A class of locally $\varphi $-symmetric contact metric spaces

We prove that a non-Sasakian contact metric manifold such that its characteristic vector field belongs to the (k,m)(k,\mu )-nullity distribution, is locally homogeneous. Further, these spaces are all locally j\varphi -symmetric in the strong sense, i.e., the reflections with respect to characteristic curves are local isometries.     

An Edge-Isoperimetric Problem for Powers of the Petersen Graph

In this paper we introduce a new order on the set of n-dimensional tuples and prove that this order preserves nestedness in the edge isoperimetric problem for the graph Pⁿ, defined as the nth cartesian power of the well-known Petersen graph. The cutwidth and wirelength of Pⁿ are also derived. These results are then generalized for the cartesian product of Pⁿ and the m-dimensional binary hypercube.     

Combinatorics of Geometrically Distributed Random Variables: Inversions and a Parameter of Knuth

. For words of length n, generated by independent geometric random variables, we consider the mean and variance of the number of inversions and of a parameter of Knuth from permutation in situ. In this way, q-analogues for these parameters from the usual permutation model are obtained.     

Multiplicity-Free Products of Schur Functions

. We classify all multiplicity-free products of Schur functions and all multiplicity-free products of characters of SL(n, C).     

Monoids for Which Condition (P) Acts are Projective

A characterisation of monoids for which all right S-acts satisfying conditions (P) are projective is given. We also give a new characterisation of those monoids for which all cyclic right S-acts satisfying condition (P) are projective, similar in nature to recent work by Kilp [6]. In addition we give a sufficient condition for all right S-acts that satisfy condition (P) to be strongly flat and show that the indecomposable acts that satisfy condition (P) are the locally cyclic acts.     

Diffusion in Glassy Polymers: Sorption of a Finite Amount of Solvent

A family of 1-D moving boundary models describing the diffusion of a finite amount of a penetrant in a glassy polymer is studied. Local existence of a unique classical solution is obtained for a generic quasilinear model. Specific data are then chosen which can be found in the literature (cf. [6]) and global existence of the classical solution and its convergence to an equilibrium solution are proven. Finally a rigorous proof is provided for a formal perturbation argument proposed in [6] and used therein to estimate the rate of convergence of the solution towards the equilibrium.     

Multiple positive solutions for a nonlinear Schr?dinger equation

We are interested in positive entire solutions of the nonlinear Schrödinger equation -Du+(la(x)+1)u = u^p-\Delta u+(\lambda a(x)+1)u = u^p where a ≤ 0 has a potential well and p > 1 is subcritical. Using variational methods we prove the existence of multiple positive solutions which localize near the potential well int(a^-1(0)) for l\lambda large.     

The Minding formula and its applications

We apply the Minding Formula for geodesic curvature and the Gauss-Bonnet Formula to calculate the total Gaussian curvature of certain 2-dimensional open complete branched Riemannian manifolds, the M\cal M surfaces. We prove that for an M\cal M surface, the total curvature depends only on its Euler characteristic and the local behaviour of its metric at ends and branch points. Then we check that many important surfaces, such as complete minimal surfaces in \Bbb Rⁿ{\Bbb R}^n with finite total curvature, complete constant mean curvature surfaces in hyperbolic 3-space H³ (–1) with finite total curvature, are actually branch point free M\cal M surfaces. Therefore as corollaries we give simple proofs of some classical theorems such as the Chern-Osserman theorem for complete minimal surfaces in \Bbb Rⁿ{\Bbb R}^n with finite total curvature. For the reader's convenience, we also derive the Minding Formula.     

Covering degrees are determined by graph manifolds involved

W.Thurston raised the following question in 1976: Suppose that a compact 3-manifold M is not covered by (surface) ×S¹ \times S^1 or a torus bundle over S¹ S^1 . If M₁ M_1 and M₂ M_2 are two homeomorphic finite covering spaces of M, do they have the same covering degree?¶For so called geometric 3-manifolds (a famous conjecture is that all compact orientable 3-manifolds are geometric), it is known that the answer is affirmative if M is not a non-trivial graph manifold.¶In this paper, we prove that the answer for non-trivial graph manifolds is also affirmative. Hence the answer for the Thurston's question is complete for geometric 3-manifolds. Some properties of 3-manifold groups are also derived.     

