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.     

A Lower Bound for the Height of a Rational Function at <Emphasis Type="Italic">S</Emphasis>-unit Points

Let a,b be given, multiplicatively independent positive integers and let >0. In a recent paper jointly with Y. Bugeaud we proved the upper bound exp(n) for g.c.d.(aⁿ–1, bⁿ–1); shortly afterwards we generalized this to the estimate g.c.d.(u–1,v–1)v) for multiplicatively independent S-units u,vZ. In a subsequent analysis of those results it turned out that a perhaps better formulation of them may be obtained in terms of the language of heights of algebraic numbers. In fact, the purposes of the present paper are: to generalize the upper bound for the g.c.d. to pairs of rational functions other than {u–1,v–1} and to extend the results to the realm of algebraic numbers, giving at the same time a new formulation of the bounds in terms of height functions and algebraic subgroups of G_m².     

The Exceptional Compact Symmetric Spaces <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript> and <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">SO</Emphasis>(4) as Tubes

In the present paper we discuss in detail the cohomogeneity one isometric actions of the Lie groups SU(3) × SU(3) and SU(3) on the exceptional compact symmetric spaces G₂ and G₂/SO(4), respectively. We show that the principal orbits coincide with the tubular hypersurfaces around the totally geodesic singular orbits, and the symmetric spaces G₂ and G₂/SO(4) can be thought of as compact tubes around SU(3) and P², respectively. Moreover, we determine the radii of these tubes and describe the shape operators of the principal orbits. Finally, we apply these results to compute the volumes of the two symmetric spaces.The author was partially supported by the Hungarian National Science and Research Foundation OTKA T032478.     

Empirical multifractal moment measures of self-similar measure for <Emphasis Type="Italic">q</Emphasis> < 0<Superscript>*</Superscript>

For q ≥ 0, Olsen [1] has attained the exact rate of convergence of the L ^q -spectrum of a self-similar measure and showed that the so-called empirical multifractal moment measures converges weakly to the normalized multifractal measures. Unfortunately, nothing is known for q < 0. Indeed, the problem of analysing the L ^q - spectrum for q < 0 is generally considered significantly more difficult since the L ^q -spectrum is extremely sensitive to small variations of μ for q < 0. In [2] we showed that self-similar measures satisfying the Open Set Condition (OSC) are Ahlfors regular and, using this fact, we obtained the exact rate of convergence of the L ^q -spectrum of a self-similar measure satisfying the OSC for q < 0. In this paper, we apply the results from [2] to show the empirical multifractal q’th moment measures of self-similar measures satisfying the OSC converges weakly to the normalized multifractal Hausdorff measures for q < 0. Authors’ addresses: Jiaqing Xiao, School of Science, Wuhan University of Technology, Wuhan 430070, China; Wu Min, School of Mathematical Sciences, South China University of Technology, Guangzhou, 510640, China     

A lower bound for the <Emphasis Type="Italic">r</Emphasis>-order of a matrix modulo <Emphasis Type="Italic">N</Emphasis>

For a positive integer N, we define the N-rank of a non singular integer d × d matrix A to be the maximum integer r such that there exists a minor of order r whose determinant is not divisible by N. Given a positive integer r, we study the growth of the minimum integer k, such that A ^k − I has N-rank at most r, as a function of N. We show that this integer k goes to infinity faster than log N if and only if for every eigenvalue λ which is not a root of unity, the sum of the dimensions of the eigenspaces relative to eigenvalues which are multiplicatively dependent with λ and are not roots of unity, plus the dimensions of the eigenspaces relative to eigenvalues which are roots of unity, does not exceed d − r − 1. This result will be applied to recover a recent theorem of Luca and Shparlinski [6] which states that the group of rational points of an ordinary elliptic curve E over a finite field with q ⁿ elements is almost cyclic, in a sense to be defined, when n goes to infinity. We will also extend this result to the product of two elliptic curves over a finite field and show that the orders of the groups of rational points of two non isogenous elliptic curves are almost coprime when n approaches infinity. Author’s address: Dipartimento di Matematica e Informatica, Via Delle Scienze 206, 33100 Udine, Italy     

On the Theory of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>)-Banach Lattices

Given 1≤ p,q < ∞, let BL_pL_q be the class of all Banach lattices X such that X is isometrically lattice isomorphic to a band in some L_p(L_q)-Banach lattice. We show that the range of a positive contractive projection on any BL_pL_q-Banach lattice is itself in BL_pL_q. It is a consequence of this theorem and previous results that BL_pL_q is first-order axiomatizable in the language of Banach lattices. By studying the pavings of arbitrary BL_pL_q-Banach lattices by finite dimensional sublattices that are themselves in this class, we give an explicit set of axioms for BL_pL_q. We also consider the class of all sublattices of L_p(L_q)-Banach lattices; for this class (when p/q is not an integer) we give a set of axioms that are similar to Krivine’s well-known axioms for the subspaces of L_p-Banach spaces (when p/2 is not an integer). We also extend this result to the limiting case q = ∞.     

