相关论文: Ouroboros Functionals, Families of Ouroboros Funct…

In this paper, we aim to introduce the concept of the Ouroboros space and the complimentary concept of the Ouroboros function by using the Ouroboros equation [1] as our starting point. We start with a few univariate definitions, and then…

Ouroboros functions have shown some interesting properties when subjected to conventional operations. The aim of this paper is to continue our investigation and prove some additional properties of these functions. Using algebraic methods,…

Orthogonal polynomials and multiple orthogonal polynomials are interesting special functions because there is a beautiful theory for them, with many examples and useful applications in mathematical physics, numerical analysis, statistics…

In this paper we introduce and investigate a new kind of functional (including ordinary and evolutionary partial) differential equations. The main goal of this paper is to explore our new philosophy by some examples on functional ODEs and…

This paper presents three new families of fractional Sobolev spaces and their accompanying theory in one-dimension. The new construction and theory are based on a newly developed notion of weak fractional derivatives, which are natural…

Higher-order probabilistic programming languages allow programmers to write sophisticated models in machine learning and statistics in a succinct and structured way, but step outside the standard measure-theoretic formalization of…

We consider integration of functions with values in a partially ordered vector space, and two notions of extension of the space of integrable functions. Applying both extensions to the space of real valued simple functions on a measure…

This report introduces general ideas and some basic methods of the Bayesian probability theory applied to physics measurements. Our aim is to make the reader familiar, through examples rather than rigorous formalism, with concepts such as:…

Probabilistic solvers for ordinary differential equations (ODEs) have emerged as an efficient framework for uncertainty quantification and inference on dynamical systems. In this work, we explain the mathematical assumptions and detailed…

In this survey, we propose an overview on Lyapunov functions for a variety of compartmental models in epidemiology. We exhibit the most widely employed functions, together with a commentary on their use. Our aim is to provide a…

The purpose of this work is to analyse a family of mutually orthogonal polynomials on the unit ball with respect to an inner product which includes an additional term on the sphere. First, we will get connection formulas relating classical…

Outer measures can be used for statistical inference in place of probability measures to bring flexibility in terms of model specification. The corresponding statistical procedures such as Bayesian inference, estimators or hypothesis…

It is well-known that a random variable, i.e., a function defined on a probability space, with values in a Borel space, can be represented on the special probability space consisting of the unit interval with Lebesgue measure. We show an…

First, we establish the theory of fractional powers of first order differential operators with zero order terms, obtaining PDE properties and analyzing the corresponding fractional Sobolev spaces. In particular, our study shows that…

The Humbert-Bessel are multi-index functions with various applications in electromagnetism. New families of functions sharing some similarities with Bessel functions are often introduced in the mathematical literature, but at a closer…

Spectral Barron spaces, constituting a specialized class of function spaces that serve as an interdisciplinary bridge between mathematical analysis, partial differential equations (PDEs), and machine learning, are distinguished by the decay…

We introduce a new approach to the study of operational theories of physics using category theory. We define a generalisation of the (causal) operational-probabilistic theories of Chiribella et al. and establish their correspondence with…

We study differential operators associated with families of polynomials orthonormal with respect to certain measures. These operators, when applied to the Fourier transforms of such measures, produce basis functions for expansions of…

In this paper a general theory of semi-classical matrix orthogonal polynomials is developed. We define the semi-classical linear functionals by means of a distributional equation $D(u A) = u B,$ where $A$ and $B$ are matrix polynomials.…

The work in this paper is four-fold. Firstly, we introduce an alternative approach to solve fractional ordinary differential equations as an expected value of a random time process. Using the latter, we present an interesting numerical…