Fragments with finiteness conditions in particular over group rings

Abstract length functions on groups were introduced by Lyndon [5]. In [2] Chiswell showed that for a group action on an IR-tree there is a length function associated with each point of the tree, and moreover any length function can occur in this way. This paper establishes necessary and sufficient conditions for two length functions to arise from the same action of a group on a tree.     

Conjugacy Length in Group Extensions

Determining the length of short conjugators in a group can be considered as an effective version of the conjugacy problem. The conjugacy length function provides a measure for these lengths. We study the behavior of conjugacy length functions under group extensions, introducing the twisted and restricted conjugacy length functions. We apply these results to show that certain abelian-by-cyclic groups have linear conjugacy length function and certain semidirect products ?^d ? ?^k have at most exponential (if k > 1) or linear (if k = 1) conjugacy length functions.     

Nice equations for nice groups

There are various natural local zeta functions associated with groups and rings for each primep. We consider the question of how these functions behave as we vary the primep and the groups (or rings) range over a specific class of groups (or rings), e.g. finitely generated torsion-free nilpotent groups of a fixed Hirsch length orp-adic analytic groups of a fixed dimension. Using a result of Macintyre’s on the uniformity of parameterizedp-adic integrals, together with various natural parameter spaces we define for these classes of groups, we prove a strong finiteness theorem on the possible poles of these local zeta functions.     

On Twisted Fourier Analysis and Convergence of Fourier Series on Discrete Groups

We study norm convergence and summability of Fourier series in the setting of reduced twisted group C ^*-algebras of discrete groups. For amenable groups, Følner nets give the key to Fejér summation. We show that Abel-Poisson summation holds for a large class of groups, including e.g. all Coxeter groups and all Gromov hyperbolic groups. As a tool in our presentation, we introduce notions of polynomial and subexponential H-growth for countable groups w.r.t. proper scale functions, usually chosen as length functions. These coincide with the classical notions of growth in the case of amenable groups.     

On Kerckhoff Minima and Pleating Loci for Quasi-Fuchsian Groups

We show how Kerckhoff's results on minima of length functions on Teichmüller space can be used to analyse the possible bending loci of the boundary of the convex hull for quasi-Fuchsian groups near to the Fuchsian locus.     

Flatness,plateau limit and limit length of single-plateau functions

In this paper we consider iteration of single-plateau functions, an important class of continuous functions with infinitely many forts, and investigate changes of number and length of plateaux under iteration.We use the indices flatness, plateau limit and limit length to formulate those changes. Furthermore, we compute the flatness, plateau limit and limit length for all the nine types of single-plateau functions.     

Equations in a group with a length function

A simple counterexample to Lyndon's conjecture is given. There is defined a group of continuous functions with a regular Archimedean length function. Analogues of Wicks' theorem on the commutator and of Lyndon's theorem on the solutions of the equation a²b²=c² are proved for a wider class of groups with a regular Archimedean length function.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 935–942, July–August, 1991.     

Some test length bounds for nonrepeating functions in the {&;, ∨} basis

Testing relative to a nonrepeating alternative in a conjunction-disjunction basis is considered. A lower bound on the test length is established for all nonrepeating functions in this basis. A subsequence of easily testable functions is constructed and the corresponding tests are described. Individual lower test length bounds are proved for functions of a special form; minimality of the tests is established for the functions of the constructed subsequence.     

Spectral Properties of Four-Dimensional Compact Flat Manifolds

We study the spectral properties of a large class of compact flat Riemannian manifolds of dimension 4, namely, those whose corresponding Bieberbach groups have the canonical lattice as translation lattice. By using the explicit expression of the heat trace of the Laplacian acting on p-forms, we determine all p-isospectral and L-isospectral pairs and we show that in this class of manifolds, isospectrality on functions and isospectrality on p-forms for all values of p are equivalent to each other. The list shows for any p, 1 ≤ p ≤ 3, many p-isospectral pairs that are not isospectral on functions and have different lengths of closed geodesics. We also determine all length isospectral pairs (i.e. with the same length multiplicities), showing that there are two weak length isospectral pairs that are not length isospectral, and many pairs, p-isospectral for all p and not length isospectral. Mathematics Subject Classifications (2000): 58J53, 58C22, 20H15.     

Orientation-dependent section distributions for convex bodies

In the paper the orientation dependent chord length distribution functions for some bounded convex domains are obtained. In particular, formulae for the orientation dependent chord length distributions for a regular polygon and an ellipse are obtained. Explicit forms of the orientation dependent chord length distributions were known only in the cases of a disc and a triangle. We also obtain the cross-section area distribution functions for an ellipsoid and a cylinder. The cross-section area distribution function was known only in the case of a ball.     

Gaussian Cubature Arising from Hybrid Characters of Simple Lie Groups

Lie groups with two different root lengths allow two ‘mixed sign’ homomorphisms on their corresponding Weyl groups, which in turn give rise to two families of hybrid Weyl group orbit functions and characters. In this paper we extend the ideas leading to the Gaussian cubature formulas for families of polynomials arising from the characters of irreducible representations of any simple Lie group, to new cubature formulas based on the corresponding hybrid characters. These formulas are new forms of Gaussian cubature in the short root length case and new forms of Radau cubature in the long root case. The nodes for the cubature arise quite naturally from the (computationally efficient) elements of finite order of the Lie group.     