A Short Remark on Weierstrass Points at Infinity on <Emphasis Type="Italic">X</Emphasis><Subscript>0</Subscript>(<Emphasis Type="Italic">N</Emphasis>)

We point out that the formalism of the trace map and reduction modulo p can be used to give a short proof for the fact first proved by Ogg that is not a Weierstrass point on X₀(pM) where p is a prime not dividing M and the genus of X₀(M) is zero.     

Very Ampleness of <Emphasis Type="Italic">d</Emphasis>-standard Classes on Rational Surfaces

We consider a rational surface X_r, obtained by blowing up ² along a curvilinear zero-dimensional subscheme of length r of the regular locus of a reduced irreducible plane curve of degree d, with d 4; and we give sufficient conditions for d-standard classes to be very ample (resp. base point free or non special) on such a rational surface X_r.Postdoctoral Fellow of the Fund for Scientific Research-Flanders (Belgium).     

On the <Emphasis Type="Italic">DOTU</Emphasis>-Factorization Problem in Dimension Four

 We prove that any basis of a non-degenerate 4-dimensional lattice with sufficiently small (positive) homogeneous minimum can be represented in the form DOTU. This is of interest in connection with Minkowski’s conjecture about the product of inhomogeneous linear forms. Received 23 September 2001 RID="a" ID="a" Dedicated to Prof. Edmund Hlawka on the occasion of his 85th birthday     

<Emphasis Type="Italic">CR</Emphasis>-Warped Products in Complex Projective Spaces with Compact Holomorphic Factor

A submanifold of a Kaehler manifold is called a CR-warped product if it is the warped product N_T ×_fN of a complex submanifold N_T and a totally real submanifold N. There exist many CR-warped products N_T ×_fN in CP^h+p, h = dim_CN_T and p = dim_RN (see [5, 6]). In contrast, we prove in this article that the situation is quite different if the holomorphic factor N_T is compact. For such CR-wraped products in CP^m (4), we prove the following: (1) The complex dimension m of the ambient space is at least h + p + hp. (2) If m = h + p + hp, then N_T is CP^h(4). We also obtain two geometric inequalities for CR-warped products in CP^m with compact N_T.     

The Dichotomy Problem for Orbital Measures of <Emphasis Type="Italic">SU</Emphasis>(<Emphasis Type="Italic">n</Emphasis>)

For many orbital measures μ, on SU(n), we show that either μ^k ∈ L² or μ^k is singular to L¹. The least k for which μ^k ∈ L² is determined and is shown to be the minimum k for which the k-fold product of the conjugacy class supporting the measure has positive measure. It would be interesting to know if all orbital measures satisfy this dichotomy.     

(Almost) primitivity of Hecke <Emphasis Type="Italic">L</Emphasis>-functions

Let f be a cusp form of the Hecke space and let L _f be the normalized L-function associated to f. Recently it has been proved that L _f belongs to an axiomatically defined class of functions . We prove that when λ ≤ 2, L _f is always almost primitive, i.e., that if L _f is written as product of functions in , then one factor, at least, has degree zeros and hence is a Dirichlet polynomial. Moreover, we prove that if then L _f is also primitive, i.e., that if L _f = F ₁ F ₂ then F ₁ (or F ₂) is constant; for the factorization of non-primitive functions is studied and examples of non-primitive functions are given. At last, the subset of functions f for which L _f belongs to the more familiar extended Selberg class is characterized and for these functions we obtain analogous conclusions about their (almost) primitivity in .     

Zeta functions of<Emphasis Type="Italic">q</Emphasis>-perfect numbers

We introduce a notion ofq-analogue of the perfect numbers. We also define a new zeta function which we call a zeta function ofq-perfect numbers. In this paper, the properties of theq-perfect numbers and the zeta functions are studied. Especially, we determine theq-perfect numbers whenq is a root of unity.     

