P3BD‐closed sets

A concept called P3BD‐closed set is introduced to describe a set of positive integers which is both PBD‐closed and 3BD‐closed. The theory of P3BD‐closure is developed and a few examples of P3BD‐closed sets are exhibited. In the process, the existence spectrum of OLIQ ^?s (overlarge sets of idempotent quasigroups with their own idempotent orthogonal mates) is almost determined. A pair of orthogonal OLIQ ^?s is shown to asymptotically exist. The existence of OLIQ s (overlarge sets of idempotent quasigroups with their own orthogonal mates not necessarily specifying idempotency) is also established with only 10 possible exceptions remained. Copyright © 2011 Wiley Periodicals, Inc. J Combin Designs 19:407‐421, 2011     

带可分解性质的幂等拟群大集

具有幂单正交侣的幂等拟群称为可分解的. 具有幂等正交侣的幂等拟群称为几乎可分解的. 若v 元集合上的所有分量互不相同的3-向量能够分拆成互不相交(幂等3-向量除外) 的v-2 个v 阶幂等拟群, 则称之为v 阶幂等拟群大集. 本文使用t-平衡设计(t=2; 3) 的方法给出了可分解幂等拟群大集、几乎可分解幂等拟群大集及可分解对称幂等拟群大集(即可分解高尔夫设计) 的构造方法, 给出了其存在性的若干结果.     

Quasigroups and tactical systems

A quasigroupQ is a set together with a binary operation which satisfies the condition that any two elements of the equationxy =z uniquely determines the third. A quasigroup is in indempotent when any elementx satisfies the indentityxx =x. Several types of Tactical Systems are defined as arrangement of points into “blocks” in such a way as to balance the incidence of (ordered or unordered) pairs of points, and shown to be coexistent with idempotent quasigroups satisfying certain identifies. In particular the correspondences given are: 1. totally symmetric idempotent quasigroups and Steiner triple systems, 2. semi-symmetric idempotent quasigroups and directed triple systems, 3. idempotent quasigroups satisfying Schröder's Second Law, namely (xy)(yx)=x, and triple tourna-ments, and 4. idempotent quasigroups satisfying Stein's Third Law, namely (xy)(yx)=y, and directed tournaments. These correspondences are used to obtain corollaries on the existence of such quasig-roups from constructions of the Tactical Systems. In particular this provides a counterexample to an ”almost conjecture“ of Norton and Stein (1956) concerning the existence of those quasigroups in 3 and 4 above. Indeed no idempotent qnasigroups satisfying Stein's Third Law and with order divisible by four were known to N. S. Mendelsohn when he wrote a paper on such quasigroups for the Third Waterloo Conference on Combinatorics (May, 1968). Finally, a construction for triple tournaments is interpreted as a Generalized Semi-Direct Product of idempotent quasigroups.     

The spectrum of 2‐idempotent 3‐quasigroups with conjugate invariant subgroups

A ternary quasigroup (or 3‐quasigroup) is a pair (N, q) where N is an n‐set and q(x, y, z) is a ternary operation on N with unique solvability. A 3‐quasigroup is called 2‐idempotent if it satisfies the generalized idempotent law: q(x, x, y) = q(x, y, x) = q(y, x, x)=y. A conjugation of a 3‐quasigroup, considered as an OA(3, 4, n), $({{N}},{\mathcal{B}})$, is a permutation of the coordinate positions applied to the 4‐tuples of ${\mathcal{B}}$. The subgroup of conjugations under which $({{N}},{\mathcal{B}})$ is invariant is called the conjugate invariant subgroup of $({{N}},{\mathcal{B}})$. In this article, we determined the existence of 2‐idempotent 3‐quasigroups of order n, n≡7 or 11 (mod 12) and n≥11, with conjugate invariant subgroup consisting of a single cycle of length three. This result completely determined the spectrum of 2‐idempotent 3‐quasigroups with conjugate invariant subgroups. As a corollary, we proved that an overlarge set of Mendelsohn triple system of order n exists if and only if n≡0, 1 (mod 3) and n≠6. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 292–304, 2010     

