decidable sentences of church rosser congruences abstract a decision procedure for a class of true sentences of congruences generated by finite monadic church rosser systems is developed using this decision procedure it is shown that if m t is the monoid presented by such a system t then i it is decidable given t whether m t is a group ii it is decidable given t and a finite set a whether the submonoid generated by a is a group or a left right two sided ideal and iii green s relations for m t are decidable

direct method for yau filtering system with nonlinear observations for all known finite dimensional filters one always assumes that the observation terms be degree one polynomials however in practice the observation terms may be nonlinear e g tracking proble

on a necessary and sufficient condition for finite dimensionality of estimation ever since the technique of the kalman bucy filter was popularized there has been an intense interest in finding new classes of finite dimensional recursive filters in the late seventies the concept of the estimation algebra of a filtering system was introduced it has proven to be an invaluable tool in the study of nonlinear filtering problems in this paper a simple algebraic necessary and sufficient condition is established for an estimation algebra of a special class of filtering systems to be finite dimensional also presented is a rigorous proof of the wei norman program which allows one to construct finite dimensional recursive filters from finite dimensional estimation algebras

finite dimensional filters with nonlinear drift ii brockett s problem on classification of finite dimensional estimation algebras the idea of using estimation algebras to construct finite dimensional nonlinear filters was first proposed by brockett and mitter independently it turns out that the concept of estimation algebra plays a crucial role in the investigation of finite dimensional nonlinear filters in his talk at the international congress of mathematics in 1983 brockett proposed classifying all finite dimensional estimation algebras in this paper all finite dimensional algebras with maximal rank are classified if the dimension of the state space is less than or equal to two therefore from the lie algebraic point of view all finite dimensional filters are understood generically in the case where the dimension of state space is less than three

the uniform conjugacy problem for finite church rosser thue systems in np complete the uniform conjugacy problem for finite church rosser thue systems over alphabet is np complete if contains at least two letters if contains a single letter only then this problem is tractable

explicit solution of dmz equation in nonlinear filtering via solution of odes in this note we develop a real time and accurate solution for nonlinear filtering problems based on the gaussian distribution specifically we present an explicit solution of the duncan mortensen zakai dmz equation of the yau filtering system which includes the linear filtering system and exact filtering system the solution is given in terms of a solution of a system of ordinary differential equations in particular our method can be implemented in hardware the complexity of our algorithms is the same as those of kalman bucy filters in the case of linear filtering systems

approximation of the zakai equation by the splitting up method the objective of this article is to apply an operator splitting method to the time integration of zakai equation using this approach one can decompose the numerical integration into a stochastic step and a deterministic one both of them much simpler to handle than the original problem a strong convergence theorem is given in the spirit of existing results for deterministic problems

necessary and sufficient optimality conditions for a class of nonsmooth minimization problems the purpose of this paper is to derive in a unified way second order necessary and sufficient optimality criteria for four types of nonsmooth minimization problems thediscrete minimax problem thediscrete l1 approximation the minimization of theexact penalty function and the minimization of theclassical exterior penalty function our results correct and supplement conditions obtained by various authors in recent papers

classification of finite dimensional estimation algebras of maximal rank with arbitrary state space dimension and mitter conjecture in the late seventies the concept of the estimation algebra of a filtering system was introduced it was proven to be an invaluable tool in the study of non linear filtering problems in the early eighties brockett proposed to classify finite dimensional estimation algebras and mitter conjectured that all functions in finite dimensional estimation algebras are necessarily polynomials of total degree at most one despite the massive effort in understanding the finite dimensional estimation algebras the 20 year old problem of brockett and mitter conjecture remains open in this paper we give a classification of finite dimensional estimation algebras of maximal rank and solve the mitter conjecture affirmatively for finite dimensional estimation algebras of maximal rank in particular for an estimation algebra e of maximal rank we give a necessary and sufficient conditions for e to be finite dimensional in terms of the drift fi x and observation hj x as an important corollary we show that the numb

