On an Arithmetic Function

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Arithmetic tests forA-stability,A[α]-stability,and stiff-stability

Arithmetic tests forA-stability,A[]-stability, and stiff-stability are presented as special cases of a general stability test for numerical integration methods. The test evolves from extracted properties of the characteristic polynomial (in two variables) of the numerical method applied to the prototype scalar ordinary differential equation , Re {q}<0. The several steps of the test impose root clustering conditions — such as being Hurwitz — on a polynomial of one variable.Research sponsored in part by a special grant of the Syracuse University Senate Research Committee to the first author and by the National Science Foundation under Grant ENG76-21816 to the second author.The research reported herein was completed while the first author was on sabbatical leave in the Department of Electrical Engineering and Computer Sciences at the University of California in Berkeley, California.     

Arithmetic–geometric discrete systems

Analyzing a consensus in opinion dynamics one meets interesting open problems concerning the iteration of various means. Whereas in two dimensions the consensus can be computed by the arithmetic-geometric mean of Gauss not very much is known in higher dimensions.     

Arithmetic differential equations on $$GL_n$$, II: arithmetic Lie–Cartan theory

Motivated by the search of a concept of linearity in the theory of arithmetic differential equations (Buium in Arithmetic differential equations. Math. surveys and monographs, vol 118. American Mathematical Society, Providence, 2005), we introduce here an arithmetic analogue of Lie algebras, of Chern connections, and of Maurer–Cartan connections. Our arithmetic analogues of Chern connections are certain remarkable lifts of Frobenius on the p-adic completion of \(GL_n\) which are uniquely determined by certain compatibilities with the “outer” involutions defined by symmetric (respectively, antisymmetric) matrices. The Christoffel symbols of our arithmetic Chern connections will involve a matrix analogue of the Legendre symbol. The analogues of Maurer–Cartan connections can then be viewed as families of “linear” flows attached to each of our Chern connections. We will also investigate the compatibility of lifts of Frobenius with the inner automorphisms of \(GL_n\); in particular, we will prove the existence and uniqueness of certain arithmetic analogues of “isospectral flows” on the space of matrices.     

An Additive Problem with Primes in Arithmetic Progressions

In this paper, we extend a classical result of Hua to arithmetic progressions with large moduli. The result implies the Linnik Theorem on the least prime in an arithmetic progression.     

Arithmetic properties arising from Ramanujan’s theta functions

We prove some interesting arithmetic properties of theta function identities that are analogous to q-series identities obtained by Michael D. Hirschhorn. In addition, we find infinite family of congruences modulo powers of 2 for representations of a non-negative integer n as \(\triangle _1+4\triangle _2\) and \(\triangle +k\square \).     

Arithmetic properties of ℓ-regular partitions

For a given prime p, by studying p  -dissection identities for Ramanujan?s theta functions ψ(q)$\psi (q)$ and f(−q)$f(-q)$, we derive infinite families of congruences modulo 2 for some ?  -regular partition functions, where ?=2,4,5,8,13,16$?=2,4,5,8,13,16$.     

Analytic continuation ofζ 3 (S,K) to the critical strip. Arithmetic part

In this paper we study the zeta-function

Some Arithmetic on the Hardy Sums s2（h, κ） and s3（h, κ）

In this work, Rademacher‘s two questions about Dedekind sums are applied to the Hardy sums s2(h,κ) and s3(h,κ). Some relations between these sums are obtained and some inequalities are given.     

Arithmetic properties of the Shimura–Shintani–Waldspurger correspondence

We prove that the theta correspondence for the dual pair , for B an indefinite quaternion algebra over ℚ, acting on modular forms of odd square-free level, preserves rationality and p-integrality in both directions. As a consequence, we deduce the rationality of certain period ratios of modular forms and even p-integrality of these ratios under the assumption that p does not divide a certain L-value. The rationality is applied to give a direct construction of isogenies between new quotients of Jacobians of Shimura curves, completely independent of Faltings’ isogeny theorem. To my parents.     

Arithmetic properties of partitions with odd parts distinct

In this work, we consider the function pod(n), the number of partitions of an integer n wherein the odd parts are distinct (and the even parts are unrestricted), a function which has arisen in recent work of Alladi. Our goal is to consider this function from an arithmetic point of view in the spirit of Ramanujan’s congruences for the unrestricted partition function p(n). We prove a number of results for pod(n) including the following infinite family of congruences: for all α≥0 and n≥0,

Arithmetic of Rikuna’s generic cyclic polynomial and generalization of Kummer theory

In this paper we study the arithmetic of the Rikunas generic polynomial for the cyclic group of order n and obtain a generalized Kummer theory. It is useful under the condition that and k where is a primitive n-th root of unity and =+^–1. In particular, this result with k implies the classical Kummer theory.Mathematics Subject Classification (2000): 11R20, 12E10, 12G05     

Finding Sum of Powers on Arithmetic Progressions with Application of Cauchy’s Equation

In this paper we introduce a method to find the sum of powers on arithmetic progressions by using Cauchy’s equation and obtain a general formula. Then we apply our results to show how to determine some other sums of powers and sums of products. Our results are more general than those in [9]. Finally we discuss the sum of powers on arithmetic progressions in commmutative rings with characteristic 2 and find ‘full polynomials’.     

Optimization Design of Holding Poles Based on the Response Surface Methodology and the Improved Arithmetic Optimization Algorithm北大核心CSCD

抱杆优化设计需要耗费大量有限元分析计算时间,难以确定可行域.该文采用响应面法(response surface method,RSM)来模拟抱杆结构的真实响应,提出了改进的算术优化算法(improved arithmetic optimization algorithm,IAOA)对抱杆结构进行优化设计.将分数阶积分引入算术优化算法(arithmetic optimization algorithm,AOA),改善了算法的开发能力.采用拉丁超立方抽样,选取抱杆结构杆件截面试验样本,利用最小二乘法对样本点进行分析,构建了抱杆结构应力和位移关于杆件截面尺寸的二阶响应面代理模型.建立以抱杆质量最小化为优化目标,许用应力和位移为约束条件的优化模型,采用IAOA对其进行求解.结果表明:二阶响应面模型能够准确预测抱杆结构的响应值,IAOA的求解精度得到显著提升,代理模型可大幅降低有限元分析所需的计算代价,优化后抱杆结构质量减轻了8.2%.联合使用RSM和IAOA可有效求解大型空间杆系结构的优化设计问题.     

Equidistribution of Hecke Eigenforms on the Arithmetic Surface Γ0(N)\H

Given the orthonormal basis of Hecke eigenforms in S_2k(Γ(1)), Luo established an associated probability measure dμ_k on the modular surface Γ(1)\H that tends weakly to the invariant measure on Γ(1)\H. We generalize his result to the arithmetic surface Γ₀(N)\H where N?1 is square-free     

Certain Solutions of Equation f（z,0,…,0,uk,0,…,0）=0

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