DIRECTIONAL DO-NOTHING CONDITION FOR THE NAVIER-STOKES EQUATIONS

The numerical solution of flow problems usually requires bounded domains although the physical problem may take place in an unbounded or substantially larger domain. In this case, artificial boundaries are necessary. A well established artificial boundary condition for the Navier-Stokes equations diseretized by finite elements is the “do-nothing” condition. The reason for this is the fact that this condition appears automatically in the variational formulation after partial integration of the viscous term and the pressure gradient. This condition is one of the most established outflow conditions for Navier-Stokes but there are very few analytical insight into this boundary condition. We address the question of existence and stability of weak solutions for the Navier-Stokes equations with a “directional do-nothing” condition. In contrast to the usual “do-nothing” condition this boundary condition has enhanced stability properties. In the case of pure outflow, the condition is equivalent to the original one, whereas in the case of inflow a dissipative effect appears. We show existence of weak solutions and illustrate the effect of this boundary condition by computation of steady and non-steady flows.     

A THEORETICAL AND NUMERICAL STUDY FOR THE ROD-LIKE MODEL OF A POLYMERIC FLUID

The first part of this paper is concerned with the well-posedness for the rigid rod-like model in shear flow of a polymeric fluid. The constitutive relations considered in this work are motivated by the kinetic theory. The stress tensor is given by an integral which involves the solution of the Fokker-Planck equation. A novel numerical scheme for the Fokker-Planck equation is proposed, which preserves the positivity of the distribution function.Another part of this work establishes the convergence theory of the fully discretized schemes for a simple micro-macro simulation of a polymeric flow.     

Ergodicity of 3D Stochastic Burgers Equation

3D Burgers equation is an important model for turbulence. It is natural to expect the long-time behaviour for this hydrodynamics equation. However, there is no result about the long-time behaviour for this deterministic model. Surprisingly, if the system is perturbed by stochastic noise, we establish the existence and uniqueness of invariant measure for 3D stochastic Burgers equation.     

THE EXPANSION FOR THE ERROR DISTRIBUTION OF SAMPLE MEDIAN AND ITS RANDOM WEIGHTING STATISTIC

In this paper,the expansion for the error distribution of the sample median is obtained.To mimic the error distribution of the sample median,random weighting statistic is introduced.It is proved in this paper that the precision of the approximation of the error distribution for sample medisn by random weighting method is of the order of o.     

THREE WAY DECOMPOSITION FOR THE BOLTZMANN EQUATION

The initial value problem for the spatially homogeneous Boltzmann equation is considered. A deterministic numerical scheme for this problem is developed by the use of the three way decomposition of the unknown function as well as of the collision integral. On this way, almost linear complexity of the algorithm is achieved. Some numerical examples are presented.     

CONVERGENCE AND SUPERCONVERGENCE OF HERMITE BICUBIC ELEMENT FOR EIGENVALUE PROBLEM OF THE BIHARMONIC EQUATION

Eigenvalue problem for biharmonic equation is an interesting and important problem, seeCiarlet and Lions[3]. In 1979, Rannacher[8] used the Adini nonconforming finite element tosolve this problem and obtained:Recedely, Yang[6] has proved that the order of covergence of Ah is just 2. The aim of this paperis to improve the order of convergence by using Hermite bicubic element. To our knowledge,there is not any result for approximation to the eigenvalue problem by using this element inliteratu…     

A NEW DERIVATIVE FREE OPTIMIZATION METHOD BASED ON CONIC INTERPOLATION MODEL

In this paper, a new derivative free trust region method is developed basedon the conic interpolation model for the unconstrained optimization. The conic inter-polation model is built by means of the quadratic model function, the collinear scalingformula, quadratic approximation and interpolation. All the parameters in this model axedetermined by objective function interpolation condition. A new derivative free method isdeveloped based upon this model and the global convergence of this new method is provedwithout any information on gradient.     

