On a differential game in a nondamped distributed system

We study a differential game of approach in a system whose dynamics is described by a Sobolev‐type second‐order operator differential equation in Hilbert spaces. Solutions of the equation are represented by cosine and sine operator functions. To obtain solvability conditions of the game problem, we use support functionals of two sets, which defined by the behaviors of pursuer and evader. The results are applied to the investigation of conflicted‐controlled dynamics of bending waves in a rod.     

Two-level value function approach to non-smooth optimistic and pessimistic bilevel programs

ABSTRACT

The authors' paper in Dempe et al. [Necessary optimality conditions in pessimistic bilevel programming. Optimization. 2014;63:505–533], was the first one to provide detailed optimality conditions for pessimistic bilevel optimization. The results there were based on the concept of the two-level optimal value function introduced and analysed in Dempe et al. [Sensitivity analysis for two-level value functions with applications to bilevel programming. SIAM J. Optim. 22 (2012), 1309–1343], for the case of optimistic bilevel programs. One of the basic assumptions in both of these papers is that the functions involved in the problems are at least continuously differentiable. Motivated by the fact that many real-world applications of optimization involve functions that are non-differentiable at some points of their domain, the main goal of the current paper is to extend the two-level value function approach by deriving new necessary optimality conditions for both optimistic and pessimistic versions in bilevel programming with non-smooth data.     

Feedback control variables to restrain the Babesiosis disease

In this paper, we complete the study of the dynamics of a recognized continuous‐time model for the Babesiosis disease. The local and global asymptotic stability of the endemic state are established theoretically and experimentally. In addition, to restrain the disease in the original model when the endemic state exists, we propose and study the continuous model with feedback controls. The global stability of the boundary‐equilibrium point of this model is analyzed by means of rigorous mathematical methods. As an important consequence of this result, we propose a strategy to select feedback control variables in order to restrain the disease in the original model. This strategy allows us to make the disease vanish completely. In other words, the feedback controls are specially effective for restraining disease in the model. The validity of the established theoretical result is supported by a set of numerical simulations.     

On the fractional derivatives of radial basis functions: Theories and applications

The paper provides the fractional integrals and derivatives of the Riemann‐Liouville and Caputo type for the five kinds of radial basis functions, including the Powers, Gaussian, Multiquadric, Matérn, and Thin‐plate splines, in one dimension. It allows to use high‐order numerical methods for solving fractional differential equations. The results are tested by solving two test problems. The first test case focuses on the discretization of the fractional differential operator while the second considers the solution of a fractional order differential equation.     

Additive Maps Preserving Nilpotent Perturbation of Scalars

Let X be a Banach space over F(= R or C) with dimension greater than 2. Let N(X) be the set of all nilpotent operators and B_0(X) the set spanned by N(X). We give a structure result to the additive maps on FI + B_0(X) that preserve rank-1 perturbation of scalars in both directions. Based on it, a characterization of surjective additive maps on FI + B_0(X) that preserve nilpotent perturbation of scalars in both directions are obtained. Such a map Φ has the form either Φ(T) = cAT A~(-1)+ φ(T)I for all T ∈ FI + B_0(X) or Φ(T) = cAT*A~(-1)+ φ(T)I for all T ∈ FI + B_0(X), where c is a nonzero scalar,A is a τ-linear bijective transformation for some automorphism τ of F and φ is an additive functional.In addition, if dim X = ∞, then A is in fact a linear or conjugate linear invertible bounded operator.     

Restricted thresholding: recovering smoothness and preserving edges

Restricted non linear approximation is a generalization of the N‐term approximation in which a measure on the index set of the approximants controls the type, instead of the number, of elements in the approximation. Thresholding is a well‐known type of non linear approximation. We relate a generalized upper and lower Temlyakov property with the decreasing rate of the thresholding approximation. This relation is in the form of a characterization through some general discrete Lorentz spaces. Thus, not only we recover some results in the literature but find new ones. As an application of these results, we compress and reduce noise of some images with wavelets and shearlets and show, at least empirically, that the L²‐norm is not necessarily the best norm to measure the approximation error.     

Isometric Extension Problem Between Strictly Convex Two-dimensional Normed Spaces

In this paper, based on the smooth point of the unit ball and its support linear functional,we show two equivalent formulations of the isometric extension problem between the unit spheres of strictly convex two-dimensional normed spaces. We prove that these equivalent formulations have a positive answer in a special case.     

Analytic combinatorics of connected graphs

We enumerate the connected graphs that contain a number of edges growing linearly with respect to the number of vertices. So far, only the first term of the asymptotics and a bound on the error were known. Using analytic combinatorics, that is, generating function manipulations, we derive a formula for the coefficients of the complete asymptotic expansion. The same result is derived for connected multigraphs.     

Pseudo almost periodic in distribution solutions to impulsive partial stochastic functional differential equations

In this paper, we study the piecewise pseudo almost periodicity in distribution for a stochastic process. Using the analytic semigroup theory and fixed point strategy with stochastic analysis theory, we obtain the existence and the exponential stability of piecewise pseudo almost periodic in distribution mild solutions for impulsive partial neutral stochastic functional differential equations under non-Lipschitz conditions. Moreover, an example is given to illustrate the general theorems.     

基于渐进演化策略的增材制造自支撑结构拓扑优化算法

拓扑优化和增材制造分别是先进的结构设计技术和制造技术,将拓扑优化与增材制造融合能产生显著的协同效益.增材制造存在一些独特的制造约束,研究考虑结构自支撑约束的拓扑优化算法,可以降低材料和时间成本.基于SIMP方法框架,建立显式约束函数模型表征结构的自支撑特性,并发展了相应的拓扑优化流程,通过结构渐进演化实现自支撑.研究了相应的灵敏度分析方法,可实现并行计算.提出一个指向性灵敏度过滤算子促进支撑结构演化.采用3个数值算例进行分析,验证了指向性灵敏度过滤的有效性,所有拓扑优化结果均实现了结构自支撑.与典型方法相比,优化结果的可制造性更好.