一类超收敛数值差商公式研究

本文针对充分光滑函数的数值差商公式的余项问题进行研究,对王兴华等于文[1]中提出的关于超收敛数值差商公式的猜想进行了证明,推广了该文中定理的适用范围,得到了比较广泛的一类超收敛的数值差商公式余项的lagrange表示．     

用非标准离散函数和差商定义新广义函数

在非标准分析框架下，用离散函数定义新广义函数，用差商定义其导数．对Schwartz广义函数以及更广的Gevrey超广义函数，文章证明了广义导数可以用差商表示．此外还给出了此新广义函数和Sobolev理论的关系．     

问题征解

问题征解有关不等式与组合几何的稿件邮送陈计老师（３１５２１１，宁波大学应用数学系），其它稿件邮送葛军老师（２１００２４．商京师范大学数学系）．题号右上角（＊）表示问题提出时尚无解答。问题１６６．上海田廷彦与熊斌提供空间ｎ（≥２）点Ｐ１；，Ｐ２，…，Ｐ...     

无K_4-图子式的图的谱半径(英文)

Ｇ是一个无Ｋ４－图子式、顶点数为ｎ的简单图,ｐ（Ｇ）是图Ｇ的谱半径．本文得出一个关于ｐ（Ｇ）的上确界：等式成立当且仅当 Ｇ ≌Ｋ２　（ｎ－２）Ｋ１,其中 Ｇ１　Ｇ２是由 Ｇ１∪Ｇ２组成、并且Ｇ１中的第一个点和Ｇ２中的每一个点之间都有一条边相连：（ｎ－２）Ｋ１表示（ｎ－２）个孤立点的集合．     

格序代数的商及其代数性质

研究了格序代数的商及其一些重要的代数性质.给出了格序代数的商的概念,定义了商f-代数、商几乎f-代数和商d-代数,并给出其等价刻画.讨论了这三类格序代数的商的半素性和交换性质,得到了这些性质的若干刻画.     

差商BFGS方法的收敛速率

本文讨论差商BFGS方法的收敛速率,证明了当差商步长满足某个条件时,由差商BFGS方法产生的点列具有比多步二次收敛略强一些的收敛速率。     

关于Kn—H2n＋1的升分解

［１］中猜想：任意有正数条边的图都可以升分解．本文证明了Ｋｎ－Ｈ２ｎ＋１可以升分解,其中Ｈ２ｎ＋１表示至多有ｎ个顶点和２ｎ＋１条边的图,ｎ≥７．     

对包含整数的最小素因子和的估计

ｎ为自然数（ｎ＞１）令ｐ（ｎ）表示ｎ的最小素因子,（ｎ）表示。的所有的素因子的个数,ｗ（ｎ）表示ｎ的不同的素因子的个数．本文绘出了,的渐近估计式,其中ｒ＞０．它们改进并推广了张文鹏相应的结果．     

数学奥林匹克讲与练(21)

例题讲解１６１．空间中有８个点,其中任何４点不共面,在这些点之间连结１７条线段．求证：（１）至少存在一个由这些线段所构成的三角形;（２）由这些线段构成的三角形实际上不少于４个．证明　（１）由一个已知点所引的已知线段的数目,我们称为这点的“度”．取度最大的一点,设其度为ｎ,则有ｎ条线段由这点引出．如果不存在由已知线段构成的三角形,则这ｎ条线段的另外ｎ个端点之间均无已知线段相连．此外尚余（７－ｎ）个点,每点的度不超过ｎ,故每点至多引出ｎ条已知线段,因而由它们引出的已知线段不超过ｎ（７－ｎ）条,于是已…     

