On decomposable weakly conformally symmetric manifolds

The object of the present paper is to study decomposable weakly conformally symmetric Riemannian manifolds with several non-trivial examples.     

On quasi-conformally flat weakly Ricci symmetric manifolds

The object of the present paper is to study quasi-conformally flat weakly Ricci symmetric manifolds.      

On the classification of weakly symmetric Finsler spaces

In this paper, we give the classification of some special types of weakly symmetric Finsler spaces. We first present a general principle to classify weakly symmetric Finsler spaces and also give a method to figure out the Berwald spaces among the class of weakly symmetric Finsler spaces. Then we give an explicit classification of weakly symmetric Finsler spaces with reductive isometric groups as well as the left invariant weakly symmetric Finsler metrics on nilpotent Lie groups of the Heisenberg type. As an application, we obtain a large number of high-dimensional examples of reversible Finsler spaces which are non-Berwaldian and with vanishing S-curvature, a kind of spaces which are sought after in an open problem of Z. Shen.     

Generalized weakly symmetric rings

A ring R is defined to be GWS   if abc=0$abc=0$ implies bac⊆N(R)$bac\subseteq N(R)$ for a,b,c∈R$a,b,c\in R$, where N(R)$N(R)$ stands for the set of nilpotent elements of R. Since reduced rings and central symmetric rings are GWS, we study sufficient conditions for GWS rings to be reduced and central symmetric. We prove that a ring R is GWS   if and only if the n×n$n\times n$ upper triangular matrices ring _Un(R,R)$U_n(R,R)$ is GWS for any positive integer n. It is proven that GWS rings are directly finite and left min-abel. For a GWS ring R, R is a strongly regular ring if and only if R is a von Neumann regular ring if and only if R is a left SF   ring and J(R)=0$J(R)=0$; R is an exchange ring if and only if R is a clean ring. Finally, we show that GWS exchange rings have stable range 1 and a GWS semiperiodic ring R   with N(R)≠J(R)$N(R)\ne J(R)$ is commutative.     

Pseudo-Riemannian weakly symmetric manifolds

There is a well-developed theory of weakly symmetric Riemannian manifolds. Here it is shown that several results in the Riemannian case are also valid for weakly symmetric pseudo-Riemannian manifolds, but some require additional hypotheses. The topics discussed are homogeneity, geodesic completeness, the geodesic orbit property, weak symmetries, and the structure of the nilradical of the isometry group. Also, we give a number of examples of weakly symmetric pseudo-Riemannian manifolds, some mirroring the Riemannian case and some indicating the problems in extending Riemannian results to weakly symmetric pseudo-Riemannian spaces.     

On tame weakly symmetric algebras having only periodic modules

We classify (up to Morita equivalence) all tame weakly symmetric finite dimensional algebras over an algebraically closed field having simply connected Galois coverings, nonsingular Cartan matrices and the stable Auslander-Reiten quivers consisting only of tubes. In particular, we prove that these algebras have at most four simple modules.Received: 25 February 2002     

On weakly symmetric graphs of order twice a prime

A graph is weakly symmetric if its automorphism group is both vertex-transitive and edge-transitive. In 1971, Chao characterized all weakly symmetric graphs of prime order and showed that such graphs are also transitive on directed edges. In this paper we determine all weakly symmetric graphs of order twice a prime and show that these graphs too are directed-edge transitive.     

Semisimple weakly symmetric pseudo-Riemannian manifolds

We develop the classification of weakly symmetric pseudo-Riemannian manifolds G / H where G is a semisimple Lie group and H is a reductive subgroup. We derive the classification from the cases where G is compact, and then we discuss the (isotropy) representation of H on the tangent space of G / H and the signature of the invariant pseudo-Riemannian metric. As a consequence we obtain the classification of semisimple weakly symmetric manifolds of Lorentz signature \((n-1,1)\) and trans-Lorentzian signature \((n-2,2)\).     

On symmetric stochastic games of resource extraction with weakly continuous transitions

We study stochastic games of resource extraction, in which the players have identical preferences. The transition probability is either non-atomic or a convex combination of transition probabilities depending on the investment with coefficients also dependent on the investment. Our approach covers the unbounded utility case, which was not examined in this class of games beforehand. We prove the existence of a stationary Markov perfect equilibrium in a non-randomised class of strategies.     

On symmetric and nonsymmetric blowup for a weakly quasilinear heat equation

We construct blow-up patterns for the quasilinear heat equation (QHE) $$u_t = \nabla \cdot (k(u)\nabla u) + Q(u)$$ in Ω×(0,T), Ω being a bounded open convex set in ? ^N with smooth boundary, with zero Dirichet boundary condition and nonnegative initial data. The nonlinear coefficients of the equation are assumed to be smooth and positive functions and moreoverk(u) andQ(u)/u ^p with a fixedp>1 are of slow variation asu→∞, so that (QHE) can be treated as a quasilinear perturbation of the well-known semilinear heat equation (SHE) $$u_t = \nabla u) + u^p .$$ We prove that the blow-up patterns for the (QHE) and the (SHE) coincide in a structural sense under the extra assumption $$\smallint ^\infty k(f(e^s ))ds = \infty ,$$ wheref(v) is a monotone solution of the ODEf′(v)=Q(f(v))/v ^p defined for allv?1. If the integral is finite then the (QHE) is shown to admit an infinite number of different blow-up patterns.     

New examples of weakly symmetric spaces

We provide some new examples of weakly symmetric spaces inside the class of complete, simply connected Riemannian manifolds equipped with a complete unit Killing vector field such that the reflections with respect to its flow lines are global isometries.Supported by the Consejería de Educación del Gobierno de Canarias     

Highway games on weakly cyclic graphs

A highway problem is determined by a connected graph which provides all potential entry and exit vertices and all possible edges that can be constructed between vertices, a cost function on the edges of the graph and a set of players, each in need of constructing a connection between a specific entry and exit vertex. Mosquera (2007) introduce highway problems and the corresponding cooperative cost games called highway games to address the problem of fair allocation of the construction costs in case the underlying graph is a tree. In this paper, we study the concavity and the balancedness of highway games on weakly cyclic graphs. A graph G is called highway-game concave if for each highway problem in which G is the underlying graph the corresponding highway game is concave. We show that a graph is highway-game concave if and only if it is weakly triangular. Moreover, we prove that highway games on weakly cyclic graphs are balanced.     

Higher order weakly over-penalized symmetric interior penalty methods

In this paper we study higher order weakly over-penalized symmetric interior penalty methods for second-order elliptic boundary value problems in two dimensions. We derive h$h$–p$p$ error estimates in both the energy norm and the _L2$L_2$ norm and present numerical results that corroborate the theoretical results.     

The representation dimension of domestic weakly symmetric algebras

Auslander’s representation dimension measures how far a finite dimensional algebra is away from being of finite representation type. In [1], M. Auslander proved that a finite dimensional algebra A is of finite representation type if and only if the representation dimension of A is at most 2. Recently, R. Rouquier proved that there are finite dimensional algebras of an arbitrarily large finite representation dimension. One of the exciting open problems is to show that all finite dimensional algebras of tame representation type have representation dimension at most 3. We prove that this is true for all domestic weakly symmetric algebras over algebraically closed fields having simply connected Galois coverings.