Beispiele endlicher und unendlicher K-Loops

In this note examples are given for non trivial K-loops. There are commutative examples as well as non commuative, finite examples as well as infinite. Furthermore it will be shown that under an additional condition K-loops and Brück loops coincide.     

The Campbell- Hausdorff Series of Local Analytic Bruck Loops

The class of local analyitic Bruck loops (or equivalently K-loops) is strongly related to locally symmetric spaces. In particular, both have Lie triple systems as their tangent algebra. In this paper, we consider the existence and some properties of the Campbell-Hausdorff series of local analytic Bruck loops (K-loops). This formula can be used to determine the local symmetries of the associated symmetric space.     

Construction of a class of loops via graphs

Starting from an involutorial difference loop (P, +) of order n we construct a new loop ${(L, \oplus)}$ in the same class and possessing 2n elements. The construction was induced by studying a correspondence connecting such loops with complete graphs with parallelism and regular involution sets (see “Basic notions and known results”). We discuss conditions on (P, + ) ensuring that the loop ${(L, \oplus)}$ is a K-loop and we give explicit examples of K-loops obtained with this method. Further generalizations of this technique are proposed as well.     

K-loops from classical groups over ordered fields

K-loops will be constructed as transversals in classical groups over an ordered field. Manywell-known examples are subsumed in the present approach. Special attention is payed to the question which of these K-loops have fixed point free left inner mappings. Some new such examples are given. Isomorphisms between some of the K-loops are established as well.Dedicated to Professor Dr. Dr. h. c. Helmut Karzel on the occasion of his seventieth birthday     

On Algebraic Structures Related to Beltrami–Klein Model of Hyperbolic Geometry

A new approach to the algebraic structures related to hyperbolic geometry comes from Einstein’s special theory of relativity in 1988 (cf. Ungar, in Found Phys Lett 1:57–89, 1988). Ungar employed the binary operation of Einsteins velocity addition to introduce into hyperbolic geometry the concepts of vectors, angles and trigonometry in full analogy with Euclidean geometry (cf. Ungar, in Math Appl 49:187–221, 2005). Another approach is from Karzel for algebraization of absolute planes in the sense of Karzel et al. (Einführung in die Geometrie, 1973). In this paper we are going to develop a formulary for the Beltrami–Klein model of hyperbolic plane inside the unit circle ${\mathbb D}$ of the complex numbers ${\mathbb C}$ with geometric approach of Karzel.     

Minimum rank problems

A graph describes the zero-nonzero pattern of a family of matrices, with the type of graph (undirected or directed, simple or allowing loops) determining what type of matrices (symmetric or not necessarily symmetric, diagonal entries free or constrained) are described by the graph. The minimum rank problem of the graph is to determine the minimum among the ranks of the matrices in this family; the determination of maximum nullity is equivalent. This problem has been solved for simple trees [P.M. Nylen, Minimum-rank matrices with prescribed graph, Linear Algebra Appl. 248 (1996) 303-316, C.R. Johnson, A. Leal Duarte, The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree, Linear and Multilinear Algebra 46 (1999) 139-144], trees allowing loops [L.M. DeAlba, T.L. Hardy, I.R. Hentzel, L. Hogben, A. Wangsness. Minimum rank and maximum eigenvalue multiplicity of symmetric tree sign patterns, Linear Algebra Appl. 418 (2006) 389-415], and directed trees allowing loops [F. Barioli, S. Fallat, D. Hershkowitz, H.T. Hall, L. Hogben, H. van der Holst, B. Shader, On the minimum rank of not necessarily symmetric matrices: a preliminary study, Electron. J. Linear Algebra 18 (2000) 126-145]. We survey these results from a unified perspective and solve the minimum rank problem for simple directed trees.     

抛物问题非协调元多重网格法

抛物问题非协调元多重网格法周叔子,文承标（湖南大学应用数学系）ＮＯＮＣＯＮＦＯＲＭＩＮＧＥＬＥＭＥＮＴＭＵＬＴＩＧＲＩＤＭＥＴＨＯＤＦＯＲＭＲＡＢＯＬＩＣＥＱＵＡＴＩＯＮＳ￥ＺｈｏｕＳｈｕ－ｚｉ;ＷｅｎＣｈｅｎｇ－ｂｉａｏ（ＨｕｎａｎＵｎｉｖｅｒｓｉ...     

