a class of global fractional order projective dynamical systems involving set valued perturbations this paper studies a class of global fractional order projective dynamical systems involving set valued perturbations in real separable hilbert spaces we prove that the set of solutions for this type of systems is nonempty and closed under some suitable conditions furthermore we show that the set of solutions is continuous with respect to initial value in the sense of hausdorff metric finally an interesting numerical example is given to illustrate the validity of the main theorem presented in this paper

global asymptotic and exponential stability of a dynamic neural system with asymmetric connection weights recently a dynamic neural system was presented and analyzed due to its good performance in optimization computation and low complexity for implementation the global asymptotic stability of such a dynamic neural system with symmetric connection weights was well studied in this note based on a new lyapunov function we investigate the global asymptotic stability of the dynamic neural system with asymmetric connection weights since the dynamic neural system with asymmetric weights is more general than that with symmetric ones the new results are significant in both theory and applications specially the new result can cover the asymptotic stability results of linear systems as special cases

a novel neural network for nonlinear convex programming in this paper we present a neural network for solving the nonlinear convex programming problem in real time by means of the projection method the main idea is to convert the convex programming problem into a variational inequality problem then a dynamical system and a convex energy function are constructed for resulting variational inequality problem it is shown that the proposed neural network is stable in the sense of lyapunov and can converge to an exact optimal solution of the original problem compared with the existing neural networks for solving the nonlinear convex programming problem the proposed neural network has no lipschitz condition no adjustable parameter and its structure is simple the validity and transient behavior of the proposed neural network are demonstrated by some simulation results

encoding information in chemical chaos by controlling symbolic dynamics in this paper we describe a technique of encoding and then decoding symbol sequences containing information into chaotic oscillations of the belousov zhabotinsky reaction the encoding technique is based on controlling the chaotic oscillations by applying small parameter perturbations and on learning the grammar of corresponding symbol dynamics produced by the free running chaotic system the use of small parameter perturbations requires that we respect the grammar of the symbol dynamics which represents the physical dynamical system we present a method for learning the grammar of a symbol dynamics in terms of allowed transitions between the bins defined by the symbol generating partition the encoding technique can be easily utilized for targeting and stabilization of any unstable periodic orbit s1063 651x 97 10904 7

a capable neural network model for solving the maximum flow problem this paper presents an optimization technique for solving a maximum flow problem arising in widespread applications in a variety of settings on the basis of the karush kuhn tucker kkt optimality conditions a neural network model is constructed the equilibrium point of the proposed neural network is then proved to be equivalent to the optimal solution of the original problem it is also shown that the proposed neural network model is stable in the sense of lyapunov and it is globally convergent to an exact optimal solution of the maximum flow problem several illustrative examples are provided to show the feasibility and the efficiency of the proposed method in this paper

one layer continuous and discrete time projection neural networks for solving variational inequalities and related optimization problems this paper presents one layer projection neural networks based on projection operators for solving constrained variational inequalities and related optimization problems sufficient conditions for global convergence of the proposed neural networks are provided based on lyapunov stability compared with the existing neural networks for variational inequalities and optimization the proposed neural networks have lower model complexities in addition some improved criteria for global convergence are given compared with our previous work a design parameter has been added in the projection neural network models and it results in some improved performance the simulation results on numerical examples are discussed to demonstrate the effectiveness and characteristics of the proposed neural networks

a novel neural network for variational inequalities with linear and nonlinear constraints variational inequality is a uniform approach for many important optimization and equilibrium problems based on the sufficient and necessary conditions of the solution this paper presents a novel neural network model for solving variational inequalities with linear and nonlinear constraints three sufficient conditions are provided to ensure that the proposed network with an asymmetric mapping is stable in the sense of lyapunov and converges to an exact solution of the original problem meanwhile the proposed network with a gradient mapping is also proved to be stable in the sense of lyapunov and to have a finite time convergence under some mild conditions by using a new energy function compared with the existing neural networks the new model can be applied to solve some nonmonotone problems has no adjustable parameter and has lower complexity thus the structure of the proposed network is very simple since the proposed network can be used to solve a broad class of optimization problems it has great application potential the validity and transient behavior of the proposed neural network are demonstrated by several numerical examples

