Strong Uniqueness of Best Approximations from RS-sets in Banach Spaces

The problem of strong uniqueness of best approximation from an RS set in a Banach space is considered. For a fixed RS set G and an element x∈X , we proved that the best approximation g * to x from G is strongly unique.

Operators of Approximations and Approximate Power Set Spaces

Boundary inner and outer operators are introduced, and union, intersection, complement operators of approximations are redefined. The approximation operators have a good property of maintaining union, intersection, complement operators, so the rough set theory has been enriched from the operator-oriented and set-oriented views. Approximate power set spaces are defined, and it is proved that the approximation operators are epimorphisms from power set space to approximate power set spaces. Some basic properties of approximate power set space are got by epimorphisms in contrast to power set space.

Liquid Circulation in a Multi-tube Air-lift Loop Reactor

A multi-tube air-lift loop reactor (MT-ALR) is presented in this paper. Based on the energy conservation, a mathematical model describing the liquid circulation flow rate was developed, which was determined by gas velocity, the cross areas of riser and downcomer, gas hold-up and the local frictional loss coefficient. The experimental data indicate that either increase of gas flow rate or reduction of the downcomer diameter contributes to higher liquid circulation rate. The correlation between total and the local frictional loss coefficients was also established.Effects of gas flowrate in two risers and diameter of downcomer on the liquid circulation rate were examined. The value of total frictional loss coefficient was measured as a function of the cross area of downcomer and independent of the gas flow rate. The calculated results of liquid circulation rates agreed well with the experimental data with an average relative error of 9.6%.

POINTWISE CONVERGENCE RATE OF VANISHING VISCOSITY APPROXIMATIONS FOR SCALAR CONSERVATION LAWS WITH BOUNDARY

This article is concerned with the pointwise error estimates for vanishing vis- cosity approximations to scalar convex conservation laws with boundary.By the weighted error function and a bootstrap extrapolation technique introduced by Tadmor-Tang,an optimal pointwise convergence rate is derived for the vanishing viscosity approximations to the initial-boundary value problem for scalar convex conservation laws,whose weak entropy solution is piecewise C 2 -smooth with interaction of elementary waves and the ...

Generalized Galerkin Approximations for Pseudoinverses and Operator Equations of the First Kind

The main result of this paper is a basic theorem about generalized Galerkin approximations for pseudoinverses and operator equations of the first kind, which is presented as follows ：Let H be a Hilbert space, { Hn } a sequence of closed subspaces of H, Pn the orthogonal projection of H onto Hn, A∈B（H） and An∈B（Hn）. Suppose s-lim↑n→∞Hn=H, lim↑n→∞｜｜Pn°（A-An） ｜｜n=0,-↑R（An）=R（An）（n∈N）. Then the following four propositions are equivalent ： （a） sup↑n∈Nv∈An^-1 inf ｜｜υ｜｜〈∞ if un∈R（An） and lim↑n→∞un=0; （b） sup↑n∈N｜｜ An ｜｜ 〈∞; （c） if un∈R（An） and lim↑n→∞ un=u, then u∈R（A） and s-lim↑n→∞An^-1（un）=A^-1（u）; （d） if un∈R（An） and lim↑n→∞un=u.then u∈R（A） and lim↑n→∞Au^＋（un）=A^＋（u）. Furtherrnore, if any of the above propositions holds, we have thin N（A）=s-lim↑n→∞N（An ）,R（A） = s-lim↑n→∞R（An ）, -↑R（A） =R（A）.

Convergence of Invariant Measures of Truncation Approximations to Markov Processes

Let Q be the Q-matrix of an irreducible, positive recurrent Markov process on a countable state space. We show that, under a number of conditions, the stationary distributions of the n × n north-west corner augmentations of Q converge in total variation to the stationary distribution of the process. Two conditions guaranteeing such convergence include exponential ergodicity and stochastic monotonicity of the process. The same also holds for processes dominated by a stochastically monotone Markov process. In addition, we shall show that finite perturbations of stochastically monotone processes may be viewed as being dominated by a stochastically monotone process, thus extending the scope of these results to a larger class of processes. Consequently, the augmentation method provides an attractive, intuitive method for approximating the stationary distributions of a large class of Markov processes on countably infinite state spaces from a finite amount of known information.

A meshless algorithm with moving least square approximations for elliptic Signorini problems

Based on the moving least square （MLS） approximations and the boundary integral equations （BIEs）, a meshless algorithm is presented in this paper for elliptic Signorini problems. In the algorithm, a projection operator is used to tackle the nonlinear boundary inequality conditions. The Signorini problem is then reformulated as BIEs and the unknown boundary variables are approximated by the MLS approximations. Accordingly, only a nodal data structure on the boundary of a domain is required. The convergence of the algorithm is proven. Numerical examples are given to show the high convergence rate and high computational efficiency of the presented algorithm.

Well-Posedness and Finite Element Approximations for Elliptic SPDEs with Gaussian Noises

The paper studies the well-posedness and optimal error estimates of spectral finite element approximations for the boundary value problems of semi-linear elliptic SPDEs driven by white or colored Gaussian noises.The noise term is approximated through the spectral projection of the covariance operator,which is not required to be commutative with the Laplacian operator.Through the convergence analysis of SPDEs with the noise terms replaced by the projected noises,the well-posedness of the SPDE is established under certain covariance operator-dependent conditions.These SPDEs with projected noises are then numerically approximated with the finite element method.A general error estimate framework is established for the finite element approximations.Based on this framework,optimal error estimates of finite element approximations for elliptic SPDEs driven by power-law noises are obtained.It is shown that with the proposed approach,convergence order of white noise driven SPDEs is improved by half for one-dimensional problems,and by an infinitesimal factor for higher-dimensional problems.

