Related papers: Univariate Real Analysis
We prove that several results in different areas of number theory such as the divergent series, summation of arithmetic functions, uniform distribution modulo one and summation over prime numbers which are currently considered to be…
This work establishes the existence of addition theorems and double-angle formulas for Ck real scalar functions. Moreover, we determine necessary and sufficient conditions for a bivariate function to be an addition formula for a Ck real…
The present article is devoted to the description of further investigations of the author of this article. These investigations (in terms of various representations of real numbers) include the generalized Salem functions and…
The paper is devoted to a comprehensive second-order study of a remarkable class of convex extended-real-valued functions that is highly important in many aspects of nonlinear and variational analysis, specifically those related to…
We summarize four different versions of our course notes on the limits of mathematics.
We present a simplified integral of functions of several variables. Although less general than the Riemann integral, most functions of practical interest are still integrable. On the other hand, the basic integral theorems can be obtained…
Primitive recursion, mu-recursion, universal object and universe theories, complexity controlled iteration, code evaluation, soundness, decidability, G\"odel incompleteness theorems, inconsistency provability for set theory, constructive…
In this article we present certain formulas involving arithmetical functions. In the first part we study properties of sums and product formulas for general type of arithmetic functions. In the second part we apply these formulas to the…
In this article, we consider a simple representation for real numbers and propose top-down procedures to approximate various algebraic and transcendental operations with arbitrary precision. Detailed algorithms and proofs are provided to…
These lecture notes were written with the aim to provide an accessible though technically solid introduction to the logic of systematical analyses of statistical data to both undergraduate and postgraduate students, in particular in the…
The book is designed for a semester-long course in Foundations of Geometry and meant to be rigorous, conservative, elementary and minimalist. List of topics: Euclidean geometry: The Axioms / Half-planes / Congruent triangles / Perpendicular…
Some aspects of analysis involving fields with absolute value functions are discussed, which includes the real or complex numbers with their standard absolute values, as well as ultrametric situations like the p-adic numbers.
This is the Habilitation Thesis manuscript presented at Besan\c{c}on on January 5, focusing on Matrix Analysis, Matrix Inequalities and Matrix Decompositions. There are also some topics in (Hilbert space) Operator Theory. The text should be…
A rational homogeneous (of degree one) positive real matrix-valued function is presented as the Schur complement of a block of the linear pencil with positive semidefinite matrix coefficients. The partial derivative numerators of a rational…
This note presents a unified theorem of the alternative that explicitly allows for any combination of equality, componentwise inequality, weak dominance, strict dominance, and nonnegativity relations. The theorem nests 60 special cases,…
In the preceding articles we considered fractional integral transforms involving one real scalar variable, one real matrix variable and real scalar multivariable case. In the present paper we consider the multivariable case when the…
This book provides a gentle introduction to fractional Sobolev spaces, which play a central role in the calculus of variations, partial differential equations, and harmonic analysis. The first part deals with fractional Sobolev spaces of…
We provide a clear and concise introduction to the subjects of inverse problems and data assimilation, and their inter-relations. The first part of our notes covers inverse problems; this refers to the study of how to estimate unknown model…
In the context of the correspondence between real functions on the unit circle and inner analytic functions within the open unit disk, that was presented in previous papers, we show that the constructions used to establish that…
This work connects two mathematical fields - computational complexity and interval linear algebra. It introduces the basic topics of interval linear algebra - regularity and singularity, full column rank, solving a linear system, deciding…