Let W be a nite Weyl group with root system , simple roots 1 coordinates for the sum of all positive roots, and for w 2 W let`(w) denote the length and D(w) = fi : w ?1 i < 0g the descent set of w. By analyzing the structure of the corresponding aane Weyl group, we prove that P w2W q (w)?`(w) = f Q n i=1 (1 ? q b i)=(1 ? q e i), where (w) = P i2D(w) b i .

It is well known that a group is free if and only if it acts freely without inversions on a tree. We prove a generalisation of this fact by deening a quasi-tree to be a graph with a bound on the size of its simple loops. It is shown that a nitely generated group acting freely on such a graph is isomorphic to a free product of free groups and nite groups.

This paper presents some methods of representing canonical commutation relations in terms of hyper nite-dimensional matrices, which are constructed by nonstandard analysis. The rst method uses representations of a nonstandard extension of nite Heisenberg group, called hyper nite Heisenberg group. The second is based on hyper nite-dimensional representations of so(3). Then, the cases of in nite ...

In this paper, we present a novel algorithmic approach , the hybrid matrix geometric/invariant subspace method, for nding the stationary probability distribution of the nite QBD process which arises in performance analysis of computer and communication systems. Assuming that the QBD state space is deened in two dimensions with m phases and K + 1 levels , the solution vector for level k, k ; 0 k...

In this paper, we present a novel algorithmic approach , the hybrid matrix geometric/invariant subspace method, for nding the stationary probability distribution of the nite QBD process which arises in performance analysis of computer and communication systems. Assuming that the QBD state space is deened in two dimensions with m phases and K + 1 levels , the solution vector for level k, k ; 0 k...

vasil&apos;ev posed problem 16.26 in [the kourovka notebook: unsolved problems in group theory, 16th ed.,sobolev inst. math., novosibirsk (2006).] as follows:does there exist a positive integer $k$ such that there are no $k$ pairwise nonisomorphicnonabelian finite simple groups with the same graphs of primes? conjecture: $k = 5$.in [zvezdina, on nonabelian simple groups having the same prime gr...

We consider nite separable coverings of curves f : X → Y over a eld of characteristic p ≥ 0. We are interested in describing the possible monodromy groups of this cover if the genus of X is xed. There has been much progress on this problem over the past decade in characteristic zero. Recently Frohardt and Magaard completed the nal step in resolving the Guralnick Thompson conjecture showing that...

Let ? be a lattice in a simple higher rank Lie group G. We describe all locally compact (not necessarily Lie) groups H in which ? can be embedded as a lattice. For lattices ? in rank one groups G (with the only exception of non-uniform lattices in G ' SL 2 (R), which are virtually free groups) we give a similar description of all possible locally compact groups H, in which ? can be embedded as ...

The Independence Theorem for the congruence lattice and the auto-morphism group of a nite lattice was proved by V. A. Baranski and A. Urquhart. Both proofs utilize the characterization theorem of congruence lattices of nite lattices (as nite distributive lattices) and the characterization theorem of auto-morphism groups of nite lattices (as nite groups). In this paper, we introduce a new, stron...