Derived subgroups of fixed points

LetA be an elementary abelianq-group acting on a finiteq′-groupG. We show that ifA has rank at least 3, then properties ofC _G(a)′, 1 ≠a ∈A restrict the structure ofG′. In particular, we consider exponent, order, rank and number of generators. This author was supported by the NSF. This author was supported by CNP_q-Brazil.     

Estimates for the Jung constant in Banach lattices

Sharp lower estimates for the Jung constantJ(E) in Banach latticesE satisfying an upperp-estimate and a lowerq-estimate are given. Moreover, the minimal value ofJ(E) with respect to equivalent renormings ofE is calculated inE=L _p,q for finitep andq, as well as in more general spacesE. Finally, a nontrivial estimate for the radiusr _{L p,∞} (A) is obtained forA being a bounded sequence of disjointly supported functions inL _p,∞ .     

The Helmholtz decomposition of weightedLq-spaces for Muckenhoupt weights

We consider weights of Muckenhoupt classA _q, 1<q<∞. For a bounded Lipschitz domain Ω⊂ℝⁿ we prove a compact embedding and a Poincaré inequality in weighted Sobolev spaces. These technical tools allow us to solve the weak Neumann problem for the Laplace equation in weighted spaces on ℝⁿ, ℝⁿ ₊, on bounded and on exterior domains Ω with boundary of classC ¹, which will yield the Helmholtz decomposition ofL _ω ^q(Ω)ⁿ for general ω∈A _q. This is done by transferring the method of Simader and Sohr [4] to the weighted case. Our result generalizes a result of Farwig and Sohr [2] where the Helmholtz decomposition ofL _ω ^p(Ω)ⁿ is proved for an exterior domain and weights of Muckenhoupt class without singularities or degeneracies in a neighbourhood of ϖΩ.

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Sunto In questo lavoro consideriamo dei pesi della classe di MuckenhouptA _q, 1<q<∞. Per un dominio limitato lipschitziano Ω⊂ℝⁿ, dimostriamo una immersione compatta ed una disuguaglianza di Poincaré in spazi di Sobolev con peso. Questa tecnica ci consente di risolvere il problema debole di Neumann per l’equazione di Laplace in spazi pesati in ℝⁿ, ℝⁿ ₊ in domini limitati ed in domini esterni con frontiera di classeC ¹, che conduce alla decomposizione di Helmholtz diL _ω ^q(Ω)ⁿ per un qualsiasi ω∈A _q. Il risultato è ottenuto trasferendo il metodo di Simader e Sohr [4] al caso pesato. Quello qui presente estende un risultato di Farwig e Sohr [2] dove la decomposizione di Helmholtz diL _ω ^q(Ω)ⁿ è dimostrata per domini esterni e pesi della classe di Muckenhoupt privi di singolarità in un intorno di ϖΩ.

     

Empirical multifractal moment measures of self-similar measure for q < 0*

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.     

A generalization of a result of R. Lyons about measures on [0, 1)

Letμ be a probability measure on [0, 1), invariant underS:x ↦px mod 1, and for which almost every ergodic component has positive entropy. Ifq is a real number greater than 1 for which logq/ logp is irrational, andT _n sendsx toq ⁿx mod 1, then for any ε>0 the measureμT _n ⁻¹ will — for a set ofn of positive lower density — be within ε of Lebesgue measure.     

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     

On the Carleson Measure Criterion in Linear Systems Theory

In Ho and Russell (SIAM J Control Optim 21(4):614–640, 1983), and Weiss (Syst Control Lett 10(1): 79–82, 1988), a Carleson measure criterion for admissibility of one-dimensional input elements with respect to diagonal semigroups is given. We extend their results from the Hilbert space situation (L ²-admissibility on the state space ℓ ₂) to the more general situation of L ^p -admissibility on the state space ℓ _q . For analytic diagonal semigroups we present a new result that does not rely on Laplace transform methods. A comparison of both criteria leads to a result on L ^p -admissibility for reciprocal systems in the sense of Curtain (Syst Control Lett 49(2):81–89, 2003).     

