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A Complete diophantine Characterization of the Rational Torsion of an Elliptic Curve

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Acta Mathematica Sinica,English Series 2012年第1期28卷 83-96页

作者： Irene GARCíA-SELFA Jos M.TORNERO Departamento de Matemáticas Universidad de HuelvaFacultad de Ciencias ExperimentalesCampus de El CarmenAvenida de las Fuerzas Armadass/n21071 HuelvaSpain Departamento de ' Algebra Universidad de SevillaP.O.116041080 SevillaSpain

We give a complete characterization for the rational torsion of an elliptic curve in terms of the （non-）existence of integral solutions of a system of diophantine equations.

关键词： Elliptic curves diophantine equations

Small Prime Solutions of a Pair of Linear equations in Five Prime Variables

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Acta Mathematica Sinica,English Series 2010年第3期26卷 569-578页

作者： Zhen CUI Hong Ze LI Department of Mathematics Shanghai Jiao Tong University Shanghai 200240 P. R. China

Assume an additional congruent condition on the coefficients. We prove that the pair 5 of linear equations ∑j=1^5 αλjpj = bλ （λ= 1, 2） has solutions in primes pj satisfying pj 〈〈（｜b1｜＋｜b2｜＋1） maxλ,j ｜αλj｜^2318＋ε. This improves the exponent 79680 without assuming the additional condition of the second author＇s.

关键词： diophantine equations Deuring-Heilbronn phenomenon large sieve inequality zero-free region

General solution to diagonal model matching control of multiple-output-delay systems and its applications in adaptive scheme

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Progress in Natural Science:Materials International 2009年第1期19卷 79-90页

作者： Yingmin Jia The Seventh Research Division,Beihang University,Beijing 100083,China The Seventh Research Division Beihang University Beijing 100083 China

This paper mainly studies the model matching problem of multiple-output-delay systems in which the reference model is assigned to a diagonal transfer function matrix.A new model matching controller structure is first developed,and then,it is shown that the controller is feasible if and only if the sets of diophantine equations have common *** obtained controller allows a parametric representation,which shows that an adaptive scheme can be used to tolerate parameter variations in the *** resulting adaptive law can guarantee the global stability of the closed-loop systems and the convergence of the output error.

关键词： Model matching control Decoupling Adaptation Time delays Multivariable systems diophantine equations

On the 2-primary Part of Tame Kernels of Real Quadratic Fields

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Acta Mathematica Sinica,English Series 2016年第7期32卷 807-812页

作者： Xiao Yun CHENG Xue Jun GUO School of Science Nanjing University of Aeronautics and Astronautics Department of Mathematics Nanjing University

Let F = Q（x/P）, where p = 8t ＋ 1 is a prime. In this paper, we prove that a speclm case of Qin＇s conjecture on the possible structure of the 2-primary part of K2OF up to 8-rank is a consequence of a conjecture of Cohen and Lagarias on the existence of governing fields. We also characterize the 16-rank of K2OF, which is either 0 or 1, in terms of a certain equation between 2-adic Hilbert symbols being satisfied or not.

关键词： Tame kernels class groups diophantine equations

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Erds猜想,Pell序列和丢番图方程

哈尔滨工业大学学报 1987年第2期 122-124页

作者：曹珍富哈尔滨工业大学数学系

1. Introduction and Results 1.1 Erds conjecture In 1939, Erds conjectured that the equation （mn）=yk, n>m>1, k≥3, （1）has no integer solution, where（mn）is a combinatorial number. In 1951, Erds himself proved that the conjecture is right when m>3, leaving the cases m=2 and m=3 unsolved （see Ref.[1]）. In 1984, we proved that （2n）（n>2） is never a 2kth power if k>1. Since equation

关键词： diophantine equations sequence combinatorial number

Computational Resolution of a Boolean Equation of 21 Variables

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American Journal of Operations Research 2022年第5期12卷 157-178页

作者： Esther Claudine Bitye Mvondo Department of Mathematics University of Yaounde I Yaounde Cameroon

The Alienor method has been elaborated at the beginning of the 1980s by Yves Cherruault and Arthur Guillez (1983). The following people have also greatly contributed to the improvement of this new optimization method: Blaise Somé, Gaspar Mora, Balira Konfé, Jean Claude Mazza and Esther Claudine Bityé Mvondo. The basic idea consists in using a reducing transformation allowing us to simplify a multivariable optimization problem to a new optimization problem according to a single variable. The rational gestion of enterprises leads generally to the use of Operational Research, often called management science. The term Operational Research means a scientific approach to decision making, that seeks optimization in a system. Consequently, it is better to take the right decisions. Otherwise, fatal consequences can occur instantaneously [1]. Generally, we have to maximize the global profit margin, taking into account some constraints. For instance, in an integer programming problem, some or all the variables are required to be nonnegative integers. In this paper, we present new reducing transformations for global optimization in integer, binary and mixed variables as well as the applications in Boolean algebra by solving a Boolean Equation of 21 variables. The applications in Operational Research are presented on various examples, resolved by using the tabulator Excel of Microsoft.

