29,033 research outputs found

    Thread-safe lattice Boltzmann for high-performance computing on GPUs

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    We present thread-safe, highly-optimized lattice Boltzmann implementations, specifically aimed at exploiting the high memory bandwidth of GPU-based architectures. At variance with standard approaches to LB coding, the proposed strategy, based on the reconstruction of the post-collision distribution via Hermite projection, enforces data locality and avoids the onset of memory dependencies, which may arise during the propagation step, with no need to resort to more complex streaming strategies. The thread-safe lattice Boltzmann achieves peak performances, both in two and three dimensions and it allows to sensibly reduce the allocated memory ( tens of GigaBytes for order billions lattice nodes simulations) by retaining the algorithmic simplicity of standard LB computing. Our findings open attractive prospects for high-performance simulations of complex flows on GPU-based architectures

    Soliton Gas: Theory, Numerics and Experiments

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    The concept of soliton gas was introduced in 1971 by V. Zakharov as an infinite collection of weakly interacting solitons in the framework of Korteweg-de Vries (KdV) equation. In this theoretical construction of a diluted soliton gas, solitons with random parameters are almost non-overlapping. More recently, the concept has been extended to dense gases in which solitons strongly and continuously interact. The notion of soliton gas is inherently associated with integrable wave systems described by nonlinear partial differential equations like the KdV equation or the one-dimensional nonlinear Schr\"odinger equation that can be solved using the inverse scattering transform. Over the last few years, the field of soliton gases has received a rapidly growing interest from both the theoretical and experimental points of view. In particular, it has been realized that the soliton gas dynamics underlies some fundamental nonlinear wave phenomena such as spontaneous modulation instability and the formation of rogue waves. The recently discovered deep connections of soliton gas theory with generalized hydrodynamics have broadened the field and opened new fundamental questions related to the soliton gas statistics and thermodynamics. We review the main recent theoretical and experimental results in the field of soliton gas. The key conceptual tools of the field, such as the inverse scattering transform, the thermodynamic limit of finite-gap potentials and the Generalized Gibbs Ensembles are introduced and various open questions and future challenges are discussed.Comment: 35 pages, 8 figure

    Tonelli Approach to Lebesgue Integration

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    Leonida Tonelli devised an interesting and efficient method to introduce the Lebesgue integral. The details of this method can only be found in the original Tonelli paper and in an old italian course and solely for the case of the functions of one variable. We believe that it is woth knowing this method and here we present a complete account for functions of every number of variables

    Finding and Counting Patterns in Sparse Graphs

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    Percentage ratios of cutting forces during high-reed face milling

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    This research paper is concerned with the experimental study of high-feed end milling of 1.4541 (X6CrNiTi18-10) stainless steel with replaceable cermet plates. Several machining operations were performed under different cutting conditions. The variable values were depth of cut, feed per tooth and cutting speed. The results were analyzed, and cutting forces were evaluated for dependence on cutting conditions (cutting speed, depth of cut, feed per tooth). The obtained data were statistically processed and plotted in graphs. It was found that the percentage distribution of cutting forces changed as the tool load increased. The ratio of forces acting in individual axes also changed with varying trends. An increasing trend was recorded in the x and y axes, while a decreasing trend was recorded in the z axis. Measured change, approximately 10%, can no longer be neglected as it can significantly influence the clamping stability of a part.IGA/FT/2023/005TBU of the Zlin Internal Grant Agency [IGA/FT/2023/005

    Central Values of Degree Six L-functions: The Case of Hilbert Modular Forms

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    In this paper we give a formula for the central value of the completed LL-function L(s,Sym2g×f)L(s,Sym^{2} g\times f), where ff and gg are Hilbert newforms, by explicitly computing the local integrals appearing in the refined Gan-Gross-Prasad conjecture for SL2×SL2~SL_{2}\times\tilde{SL_{2}}. We also work out the rationality of this value in some special cases and give a conjecture for the general case

    Isotopic piecewise affine approximation of algebraic or C1C^1 varieties

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    We propose a novel sufficient condition establishing that a piecewise affine variety has the same topology as a variety of the sphere Sn\mathbb{S}^n defined by positively homogeneous C1C^1 functions. This covers the case of C1C^1 varieties in the projective space Pn\mathbb{P}^n. We prove that this condition is sufficient in the case of codimension one and arbitrary dimension. We describe an implementation working for homogeneous polynomials in arbitrary dimension and codimension and give experimental evidences that our condition might still be sufficient in codimension greater than one

    Efficient computation of the overpartition function and applications

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    In this paper we develop a method to calculate the overpartition function efficiently using a Hardy-Rademacher-Ramanujan type formula, and we use this method to find many new Ramanujan-style congruences whose existence is predicted by Treneer and a few of which were first discovered by Ryan, Scherr, Sirolli and Treneer

    Genus character LL-functions of quadratic orders in an adelic way and maximal orders of matrix algebras

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    For a quadratic extension KK of Q{\mathbb Q}, we consider orders OO in KK that are not necessarily maximal and the ideal class group Cl+(O)Cl^+(O) in the narrow sense of proper ideals of OO. Characters of Cl+(O)Cl^+(O) of order at most two are traditionally called genus characters. Explicit description of such characters is known classically, but explicit LL-functions associated to those characters are only recently obtained partially by Chinta and Offen and completely by Kaneko and Mizuno. As remarked in the latter paper, the present author also obtained the formula of such LL-functions independently. Indeed, here we will give a simple and transparent alternative proof of the formula by rewriting explicit genus characters and their values in an adelic way starting from scratch. We also add an explicit formula for the genus number in the wide sense, which is maybe known but rarely treated. As an appendix we give an ideal-theoretic characterization of isomorphism classes of maximal orders of the matrix algebras Mn(F)M_n(F) over a number field FF up to GLn(F)GL_n(F) and GLn+(F)GL_n^+(F) conjugation respectively, and apply genus numbers to count them when n=2n=2 and FF is quadratic. To avoid any misconception, we include some easy known details.Comment: 25 page

    Computational approach to the Schottky problem

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    We present a computational approach to the classical Schottky problem based on Fay's trisecant identity for genus g4g\geq 4. For a given Riemann matrix BHg\mathbb{B}\in\mathbb{H}^{g}, the Fay identity establishes linear dependence of secants in the Kummer variety if and only if the Riemann matrix corresponds to a Jacobian variety as shown by Krichever. The theta functions in terms of which these secants are expressed depend on the Abel maps of four arbitrary points on a Riemann surface. However, there is no concept of an Abel map for general BHg\mathbb{B} \in \mathbb{H}^{g}. To establish linear dependence of the secants, four components of the vectors entering the theta functions can be chosen freely. The remaining components are determined by a Newton iteration to minimize the residual of the Fay identity. Krichever's theorem assures that if this residual vanishes within the finite numerical precision for a generic choice of input data, then the Riemann matrix is with this numerical precision the period matrix of a Riemann surface. The algorithm is compared in genus 4 for some examples to the Schottky-Igusa modular form, known to give the Jacobi locus in this case. It is shown that the same residuals are achieved by the Schottky-Igusa form and the approach based on the Fay identity in this case. In genera 5, 6 and 7, we discuss known examples of Riemann matrices and perturbations thereof for which the Fay identity is not satisfied
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