Absolute summability of infinite series

It is shown in [4] that if a normal matrix,A satisfies some conditions then |C,1| _k summability implies |A| _k summability wherek≥1. In the present paper, we consider the converse implication.     

On automorphisms of finite Abelian <Emphasis Type="Italic">p</Emphasis>-groups

Let A be a finitely generated abelian group. We describe the automorphism group Aut(A) using the rank of A and its torsion part p-part A _p . For a finite abelian p-group A of type (k ₁, ..., k _n ), simple necessary and sufficient conditions for an n × n-matrix over integers to be associated with an automorphism of A are presented. Then, the automorphism group Aut(A) for a finite p-group A of type (k ₁, k ₂) is analyzed. This work has begin during the visit of the second author to the Faculty of Mathematics and Computer Science, Nicolaus Copernicus University during the period July 31–August 13, 2005. This visit was supported by the Nicolaus Copernicus University and a grant from Cnpq.     

Balancing the n-Cube: A Census of Colorings

Weights of 1 or 0 are assigned to the vertices of the n-cube in n-dimensional Euclidean space. Such an n-cube is called balanced if its center of mass coincides precisely with its geometric center. The seldom-used n-variable form of Pólya's enumeration theorem is applied to express the number N _{n, 2k} of balanced configurations with 2k vertices of weight 1 in terms of certain partitions of 2k. A system of linear equations of Vandermonde type is obtained, from which recurrence relations are derived which are computationally efficient for fixed k. It is shown how the numbers N _{n, 2k} depend on the numbers A _{n, 2k} of specially restricted configurations. A table of values of N _{n, 2k} and A _{n, 2k} is provided for n = 3, 4, 5, and 6. The case in which arbitrary, nonnegative, integral weights are allowed is also treated. Finally, alternative derivations of the main results are developed from the perspective of superposition.     

Algebras almost commuting with Clifford algebras in a II∞ factor

We are concerned here with certain Banach algebras of operators contained within a fixed II factor N. These algebras may be thought of as noncommutative classifying spaces for the functor Ext _* ^N The basic objects of study are the algebras A _kN (for n=1, 2,...). Here, we are given an essentially unique representation of the complex Clifford algebra C _k N and the elements of A _k are those operators in N which exactly commute with the first k–1 generators of C _k and also commute with the kth generator modulo a symmetric ideal N. Up to isomorphism, these algebras are periodic with period 2.We determine completely the homotopy types of A ₁ ^–1 and A ₂ ^–1 Here, A ₁ ^–1 is homotopy equivalent to the space of (Breuer) Fredholm operators in N, while A ₂ ^–1 is homotopy equivalent to the group K _N ^–1 ={x N^–1¦ x=1+k, k K_N}. We use these results to compute the K-theory of A ₁ and A ₂.For a fixed C ^*-algebra A, we define abelian groups G _k,N(A) of equivalence classes of homomorphisms AA _k. Letting N = M (H) for a II₁ factor M we define similar abelian groups G _k(A, M) where we replace N by L(E) for countably generated right Hilbert M-modules E with (left) actions C _k L(E). Using ideas of Skandalis, we show that G _k,NG_k(A, M) so that the G _k,N are stable half-exact homotopy functors because the G _k(·, M) are such.In general, we show that G _k(A, M)KK ^k(A, M) and so our theory fits neatly into Kasparov KK-theory. We investigate many interesting examples from our point of view.     

The Yoneda Algebra of a Graded Ore Extension

Let A be a connected-graded algebra with trivial module 𝕜, and let B be a graded Ore extension of A. We relate the structure of the Yoneda algebra E(A): = Ext _A (𝕜, 𝕜) to E(B). Cassidy and Shelton have shown that when A satisfies their 𝒦₂ property, B will also be 𝒦₂. We prove the converse of this result.     