Purely tetrahedral quadruple systems

An oriented tetrahedron is a set of four vertices and four cyclic triples with the property that any ordered pair of vertices is contained in exactly one of the cyclic triples. A tetrahedral quadruple system of order n (briefly TQS(n)) is a pair (X,B), where X is an nelement set and B is a set of oriented tetrahedra such that every cyclic triple on X is contained in a unique member of B. A TQS(n) (X,B) is pure if there do not exist two oriented tetrahedra with the same vertex set. In this paper, we show that there is a pure TQS(n) if and only if n = 2,4 (mod 6), n > 4, or n = 1,5 (mod 12). One corollary is that there is a simple two-fold quadruple system of order n if and only if n = 2,4 (mod 6) and n > 4, or n = 1,5 (mod 12). Another corollary is that there is an overlarge set of pure Mendelsohn triple systems of order n for n=1,3 (mod 6), n > 3, or n =0,4 (mod 12).     

偏序集的关联环的同构问题

设Ｘ是局部有限偏序集（或拟序集），Ｒ是含１的结合环，Ⅰ（Ｘ，Ｒ）是Ｒ上Ｘ的关联环，关联环的同构问题是指：问题１：怎样的环，能使环同构Ⅰ（Ｘ，Ｒ）Ⅰ（Ｘ，Ｒ）推出偏序集之间的同构Ｘ芒Ｘ’？问题２：怎样的环或偏序集，能使环同构Ⅰ（Ｘ，Ｒ）Ⅰ（Ｘ，Ｓ）推出Ｒ　Ｓ？本文证明了对唯一幂等元环（非交换），问题１有正面回答；对问题２，我们证明了对交换不可分解环Ｒ、Ｓ，由环同构Ⅰ（Ｘ，Ｒ）Ⅰ（Ｘ，Ｒ）可得到Ｒ＝Ｓ，Ｘ＝Ｘ’。     

带共轭性质拟群的计数

由-个拟群(Q,(×))可以定义出6个共轭拟群,这6个共轭拟群不一定互不相同,其构成的集合C(Q,(×))的基数t可能的取值是1,2,3或6.记q(n,t)是所有满足|C(Q,(×))|=t的n阶拟群的个数,本文将给出q(n,2)和q(n,6)的计数问题.     

Banach空间中的平均非扩张映象:不动点的存在定理

<正> 设X是Banach空间,E是X中的集合,T是映集合E到自身的映象.若T满足条件(称为平均非扩张条件)其中x,y∈E,a,b,c≥0且a+2b+2c≤1,则称T是平均非扩张映象. 文[1]概括了近年来研究关于平均非扩张映象不动点的一些主要结果.本文进     

Idempotent matrix lattices over distributive lattices

In this paper, the partially ordered set of idempotent matrices over distributive lattices with the partial order induced by a set of lattice matrices is studied. It is proved that this set is a lattice; the formulas for meet and join calculation are obtained. In the lattice of idempotent matrices over a finite distributive lattice, all atoms and coatoms are described. We prove that the lattice of quasi-orders over an n-element set Qord(n) is not graduated for n ≥ 3 and calculate the greatest and least lengths of maximal chains in this lattice. We also prove that the interval ([I, J]_≤, ≤) of idempotent (n × n)-matrices over {ie879-01}-lattices is isomorphic to the lattice of quasi-orders Qord(n). Using this isomorphism, we calculate the lattice height of idempotent {ie879-02}-matrices. We obtain a structural criterion of idempotent matrices over distributive lattices. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 4, pp. 121–144, 2007.     

