The Similarity Between Finitary and Artinian-Finitary Groups

Let M be a module over the commutative ring R. The finitary automorphism group of M over R is FAut_RM = {g ? Aut_RM :M(g-1) is R-Noetherian}{\rm FAut}_RM =\{g\in{\rm Aut}_RM :M(g-1)\ {\rm is}\ R\hbox{-}{\rm Noetherian}\} and the Artinian-finitary automorphism group of M over R is F₁Aut_RM = {g ? Aut_RM : M(g-1) is R-Artinian}.{\rm F}_1{\rm Aut}_RM = \{g\in{\rm Aut}_RM : M(g-1)\ {\rm is}\ R\hbox{-}{\rm Artinian}\}. We investigate further the surprisingly close relationship between these two types of automorphism groups. Their group theoretic properties seem practically identical.     

Topological dimensions and multifractal Rényi dimensions of Polish spaces: a multifractal Szpilrajn type theorem

In this paper we consider the relationship between the topological dimension and the lower and upper q-Rényi dimensions and of a Polish space X for q ∈ [1, ∞]. Let and denote the Hausdorff dimension and the packing dimension, respectively. We prove that for all analytic metric spaces X (whose upper box dimension is finite) and all q ∈ (1, ∞); of course, trivially, for all q ∈ [1, ∞]. As a corollary to this we obtain the following result relating the topological dimension and the lower and upper q-Rényi dimensions: for all Polish spaces X and all q ∈ [1, ∞]; in (1) and (2) we have used the following notation, namely, for two metric spaces X and Y, we write X ∼ Y if and only if X is homeomorphic to Y. Equality (1) has recently been proved for q = ∞ by Myjak et al. Author’s address: Department of Mathematics, University of St. Andrews, St. Andrews, Fife KY16 9SS, Scotland     

An iterated logarithm law related to decimal and continued fraction expansions

For an irrational number x and n ≥ 1, we denote by k _n (x) the exact number of partial quotients in the continued fraction expansion of x given by the first n decimals of x. G. Lochs proved that for almost all x, with respect to the Lebesgue measure In this paper, we prove that an iterated logarithm law for {k _n (x): n ≥ 1}, more precisely, for almost all x, for some constant σ > 0. Author’s address: Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, P.R. China     

Typical <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>-Dimensions of Measures

For a probability measure μ on a subset of , the lower and upper L^q-dimensions of order are defined by We study the typical behaviour (in the sense of Baire’s category) of the L^q-dimensions and . We prove that a typical measure μ is as irregular as possible: for all q ≥ 1, the lower L^q-dimension attains the smallest possible value and the upper L^q-dimension attains the largest possible value.     

Arithmetic mean of differences of Dedekind sums

Recently, Girstmair and Schoissengeier studied the asymptotic behavior of the arithmetic mean of Dedekind sums , as N → ∞. In this paper we consider the arithmetic mean of weighted differences of Dedekind sums in the form , where is a continuous function with , runs over , the set of Farey fractions of order Q in the unit interval [0,1] and are consecutive elements of . We show that the limit lim _Q→∞ A _h (Q) exists and is independent of h.     

Asymptotic behaviour of certain sets of associated prime ideals of Ext-modules

Let R be a commutative Noetherian ring, be an ideal of R and M be a finitely generated R-module. Melkersson and Schenzel asked whether the set becomes stable for a fixed integer i and sufficiently large j. This paper is concerned with this question. In fact, we prove that if s ≥ 0 and n ≥ 0 such that for all i with i < n, then is finite for all i with i < n, and is finite for all i with i ≤ n, where for a subset T of Spec(R), we set . Also, among other things, we show that if n ≥ 0, R is semi-local and is finite for all i with i < n, then is finite for all i with i ≤ n. K. Khashyarmanesh was partially supported by a grant from Institute for Studies in Theoretical Physics and Mathematics (IPM) Iran (No. 86130027).     

Probability measures on [SIN] groups and some related ideals in group algebras

In 1921, Blichfeldt gave an upper bound on the number of integral points contained in a convex body in terms of the volume of the body. More precisely, he showed that , whenever is a convex body containing n + 1 affinely independent integral points. Here we prove an analogous inequality with respect to the surface area F(K), namely . The proof is based on a slight improvement of Blichfeldt’s bound in the case when K is a non-lattice translate of a lattice polytope, i.e., K = t + P, where and P is an n-dimensional polytope with integral vertices. Then we have . Moreover, in the 3-dimensional case we prove a stronger inequality, namely . Authors’ addresses: Martin Henk, Institut für Algebra und Geometrie, Universit?t Magdeburg, Universit?tsplatz 2, D-39106 Magdeburg, Germany; J?rg M. Wills, Mathematisches Institut, Universit?t Siegen, ENC, D-57068 Siegen, Germany     

Smooth Universal Taylor Series

Let be a simply connected domain in , such that is connected. If g is holomorphic in Ω and every derivative of g extends continuously on , then we write g ∈ A^∞ (Ω). For g ∈ A^∞ (Ω) and we denote . We prove the existence of a function f ∈ A^∞(Ω), such that the following hold:

Automorphism groups of semidirect products

This paper shows that if is a semidirect product of finite groups, then if and only if and for all . As an application, we investigate the automorphism group of a split metacyclic p-group for odd p. The second author is supported by the Natural Science Foundation of China (10671058).     

