Sharpness of the Korányi approach region

We prove a Littlewood-type theorem which shows the sharpness of the Korányi approach region for the boundary behavior of Poisson-Szegö integrals on the unit ball of org/proc/2005-133-08/S0002-9939-05-08020-2/gif-abstract0/img1.gif" ALT="$\mathbb{C} ^n$">. Our result is stronger than Hakim and Sibony (1983).

     

Positivity preserving transformations for -binomial coefficients

Several new transformations for org/tran/2005-357-06/S0002-9947-04-03680-3/gif-abstract0/img1.gif" ALT="$q$">-binomial coefficients are found, which have the special feature that the kernel is a polynomial with nonnegative coefficients. By studying the group-like properties of these positivity preserving transformations, as well as their connection with the Bailey lemma, many new summation and transformation formulas for basic hypergeometric series are found. The new org/tran/2005-357-06/S0002-9947-04-03680-3/gif-abstract0/img2.gif" ALT="$q$">-binomial transformations are also applied to obtain multisum Rogers-Ramanujan identities, to find new representations for the Rogers-Szegö polynomials, and to make some progress on Bressoud's generalized Borwein conjecture. For the original Borwein conjecture we formulate a refinement based on new triple sum representations of the Borwein polynomials.

     

An old conjecture of Erdos-Turán on additive bases

There is a 1941 conjecture of Erdos and Turán on what is now called additive basis that we restate:

Conjecture 0.1(Erdos and Turán). Suppose that org/mcom/2006-75-253/S0025-5718-05-01777-1/gif-abstract0/img1.gif" ALT="$0=\delta_0<\delta_1<\delta_2<\delta_3\cdots$"> is an increasing sequence of integers and

org/mcom/2006-75-253/S0025-5718-05-01777-1/gif-abstract0/img2.gif" ALT="\begin{displaymath}s(z) : = \sum_{i=0}^\infty z^{\delta_i}. \end{displaymath}">

Suppose that

org/mcom/2006-75-253/S0025-5718-05-01777-1/gif-abstract0/img3.gif" ALT="\begin{displaymath}s^2(z) := \sum_{i=0}^\infty b_i z^i. \end{displaymath}">

If org/mcom/2006-75-253/S0025-5718-05-01777-1/gif-abstract0/img4.gif" ALT="$b_i>0$"> for all org/mcom/2006-75-253/S0025-5718-05-01777-1/gif-abstract0/img5.gif" ALT="$i$">, then org/mcom/2006-75-253/S0025-5718-05-01777-1/gif-abstract0/img6.gif" ALT="$\{b_n\}$"> is unbounded.

Our main purpose is to show that the sequence org/mcom/2006-75-253/S0025-5718-05-01777-1/gif-abstract0/img7.gif" ALT="$\{b_n\}$"> cannot be bounded by org/mcom/2006-75-253/S0025-5718-05-01777-1/gif-abstract0/img8.gif" ALT="$7$">. There is a surprisingly simple, though computationally very intensive, algorithm that establishes this.

     

LITTLEWOOD-PALEY THEOREM FOR SCHR(O)DINGER OPERATORS

Let H be a Schr(o)dinger operator on Rn. Under a polynomial decay condition for the kernel of its spectral operator, we show that the Besov spaces and Triebel-Lizorkin spaces associated with H are well defined. We further give a Littlewood-Paley characterization of Lp spaces in terms of dyadic functions of H. This generalizes and strengthens the previous result when the heat kernel of H satisfies certain upper Gaussian bound.     

Symmetry classes of tensors and semi-direct product of finite abelian groups

In the study of symmetry classes of tensors, finding examples of symmetry classes of tensors that possess an o*-basis is of considerable interest. There are only few classes of groups that have been provided a necessary and sufficient condition for having such a basis. There is no general criterion for any finite group yet. In this note, we provide a necessary and sufficient condition for the existence of o*-basis of symmetry classes of tensors associated with semi-direct product of some finite abelian groups and, consequently, their wreath product.     

The dynamic behavior of deterministic and stochastic delayed SIQS model

In this paper, we present the deterministic and stochastic delayed SIQS epidemic models. For the deterministic model, the basic reproductive number $R_{0}$ is given. Moreover, when $R_{0}<1$, the disease-free equilibrium is globally asymptotical stable. When $R_{0}>1$ and additional conditions hold, the endemic equilibrium is globally asymptotical stable. For the stochastic model, a sharp threshold $\overset{\wedge }{R}_{0}$ which determines the extinction or persistence in the mean of the disease is presented. Sufficient conditions for extinction and persistence in the mean of the epidemic are established. Numerical simulations are also conducted in the analytic results.     

Dynamical behaviour and exact solutions of thirteenth order derivative nonlinear Schr\"{o}dinger equation

In this paper, we considered the model of the thirteenth order derivatives of nonlinear Schr\"{o}dinger equations. It is shown that a wave packet ansatz inserted into these equations leads to an integrable Hamiltonian dynamical sub-system. By using bifurcation theory of planar dynamical systems, in different parametric regions, we determined the phase portraits. In each of these parametric regions we obtain possible exact explicit parametric representation of the traveling wave solutions corresponding to homoclinic, hetroclinic and periodic orbits.     

Erratum to "More Results on Hermite-Hadamard type inequality through (α,m)-preinvexity"

In this paper, we present some corrections to definitions of $m$-preinvex, $\left( \alpha,m\right) $-preinvex functions and statements of the theorems of the results proved in [7].     

Uniform bounds on growth in o‐minimal structures

We prove that a function definable with parameters in an o‐minimal structure is bounded away from ∞ as its argument goes to ∞ by a function definable without parameters, and that this new function can be chosen independently of the parameters in the original function. This generalizes a result in [1]. Moreover, this remains true if the argument is taken to approach any element of the structure (or ±∞), and the function has limit any element of the structure (or ±∞) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Estimates of commutators for multilinear operators on Herz-type spaces

In this paper, the boundedness of mulitilinear commutator [b,T] on Herz-type space is considered, where T is a standard Calderón-Zygmund singular operator and b ∈ (BMO(Rn))m.