Structural results for partially observed control models

A general partially observed control model with discrete time parameter is investigated. Our main interest concerns monotonicity results and bounds for the value functions and for optimal policies. In particular, we show how the value functions depend on the observation kernels and we present conditions for a lower bound of an optimal policy. Our approach is based on two multivariate stochastic orderings: theTP ₂ ordering and the Blackwell ordering.Dedicated to Prof. Dr. K. Hinderer on the occassion of his 60^th birthday     

Combinatorial interpretations of the q-Faulhaber and q-Salié coefficients

Recently, Guo and Zeng discovered two families of polynomials featuring in a q-analogue of Faulhaber's formula for the sums of powers and a q-analogue of Gessel-Viennot's formula involving Salié's coefficients for the alternating sums of powers. In this paper, we show that these are polynomials with symmetric, nonnegative integral coefficients by refining Gessel-Viennot's combinatorial interpretations.     

n-Colour self-inverse compositions

MacMahon’s definition of self-inverse composition is extended ton-colour self-inverse composition. This introduces four new sequences which satisfy the same recurrence relation with different initial conditions like the famous Fibonacci and Lucas sequences. For these new sequences explicit formulas, recurrence relations, generating functions and a summation formula are obtained. Two new binomial identities with combinatorial meaning are also given.     

Numerical and empirical approach in predicting the penetration of a concrete target by an ogive-nosed projectile

This paper demonstrates the application of both numerical simulation and empirical equation in predicting the penetration of a concrete target by an ogive-nosed projectile. The results from the experiment performed by Gran and Frew [In-target radial stress measurements from penetration experiments into concrete by ogive-nose steel projectiles, Int. J. Impact Eng. 19 (8) (1997) 715–726] are used as a benchmark for comparison. In the numerical simulations a 3.0-caliber radius-head steel ogival-nose projectile with a mass of 2.3 kg is fired against cylindrical concrete target with a striking velocity of 315 m/s. The simulation, performed using AUTODYN 2-D, assesses three numerical schemes, namely Langrange, Euler–Lagrange coupling and smooth particles hydrodynamics SPH–Lagrange coupling, in predicting the maximum depth of penetration and the radial stress–time response of the concrete target. When assessing the three solution techniques we hypothesize that the effect of strain rate on strength for the concrete target does not adversely affect the prediction on the maximum depth of penetration and the radial stress–time response of the concrete target. In the empirical approach the penetration equation developed by Forrestal et al. [An empirical equation for penetration depth of ogive-nose projectiles into concrete targets, Int. J. Impact Eng. 15 (4) (1994) 395–405] is used to determine the maximum depth of penetration and the deceleration–time response. The deceleration–time response for the projectile using the empirical approach is compared with those obtained from the numerical simulations. Results from both the numerical and empirical approaches are consistent. The calculated depth of penetration from both approaches yield relatively good agreement with that obtained from the experiment. The numerical simulations using each of the three numerical schemes are also able to reproduce the profiles from the radial stress measurements. Simulations using the SPH numerical scheme give the best overall agreement. The good overall agreement with the experimental radial stress measurements and consistent results between both empirical and numerical approach, enhanced the confidence in engineers and ballisticians when using these two approaches in complementing full-scale testing.     

Laguerre expansion on the Heisenberg group and Fourier-Bessel transform on ?n

Given a principal value convolution on the Heisenberg group H _n = ℂ ⁿ × ℝ, we study the relation between its Laguerre expansion and the Fourier-Bessel expansion of its limit on ℂ ⁿ . We also calculate the Dirichlet kernel for the Laguerre expansion on the group H _n . Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

The Ansatz Method for Analyzing Schrödinger’s Equation with Three Anharmonic Potentials in D Dimensions

In this letter, by applying a suitable ansatz to the wave functions, the solutions of the D-dimensional radial Schrödinger equation with some anharmonic potentials are obtained.     

多模玻色二次多项式型系统的特性函数和准概率分布函数

运用多模玻色系统广义线性量子变换的普遍理论，对多模玻色二次多项式型系统进行了研究.给出了多模玻色二次多项式型系统的正规、反正规、Wigner特性函数、P表示和Q表示.举例讨论了该系统的压缩性质. 关键词： 多模玻色二次多项式型系统 特性函数 准概率分布函数     

On q-orthogonal polynomials over a collection of complex origin intervals related to little q-Jacobi polynomials

We construct the sequence of orthogonal polynomials with respect to an inner product which is defined by q-integrals over a collection of intervals in the complex plane. We prove that they are connected with little q-Jacobi polynomials. For such polynomials we discuss a few representations, a recurrence relation, a difference equation, a Rodrigues-type formula and a generating function. 2000 Mathematics Subject Classification Primary—33D45, 42C05     

二能级原子与逆场算符耦合系统的J-C模型

利用表象变换方法,对含逆场算符的J-C模型,分别在相互作用表象中和薛定谔表象中求出了系统的演化波函数,并讨论了失谐量为零时的拉比振动.     

Tail Bounds for the Supremums of Empirical Processes over Unbounded Classes of Functions

So far the study of exponential bounds of an empirical process has been restricted to a bounded index class of functions. The case of an unbounded index class of functions is now studied on the basis of a new symmetrization idea and a new method of truncating the original probability space; the exponential bounds of the tail probabilities for the supremum of the empirical process over an unbounded class of functions are obtained. The exponential bounds can be used to establish laws of the logarithm for the empirical processes over unbounded classes of functions.