Dirichlet and Bergman Spaces of Holomorphic Functions on the Unit Ball of ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let B denote the unit ball in ⁿ, n 1, and let and denote the volume measure and gradient with respect to the Bergman metric on B. In the paper we consider the weighted Dirichlet spaces , , and weighted Bergman spaces , , , of holomorphic functions f on B for which and respectively are finite, where and The main result of the paper is the following theorem.Theorem 1. Let f be holomorphic on B and .(a) If for some , then for all p, , with .(b) If for some p, , then for all with . Combining Theorem 1 with previous results of the author we also obtain the following.Theorem 2. Suppose is holomorphic in B. If for some p, , and , then . Conversely, if for some p, , then the series in * converges.     

Book Reviews

For , let E(λ_*, λ^*) be the set It has been proved in [1] and [3] that E(λ_*, λ^*) is an uncountable set. In the present paper, we strengthen this result by showing that where dim denotes the Hausdorff dimension.     

Book Reviews

For , let E(λ_*, λ^*) be the set It has been proved in [1] and [3] that E(λ_*, λ^*) is an uncountable set. In the present paper, we strengthen this result by showing that where dim denotes the Hausdorff dimension.     

Remarks on balance in generalized Tate cohomology

We consider two pairs of complete hereditary cotorsion theories on the category of left R-modules, such that We prove that for any left R-modules M, N and for any n ≧ 1, the generalized Tate cohomology modules can be computed either using a left of M and a left of M or using a right a right of N. Received: 17 December 2004     

Geometric progressions in sumsets over finite fields

Given two sets , the set of d dimensional vectors over the finite field with q elements, we show that the sumset contains a geometric progression of length k of the form vΛ ^j , where j = 0,…, k − 1, with a nonzero vector and a nonsingular d × d matrix Λ whenever . We also consider some modifications of this problem including the question of the existence of elements of sumsets on algebraic varieties.     

Inequalities for irregular Gabor frames

In [C.K. Chui and X.L. Shi, Inequalities of Littlewood-Paley type for frames and wavelets, SIAM J. Math. Anal., 24 (1993), 263–277], the authors proved that if is a Gabor frame for with frame bounds A and B, then the following two inequalities hold: and . In this paper, we show that similar inequalities hold for multi-generated irregular Gabor frames of the form , where Δ _k and Λ _k are arbitrary sequences of points in and , 1 ≤ k ≤ r. Corresponding author for second author Authors’ address: Lili Zang and Wenchang Sun, Department of Mathematics and LPMC, Nankai University, Tianjin 300071, China     

Sums of the error term function in the mean square for ζ(s)

Sums of the form are investigated, where is the error term in the mean square formula for . The emphasis is on the case k = 1, which is more difficult than the corresponding sum for the divisor problem. The analysis requires bounds for the irrationality measure of e^2πm and for the partial quotients in its continued fraction expansion. Authors’ addresses: Y. Bugeaud, Université Louis Pasteur, Mathématiques, 7 rue René Descartes, F-67084 Strasbourg cedex, France; A. Ivić, Katedra Matematike RGF-a, Universitet u Beogradu, Đušina 7, 11000 Beograd, Serbia     

Finding the Best Face on a Voronoi Polyhedron – The Strong Dodecahedral Conjecture Revisited

In this paper we prove the following theorem. The surface area density of a unit ball in any face cone of a Voronoi cell in an arbitrary packing of unit balls of Euclidean 3-space is at most and so the surface area of any Voronoi cell in a packing with unit balls in Euclidean 3-space is at least This result and the ideas of its proof support the Strong Dodecahedral Conjecture according to which the surface area of any Voronoi cell in a packing with unit balls in Euclidean 3-space is at least as large as 16.6508..., the surface area of a regular dodecahedron of inradius 1.The authors were partially supported by the Hung. Nat. Sci. Found (OTKA), grant no. T043556.Supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant.     

Relaxation of variational problems in two-dimensional nonlinear elasticity

Consider a plate occupying in a reference configuration a bounded open set Ω ⊂ ℝ ² , and let be its stored-energy function. In this paper we are concerned with relaxation of variational problems of type:

A policy iteration algorithm for fixed point problems with nonexpansive operators

The aim of this paper is to solve the fixed point problems:

Orbits of Curves on Certain K3 Surfaces

In this paper, we study the family of algebraic K3 surfaces generated by the smooth intersection of a (1, 1) form and a (2, 2) form in defined over and with Picard number 3. We describe the group of automorphisms on V. For an ample divisor D and an arbitrary curve C ₀ on V, we investigate the asymptotic behavior of the quantity . We show that the limit