On products of <Emphasis Type="Italic">k</Emphasis> atoms

Let H be an atomic monoid. For let denote the set of all with the following property: There exist atoms (irreducible elements) u ₁, …, u _k , v ₁, …, v _m ∈ H with u ₁· … · u _k = v ₁ · … · v _m . We show that for a large class of noetherian domains satisfying some natural finiteness conditions, the sets are almost arithmetical progressions. Suppose that H is a Krull monoid with finite cyclic class group G such that every class contains a prime (this includes the multiplicative monoids of rings of integers of algebraic number fields). We show that, for every , max which settles Problem 38 in [4]. Authors’ addresses: W. Gao, Center for Combinatorics, Nankai University, Tianjin 300071, P.R. China; A. Geroldinger, Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universit?t Graz, Heinrichstra?e 36, 8010 Graz, Austria     

Line partitions of internal points to a conic in <Emphasis Type="Italic">PG</Emphasis>(2,<Emphasis Type="Italic">q</Emphasis>)

All sets of lines providing a partition of the set of internal points to a conic C in PG(2,q), q odd, are determined. There exist only three such linesets up to projectivities, namely the set of all non-tangent lines to C through an external point to C, the set of all non-tangent lines to C through a point in C, and, for square q, the set of all non-tangent lines to C belonging to a Baer subplane PG(2,√q) with √q+1 common points with C. This classification theorem is the analogous of a classical result by Segre and Korchmáros [9] characterizing the pencil of lines through an internal point to C as the unique set of lines, up to projectivities, which provides a partition of the set of all non-internal points to C. However, the proof is not analogous, since it does not rely on the famous Lemma of Tangents of Segre which was the main ingredient in [9]. The main tools in the present paper are certain partitions in conics of the set of all internal points to C, together with some recent combinatorial characterizations of blocking sets of non-secant lines, see [2], and of blocking sets of external lines, see [1].     

A study of (<Emphasis Type="Italic">x</Emphasis>(<Emphasis Type="Italic">q</Emphasis> + 1), <Emphasis Type="Italic">x</Emphasis>; 2, <Emphasis Type="Italic">q</Emphasis>)-minihypers

In this paper, we study the weighted (x(q + 1), x; 2, q)-minihypers. These are weighted sets of x(q + 1) points in PG(2, q) intersecting every line in at least x points. We investigate the decomposability of these minihypers, and define a switching construction which associates to an (x(q + 1), x; 2, q)-minihyper, with x ≤ q ² − q, not decomposable in the sum of another minihyper and a line, a (j(q + 1), j; 2, q)-minihyper, where j = q ² − q − x, again not decomposable into the sum of another minihyper and a line. We also characterize particular (x(q + 1), x; 2, q)-minihypers, and give new examples. Additionally, we show that (x(q + 1), x; 2, q)-minihypers can be described as rational sums of lines. In this way, this work continues the research on (x(q + 1), x; 2, q)-minihypers by Hill and Ward (Des Codes Cryptogr 44:169–196, 2007), giving further results on these minihypers.     

Mappings of Ruled Surfaces in Simply Isotropic Space <Emphasis Type="Italic">I</Emphasis><Subscript>3</Subscript><Superscript>1</Superscript> that Preserve the Generators

In this paper we study some mappings of skew ruled surfaces in simply isotropic space which preserve the generators. We study isometries, conformal mappings and mappings which preserve the area. Furthermore, we study mappings of surfaces in I ₃ ¹ which preserve the asymptotic lines.Received December 18, 2001; in revised form July 12, 2002 Published online April 4, 2003     

The Number of Linearly Independent Binary Vectors with Applications to the Construction of Hypercubes and Orthogonal Arrays,Pseudo (<Emphasis Type="Italic">t</Emphasis>, <Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">s</Emphasis>)-Nets and Linear Codes

We study formulae to count the number of binary vectors of length n that are linearly independent k at a time where n and k are given positive integers with 1kn. Applications are given to the design of hypercubes and orthogonal arrays, pseudo (t, m, s)-nets and linear codes.This revised version was published online in October 2004 with a corrected Received date.     

Irreducible <Emphasis Type="Italic">p</Emphasis>-Brauer characters of <Emphasis Type="Italic">p</Emphasis>-power degree for the symmetric and alternating groups

Starting from the question when all irreducible p-Brauer characters for a symmetric or an alternating group are of p-power degree, we classify the p-modular irreducible representations of p-power dimension in some families of representations for these groups. In particular, this then allows to confirm a conjecture by W. Willems for the alternating groups. Received: 14 June 2006     

Pointwise convergence for semigroups in vector-valued <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

Suppose that {T _t  : t  ≥  0} is a symmetric diffusion semigroup on L ²(X) and denote by its tensor product extension to the Bochner space , where belongs to a certain broad class of UMD spaces. We prove a vector-valued version of the Hopf–Dunford–Schwartz ergodic theorem and show that this extends to a maximal theorem for analytic continuations of on . As an application, we show that such continuations exhibit pointwise convergence.