Martingales and the Robbins-Monro procedure in D[0, 1]

The Robbins-Monro procedure for recursive estimation of a zero point of a regression function f is investigated for the case f defined on and with values in the space D[0, 1] of real-valued functions on [0, 1] that are right-continuous and have left-hand limits, endowed with Skorohod's J₁-topology. There are proved an a.s. convergence result and an invariance principle where the limit process is a Gaussian Markov process with paths in the space of continuous C[0, 1]-valued functions on [0, 1]. At first the case f(x) ≡ x, i.e., the case of a martingale in D[0, 1], is treated and by this then the general case. An application to an initial value problem with only empirically available function values is sketched.     

Asymptotic Independence and Additive Functionals

A strong approximation result is proved for the partial sum process of i.i.d. sequence of vectors having dependent components, where the components of the approximating process are independent. This result is applied for additive functionals of random walks in one and two dimensions.     

Three nearly scaling-invariant versions of an exterior point algorithm for linear programming

In this paper, we describe three versions of a primal exterior point Simplex type algorithm for solving linear programming problems. Also, these algorithms are not affected mainly by scaling techniques. We compare their practical effectiveness versus the revised primal Simplex algorithm (our implementation) and the MATLAB’s implementations of Simplex and Interior Point Method. A computational study on randomly generated sparse linear programs is presented to establish the practical value of the proposed versions. The results are very encouraging and verify the superiority of the exterior point versions over the other algorithms either using scaling techniques or not.     

相对论力学系统的形式不变性与Lie对称性

研究相对论力学系统的形式不变性和Lie对称性.给出相对论力学系统在无限小变换下形式不变性和Lie对称性的定义、判据和守恒量,得到形式不变性和Lie对称性的关系,并举例说明结果的应用. 关键词： 相对论 力学系统 形式不变性 Lie对称性     

Parameter Invariance in Field Theory and the Hamiltonian Formalism

The deparametrization problem for parameter‐invariant Lagrangian densities defined over J¹(N, F), is solved in terms of a projection onto a suitable jet bundle. The Hamilton‐Cartan formalism for such Lagrangians is then introduced and the pre‐symplectic structure of such variational problems is proved to be projectable through the aforementioned projection. Specific examples with physical meaning are also analyzed. 1998 PACS codes. 02.20.Tw Infinite‐dimensional Lie groups, 02.30.Wd Calculus of variations and optimal control, 02.40.Ky Riemannian geometries, 02.40.Ma Global differential geometry, 02.40.Vh Global analysis and analysis on manifolds, 04.20.Fy Canonical formalism, Lagrangians, and variational principles, 11.10.Ef Lagrangian and Hamiltonian approach, 11.10.Kk Field theories in dimensions other than four, 11.25.Sq Nonperturbative techniques; string field theory. 1991 Mathematics Subject Classification. Primary: 58E30 Variational principles; Secondary: 53B20 Local Riemannian geometry, 58A20 Jets, 58E12 Applications to minimal surfaces (problems in two independent variables), 58G35 Invariance and symmetry properties, 81S10 Geometric quantization, symplectic methods, 83E30 String and superstring theories.     

Shape invariance in phase space

Shape invariance is a powerful solvability condition, that allows for complete knowledge of the energy spectrum, and eigenfunctions of a system. After a short introduction into the deformation quantization formalism, this paper explores the implications of the supersymmetric quantum mechanics and shape invariance techniques to the phase space formalism. We show that shape invariance induces a new set of relations between the Wigner functions of the system, that allows for their direct calculation, once we know one of them. The simple harmonic oscillator and the Morse potential are solved as examples.