A Note on Ultrametric Matrices

It is proved in this paper that special generalized ultrametric and special matrices are, in a sense, extremal matrices in the boundary of the set of generalized ultrametric and matrices, respectively. Moreover, we present a new class of inverse M-matrices which generalizes the class of matrices.     

Generalized Cardinal Properties of Lattices and Lattice Ordered Groups

We denote by K the class of all cardinals; put K = K {}. Let be a class of algebraic systems. A generalized cardinal property f on is defined to be a rule which assings to each A an element fA of K such that, whenever A₁, A₂ and A₁ A₂, then fA ₁ = fA ₂. In this paper we are interested mainly in the cases when (i) is the class of all bounded lattices B having more than one element, or (ii) is a class of lattice ordered groups.     

A Note on the Oscillation of Second Order Differential Equations

We give a sufficient condition for the oscillation of linear homogeneous second order differential equation y + p(x)y + q(x)y = 0, where and is positive real number.     

Integral varieties of the canonical cone structure on <Emphasis Type="Italic">G</Emphasis>/<Emphasis Type="Italic">P</Emphasis>

The canonical cone structure on a compact Hermitian symmetric space G/P is the fiber bundle where is the cone of the highest weight vectors under the action of the reductive part of P. It is known that the cone coincides with the cone of the vectors tangent to the lines in G/P passing through x, when we consider G/P as a projective variety under its homogeneous embedding into the projective space of the irreducible representation space V of G with highest weight associated to P. A subvariety X of G/P is said to be an integral variety of at all smooth points xG/P. Equivalently, an integral variety of is a subvariety of G/P whose embedded projective tangent space at each smooth point is a linear space We prove a kind of rigidity of the integral varieties under some dimension condition. After making a uniform setting to study the problem, we apply the theory of Lie algebra cohomology as a main tool. Finally we show that the dimension condition is necessary by constructing counterexamples.     

Isoperimetric inequalities and random walks on quotients of graphs and buildings

Let be a Euclidean or hyperbolic building and let GAut be a locally compact unimodular group, which acts strongly transitively on . We use graphs , quasi-isometric to , to study asymptotic properties of quotients , where is a discrete subgroup of G. If G has Kazhdans property (T) we show that such quotients satisfy strong isoperimetric inequalities. This yields new examples of graphs with positive Cheeger constant. Such graphs cannot be bi-Lipschitz embedded into Hilbert space. Moreover, simple random walks on such quotients are shown to be recurrent if and only if is a uniform lattice in G.Mathematics Subject Classification (1991): 11E95, 22E40, 22E50, 51E24, 60G50in final form: 10 October 2003     

Diophantine inequalities involving several power sums

Let denote the ring of power sums, i.e. complex functions of the form for some and _iA, where is a multiplicative semigroup. Moreover, let We consider Diophantine inequalities of the form where >1 is a quantity depending on the dominant roots of the power sums appearing as coefficients in F(n,y), and show that all its solutions have y parametrized by some power sums from a finite set. This is a continuation of the work of Corvaja and Zannier [4–6] and of the authors [10, 18] on such problems.Mathematics Subject Classification (2000):11D45,11D61Revised version: 6 May 2004     

Isomorphisms of algebras associated with directed graphs

Given countable directed graphs G and G, we show that the associated tensor algebras (G) and (G) are isomorphic as Banach algebras if and only if the graphs G are G are isomorphic. For tensor algebras associated with graphs having no sinks or no sources, the graph forms an invariant for algebraic isomorphisms. We also show that given countable directed graphs G, G, the free semigroupoid algebras and are isomorphic as dual algebras if and only if the graphs G are G are isomorphic. In particular, spatially isomorphic free semigroupoid algebras are unitarily isomorphic. For free semigroupoid algebras associated with locally finite directed graphs with no sinks, the graph forms an invariant for algebraic isomorphisms as well.Mathematics Subject Classification (2000): 47L80, 47L55, 47L40Acknowledgments. We would like to thank the referee for several constructive suggestions on the initial draft and for bringing to our attention the work in [8,9]. The first author was partially supported by a research grant from ECU and the second author by an NSERC research grant and start up funds from the University of Guelph. We thank David Pitts for enlightening conversations and Alex Kumjian for helpful comments on the literature.     

Changement de base pour les foncteurs <Emphasis Type="Italic">Tor</Emphasis>

Il existe un théorème de changement de base pour les foncteurs dérivés Ext (voir par exemple [5]) et le but de cet article est, dans un premier temps, de démontrer un théorème analogue pour les foncteurs dérivés Tor. Il donne un isomorphisme de la forme Tor_*(F,R(G)) Tor_*(L(F),G), où (L,R) est un couple de foncteurs obtenus à partir dun - bimodule, F un ^op-module et G un -module. On définit ensuite la notion de catégorie croisée, qui donne naissance à des exemples de ce type. Le théorème de Pirashvili et Richter sur lhomologie de Hochschild et lhomologie cyclique démontré dans [6] en est une conséquence immédiate.Mathematics Subject Classification (2000): 18G15, 16E40     

Topologically transitive extensions of bounded operators

Let X be any Banach space and T a bounded operator on X. An extension of the pair (X,T) consists of a Banach space in which X embeds isometrically through an isometry i and a bounded operator on such that When X is separable, it is additionally required that be separable. We say that is a topologically transitive extension of (X, T) when is topologically transitive on , i.e. for every pair of non-empty open subsets of there exists an integer n such that is non-empty. We show that any such pair (X,T) admits a topologically transitive extension , and that when H is a Hilbert space, (H,T) admits a topologically transitive extension where is also a Hilbert space. We show that these extensions are indeed chaotic.Mathematics Subject Classification (2000): 47 A 16     

Almost periodic compactifications of group extensions

Let and be groups and let be an extension of by . Given a property of group compactifications, one can ask whether there exist compactifications and of N and K such that the universal -compactification of G is canonically isomorphic to an extension of by . We prove a theorem which gives necessary and sufficient conditions for this to occur for general properties and then apply this result to the almost periodic and weakly almost periodic compactifications of G.     

