A STONE-WEIERSTRASS THEOREM FOR CERTAIN DISCONTINUOUS FUNCTIONS

La Abe point separating uiatal subalgebra. of C(T) where T is a compact metric space. For each bounded function f:T which is continuous on the complement of a meagre subset of T there exists a sequence (wn) of elements of the algebra A such that the sequence (w) converges uniformly to the function f on each compact subset of the interior of the continuity points of the function f.

ROBUST ANALYSIS AND GLOBAL MINIMIZATION OF A CLASS OF DISCONTINUOUS FUNCTIONS (Ⅱ)

In this Paper we continue to investigate global minimization problems. An integral approach is applied to treat a global minimization problem of a discontinuous function. With the help ofthe theory of measure (Q-measure) and integration, optimality conditions of a robust function over arobust set are derived. Algorithms and their implementations for finding global minima are proposed.Numerical tests and applications show that the algorithms are effective.

Integral Global Minimization of Constrained Problems with Discontinuous Penalty Functions

A class of discontinuous penalty functions was proposed to solve constrained minimization problems with the integral approach to global optimization, m-mean value and v-variance optimality conditions of a constrained and penalized minimization problem were investigated. A nonsequential algorithm was proposed. Numerical examples were given to illustrate the effectiveness of the algorithm.

Multistability of delayed complex-valued recurrent neural networks with discontinuous real-imaginarytype activation functions

In this paper, the multistability issue is discussed for delayed complex-valued recurrent neural networks with discontinuous real-imaginary-type activation functions. Based on a fixed theorem and stability definition, sufficient criteria are established for the existence and stability of multiple equilibria of complex-valued recurrent neural networks. The number of stable equilibria is larger than that of real-valued recurrent neural networks, which can be used to achieve high-capacity associative memories. One numerical example is provided to show the effectiveness and superiority of the presented results.

Interaction of elementary waves of scalar conservation laws with discontinuous flux function

In this paper, the Riemann solutions for scalar conservation laws with discontinuous flux function were constructed. The interactions of elementary waves of the conservation laws were concerned, and the numerical simulations were given.

Robust stability of mixed Cohen–Grossberg neural networks with discontinuous activation functions

Purpose–The purpose of this paper is to develop a method for the existence,uniqueness and globally robust stability of the equilibrium point for Cohen–Grossberg neural networks with time-varying delays,continuous distributed delays and a kind of discontinuous activation functions.Design/methodology/approach–Basedonthe Leray–Schauderalternativetheoremand chainrule,by using a novel integral inequality dealing with monotone non-decreasing function,the authors obtain a delay-dependent sufficient condition with less conservativeness for robust stability of considered neural networks.Findings–Itturns out thattheauthors’delay-dependent sufficientcondition canbeformed intermsof linear matrix inequalities conditions.Two examples show the effectiveness of the obtained results.Originality/value–The novelty of the proposed approach lies in dealing with a new kind of discontinuous activation functions by using the Leray–Schauder alternative theorem,chain rule and a novel integral inequality on monotone non-decreasing function.

μ-stability of multiple equilibria in Cohen-Grossberg neural networks and its application to associative memory

In this paper, the μ-stability of multiple equilibrium points(EPs) in the Cohen-Grossberg neural networks(CGNNs) is addressed by designing a kind of discontinuous activation function(AF). Under some criteria, CGNNs with this AF are shown to possess at least 5^(n)EPs, of which 3^(n)EPs are locally μ-stable. Compared with the saturated AF or the sigmoidal AF, CGNNs with the designed AF can produce many more total/stable EPs. Therefore, when CGNNs with the designed discontinuous AF are applied to associative memory, they can store more prototype patterns. Moreover, the AF is expanded to a more general version to further increase the number of total/stable equilibria. The CGNNs with the expanded AF are found to produce(2k+3)^(n)EPs, of which (k+2)^(n)EPs are locally μ-stable. By adjusting two parameters in the AF, the number of sufficient conditions ensuring the μ-stability of multiple equilibria can be decreased. This finding implies that the computational complexity can be greatly reduced.Two numerical examples and an application to associative memory are illustrated to verify the correctness of the obtained results.

The numerical simulation of interface crack propagation without re-meshing

Based on the framework of the extended finite element method (XFEM), the enriched exponent discontinuous function is modified properly by introducing the rigidity ratio of two sides materials of interface crack, and the portion integral scheme is adopted for interface elements containing two materials. To embody the singularity of the crack tip, the triangle function is introduced directly. What’s more, the maximum loop stress fracture criterion is adopted to determine the extension direction in extended material domains, and the true extension distance for each load step is determined by reducing or increasing half the current trial extension distance until the equivalent stress intensity factor reaches the type I fracture toughness of material. Finally, with the improved XFEM, the interface crack propagation in a cantilever deep beam and concrete gravity dam are simulated without re-meshing respectively and their failure modes are also analyzed.