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.     

On the number of generators needed for free profinite products of finite groups

We provide lower estimates on the minimal number of generators of the profinite completion of free products of finite groups. In particular, we show that if C ₁, ..., C _n are finite cyclic groups then there exists a finite group G which is generated by isomorphic copies of C ₁, ..., C _n and the minimal number of generators of G is n. The first author’s research is partially supported by NSF grant DMS-0701105. The second author’s research is partially supported by OTKA grant T38059 and the Magyary Zoltán Postdoctoral Fellowship.     

On the value of the critical point in fractal percolation

We derive a new lower bound p_c>0.8107 for the critical value of Mandelbrot's dyadic fractal percolation model. This is achieved by taking the random fractal set (to be denoted A_∞) and adding to it a countable number of straight line segments, chosen in a certain (nonrandom) way as to simplify greatly the connectivity structure. We denote the modified model thus obtained by C_∞, and write C_n for the set formed after n steps in its construction. Now it is possible, using an iterative technique, to compute the probability of percolating through C_n for any parameter value p and any finite n. For p=0.8107 and n=360 we obtain a value less than 10^?5; using some topological arguments it follows that 0.8107 is subcritical for C_∞ and hence (since _∞ dominates A_∞) for A_∞. ©2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 332–345, 2001.     

Farrell cohomology and Brown theorems for profinite groups

LetG be a profinite group which has an open subgroupH such that the cohomologicalp-dimensiond≔cd_p(H) is finite (p is a fixed prime). The main result of this paper expresses thep-primary part of high degree cohomology ofG in terms of the elementary abelianp-subgroups ofG: From the latter one constructs a natural profinite simplicial setA G, on whichG acts by conjugation. ThenH ⁿ(G,M)≅H _G ⁿ (AG,M) holds forn≧d+r and everyp-primary discreteG-moduleM (r≔p-rank ofG). If one uses profinite Farrell cohomology, which is introduced in this paper, the analogous fact holds in all degrees. These results are the profinite analogues of theorems by K.S. Brown for discrete groups.     

Integral Lie powers of a lattice for the cyclic group of order 2

Let L_n denote the n-th homogeneous component of the free Lie ring L(W) on a given \Bbb ZC₂{{\Bbb Z}}C_{2}-lattice W. This paper gives explicit formulae for the multiplicities of the three indecomposable \Bbb ZC₂{{\Bbb Z}}C_{2}-lattices in a Krull-Schmidt decomposition of L_n. In the case where W is a free \Bbb ZC₂{{\Bbb Z}}C_{2}-lattice, L_n is shown to have no non-zero direct summand on which C₂ acts trivially - this extends a result of R. M. Bryant for the special case where W is the regular \Bbb ZC₂{{\Bbb Z}}C_{2}-lattice. As an application, the structure of the higher dimensional modules associated to a non-cyclic free presentation of C₂ is determined.     

Iterating Brownian Motions,Ad Libitum

Let B ₁,B ₂,… be independent one-dimensional Brownian motions parameterized by the whole real line such that B _i (0)=0 for every i≥1. We consider the nth iterated Brownian motion W _n (t)=B _n (B _n?1(?(B ₂(B ₁(t)))?)). Although the sequence of processes (W _n ) _n≥1 does not converge in a functional sense, we prove that the finite-dimensional marginals converge. As a consequence, we deduce that the random occupation measures of W _n converge to a random probability measure μ _∞. We then prove that μ _∞ almost surely has a continuous density which should be thought of as the local time process of the infinite iteration W _∞ of independent Brownian motions. We also prove that the collection of random variables (W _∞(t),t∈??{0}) is exchangeable with directing measure μ _∞.     

Sharpening of the estimates for relative widths of classes of differentiable functions

We improve the earlier obtained upper estimates for the least value of the coefficient M for which the Kolmogorov widths d _n (W _C ^r , C) of the function class W _C ^r are equal to the relative widths K _n (W _C ^r , MW _C ^j , C) of the class W _C ^r with respect to the class MW _C ^j , j < r.     

Identifiability at the boundary for first-order terms

Let Ω be a domain in R ⁿ whose boundary is C ¹ if n≥3 or C ^1,β if n=2. We consider a magnetic Schrödinger operator L _{W , q} in Ω and show how to recover the boundary values of the tangential component of the vector potential W from the Dirichlet to Neumann map for L _{W , q} . We also consider a steady state heat equation with convection term Δ+2W·? and recover the boundary values of the convection term W from the Dirichlet to Neumann map. Our method is constructive and gives a stability result at the boundary.     

Writing Commutators of Commutators as Products of Cubes in Groups

We study the cube length of certain elements of the derived subgroup of a group G. By the cube length Cu(γ) of an element γ of a group G, we mean the least natural number k such that γ is a products of k cubes. We find an upper bound for the cube length of a commutator of commutators. If W = F?C _∞ is the wreath product of a free group F by the infinite cyclic group, we show that every element of W″ is a product of at most three cubes in W.     

A direct proof of Gromov’s theorem

A new proof of a theorem by Gromov is given: for any positive C and any integer n greater than 1, there exists a function Δ _C,n (δ) such that if the Gromov–Hausdorff distance between two complete Riemannian n-manifolds V and W is at most δ, their sectional curvaturcs |K _σ | do not exceed C, and their injectivity radii are at least 1/C, than the Lipschitz distance between V and W is less than Δ _C,n (δ), and Δ _C,n (δ) → 0 as δ → 0. Bibliography: 6 titles.     