Characterization of varietal covering and stem groups

Let A _m be the variety of abelian groups of exponent m, where mis a square free positive integer. It is shown that every group G has an A _m -covering group, which generalizes the work of I. Schur (1904) and M.R. Jones (1973). Also, similar to the work of Yamazaki (1964) in the variety of abelian groups, if 1→ A → H → G → 1 is an A _m -stem extension, then we show that H is a homomorphic image of an A _m -covering group of G. Finally,in this variety some results of Hall's type will be proved.     

Dimension Theory and Nonstable K1 of Quadratic Modules

Employing Bak's dimension theory, we investigate the nonstable quadratic K-group K _1,2n (A, ) = G _2n (A, )/E _2n (A, ), n 3, where G _2n (A, ) denotes the general quadratic group of rank n over a form ring (A, ) and E _2n (A, ) its elementary subgroup. Considering form rings as a category with dimension in the sense of Bak, we obtain a dimension filtration G _2n (A, ) G _2n ⁰(A, ) ; G _2n ¹(A, ) ... E _2n (A, ) of the general quadratic group G _2n (A, ) such that G _2n (A, )/G _2n ⁰(A, ) is Abelian, G _2n ⁰(A, ) G _2n ¹(A, ) ... is a descending central series, and G _2n ^d(A)(A, ) = E _2n (A, ) whenever d(A) = (Bass–Serre dimension of A) is finite. In particular K _1,2n (A, ) is solvable when d(A) < .     

Reliability Approximation for Markov Chain Imbeddable Systems

In the present article, a simple method is developed for approximating the reliability of Markov chain imbeddable systems. The approximating formula reduces the problem to the reliability assessment of smaller systems with structure similar to the original systems. Two specific reliability structures which have attracted considerable research interest recently (r-within-consecutive-k-out-of-n system and two dimensional r-within-k ₁ × k ₂-out-of-n ₁ × n ₂ system) are studied by the new approach and numerical calculations are carried out, which reveal the high quality of our approximations. Several possible extensions and generalizations are also presented in brief.     

The postage stamp problem and arithmetic in base <Emphasis Type="Italic">r</Emphasis>

Let h, k be fixed positive integers, and let A be any set of positive integers. Let hA ≔ {a ₁ + a ₂ + ... + a _r : a _i ∈ A, r ⩽ h} denote the set of all integers representable as a sum of no more than h elements of A, and let n(h, A) denote the largest integer n such that {1, 2,...,n} ⊆ hA. Let n(h, k) := : n(h, A), where the maximum is taken over all sets A with k elements. We determine n(h, A) when the elements of A are in geometric progression. In particular, this results in the evaluation of n(h, 2) and yields surprisingly sharp lower bounds for n(h, k), particularly for k = 3.     

Kp,q-factorization of complete bipartite graphs

Let K _m,nbe a complete bipartite graph with two partite sets having m and n vertices, respectively. A K _p,q-factorization of K _m,n is a set of edge-disjoint K _p,q-factors of K _m,n which partition the set of edges of K _m,n. When p = 1 and q is a prime number, Wang, in his paper “On K _1,k -factorizations of a complete bipartite graph” (Discrete Math, 1994, 126: 359—364), investigated the K _1,q -factorization of K _m,nand gave a sufficient condition for such a factorization to exist. In the paper “K _1,k -factorizations of complete bipartite graphs” (Discrete Math, 2002, 259: 301—306), Du and Wang extended Wang’s result to the case that q is any positive integer. In this paper, we give a sufficient condition for K _m,n to have a K _p,q-factorization. As a special case, it is shown that the Martin’s BAC conjecture is true when p : q = k : (k+ 1) for any positive integer k.     

The Flag-Transitive C 3-Geometries of Finite Order

It is shown that a flag-transitive C ₃-geometry of finite order (x, y) with x2 is either a finite building of type C ₃ (and hence the classical polar space for a 6-dimensional symplectic space, a 6-dimensional orthogonal space of plus type, a 6- or 7-dimensional hermitian space, a 7-dimensional orthogonal space, or an 8-dimensional orthogonal space of minus type) or the sporadic A ₇-geometry with 7 points.     