关键词： Alpha-Dense Curves Boolean equations diophantine equations Global Optimization and Operational Research

Tiling a Plane with Semi-Regular Equilateral Polygons with 2m-Sides

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Open Journal of Discrete Mathematics 2021年第1期11卷 13-30页

作者： Nenad Stojanovic Department of Mathematics Faculty of Agriculture University of Banja Luka Banja Luka Bosnia and Herzegovina

In this paper, tiling a plane with equilateral semi-regular convex polygons is considered, and, that is, tiling with equilateral polygons of the same type. Tiling a plane with semi-regular polygons depends not only on the type of a semi-regular polygon, but also on its interior angles that join at a node. In relation to the interior angles, semi-regular equilateral polygons with the same or different interior angles can be joined in the nodes. Here, we shall first consider tiling a plane with semi-regular equilateral polygons with 2m-sides. The analysis is performed by determining the set of all integer solutions of the corresponding Diophantine equation in the form of , whereare the non-negative integers which are not equal to zero at the same time, and are the interior angles of a semi-regular equilateral polygon from the characteristic angle. It is shown that of all semi-regular equilateral polygons with 2m-sides, a plane can be tiled only with the semi-regular equilateral quadrilaterals and semi-regular equilateral hexagons. Then, the problem of tiling a plane with semi-regular equilateral quadrilaterals is analyzed in detail, and then the one with semi-regular equilateral hexagons. For these semi-regular polygons, all possible solutions of the corresponding Diophantine equations were analyzed and all nodes were determined, and then the problem for different values of characteristic elements was observed. For some of the observed cases of tiling a plane with these semi-regular polygons, some graphical presentations of tiling constructions are also given.

关键词： Tiling a Plane Semi-Regular Plane Tiling diophantine equations

Relating Optimization Problems to Systems of Inequalities and Equalities

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American Journal of Operations Research 2020年第6期10卷 284-298页

作者： H. W. Corley E. O. Dwobeng Center on Stochastic Modeling Optimization and Statistics (COSMOS) The University of Texas at Arlington Arlington TX USA

In quantitative decision analysis, an analyst applies mathematical models to make decisions. Frequently these models involve an optimization problem to determine the values of the decision variables, a system S of possibly non- linear inequalities and equalities to restrict these variables, or both. In this note, we relate a general nonlinear programming problem to such a system S in such a way as to provide a solution of either by solving the other—with certain limitations. We first start with S and generalize phase 1 of the two-phase simplex method to either solve S or establish that a solution does not exist. A conclusion is reached by trying to solve S by minimizing a sum of artificial variables subject to the system S as constraints. Using examples, we illustrate how this approach can give the core of a cooperative game and an equilibrium for a noncooperative game, as well as solve both linear and nonlinear goal programming problems. Similarly, we start with a general nonlinear p

关键词： Optimization Inequalities and Equalities Goal Programming Games diophantine equations

On Ellipsoids Attached to Root Systems

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Journal of Applied Mathematics and Physics 2016年第8期4卷 1513-1521页

作者： Anatoli Loutsiouk Department of Civil Engineering King Mongkut University of Technology Thonburi BangkokThailand

For any finite-dimensional complex semisimple Lie algebra, two ellipsoids (primary and secondary) are considered. The equations of these ellipsoids are diophantine equations, and the Weyl group acts on the sets of all their diophantine solutions. This provides two realizations (primary and secondary) of the Weyl group on the sets of diophantine solutions of the equations of the ellipsoids. The primary realization of the Weyl group suggests an order on the Weyl group, which is stronger than the Chevalley-Bruhat ordering of the Weyl group, and which provides an algorithm for the Chevalley-Bruhat ordering. The secondary realization of the Weyl group provides an algorithm for constructing all reduced expressions for any of its elements, and thus provides another way for the Chevalley-Bruhat ordering of the Weyl group.

关键词： Complex Semisimple Lie Algebra Cartan Subalgebra Weyl Group Cartan Matrix Primary and Secondary Ellipsoids diophantine equations Geometric Realizations Coxeter Relations Dynkin Diagram Chevalley-Bruhat Ordering Reduced Expressions