Related papers: Distribution Theory by Riemann Integrals
We show that there is a way to unify distribution functions that describe simultaneously a signal in space and (spatial) frequency. Probably the most known of them is the Wigner distribution function. Here we show how to unify functions of…
Sampling theory in spaces other than the space of band-limited functions has recently received considerable attention. This is in part because the band-limitedness assumption is not very realistic in many applications. In addition,…
Conditional diffusion models serve as the foundation of modern image synthesis and find extensive application in fields like computational biology and reinforcement learning. In these applications, conditional diffusion models incorporate…
The idea of generating integrals analogous to generating functions is first introduced in this paper. A new proof of the well-known Finite Harmonic Series Theorem in Analysis and Analytical Number Theory is then obtained by the method of…
This is a gentle introduction to Colombeau nonlinear generalized functions, a generalization of the concept of distributions such that distributions can freely be multiplied. It is intended to physicists and applied mathematicians who…
Margin enlargement over training data has been an important strategy since perceptrons in machine learning for the purpose of boosting the robustness of classifiers toward a good generalization ability. Yet Breiman (1999) showed a dilemma…
Two ubiquitous aspects of large-scale data analysis are that the data often have heavy-tailed properties and that diffusion-based or spectral-based methods are often used to identify and extract structure of interest. Perhaps surprisingly,…
Distribution regression, where the goal is to predict a scalar response from a distribution-valued predictor, arises naturally in settings where observations are grouped and outcomes depend on group-level characteristics rather than on…
Probability distribution theory helps in studying the impact of various dimensions in life while the Mittag-Leffler function and bicomplex are used in electromagnetism, quantum mechanics, and signal theory. Considering the importance of…
The extension of the Dirac Delta distribution (DD) to the complex field is needed for dealing with the complex-energy solutions of the Schr\"odinger equation, typically when calculating their inner products. In quantum scattering theory the…
Distributions, i.e., subsets of tangent bundles formed by piecing together subspaces of tangent spaces, are commonly encountered in the theory and application of differential geometry. Indeed, the theory of distributions is a fundamental…
Roughly speaking, functional analysis is the study of vector spaces of arbitrary dimension over the field of real or complex numbers, and the continuous linear mappings between such spaces. Naturally, the notion of continuity requires a…
It is well-known that the Lebesgue integral generalises the Riemann integral. However, as is also well-known but less frequently well-explained, this generalisation alone is not the reason why the Lebesgue integral is important and needs to…
We develop a new algebraic setting for treating piecewise functions and distributions together with suitable differential and Rota-Baxter structures. Our treatment aims to provide the algebraic underpinning for symbolic computation systems…
We give a rigorous derivation of the Fourier transform of the Heaviside function within a framework for tempered distributions that is suitable for undergraduate engineering and mathematics students. The proofs rely on fundamental concepts…
Within the expansive domain of optical sciences, achieving the precise characterization of light beams stands as a fundamental pursuit, pivotal for various applications, including telecommunications and imaging technologies. This study…
Evaluating side-channel analysis (SCA) security is a complex process, involving applying several techniques whose success depends on human engineering. Therefore, it is crucial to avoid a false sense of confidence provided by non-optimal…
Errors in the data and the forward operator of an inverse problem can be handily modelled using partial order in Banach lattices. We present some existing results of the theory of regularisation in this novel framework, where errors are…
In cond-mat/0103603 Diffusion algebras have been introduced in the context of one-dimensional stochastic processes with exclusion in statistical mechanics. While this reference is focused on the needs of the physicist reader and thus states…
Diffusion models are learning pattern-learning systems to model and sample from data distributions with three functional components namely the forward process, the reverse process, and the sampling process. The components of diffusion…