p-Adic framed braids

In this paper we define the p-adic framed braid group F∞,n, arising as the inverse limit of the modular framed braids. An element in F∞,n can be interpreted geometrically as an infinite framed cabling. F∞,n contains the classical framed braid group as a dense subgroup. This leads to a set of topological generators for F∞,n and to approximations for the p-adic framed braids. We further construct a p-adic Yokonuma-Hecke algebra Y∞,n(u) as the inverse limit of a family of classical Yokonuma-Hecke algebras. These are quotients of the modular framed braid groups over a quadratic relation. Finally, we give topological generators for Y∞,n(u).     

Schläfli modular equations for generalized Weber functions

Sets of appropriately normalized eta quotients, that we call level n Weber functions, are defined, and certain identities generalizing Weber function identities are proved for these functions. Schläfli type modular equations are explicitly obtained for Generalized Weber Functions associated with a Fricke group Γ⁰(n)⁺, for n=2,3,5,7,11,13 and 17.     

Torus quotients of homogeneous spaces

We study torus quotients of principal homogeneous spaces. We classify the Grassmannians for which semi-stable=stable and as an application we construct smooth projective varieties as torus quotients of certain homogeneous spaces. We prove the finiteness of the ring ofT invariants of the homogeneous co-ordinate ring of the GrassmannianG _2,n (n odd) over the ring generated byR ₁, the first graded part of the ring ofT invariants.     

Continued fractions for some alternating series

We discuss certain simple continued fractions that exhibit a type of “self-similar” structure: their partial quotients are formed by perturbing and shifting the denominators of their convergents. We prove that all such continued fractions represent transcendental numbers. As an application, we prove that Cahen's constant $$C = \sum\limits_{i \geqslant 0} {\frac{{( - 1)^i }}{{S_i - 1}}}$$ is transcendental. Here (S _n ) isSylvester's sequence defined byS ₀=2 andS _n+1 =S _n ² ?S _n +1 forn≥0. We also explicitly compute the continued fraction for the numberC; its partial quotients grow doubly exponentially and they are all squares.     

Continuity of topological entropy for perturbation of time-one maps of hyperbolic flows

In this paper we derive necessary and sufficient homological and cohomological conditions for profinite groups and modules to be of type FP_n over a profinite ring R, analogous to the Bieri–Eckmann criteria for abstract groups. We use these to prove that the class of groups of type FP_n is closed under extensions, quotients by subgroups of type FP_n, proper amalgamated free products and proper HNN-extensions, for each n. We show, as a consequence of this, that elementary amenable profinite groups of finite rank are of type FP_∞ over all profinite R. For any class C of finite groups closed under subgroups, quotients and extensions, we also construct pro-C groups of type FP_n but not of type FP_n+1 over Z_? for each n. Finally, we show that the natural analogue of the usual condition measuring when pro-p groups are of type FP_n fails for general profinite groups, answering in the negative the profinite analogue of a question of Kropholler.     

Transcendence of Thue–Morse p-Adic Continued Fractions

It has been proved that real numbers defined as a limit of continued fractions with a Thue–Morse sequence as the sequence of positive integer partial quotients are transcendental. The same holds for more general sequences of partial quotients. Moreover, transcendence results have been proved for p-adic numbers defined by Hensel development. Here, we tackle the transcendence question in case of p-adic numbers defined as a limit of continued fractions by proving a similar result as the first one.     

Stabilization of statistics in wave and Klein-Gordon equations with mixing. Scattering theory for infinite energy solutions

We consider wave and Klein-Gordon equations in the whole space ?ⁿ with arbitraryn≥2. We assume initial data to be homogeneous random functions in ?ⁿ with zero expectation and finite mean density of energy. Moreover, we assume initial data fit mixing condition of Ibragimov-Linnik type. We consider the distributions of the random solution at the moment of timet. The main results mean the convergence of this distribution to some Gaussian measure ast→∞. This is a central limit theorem for wave and Klein-Gordon equations. The limit Gaussian measures are invariant measures for equations considered. Corresponding stationary random solutions are ergodic and mixing in time. The results are inspired by mathematical problems of statistical physics.     