a general methodology for designing globally convergent optimization neural networks we present a general methodology for designing optimization neural networks we prove that the neural networks constructed by using the proposed method are guaranteed to be globally convergent to solutions of problems with bounded or unbounded solution sets in contrast with the gradient methods whose convergence is not guaranteed we show that the proposed method contains both the gradient methods and nongradient methods employed in existing optimization neural networks as special cases based on the theoretical results of the proposed method we study the convergence and stability of general gradient models in the case of unisolated solutions using the proposed method we derive some new neural network models for a very large class of optimization problems in which the equilibrium points correspond to exact solutions and there is no variable parameter finally some numerical examples show the effectiveness of the method

on comparing dynamical systems by defective conjugacy a symbolic dynamics interpretation of commuter functions abstract while the field of dynamical systems has been focused on properties which are invariant to good change of variables namely conjugacy which is an equivalence relationship when using dynamical systems methods in science and modeling there lacks a dynamical way to compare dynamical systems even when they are in some sense close in skufca and bolt 2007 7 and skufca and bolt 2008 8 we introduced mathematics to support a philosophy that two dynamical systems should be compared through a change of coordinates between them that is a commuter between them which may fail to be a homeomorphism the progressive degree to which the commuter fails to be a homeomorphism defines what we call a homeomorphic defect however at the time of publication of these papers there were limits in the mathematical technology requiring that the transformations be one dimensional mappings and flows which are well described for construction of the commuters by fixed point iteration and further difficulties in numerically computing defects in the more complicated one dimensional cases and further limits to higher dimensional problems therefore here we extend the theory to allow for multivariate transformations with construction methods separate from the fixed point iteration and new methods to compute defect in the course of this work we introduce several new technical innovations in order to cope with much more general problems we introduce assignment mappings to understand and illustrate commuters in a broader setting we discuss the role of symbolic dynamics and coding as related to commuters as well as defect measure further we discuss refinement and convergence of a nested refinement of commuter representations this work represents a step forward in the possibility of using the commuter and defects to judge model quality in those dynamical systems for which a symbolic dynamics and hence a generating partition may be available while finding a generating partition is a problem in its own right we offer this work as further perspective for interpretation of the meaning of commuters and defect measure

common nature of learning between back propagation and hopfield type neural networks for generalized matrix inversion with simplified models in this paper two simple structure neural networks based on the error back propagation bp algorithm i e bp type neural networks bpnns are proposed developed and investigated for online generalized matrix inversion specifically the bpnn l and bpnn r models are proposed and investigated for the left and right generalized matrix inversion respectively in addition for the same problem solving task two discrete time hopfield type neural networks hnns are developed and investigated in this paper similar to the classification of the presented bpnn l and bpnn r models the presented hnn l and hnn r models correspond to the left and right generalized matrix inversion respectively comparing the bpnn weight updating formula with the hnn state transition equation for the specific i e left or right generalized matrix inversion we show that such two derived learning expressions turn out to be the same in mathematics although the bp and hopfield type neural networks are evidently different from each other a great deal in terms of network architecture physical meaning and training patterns numerical results with different illustrative examples further demonstrate the efficacy of the presented bpnns and hnns for online generalized matrix inversion and more importantly their common natures of learning

a homomorphic characterization of principal semiafls without using intersection with regular sets abstract if one is asked to consider which one of the semiafl or afl operations differs most from the remaining ones a set theoretically well grounded conclusion would be intersection with regular sets inspired by the result of culik fich and salomaa 1 we shall show that this operation can be replaced by catenation with an endmarker the basis of the paper is the result that any free principal semiafl m l can be presented in the form m l h 4 h 3 1 h 2 h 1 1 l each h 1 a homomorphism h 2 and h 3 free h 4 length preserving where is a symbol not in the alphabet of l if m l is not free we get the same representation but now h 4 erases the last letter from each word in h 3 1 h 2 1 h 1 1 l and is length preserving for other letters for the family of free regular languages we may choose l the last section contains a homomorphic characterization of regular star languages

