Fixed point theorems for nonexpansive maps in Banach spaces

In this work, we present some new versions of fixed point theorems for nonexpansive maps and 1-set contractions defined on closed, convex, not necessarily bounded subsets of Banach spaces. Our proofs rely on a compactness result for an approximate fixed point set. The Kuratowski measure of noncompactness is used throughout. To illustrate the results obtained, some applications to Banach algebras and Hammerstein integral equations are provided.     

Local Lp-saturation of positive linear convolution operators

The approximation of unbounded functions by positive linear operators under multiplier enlargement is investigated. It is shown that a very wide class of positive linear operators can be used to approximate functions with arbitrary growth on the real line. Estimates are given in terms of the usual quantities which appear in the Shisha-Mond theorem. Examples are provided.     

G-convergence,Dirichlet to Neumann maps and invisibility

We establish optimal conditions under which the G-convergence of linear elliptic operators implies the convergence of the corresponding Dirichlet to Neumann maps. As an application we show that the approximate cloaking isotropic materials from [19] are independent of the source.     

Semicontinuity of the Approximate Solution Sets of Multivalued Quasiequilibrium Problems

We consider two kinds of approximate solutions and approximate solution sets to multivalued quasiequilibrium problems. Sufficient conditions for the lower semicontinuity, Hausdorff lower semicontinuity, upper semicontinuity, Hausdorff upper semicontinuity, and closedness of these approximate solution sets are established. Applications in approximate quasivariational inequalities, approximate fixed points, and approximate quasioptimization problems are provided.     

An approximate algorithm for prognostic modelling using condition monitoring information

Established condition based maintenance modelling techniques can be computationally expensive. In this paper we propose an approximate methodology using extended Kalman-filtering and condition monitoring information to recursively establish a conditional probability density function for the residual life of a component. The conditional density is then used in the construction of a maintenance/replacement decision model. The advantages of the methodology, when compared with alternative approaches, are the direct use of the often multi-dimensional condition monitoring data and the on-line automation opportunity provided by the computational efficiency of the model that potentially enables the simultaneous condition monitoring and associated inference for a large number of components and monitored variables. The methodology is applied to a vibration monitoring scenario and compared with alternative models using the case data.     

A General Cavitation Model for the Highly Nonlinear Mie-Grüneisen Equation of State

A general one-fluid cavitation model is proposed for a family of Mie-Grüneisen equations of state (EOS), which can provide a wide application of cavitation flows, such as liquid-vapour transformation and underwater explosion. An approximate Riemann problem and its approximate solver for the general cavitation model are developed. The approximate solver, which provides the interface pressure and normal velocity by an iterative method, is applied in computing the numerical flux at the phase interface for our compressible multi-medium flow simulation on Eulerian grids. Several numerical examples, including Riemann problems and underwater explosion applications, are presented to validate the cavitation model and the corresponding approximate solver.     

A multivariate version of Hoeffding’s Phi-Square

A multivariate measure of association is proposed, which extends the bivariate copula-based measure Phi-Square introduced by Hoeffding [22]. We discuss its analytical properties and calculate its explicit value for some copulas of simple form; a simulation procedure to approximate its value is provided otherwise. A nonparametric estimator for multivariate Phi-Square is derived and its asymptotic behavior is established based on the weak convergence of the empirical copula process both in the case of independent observations and dependent observations from strictly stationary strong mixing sequences. The asymptotic variance of the estimator can be estimated by means of nonparametric bootstrap methods. For illustration, the theoretical results are applied to financial asset return data.     

N-S方程一般近似惯性流形构造和逼近

1．引言为了用有限维常微分方程来研究Navier七lobes（N七）方程的长时间动力学行为,Foias,Sell和Temaml］引入了耗散系统惯性流形的概念．但惯性流形存在的一个相当苛刻的条件一谱间隔条件一是包括N习方程在内的很多耗散系统无法满足的．因此,Foias,Manlea和TemamZ］随后又提出了近似惯性流形的概念．近似惯性流形也是一个光滑的Lipschitz流形,所有原方程的解在时间充分大时,将被吸引进入该流形的一个三邻域中．因其存在性不需要谱间隔条件来保证,从而可证明包括N－S方程在内的一大类耗散系统存在近似惯性流形．利用近似惯性…     

Approximate Controllability for Integrodifferential Equations with Multiple Delays

This paper considers the approximate controllability for a class of control systems governed by semilinear delay integrodifferential equations with multiple delays. Sufficient conditions for approximate controllability are established by using the Schauder fixed-point theorem. The results obtained improve some analogous existing results. Several examples are provided to illustrate the application of the approximate controllability result.     

Controllability Properties of Linear Mean-Field Stochastic Systems

We study some controllability properties for linear stochastic systems of mean-field type. First, we give necessary and sufficient criteria for exact terminal-controllability. Second, we characterize the approximate and approximate null-controllability via duality techniques. Using Riccati equations associated to linear quadratic problems in the control of mean-field systems, we provide a (conditional) viability criterion for approximate null-controllability. In the classical diffusion framework, approximate and approximate null-controllability are equivalent. This is no longer the case for mean-field systems. We provide sufficient (algebraic) invariance conditions implying approximate null-controllability. We also present a general class of systems for which our criterion is equivalent to approximate null-controllability property. We also introduce some rank conditions under which approximate and approximate null-controllability are equivalent. Several examples and counter-examples as well as a partial algorithm are provided.     

