Strong annihilating pairs for the Fourier-Bessel transform

The aim of this paper is to prove two new uncertainty principles for the Fourier-Bessel transform (or Hankel transform). The first of these results is an extension of a result of Amrein, Berthier and Benedicks, it states that a non-zero function f and its Fourier-Bessel transform Fα(f) cannot both have support of finite measure. The second result states that the supports of f and Fα(f) cannot both be (ε,α)-thin, this extending a result of Shubin, Vakilian and Wolff. As a side result we prove that the dilation of a C₀-function are linearly independent. We also extend Faris's local uncertainty principle to the Fourier-Bessel transform.     

On the infinite divisibility of the product of two г-distributed stochastical variables

The authors give a positive answer to the question “if X_α is г-distributed of order α, and X_β of order β, with X_α and X_β independent, is X_αX_β infinitely divisible?”. This question, posed by Steutel in Ref. 1, has not been answered up to now, so far as they can find in the literature. In addition they show that the distribution of X_αX_β is a generalized г-distribution.     

A lower bound for the complexity of Craig's interpolants in sentential logic

For any sentenceα (in sentential logic) letd _α be the delay complexity of the boolean functionf _α represented byα. We prove that for infinitely manyd (and starting with somed<620) there exist valid implicationsα→β withd _α,d _β≦d such that any Craig's interpolantx has its delay complexityd _χ greater thand+(1/3)·log(d/2). This is the first (non-trivial) known lower bound on the complexity of Craig's interpolants in sentential logic, whose general study may well have an impact on the central problems of computation theory.     

A note on the circuit polynomials and characteristic polynomials of wheels and ladders

Let G be a graph. The circuit polynomial of G is the polynomial Σ Π_αw_α, where w_α is a weight given to the circuit α in G, Π_αw_α is the weight of a spanning subgraph of G, consisting of circuits, nodes and edges, and the summation is taken over all such spanning subgraphs in G. The circuit polynomial of a graph is a generalization of its characteristic polynomial. The characteristic polynomials of wheels and ladders are deduced from their corresponding circuit polynomials.     

Multipliers of the Hölder classes

Following the lines of [13] we introduce the classes of mixed smoothness $\text{L}$α_p(Rⁿ), $\text{L}$α_p(Zⁿ) for a multi-index α = (α₁,…, α_n). Such classes are naturally tied up with the study of semi-elliptic differential and difference equations.Besides a brief presentation of such classes, we concentrate our research on the study of mixed homogeneous multipliers with homogeneity β = (β₁, …, β_n) and their preservation of mixed homogeneous Hölder classes $\text{L}$_α^∞, for a different multi-index α.In the last paragraph we apply the results to produce various improvements of the classical Schauder's estimates, for differential and difference equations, in the parabolic and elliptic case.     

Bounding the sum of powers of the Laplacian eigenvalues of graphs

For a non-zero real number α, let s _α (G) denote the sum of the αth power of the non-zero Laplacian eigenvalues of a graph G. In this paper, we establish a connection between s _α (G) and the first Zagreb index in which the Hölder’s inequality plays a key role. By using this result, we present a lot of bounds of s _α (G) for a connected (molecular) graph G in terms of its number of vertices (atoms) and edges (bonds). We also present other two bounds for s _α (G) in terms of connectivity and chromatic number respectively, which generalize those results of Zhou and Trinajsti? for the Kirchhoff index [B Zhou, N Trinajsti?. A note on Kirchhoff index, Chem. Phys. Lett., 2008, 455: 120–123].     

Two families of functions related to the fractional powers of generators of strongly continuous contraction semigroups

Two families of functions on (0,∞) are related to the theory of fractional powers of generators of strongly continuous semigroups—namely the family (σ_α(·,t))_t>0 of density functions of the one-sided stable semigroup of order α∈(0,1), and a family (e_α(·,μ))_μ>0 of Mittag-Leffler type functions. For the latter family we make this relation visible. We collect some important properties of these functions, and we improve a known result on the Laplace transform of the functions e_α(·,μ) and their derivatives in the sense that we enlarge the domains of the Laplace variable. We furthermore find an expression for the Laplace transform of the functions t?σ_α(x,t), x>0, in terms of the derivative e′_α(·,μ).     

Some new almost sure results on the functional increments of the uniform empirical process

Given an observation of the uniform empirical process α_n, its functional increments α_n(u+a_n⋅)−α_n(u) can be viewed as a single random process, when u is distributed under the Lebesgue measure. We investigate the almost sure limit behaviour of the multivariate versions of these processes as n→∞ and a_n↓0. Under mild conditions on a_n, a convergence in distribution and functional limit laws are established. The proofs rely on a new extension of the usual Poissonisation tools for the local empirical process.     

A local central limit theorem on the Laguerre hypergroup

We consider here the Laguerre hypergroup (K,α_*), where K=[0,+∞[×R and α_* a convolution product on K coming from the product formula satisfied by the Laguerre functions (m∈N, α?0). We set on this hypergroup a local central limit theorem which consists to give a weakly estimate of the asymptotic behavior of the convolution powers μα_*k=μα_*?α_*μ (k times), μ being a given probability measure satisfying some regularity conditions on this hypergroup. It is also given a central local limit theorem for some particular radial probability measures on the (2n+1)-dimensional Heisenberg group Hn.     

Note on the point character of l 1-spaces

We prove that for any ordinal α, any integer t ≥ 0, the point character of the space l ₁(ω _{α + t} ) is no more than ω _α . Combined with an earlier result from [5], this yields that for any infinite cardinal κ the point character of l ₁(κ) is the largest cardinal ω _α ≤ κ where α = 0 or a limit ordinal.     