nonlinear day to day traffic dynamics with driver experience delay modeling stability and bifurcation analysis abstract in day to day traffic assignment problems travelers past experiences have important impact on their cost prediction which influences their route choice and consequently the arising flow patterns in the network many travelers execute the same trip in every few days not daily which leads to time delays in the system in this paper we propose a nonlinear discrete time model with driver experience delay the linear stability of the stochastic user equilibrium is analyzed by studying the eigenvalues of the jacobian matrix of the system while the nonlinear oscillations arising at the bifurcations are investigated by normal form calculations numerical continuation and simulation the methods are demonstrated on a two route example by applying rigorous analysis we show that the linearly unstable parameter domain as well as the period of arising oscillations increase with the delay moreover delays and nonlinearities result in an extended domain of bistability where the stochastic user equilibrium coexists with stable and unstable oscillations this study explains the influence of initial conditions on the dynamics of transportation networks and may provide guidance for network design and management

stability of the modified projected dynamical systems abstract in this paper we propose and analyse a modified projection type dynamical system associated with variational inequalities by using the technique of updating the solution we prove that the globally asymptotic stability of this dynamical system requires only the pseudomonotonicity of the underlying operator which is a weaker condition than monotonicity the results obtained in this paper represent a significant improvement of the previously known results

dynamic system methods for solving mixed linear matrix inequalities and linear vector inequalities and equalities a novel idea is proposed for solving a system of mixed linear matrix inequalities and linear vector inequalities and equalities first the problem is converted into an unconstrained minimization problem with a continuously differentiable convex objective function then a continuous time dynamic system and a discrete time dynamic system are proposed for solving it under some mild conditions the proposed dynamic systems are shown to be globally convergent to a solution of the problem the merits of the methods refer to their simple numerical implementations and capability for handling nonstrict lmis easily in addition the methods are promising in neural circuits realization and therefore have potential applications in many online control problems several numerical examples are presented to illustrate the performance of the methods and substantiate the theoretical results

an efficient dynamic model for solving a portfolio selection with uncertain chance constraint models this paper presents a neural network model for solving maximization programming model with chance constraint in which the security returns are uncertain variables the main idea is to replace the portfolio selection models when the uncertain returns are chosen as some special cases such as linear uncertain variables trapezoidal uncertain variables and normal uncertain variables with a linear programming lp problem according to the saddle point theorem optimization theory convex analysis theory lyapunov stability theory and lasalle invariance principle the equilibrium point of the proposed neural network is proved to be equivalent to the optimal solution of the original problem it is also shown that the proposed neural network model is stable in the sense of lyapunov and it is globally convergent to an exact optimal solution of the portfolio selection problem with uncertain returns two illustrative examples are provided to show the feasibility and the efficiency of the proposed method in this paper

implicit dynamical systems and quasi variational inequalities in this paper we propose and analyze some implicit projected dynamical systems associated with quasi variational inequalities by using the techniques of the projection and the wiener hopf equations we prove that the globally asymptotic stability of these dynamical systems requires only the pseudomonotonicity of the operator which is a weaker condition than monotonicity we also discuss some special cases which can be obtained from our main results the results obtained in this paper represent a significant improvement of the previously known results

refining the hierarchy of blind multicounter languages and twist closed trios we introduce the new families k r rbc of languages accepted in quasi realtime by one way counter automata having k blind counters of which at least r are reversal bounded it is proved that these families form a strict and linear hierarchy of semi afls within the the family blind m c1 of blind multicounter languages with generator c1 w a1 b1 w a1 w b1 this thereby combines the families blind and rbc from 13 to one strict hierarchy and generalizes and sharpens greibachs results the strict inclusions between the k counter families k r rbc are proved using linear algebra techniques we also study the language theoretic monadic operation twist 18 20 in connection with the semi afls of languages accepted by multicounter and multipushdown acceptors all restricted to reversal bounded behavior it is verified that the family k r rbc is twist closed if and only if r 0 in which case k 0 rbc m ck ck being the k fold shuffle of disjoint copies of c1 we characterize the family m pal of languages accepted in quasi realtime by nondeterministic one way reversal bounded multipushdown acceptors as the least twist closed trio mtwist pal generated by the set of palindromes pal w a b w wrev

