization have been studied in recent literature. The family of all integral lattice-free polyhedra that are not properly contained in another integral …
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Computing an LLL-reduced Basis of the Orthogonal Latice Open
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Integer lattice gas with a sampling collision operator for the fluctuating diffusion equation Open
We developed an integer lattice gas method for the fluctuating diffusion equation. Such a method is unconditionally stable and able to recover the Poisson distribution for the microscopic densities. A key advance for integer lattice gases …
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Generalized visibility of lattice points in higher dimensions Open
For any k≥2 and fixed b=(b1,⋯,bk)∈Nk, this paper concerns the number of integer lattice points in Zk which are b-visible from a set of N watch-points. Moreover, for N=2 and b˜=(b˜1,⋯,b˜k)∈Nk with b˜≠b, we consider the mixed visibility, tha…