Related papers: x-area
We give an elementary and self-contained proof of the uniformization theorem for non-compact simply-connected Riemann surfaces.
We illustrate the experimental, empirical, approach to mathematics (that contrary to popular belief, is often rigorous), by using parking functions and their "area" statistic, as a case study. Our methods are purely finitistic and…
A very simple and short proof of the polynomial matrix spectral factorization theorem (on the unit circle as well as on the real line) is presented, which relies on elementary complex analysis and linear algebra.
We introduce the concept of shifting distance functions, and we establish a new fixed point theorem which generalizes the Banach contraction principle. An example is provided to illustrate our result.
We prove a generalization of one of Lie's Theorems in the context of Lie-like algebras$^{2-nd}$.
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…
We prove a sharp version of Hal\'asz's theorem on sums $\sum_{n \leq x} f(n)$ of multiplicative functions $f$ with $|f(n)|\le 1$. Our proof avoids the "average of averages" and "integration over $\alpha$" manoeuvres that are present in many…
Euler defines a function f(x) somehow as an infinite product and a generalization of [x], where [x] ist, what we now call following Legendre the Gamma-Funktion. He gets some recursive relationships for f(x), by applying some very nice…
Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we develop general theorems on permutation polynomials over finite fields. As a…
In Diophantine approximation, inhomogeneous problems are linked with homogeneous ones by means of the so-called Transference Theorems. We revisit this classical topic by introducing new exponents of Diophantine approximation. We prove that…
A generalization of the generating function for Gegenbauer polynomials is introduced whose coefficients are given in terms of associated Legendre functions of the second kind. We discuss how our expansion represents a generalization of…
We establish an effective version of Schmidt's subspace theorem on a smooth projective variety $\mathcal{X}$ over function fields of characteristic zero for hypersurfaces located in m-subgeneral position with respect to $\mathcal{X}$. Our…
We provide a new proof of Alesker's Irreducibility Theorem. We first introduce a new localization technique for polynomial valuations on convex bodies, which we use to independently prove that smooth and translation invariant valuations are…
We prove equidistribution of a generic net of small points in a projective variety X over a function field K. For an algebraic dynamical system over K, we generalize this equidistribution theorem to a small generic net of subvarieties. For…
We establish sharp estimates that adapt the polynomial method to arbitrary varieties. These include a partitioning theorem, estimates on polynomials vanishing on fixed sets and bounds for the number of connected components of real algebraic…
A generalization of metric space is presented which is shown to admit a theory strongly related to that of ordinary metric spaces. To avoid the topological effects related to dropping any of the axioms of metric space, first a new, and…
We provide a geometric interpretation for a normalized version of the minimal denominator function, $$q_{\min}(x,\delta)=\min\left\{q\in \mathbb{N}: \text{ there exists } p\in\mathbb{Z} \text{ such that } \frac{p}{q}\in…
In this article we show how the Dedekind-Hasse criterion may be applied to prove a simple result about quadratic number fields that usually is derived as a consequence of the theory of ideals and ideal classes.
In this paper we give an additive representation of the factorial, which can be proven by a simple quick analytical argument. We also present some generalizations, which are linked, on the one hand to an arithmetical theorem proven by Euler…
This paper states and proves a generalization of the well-known Desargues involution theorem from plane projective geometry.