dynamic network traffic assignment considered as a continuous time optimal control problem two continuous time formulations of the dynamic traffic assignment problem are considered one that corresponds to system optimization and the other to a version of user optimization on a single mode network using optimal control theory pontryagin s necessary conditions are analyzed and given economic interpretations that correspond to intuitive notions regarding dynamic system optimized and dynamic user optimized traffic flow patterns notably we offer the first dynamic generalization of beckmann s equivalent optimization problem for static user optimized traffic assignment in the form of an optimal control problem the analysis further establishes that a constraint qualification and convexity requirements for the hamiltonian which together ensure that the necessary conditions are also sufficient are satisfied under commonly encountered regularity conditions

solving complementarity and variational inequalities problems using neural networks in this paper we propose a recurrent neural network model for solving a class of monotone variational inequalities problem with linear constraints the neural network is stable in the sense of lyapunov and globally convergent to an optimal solution compared with the existing convergence results the present proof do not require lipschitz continuity condition on the objective function this neural network model has no adjustable parameter thus its structure is very simple variational inequalities problem with general set of constraints plus a general form of the complementarity problems are solved using the proposed neural networks some examples demonstrated to show the applicability of the proposed neural networks to solve various nonlinear optimization problems numerically

a system of fractional order interval projection neural networks in this paper a system of fractional order interval projection neural networks is introduced and investigated under some suitable assumptions the existence and uniqueness of the equilibrium point of this type of interval projection neural networks is proved moreover exponential stability of this type of neural networks is obtained also in the last section we give two numerical examples to demonstrate the validity of our results

a novel neural network for generally constrained variational inequalities this paper presents a novel neural network for solving generally constrained variational inequality problems by constructing a system of double projection equations by defining proper convex energy functions the proposed neural network is proved to be stable in the sense of lyapunov and converges to an exact solution of the original problem for any starting point under the weaker cocoercivity condition or the monotonicity condition of the gradient mapping on the linear equation set furthermore two sufficient conditions are provided to ensure the stability of the proposed neural network for a special case the proposed model overcomes some shortcomings of existing continuous time neural networks for constrained variational inequality and its stability only requires some monotonicity conditions of the underlying mapping and the concavity of nonlinear inequality constraints on the equation set the validity and transient behavior of the proposed neural network are demonstrated by some simulation results

exponential stability of globally projected dynamic systems in this paper we further analyze and prove the stability and convergence of the dynamic system proposed by friesz et al 1994 whose equilibria solve the associated variational inequality problems two sufficient conditions are provided to ensure the asymptotic stability of this system with a monotone and asymmetric mapping by means of an energy function meanwhile this system with a monotone and gradient mapping is also proved to be asymptotically stable using another energy function furthermore the exponential stability of this system is also shown under strongly monotone condition some obtained results improve the existing ones and the given conditions can be easily checked in practice since this dynamic system has wide applications the obtained results are significant in both theory and applications

a new recurrent neural network for solving convex quadratic programming problems with an application to the k winners take all problem in this paper a new recurrent neural network is proposed for solving convex quadratic programming qp problems compared with existing neural networks the proposed one features global convergence property under weak conditions low structural complexity and no calculation of matrix inverse it serves as a competitive alternative in the neural network family for solving linear or quadratic programming problems in addition it is found that by some variable substitution the proposed network turns out to be an existing model for solving minimax problems in this sense it can be also viewed as a special case of the minimax neural network based on this scheme a k winners take all k wta network with o n complexity is designed which is characterized by simple structure global convergence and capability to deal with some ill cases numerical simulations are provided to validate the theoretical results obtained more importantly the network design method proposed in this paper has great potential to inspire other competitive inventions along the same line

