Sharp Observability Inequalities for the 1-D Plate Equation with a Potential

This paper deals with the problem of sharp observability inequality for the 1-D plate equation wtt + wxxxx + q(t,x)w = 0 with two types of boundary conditions w = wxx = 0 or w = wx = 0,and q(t,x) being a suitable potential.The author shows that the sharp observability constant is of order exp(C q ∞27) for q ∞≥ 1.The main tools to derive the desired observability inequalities are the global Carleman inequalities,based on a new point wise inequality for the fourth order plate operator.     

On statistical convergence of vector-valued sequences associated with multiplier sequences

We introduce vector-valued sequence spaces w _∞(F, Q, p, u), w ₁(F, Q, p, u), w ₀(F, Q, p, u), S _u ^q , and S _0u ^q using a sequence of modulus functions and a multiplier sequence u = (u _k ) of nonzero complex numbers. We give some relations for these sequence spaces. It is also shown that if a sequence is strongly u _q -Cesàro summable with respect to the modulus function, then it is u _q -statistically convergent. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 1, pp. 125–131, January, 2006.     

Strongly singular convolution operators on the weighted Herz-type Hardy spaces

Let 0<p≤1<q<0, andw ₁ ,w ₂ ∈ A ₁ (Muckenhoupt-class). In this paper the authors prove that the strongly singular convolution operators are bounded from the homogeneous weighted Herz-type Hardy spacesH K^{α, p} _q(w₁; w₂) to the homogeneous weighted Herz spacesK ^{α, p} _q (w₁; w₂), provided α=n(1−1/q). Moreover, the boundedness of these operators on the non-homogeneous weighted Herz-type Hardy spacesH K ^{α, p} _q (w ₁;w ₂) is also investigated. Supported by the National Natural Science Foundation of China     

Necessary and sufficient conditions for boundedness of commutators of strongly singular integral operators with weighted Lipschitz functions

In this paper, we prove the commutator T _b generated by the strongly singular integral operator T and the function b is bounded from L ^p (w) to L ^q (w ^1−q ) if and only if b ∈ Lip _β (w), where w ∈ A ₁, 0 < β < 1, 1 < p < n/β and 1/q = 1/p − β/n. To do this, we first show a maximal function estimate for the commutator.     

Wiener��s Lemma for Infinite Matrices II

In this paper, we introduce a class of infinite matrices related to the Beurling algebra of periodic functions, and we show that it is an inverse-closed subalgebra of B(l^q_w){\mathcal{B}}(\ell^{q}_{w}), the algebra of all bounded linear operators on the weighted sequence space l^q_w\ell^{q}_{w}, for any 1≤q<∞ and any discrete Muckenhoupt A _q -weight w.     

Life span of solutions of a weakly coupled parabolic system

We consider the Cauchy problem for the weakly coupled parabolic system ∂ _t w _λ−Δ w _λ = F(w _λ) in R ^N , where λ > 0, w _λ = (u _λ, v _λ), F(w _λ) = (v _λ ^p , u _λ ^q ) for some p, q ≥ 1, pq > 1, and w_l(0) = (lj₁, l^\fracq+1p+1j₂)w_{\lambda}(0) = ({\lambda}{\varphi}_1, {\lambda}^{\frac{q+1}{p+1}}{\varphi}_2), for some nonnegative functions φ₁, φ₂ ?\in C ₀(R ^N ). If (p, q) is sub-critical or either φ₁ or φ₂ has slow decay at ∞, w _λ blows up for all λ > 0. Under these conditions, we study the blowup of w _λ for λ small.     

Entropy numbers of embeddings of weighted Besov spaces III. Weights of logarithmic type

We determine the exact asymptotic order of the entropy numbers of compact embeddings of weighted Besov spaces in the case where the ratio of the weights w(x) = w ₁(x)/w ₂(x) is of logarithmic type. This complements the known results for weights of polynomial type. The estimates are given in terms of the number 1/p = 1/p ₁ − 1/p ₂ and the function w(x). We find an interesting new effect: if the growth rate at infinity of w(x) is below a certain critical bound, then the entropy numbers depend only on w(x) and no longer on the parameters of the two Besov spaces. All results remain valid for Triebel–Lizorkin spaces as well.     

