Compactness of the product topology and the solvability of infinite systems of infinite linear equations

If every finite subsystem of an infinite system of linear equations (say, over the field of real numbers) each with finitely many unknowns has a solution then the entire system has a solution. The situation is not so if the equations contain infinitely many unknowns. In this case, as shown below, the solvability of every finite subsystem implies the solva. bility of the entire system provided finite subsystems have solution with common upper and lower bounds and the coefficients of ever equation satisfy some boundedness or convergence conditions. The passage from the solvability of finite subsystem to the solvability of the entire system is achieved based on Tychnoff’s theorem stating that any product of compact topological spaces is compact in their product topology.     

∑_(i=1)~n x_i／d_i表示的最小整数及其应用

设I(d1,…,dn)为方程∑i=1n xi/di≡0(mod 1),1≤xi≤di-1的解的个数．若I(d1,…,dn)>0,令L(d1,…,dn)为∑i=1n xi/di(1≤xi≤di-1)表示的最小整数．I(d1,…,dn),L(d1,…,dn)及它们的估计在有限域上对角方程解数的研究中具有重要作用．本文给出了等式成立的若干充分条件,这里ωi=gcd(di,lcm[dj:j≠i]),i=1,…,n．     

A Note on Semigroup Algebras of Permutable Semigroups

Let S be a semigroup and 𝔽 be a field. For an ideal J of the semigroup algebra 𝔽[S] of S over 𝔽, let ?_J denote the restriction (to S) of the congruence on 𝔽[S] defined by the ideal J. A semigroup S is called a permutable semigroup if α ○ β = β ○ α is satisfied for all congruences α and β of S. In this paper we show that if S is a semilattice or a rectangular band then φ_{{S; 𝔽}}: J → ?_J is a homomorphism of the semigroup (Con(𝔽[S]); ○ ) into the relation semigroup (?_S; ○ ) if and only if S is a permutable semigroup.     

Directly Indecomposables in Semidegenerate Varieties of Connected po-Groupoids

We study varieties with a term-definable poset structure, po-groupoids. It is known that connected posets have the strict refinement property (SRP). In Sánchez Terraf and Vaggione (Trans Am Math Soc, in press) it is proved that semidegenerate varieties with the SRP have definable factor congruences and if the similarity type is finite, directly indecomposables are axiomatizable by a set of first-order sentences. We obtain such a set for semidegenerate varieties of connected po-groupoids and show its quantifier complexity is bounded in general. Supported by Conicet.     

单模数多变量二次同余方程组的求解及其在信息安全中的应用

根据二次同余理论,利用当P=5mod8,n^（p-1）/4=±1modp时,方程X^2=nmodP的解有具体表达式这一事实,设计了基于多变量二次同余方程组实现的密码体系。数值算例证明,该类密码体系是可行的,体系信息冗余且是一次一密的,要依赖于解密策略的交互确认,所以该类密码体系是安全的。     

有限网络的一个性质

借助实例提出一个有关有限网络研究的基本问题：对装置的任意电路网络是否总能经有限次改变状态后,从全＂关闭＂状态变为全＂开启＂状态.利用无向图作为网络的数学模型,并利用邻接矩阵,可把该基本问题归结为证明二元域Z2上一个特殊的线性方程组是否有解的问题.基于二元域Z2的运算性质及此域上线性方程组的理论,可严格证明上述线性方程组...     

正则带的半格结构

Petrich解决了一般带的构造定理（见[1]或[2]），在此基础上，我们将证明正则带（满足等式axya=axaya的带）的一些特征，并给出一个带为正则带或右似正规带（满足等式xya=xaya的带）的充分必要条件，这些结果是Yamada和Kimura的关于正规带（满足等式axya=ayxa的带）的结果的推广，正规带被他们描述为矩形带的强半格（见[1]或[3]）。     

Solving linear equations in a vector space over a finite field

We study the maximum possible size of a subset in a vector space over a finite field which contains no solution of a given linear equation (or a system of linear equations). This is a finite field version of Ruzsa's work [7].     

Farkas' Lemma, other theorems of the alternative, and linear programming in infinite-dimensional spaces: a purely linear-algebraic approach

We briefly consider several formulations of Farkas' Lemma first. Then we assume the setting of two vector spaces, one of them being linearly ordered, over a linearly ordered field till the end of this article. In this setting, we state a generalized version of Farkas' Lemma and prove it in a purely linear-algebraic way. Afterwards, we present Theorems of Motzkin, Tucker, Carver, Dax, and some other theorems of the alternative that characterize consistency of a finite system of linear inequalities. We also mention the Key Theorem, which is a related result. Finally, we use Farkas' Lemma to prove the Duality Theorem for linear programming (with a finite number of linear constraints). The Duality Theorem that is proved here covers, among others, linear programming in a real vector space of finite or infinite dimension and lexicographic linear programming.     

HOMOMORPHISMS BETWEEN CHEVALLEY GROUPS OF TYPES C_nAND G_2 OVER FINITE FIELDS

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