On perturbation theory for points of discrete spectrum of dissipative operator functions

We consider the operator function L(α, θ) = A(α) ? θR of two complex arguments, where A(α) is an analytic operator function defined in some neighborhood of a real point α ₀ ∈ ? and ranging in the space of bounded operators in a Hilbert space ?. We assume that A(α) is a dissipative operator for real α in a small neighborhood of the point α ₀ and R is a bounded positive operator; moreover, the point α ₀ is a normal eigenvalue of the operator function L(α, θ ₀) for some θ ₀ ∈ ?, and the number θ ₀ is a normal eigenvalue of the operator function L(α ₀ θ). We obtain analogs and generalizations of well-known results for self-adjoint operator functions A(α) on the decomposition of α- and θ-eigenvalues in neighborhoods of the points α ₀ and θ ₀, respectively, and on the representation of the corresponding eigenfunctions by series.     

On the non-Archimedean metric Mahler measure

Recently, Dubickas and Smyth constructed and examined the metric Mahler measure and the metric naïve height on the multiplicative group of algebraic numbers. We give a non-Archimedean version of the metric Mahler measure, denoted M_∞, and prove that M_∞(α)=1 if and only if α is a root of unity. We further show that M_∞ defines a projective height on as a vector space over Q. Finally, we demonstrate how to compute M_∞(α) when α is a surd.     

On the sum ∑k=13 | 2πk − αk |, αk algebraic numbers

If α₁, α₂, α₃ are algebraic numbers satisfying (i) the height of α₁, α₂, α₃ do not exceed H (ii) the degree of the field generated by α₁, α₂, α₃ over the field of rational numbers do not exceed D, then a positive lower bound for

$\text{∑}\text{k=1}\text{3}\text{|2π}{}^{\mathrm{k}}\text{?αk|}$

is determined explicitly (except for an absolute constant) in terms of D and H.     

On a class of polynomials obtained from the circuits in a graph and its application to characteristics polynomials of graphs

Let G be a graph. With each circuit α in G, we can associate a weight w_α, A circuit cover of G is a spanning subgraph of G in which every component is a circuit. With every circuit cover of G, we can associate the monomial Π_αw_α, where the product is taken over all components of the cover. The circuit polynomial of G is ΣΠ_αw_α, where the summation is taken over all circuit covers in G. We show that the characteristics polynomial of G is a special case of its circuit polynomial. Previously obtained and also new results for characteristic polynomials are easily deduced. We also derive the circuit polynomials of various classes of graphs.     

Uniqueness of positive radial solutions for Dirichlet problems on annular domains

This paper deals with the uniqueness of positive radial solutions to Dirichlet problems on annular domains in Rn, n?3. As an application we can obtain the results to equation Δu+up−α₁u−α₀=0, where p>1, α₁?0, α₀?0 and α₁+α₀>0.     

On the power of the zeros of Bessel functions

For ν≥0 let c_νk be the k-th positive zero of the cylinder functionC _v(t)=J _v(t)cosα-Y _v(t)sinα, 0≤α>π whereJ _ν(t) andY _ν(t) denote the Bessel functions of the first and the second kind, respectively. We prove thatC _v,k ^1+H(x) is convex as a function of ν, ifc _νk≥x>0 and ν≥0, whereH(x) is specified in Theorem 1.1.     

On the Best Pinching Constant of Conformal Metrics on \mathbb {S}^{2} with One and Two Conical Singularities

We answer a long-standing open question asked by Thurston (The Geometry and Topology of Three-Manifolds. Princeton University Press, Princeton, 1978) concerning the best pinching constant for conformal metrics on $\mathbb {S}^{2}$ with one and two conical singularities of angles 2π(1+α ₁) and 2π(1+α ₁),2π(1+α ₂) in case α ₁∈(?1,0) and ?1<α ₁<α ₂<0, respectively. The case of one conical singularity is a corollary of a result in Chen and Lin (Commun. Anal. Geom. 6(1):1–19, 1998) concerning the curvature of conformal metrics on ?² with bounded Gaussian curvature 0<a≤K≤b<+∞. The case with two conical singularities is worked out by a generalization of that result.     

A note on the transcendence of infinite products

The paper deals with several criteria for the transcendence of infinite products of the form $\prod\limits_{n = 1}^\infty {[{b_n}{a^{{a_n}}}]/{b_n}{a^{{a_n}}}} $ where α > 1 is a positive algebraic number having a conjugate α* such that α ≠ |α*| > 1, {a _n } _n=1 ^∞ and {b _n } _n=1 ^∞ are two sequences of positive integers with some specific conditions. The proofs are based on the recent theorem of Corvaja and Zannier which relies on the Subspace Theorem (P.Corvaja, U.Zannier: On the rational approximation to the powers of an algebraic number: solution of two problems of Mahler and Mend`es France, Acta Math. 193, (2004), 175–191).     

The joint universality theorem for a pair of Hurwitz zeta functions

In this paper we investigate the joint functional distribution for a pair of Hurwitz zeta functions ζ(s,αj) (j=1,2) in the case that real transcendental numbers α₁ and α₂ satisfy α₂∈Q(α₁). Especially we establish the joint universality theorem for these zeta functions.     

On the spectral height of F-compact spaces

We prove that given an ordinal α with 0 < α ≤ ω ₁ and α ≠ β+1, where β is a limit ordinal, there exists an F-compact space of spectral height α.