Regularity and Green's Relations for Semigroups of Transformations Preserving Orientation and an Equivalence

Let be the full transformation semigroup on a set X. For a non-trivial equivalence E on X, let

Green's Equivalences on Semigroups of Transformations Preserving Order and an Equivalence Relation

Let ${\cal T}_X$ be the full transformation semigroup on the set $X$, \[ T_{E}(X)=\{f\in {\cal T}_X\colon \ \forall(a,b)\in E,(f(a),f(b))\in E\} \] be the subsemigroup of ${\cal T}_X$ determined by an equivalence $E$ on $X$. In this paper the set $X$ under consideration is a totally ordered set with $mn$ points where $m\geq 2$ and $n\geq 3$. The equivalence $E$ has $m$ classes each of which contains $n$ consecutive points. The set of all order preserving transformations in $T_{E}(X)$ forms a subsemigroup of $T_E(X)$ denoted by \[ {\cal O}_{E}(X)=\{f\in T_{E}(X)\colon \ \forall\, x, y\in X, \ x\leq y \mbox{ implies } f(x)\leq f(y)\}. \] The nature of regular elements in ${\cal O}_{E}(X)$ is described and the Green's equivalences on ${\cal O}_{E}(X)$ are characterized completely.     

双对称非负定阵一类逆特征值问题的最小二乘解

１．引言 逆特征值问题在工程中有广泛的应用,其研究已有一些很好的结果［１－５］．最近,文［６］还研究了双对称矩阵逆特征值问题,即研究了如下两个问题： 问题Ａ．已知Ｘ∈Ｒｎｘｍ,Ａ＝ｄｉａｇ（λ１…,λｍ）,求Ａ∈ＢＳＲｎｘｎ使 ＡＸ＝ＸＡ,其中 Ｒｎｘｍ表示全体 ｎ ｘ ｍ实矩阵集合, ＢＳＲｎｘｎ表示全体 ｎ ｘ ｎ双对称阵集合． 问题Ｂ．已知Ａ＊ＥＲｎｘｎ,求Ａ∈ＳＥ使 ｜｜Ａ＊－Ａ｜｜＝ ｉｎｆ ｜｜Ａ＊－Ａ｜｜ ＡＦＳＥ其中 ＳＥ是问题 Ａ的解集合,｜｜． ｜｜表示 Ｆｒｏｂｅｎｉｕｓ范数． 在实际问题中, …     

Vector subdivision schemes in (Lp(Rs))r(1 ≤ p ≤∞) spaces

The purpose of this paper is to investigate the refinement equations of the form ψ(x) = ∑α∈Zs a(α)ψ(Mx - α), x ∈ Rs,where the vector of functions ψ=(ψ1,…,ψr)T is in (Lp(Rs))r, 1≤p≤∞,a(α),α∈Zs,is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix suchthat lim n→∞ M-n = 0. In order to solve the refinement equation mentioned above, we start with a vectorof compactly supported functions ψ0 ∈ (Lp(Rs))r and use the iteration schemes fn := Qnaψ0,n = 1,2,…,where Qa is the linear operator defined on (Lp(Rs))r given by Qaψ:= ∑α∈Zs a(α)ψ(M·- α),ψ∈ (Lp(Rs))r. This iteration scheme is called a subdivision scheme or cascade algorithm. In this paper, we characterize the Lp-convergence of subdivision schemes in terms of the p-norm joint spectral radius of a finite collection of somelinear operators determined by the sequence a and the set B restricted to a certain invariant subspace, wherethe set B is a complete set of representatives of the distinct cosets of the quotient group Zs/MZs containing 0.     

一类变换半群的正则元和Green关系

设T_X为X上的全变换半群,E为X上的等价关系,令T_E(X)={f∈T_X:■(x,y)∈E,(f(x),f(y))∈E},则T_E(X)是T_X的子半群,如果X是一个全序集,E是X上的一个凸等价关系,设OP_E(X)为T_E(X)中所有保向映射作成的半群。对于有限全序集X上一类特殊的凸等价关系E,本文刻画了半群OP_E(X)的正则元的特征,并且描述了这个半群上的Green关系。     

