Tame combing and almost convexity conditions

We give the first examples of groups which admit a tame combing with linear radial tameness function with respect to any choice of finite presentation, but which are not minimally almost convex on a standard generating set. Namely, we explicitly construct such combings for Thompson??s group F and the Baumslag?CSolitar groups BS(1, p) with p ??? 3. In order to make this construction for Thompson??s group F, we significantly expand the understanding of the Cayley complex of this group with respect to the standard finite presentation. In particular we describe a quasigeodesic set of normal forms and combinatorially classify the arrangements of 2-cells adjacent to edges that do not lie on normal form paths.     

Reduction of almost Poisson brackets for nonholonomic systems on Lie groups

We present a systematic geometric construction of reduced almost Poisson brackets for nonholonomic systems on Lie groups with invariant kinetic energy metric and constraints. Our construction is of geometric interest in itself and is useful in the hamiltonization of some classical examples of nonholonomic mechanical systems.      

Growth functions,rewriting systems,and the Euler characteristic

We present a formula that expresses the number of admissible words for an arbitrary system of prohibitions in terms of the generating series. We also indicate applications of this formula to the calculation of the topological entropy of symbolic systems and to the calculation of growth functions of groups and algebras. Translated fromMatematicheskie Zametki, Vol. 58, No. 5, pp. 653–668, November, 1995. This research was partially supported by the Russian Foundation for Basic Research under grant No. 94-01-00820 and by the International Science Foundation grant MV1000.     

Free operated monoids and rewriting systems

The construction of bases for quotients is an important problem. In this paper, applying the method of rewriting systems, we give a unified approach to construct sections—an alternative name for bases in semigroup theory—for quotients of free operated monoids. As applications, we capture sections of free \(*\)-monoids and free groups, respectively.     

Almost convexity on Abelian groups

The notion of almost convexity is studied and an extension of Kuczma’s theorem, originally proved in finite-dimensional spaces, is presented. The phrase “almost” is meant in the sense of abstract σ-ideals. The main result also generalizes the theorem proved in Jarczyk and Laczkovich (Math Inequal Appl 13:217–225, 2009).     

The Connes-Kasparov conjecture for almost connected groups and for linear p-adic groups

Let G be a locally compact group with cocompact connected component. We prove that the assembly map from the topological K-theory of G to the K-theory of the reduced C^*-algebra of G is an isomorphism. The same is shown for the groups of k-rational points of any linear algebraic group over a local field k of characteristic zero. Dedicated to the memory of Peter Slodowy     

Dynamic convexity for natural thermostatted systems

Natural thermostatted systems are mechanical systems whose Lagrangian is the difference of a kinetic and a potential energy, subjected to the nonholonomic constraint of a constant kinetic energy. When any two points of the configuration space are joined by a thermostatted motion, we say that the system is dynamically convex. A thermostatted charged particle on the plane with a constant electric field is not a dynamically convex system. We prove a general sufficient condition for dynamic convexity, from which whole classes of examples are easily constructed.     

概周期系统概周期解的存在唯一性

运用Ｌｉａｐｕｎｏｖ函数，研究了概周期系统概周期解的存在唯一性，得到了一个方便应用的判定定理．     

Torsion units for some almost simple groups

We investigate the Zassenhaus conjecture regarding rational conjugacy of torsion units in integral group rings for certain automorphism groups of simple groups. Recently, many new restrictions on partial augmentations for torsion units of integral group rings have improved the effectiveness of the Luther-Passi method for verifying the Zassenhaus conjecture for certain groups. We prove that the Zassenhaus conjecture is true for the automorphism group of the simple group PSL(2, 11). Additionally we prove that the Prime graph question is true for the automorphism group of the simple group PSL(2, 13).     

Tame measures and Raikov systems on the circle

This paper constructs a tame continuous probability measure concentrated on a proper Raikov System on the circle group $\text{T}$.     

Coxeter groups are almost convex

In [C] Cannon introduced the notion of almost convexity for the Cayley graph of a finitely generated group. In this paper, we observe that standard facts about Coxeter groups imply that the Cayley graph associated to any Coxeter system is almost convex.Partially supported by NSF grants DMS-8905378 and DMS-9007959.     

Kostant's convexity theorem and the compact classical groups

The relationship between the classical Schur-Horn's theorem on the diagonal elements of a Hermitian matrix with prescribed eigenvalues and Kostant's convexity theorem in the context of Lie groups. By using Kostant's convexity theorem, we work out the statements on the special orthogonal group and the symplectic group explicitly. Schur-Horn's result can be stated in terms of a set of inequalities. The counterpart in the Lie-theoretic context is related to a partial ordering, introduced by Atiyah and Bott, defined on the closed fundamental Weyl chamber. Some results of Thompson on the diagonal elements of a matrix with prescribed singular values are recovered. Thompson-Poon's theorem on the convex hull of Hermitian matrices with prescribed eigenvalues is also generalized. Then a result of Atiyah-Bott is recovered.     

Actions of almost simple groups on compact eight-dimensional projective planes are classical,almost

We show that the existence of an almost simple group of automorphisms of dimension greater than 10 characterizes the Hughes planes (including the quarternion plane) among the 8-dimensional compact projective planes.Dedicated to Prof. Helmut R. Salzmann on his 65th birthday