Rectifications of A∞-Algebras

In this article, in the setting of connected DG-modules, we prove that, for any A _∞-algebra (M, {m _i } _i≥1), there is a chain contraction from a DG-algebra A _M onto the DG-module M such that the A _∞-algebra structure induced by perturbation theory on M is precisely the original one. In fact, the mentioned DG-algebra can be considered a rectification of the A _∞-algebra in the sense of (Boardman and Vogt, 1973 Boardman , J. M. , Vogt , R. M. ( 1973 ). Homotopy Invariant Algebraic Structures on Topological Spaces . Lecture Notes in Mathematics . Springer-Verlag , pp. 347 . [Google Scholar]). Appropiate dual results are given for A _∞-coalgebras.     

On the Spectral Analysis of Direct Sums of Riemann-Liouville Operators in Sobolev Spaces of Vector Functions

Let J ^α _k be a real power of the integration operator J _k defined on the Sobolev space W ^k _p [0, 1]. We investigate the spectral properties of the operator defined on . Namely, we describe the commutant {A _k }′, the double commutant and the algebra Alg A _k . Moreover, we describe the lattices Lat A _k and HypLat A _k of invariant and hyperinvariant subspaces of A _k , respectively. We also calculate the spectral multiplicity  of A _k and describe the set Cyc A _k of its cyclic subspaces. In passing, we present a simple counterexample for the implication

Map Preserving Local Properties of a Digital Image

In order to study digital topological properties of a k-surface in Z ⁿ , we generalize the topological number in Bertrand (Pattern Recogn. Lett. 15:1003–1011, 1994). Furthermore, we show that a local (k ₀,k ₁)-isomorphism preserves some digital-topological properties, such as a generalized topological number and a simple k ₀-point, and prove that a local (k ₀,k ₁)-isomorphism takes a simple k ₀-surface in into a simple k ₁-surface in .      

On nonautonomous linear systems of differential and difference equations with -symmetric coefficient matrices

Let denote the set of continuous n×n matrices on an interval . We say that is a nontrivial k-involution if where ζ=e^-2πi/k, d₀+d₁++d_k-1=n, and with . We say that is R-symmetric if R(t)A(t)R^-1(t)=A(t), , and we show that if A is R-symmetric then solving x^′=A(t)x or x^′=A(t)x+f(t) reduces to solving k independent d_ℓ×d_ℓ systems, 0ℓk-1. We consider the asymptotic behavior of the solutions in the case where . Finally, we sketch analogous results for linear systems of difference equations.     

<Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-nearest neighbor estimators of entropy

For estimating the entropy of an absolutely continuous multivariate distribution, we propose nonparametric estimators based on the Euclidean distances between the n sample points and their k _n -nearest neighbors, where {k _n : n = 1, 2, …} is a sequence of positive integers varying with n. The proposed estimators are shown to be asymptotically unbiased and consistent.      

F_2+vF_2上的循环码

本文研究了环F_2+vF_2上的循环码.利用标准形生成元集刻画了环F_2+vF_2上的循环码的代数结构,证明了环F_2+vF_2上的每一个非零的循环码均有唯一的标准形生成元集,进而得到了每一个循环码均是由一个多项式生成的.     

New applications of Picard?s successive approximations

The iterative method of successive approximations, originally introduced by Émile Picard in 1890, is a basic tool for proving the existence of solutions of initial value problems regarding ordinary first order differential equations. In the present paper, it is shown that this method can be modified to get estimates for the growth of solutions of linear differential equations of the typef(k)+Ak−1(z)f(k−1)+?+A₁(z)f^′+A₀(z)f=0 with analytic coefficients. A short comparison to the growth results in the literature, obtained by means of different methods, is also given. It turns out that many known results can be proved by applying Picard?s successive approximations in an effective way. Self-contained considerations are carried out in the complex plane and in the unit disc, and some remarks about solutions of real linear differential equations are made.