The asymptotic number of labeled connected graphs with a given number of vertices and edges

Let c(n, q) be the number of connected labeled graphs with n vertices and q ≤ N = (2n ) edges. Let x = q/n and k = q ? n. We determine functions w_k ? 1. a(x) and φ(x) such that c(n, q) ? w_k(q^N)eⁿφ(x)+a(x) uniformly for all n and q ≥ n. If ? > 0 is fixed, n→ ∞ and 4q > (1 + ?)n log n, this formula simplifies to c(n, q) ? (N^q) exp(–ne^?2q/n). on the other hand, if k = o(n^1/2), this formula simplifies to c(n, n + k) ? 1/2 w_k (3/π)^1/2 (e/12k)^k/2nⁿ?^(3k?1)/2.     

Life span of solutions of a weakly coupled parabolic system

We consider the Cauchy problem for the weakly coupled parabolic system ∂ _t w _λ−Δ w _λ = F(w _λ) in R ^N , where λ > 0, w _λ = (u _λ, v _λ), F(w _λ) = (v _λ ^p , u _λ ^q ) for some p, q ≥ 1, pq > 1, and , for some nonnegative functions φ₁, φ₂ C ₀(R ^N ). If (p, q) is sub-critical or either φ₁ or φ₂ has slow decay at ∞, w _λ blows up for all λ > 0. Under these conditions, we study the blowup of w _λ for λ small.      

Two modular equations for squares of the Rogers–Ramanujan functions with applications

Two modular identities of Gordon, McIntosh, and Robins are shown to be connected to the Rogers–Ramanujan continued fraction R(q), and in particular to Ramanujan’s parameter k:=R(q)R ²(q ²). Using this connection, we give new modular relations for R(q), and offer new and uniform proofs of several results of Ramanujan. In particular, we give a new proof of a famous and fundamental modular identity satisfied by the Rogers–Ramanujan continued fraction. We furthermore show that many analogous results hold for Ramanujan’s parameters μ:=R(q)R(q ⁴) and ν:=R ²(q ^1/2)R(q)/R(q ²). New proofs are offered for modular relations connecting R(q) to R(−q), R(q ²), and R(q ⁴), and new relations connecting R(q) at these arguments are offered. Eleven identities for the Rogers–Ramanujan functions are proved, including four new identities.      

Boundedness of maximal,potential type,and singular integral operators in the generalized variable exponent Morrey type spaces

We consider generalized Morrey type spaces M^{p( ·),q( ·),w( ·)}( W) {\mathcal{M}^{p\left( \cdot \right),\theta \left( \cdot \right),\omega \left( \cdot \right)}}\left( \Omega \right) with variable exponents p(x), θ(r) and a general function ω(x, r) defining a Morrey type norm. In the case of bounded sets W ì \mathbbRⁿ \Omega \subset {\mathbb{R}^n} , we prove the boundedness of the Hardy–Littlewood maximal operator and Calderón–Zygmund singular integral operators with standard kernel. We prove a Sobolev–Adams type embedding theorem M^{p( ·),q₁( ·),w₁( ·)}( W) ? M^{q( ·),q₂( ·),w₂( ·)}( W) {\mathcal{M}^{p\left( \cdot \right),{\theta_1}\left( \cdot \right),{\omega_1}\left( \cdot \right)}}\left( \Omega \right) \to {\mathcal{M}^{q\left( \cdot \right),{\theta_2}\left( \cdot \right),{\omega_2}\left( \cdot \right)}}\left( \Omega \right) for the potential type operator I ^α(·) of variable order. In all the cases, we do not impose any monotonicity type conditions on ω(x, r) with respect to r. Bibliography: 40 titles.     

Orthogonal graphs of characteristic 2 and their automorphisms

The (singular) orthogonal graph O(2ν + δ,q) over a field with q elements and of characteristic 2 (where ν 1, and δ = 0,1 or 2) is introduced. When ν = 1, O(2 · 1,q), O(2 · 1 + 1,q) and O(2 · 1 + 2,q) are complete graphs with 2, q + 1 and q2 + 1 vertices, respectively. When ν 2, O(2ν + δ,q) is strongly regular and its parameters are computed. O(2ν + 1,q) is isomorphic to the symplectic graph Sp(2ν,q). The chromatic number of O(2ν + δ,q) except when δ = 0 and ν is odd is computed and the group of graph automo...     

