Structural Formulas for Orthogonal Matrix Polynomials Satisfying Second-Order Differential Equations,II

We develop structural formulas satisfied by some families of orthogonal matrix polynomials of size 2 × 2 satisfying second-order differential equations with polynomial coefficients. We consider here three one-parametric families of weight matrices, namely,

Structural Formulas for Orthogonal Matrix Polynomials Satisfying Second-Order Differential Equations, I

We develop structural formulas satisfied by some families of orthogonal matrix polynomials of size $2\times 2$ satisfying second-order differential equations with polynomial coefficients. We consider here two one-parametric families of weight matrices, namely \[ H_{a,1}(t)\;=\;e^{-t^2} \left( \begin{array}{@{}cc@{}} 1+\vert a\vert ^2t^2 & at \\bar at & 1 \end{array} \right) \quad {\rm and} \quad H_{a,2}(t)\;=\;e^{-t^2} \left( \begin{array} {@{}cc@{}} 1+\vert a\vert ^2t^4 & at^2 \\bar at^2 & 1 \end{array} \right), \] $a\in \mbox{\bf C} $ and $t\in \mbox{\bf R} $, and their corresponding orthogonal polynomials.     

On sum-product bases

We investigate additive-multiplicative bases in . Let , s>2, and . It is proved that , provided min {|B| ^s/2|A|^(s−2)/2,|A| ^s/2|B|^(s−2)/2}>p ^s/2. This note is supported by “Balaton Program Project” and OTKA grants K 61908, K 67676.     

Impulsive fractional boundary value problem with <Emphasis Type="Italic">p</Emphasis>-Laplace operator

In the present paper, we deal with the existence and multiplicity of solutions for the following impulsive fractional boundary value problem

$$\begin{aligned} {_{t}}D_{T}^{\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t)\right) + a(t)|u(t)|^{p-2}u(t)= & {} f(t,u(t)),\;\;t\ne t_j,\;\;\hbox {a.e.}\;\;t\in [0,T],\\ \Delta \left( {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t_j)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t_j)\right) \right)= & {} I_j(u(t_j))\;\;j=1,2,\ldots ,n,\\ u(0)= & {} u(T) = 0. \end{aligned}$$

where \(\alpha \in (1/p, 1]\), \(1<p<\infty \), \(0 = t_0<t_1< t_2< \cdots< t_n < t_{n+1} = T\), \(f:[0,T]\times \mathbb {R} \rightarrow \mathbb {R}\) and \(I_j : \mathbb {R} \rightarrow \mathbb {R}\), \(j = 1, \ldots , n\), are continuous functions, \(a\in C[0,T]\) and

$$\begin{aligned} \Delta \left( {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t_j)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t_j)\right) \right)= & {} {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u\left( t_j^+\right) \right| ^{p-2}{_{0}}D_{t}^{\alpha }u\left( t_j^+\right) \right) \\&- {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t_j^-)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u\left( t_j^-\right) \right) ,\\ {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u\left( t_j^+\right) \right| ^{p-2}{_{0}}D_{t}^{\alpha }u\left( t_j^+\right) \right)= & {} \lim _{t \rightarrow t_j^+} {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t)\right) ,\\ {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t_j^-)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t_j^-)\right)= & {} \lim _{t\rightarrow t_j^-}{_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t)\right) . \end{aligned}$$

By using variational methods and critical point theory, we give some criteria to guarantee that the above-mentioned impulsive problems have at least one weak solution and a sequences of weak solutions.

     

Two new families of <Emphasis Type="Italic">q</Emphasis>-positive integers

Let n,p and k be three non negative integers. We prove that the apparently rational fractions of q:

Discrete Entropies of Orthogonal Polynomials

Given a nontrivial Borel measure on ℝ, let p _n be the corresponding orthonormal polynomial of degree n whose zeros are λ _j ⁽ⁿ⁾, j=1,…,n. Then for each j=1,…,n,

Sharp estimates for intrinsic ultracontractivity on C 1,α-domains

Let be the first Dirichlet eigenfunction on a connected bounded C ^1,α-domain in and the corresponding Dirichlet heat kernel. It is proved that where λ₂ > λ₁ are the first two Dirichlet eigenvalues. This estimate is sharp for both short and long times. Bounded Lipschitz domains, elliptic operators on manifolds, and a general framework are also discussed. Supported in part by Creative Research Group Fund of the National Foundation of China (no. 10121101), the 973-Project in China and RFDP(20040027009).     

Quantitative Robust Uncertainty Principles and Optimally Sparse Decompositions

In this paper we develop a robust uncertainty principle for finite signals in which states that, for nearly all choices such that

Stably isomorphic dual operator algebras

We prove that two unital dual operator algebras A, B are stably isomorphic if and only if they are Δ-equivalent (Eleftherakis in J Pure Appl Algebra, ArXiv:math. OA/0607489v4, 2007), if and only if they have completely isometric normal representations α,β on Hilbert spaces H, K respectively and there exists a ternary ring of operators such that and This project is cofunded by European Social Fund and National Resources—(EPEAEK II) “Pyhtagoras II” grant No. 70/3/7997.     