矩阵方程A~TXA=D的双对称最小二乘解

１．引 言 本文用 Ｒｎ×ｍ表示全体 ｎ×ｍ实矩阵集合，用 ＳＲｎ×ｎ（ＳＲ０ｎ×ｎ）表示全体 ｎ× ｎ实对称（实对称半正定）矩阵集合，ＯＲｎ×ｎ表示全体 ｎ× ｎ实正交矩阵集合，ＢＳＲｎ×ｎ表示全体ｎ×ｎ双对称实矩阵集合．这里，一个实对称矩阵Ａ＝（ａｉｊ）ｎ×ｎ被称为双对称矩阵，如果对所有的 　　　　　　　 　　　　　　　　　　　　　　　用Ａ×Ｂ表示矩阵 Ａ与 Ｂ的Ｈａｄａｍａｒｄ乘积，Ｉｋ表示 ｋ× ｋ阶单位矩阵，Ｏ表示零矩阵，Ｓｋ＝（ｅｋ，…，ｅ２，ｅ１）∈ Ｒｋ×ｋ，其中ｅｉ表示Ｉｋ的第ｉ列． 矩阵方程…     

Characterization of polynomials and divided difference

For distinct points x₁,x₂,…,x_n in ℛ (the reals), letϕ[x₁, x₂,…,x_n] denote the divided difference ofϕ. In this paper, we determine the general solutionϕ,g: ℛ → ℛ of the functional equationϕ[x₁,x₂,…,x_n] =g(x₁,+ x₂ + … + x_n) for distinct x₁,x₂,…, x_n in ℛ without any regularity assumptions on the unknown functions.     

Congruences on generalized inverse semigroups

A generalized inverse semigroup is a regular semigroup whose idempotents satisfy a permutation identity X₁ X₂...X_n=X_p1 X_p2...X_pn, where (P₁, P₂..., P_n) is a nontrivial permutation of (1, 2,..., n). Yamada [4] has given a complete classification of generalized inverse semigroups in terms of inverse semigroups, left normal bands, and right normal bands. In this paper we show that every congruence on a generalized inverse semigroup is uniquely determined by a congruence on its associated inverse semigroup, left normal band, and right normal band. A converse is also provided. This paper is extracted from the doctoral thesis of the author written at Monash University under the direction of Professor G. B. Preston. The research was carried out while the author held a Commonwealth Postgraduate Award.     

A problem on semigroups satisfying permutation identities

Let σ be a nontrivial permutation of ordern. A semigroupS is said to be σ-permutable ifx ₁ x ₂ ...x _n =x _σ(1) x _σ(2) ...x _σ(n) , for every (x ₁ ,x ₂,...,x _n )∈S ⁿ . A semigroupS is called(r,t)-commutative, wherer,t are in ℕ^*, ifx ₁ ...x _r x _r+1 ...x _r+t =x _r+1 ...x _r+t x ₁ ...x _r , for every (x ₁ ,x ₂,...,x _r+t ∈S ^r+t . According to a result of M. Putcha and A. Yaqub ([11]), if σ is a fixed-point-free permutation andS is a σ-permutable semigroup then there existsk ∈ ℕ^* such thatS is (1,k)-commutative. In [8], S. Lajos raises up the problem to determine the leastk=k(n) ∈ ℕ^* such that, for every fixed-point-free permutation σ of ordern, every σ-permutable semigroup is also (1,k)-commutative. In this paper this problem is solved for anyn less than or equal to eight and also whenn is any odd integer. For doing this we establish that if a semigroup satisfies a permutation identity of ordern then inevitably it also satisfies some permutation identities of ordern+1.     

Permutations with Low Discrepancy Consecutive k-sums

Consider the permutation π=(π₁,…, π_n) of 1,2,…, n as being placed on a circle with indices taken modulo n. For given kn there are n sums of k consecutive entries. We say the maximum difference of any consecutive k-sum from the average k-sum is the discrepancy of the permutation. We seek a permutation of minimum discrepancy. We find that in general the discrepancy is small, never more than k+6, independent of n. For g= gcd(n,k)>1, we show that the discrepancy is . For g=1 it is more complicated. Our constructions show that the discrepancy never exceeds k/2 by more than 9 for large n, while it is at least k/2 for infinitely many n.We also give an analysis for the easier case of linear permutations, where we view the permutation as written on a line. The analogous discrepancy is at most 2 for all n,k.     