Relations among certain classes of incidence loops, codes and chain structures

In a loop (L, +) for every element a ∈ L* := L\{0} the set Z(a) ? {x ∈ L ¦ a+x = x+a} is defined. We focus our attention on those classes of abstract loops for which Z(a) is a subgroup of (L, +) for all a ∈ L* and for which the set is a 0-bundle (for the definition see §1). The first step in this direction is given by the case in which Z(a) ? Z₂ for all a ∈ L*. This condition implies that (L, +) is of exponent 2, i.e. a + a = 0 ?a ∈ L. In the present note we connect left (right) loops of exponent 2 to certain classes of codes (via chain structures) and relate loop automorphisms to a class of code automorphisms. We also exhibit a procedure to obtain loops in which Z(a) ? Z₂ by means of a class of codes.     

When is an L(B, m, n, r, s, t, z, w) loop SRAR?

In [5, 6], the second author and D. A. ROBINSON initiated a study of non-Moufang Bol loops with the property that over a field, necessarily of characteristic 2, their loop rings satisfy the right, but not the left, Bol identity. They called such loops SRAR and showed that the family of SRAR loops includes those Bol loops which have a unique non-identity commutator/associator. In [4, 2], the current authors presented a construction for a new class of Bol loops denoted L(B,m,n,r,s,t,z,w) with initial data a given (possibly associative) Bol loop B, elements, r, s, t, z and w in the centre of B, and integers m and n.     

Loops mit distributiven und geometrischen Unterloopverbänden

The following facts are shown: A loop with a finite distributive subloop lattice is finite, monogenic and all its subloops are monogenic. Therefore, power-associative loops having finite distributive subloop lattices are cyclic groups. A loop G with its subloop lattice L(G) being a finite n-dimensional projective geometry is generated by at most n+1 elements. For all n IN, n4, there are power-associative loops whose subloop lattices are projective lines with n points. Furthermore, for a given projective planeP _n (desarguesian or non-desarguesian) of order n there exists a power-associative loopG with L(G) -P _n.     

Defect and Area in Beltrami–Klein Model of Hyperbolic Geometry

Ungar (Beyond the Einstein addition law and its gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrouector Spaces, 2001; Comput Math Appl 49:187–221, 2005; Comput Math Appl 53, 2007) introduced into hyperbolic geometry the concept of defect based on relativity addition of A. Einstein. Another approach is from Karzel (Resultate Math. 47:305–326, 2005) for the relation between the K-loop and the defect of an absolute plane in the sense (Karzel in Einführung in die Geometrie, 1973). Our main concern is to introduce a systematical exact definition for defect and area in the Beltrami–Klein model of hyperbolic geometry. Combining the ideas and methods of Karzel and Ungar give an elegant concept for defect and area in this model. In particular we give a rigorous and elementary proof for the defect formula stated (Ungar in Comput Math Appl 53, 2007). Furthermore, we give a formulary for area of circle in the Beltrami–Klein model of hyperbolic geometry.     

On the Structure of Alternating Direction Implicit (A.D.I.) and Locally One Dimensional (L.O.D.) Difference Methods

The general structure of A.D.I. and L.O.D. difference schemesis considered with regard to their construction for time dependentproblems in two and three space dimensions. By considering approximationsto exp {k(L+M)} where L and M are differential operators inthe space variables and k is the time step, we show how severalknown schemes can be viewed as having come from this type ofapproximation. In addition several new schemes based on thistype of approximation are suggested. The arguments used areentirely informal and no attempt is made to prove the stabilityor convergence of the various schemes. Our aim is merely topoint out a possible structure for the generation of A.D.I.and L.O.D. difference schemes.     

Global dynamics of the periodic logistic system with periodic impulsive perturbations

This paper studies the global behaviors of the periodic logistic system with periodic impulsive perturbations. The results of D.D. Bainov and P.S. Simeonov (1993) are extended and dynamics different from the corresponding continuous system are found. It is shown that the system may have a unique positive periodic solution which is globally asymptotically stable, or go extinct when the two periods are rational dependent. When they are rational independent, the system has no periodic solutions, however, still has a global attractor or go extinct under some conditions.     