Notes on Rough Set Approximations and Associated Measures

We review and compare two definitions of rough set approximations.One is defined by a pair of sets in the universe and the other by a pair of sets in the quotient universe.The latter definition,although less studied,is semantically superior for interpreting rule induction and is closely related to granularity switching in granular computing.Numerical measures about the accuracy and quality of approximations are examined.Several semantics difficulties are commented.

PROJECTION METHODS AND APPROXIMATIONS FOR ORDINARY DIFFERENTIAL EQUATIONS

A formulation of a differential equation as projection and fixed point pi-Mem alloivs approximations using general piecnvise functions. We prone existence and uniqueness of the up proximate solution* convergence in the L2 norm and nodal supercnnvergence. These results generalize those obtained earlier by Hulme for continuous piecevjise polynomials and by Delfour-Dubeau for discontinuous pieceuiise polynomials. A duality relationship for the two types of approximations is also given.

The Influence of the Exchange-Correlation Functional on the Non-Interacting Kinetic Energy and Its Implications for Orbital-Free Density Functional Approximations

In this work it is shown that the kinetic energy and the exchange-correlation energy are mutual dependent on each other.This aspect is first derived in an orbital-free context.It is shown that the total Fermi potential depends on the density only,the individual parts,the Pauli kinetic energy and the exchange-correlation energy,however,are orbital dependent and as such mutually influence each other.The numerical investigation is performed for the orbital-based non-interacting Kohn-Sham system in order to avoid additional effects due to further approximations of the kinetic energy.The numerical influence of the exchange-correlation functional on the non-interacting kinetic energy is shown to be of the orderof a few Hartrees.For chemical purposes,however,the energetic performance as a function of the nuclear coordinates is much more important than total energies.Therefore,the effect on the bond dissociation curve was studied exemplarily for the carbon monoxide.The data reveals that,the mutual influence between the exchange-correlation functional and the kinetic energy has a significant influence on bond dissociation energies and bond distances.Therefore,the effect of the exchange-correlation treatment must be considered in the design of orbital-free density functional approximations for the kinetic energy.

A Study of Weighted Polynomial Approximations with Several Variables (II)

In this paper we investigate weighted polynomial approximations with several variables. Our study relates to the approximation for by weighted polynomial. Then we will give some results relating to the Lagrange interpolation, the best approximation, the Markov-Bernstein inequality and the Nikolskii- type inequality.

On M–Term Approximations of the Nikol'skii–Besov Class in the Lorentz Spaces

In this paper, we consider a Lorentz space with a mixed norm of periodic functions of many variables. We obtain the exact estimation of the best M-term approximations of Nikol＇skii＇s and Besov＇s classes in the Lorentz space with the mixed norm.

CHARACTERIZATION OF BEST APPROXIMATIONS IN METRIC LINEAR SPACES

Let (X,d) be a real metric linear space, with translation-invariant metric d and C a linear subspace of X. In this paper we use functionals in the Lipschitz dual of X to characterize those elements of G which are best approximations to elements of X.We also give simultaneous characterization of elements of best approximation and also consider elements of ε-approximation.

Liquid Flow Distribution in a New Type of Multi-tube Column

A new type of multi-tube column, in which a novel internal-structure is installed in the bed of random packing, has been invented. The internal-structure reduces the hydraulic radius of the column and adjusts the gas/liquid flow so that the liquid maldistribution is greatly abated. A mathematical model with boundary conditions based on close-open mechanism at the internal-structure was established to study the hydraulic behavior of such a column and experiments were carded out to verify the applicability of the model. Predictions from the model agreed well with experimental results. The optimal ratio of the hydraulic diameter of the new multi-tube column to the packing element diameter is found to be 17.5 to 23.3 by employing the optimization method of nonlinear programming. And this new type of multi-tube column opens a new way for random packing columns to be scaled up for industry.

Exponential Convergence of Finite-dimensional Approximations to Linear Bond-based Peridynamic Boundary Value Problems

In this paper, we demonstrate that the finite-dimensional approximations to the solutions of a linear bond-based peridynamic boundary value problem converge to the exact solution exponentially with the analyticity assumption of the forcing term, therefore greatly improve the convergence rate derived in literature.

On the Behavior of Combination High-Order Compact Approximations with Preconditioned Methods in the Diffusion-Convection Equation

In this paper, a family of high-order compact finite difference methods in combination preconditioned methods are used for solution of the Diffusion-Convection equation. We developed numerical methods by replacing the time and space derivatives by compact finite-difference approximations. The system of resulting nonlinear finite difference equations are solved by preconditioned Krylov subspace methods. Numerical results are given to verify the behavior of high-order compact approximations in combination preconditioned methods for stability, convergence. Also, the accuracy and efficiency of the proposed scheme are considered.

Method of Successive Approximations for a Fluid Structure Interaction Problem

In this paper, we present a method for solving coupled problem. This method is mainly based on the successive approximations method. The external force acting on the structure is replaced by λ = p (x1, H + u (x1, λ)). Then we have a nonlinear equation of unknown?λ to solve by successive approximations method. By this method, we obtain easily the analytic expression of the displacement. In addition, good results are obtained with only a few iterations.

ON POINTWISE R-SUBWEAKLY COMMUTING MAPS AND BEST APPROXIMATIONS

The main purpose of this paper is to prove some common fixed point theorems for pointwise R-subweakly commuting maps on non-starshaped domains in p-normed spaces and locally convex topological vector spaces. As applications, invariant approximation results are established. This work provides extension as well as substantial improvement of several results in the existing literature.