The convergence of the best discrete linear Lp approximation as p → 1

It is well known that the best discrete linear L_p approximation converges to a special best Chebyshev approximation as p → ∞. In this paper it is shown that the corresponding result for the case p → 1 is also true. Furthermore, the special best L₁ approximation obtained as the limit is characterized as the unique solution of a nonlinear programming problem on the set of all L₁ solutions.     

On the relationship between the moving finite-element procedure and best piecewise L2 fits with adjustable nodes

It is shown that, for a class of time-dependent partial differential equations of the form u_t = ??u, one step of the moving finite-element (MFE) procedure corresponds to one iteration of an algorithm for obtaining best L₂ fits with adjustable nodes to continuous functions. In the steady-state limit the MFE procedure gives the best fit of ??u, with adjustable nodes, to the null function. For first-order partial differential equations, the MFE procedure moves nodes with approximate characteristic nodal speeds. We identify an additional speed component arising directly from the L₂ projection which seeks a best fit in the sense described above. © 1994 John Wiley & Sons, Inc.     

Fuzzy system reliability analysis by the use of Tω (the weakest t-norm) on fuzzy number arithmetic operations

In general, the sup-min convolution has been used for fuzzy arithmetic to analyze fuzzy system reliability, where the reliability of each system component is represented by fuzzy numbers. It is well known that T_ω-based addition preserves the shape of L-R type fuzzy numbers. In this paper, we show T_ω-based multiplication also preserves the shape of L-R type fuzzy numbers. We then apply T_ω-based arithmetic operations to fuzzy system reliability analysis. In fact, we show that we can simplify fuzzy arithmetic operations and even get the exact solutions for L-R type fuzzy system reliability, while others [Singer, Fuzzy Sets Syst. 34 (1990) 145; Cheng and Mon, Fuzzy Sets Syst. 56 (1993) 29; Chen, Fuzzy Sets Syst. 64 (1994) 31] have got the approximate solutions using sup-min convolution for evaluating fuzzy system reliability.     

Weak and Strong Solutions for the Stokes Approximation of Non-homogeneous Incompressible Navier-Stokes Equations

In this paper,the Dirichlet problem of Stokes approximate of non-homogeneous incompressibleNavier-Stokes equations is studied.It is shown that there exist global weak solutions as well as global andunique strong solution for this problem,under the assumption that initial density ρ_0(x)is bounded away from0 and other appropriate assumptions(see Theorem 1 and Theorem 2).The semi-Galerkin method is applied toconstruct the approximate solutions and a prior estimates are made to elaborate upon the compactness of theapproximate solutions.     

Waiting-Time Behavior in a Nonlinear Diffusion Equation

We study the nonlinear diffusion equation u_t*=(uⁿu_x)_x, which occurs in the study of a number of problems. Using singular-perturbation techniques, we construct approximate solutions of this equation in the limit of small n. These approximate solutions reveal simply the consequences of this variable diffusion coefficient, such as the finite propagation speed of interfaces and waiting-time behavior (when interfaces wait a finite time before beginning to move), and allow us to extend previous results for this equation.     

A modified truncated singular value decomposition method for discrete ill‐posed problems

Truncated singular value decomposition is a popular method for solving linear discrete ill‐posed problems with a small to moderately sized matrix A. Regularization is achieved by replacing the matrix A by its best rank‐k approximant, which we denote by A_k. The rank may be determined in a variety of ways, for example, by the discrepancy principle or the L‐curve criterion. This paper describes a novel regularization approach, in which A is replaced by the closest matrix in a unitarily invariant matrix norm with the same spectral condition number as A_k. Computed examples illustrate that this regularization approach often yields approximate solutions of higher quality than the replacement of A by A_k.Copyright © 2014 John Wiley & Sons, Ltd.     

ON MULTI-DIMENSIONAL SONIC-SUBSONIC FLOW

In this paper, a compensated compactness framework is established for sonicsubsonic approximate solutions to the n-dimensional (n ≥ 2) Euler equations for steady irrotational flow that may contain stagnation points. This compactness framework holds provided that the approximate solutions are uniformly bounded and satisfy H 1 loc (Ω) compactness conditions. As illustration, we show the existence of sonic-subsonic weak solution to n-dimensional (n ≥ 2) Euler equations for steady irrotational flow past obstacles or through an infinitely long nozzle. This is the first result concerning the sonic-subsonic limit for n-dimension (n ≥ 3).     

Approximation with sums of exponentials in Lp[0, ∞)

We consider the problem of approximating a given f from L_p [0, ∞) by means of the family V_n(S) of exponential sums; V_n(S) denotes the set of all possible solutions of all possible nth order linear homogeneous differential equations with constant coefficients for which the roots of the corresponding characteristic polynomials all lie in the set S. We establish the existence of best approximations, show that the distance from a given f to V_n(S) decreases to zero as n becomes infinite, and characterize such best approximations with a first-order necessary condition. In so doing we extend previously known results that apply in L_p[0, 1].     

The computation of a best monotone L p approximation for 1 ≤ p < ∞

An algorithm for the computation of a best monotone L _p approximate 1≤p<∞ is presented and its convergence theorems are established.     

Continuous approximate solutions in parametric optimization

The question of the existence of approximate solutions in parametric optimization is considered. Most results show that (under hypotheses) if a certain optimization problem has an approximate solution x ₀ for a value p ₀ of a parameter, then an approximate solution x=b(p) can be found for p in P, with b continuous, b(p ₀)=x₀, and any two such bs are homotopic. Some topological methods (use of fibrations) are used to weaken the usual convex hypotheses of such results. An equisemicontinuity condition (relative to a constraint) is introduced to allow some noncompactness. The results are applied to get approximate Nash equilibrium results for games with some nonconvexity in the strategy sets.     