一类凸规划的多项式预估校正内点法

1、引言 1990年由Mehrotra对线性规划问题提出了一个称为预估校正的方法,并在1992年给出了其数值算法．1993年Mizuno,Todd和Y．Ye．给出了改进的预估校正内点法,使得一个预估步后只跟一个校正步．1994年F．A．Potra给出了不可行预估校正内点法,使得可以从一个不可行的初始点开始算法的迭代,并证明了其为二次收敛．     

Characteristic for reflexive relations

Poincaré characteristic for reflexive relations (oriented graphs) is defined in terms of homology and is not invariant under transitive closure. Formulas for the Poincaré characteristic of products, joins, and bounded products are given. Euler's definition of characteristic extends to certain filtrations of reflexive relations which exist iff there are no oriented loops. Euler characteristic is independent of filtration, agrees with Poincaré characteristic, and is unique. One-sided Möbius characteristic is defined as the sum of all values of a one-sided inverse of the zeta function. Such one-sided inverses exist iff there are no local oriented loops (although there may be global oriented loops). One-sided Möbius characteristic need not be Poincaré characteristic, but it is when a one-sided local transitivity condition is satisfied. A two-sided local transitivity condition insures the existence of the Möbius function but Möbius inversion fails for non-posets.     

Preconditioners for the pressure-correction method applied to the unsteady stokes problem

In this paper, the unsteady Stokes problem is considered and also the pressure-correction method for the problem is described. At a fixed time level, we reduce the problem to two summetric positive definite problems which depend on a time step parameter. Linear systems that arise from the problems are large, sparse, symmetric, positive definite and ill-conditioned as the time step tends to zero. Preconditioned problems based on an additive Schwarz method for solving the symmetric positive definite problems are derived and preconditioners are defined implicitly. It will be shown that the rate of convergence is independent of the mesh parameters as well as the time step size     

A Schwarz domain decomposition method with gradient projection for optimal control governed by elliptic partial differential equations

A domain decomposition method (DDM) is presented to solve the distributed optimal control problem. The optimal control problem essentially couples an elliptic partial differential equation with respect to the state variable and a variational inequality with respect to the constrained control variable. The proposed algorithm, called SA-GP algorithm, consists of two iterative stages. In the inner loops, the Schwarz alternating method (SA) is applied to solve the state and co-state variables, and in the outer loops the gradient projection algorithm (GP) is adopted to obtain the control variable. Convergence of iterations depends on both the outer and the inner loops, which are coupled and affected by each other. In the classical iteration algorithms, a given tolerance would be reached after sufficiently many iteration steps, but more iterations lead to huge computational cost. For solving constrained optimal control problems, most of the computational cost is used to solve PDEs. In this paper, a proposed iterative number independent of the tolerance is used in the inner loops so as to save a lot of computational cost. The convergence rate of L²-error of control variable is derived. Also the analysis on how to choose the proposed iteration number in the inner loops is given. Some numerical experiments are performed to verify the theoretical results.     

A geometric interpretation of Ungar's addition and of gyration in the hyperbolic plane

We present a geometric interpretation of the operation a⊕b and the gyration on the unit-disc as defined by A.A. Ungar. Using this geometric interpretation we show that the two known generalizations to the n-dimensional unit ball are identical. The interpretation in the plane leads us to the notion of outer-median of a triangle and we discuss some possible properties of this median.     

有向图的L(j,k)数的上界

给定正整数j≥k,有向图D的一个L(j,k)-标号是指从V(D)到非负整数集的一个函数f,使得当x在D中邻接到y时|f(x)-f(y)|≥j,当x在D中到y距离为二时|f(x)-f(y)|≥k.f的像元素称为标号.L(j,k)一标号问题就是确定(?)j,k-数(?)j,k(D),这个参数等于(?) max{f(x)|x∈V(D)},这里f取遍D的所有L(j,k)-标号.本文根据有向图的有向着色数及最长有向路的长度来研究(?)j,k-数,证明了:(1)对任何有向着色数为(?)(D)的有向图D,(?)j,k(D)≤((?)(D)-1)j;(2)对任何最长有向路的长度为l的有向图D,如果不含有向圈或者D中最长有向圈长度为l 1,则(?)j,k(D)≤lj.并且这两个界都是可达的.最后我们对l=3的有向图给出了3j-L(j,k)-labelling的一个有效算法.     

A generalization of the rotation matrix and related results

A new generalization of the rotation group involving a skew circulant matrix is given. Using the exponential map, a unified treatment is given to this generalization and to one due to Ungar. The special functions associated with the corresponding Lie groups are the trigonometric and hyperbolic functions of order n. Infinitesimal generators and invariants under the corresponding transformations are also obtained. A general theorem on linear transformations involving circulant and skew circulant matrices is also given.