The Vector-Valued Big <Emphasis Type="Italic">q</Emphasis>-Jacobi Transform

Big q-Jacobi functions are eigenfunctions of a second-order q-difference operator L. We study L as an unbounded self-adjoint operator on an L ²-space of functions on ℝ with a discrete measure. We describe explicitly the spectral decomposition of L using an integral transform ℱ with two different big q-Jacobi functions as a kernel, and we construct the inverse of ℱ.      

On the Frattini normal embeddability of products ofp-groups

ItH _i is a finite non-abelianp-group with center of orderp, for 1≦j≦R, then the direct product of theH _i does not occur as a normal subgroup contained in the Frattini subgroup of any finitep-group. If the Frattini subgroup Φ of a finitep-groupG is cyclic or elementary abelian of orderp ², then the centralizer of Φ inG properly contains Φ. Non-embeddability properties of products of groups of order 16 are established.     

Models with a Kronecker product covariance structure: Estimation and testing

In this article we consider a pq-dimensional random vector x distributed normally with mean vector θ and covariance matrix Λ assumed to be positive definite. On the basis of N independent observations on the random vector x, we want to estimate parameters and test the hypothesis H: Λ = Ψ ⊗ Σ, where Ψ = (ψ _ij ): q × q, ψ _qq = 1, and Σ = (σ _ij ): p × p, and Λ = (ψ _ij Σ), the Kronecker product of Ψ and Σ. That is instead of 1/2pq(pq + 1) parameters, it has only 1/2p(p + 1) + 1/2q(q + 1) − 1 parameters. A test based on the likelihood ratio is given to check if this model holds. And, when this model holds, we test the hypothesis that Ψ is a matrix with intraclass correlation structure. The maximum likelihood estimators (MLE) are obtained under the hypothesis as well as under the alternatives. Using these estimators the likelihood ratio tests (LRT) are obtained. One of the main objects of the paper is to show that the likelihood equations provide unique estimators.      

On a Furstenberg-Katznelson-Weiss Type Theorem over Finite Fields

Using Fourier analysis, Covert, Hart, Iosevich, and Uriarte-Tuero (2008) showed that if the cardinality of a subset of the 2-dimensional vector space over a finite field with q elements is ≥ ρq ², with q^-1/2 << r ￡ 1{q^{-1/2} \ll \rho \leq 1} then it contains an isometric copy of ≥ cρq ³ triangles. In this note, we give a graph theoretic proof of this result.     

On the singularities of foliations and of vector bundle maps

LetF be a (smooth) Γ _q ^∞ -stucture (often called a codimension-q Haefliger structure) on a compact manifoldX ⁿ . Cohomological invariants associated to the singularities ofF are defined whose vanishing is shown to be a necessary condition for deformingF to a codimension-q foliation onX ⁿ . An analagous approach to vector bundle maps is then utilized to prove a general theorem concerning the possibility of embedding a vector bundle in the tangent bundle ofX ⁿ , and applications to the planefield problem are given. In the final section geometric realizations of the singularity classes associated toF are constructed.     

Projecting the one-dimensional Sierpinski gasket

LetS⊂ℝ² be the Cantor set consisting of points (x,y) which have an expansion in negative powers of 3 using digits {(0,0), (1,0), (0,1)}. We show that the projection ofS in any irrational direction has Lebesgue measure 0. The projection in a rational directionp/q has Hausdorff dimension less than 1 unlessp+q ≡ 0 mod 3, in which case the projection has nonempty interior and measure 1/q. We compute bounds on the dimension of the projection for certain sequences of rational directions, and exhibit a residual set of directions for which the projection has dimension 1. This work was partially completed while the author was at the Institut Fourier, Grenoble, France.     