On signed majority total domination in graphs

We initiate the study of signed majority total domination in graphs. Let G = (V, E) be a simple graph. For any real valued function f: V and S V, let . A signed majority total dominating function is a function f: V {–1, 1} such that f(N(v)) 1 for at least a half of the vertices v V. The signed majority total domination number of a graph G is = min{f(V): f is a signed majority total dominating function on G}. We research some properties of the signed majority total domination number of a graph G and obtain a few lower bounds of .This research was supported by National Natural Science Foundation of China.     

Projectability and weak homogeneity of pseudo effect algebras

In this paper we deal with a pseudo effect algebra possessing a certain interpolation property. According to a result of Dvurečenskij and Vettterlein, can be represented as an interval of a unital partially ordered group G. We prove that is projectable (strongly projectable) if and only if G is projectable (strongly projectable). An analogous result concerning weak homogeneity of and of G is shown to be valid. This work has been partially supported by the Slovak Academy of Sciences via the project Center of Excellence—Physics of Information (grant I/2/2005).     

q-Skew Derivations and Polynomial Identities

Let R be a prime ring and its left Martindale quotient ring. Assume that a q-skew -derivation of R satisfies the identity relationfor all x R, where the subring of constants of on R. It is proved that if R⁽⁾ satisfies nontrivial polynomial identities, then so does R. This answers affirmatively a problem raised in Bergen and Grzeszczuk [2] by removing the assumption on the algebraicity of .Mathematics Subject Classification (2000): 16W20, 16W25, 16W55Members of Mathematics Division, National Center for Theoretical Sciences at Taipei.Acknowledgement The authors are thankful to the referee for her/his useful suggestions and comments. This research was supported by the National Science Council of Taiwan.     

On an eigenvalue problem related to the critical exponent

In this paper we study the eigenvalue problemwhere is a smooth bounded domain, and u is a positive solution of the problemsuch thatwhere S is the best Sobolev constant for the embedding of H¹₀() into L²*(), We prove several estimates for the eigenvalues _i, of (I), i=2,..,N+2 and some qualitative properties of the corresponding eigenfunctions.Supported by M.I.U.R., project Variational methods and nonlinear differential equations.     

Modified Ariki-Koike algebras and cyclotomic <Emphasis Type="Italic">q</Emphasis>-Schur algebras

The modified Ariki-Koike algebra is a variation of the original Ariki-Koike algebra over an integral domain R. When R is a rational function field over the independent parameters, But for general R, is not isomorphic to , and has a simpler structure than . In this paper, we construct a cellular basis of which has a similar property as the cellular basis of introduced by Dipper-James-Mathas. By comparing these two cellular bases, we obtain some estimate on the decomposition numbers of in terms of the decomposition numbers of . We also prove the integral form of the Schur-Weyl reciprocity between a certain quantum algebra U_q and on the tensor space      

Associated primes and cofiniteness of local cohomology modules

Let be an ideal of Noetherian ring R and let s be a non-negative integer. Let M be an R-module such that is finite R-module. If s is the first integer such that the local cohomology module is non -cofinite, then we show that is finite. In particular, the set of associated primes of is finite. Let be a local Noetherian ring and let M be a finite R-module. We study the last integer n such that the local cohomology module is not -cofinite and show that n just depends on the support of M.The research of the first author was supported in part by a grant from IPM (No. 83130114).The second author was supported by a grant from University of Tehran (No. 6103023/1/01).     

Analysis of the -Neumann problem along a straight edge

We show there exists an L^p solution, for p (2,), to the -Neumann problem on an edge domain in ² for (0,1)-forms, and we explicitly compute the singularities, which are of complex logarithmic and arctangent type, along the edge, of the solution.Mathematics Subject Classification (2000): 32W05, 35B65     

On Cross-intersecting Families of Sets

A family of -element subsets and a family of k-element subsets of an n-element set are cross-intersecting if every set from has a nonempty intersection with every set from . We compare two previously established inequalities each related to the maximization of the product , and give a new and short proof for one of them. We also determine the maximum of for arbitrary positive weights ,_k.     

Bounded trace <Emphasis Type="Italic">C</Emphasis>*-algebras and integrable actions

Let G be a second countable group, A be a separable C*-algebra with bounded trace and a strongly continuous action of G on A. Suppose that the action of G on induced by is free and the G-orbits are locally closed. We show that the crossed product A×G has bounded trace if and only if G acts integrably (in the sense of Rieffel and an Huef) on . In the course of this, we show that the extent of non-properness of an integrable action gives rise to a lower bound for the size of the (finite) upper multiplicities of the irreducible representations of the crossed product.Mathemactics Subject Classification (1991): 46L55