Related papers: Hypergeometric representations and differential-di…

Hypergeometric functions and their generalizations play an important r\^{o}les in diverse applications. Many authors have been established generalizations of hypergeometric functions by a number ways. In this paper, we aim at establishing…

Hypergeometric class equations are given by second order differential operators in one variable whose coefficient at the second derivative is a polynomial of degree $\leq2$, at the first derivative of degree $\leq1$ and the free term is a…

We consider the generators of gauge transformations with test functions which do not vanish on the boundary of a spacelike region of interest. These are known to generate the edge degrees of freedom in a gauge theory. In this paper, we…

In this position paper, we promote the study of function spaces parameterized by machine learning models through the lens of algebraic geometry. To this end, we focus on algebraic models, such as neural networks with polynomial activations,…

A review is made of the basic tools used in mathematics to define a calculus for pseudodifferential operators on Riemannian manifolds endowed with a connection: esistence theorem for the function that generalizes the phase; analogue of…

Various methods in statistical learning build on kernels considered in reproducing kernel Hilbert spaces. In applications, the kernel is often selected based on characteristics of the problem and the data. This kernel is then employed to…

We study a 3-parametric family of stochastic point processes on the one-dimensional lattice originated from a remarkable family of representations of the infinite symmetric group. We prove that the correlation functions of the processes are…

The use of covariance kernels is ubiquitous in the field of spatial statistics. Kernels allow data to be mapped into high-dimensional feature spaces and can thus extend simple linear additive methods to nonlinear methods with higher order…

These notes provide a self-contained introduction to kernel methods and their geometric foundations in machine learning. Starting from the construction of Hilbert spaces, we develop the theory of positive definite kernels, reproducing…

Kernel methods in machine learning use a kernel function that takes two data points as input and returns their inner product after mapping them to a Hilbert space, implicitly and without actually computing the mapping. For many kernel…

Kernel theorems, in general, provide a convenient representation of bounded linear operators. For the operator acting on a concrete function space, this means that its action on any element of the space can be expressed as a generalised…

Mathematical functions, which often appear in mathematical analysis, are referred to as special functions and have been studied over hundreds of years. Many books and dictionaries are available that describe their properties and serve as a…

In this paper we consider the problems of supervised classification and regression in the case where attributes and labels are functions: a data is represented by a set of functions, and the label is also a function. We focus on the use of…

With view to applications in stochastic analysis and geometry, we introduce a new correspondence for positive definite kernels (p.d.) $K$ and their associated reproducing kernel Hilbert spaces. With this we establish two kinds of…

Recently, there emerges different versions of beta function and hypergeometric functions containing extra parameters. Gaining enlightenment from these ideas, we will first introduce a new extension of generalized hypergeometric function and…

In this article, we study a large class of radial probability density functions defined on the p-adic numbers from which it is possible to obtain certain non-archimedean pseudo-differential operators. These operators are associated with…

Gaussian processes are flexible function approximators, with inductive biases controlled by a covariance kernel. Learning the kernel is the key to representation learning and strong predictive performance. In this paper, we develop…

In this article we shall study the analytic theory and the representation theoretic interpretations of Hankel transforms and fundamental Bessel kernels of an arbitrary rank over an archimedean field.

Geometric and Topological Deep Learning are rapidly growing research areas that enhance machine learning through the use of geometric and topological structures. Within this framework, Group Equivariant Non-Expansive Operators (GENEOs) have…

We study a reproducing kernel Hilbert space of functions defined on the positive integers and associated to the binomial coefficients. We introduce two transforms, which allow us to develop a related harmonic analysis in this Hilbert space.…