A note on unimodular congruence of the Laplacian matrix of a graph

Let G be a simple connected graph and L(G) be its Laplacian matrix. In this note, we prove that L(G) is congruent by a unimodular matrix to its Smith normal form if and only if G is a tree.     

A Characterization of Finitely Decidable Congruence Modular Varieties

For every finitely generated, congruence modular variety of finite type we find a finite family of finite rings such that the variety is finitely decidable if and only if is congruence permutable and residually small, all solvable congruences in finite algebras from are Abelian, each congruence above the centralizer of the monolith of a subdirectly irreducible algebra from is comparable with all congruences of , each homomorphic image of a subdirectly irreducible algebra with a non-Abelian monolith has a non-Abelian monolith, and, for each ring from , the variety of -modules is finitely decidable.

     

A Lower Bound for Congruence Representations

G. Grätzer, H. Lakser, and E. T. Schmidt proved that every finite distributive lattice D with n join-irreducible elements can be represented as the congruence lattice of a lattice L with O(n2) elements. G. Grätzer, I. Rival, and N. Zaguia gave kn, < 2, as a lower bound for the size of such a lattice L; a sharper form, 1/64(n/log2n)2, of this result was given by Y. Zhang.In this note, we apply a recent result of R. Freese, to obtain 1/16 n2/log2n as a lower bound. We also give a direct proof of Freese's result.     

Schubert cycles, differential forms and D-modules on varieties of unseparated flags

Proper homogeneous G-spaces (where G is semisimple algebraic group) over positive characteristic fields k can be divided into two classes, the first one being the flag varieties G/P and the second one consisting of varieties of unseparated flags (proper homogeneous spaces not isomorphic to flag varieties as algebraic varieties). In this paper we compute the Chow ring of varieites of unseparated flags, show that the Hodge cohomology of usual flag varieties extends to the general setting of proper homogeneous spaces and give an example showing (by geometric means) that the D -affinity of Beilinson and Bernstein fails for varieties of unseparated flags.     

一类格序半群的同余

本文讨论了一类可换格序半群的同余性质。给出了这类格序半群同余的表示和同态分解定理。这个结论一般化了［１］和［２］中的结果。     

一类逆半群的亚直可约性

本文利用逆半群上的同余扩张，讨论了一类逆半群的亚直可约性，并刻划了这类逆半群的幂等元集的特征。     

Finiteness conditions and distributive laws for Boolean algebras

We compare diverse degrees of compactness and finiteness in Boolean algebras with each other and investigate the influence of weak choice principles. Our arguments rely on a discussion of infinitary distributive laws and generalized prime elements in Boolean algebras. In ZF set theory without choice, a Boolean algebra is Dedekind finite if and only if it satisfies the ascending chain condition. The Denumerable Subset Axiom (DS) implies finiteness of Boolean algebras with compact top, whereas the converse fails in ZF. Moreover, we derive from DS the atomicity of continuous Boolean algebras. Some of the results extend to more general structures like pseudocomplemented semilattices (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Necessary and Sufficient Conditions for Mcshane’s Identity and Variations

Greg McShane introduced a remarkable identity for lengths of simple closed geodesics on the once punctured torus with a complete, finite volume hyperbolic structure. Bowditch later generalized this and gave sufficient conditions for the identity to hold for general type-preserving representations of a free group on two generators Γ to SL(2,C), this was further generalized by the authors to obtain sufficient conditions for a generalized McShane’s identity to hold for arbitrary (not necessarily type-preserving) non-reducible representations in Tan et al. (Submitted). Here we extend the above by giving necessary and sufficient conditions for the generalized McShane identity to hold (Akiyoshi, Miyachi and Sakuma had proved it for type-preserving representations). We also give a version of Bowditch’s variation of McShane’s identity to once-punctured torus bundles, in the case where the monodromy is generated by a reducible element, and provide necessary and sufficient conditions for the variations to hold.     

A Canonical Compatible Metric for Geometric Structures on Nilmanifolds

Let (N, γ) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their ‘almost’ versions). We define a left invariant Riemannian metric on N compatible with γ to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. We prove that minimal metrics (if any) are unique up to isometry and scaling, they develop soliton solutions for the ‘invariant Ricci’ flow and are characterized as the critical points of a natural variational problem. The uniqueness allows us to distinguish two geometric structures with Riemannian data, giving rise to a great deal of invariants.Our approach proposes to vary Lie brackets rather than inner products; our tool is the moment map for the action of a reductive Lie group on the algebraic variety of all Lie algebras, which we show to coincide in this setting with the Ricci operator. This gives us the possibility to use strong results from geometric invariant theory.Communicated by: Nigel Hitchin (Oxford) Mathematics Subject Classifications (2000): Primary: 53D05, 53D55; Secondary: 22E25, 53D20, 14L24, 53C30.