On the length of Boolean functions in the class of exclusive-OR sums of pseudoproducts

Representations of Boolean functions by exclusive-OR sums (modulo 2) of pseudoproducts is studied. An ExOR-sum of pseudoproducts (ESPP) is the sum modulo 2 of products of affine (linear) Boolean functions. The length of an ESPP is defined as the number of summands in this form, and the length of a Boolean function in the class of ESPPs is defined as the minimum length of an ESPP representing this function. The Shannon function L ^ESPP(n) of the length of Boolean functions in the class of ESPPs is considered, which equals the maximum length of a Boolean function of n variables in this class. Lower and upper bounds for the Shannon function L ^ESPP(n) are found. The upper bound is proved by using an algorithm which can be applied to construct representations by ExOR-sums of pseudoproducts for particular Boolean functions.     

Hermite Interpolation in Loop Groups and Conjugate Quadrature Filter Approximation

A classical result of Weierstrass ensures that any continuous finite length trajectory in a vector space can be uniformly approximated by one whose coordinates are trigonometric functions. We derive an analogous result for trajectories in spheres and apply it to show that a continuous frequency response of a conjugate quadrature filter can be uniformly approximated by the frequency response of a finitely supported conjugate quadrature filter. We also extend this result, so as to preserve specified roots of the frequency response, and derive an approximation result for refinable functions whose integer translates are orthonormal. Our methods utilize properties of loop groups, jets, and the Brouwer topological degree. Mathematics Subject Classifications (2000) 22E67(Primary), 41A29(Secondary), 42A10, 42A11, 42C40, 47H10, 47H11.     

The marked length spectrum vs. the Laplace spectrum on forms on Riemannian nilmanifolds

The subject of this paper is the relationships among the marked length spectrum, the length spectrum, the Laplace spectrum on functions, and the Laplace spectrum on forms on Riemannian nilmanifolds. In particular, we show that for a large class of three-step nilmanifolds, if a pair of nilmanifolds in this class has the same marked length spectrum, they necessarily share the same Laplace spectrum on functions. In contrast, we present the first example of a pair of isospectral Riemannian manifolds with the same marked length spectrum but not the same spectrum on one-forms. Outside of the standard spheres vs. the Zoll spheres, which are not even isospectral, this is the only example of a pair of Riemannian manifolds with the same marked length spectrum, but not the same spectrum on forms. This partially extends and partially contrasts the work of Eberlein, who showed that on two-step nilmanifolds, the same marked length spectrum implies the same Laplace spectrum both on functions and on forms. Research at MSRI supported in part by NSF grant DMS-9022140. Research at MSRI and Texas Tech supported in part by NSF grant DMS-9409209.     

Order preserving reductions and polynomial improving paths

This paper shows that neighborhood transformations and data-independent order transformations preserve the length of improving paths and order of local optima of neighborhood functions. These results imply that finding effective neighborhood functions for Zero-One IP is at least as hard as finding effective neighborhood functions for any other NPO problem.     

Read-once functions with hard-to-test projections

An effect of an increase in minimum test length for functions under constant substitutions of constants instead of variables in a checking test problem for read-once functions is described. A family of bases is described, and sequences of functions that are read-once in these bases and have projections whose testing requires more vectors than these functions themselves are constructed.     

Sooner and later waiting time problems for success and failure runs in higher order Markov dependent trials

The probability generating functions of the waiting times for the first success run of length k and for the sooner run and the later run between a success run of length k and a failure run of length r in the second order Markov dependent trials are derived using the probability generating function method and the combinatorial method. Further, the systems of equations of 2^.m conditional probability generating functions of the waiting times in the m-th order Markov dependent trials are given. Since the systems of equations are linear with respect to the conditional probability generating functions, they can be solved exactly, and hence the probability generating functions of the waiting time distributions are obtained. If m is large, some computer algebra systems are available to solve the linear systems of equations.This research was partially supported by the Natural Sciences and Engineering Research Council of Canada.     

Polynomial Dehn Functions and the Length of Asynchronously Automatic Structures

We extend the range of observed behaviour among length functionsof optimal asynchronously automatic structures. We do so bymeans of a construction that yields asynchronously automaticgroups with finite aspherical presentations where the Dehn functionof the group is polynomial of arbitrary degree. Many of thesegroups can be embedded in the automorphism group of a free group.Moreover, the fact that the groups have aspherical presentationsmakes them useful tools in the search to determine the spectrumof exponents for second order Dehn functions. We contributeto this search by giving the first exact calculations of groupswith quadratic and superquadratic exponents. 2000 Mathematical Subject Classification: 20F06, 20F65, 20F69.     

A finitary Tits' alternative

It is shown that for any family of finite groups of uniformly bounded rank, either (i) a subdirect product of these groups contains a non-cyclic free group, or (ii) there exists a single word w which is a law in each group, and moreover, if N is the length of the word, and r the maximal rank of each finite group, then each group is nilpotent-of-bounded class-by-abelian-by-bounded-index, with the bounds being functions of N and r alone. Additionally, various corollaries are derived from this result.     

Trees,tight extensions of metric spaces,and the cohomological dimension of certain groups: A note on combinatorial properties of metric spaces

The concept of tight extensions of a metric space is introduced, the existence of an essentially unique maximal tight extension T_x—the “tight span,” being an abstract analogon of the convex hull—is established for any given metric space X and its properties are studied. Applications with respect to (1) the existence of embeddings of a metric space into trees, (2) optimal graphs realizing a metric space, and (3) the cohomological dimension of groups with specific length functions are discussed.