Bessel Convolutions on Matrix Cones: Algebraic Properties and Random Walks

Bessel-type convolution algebras of measures on the matrix cones of positive semidefinite q×q-matrices over ?,?,? were introduced recently by Rösler. These convolutions depend on a continuous parameter, generate commutative hypergroups, and have Bessel functions of matrix argument as characters. In this paper, we study the algebraic structure of these hypergroups. In particular, the subhypergroups, quotients, and automorphisms are classified. The algebraic properties are partially related to the properties of random walks on these matrix Bessel hypergroups. In particular, known properties of Wishart distributions, which form Gaussian convolution semigroups on these hypergroups, are put into a new light. Moreover, limit theorems for random walks are presented. In particular, we obtain strong laws of large numbers and a central limit theorem with Wishart distributions as limits.     

Codimension and pseudometric in co-Heyting algebras

In this paper, we introduce a notion of dimension and codimension for every element of a bounded distributive lattice L. These notions prove to have a good behavior when L is a co-Heyting algebra. In this case the codimension gives rise to a pseudometric on L which satisfies the ultrametric triangle inequality. We prove that the Hausdorff completion of L with respect to this pseudometric is precisely the projective limit of all its finite dimensional quotients. This completion has some familiar metric properties, such as the convergence of every monotonic sequence in a compact subset. It coincides with the profinite completion of L if and only if it is compact or equivalently if every finite dimensional quotient of L is finite. In this case we say that L is precompact. If L is precompact and Hausdorff, it inherits many of the remarkable properties of its completion, specially those regarding the join/meet irreducible elements. Since every finitely presented co-Heyting algebra is precompact Hausdorff, all the results we prove on the algebraic structure of the latter apply in particular to the former. As an application, we obtain the existence for every positive integers n, d of a term t _{n, d} such that in every co-Heyting algebra generated by an n-tuple a, t _{n, d} (a) is precisely the maximal element of codimension d.     

Permanental compounds and permanents of (0, 1)-circulants

It was shown by the author in a recent paper that a recurrence relation for permanents of (0, 1)-circulants can be generated from the product of the characteristic polynomials of permanental compounds of the companion matrix of a polynomial associated with (0, 1)-circulants of the given type. In the present paper general properties of permanental compounds of companion matrices are studied, and in particular of convertible companion matrices, i.e., matrices whose permanental compounds are equal to the determinantal compounds after changing the signs of some of their entries. These results are used to obtain formulas for the limit of the nth root of the permanent of the n × n (0, 1)-circulant of a given type, as n tends to infinity. The root-squaring method is then used to evaluate this limit for a wide range of circulant types whose associated polynomials have convertible companion matrices.     

Markov classes in certain finite quotients of artin’s braid group

This paper studies three finite quotients of the sequence of braid groups {B _n;n = 1,2,…}. Each has the property that Markov classes in {ie160-1} = ∐B _n pass to well-defined equivalence classes in the quotient. We are able to solve the Markov problem in two of the quotients, obtaining canonical representatives for Markov classes and giving a procedure for reducing an arbitrary representative to the canonical one. The results are interpreted geometrically, and related to link invariants of the associated links and the value of the Jones polynomial on the corresponding classes. This material is based upon work partially supported by the National Science Foundation under Grant No. DMS-8503758.     

Convergence in distribution of quotients of order statistics

Let X₁, X₂,… be i.i.d. random variables with continuous distribution function F < 1. It is known that if 1 - F(x) varies regularly of order - p, the successive quotients of the order statistics in decreasing order of X₁,…,X_n are asymptotically independent, as n→∞, with distribution functions x^kp, k = 1, 2, …. A strong converse is proved, viz. convergence in distribution of this type of one of the quotients implies regular varation of 1 - F(x).     

