Some identities and congruences concerning Euler numbers and polynomials

In this paper, using the properties of the moments of p-adic measures, we establish some identities and Kummer likewise congruences concerning Euler numbers and polynomials. In the preliminaries, we introduce the Laplace transform which is an important tool for the determination of the moments of p-adic measures. We also give a sequence n(dn) linked to Euler numbers and which satisfies the same type of congruences and identities as the Euler numbers. At the end, for p=2, we give congruences on Euler numbers involving the sequence n(dn).     

Semidefinite representation of convex sets

Let be a semialgebraic set defined by multivariate polynomials g _i (x). Assume S is convex, compact and has nonempty interior. Let , and ∂ S (resp. ∂ S _i ) be the boundary of S (resp. S _i ). This paper, as does the subject of semidefinite programming (SDP), concerns linear matrix inequalities (LMIs). The set S is said to have an LMI representation if it equals the set of solutions to some LMI and it is known that some convex S may not be LMI representable (Helton and Vinnikov in Commun Pure Appl Math 60(5):654–674, 2007). A question arising from Nesterov and Nemirovski (SIAM studies in applied mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1994), see Helton and Vinnikov in Commun Pure Appl Math 60(5):654–674, 2007 and Nemirovski in Plenary lecture, International Congress of Mathematicians (ICM), Madrid, Spain, 2006, is: given a subset S of , does there exist an LMI representable set Ŝ in some higher dimensional space whose projection down onto equals S. Such S is called semidefinite representable or SDP representable. This paper addresses the SDP representability problem. The following are the main contributions of this paper: (i) assume g _i (x) are all concave on S. If the positive definite Lagrange Hessian condition holds, i.e., the Hessian of the Lagrange function for optimization problem of minimizing any nonzero linear function ℓ ^T x on S is positive definite at the minimizer, then S is SDP representable. (ii) If each g _i (x) is either sos-concave ( − ∇² g _i (x) = W(x) ^T W(x) for some possibly nonsquare matrix polynomial W(x)) or strictly quasi-concave on S, then S is SDP representable. (iii) If each S _i is either sos-convex or poscurv-convex (S _i is compact convex, whose boundary has positive curvature and is nonsingular, i.e., ∇g _i (x) ≠ 0 on ∂ S _i ∩ S), then S is SDP representable. This also holds for S _i for which ∂ S _i ∩ S extends smoothly to the boundary of a poscurv-convex set containing S. (iv) We give the complexity of Schmüdgen and Putinar’s matrix Positivstellensatz, which are critical to the proofs of (i)–(iii).      

A noncommutative moment problem

We prove a noncommutative moment theorem and relate it to Connes' problem of embedding finite factor von Neumann algebras into an ultraproduct of the hyperfinite factor. We include a linear-algebraic equivalent of Connes' problem, which asks for a characterization of all noncommutative polynomials which have positive trace when the variables are replaced by contractive hermitian matrices.

     

Approximations for solute transport through porous media with flow transverse to layering

Analytical approaches for the prediction of solute transport in layered porous media are investigated for the case of flow perpendicular to the direction of layering. One approach involves the use of averaging techniques to treat the profile as an equivalent homogeneous medium. The method is demonstrated on hypothetical and laboratory-measured data sets and a criterion for validity of the method is given. The second approach involves the use of time convolution to predict breakthrough curves for layered systems on the assumption that layer interactions have no significant effect on transport. Accuracy criteria are derived by comparing moments of the exact and approximate solutions and it is found that the convolution method has broader applicability than the equivalent single-layer analysis. An extension of the convolution method to include consideration of nonequilibrium transport due to the presence of mobile-immobile regions is presented and demonstrated by analysis of laboratory breakthrough data from a two-layer system exhibiting mobile-immobile regions.     

Remark on the moment expansion of total π-electron energy

Summary The truncated expansion of the function |x| was frequently used to express the total Hückel -electron energy (E) in terms of moments. We now present an identity which connectsE with an infinite series of moments. This series is convergent. Lower and upper bounds forE are obtained, based on the same infinite moment expansion.     

Bounds for moments of cubic and quartic Dirichlet L-functions

We study the $2k$-th moment of central values of the family of primitive cubic and quartic Dirichlet $L$-functions. We establish sharp lower bounds for all real $k\ge 1/2$ unconditionally for the cubic case and under the Lindelöf hypothesis for the quartic case. We also establish sharp lower bounds for all real $0\le k<1/2$ and sharp upper bounds for all real $k\ge 0$ for both the cubic and quartic cases under the generalized Riemann hypothesis (GRH). As an application of our results, we establish quantitative non-vanishing results for the corresponding $L$-values.     

随机环境中随机指标分枝过程矩的渐近性

引入了随机环境中随机指标分枝过程模型,证明了该模型矩的渐近性。     

关于sup(n≥1)|S_n/n(1/r)|(0〈r〈2)和sup(n≥1)|S_n/■|的矩

设 {Xn,n≥ 1 }是 φ-混合的同分布的随机变量序列 ,记 Sn=∑ni=1Xi( n≥ 1 ) .该文的目的是要在一定的矩条件和混合速度限制下 ,讨论了 supn≥ 1| Snn1/ r| ( 0 相似文献   

Correlated electron-pair properties of the Li atom in momentum space

On the basis of multiconfiguration Hartree–Fock calculations, correlated electron-pair intracule (relative motion) and extracule (center-of-mass motion) properties are reported for the Li atom in momentum space. The present results are more accurate and consistent than those in the literature. Received: 10 September 2001 / Accepted: 11 December 2001 / Published online: 22 March 2002     

Integral Representations of Increments of Stochastic Processes

A formulation is presented in which the increment of a stochastic process is represented as an integral of the derivative of the process. It is shown that this representation is an alternative to the more common approach of writing equations for the differentials of stochastic processes. A possible advantage of the integral formulation is that its reliance on derivatives, rather than differentials, makes the operations of stochastic calculus more closely resemble those of ordinary deterministic calculus. This similarity to well-known mathematics may serve to make stochastic calculus accessible to a broader audience than in the past. The integral formulation is herein shown to be compatible with the Itô differential rule for non-Gaussian processes and is used to describe the increment of the nonstationary response of a system governed by a vector stochastic equation with parametric delta-correlated excitation.