a recurrent neural network for solving bilevel linear programming problem in this brief based on the method of penalty functions a recurrent neural network nn modeled by means of a differential inclusion is proposed for solving the bilevel linear programming problem blpp compared with the existing nns for blpp the model has the least number of state variables and simple structure using nonsmooth analysis the theory of differential inclusions and lyapunov like method the equilibrium point sequence of the proposed nns can approximately converge to an optimal solution of blpp under certain conditions finally the numerical simulations of a supply chain distribution model have shown excellent performance of the proposed recurrent nns

a new projection based neural network for constrained variational inequalities this paper presents a new neural network model for solving constrained variational inequality problems by converting the necessary and sufficient conditions for the solution into a system of nonlinear projection equations five sufficient conditions are provided to ensure that the proposed neural network is stable in the sense of lyapunov and converges to an exact solution of the original problem by defining a proper convex energy function the proposed neural network includes an existing model and can be applied to solve some nonmonotone and nonsmooth problems the validity and transient behavior of the proposed neural network are demonstrated by some numerical examples

on a system of fuzzy fractional differential inclusions with projection operators abstract the aim of this paper is to introduce and study a class of fuzzy fractional differential inclusions with projection operators which captures the desired features of both fuzzy differential inclusions and fractional order projective dynamical systems within the same framework the existence of solutions for the open situation is proved by using continuous selection theorem moreover the existence of solutions for the closed situation is showed by employing lipschitzean selection theorem and fixed point theorem respectively furthermore an application and a numerical example are also given

a neural network for a class of convex quadratic minimax problems with constraints in this paper we propose a neural network for solving a class of convex quadratic minimax problems with constraints four sufficient conditions are provided to ensure the asymptotic stability of the proposed network furthermore the exponential stability of the proposing network is also proved under certain conditions the results obtained here can be further extended to the globally projected dynamical system in addition some new stability conditions for the system are also obtained since our stability conditions can be easily checked in practice these results becomes more attractive in real applications

neural network for quadratic optimization with bound constraints a recurrent neural network is presented which performs quadratic optimization subject to bound constraints on each of the optimization variables the network is shown to be globally convergent and conditions on the quadratic problem and the network parameters are established under which exponential asymptotic stability is achieved through suitable choice of the network parameters the system of differential equations governing the network activations is preconditioned in order to reduce its sensitivity to noise and to roundoff errors the optimization method employed by the neural network is shown to fall into the general class of gradient methods for constrained nonlinear optimization and in contrast with penalty function methods is guaranteed to yield only feasible solutions

a nonfeasible gradient projection recurrent neural network for equality constrained optimization problems in this paper a recurrent neural network for both convex and nonconvex equality constrained optimization problems is proposed which makes use of a cost gradient projection onto the tangent space of the constraints the proposed neural network constructs a generically nonfeasible trajectory satisfying the constraints only as t rarr infin local convergence results are given that do not assume convexity of the optimization problem to be solved global convergence results are established for convex optimization problems an exponential convergence rate is shown to hold both for the convex case and the nonconvex case numerical results indicate that the proposed method is efficient and accurate

a delayed neural network for solving linear projection equations and its analysis in this paper we present a delayed neural network approach to solve linear projection equations the lyapunov krasovskii theory for functional differential equations and the linear matrix inequality lmi approach are employed to analyze the global asymptotic stability and global exponential stability of the delayed neural network compared with the existing linear projection neural network theoretical results and illustrative examples show that the delayed neural network can effectively solve a class of linear projection equations and some quadratic programming problems

solving generally constrained generalized linear variational inequalities using the general projection neural networks generalized linear variational inequality glvi is an extension of the canonical linear variational inequality in recent years a recurrent neural network nn called general projection neural network gpnn was developed for solving glvis with simple bound often box type or sphere type constraints the aim of this paper is twofold first some further stability results of the gpnn are presented second the gpnn is extended for solving glvis with general linear equality and inequality constraints a new design methodology for the gpnn is then proposed furthermore in view of different types of constraints approaches for reducing the number of neurons of the gpnn are discussed which results in two specific gpnns moreover some distinct properties of the resulting gpnns are also explored based on their particular structures numerical simulation results are provided to validate the results

