Related papers: P-adic numbers and kernels

Let $p$ be a prime. We discuss $p$-adic properties of various arithmetical functions related to the coefficients of modular form and generating functions. Modular forms are considered as a tool of solving arithmetical problems. Examples of…

We consider the topology of simplicial complexes with vertices the points of a random point process and faces determined by distance relationships between the vertices. In particular, we study the Betti numbers of these complexes as the…

Generalizations and variations of the fundamental lemma by Willems et al. are an active topic of recent research. In this note, we explore and formalize the links between kernel regression and some known nonlinear extensions of the…

Prime numbers are the building blocks of our arithmetic, however, their distribution still poses fundamental questions. Bernhard Riemann showed that the distribution of primes could be given explicitly if one knew the distribution of the…

We derive the P-finite recurrences for classes of sequences with ordinary generating function containing roots of polynomials. The focus is on establishing the D-finite differential equations such that the familiar steps of reducing their…

We present a simple numerical scheme for perturbation theory (PT) calculations of large-scale structure. Solving the evolution equations for perturbations numerically, we construct the PT kernels as building blocks of statistical…

By using a simple interpolation argument, in previous work we have proven the existence of the thermodynamic limit, for mean field disordered models, including the Sherrington-Kirkpatrick model, and the Derrida p-spin model. Here we extend…

We introduce a Cherednik kernel and a hypergeometric function for integral root systems and prove their relation to spherical functions associated with Riemannian symmetric spaces of reductive Lie groups. Furthermore, we characterize the…

In this article, we discuss the remarkable connection between two very different fields, number theory and nuclear physics. We describe the essential aspects of these fields, the quantities studied, and how insights in one have been…

Atomistic simulations are a powerful tool for studying the dynamics of molecules, proteins, and materials on wide time and length scales. Their reliability and predictiveness, however, depend directly on the accuracy of the underlying…

The success of kernel-based learning methods depend on the choice of kernel. Recently, kernel learning methods have been proposed that use data to select the most appropriate kernel, usually by combining a set of base kernels. We introduce…

Kernel ridge regression (KRR) is a popular class of machine learning models that has become an important tool for understanding deep learning. Much of the focus thus far has been on studying the proportional asymptotic regime, $n \asymp d$,…

Following an original idea of F. Guerra, in this notes we analyze the Sherrington-Kirkpatrick model from different perspectives, all sharing the underlying approach which consists in linking the resolution of the statistical mechanics of…

Let p be a prime number. In this paper we use an old technique of Ore, based on Newton polygons, to construct in an efficient way p-integral bases of number fields defined by a p-regular equation. To illustrate the potential applications of…

Statistical learning theory provides bounds of the generalization gap, using in particular the Vapnik-Chervonenkis dimension and the Rademacher complexity. An alternative approach, mainly studied in the statistical physics literature, is…

Z^d-extensions of probability-preserving dynamical systems are themselves dynamical systems preserving an infinite measure, and generalize random walks. Using the method of moments, we prove a generalized central limit theorem for additive…

The primary purpose of this paper is to generalize the classical Riemann zeta function to the setting of Krull monoids with torsion class groups. We provide a first study of the same generalization by extending Euler's classical product…

In a recent paper (Tran et al., Ann.Phys.311(2004)204), some asymptotic number theoretical results on the partitioning of an integer were derived exploiting its connection to the quantum density of states of a many-particle system. We…

We report a rigorous theory to show the origin of the unexpected periodic behavior seen in the consecutive differences between prime numbers. We also check numerically our findings to ensure that they hold for finite sequences of primes,…

The current research regarding the Riemann zeros suggests the existence of a non-trivial algebraic/analytic structure on the set of Riemann zeros. The duality between primes and Riemann zeta function zeros suggests some new goals and…