Some intriguing upper bounds for separating hash families

An N ×n matrix on q symbols is called {w_1,...,w_t}-separating if for arbitrary t pairwise disjoint column sets C_1,..., C_t with |C_i|=w_i for 1 ≤i≤t, there exists a row f such that f(C_1),...,f(C_t) are also pairwise disjoint, where f(C_i) denotes the collection of componentn of C_i restricted to row f. Given integers N, q and w_1,...,w_t, denote by C(N,q,{w_1,...,w_t}) the maximal a such that a corresponding matrix does exist.The determination of C(N,q,{w_1,...,w_t}) has received remarkable attention during the recent years. The main purpose of this paper is to introduce two novel methodologies to attack the upper bound of C(N, q, {w_1,...,w_t}).The first one is a combination of the famous graph removal lemma in extremal graph theory and a Johnson-type recursive inequality in coding theory, and the second onc is the probabilistic method. As a consequence, we obtain several intriguing upper bounds for some parameters of C(N,q,{w_1,...,w_t}), which significantly improve the previously known results.     

Leading coefficients of Kazhdan–Lusztig polynomials for Deodhar elements

We show that the leading coefficient of the Kazhdan–Lusztig polynomial P _x,w (q) known as μ(x,w) is always either 0 or 1 when w is a Deodhar element of a finite Weyl group. The Deodhar elements have previously been characterized using pattern avoidance in Billey and Warrington (J. Algebraic Combin. 13(2):111–136, [2001]) and Billey and Jones (Ann. Comb. [2008], to appear). In type A, these elements are precisely the 321-hexagon avoiding permutations. Using Deodhar’s algorithm (Deodhar in Geom. Dedicata 63(1):95–119, [1990]), we provide some combinatorial criteria to determine when μ(x,w)=1 for such permutations w. The author received support from NSF grants DMS-9983797 and DMS-0636297.     

Optimal constant weight covering codes and nonuniform group divisible 3-designs with block size four

Let K _q (n, w, t, d) be the minimum size of a code over Z _q of length n, constant weight w, such that every word with weight t is within Hamming distance d of at least one codeword. In this article, we determine K _q (n, 4, 3, 1) for all n ≥ 4, q = 3, 4 or q = 2 ^m  + 1 with m ≥ 2, leaving the only case (q, n) = (3, 5) in doubt. Our construction method is mainly based on the auxiliary designs, H-frames, which play a crucial role in the recursive constructions of group divisible 3-designs similar to that of candelabra systems in the constructions of 3-wise balanced designs. As an application of this approach, several new infinite classes of nonuniform group divisible 3-designs with block size four are also constructed.     

On an Isomorphism Problem of Some Dual Hyperovals in PG(2d + 1, q) with q even

Let q = 2^l with l≥ 1 and d ≥ 2. We prove that any automorphism of the d-dimensional dual hyperoval over GF(q), constructed in [3] for any (d + 1)-dimensional GF(q)-vector subspace V in GF(qⁿ) with n≥ d + 1 and for any generator σ of the Galois group of GF(qⁿ) over GF(q), always fixes the special member X(∞). Moreover, we prove that, in case V = GF(q^d+1), two dual hyperovals and in PG(2d + 1,q), where σ and τ are generators of the Galois group of GF(q^d+1) over GF(q), are isomorphic if and only if (1) σ = τ or (2) σ τ = id. Therefore, we have proved that, even in the case q > 2, there exist non isomorphic d-dimensional dual hyperovals in PG(2d + 1,q) for d ≥ 3.     

Nonuniqueness in the quenching problem

This paper deals with nonnegative solutions of for

On recognizability of some finite simple orthogonal groups by spectrum

It is proved that, if G is a finite group that has the same set of element orders as the simple group D _p (q), where p is prime, p ≥ 5 and q ∈ {2, 3, 5}, then the commutator group of G/F(G) is isomorphic to D _p (q), the subgroup F(G) is equal to 1 for q = 5 and to O _q (G) for q ∈ {2, 3}, F(G) ≤ G′, and |G/G′| ≤ 2.