a delayed projection neural network for solving linear variational inequalities in this paper a delayed projection neural network is proposed for solving a class of linear variational inequality problems the theoretical analysis shows that the proposed neural network is globally exponentially stable under different conditions by the proposed linear matrix inequality lmi method the monotonicity assumption on the linear variational inequality is no longer necessary by employing lagrange multipliers the proposed method can resolve the constrained quadratic programming problems finally simulation examples are given to demonstrate the satisfactory performance of the proposed neural network

design of recurrent neural networks for solving constrained least absolute deviation problems recurrent neural networks for solving constrained least absolute deviation lad problems or l1 norm optimization problems have attracted much interest in recent years but so far most neural networks can only deal with some special linear constraints efficiently in this paper two neural networks are proposed for solving lad problems with various linear constraints including equality two sided inequality and bound constraints when tailored to solve some special cases of lad problems in which not all types of constraints are present the two networks can yield simpler architectures than most existing ones in the literature in particular for solving problems with both equality and one sided inequality constraints another network is invented all of the networks proposed in this paper are rigorously shown to be capable of solving the corresponding problems the different networks designed for solving the same types of problems possess the same structural complexity which is due to the fact these architectures share the same computing blocks and only differ in connections between some blocks by this means some flexibility for circuits realization is provided numerical simulations are carried out to illustrate the theoretical results and compare the convergence rates of the networks

unary multiple equality sets the languages of rational matrices unary multiple equality sets are equality sets of finite sets of homomorphisms which map into the free monoids over single letter alphabets these languages can equally well be defined by rational matrices unary multiple equality sets have extraordinary language theoretic properties they coincide with the commutative multiple equality sets each unary multiple equality set is precisely classified by the rank of its defining matrices and all fundamental decision questions are polynomially decidable the least trio containing all unary multiple equality sets equals the class of languages accepted by blind multicounter machines and by simple multihead finite automata finally crossing free unary multiple equality sets are introduced which vary from unary multiple equality sets in a similar way as the one sided dyck set varies from the two sided dyck set

global exponential stability of neural networks with globally lipschitz continuous activations and its application to linear variational inequality problem this paper investigates the existence uniqueness and global exponential stability ges of the equilibrium point for a large class of neural networks with globally lipschitz continuous activations including the widely used sigmoidal activations and the piecewise linear activations the provided sufficient condition for ges is mild and some conditions easily examined in practice are also presented the ges of neural networks in the case of locally lipschitz continuous activations is also obtained under an appropriate condition the analysis results given in the paper extend substantially the existing relevant stability results in the literature and therefore expand significantly the application range of neural networks in solving optimization problems as a demonstration we apply the obtained analysis results to the design of a recurrent neural network rnn for solving the linear variational inequality problem vip defined on any nonempty and closed box set which includes the box constrained quadratic programming and the linear complementarity problem as the special cases it can be inferred that the linear vip has a unique solution for the class of lyapunov diagonally stable matrices and that the synthesized rnn is globally exponentially convergent to the unique solution some illustrative simulation examples are also given

equilibrium routing under uncertainty we provide a brief introduction to the basic models used to describe traffic on congested networks both in urban transport and telecommunications we discuss traffic equilibrium models covering atomic and non atomic routing games with emphasis on situations where the travel times are subject to random fluctuations we use convex optimization to present the models in a unified framework that stresses the common underlying structures as a prototypical example of traffic equilibrium with elastic demands we discuss some models for routing and congestion control in telecommunications we also describe a class of stochastic dynamics that model the adaptive behavior of agents and which provides a plausible micro foundation for the equilibrium finally we present some recent ideas on how risk averse behavior might be incorporated in the equilibrium models

day to day dynamic network disequilibria and idealized traveler information systems in this paper we present tatonnement models for calculating static wardropian user equilibria on congested networks with fully general demand and cost structures we present both a qualitative analysis of stability and numerical studies which show that such an approach provides a reliable means for determining static user equilibria we also describe circumstances for which these models depict day to day adjustments from one realizable disequilibrium state to another and how these adjustment processes differ depending on the quality of the information being provided by abstract traveler information systems specifically we demonstrate that such dynamic adjustment processes settle down to equilibria both when information is complete and when it is incomplete

