Related papers: The Mean Value Theorem: Analytical Proof and Compu…

In this short note, we give two proofs of the infinitude of primes via valuation theory and give a new proof of the divergence of the sum of prime reciprocals by Roth's theorem and Euler-Legendre's theorem for arithmetic progressions.

We develop a theory of Valuation Hilbert Modules and prove a version of Beurling's theorem for these. Then we apply our version of Beurling's theorem to obtain complete descriptions of the closed invariant subspaces of a number of Hilbert…

The First and Second Liouville's Theorems provide correspondingly criterium for integrability of elementary functions "in finite terms" and criterium for solvability of second order linear differential equations by quadratures. The…

The familiar second derivative test for convexity, combined with resolvent calculus, is shown to yield a useful tool for the study of convex matrix-valued functions. We demonstrate the applicability of this approach on a number of theorems…

We present a novel approach, which is based on multiple-valued logic (MVL), to the verification and analysis of digital hardware designs, which extends the common ternary or quaternary approaches for simulations. The simulations which are…

We give a characterization of smooth, rotation and dually epi-translation invariant valuations and use this result to obtain a new proof of the Hadwiger theorem on convex functions. We also give a description of the construction of the…

An econometric or statistical model may undergo a marginal gain if we admit a new variable to the model, and a marginal loss if we remove an existing variable from the model. Assuming equality of opportunity among all candidate variables,…

As the complexity of machine learning (ML) models increases, resulting in a lack of prediction explainability, several methods have been developed to explain a model's behavior in terms of the training data points that most influence the…

A differential algebra of nonlinear generalized functions is presented as a tool for a wide range of nonsmooth nonlinear problems. The power of the differential algebra is used to do mathematical calculations or proofs; then the final…

Metric regularity is among the central concepts of nonlinear and variational analysis, constrained optimization, and their numerous applications. However, metric regularity can be elusive for some important ill-posed classes of problems…

The classical theory for the meta-analysis of $p$-values is based on the assumption that if the overall null hypothesis is true, then all $p$-values used in a chosen combined test statistic are genuine, i.e., are observations from…

In this work, we show that uniform integrability is not a necessary condition for central limit theorems (CLT) to hold for normalized multilevel Monte Carlo (MLMC) estimators and we provide near optimal weaker conditions under which the CLT…

In a recent paper [5] a smooth function f : [0; 1] --> R with all derivatives vanishing at 0 has been considered and a global condition, showing that f is indeed identically 0, has been presented. The purpose of this note is to replace the…

Several matrix/operator inequalies are given. Most of them are unexpected extensions of the Araki Log-majorization theorem, obtained thanks to a new log-majorization for positive linear maps and normal operators (Theorem 2.9). The main idea…

We establish a connection between the ratios conjecture for the Riemann zeta-function and a conjecture concerning correlations of convolutions of M\"{o}bius and divisor functions. Specifically, we prove that the ratios conjecture and an…

Differential Calculus is a staple of the college mathematics major's diet. Eventually one becomes tired of the same routine, and wishes for a more diverse meal. The college math major may seek to generalize applications of the derivative…

Variational analysis provides the theoretical foundations and practical tools for constructing optimization algorithms without being restricted to smooth or convex problems. We survey the central concepts in the context of a concrete but…

Across machine learning (ML) sub-disciplines researchers make mathematical assumptions to facilitate proof-writing. While such assumptions are necessary for providing mathematical guarantees for how algorithms behave, they also necessarily…

This paper is an introduction to the theory of multivector functions of a real variable. The notions of limit, continuity and derivative for these objects are given. The theory of multivector functions of a real variable, even being similar…

Multivariate functional data has received considerable attention but testing for equality of mean surfaces and its profile has limited progress. The existing literature has tested equality of either mean curves of univariate functional…