Optimal estimates for lower and upper bounds of approximation errors in the p-version of the finite element method in two dimensions

Dedicated to Prof. O. Widlund on the occasion of his 60th birthday Correspondence to: B. Guo Summary. This paper will analyze the lower and upper error bounds of the finite element solution of the p-version for linear elliptic problems in polygonal domains. The optimal rate of convergence is rigorously proved based on the sharp estimates of lower and upper bounds of the approximation error.     

Lagrangian Matroids and Cohomology

We prove that \bigtriangleup \bigtriangleup -matroids associated with maps on compact closed surfaces are representable, with the space of representation provided by cohomology of the surface with punctured points.     

The semigroup of singular endomorphisms

Let V be a vector space of dimension n over a field K. Here we denote by Sn the set of all singular endomorphisms of V. Erdos [5], Dawlings [4] and Thomas J. Laffey [6] have shown that Sn is an idempotent generated regular semigroup. In this paper we apply the theory of inductive groupoids, in particular the construction of the idempotent generated regular semigroup given in §6 of [8] to detemine some combinatorial properties of the semigroup Sn.     

Centralizers and Central Idempotents of Semigroup Rings

. Let A be a nonempty subset of an associative ring R . Call the subring CR(A)={r] R\mid ra=ar \quadfor all\quad a] A} of R the centralizer of A in R . Let S be a semigroup. Then the subsemigroup S'= {s] S\mid sa=sb \quador\quad as=bs \quadimplies\quad a=b \quadfor all a,b] S} of S is called the C -subsemigroup. In this paper, the centralizer CR[S](R[M]) for the semigroup ring R[S] will be described, where M is any nonempty subset of S' . An non-zero idempotent e is called the central idempotent of R[S] if e lies in the center of R[S] . Assume that S\backslash S' is a commutative ideal of S and Annl(R)=0 . Then we show that the supporting subsemigroup of any central idempotent of R[S] must be finite.     

Guest Editor's Foreword

No .     

Subtree Transfer Operations and Their Induced Metrics on Evolutionary Trees

. Leaf-labelled trees are widely used to describe evolutionary relationships, particularly in biology. In this setting, extant species label the leaves of the tree, while the internal vertices correspond to ancestral species. Various techniques exist for reconstructing these evolutionary trees from data, and an important problem is to determine how "far apart" two such reconstructed trees are from each other, or indeed from the true historical tree. To investigate this question requires tree metrics, and these can be induced by operations that rearrange trees locally. Here we investigate three such operations: nearest neighbour interchange (NNI), subtree prune and regraft (SPR), and tree bisection and reconnection (TBR). The SPR operation is of particular interest as it can be used to model biological processes such as horizontal gene transfer and recombination. We count the number of unrooted binary trees one SPR from any given unrooted binary tree, as well as providing new upper and lower bounds for the diameter of the adjacency graph of trees under SPR and TBR. We also show that the problem of computing the minimum number of TBR operations required to transform one tree to another can be reduced to a problem whose size is a function just of the distance between the trees (and not of the size of the two trees), and thereby establish that the problem is fixed-parameter tractable.     

A Remark on non-separable solutions to the Schroeder-Bernstein Problem for Banach spaces

We show that the geometric structure of Banach spaces which are solutions to the Schroeder-Bernstein Problem is very complex. More precisely, we prove that there exists a non-separable solution E to this problem such that

1. E is isomorphic to each one of its finite codimensional subspaces.
2. E has no complemented Hereditarily Indecomposable subspace.
3. E has no complemented subspace isomorphic to its square.
4. E has no non-trivial divisor.

     

Remarks on the Article “On the Stability of Wavelet and Gabor Frames (Riesz Bases)” by J. Zhang

This article consists of some remarks on stability results forGabor and wavelet frames fromZhang Jing (J. Fourier Anal. Appl.5, (1999), 105–125),as well as a correction of a proof of a multivariate version of theKadec's 1/4-theorem in the same article.     

On Definability of Hypergraphs by Their Semigroups of Homomorphisms

. No Abstract.     

Book Review

. No Abstract.     

The Classification of Certain Four-Weight Spin Models

In this paper we show that if one of the matrices {Wi, 1 h i h 4} of a four-weight spin model (X, W1, W2, W3, W4; D) is equivalent to the matrix of a Potts model or a cyclic model as type II matrix and |X| S 5, then the spin model is gauge equivalent to a Potts model or a cyclic model up to simultaneous permutations on rows and columns. Using this fact and Nomura's result [12] we show that every four-weight spin model of size |X| = 5 is gauge equivalent to either a Potts model or a cyclic model up to simultaneous permutations on rows and columns.