Related papers: Potpourri, 8
These informal notes are concerned with spaces of functions in various situations, including continuous functions on topological spaces, holomorphic functions of one or more complex variables, and so on.
In this paper, we introduce a new function related to the sum of element orders of finite groups. It is used to give some criteria for a finite group to be cyclic, abelian, nilpotent, supersolvable and solvable, respectively.
We introduce a general notion of covering property, of which many classical definitions are particular instances. Notions of closure under various sorts of convergence, or, more generally, under taking kinds of accumulation points, are…
These notes concern linear transformations on R^n and C^n, exponentials of linear transformations, and some related geometric questions.
We look at a class of transcendental real numbers xi which, together with their square, satisfy some extremal property of simultaneous approximation by rational numbers with the same denominator. We give a sufficient condition for such a…
In literature, the central limit theorems for the product of sums of various random variables have studied. The purpose of this note is to show that this kind of results are corollary of the invariance principle.
Sum rules are elegant formulas that relate entropy functionals to coefficients associated with orthogonal polynomials [Sim11]. In a series of paper (see for example [GNR16], [GNR17], [BSZ18a], [BSZ18b]), interesting connections have been…
These lecture notes provide an informal introduction to the theory of nonnegative polynomials and sums of squares. We highlight the history and some recent developments, especially the new connections with classical (complex) algebraic…
In this note, we propose simple summations for primes, which involve two finite nested sums and Bernoulli numbers. The summations can also be expressed in terms of Bernoulli polynomials.
These notes provide a quick introduction to the Coq system and show how it can be used to define logical concepts and functions and reason about them. It is designed as a tutorial, so that readers can quickly start their own experiments,…
These notes are devoted to the theory of exponential sums over finite fields. The first chapter recalls some of the number-theoretic interest of such sums. The second chapter discusses the $L$-functions attached to such sums, the "Weil…
This paper is concerned with the study of the fractional finite sums theory. We present the classes of functions for which it is possible to characterize the constant related to the derivative of fractional sums (denominated by essence of a…
In this paper we present some interesting results involving summation of series in particular trigonometric ones. We failed to locate these results in existing literature or in the web like MathWorld (http://mathworld.wolfram.com/) nor…
Many generating series of combinatorially interesting numbers have the property that the sum of the terms of order $<p$ at some suitable point is congruent to a zero of a zeta-function modulo infinitely many primes $p$. Surprisingly, very…
In these notes we focus on commutative finite-dimensional normed algebras and some basic examples.
This note provides new closed forms evaluations of a few classes of exponential sums associated with elliptic curves and hyperelliptic curves.
Large collections of videos are grouped into clusters by a topic keyword, such as Eiffel Tower or Surfing, with many important visual concepts repeating across them. Such a topically close set of videos have mutual influence on each other,…
In this article, we extend some results about algebra $A$ with the group of units $U(A)$ having a special polynomial identity, Laurent polynomial. And we present a new version of B. Hartley Conjecture with these identities.
This note quantifies, via a sharp inequality, an interplay between (a) the characteristic rank of a vector bundle over a topological space X, (b) the Z/2Z-Betti numbers of X, and (c) sums of the numbers of certain partitions of integers. In…
Despite the great development of document summarisation techniques nowadays, factual inconsistencies between the generated summaries and the original texts still occur from time to time. This study explores the possibility of adopting…