A new chaotic Hopfield network with piecewise linear activation function

This paper presents a new chaotic Hopfield network with a piecewise linear activation function. The dynamic of the network is studied by virtue of the bifurcation diagram, Lyapunov exponents spectrum and power spectrum. Numerical simulations show that the network displays chaotic behaviours for some well selected parameters.

On Characterization of Poised Nodes for a Space of Bivariate Functions

There are several examples of spaces of univariate functions for which we have a characterization of all sets of knots which are poised for the interpolation problem. For the standard spaces of univariate polynomials, or spline functions the mentioned results are well-known. In contrast with this, there are no such results in the bivariate case. As an exception, one may consider only the Pascal classic theorem, in the interpolation theory interpretation. In this paper, we consider a space of bivariate piecewise linear functions, for which we can readily find out whether the given node set is poised or not. The main tool we use for this purpose is the reduction by a basic subproblem, introduced in this paper.

Piecewise linear programming approach to solve multi-objective matrix games with I-fuzzy goals

The intuitionistic fuzzy set(I-fuzzy set)plays an effective role in game theory when players face‘neither this nor that’situation to set their goals.This study presents a maxmin–minmax solution to multi-objective two person zero-sum matrix games with I-fuzzy goals.In this article,a class of piecewise linear membership and non-membership functions for I-fuzzy goals is constructed.These functions are more effective in real games because marginal rate of increase(decrease)of such membership functions(non-membership functions)is different in different intervals of tolerance errors.Finally,one numerical example is given to examine the effectiveness and advantages of the proposed results.

RELU DEEP NEURAL NETWORKS AND LINEAR FINITE ELEMENTS

In this paper,we investigate the relationship between deep neural net works(DNN)with rectified linear unit(ReLU)function as the activation function and continuous piecewise linear(CPWL)functions,especially CPWL functions from the simplicial linear finite element method(FEM).We first consider the special case of FEM.By exploring the DNN representation of its nodal basis functions,we present a ReLU DNN representation of CPWL in FEM.We theoretically establish that at least 2 hidden layers are needed in a ReLU DNN to represent any linear finite element functions inΩ■R^2 when d≥2.Consequently,for d=2,3 which are often encountered in scientific and engineering computing,the minimal number of two hidden layers are necessary and sufficient for any CPWL function to be represented by a ReLU DNN.Then we include a detailed account on how a general CPWL in R^d can be represented by a ReLU DNN with at most[log2(d+1)]|hidden layers and we also give an estimation of the number of neurons in DNN that are needed in such a represe ntation.Furthermore,using the relationship bet ween DNN and FEM,we theoretically argue that a special class of DNN models with low bit-width are still expected to have an adequate representation power in applications.Finally,as a proof of concept,we present some numerical results for using ReLU DNNs to solve a two point boundary problem to demonstrate the potential of applying DNN for numerical solution of partial differential equations.

When Does the Equality J（X＊） = J（X） Hold for a Two-dimensional Banach Space X ？

In this paper, we consider the following problem about the James constant： When does the equality J（X＊） = J（X） hold for a Banach space X ？ It is known that the James constant of a Banach space does not coincide with that of its dual space in general. In fact, we already have counterexamples of two-dimensional normed spaces that are equipped with either symmetric or absolute norms. However,we show that if the norm on a two-dimensional space X is both symmetric and absolute, then the equality J（X＊） = J（X） holds. This provides a global answer to the problem in the two-dimensional case.