On the Dimension of Coinvariants of Permutation Representations

Let be a univariate, separable polynomial of degree n with roots x ₁,…,x _n in some algebraic closure of the ground field . It is a classical problem of Galois theory to find all the relations between the roots. It is known that the ideal of all such relations is generated by polynomials arising from G-invariant polynomials, where G is the Galois group of f(Z). Namely: The action of G on the ordered set of roots induces an action on by permutation of the coordinates and each defines a relation P − P(x ₁,…,x _n ) called a G-invariant relation. These generate the ideal of all relations. In this note we show that the ideal of relations admits an H-basis of G-invariant relations if and only if the algebra of coinvariants has dimension ‖G‖ over . To complete the picture we then show that the coinvariant algebra of a transitive permutation representation of a finite group G has dimension ‖G‖ if and only if G = Σ _n acting via the tautological permutation representation.     

An upper bound on the number of high-dimensional permutations

What is the higher-dimensional analog of a permutation? If we think of a permutation as given by a permutation matrix, then the following definition suggests itself: A d-dimensional permutation of order n is an n×n×...×n=[n] ^d+1 array of zeros and ones in which every line contains a unique 1 entry. A line here is a set of entries of the form {(x ₁,...,x _i?1,y,x _i+1,...,x _d+1)|n≥y≥1} for some index d+1≥i≥1 and some choice of x _j ∈ [n] for all j ≠ i. It is easy to observe that a one-dimensional permutation is simply a permutation matrix and that a two-dimensional permutation is synonymous with an order-n Latin square. We seek an estimate for the number of d-dimensional permutations. Our main result is the following upper bound on their number $$\left( {(1 + o(1))\frac{n} {{e^d }}} \right)^{n^d } .$$ We tend to believe that this is actually the correct number, but the problem of proving the complementary lower bound remains open. Our main tool is an adaptation of Brégman’s [1] proof of the Minc conjecture on permanents. More concretely, our approach is very close in spirit to Schrijver’s [11] and Radhakrishnan’s [10] proofs of Brégman’s theorem.     

The form of solutions and periodicity for some systems of third‐order rational difference equations

In this paper, we deal with the system that has solutions and the periodicity character of the following systems of rational difference equations with order three with initial conditions x_?2,x_?1,x₀,y_?2,y_?1, and y₀ that are arbitrary nonzero real numbers. Some numerical examples will be given to illustrate our results. Copyright © 2015 John Wiley & Sons, Ltd.     

Finite‐difference approximation for the u(k)‐derivative with O(hM−k+1) accuracy: An analytical expression

An approximation of function u(x) as a Taylor series expansion about a point x₀ at M points x_i, ～ i = 1,2,…,M is used where x_i are arbitrary‐spaced. This approximation is a linear system for the derivatives u^(k) with an arbitrary accuracy. An analytical expression for the inverse matrix A ^?1 where A = [A_ik] = (x_i ? x₀)^k is found. A finite‐difference approximation of derivatives u^(k) of a given function u(x) at point x₀ is derived in terms of the values u(x_i). © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Epimorphisms, dominions and permutative semigroups

We show that any right semicommutative semigroup satisfying the minimum condition on principal right ideals is saturated. Further we show that if a supersaturated semigroup S has a globally idempotent ideal U which satisfies a nontrivial permutation identity $x_{1}x_{2}\ldots x_{n}=x_{i_{1}}x_{i_{2}}\ldots x_{i_{n}}We show that any right semicommutative semigroup satisfying the minimum condition on principal right ideals is saturated. Further we show that if a supersaturated semigroup S has a globally idempotent ideal U which satisfies a nontrivial permutation identity x₁x₂?x_n=x_i1x_i2?x_inx_{1}x_{2}\ldots x_{n}=x_{i_{1}}x_{i_{2}}\ldots x_{i_{n}} such that i ₁=1 and i _n ≠n, then U is also supersaturated.     

Inverse problems in underwater acoustics and approximation of the velocity field for signal propagation

A new approach is suggested for solving the inverse problem in underwater acoustics — the determination of the velocity of signal propagation in an inhomogeneous medium. The unknown velocity u(x(t), t) is assumed to be independent of the time t and is sought in the form of a polynomial with respect to the degree of the space coordinate x ₁, x₂, x₃.Translated fromVychislitel'naya i Prikladnaya Matematika, No. 69, pp. 102–105, 1989.