representations of language families by homomorphic equality operations and generalized equality sets abstract in this paper the operations of homomorphic equality and inverse homomorphic equality are introduced and are studied in all detail as part of formal language theory these operations are defined from n tuples of homomorphisms by a matching of the homomorphic images they model a simple and powerful equality mechanism and incorporate the notion of generalized equality sets single homomorphisms on free monoids are natural and simple operations however combining n tuples of homomorphisms into a homomorphic equality or inverse homomorphic equality may result in very complex morphic mappings which map e g to the set of solutions of an instance of the post correspondence problem in this paper particular emphasis is laid upon strong representation theorems in the general case every recursively enumerable set can be obtained from the trivial set by applying an inverse homomorphism a inverse homomorphic equality of two nonerasing homomorphisms and an erasing homomorphism this is a uniform morphic representation of the recursively enumerable sets with special regular sets as a basis and it is minimal in its constituents hence the recursively enumerable sets are the smallest class containing all regular sets of the form and closed under homomorphic equality or under inverse homomorphic equality and homomorphism every nonempty language is a generator of the recursively enumerable sets when the full trio operations are extended to include homomorphic equality or inverse homomorphic equality in a similar manner the regular sets can be characterized in terms of homomorphic equality and inverse homomorphic equality operations with bounded balance finally in the nonerasing case the target are classes of the form h l bnp for a class of languages l h l bnp is the smallest class containing l and closed under nonerasing homomorphic equality or under inverse homomorphic equality and nonerasing homomorphism it includes the intersection closure of l every language in h l bnp can be obtained from a language in l by a nonerasing homomorphic equality or an inverse homomorphic equality of three homomorphisms and followed by a nonerasing homomorphism provided that l is closed under inverse homomorphism and endmarking this representation goes through to trio generators of l which are the generators of h l bnp if the trio operations are extended to include nonerasing homomorphic equality or inverse homomorphic equality the operations of homomorphic equality and inverse homomorphic equality are based on generalized equality sets these are equality sets of finite sets of homomorphisms which are used forwards or backwards we investigate basic language theoretic properties of these sets including the star event property commutativity and hardest sets and provide a complete classification of the generalized equality sets according to the numbers of homomorphisms that are used forwards or backwards

a discrete time neural network for optimization problems with hybrid constraints recurrent neural networks have become a prominent tool for optimizations including linear or nonlinear variational inequalities and programming due to its regular mathematical properties and well defined parallel structure this brief presents a general discrete time recurrent network for linear variational inequalities and related optimization problems with hybrid constraints in contrary to the existing discrete time networks this general model can operate not only on bound constraints but also on hybrid constraints comprised of inequality equality and bound constraints the model has dynamical properties of global convergence asymptotical and exponential convergences under some weaker conditions numerical examples demonstrate its efficacy and performance

a feedback neural network for solving convex constraint optimization problems in this paper a feedback neural network model is presented by two steps firstly a convex sub optimization problem with bound constraints is established by introducing an energy function and the neural subnetwork for solving the sub optimization problem is constructed based on the projection method secondly a feedback neural network is proposed by using the subnetwork and can converge to an exact optimal solution of primal optimization problem the distinguishing features of the proposed feedback network are no lagrange multipliers no dual variables and no penalty parameters it has the least number of state variables simple structure and is suitable for hardware implementation two simulation examples are provided to show the feasibility and efficiency of proposed method in the paper

relaxing conjugacy to fit modeling in dynamical systems we address a fundamental modeling issue in science as related to the field of dynamical systems when is a model of a physical system a ogoodo representation conjugacy provides a means to determine if two systems are dynamically equivalent we develop mathematical technology to decide when dynamics of a otoyo model are like although not identical to dynamics of the physical system since the concept of conjugacy is too rigid for such cases we contrast the usual methodology where model quality is measured in a banach space to our dynamically motivated notion of matching orbits as best as possible we highlight our methods with a lower ordered model of a noisy logistic map and also a simplified model of a lorenz system such that the usual one dimensional map model is not exactly justified in the traditional sense