Related papers: Convergence and divergence testing theory and appl…
In this paper, we present the foundations of Summability Calculus, which places various established results in number theory, infinitesimal calculus, summability theory, asymptotic analysis, information theory, and the calculus of finite…
There are two distinct definitions of 'P-value' for evaluating a proposed hypothesis or model for the process generating an observed dataset. The original definition starts with a measure of the divergence of the dataset from what was…
The method of brackets is an efficient method for the evaluation of a large class of definite integrals on the half-line. It is based on a small collection of rules, some of which are heuristic. The extension discussed here is based on the…
We define an integral of real-valued functions with respect to a measure that takes its values in the extended positive cone of a partially ordered vector space $E$. The monotone convergence theorem, Fatou's lemma, and the dominated…
The one-sided and full Hilbert transforms are evaluated exactly by means of the method of finite-part integration [E.A. Galapon, \textit{Proc. Roy. Soc. A} \textbf{473}, 20160567 (2017)]. In general, the result consists of two terms -- the…
Cut-elimination is the bedrock of proof theory with a multitude of applications from computational interpretations to proof analysis. It is also the starting point for important meta-theoretical investigations including decidability,…
Fundamental to the theory of continued fractions is the fact that every infinite continued fraction with positive integer coefficients converges; however, it is unknown precisely which continued fractions with integer coefficients (not…
It is shown how to extend the formal variational calculus in order to incorporate integrals of divergences into it. Such a generalization permits to study nontrivial boundary problems in field theory on the base of canonical formalism.
Recent years have witnessed a wave of research activities in systems science toward the study of population systems. The driving force behind this shift was geared by numerous emerging and ever-changing technologies in life and physical…
Hypothesis testing in high dimensional data is a notoriously difficult problem without direct access to competing models' likelihood functions. This paper argues that statistical divergences can be used to quantify the difference between…
The deviation test belong to core tools in point process statistics, where hypotheses are typically tested considering differences between an empirical summary function and its expectation under the null hypothesis, which depend on a…
This review is focused on the borderline region of theoretical physics and mathematics. First, we describe numerical methods for the acceleration of the convergence of series. These provide a useful toolbox for theoretical physics which has…
Diagnostic testing is germane to a variety of scenarios in medicine, pandemic tracking, threat detection, and signal processing. This is an expository paper with some original results. Here we first set up a mathematical architecture for…
Some limit theorems of the type $\int_{\Omega}f_n dm_n -- --> \int_{\Omega}f dm$ are presented for scalar, (vector), (multi)-valued sequences of m_n-integrable functions f_n. The convergences obtained, in the vector and multivalued…
The concept of boundary plays an important role in several branches of general relativity, e.g., the variational principle for the Einstein equations, the event horizon and the apparent horizon of black holes, the formation of trapped…
A rigorous connection between large deviations theory and Gamma-convergence is established. Applications include representations formulas for rate functions, a contraction principle for measurable maps, a large deviations principle for…
One of the basic principles of Approximation Theory is that the quality of approximations increase with the smoothness of the function to be approximated. Functions that are smooth in certain subdomains will have good approximations in…
Score-based methods have recently seen increasing popularity in modeling and generation. Methods have been constructed to perform hypothesis testing and change-point detection with score functions, but these methods are in general not as…
We consider a functional being a difference of two differentiable convex functionals on a closed ball. Existence and multiplicity of critical points is investigated. Some applications are given.
We consider nonlinear, or "event-dependent", sampling, i.e. such that the sampling instances {tk} depend on the function being sampled. The use of such sampling in the construction of Lebesgue's integral sums is noted and discussed as…