Fluxes Across Parts of Fractal Boundaries

This paper determines the flux F (q,T) of a physical quantity across a part T of the boundary of a ‘rough body.’ The latter term means that the (measure theoretic) boundary ∂B of the body B is fractal in the sense that the outer normal n to B is not defined for almost every point of ∂B with respect to the n − 1 dimensional Hausdorff measure . The quantity is represented by a bounded measurable flux vectorfield q on with bounded distributional divergence. Cauchy’s formula for regular surfaces T,

Identities involving Farey fractions

The rational numbers a/q in [0, 1] can be counted by increasing height H(a/q) = max(a, q), or ordered as real numbers. Franel’s identity shows that the Riemann hypothesis is equivalent to a strong bound for a measure of the independence of these two orderings. We give a proof using Dedekind sums that allows weights w(q). Taking w(q) = χ(q) we find an extension to Dirichlet L-functions.     

Gaussian partitions

Let V=V(n,q) denote the finite vector space of dimension n over the finite field with q elements. A subspace partition of V is a collection Π of subspaces of V such that each 1-dimensional subspace of V is in exactly one subspace of Π. In a recent paper, we proved some strong connections between the lattice of the subspace partitions of V and the lattice of the set partitions of n={1,…,n}. We now define a Gaussian partition of [n] _q =(q ⁿ −1)/(q−1) to be a nonincreasing sequence of positive integers formed by ordering all elements of some multiset {dim(W):W∈Π}, where Π is a subspace partition of V. The Gaussian partition function gp(n,q) is then the number of all Gaussian partitions of [n] _q , and is naturally analogous to the classical partition function p(n). In this paper, we initiate the study of gp(n,q) by exhibiting all Gaussian partitions for small n. In particular, we determine gp(n,q) as a polynomial in q for n≤5, and find a lower bound for gp(6,q).     

Jordan decompositions in alternative loop rings

It is observed that the additive as well as multiplicative Jordan decompositions hold in alternative loop algebras of finiteRA loops and theRA loops for which the additive Jordan decomposition holds in the integral loop ring are characterized. Multiplicative Jordan decomposition (MJD) inZL, whereL is a finiteRA loop with cyclic centre is analysed, besides settling MJD for integral loop rings of allRA loops of order ≤32. It is also shown that for any finiteRA loopL,U (ZL) is an almost splittable Moufang loop. Research of the second author is supported by CSIR.     

The Structure of the Closure of the Rational Functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>(μ)

Let K be a compact subset in the complex plane and let A(K) be the uniform closure of the functions continuous on K and analytic on K°. Let μ be a positive finite measure with its support contained in K. For 1 ≤ q < ∞, let A^q(K, μ) denote the closure of A(K) in L^q(μ). The aim of this work is to study the structure of the space A^q(K, μ). We seek a necessary and sufficient condition on K so that a Thomson-type structure theorem for A^q(K, μ) can be established. Our theorem deduces J. Thomson’s structure theorem for P^q(μ), the closure of polynomials in L^q(μ), as the special case when K is a closed disk containing the support of μ.     

Irreducible subrepresentations of the conjugation representation of finitep-groups

For finitep-groupsG we study the conjugation representation γG which is defined by lettingG act on itself by conjugation. Roth conjectured that every irreducible representation ofG which is trivial on the center ofG, occurs in γG. However, this is not true in general. We construct minimal counterexamples and verify Roth's conjecture for various classes of finitep-groups, for instance those of maximal class. Dedicated to Professor George Maltese on the occasion of his 60th birthday     

Computer search in projective planes for the sizes of complete arcs

The spectrum of possible sizes k of complete k-arcs in finite projective planes PG(2, q) is investigated by computer search. Backtracking algorithms that try to construct complete arcs joining the orbits of some subgroup of collineation group PΓ L (3, q) and randomized greedy algorithms are applied. New upper bounds on the smallest size of a complete arc are given for q = 41, 43, 47, 49, 53, 59, 64, 71 ≤ q ≤ 809, q ≠ 529, 625, 729, and q = 821. New lower bounds on the second largest size of a complete arc are given for q = 31, 41, 43, 47, 53, 125. Also, many new sizes of complete arcs are obtained for 31 ≤ q ≤ 167.