IMPLICIT DIFFERENCE APPROXIMATION FOR A TIME FRACTIONAL ADVECTION-DISPERSION EQUATION

In this paper, a time fractional advection-dispersion equation is considered. From the relationship between the Caputo derivative and the Grunwald derivative, the Caputo derivative is approximated by using the Griinwald derivative. An implicit difference approximation for this equation is proposed. We prove that this approximation is unconditionally stable and convergent. Finally, numerical examples are given.     

具有内生基础设施分配的城市经济增长模型

In this paper, an urban economic growth model with endogenous infrastructure allocation is given by introducing the two-variable utility function for city＇s inhabitant. A twodimensional dynamical system is obtained by solving the utility maximization problem and it is proved that this system has the unique non-zero equilibrium which is a saddle. The model has the unique optimal growth and an optimal rate of infrastructure allocation.     

Nowhere-zero 15-flow in 3-edge-connected bidirected graphs

It was conjectured by Bouchet that every bidirected graph which admits a nowhere-zero κ flow will admit a nowhere-zero 6-flow. He proved that the conjecture is true when 6 is replaced by 216. Zyka improved the result with 6 replaced by 30. Xu and Zhang showed that the conjecture is true for 6-edge-connected graphs. And for 4-edge-connected graphs, Raspaud and Zhu proved it is true with 6 replaced by 4. In this paper, we show that Bouchet＇s conjecture is true with 6 replaced by 15 for 3-edge-connected graphs.     

The Betti reciprocity principle and the normal boundary component control problem for linear elastic systems

The background for this article is the question of modification of the geometric configuration of an elastic structure by means of “volume” type actuation. In this actuation mode stresses are applied to the elastic body by injection/extraction of a fluid into, or from, a large number of vacuoles in the elastic “matrix” material. Previous articles by the author, and others, have examined this process and studied its effectiveness in the context of a “naive” continuous model. The present paper continues along these lines, exploring “normal boundary component controllability” criterion for determining achievable configurations for the controlled system in the two-dimensional case. Connections with conformal mapping lead to affirmative results for approximate controllability in this sense and Fourier series techniques provide exact controllability results for the case wherein the domain of the uncontrolled system is a two-dimensional disk.      

Differentiable structure of ω-subsets of symplectic groups

In this paper, we study the differentiable structure of the ω-subset of Sp(2n), which is formed by all matrices in Sp(2n) possessing ω as an eigenvalue, for ω on the unit circle in the complex plane. Based on this result the ω -index theory parametrized by all ω on the unit circle for arbitrary symplectic paths is defined.     

Automorphisms of Maps with a Given Underlying Graph and Their Application to Enumeration

A graph is called a semi-regular graph if its automorphism group action on its ordered pair of adjacent vertices is semi-regular. In this paper, a necessary and sufficient condition for an automorphism of the graph F to be an automorphism of a map with the underlying graph F is obtained. Using this result, all orientation-preserving automorphisms of maps on surfaces (orientable and non-orientable) or just orientable surfaces with a given underlying semi-regular graph F are determined. Formulas for the numbers of non-equivalent embeddings of this kind of graphs on surfaces (orientable, non-orientable or both) are established, and especially, the non-equivalent embeddings of circulant graphs of a prime order on orientable, non-orientable and general surfaces are enumerated.     

DOUBLE Φ-FUNCTION INEQUALITY FOR NONNEGATIVE SUBMARTINGALES

The authors establish a kind of inequalities for nonnegative submartingales which depend on two functions Φ and ψ, and obtain the equivalent conditions for Φ and ψ such that this kind of inequalities holds. In the casen Φ =ψ∈Δ2, it is proved that this necessary and sufficient condition is equivalent to qΦ > 1.     

A consistency result on thin-very tall Boolean algebras

It was proved by Baumgartner and Shelah that Con (ZFC)→Con (ZFC + “there is a superatomic Boolean algebra of width ω and height ω₂”). In this paper we improve Baumgartner-Shelah’s theorem, showing that Con (ZFC)→Con (ZFC+“for every α<ω₃ there is a superatomic Boolean algebra of width ω and height α”). The preparation of this paper was supported by DGICYT Grant PB98-1231.     