finite dimensional filters with non linear drift ix construction of finite dimensional estimation algebras of non maximal rank the idea of using estimation algebra to construct finite dimensional nonlinear filters was first proposed by brockett and mitter independently it has proven to be an invaluable tool in the study of nonlinear filtering problem in 1983 brockett proposed to classify all finite dimensional estimation algebras in this paper we give the construction of finite dimensional estimation algebras of non maximal rank these non maximal rank finite dimensional estimation algebras play an important role in brockett s classification problem

a counterexample of comments on stability margin evaluation for uncertain linear systems in the original paper c gong and s thompson ibid vol 39 p 548 50 1994 there is given what is claimed to be the tightest bound for all unstructured pertubations such that a specified system keeps its asymptotic stability j h su ibid vol 39 p 2523 4 1994 showed that it is not the tightest bound and gives another bound that bound is shown by a counterexample to be incorrect

set valued approximations and newton s methods to rm under the assumption of semi smoothness of the mapping we prove that the approximation can be obtained through the clarke generalized jacobian ioffe ralph generalized jacobian b subdifferential and their approximations as an application we propose a generalized newton s method based on the point based set valued approximation for solving nonsmooth equations we show that the proposed method converges locally superlinearly without the assumption of semi smoothness finally we include some well known generalized newton s methods in our method and consolidate the convergence results of these methods

sufficient conditions for the generalized problem of bolza this paper presents sufficient conditions for strong local optimality in the generalized problem of bolza these conditions represent a unification in the sense that they can be applied to both the calculus of variations and to optimal control problems as well as problems with nonsmooth data also this paper brings to light a new point of view concerning the jacobi condition in the classical calculus of variations showing that it can be considered as a condition which guarantees the existence of a canonical transformation which transforms the original hamiltonian to a locally concave convex hamiltonian

an o t 3 algorithm for testing the church rosser property of thue systems abstract we give an o t 3 algorithm for testing the church rosser property of thue systems using the linear string matching algorithm by knuth morris and pratt 1977 this improves the earlier bound of o t 6 given by book and o dunlaing 1981 the proposed algorithm uses a reduction algorithm for finding a normal form of a string which is based on building a trie for matching a finite set of patterns as proposed by aho and corasick 1975

structure theorem for five dimensional estimation algebras abstract the problem of classification of finite dimensional estimation algebras was formally proposed by brockett in his lecture at international congress of mathematicians in 1983 due to the difficulty of the problem in the early 1990s brockett suggested that one should understand the low dimensional estimation algebras first in this paper we give classification of estimation algebras of dimension at most five although the classification of finite dimensional estimation algebra of maximal rank was completed by yau and his coworkers chen chiou hu wong and wu the problem of classification of non maximal rank finite dimensional estimation algebra is still wide open except for the case of state space dimension 2 hopefully the result of this paper will shed some light on the non maximal rank estimation algebras

complexity results on the conjugacy problem for monoids abstract the complexity of the conjugacy problem cp m for monoids given by presentations of the form t is investigated here t denotes a possibly infinite thue system over the alphabet the following results are obtained 1 1 if t is finite and church rosser then cp m is in n time n if in addition t is special then cp m is in p 2 2 if t is infinite and church rosser but of the form i i d i r 1 where i is finite and each d i is in np then cp m is in np however if i is infinite then cp m can be undecidable even if t is recursive 3 3 if t is finite and almost confluent then cp m can be decidable with any degree in the grzegorczyk hierarchy or it can be undecidable with any recursively enumerable degree of unsolvability

some polynomial time algorithms for finite monadic church rosser thue systems abstract based on a careful analysis of reduction sequences in monadic thue systems we show that some uniform decision problems among them the uniform conjugacy problem are decidable in polynomial time for finite monadic church rosser thue systems on the other hand an example of a decision problem is exhibited that is undecidable even for this class of thue systems

on generalized second order derivatives and taylor expansions in nonsmooth optimization a representation of cominetti and correa s generalized second order directional derivative siam j control optim 28 1990 pp 789 809 is given and then applied to obtain a taylor theorem type result a conjecture in siam j control optim 28 1990 pp 789 809 concerning functions of the form max 1 leq i leq n g i x is proved under a strengthened assumption but not true otherwise

