相关论文: Some divergent trigonometric integrals
Perhaps the most significant, if not the most important, achievements in chemistry and physics are the Periodic Table of the Elements in Chemistry and the Standard Model of Elementary Particles in Physics. A comparable achievement in…
The author's method (math-ph/9804010) that uses the Laplace transform to find exact values for a large class of convergent series is extended to trigonometric series.
Variational and divergence symmetries are studied in this paper for the whole class of linear and nonlinear equations of maximal symmetry, and the associated first integrals are given in explicit form. All the main results obtained are…
In this paper the double-sided Talor's approximations are used to obtain generalisations and improvements of some trigonometric inequalities.
Algebraic dichotomy is a generalization of an exponential dichotomy (Lin, JDE2009). This paper gives a version of Hartman-Grobman linearization theorem assuming that linear system admits an algebraic dichotomy, which generalizes the…
We state some elementary problems concerning the relation between difference calculus and differential calculus, and we try to convince the reader that, in spite of the simplicity of the statements, a solution of these problems would be a…
In one of the three 2010/2011 issues of the journal 'MathematicalSpectrum', this author gave a three-parameter description of the entire set of integral triangles(i.e. triangles with integer side lengths)and with a 120 degree angle.This…
Fundamental to the theory of continued fractions is the fact that every infinite continued fraction with positive integer coefficients converges; however, it is unknown precisely which continued fractions with integer coefficients (not…
We prove an interesting identity for the sum of determinants, which is a generalization of the sum of a geometric progression. The proof is quite long and a number of other identities are proved along the way. Some of the more elementary…
Integration at a point is a new kind of integration derived from integration over an interval in infinitesimal and infinity domains which are spaces larger than the reals. Consider a continuous monotonic divergent function that is…
In this paper, we consider a discrete version of iterated integrals by the naive (equally divided) Riemann sum. In particular, basic three formulas for usual iterated integrals are discritized. Moreover, we proved cyclic sum formulas for…
We introduce, and partially resolve, a conjecture that brings a three-centuries-old derangements phenomenon and its much younger two-decades-old analogue under the same umbrella. Through a graph-theoretic lens, a derangement is a perfect…
In this paper, we establish several new inequalities for some twice differantiable mappings. Then, we apply these inequalities to obtain new midpoint, trapezoid and perturbed trapezoid rules. Finally, some applications for special means of…
We believe we have made progress in the age-old problem of divisibility rules for integers. Universal divisibility rule is introduced for any divisor in any base number system. The divisibility criterion is written down explicitly as a…
Inequalities play important roles not only in mathematics, but also in other fields, such as economics and engineering. Even though many results are published on Hermite-Hadamard (H-H) type inequalities, new researcher to this fields often…
In this paper we present a combinatorial proof of Selberg's integral formula. We start by giving a bijective proof of a Theorem about the number of topological orders of a certain related directed graph. Selberg's Integral Formula then…
The purpose of this paper is to study a class of ill-posed differential equations. In some settings, these differential equations exhibit uniqueness but not existence, while in others they exhibit existence but not uniqueness. An example of…
Classical mathematics are founded within set theory, but sets don't have \emph{symmetries}. We conjecture that if we allow sets with symmetries, then many problems such as \emph{Mirror symmetry} or \emph{Homological mirror symmetry} can be…
The original version of this paper contains an error; when this is corrected the basic conclusion changes. A revised manuscript will be submitted shortly.
Classical results on aperiodic tilings are rather complicated and not widely understood. Below, an alternative approach is discussed in hope to provide additional intuition not apparent in classical works.