On positivity of solutions of degenerate boundary value problems for second-order elliptic equations

In this paper we study thedegenerate mixed boundary value problem:Pu=f in Ω,B _u =gon Ω∂Г where ω is a domain in ℝ ⁿ ,P is a second order linear elliptic operator with real coefficients, Γ⊆∂Ω is a relatively closed set, andB is an oblique boundary operator defined only on ∂Ω/Γ which is assumed to be a smooth part of the boundary. The aim of this research is to establish some basic results concerning positive solutions. In particular, we study the solvability of the above boundary value problem in the class of nonnegative functions, and properties of the generalized principal eigenvalue, the ground state, and the Green function associated with this problem. The notion of criticality and subcriticality for this problem is introduced, and a criticality theory for this problem is established. The analogs for the generalized Dirichlet boundary value problem, where Γ=∂Ω, were examined intensively by many authors.     

Saddle-Point Criteria in an η-Approximation Method for Nonlinear Mathematical Programming Problems Involving Invex Functions

In this paper, the η-approximation method introduced by Antczak (Ref. 1) for solving a nonlinear constrained mathematical programming problem involving invex functions with respect to the same function η is extended. In this method, a so-called η-approximated optimization problem associated with the original mathematical programming problems is constructed; moreover, an η-saddle point and an η-Lagrange function are defined. By the help of the constructed η-approximated optimization problem, saddle-point criteria are obtained for the original mathematical programming problem. The equivalence between an η-saddle point of the η-Lagrangian of the associated η-approximated optimization problem and an optimal solution in the original mathematical programming problem is established.     

Constrained energy problems with applications to orthogonal polynomials of a discrete variable

Given a positive measure Σ with gs > 1, we write Με ℳ^Σ if Μ is a probability measure and Σ—Μ is a positive measure. Under some general assumptions on the constraining measure Σ and a weight functionw, we prove existence and uniqueness of a measure λ^Σ _w that minimizes the weighted logarithmic energy over the class ℳ^Σ. We also obtain a characterization theorem, a saturation result and a balayage representation for the measure λ^Σ _w As applications of our results, we determine the (normalized) limiting zero distribution for ray sequences of a class of orthogonal polynomials of a discrete variable. Explicit results are given for the class of Krawtchouk polynomials. The research done by this author is in partial fulfillment of the Ph.D. requirements at the University of South Florida. The research done by this author was supported, in part, by U.S. National Science Foundation under grant DMS-9501130.     

Computation of exact confidence limits from discrete data

Suppose that the data have a discrete distribution determined by (∞, ψ) where θ is the scalar parameter of interest and ψ is a nuisance parameter vector. The Buehler 1 - α upper confidence limit for θ is as small as possible, subject to the constraints that (a) its coverage probability is at least 1 - α and (b) it is a nondecreasing function of a pre-specified statisticT. This confidence limit has important biostatistical and reliability applications. The main result of the paper is that for a wide class of models (including binomial and Poisson), parameters of interest 9 and statisticsT (which possess what we call the “logical ordering” property) there is a dramatic increase in the ease with which this upper confidence limit can be computed. This result is illustrated numerically for θ a difference of binomial probabilities. Kabaila & Lloyd (2002) also show that ifT is poorly chosen then an assumption required for the validity of the formula for this confidence limit may not be satisfied. We show that for binomial data this assumption must be satisfied whenT possesses the “logical ordering” property.     

Multidimensional Lipschitz global optimization based on efficient diagonal partitions

This is a summary of the author’s PhD thesis, supervised by Yaroslav D. Sergeyev and defended on May 5, 2006, at the University of Rome “La Sapienza”. The thesis is written in English and is available from the author upon request. In this work, the global optimization problem of a multidimensional “black-box” function satisfying the Lipschitz condition over a hyperinterval with an unknown Lipschitz constant is considered. The objective function is assumed hard to evaluate. A new efficient diagonal scheme for constructing fast algorithms for solving this problem is examined and illustrated by developing several powerful global optimization methods. A deep theoretical study is performed which highlights the benefit of the approach introduced over traditionally used diagonal algorithms. Theoretical conclusions are confirmed by results of extensive numerical experiments.