On relative widths of classes of differentiable functions. II

We obtain an upper bound for the least value of the factor M for which the Kolmogorov widths d _n (W _C ^r , C) are equal to the relative widths K _n (W _C ^r , MW _C ^j , C) of the class of functions W _C ^r with respect to the class MW _C ^j , provided that j > r. This estimate is also true in the case where the space L is considered instead of C.     

Rate of Convergence of Chlodowsky-Type Bernstein Operators for Functions of Bounded Variation

In the current paper, we estimate the rate of pointwise convergence of the Chlodowsky-type Bernstein operators C _n (f,x) for functions defined on the interval [0,b _n ], for b _n  → ∞ as n → ∞, which are of bounded variation on [0, ∞). To prove our main result, we have used some methods and techniques from probability theory.     

On some subclasses of well-covered graphs

A set of points in a graph is independent if no two points in the set are adjacent. A graph is well covered if every maximal independent set is a maximum independent set or, equivalently, if every independent set is contained in a maximum independent set. The well-covered graphs are classified by the W_n property: For a positive integer n, a graph G belongs to class W_n if ≥ n and any n disjoint independent sets are contained in n disjoint maximum independent sets. Constructions are presented that show how to build infinite families of W_n graphs containing arbitrarily large independent sets. A characterization of W_n graphs in terms of well-covered subgraphs is given, as well as bounds for the size of a maximum independent set and the minimum and maximum degrees of points in W_n graphs.     

Type II Matrices of Size Five

. A type II matrix is an n×n matrix W with non-zero entries W _i,j which satisfies , i, j=1, …, n. Two type II matrices W, W′ are said to be equivalent if W′=P ₁Δ₁ WΔ₂ P ₂ holds for some permutation matrices P ₁, P ₂ and for some non-singular diagonal matrices Δ₁, Δ₂. In the present paper, it is shown that there are up to equivalence exactly three type II matrices in M ₅(C). Received: August 15, 1996 Revised: May 16, 1997     

Normal subgroups of the algebraic fundamental group of affine curves in positive characteristic

Let π₁(C) be the algebraic fundamental group of a smooth connected affine curve, defined over an algebraically closed field of characteristic p > 0 of countable cardinality. Let N be a normal (respectively, characteristic) subgroup of π ₁(C). Under the hypothesis that the quotient π ₁(C)/N admits an infinitely generated Sylow p-subgroup, we prove that N is indeed isomorphic to a normal (respectively, characteristic) subgroup of a free profinite group of countable cardinality. As a consequence, every proper open subgroup of N is a free profinite group of countable cardinality. Amílcar Pacheco and Pavel Zalesskii were partially supported by CNPq research grants 305731/2006-8 and 307823/2006-7, respectively. They were also supported by Edital Universal CNPq 471431/2006-0.     

Maximality properties for isometric interpolating sequences and sequences of trivial points in the spectrum of H∞

Let (x_n) be an isometric interpolating sequence or a sequence of trivial points in the spectrum of H^∞. It is shown that either every cluster point of that sequence has a maximal support set or there exists y ∈ M(H^∞+C) such that the support of x_n is contained in the support of y for infinitely many n. Similar results for Gleason parts are obtained, too. We also investigate the H^∞‐convex hulls of countable unions of support sets and show that whenever supp x ? supp y and x /∈ , then the H^∞‐convex hull of supp x does not meet . (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Projective normality of Weyl group quotients

In this note, we prove that for the standard representation Vof the Weyl group W of a semi-simple algebraic group of type A _n , B _n , C _n , D _n , F ₄ and G ₂ over \mathbbC\mathbb{C}, the projective variety \mathbbP(V^m)/W\mathbb{P}(V^m)/W is projectively normal with respect to the descent of O(1)^?|W|\mathcal{O}(1)^{\otimes |W|}, where V ^m denote the direct sum of m copies of V.     

The Ramsey numbers for cycles versus wheels of odd order

For two given graphs G₁ and G₂, the Ramsey number R(G₁,G₂) is the smallest integer n such that for any graph G of order n, either G contains G₁ or the complement of G contains G₂. Let C_n denote a cycle of order n and W_m a wheel of order m+1. It is conjectured by Surahmat, E.T. Baskoro and I. Tomescu that R(C_n,W_m)=2n−1 for even m≥4, n≥m and (n,m)≠(4,4). In this paper, we confirm the conjecture for n≥3m/2+1.     

Holomorphic Dependence of the Heins Map

Let D be a bounded convex domain and Hol _c (D,D) the set of holomorphic maps from D to C ⁿ with image relatively compact in D. Consider Hol _c (D,D) as a open set in the complex Banach space H ^∞ _n (D) of bounded holomorphic maps from D to C ⁿ . We show that the map τ: Hol _c (D,D) → D (called the Heins map for D equals to the unit disc of C) which associates to ? ∈ Hol _c (D,D) its unique fixed point τ? ∈ D is holomorphic and its differential is given by dτ_?(v) = (Id-df_τ(?))^?1 v(τ(?)) for v ∈ H ^∞ _n (D).