URS' For Weierstrass Products without Exponential Factors

Let $ \cal W $ be the set of entire functions equal to a Weierstrass product of the form $ {f(x)= Ax^q\lim_{r \to \infty} \prod_{|a_j|\leq r}{(1- \fraca {x} {a_j})}} $ where the convergence is uniform in all bounded subsets of $ {\shadC} $ , let $ \cal V $ be the set of $ f\in {\cal W} $ such that $ {\shadC} [\,f]\subset {\cal W} $ , and let $ {\cal H} $ be the $ {\shadC} $ -algebra of entire functions satisfying $ { {\lim_{r\to \infty } } ({\ln M(r,f) / r})=0} $ . Then $ \cal H $ is included in $ {\cal V} $ and strictly contains the set of entire functions of genus zero, (which, itself, strictly contains the $ {\shadC} $ -algebra of entire functions of order 𝜌 < 1). Let $ n, m\in {\shadN} ^* $ satisfy n > m S 3. Let $ a\in {\shadC}^* $ satisfies $ {a^n\not = \fraca{n^n}{(m^m(n-m)^{n-m}})} $ and assume that for every ( n m m )-th root ξ of 1 different from m 1, a satisfies further $ {a^{n}\neq (1+\xi )^{n-m} (\fraca{n^n}{((n-m)^{n-m}m^m}))} $ . Let P ( X ) = X n m aX m + 1 and let T n,m ( a ) be the set of its zeros. Then T n,m ( a ) has n distinct points and is a urs for $ {\cal V} $ . In particular this applies to functions such as sin x and cos x .     

A graph-theoretic approach to quasigroup cycle numbers

Norton and Stein associated a number with each idempotent quasigroup or diagonalized Latin square of given finite order n, showing that it is congruent mod 2 to the triangular number T(n). In this paper, we use a graph-theoretic approach to extend their invariant to an arbitrary finite quasigroup. We call it the cycle number, and identify it as the number of connected components in a certain graph, the cycle graph. The congruence obtained by Norton and Stein extends to the general case, giving a simplified proof (with topology replacing case analysis) of the well-known congruence restriction on the possible orders of general Schroeder quasigroups. Cycle numbers correlate nicely with algebraic properties of quasigroups. Certain well-known classes of quasigroups, such as Schroeder quasigroups and commutative Moufang loops, are shown to maximize the cycle number among all quasigroups belonging to a more general class.     

某类常微系统的一个基本性质

<正> 简介 考虑一n维紧致的C~∞型Riemann流形M~n(n≥2)上,由所有的C~1型常微系统就C~1度量作成的空间.这里为简便,一系统∈将暂称为A_o-系统,如果它只具有有限多个奇点和至多可数多个周期轨道;将暂称为A_1-系统,如果它只具有有限多个奇点和有限多个周期轨道.依据周知的结果,一般绝不是中每一系统都有任意小的邻     

无爪3 -正则图的独立数

如果图G的一个集合X中任两个点不相邻, 则称 X 为独立集合. 如果 N[X]=V(G), 则称X是一个控制集合. i(G)(β(G))分别表示所有极大独立集合的最小(最大)基数. γ(G)(Γ(G))表示所有极小控制集合的最小(最大)基数. 在这篇论文中, 作者证明如下结论: (1) 如果 G ∈R 且G 是n阶3 -正则图, 则 γ(G)= i(G), β(G)=n/3. (2) 每个n阶连通无爪3 -正则图 G, 如果 G(G≠ K₄) 且不含诱导子图K₄-e, 则 β(G) =n/3.     

保持序和等价关系的自然偏序变换半群

设X为一个集合,■_X为X上的全变换半群.设E是X上的一个等价关系,定义T_E(X)={f∈■_X:■(x,y)∈E,(f(x),f(y))∈E},则T_E(X)是由等价关系E所确定的■_X的子半群.本文中,所考虑的集合X是一个有限全序集,同时E是非平凡的且所有的E-类都是凸集.显然■_E(X)={f∈T_E(X):■_x,y∈X,x≤y蕴涵f(x)≤f(y)}是T_E(X)的一个子半群.我们赋予■_E(X)自然偏序并讨论何时■_E(X)中的两个元素是关于这个偏序是相关的,然后确定■_E(X)中那些关于≤是相容的元素.此外,还描述了极大(极小)元和覆盖元.     

A Characterization of Topologically Transitive Attributes for a Class of Dynamical Systems

In this work,by virtue of the properties of weakly almost periodic points of a dynamical system (X,T) with at least two points,the authors prove that,if the measure center M(T) of T is the whole space,...