A Generalization of a Two-Parameter Quantization for the Group GL 2(k)

We generalize a well-known two-parameter quantization for the group GL ₂(k) (over an arbitrary field k). Specifically, a certain class of Hopf algebras is constructed containing that quantization. The algebras are constructed given an arbitrary coalgebra and an arbitrary pair of its commuting anti-isomorphisms, and are defined by quadratic relations. They are densely linked to the compact quantum groups introduced by Woronowicz. We give examples of Hopf algebras that can be rowed up to the two-parameter quantization for GL ₂(k).     

Problem of the free oscillations of a capillary bridge

The free oscillations of a capillary bridge whose equilibrium shape is determined by the surface tension forces and the static gravity field are investigated. The values of 25 “lower” levels of the spectrum of natural oscillations of the capillary bridge are found for various control parameters in accordance with the experimental conditions.     

Spline Techniques for Solving Singularly-Perturbed Nonlinear Problems on Nonuniform Grids

A numerical method based on cubic splines with nonuniform grid is given for singularly-perturbed nonlinear two-point boundary-value problems. The original nonlinear equation is linearized using quasilinearization. Difference schemes are derived for the linear case using a variable-mesh cubic spline and are used to solve each linear equation obtained via quasilinearization. Second-order uniform convergence is achieved. Numerical examples are given in support of the theoretical results.     

Ergodic theory of amenable semigroup actions

In this paper, among other things, we state and prove the mean ergodic theorem for amenable semigroup algebras.     

Circuit-switching networks with FIFO plus justifying blockout service on infinite graphs

A class of circuit-switching open queueing networks is discussed. The main result of the paper is that if extra message flows are not too intensive and the path distribution is mainly concentrated on the paths of (graph) distance 1 (nearest neighbour connections), then the network has a unique stationary working regime, no matter how large the configuration graph of the network is. Standard properties of this regime are established such as decay of correlation and continuity.     

Disintegration and perfectness of measure spaces

For every finite measure space (X,A,P) we find a unique representation P=Q₁+Q₂+Q₃ such that Q₁ is compact, Q₂ is perfect and purely noncompact and Q₃ is purely nonperfect. We show that every Pachl-_O-disintegrable probability space is Ramachandran-_O-disintegrable and therefore perfect and under a certain condition we prove the equivalence between compactness and Ramachandran-_O-disintegrability.     

Small random perturbations of finite- and infinite-dimensional dynamical systems: Unpredictability of exit times

We apply previous results on the pathwise exponential loss of memory of the initial condition for stochastic differential equations with small diffusion to the problem of the asymptotic distribution of the first exit times from an attracted domain. We show under general hypotheses that the suitably rescaled exit time converges in the zero-noise limit to an exponential random variable. Then we extend the results to an infinite-dimensional case obtained by adding a small random perturbation to a nonlinear heat equation.     

Limiting Lagrangians: A primal approach

In this paper, we consider a convex program with either a finite or an infinite number of constraints and its formal Lagrangian dual. We show that either the primal program satisfies a general condition which implies there is no duality gap or that there is a nonzero vectord with the following properties: First, wheneverd is added to the objective function, where is a positive number not greater than one, the resulting program satisfies the general sufficient condition cited above for no duality gap. Second, the optimal value of this perturbed program is attained and tends to the optimal value of the original program as tends to zero. Third, the optimal solutions of the perturbed programs form a minimizing sequence of the original program. As a consequence of the above, we derive the limiting Lagrangian theory of Borwein, Duffin, and Jeroslow.The authors are indebted to an unknown referee who suggested the very short and elegant proofs of Lemma 2.3 and Theorem 2.3.This work was completed while the first author was a member of the College of Management, Georgia Institute of Technology, Atlanta, Georgia.     

A High-Order Finite Difference Scheme for 3D Unsteady Convection Diffusion Reaction Equations北大核心CSCD

针对三维非稳态对流扩散反应方程,构造了一种高精度紧致有限差分格式,对空间的离散采用四阶紧致差分方法,对时间的离散采用Taylor级数展开和余项修正技术,所提格式在时间上的精度为二阶、在空间上的精度为四阶。利用Fourier稳定性分析法证明了该格式是无条件稳定的。最后给出数值算例验证了理论结果。     

On the global optimization properties of finite-difference local descent algorithms

The paper is devoted to the convergence properties of finite-difference local descent algorithms in global optimization problems with a special -convex structure. It is assumed that the objective function can be closely approximated by some smooth convex function. Stability properties of the perturbed gradient descent and coordinate descent methods are investigated. Basing on this results some global optimization properties of finite-difference local descent algorithms, in particular, coordinate descent method, are discovered. These properties are not inherent in methods using exact gradients.The paper was presented at the II. IIASA-Workshop on Global Optimization, Sopron (Hungary), December 9–14, 1990.