Central limit asymptotics for shifts of finite type

We study the rate of convergence and asymptotic expansions in the central limit theorem for the class of Hölder continuous functions on a shift of finite type endowed with a stationary equilibrium state. It is shown that the rate of convergence in the theorem isO(n ^?1/2) and when the function defines a non-lattice distribution an asymptotic expansion to the order ofo(n ^?1/2) is given. Higher-order expansions can be obtained for a subclass of functions. We also make a remark on the central limit theorem for (closed) orbital measures.     

Differentiability at the edge of the percolation cone and related results in first-passage percolation

We study first-passage percolation in two dimensions, using measures μ on passage times with b: = inf  supp(μ) > 0 and ${\mu(\{b\})=p\geq \vec p_c}$ , the threshold for oriented percolation. We first show that for each such μ, the boundary of the limit shape for μ is differentiable at the endpoints of flat edges in the so-called percolation cone. We then conclude that the limit shape must be non-polygonal for all of these measures. Furthermore, the associated Richardson-type growth model admits infinite coexistence and if μ is not purely atomic the graph of infection has infinitely many ends. We go on to show that lower bounds for fluctuations of the passage time given by Newman–Piza extend to these measures. We establish a lower bound for the variance of the passage time to distance n of order log n in any direction outside the percolation cone under a condition of finite exponential moments for μ. This result confirms a prediction of Newman and Piza (Ann Probab 23:977–1005, 1995) and Zhang (Ann Probab 36:331–362, 2008). Under the assumption of finite radius of curvature for the limit shape in these directions, we obtain a power-law lower bound for the variance and an inequality between the exponents χ and ξ.     

A property of Pisot numbers and Fourier transforms of self-similar measures

For any Pisot number β it is known that the set F (β)={t:lim n→∞‖tβ n‖= 0} is countable,where a is the distance between a real number a and the set of integers.In this paper it is proved that every member in this set is of the form cβ n,where ‖n‖ is a nonnegative integer and c is determined by a linear system of equations.Furthermore,for some self-similar measures μ associated with β,the limit at infinity of the Fourier transforms lim n→∞μ(tβ n)≠0 if and only if t is in a certain subset of F (β).This generalizes a similar result of Huang and Strichartz.     

Ideals with linear quotients

We study basic properties of monomial ideals with linear quotients. It is shown that if the monomial ideal I has linear quotients, then the squarefree part of I and each component of I as well as mI have linear quotients, where m is the graded maximal ideal of the polynomial ring. As an analogy to the Rearrangement Lemma of Björner and Wachs we also show that for a monomial ideal with linear quotients the admissible order of the generators can be chosen degree increasingly.     

Full residual finiteness growths of nilpotent groups

Full residual finiteness growth of a finitely generated group G measures how efficiently word metric n-balls of G inject into finite quotients of G. We initiate a study of this growth over the class of nilpotent groups. When the last term of the lower central series of G has finite index in the center of G we show that the growth is precisely n^b, where b is the product of the nilpotency class and dimension of G. In the general case, we give a method for finding an upper bound of the form n^b where b is a natural number determined by what we call a terraced filtration of G. Finally, we characterize nilpotent groups for which the word growth and full residual finiteness growth coincide.     

Measuring the interactions among variables of functions over the unit hypercube

By considering a least squares approximation of a given square integrable function f:n[0,1]→R by a multilinear polynomial of a specified degree, we define an index which measures the overall interaction among variables of f. This definition extends the concept of Banzhaf interaction index introduced in cooperative game theory. Our approach is partly inspired from multilinear regression analysis, where interactions among the independent variables are taken into consideration. We show that this interaction index has appealing properties which naturally generalize several properties of the Banzhaf interaction index. In particular, we interpret this index as an expected value of the difference quotients of f or, under certain natural conditions on f, as an expected value of the derivatives of f. Finally, we discuss a few applications of the interaction index in aggregation function theory.     

Some limit theorems for negatively associated random variables

Let {X _n , n ≥ 1} be a sequence of negatively associated random variables. The aim of this paper is to establish some limit theorems of negatively associated sequence, which include the L ^p -convergence theorem and Marcinkiewicz–Zygmund strong law of large numbers. Furthermore, we consider the strong law of sums of order statistics, which are sampled from negatively associated random variables.