N-waves in elasticity

We study the large time behavior of solutions of the one-dimensional equations of elasticity, typically a nonconvex system of conservation laws. We show that solutions with compact initial support converge in L₁ to a superposition of N-waves, at algebraic rate. Likewise, we show that the perturbation expansion of weakly nonlinear geometric optics is uniformly valid to second order. Our analysis uses approximate scalar laws, together with L₁ stability of scalar solutions and decay in total variation of the wave speed.     

Free/Congested Two-Phase Model from Weak Solutions to Multi-Dimensional Compressible Navier-Stokes Equations

We approximate a two–phase model by the compressible Navier-Stokes equations with a singular pressure term. Up to a subsequence, these solutions are shown to converge to a global weak solution of the compressible system with the congestion constraint studied for instance by Lions and Masmoudi. The paper is an extension of the previous result obtained in one-dimensional setting by Bresch et al. to the multi-dimensional case with heterogeneous barrier for the density.     

A class of functional equation and fractal interpolation functions

A new class of functional equation in C0(I) is investigated. It is proved that some class of FIF satisfies the functional equation. Another functional equation is constructed. Theirsolutions can approximate FIF arbitrarily. And a new approximate estimate between FIF andinterpolated function is given.     

Constrained Lp-Approximation by Generalized n-Convex Functions Induced by ECT-Systems

The problem of finding a best L_p-approximation (1 ≤ p < ∞) to a function in L_p from a special subcone of generalized n-convex functions induced by an ECT-system is considered. Tchebycheff splines with a countably infinite number of knots are introduced and best approximations are characterized in terms of local best approximations by these splines. Various properties of best approximations and their uniqueness in L₁ are investigated. Some special results for generalized monotone and convex cases are obtained.     

Homotopy analysis method for multiple solutions of the fractional Sturm-Liouville problems

In this paper, Homotopy Analysis Method (HAM) is applied to numerically approximate the eigenvalues of the fractional Sturm-Liouville problems. The eigenvalues are not unique. These multiple solutions, i.e., eigenvalues, can be calculated by starting the HAM algorithm with one and the same initial guess and linear operator L\mathcal{L}. It can be seen in this paper that the auxiliary parameter (^h/_2p),\hbar, which controls the convergence of the HAM approximate series solutions, has another important application. This important application is predicting and calculating multiple solutions.     

Global optimal approximate solutions

Given non-void subsets A and B of a metric space and a non-self mapping T:A? B{T:A\longrightarrow B}, the equation T x = x does not necessarily possess a solution. Eventually, it is speculated to find an optimal approximate solution. In other words, if T x = x has no solution, one seeks an element x at which d(x, T x), a gauge for the error involved for an approximate solution, attains its minimum. Indeed, a best proximity point theorem is concerned with the determination of an element x, called a best proximity point of the mapping T, for which d(x, T x) assumes the least possible value d(A, B). By virtue of the fact that d(x, T x) ≥ d(A, B) for all x in A, a best proximity point minimizes the real valued function x? d(x, T x){x\longrightarrow d(x, T\,x)} globally and absolutely, and therefore a best proximity in essence serves as an ideal optimal approximate solution of the equation T x = x. The aim of this article is to establish a best proximity point theorem for generalized contractions, thereby producing optimal approximate solutions of certain fixed point equations. In addition to exploring the existence of a best proximity point for generalized contractions, an iterative algorithm is also presented to determine such an optimal approximate solution. Further, the best proximity point theorem obtained in this paper generalizes the well-known Banach’s contraction principle.     

Best simultaneous L1 approximations

Three possible definitions are proposed for best simultaneous L₁ approximation to n continuous real-valued functions, and the relation between best simultaneous approximations and best L₁ approximations to the arithmetic mean of the n functions is discussed.     

Robust M-estimators of location vectors

Let X₁,…,X_n be i.i.d. random vectors in R^m, let θεR^m be an unknown location parameter, and assume that the restriction of the distribution of X₁−θ to a sphere of radius d belongs to a specified neighborhood of distributions spherically symmetric about 0. Under regularity conditions on and d, the parameter θ in this model is identifiable, and consistent M-estimators of θ (i.e., solutions of Σ_i=1ⁿψ(|X_i− |)(X_i− )=0) are obtained by using “re-descenders,” i.e., ψ's wh satisfy ψ(x)=0 for x≥c. An iterative method for solving for is shown to produce consistent and asymptotically normal estimates of θ under all distributions in . The following asymptotic robustness problem is considered: finding the ψ which is best among the re-descenders according to Huber's minimax variance criterion.     

Analysis of the behavior of a weak solution to <Emphasis Type="Italic">m</Emphasis>-Hessian equations near the boundary of a domain

We consider the Dirichlet problem for the m-Hessian equations F _m [u] = f in a domain Ω and analyze the behavior of approximate solutions at the boundary of Ω. We show that the growth rate for weak solutions towards to the boundary locally depends on the summability exponent of f or on the fact whether f belongs to a certain Morrey type space near the boundary. The result obtained can be used for estimating the H?lder constant for weak solutions in the closed domain. Bibliography: 11 titles.     

Best proximity points: approximation and optimization

A best proximity point theorem explores the existence of an optimal approximate solution, known as a best proximity point, to the equations of the form Tx = x where T is a non-self mapping. The purpose of this article is to establish some best proximity point theorems for non-self non-expansive mappings, non-self Kannan- type mappings and non-self Chatterjea-type mappings, thereby producing optimal approximate solutions to some fixed point equations. Also, algorithms for determining such optimal approximate solutions are furnished in some cases.