Normal Form For Elements Research Articles

Unified theory for finite Markov chains

We provide a unified framework to compute the stationary distribution of any finite irreducible Markov chain or equivalently of any irreducible random walk on a finite semigroup S. Our methods use geometric finite semigroup theory via the Karnofsky–Rhodes and the McCammond expansions of finite semigroups with specified generators; this does not involve any linear algebra. The original Tsetlin library is obtained by applying the expansions to P(n), the set of all subsets of an n element set. Our set-up generalizes previous groundbreaking work involving left-regular bands (or R-trivial bands) by Brown and Diaconis, extensions to R-trivial semigroups by Ayyer, Steinberg, Thiéry and the second author, and important recent work by Chung and Graham. The Karnofsky–Rhodes expansion of the right Cayley graph of S in terms of generators yields again a right Cayley graph. The McCammond expansion provides normal forms for elements in the expanded S. Using our previous results with Silva based on work by Berstel, Perrin, Reutenauer, we construct (infinite) semaphore codes on which we can define Markov chains. These semaphore codes can be lumped using geometric semigroup theory. Using normal forms and associated Kleene expressions, they yield formulas for the stationary distribution of the finite Markov chain of the expanded S and the original S. Analyzing the normal forms also provides an estimate on the mixing time.

Study of the algebra of smooth integro-differential operators with applications

We study the algebra of integro-differential operators with smooth coefficients and kernels on a subspace of [Formula: see text]. We find a normal form for elements of this algebra and determine its unit group. The formulation of inverses gives explicit solutions of inhomogeneous linear Volterra integro-differential equations and Volterra integral equations of first kind with smooth kernels.

Growth series of crossed and two-sided crossed products of cyclic groups

Abstract We recall that the two-sided crossed product of finite cyclic groups is actually a generalization of the crossed product construction of the same type of groups (cf. [10]). In this paper, by considering the crossed and two-sided crossed products obtained from both finite and infinite cyclic groups, we first present the complete rewriting systems and normal forms of elements over crossed products. (We should note that the complete rewriting systems and normal forms of elements over two-sided crossed products have been recently defined in [10]). In the crossed product case, we will consider their presentations that were given in [2]. As a next step, by using the normal forms of elements of these two products, we calculate the growth series of the crossed product of different combinations of finite and infinite cyclic groups as well as the growth series of two-sided crossed product of finite cyclic groups.

Gröbner-Shirshov bases for extended modular, extended Hecke, and Picard groups

In this paper, Grobner-Shirshov bases (noncommutative) for extended modular, extended Hecke and Picard groups are considered. A new algorithm for obtaining normal forms of elements and hence solving the word problem in these groups is proposed.

Partially commutative Artin–Coxeter groups and their arboreal structure

Given a group G and a set S⊆ G of generators, set S −1={s −1 | s∈G} and S ̃ =S∪S −1 . For g∈ G, let l( g) denote the minimum length of any expression g= s 1… s d with s i∈ S ̃ . For g, h∈ G, set g⊂ h if l( g)+ l( g −1 h)= l( h). The paper is devoted to the study of the pairs ( G, S) for which 1∉S, S∩S −1=S 1 ≔ {s∈S | s 2=1} , and the partial order ⊂ satisfies the following conditions: (i) ( G,⊂) is a semilattice; denote by g∩ h the greatest lower bound w.r.t. the order ⊂ for any pair ( g, h) of elements of G, (ii) g −1( g∩ h)⊂ g −1 h for all g, h∈ G, and (iii) gh= hg is the least upper bound g∪ h w.r.t. ⊂ for the pair ( g, h) whenever g∩ h=1 and there exists u∈ G such that g⊂ u and h⊂ u. It is shown that the pairs above are exactly those for which G admits the presentation G=〈S; s 2=1 for s∈ S 1, and sts −1 t −1=1 for those s,t∈S, s≠t , for which the commuting relation st= ts holds in G〉. Call partially commutative Artin–Coxeter groups the groups defined by such presentations. The pairs ( G, S) above satisfy a “deletion condition” (D) analogous to the well-known deletion condition for Coxeter groups. It is shown that the pairs ( G, S) satisfying (D) have solvable word problem, as is the case with usual Coxeter groups. Normal forms for elements in partially commutative Artin–Coxeter groups are also described.

DECOMPOSITION OF STEINBERG GROUPS

Let A be an associative ring with identity, St_n(A) the Steinberg group of dimension n over A. The stable Steinberg group is the direct limit of the St_n(A) in a natural way. Several authors studied various normal forms for elements of St(A) and St_n(A) (cf.[ 3—7]).

Generic properties in Euclidean kinematics

The motion of a rigid body in a Euclidean space E n is represented by a path in the Euclidean isometry group E(n). A normal form for elements of the Lie algebra of this group leads to a stratification of the algebra which is shown to be Whitney regular. Translating this along invariant vector fields give rise to a stratification of the jet bundles J k (R, E(n)) for k=1, 2 and, hence, via the transversality theorem, to generic properties of rigid body motions. The relation of these to the classical centrodes and axodes of motions is described, together with applications to planar 4-bar mechanisms and the dynamics of a rigid body.

Versal deformations of elements of real classical Lie algebras

Versal deformations of normal forms of elements of real classical Lie algebras o(p,q), sp(2n,R), o*(2n), sp(2p,2q) and u(p,q) are computed. Examples involving the Lie algebras o(p,q), p+q<or=6, are considered.