HIGH ORDER FINITE DIFFERENCE/SPECTRAL METHODS TO A WATER WAVE MODEL WITH NONLOCAL VISCOSITY

In this paper, efficient numerical schemes are proposed for solving the water wave model with nonlocal viscous term that describe the propagation of surface water wave. By using the Caputo fractional derivative definition to approximate the nonlocal fractional operator, finite difference method in time and spectral method in space are constructed for the considered model. The proposed method employs known 5/2 order scheme for fractional derivative and a mixed linearization for the nonlinear term. The analysis shows that the proposed numerical scheme is unconditionally stable and error estimates are provided to predict that the second order backward differentiation plus 5/2 order scheme converges with order 2 in time, and spectral accuracy in space. Several numerical results are provided to verify the efficiency and accuracy of our theoretical claims. Finally, the decay rate of solutions is investigated.     

Asymptotic properties of particle filter-based maximum likelihood estimators for state space models

We study the asymptotic performance of approximate maximum likelihood estimators for state space models obtained via sequential Monte Carlo methods. The state space of the latent Markov chain and the parameter space are assumed to be compact. The approximate estimates are computed by, firstly, running possibly dependent particle filters on a fixed grid in the parameter space, yielding a pointwise approximation of the log-likelihood function. Secondly, extensions of this approximation to the whole parameter space are formed by means of piecewise constant functions or B-spline interpolation, and approximate maximum likelihood estimates are obtained through maximization of the resulting functions. In this setting we formulate criteria for how to increase the number of particles and the resolution of the grid in order to produce estimates that are consistent and asymptotically normal.     

线性流形上广义次对称矩阵的左右逆特征值问题

研究线性流形上广义次对称矩阵的左右逆特征值问题及其最佳逼近问题.利用广义次对称矩阵的性质及矩阵的奇异值分解得到问题的通解表达式.同时,给出其有唯一的最佳逼近解以及求最佳逼近解的算法.     

Approximate controllability of second-order distributed implicit functional systems

In this paper we study the approximate controllability of control systems with states and controls in Hilbert spaces, and described by a second-order semilinear abstract functional differential equation with infinite delay. Initially we establish a characterization for the approximate controllability of a second-order abstract linear system and, in the last section, we compare the approximate controllability of a semilinear abstract functional system with the approximate controllability of the associated linear system.     

Transitive decomposition of symmetry groups for the n-body problem

Periodic and quasi-periodic solutions of the n-body problem are critical points of the action functional constrained to the Sobolev space of symmetric loops. Variational methods yield collisionless orbits provided the group of symmetries fulfills certain conditions (such as the rotating circle property). Here we generalize such conditions to more general group types and show how to constructively classify all groups satisfying such hypothesis, by a decomposition into irreducible transitive components. As examples we show approximate trajectories of some of the resulting symmetric minimizers.     

Approximate solutions and scalarization in set-valued optimization

Abstract

Certain notions of approximate weak efficient solutions are considered for a set-valued optimization problem based on vector and set criteria approaches. For approximate solutions based on the vector approach, a characterization is provided in terms of an extended Gerstewitz’s function. For the set approach case, two notions of approximate weak efficient solutions are introduced using a lower and an upper quasi order relations for sets and further compactness and stability aspects are discussed for these approximate solutions. Existence and scalarization using a generalized Gerstewitz’s function are also established for approximate solutions, based on the lower set order relation.     

利用近似能量极小构造平面C1三次Hermite插值曲线

研究了利用近似能量极小构造平面$C^1$三次Hermite插值曲线的方法.该方法的主要的目是求出$C^1$三次Hermite插值曲线的最佳切矢.通过将应变能、曲率变化能和组合能的近似函数极小化,得到了求解最佳切矢的线性方程组.通过求解发现,近似曲率变化能极小不存在唯一解, 而近似应变能极小和近似组合能极小由于方程系统的系数矩阵为严格对角占优故都存在唯一解.最后, 通过实例表明了本文方法构造平面$C^1$三次Hermite插值曲线的有效性.     

Approximate controllability of semilinear measure driven systems

This paper investigates approximate controllability of semilinear measure driven equations in Hilbert spaces. By using the semigroup theory and Schauder fixed point theorem, sufficient conditions for approximate controllability of measure driven equations are established. The obtained results are a generalization and continuation of the recent results on this issue. Finally, an example is provided to illustrate the application of the obtained results.     

Best Lp approximate solutions of nonlinear integrodifferential equations

In this paper best L_p approximate solutions are shown to exist for a wide class of integrodifferential equations. Using approximation theory techniques, a local existence theorem for solutions is established, and the convergence of the best approximate solutions to a solution is shown.     

A Spectral Method for Neutral Volterra Integro-Differential Equation with Weakly Singular Kernel

This paper is concerned with obtaining an approximate solution and an approximate derivative of the solution for neutral Volterra integro-differential equation with a weakly singular kernel. The solution of this equation, even for analytic data, is not smooth on the entire interval of integration. The Jacobi collocation discretization is proposed for the given equation. A rigorous analysis of error bound is also provided which theoretically justifies that both the error of approximate solution and the error of approximate derivative of the solution decay exponentially in $L^∞$ norm and weighted $L^2$ norm. Numerical results are presented to demonstrate the effectiveness of the spectral method.