finite dimensional filters with nonlinear drift x explicit solution of dmz equation in this note we consider the explicit solution of duncan mortensen zakai dmz equation for the finite dimensional filtering system we show that yau s 1990 1994 filtering system spl delta f sub j spl delta x sub i spl delta f sub i spl delta x sub i c sub ij constant for all i j can be solved explicitly with an arbitrary initial condition by solving a system of ordinary differential equations and a kolmogorov type equation let n be the dimension of state space we show that we need only n sufficient statistics in order to solve the dmz equation

a generalized second order derivative in nonsmooth optimization in this work a new notion of generalized second order directional derivative and generalized hessian for nonsmooth real valued functions is studied the general properties of these mathematical objects are investigated together with some calculus rules that may facilitate their practical computation two applications of these derivatives in optimization theory are considered first to obtaining necessary and sufficient second order optimality conditions for problems with or without constraints and second to extending the newton method for the minimization of a mathcal c 1 1 function

solution of filtering problem with nonlinear observations for all known finite dimensional filters one always needs the conditon that the observation terms are degree one polynomial on the other hand in many practical examples e g tracking problem the observation terms may be nonlinear our new method in this paper can treat filtering problems with nonlinear observation terms in the first time which includes kalman bucy filter as a special case

conjugation in semigroups abstract the action of any group on itself by conjugation and the corresponding conjugacy relation play an important role in group theory there have been several attempts to extend the notion of conjugacy to semigroups in this paper we present a new definition of conjugacy that can be applied to an arbitrary semigroup and it does not reduce to the universal relation in semigroups with a zero we compare the new notion of conjugacy with existing definitions characterize the conjugacy in various semigroups of transformations on a set and count the number of conjugacy classes in these semigroups when the set is infinite

brockett s problem of classification of finite dimensional estimation algebras for nonlinear filtering systems in his talk at the international congress of mathematicians in 1982 brockett proposed to classify all finite dimensional estimation algebras arising in nonlinear filters recently chen yau and leung siam j control optim 35 1997 pp 1132 1141 claimed that they had classified all finite dimensional estimation algebras of maximal rank when the dimension of the state space is less than or equal to four in this paper we introduce a series of completely new computations about the estimation algebra and we find two sets of new equations about the omega matrix as a consequence we can prove without any dimension assumption on the state space that the omega matrix is a constant matrix and thus we succeed in classifying all finite dimensional estimation algebras of maximal rank with the state space dimension being arbitrary

constructing a sequence of discrete hessian matrices of an sc 1 function uniformly convergent to the generalized hessian matrix we construct a uniform approximation for generalized hessian matrix of an sc 1 function using the discrete gradient and the extended second order derivative we define the discrete hessian matrix we construct a sequence of sets where each set is composed of discrete hessian matrices we first show some new properties of sc 1 functions then we prove that for sc 1 functions the sequence of the set of discrete hessian matrices is uniformly convergent to the generalized hessian matrix

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

semismooth homeomorphisms and strong stability of semidefinite and lorentz complementarity problems based on an inverse function theorem for a system of semismooth equations this paper establishes several necessary and sufficient conditions for an isolated solution of a complementarity problem defined on the cone of symmetric positive semidefinite matrices to be strongly regular stable we show further that for a parametric complementarity problem of this kind if a solution corresponding to a base parameter is strongly stable then a semismooth implicit solution function exists whose directional derivatives can be computed by solving certain affine problems on the critical cone at the base solution similar results are also derived for a complementarity problem defined on the lorentz cone the analysis relies on some new properties of the directional derivatives of the projector onto the semidefinite cone and the lorentz cone

generalized hessian for c 1 1 functions in infinite dimensional normed spaces the subject of this paper is the systematic study of second order notions concerning differentiable functions with lipschitz derivative the results and notions are motivated by recent papers of cominetti correa and hiriart urruty the first goal of this paper is the comparison of several known second order directional derivatives the second goal is the introduction of a generalized hessian which is a set of certain symmetric bilinear forms the relation of this generalized hessian to other existing second order derivatives is also described

testing for the church rosser property abstract it is shown that there is a polynomial time algorithm to determine whether a finite thue system is church rosser