Dichotomy laws for the behaviour near the unit element of group representations

A well known “zero-two law" shows that if is a strongly continuous one-parameter group of bounded operators on a Banach space X, and if then Here we discuss analogous problems for general unital representations θ of a topological group G on a unital Banach algebra A. Let 1 be the unit of G, and I the unit element of A. We show that either or if, moreover, θ admits “continuous division by any positive integer”, then, either or Our argument also gives automatic continuity results for representations of abelian Baire groups on a separable Banach algebra and representations of compact non abelian groups on a Banach algebra which are locally bounded and satisfy Received: 8 June 2005; revised: 13 October 2005     

Stability of a quartic functional equation

In this paper, the authors investigate the general solution and generalized Hyers–Ulam stability of the n-dimensional quartic functional equation of the form

$$\begin{aligned} f\left( \sum _{i=1}^{n}x_i\right)&= \sum _{1 \le i<j< k< l\le n} f\left( x_i+x_j+x_k+x_l\right) +\left( -n+4\right) \nonumber \\ {}&\sum _{1 \le i< j< k \le n} f\left( x_i+x_j+x_k\right) +\left( \frac{n^2-7n+12}{2}\right) \sum _{ \begin{array}{c} 1=i;\\ i\ne j \end{array}}^{n} f\left( x_i+x_j\right) \nonumber \\&- \sum _{i=1}^{n} f\left( 2x_i\right) + \left( \frac{-n^3+9n^2-26n+120}{6}\right) \ \ \sum _{i=1}^{n}\left( \frac{f(x_i)+f(-x_i)}{2}\right) \end{aligned}$$

where n is a positive integer with \({\mathbb {N}}- \{0,1,2,3,4\}\). The stability of this quartic functional equation is introduced in Banach space using direct and fixed point methods.

     

Universal co-analytic sets

There is a universal equivalence relation. The existence of a set universal for non-Borel is independent of the usual axioms of mathematics.

     

The twistor structure of the biquaternionic projective point

Souček [1, 2] discovered an intriguing connection between the standard twistor correspondence and the biquaternionic projective line The biquaternionic projective point, also has twistor structure corresponding to the collection of α- or β-planes passing through the origin in spacetime. The duality between α- or β-planes is shown to correspond to the choice of left vs. right scalar action. Moreover, we find that is homeomorphic to the scheme      

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.     

Infinitely many sign-changing solutions of an elliptic problem involving critical Sobolev and Hardy–Sobolev exponent

We study the existence and multiplicity of sign-changing solutions of the following equation

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{lllllllll} -{\Delta} u = \mu |u|^{2^{\star}-2}u+\frac{|u|^{2^{*}(t)-2}u}{|x|^{t}}+a(x)u \quad\text{in}\, {\Omega}, \\ u=0 \quad\text{on}\quad\partial{\Omega}, \end{array}\right. \end{array} $$

where Ω is a bounded domain in \(\mathbb {R}^{N}\), 0∈?Ω, all the principal curvatures of ?Ω at 0 are negative and μ≥0, a>0, N≥7, 0<t<2, \(2^{\star }=\frac {2N}{N-2}\) and \(2^{\star }(t)=\frac {2(N-t)}{N-2}\).

     

A Numerical Method for Constructing the Hyperbolic Structure of Complex Henon Mappings

For complex parameters a,c, we consider the Henon mapping given by and its Julia set, J. In this paper we describe a rigorous computer program for attempting to construct a cone field in the tangent bundle over J, which is preserved by DH, and a continuous norm in which uniformly expands the cones (and their complements). We show a consequence of a successful construction is a proof that H is {hyperbolic} on J. We give several new examples of hyperbolic maps, produced with our computer program, Hypatia, which implements our methods.     

Some remarks on systems of elliptic equations doubly critical in the whole $${\mathbb{R}^N}$$

We study the existence of different types of positive solutions to problem

Strong Polarized Relations for the Continuum

We prove that the strong polarized relation ${\left(\begin{array}{ll} 2^\mu\\ \mu \end{array}\right)\rightarrow \left(\begin{array}{ll} 2^\mu\\ \mu \end{array}\right)^{1,1}_2}$ is consistent with ZFC. We show this for ${\mu = \aleph _0}$ , and for every supercompact cardinal???. We also characterize the polarized relation below the splitting number.     

Continuous dependence on the boundary conditions for the Wentzell Laplacian

The solution u of the well-posed problem

Iterated Logarithm Law for Anticipating Stochastic Differential Equations

We prove a functional law of iterated logarithm for the following kind of anticipating stochastic differential equations