A new generalization of Aczél’s inequality and its applications to an improvement of Bellman’s inequality

A new generalized version of Aczél’s inequality is proved. This is a unified generalization of some known results. Moreover, the result is applied to the improvement of the well-known Bellman’s inequality.     

Structure relations for monic orthogonal polynomials in two discrete variables

In this paper, extensions of several relations linking differences of bivariate discrete orthogonal polynomials and polynomials themselves are given, by using an appropriate vector–matrix notation. Three-term recurrence relations are presented for the partial differences of the monic polynomial solutions of admissible second order partial difference equation of hypergeometric type. Structure relations, difference representations as well as lowering and raising operators are obtained. Finally, expressions for all matrix coefficients appearing in these finite-type relations are explicitly presented for a finite set of Hahn and Kravchuk orthogonal polynomials.     

On Appell sequences of polynomials of Bernoulli and Euler type

A construction of new sequences of generalized Bernoulli polynomials of first and second kind is proposed. These sequences share with the classical Bernoulli polynomials many algebraic and number theoretical properties. A class of Euler-type polynomials is also presented.     

关于3阶Carmichael数

朱文余和孙琦（见《数学进展》,2004,33（4）：505-507）提出了关于3阶Carmichael数的三个问题,我们（见《四川大学学报（自然科学版）》,2006,43（6）：1197-1201）肯定地回答了问题1．本文模仿Howe的寻找严格2阶Carmichael数（见Mathematics of Computation,2000,69（232）：1711—1719）的方法,提出寻找满足某种条件的3阶Carmichael数的方法,并用这种方法确实找到了几百个这样的数,因而完全肯定地回答了问题2．     

A semi-discretization method based on quartic splines for solving one-space-dimensional hyperbolic equations

In this paper, based on C³ quartic splines, a semi-discretization method containing two schemes is constructed to solve one-space-dimensional linear hyperbolic equations. It is shown that both schemes are unconditionally stable and their approximation orders are of O(k⁵+h⁴) and of O(k⁷+h⁴) with k and h being step sizes in time and space, respectively, which are much higher than those of other published schemes. A numerical example is presented and the results are compared with other published numerical results.     

Decomposing the essential spectrum

We use C^*-algebra theory to provide a new method of decomposing the essential spectra of self-adjoint and non-self-adjoint Schrödinger operators in one or more space dimensions.     

On a deterministic linear partial differential system

We develop an Lp theory for the Cauchy problem of linear partial differential systems of the form

     

Exact asymptotics of the characteristic polynomial of the symmetric Pascal matrix

We have obtained the exact asymptotics of the determinant . Inverse symbolic computing methods were used to obtain exact analytical expressions for all terms up to relative order L⁻¹⁴ to the leading term. This determinant is known to give weighted enumerations of cyclically symmetric plane partitions, weighted enumerations of certain families of vicious walkers and it has been conjectured to be proportional to the one point function of the O(1) loop model on a cylinder of circumference L. We apply our result to the loop model and give exact expressions for the asymptotics of the average of the number of loops surrounding a point and the fluctuation in this number. For the related bond percolation model at the critical point, we give exact expressions for the asymptotics of the probability that a point is on a cluster that wraps around a cylinder of even circumference and the probability that a point is on a cluster spanning a cylinder of odd circumference.     

Asymptotic analysis of the Krawtchouk polynomials by the WKB method

We analyze the Krawtchouk polynomials K _n (x,N,p,q) asymptotically. We use singular perturbation methods to analyze them for N→∞, with appropriate scalings of the two variables x and n. In particular, the WKB method and asymptotic matching are used. We obtain asymptotic approximations valid in the whole domain [0,N]×[0,N], involving some special functions. We give numerical examples showing the accuracy of our formulas.      

Meromorphic solutions of equations over non-Archimedean fields

In this paper, we give some conditions to assure that the equation P(X)=Q(Y) has no meromorphic solutions in all K, where P and Q are polynomials over an algebraically closed field K of characteristic zero, complete with respect to a non-Archimedean valuation. In particular, if P and Q satisfy the hypothesis (F) introduced by H. Fujimoto, a necessary and sufficient condition is obtained when deg P=deg Q. The results are presented in terms of parametrization of a projective curve by three entire functions. In this way we also obtain similar results for unbounded analytic functions inside an open disk.