General Mathematics
In this work, we study the generalized k-th power symbol (a/n)_k and present a comprehensive collection of its algebraic properties. The results are classified according to their dependence on the three main parameters a, n, and k. In…
We introduce a new functional space U designed to contain all classical arithmetic functions (Mobius, von Mangoldt, Euler phi, divisor functions, Dirichlet characters, etc.). The norm of U combines a Hilbert-type component, based on square…
This work, which may be seen as a companion paper to \cite{RS2}, handles the way the intersection points made by the diagonals of a regular polygon are distributed. It was stated recently by the authors that these points lie exclusively on…
Within the theoretical framework of Optimal Recovery, one determines in this article the {\em best} procedures to learn a quantity of interest depending on a H\"older function acquired via inexact point evaluations at fixed datasites. {\em…
In this short note we are presenting a method of finding particular solutions of nonhomegeneous linear equations. This approach is different from methods of undetermined coefficients or variation of parameters presented in virtually every…
The uncertainty principle is one of the fundamental tools for time-frequency analysis in signal processing, revealing the intrinsic trade-off between time and frequency resolutions. With the continuous development of various advanced…
We introduce and analyze a Spencer-type elliptic complex on the space of differential forms valued in symmetric powers of an adjoint bundle, $\Omega^\bullet(X)\otimes \mathrm{Sym}^\bullet(G)$. The complex is governed by a total differential…
Let $x\geq 1$ be a large number, and let $1 \leq a <q $ be integers such that $\gcd(a,q)=1$ and $q=O(\log^c)$ with $c>0$ constant. This note proves that the counting function for the number of primes $p \in \{p=qn+a: n \geq1 \}$ with a…
We introduce a family of regularized functionals $g_n(x)$ that generalize the Euler--Mascheroni constant $\gamma$. They arise from a weighted regularization of Clausen-type trigonometric sums, and admit explicit integral representations,…
In his book "Mathematics Rhyme and Reason," Currie discusses what he calls a $mysterious$ $pattern$ involving the sequence $ a_{n} = 2^n \sqrt{2 - \sqrt{2 + \sqrt{2 + \cdots + \sqrt{2}}}},$ where $n$ is the number of radicals. Part of the…
This work discusses an approach to solving geometric construction problems in which the given figure is included in a set ordered by construction steps. The flow of information is carried through the chain, allowing the original problem to…
A short, fairly self-contained proof is given of the Poincar\'e Conjecture. In the previous version there was an error on Page 8. This gap has now been filled.
Based on previous work we consturct an equation (Lagrange equation) and relate it with a system of generalized integrals and differential equations in such a way to provide useful evaluations and connections between them.
We study the row completion problem of polynomial and rational matrices with partial prescription of the structural data. The prescription of the complete structural data has been solved in Amparan et al., Lin. Alg. Appl. 720 (2025)…
Matrix completion results deal with the question of when a partially specified symmetric matrix can be completed to a member of certain matrix cones. Results from positive semidefinite matrix completion and completely positive matrix…
This presentation abstract, presented at 12th Applied Inverse Problems Conference, to be held at FGV EMAp, Rio de Janeiro, Brazil, in 2025, describe the theory results of applying the Chebyshev pseudospectral method (CPM) to reconstruct the…
In this paper we prove and discuss some new $\left( H_{p},weak-L_{p}\right) $ type inequalities of maximal operators of $T$ means with respect to Vilenkin systems with monotone coefficients. We also apply these results to prove a.e.…
In this paper, we study initial value problems for a class of functional equations. We introduce the concept of appropriate initial sets to enable the unique extension of an initial function into a solution defined over larger domains. Our…
In this paper, we investigate the convergence properties of Fourier partial sums associated with general orthonormal systems, focusing on functions that belong to specific differentiable function classes. While classical Fourier analysis…
Let $p>1$ be a large prime number, let $q=O(\log\log p)$ and let $1\leq a<q$ be a pair of relatively prime integers. It is proved that there is a prime primitive root $u\ll (\log p)(\log \log p)^5$ such that $u\equiv a\bmod q$ in the prime…