Practical use of variational principles for modeling water waves

This paper describes a method for deriving approximate equations for irrotational water waves. The method is based on a ‘relaxed’ variational principle, i.e., on a Lagrangian involving as many variables as possible. This formulation is particularly suitable for the construction of approximate water wave models, since it allows more freedom while preserving a variational structure. The advantages of this relaxed formulation are illustrated with various examples in shallow and deep waters, as well as arbitrary depths. Using subordinate constraints (e.g., irrotationality or free surface impermeability) in various combinations, several model equations are derived, some being well-known, other being new. The models obtained are studied analytically and exact traveling wave solutions are constructed when possible.     

Calculation of critical exponents by self-similar factor approximants

The method of self-similar factor approximants is applied to calculating the critical exponents of the O(N)-symmetric ϕ⁴ theory and of the Ising glass. It is demonstrated that this method, being much simpler than other known techniques of series summation in calculating the critical exponents, at the same time, yields the results that are in very good agreement with those of other rather complicated numerical methods. The principal advantage of the method of self-similar factor approximants is the combination of its extraordinary simplicity and high accuracy.     

A new method for computing solenoidal vector fields on arbitrary regions

We develop a new method for the efficient calculation of solenoidal vector fields on general regions. The method takes advantage of fast direct methods and uses boundary integral equations to satisfy boundary conditions. For the latter we give an effective scheme for computing far-field boundary influences (based on discrete charges). Examples and numerical results are given. The method is applicable to incompressible Navier-Stokes calculations.     

Numerical experiments with universal barrier functions for cones of Chebyshev systems

Based on previous explicit computations of universal barrier functions, we describe numerical experiments for solving certain classes of convex optimization problems. The comparison is given of the performance of the classical affine-scaling algorithm with the similar algorithm built upon the universal barrier function.     

模糊粗糙近似算子的拓扑性质

考虑论域上一二元关系所决定的模糊粗糙近似算子的拓扑性质,证明了任一自反二元关系可以决定一模糊拓扑.并且,当二元关系自反对称时,该模糊拓扑中的元是开集当且仅当它是闭集;当二元关系自反传递时,该模糊拓扑的闭包与内部算子恰为模糊粗糙上、下近似算子.     

A primal-dual approach to inexact subgradient methods

For optimization problems with computationally demanding objective functions and subgradients, inexact subgradient methods (IXS) have been introduced by using successive approximation schemes within subgradient optimization methods (Au et al., 1994). In this paper, we develop alternative solution procedures when the primal-dual information of IXS is utilized. This approach is especially useful when the projection operation onto the feasible set is difficult. We also demonstrate its applicability to stochastic linear programs.     

Extensions of the Cauchy-Goursat Integral Theorem

It is natural to conjecture that if a function f is continuous on the closed region determined by a rectifiable 1-cycle Γ and complex-differentiable on the open region then Γ_∫f=0. The main result is an extension of the classical Cauchy-Goursat Theorem: the equality conjectured holds (with no boundary condition on f^′) under the additional hypothesis that the winding numbers of Γ define an Lp function and f satisfies a matching Hölder continuity condition near the image of Γ. (In particular, continuity suffices if p=∞.) The proof uses approximations of a rectifiable path by piecewise linear paths.     

On the “Wolfsberg–Helmholz Conjecture” of “Extended‐Hückel Theory”

After having reviewed some pioneer integral approximations closely related to Rüdenberg's expansions of one‐ and two‐electron orbital products, we apply the previously described “Implicit Multi‐Center Integration” techniques on Roothaan's “restricted” Fock‐matrix components over standard atomic orbital bases. The resulting compact forms are very similar to the well‐known “Wolfsberg–Helmholz Conjecture” of “Extended‐Hückel Theory,” which relates the various off‐diagonal matrix elements of “restricted” Fock‐type to their corresponding diagonal counterparts. In this way, a “nonempirical Extended‐Hückel Theory” can be created. © 2012 Wiley Periodicals, Inc.     

A dual strategy for the implementation of the aggregation principle in decision making under uncertainty

A solution method for stochastic programs is proposed based on the aggregation principle, which allows one to find the solution of a stochastic program by aggregating the solutions of individual deterministic scenario problems. The method concentrates on finding good estimates of the dual variables associated with the non-anticipativity constraints.     

On the rate of approximation in the central limit theorem for dependent random variables and random vectors

Large “O” and small “o” approximations of the expected value of a class of smooth functions (f C^r(R)) of the normalized partial sums of dependent random variable by the expectation of the corresponding functions of normal random variables have been established. The same types of approximations are also obtained for dependent random vectors. The technique used is the Lindberg-Levy method generalized by Dvoretzky to dependent random variables.