positive solutions for some singular critical growth nonlinear elliptic equations let omega be a bounded domain in r n n greater than or equal to 4 with smooth boundary partial derivativeomega and the origin 0 is an element of omega mu n 2 2 2 2 2n n 2 k x is a smooth function on omega and positive somewhere we obtain existence results of positive solutions to the dirichlet problem deltau mu u x 2 k x u 2 2 u f x u on omega u 0 on partial derivativeomega for various k x and suitable number mu c 2004 elsevier ltd all rights reserved

an analysis of separable least squares data driven local coordinates for maximum likelihood estimation of linear systems in this paper we study a novel parametrization for state space systems namely separable least squares data driven local coordinates slsddlc the parametrization by slsddlc has recently been successfully applied to maximum likelihood estimation of linear dynamic systems in a simulation study the use of slsddlc has led to numerical advantages in comparison to the use of more conventional parametrizations including data driven local coordinates ddlc however an analysis of properties of slsddlc which are relevant to identification has not been performed up to now in this paper we provide insights into the geometry and topology of the slsddlc construction and show a number of results which are important for actual identification in particular for maximum likelihood estimation we also prove that the separable least squares methodology is indeed guaranteed to be applicable to maximum likelihood estimation of linear dynamic systems in typical situations

balanced parametrization of classes of linear systems canonical forms and parametrizations are presented for several sets of minimal systems of given dimension asymptotically stable systems allpass systems bounded real systems positive real systems minimum phase systems and the class of all minimal systems the approach is based on balancing techniques for these classes of systems applications of these results to hankel operators and model reduction are discussed

the complexity of some polynomial network consistency algorithms for constraint satisfaction problems abstract constraint satisfaction problems play a central role in artificial intelligence a class of network consistency algorithms for eliminating local inconsistencies in such problems has previously been described we analyze the time complexity of several node arc and path consistency algorithms and prove that arc consistency is achievable in time linear in the number of binary constraints the waltz filtering algorithm is a special case of the arc consistency algorithm in the edge labelling computational vision application the constraint graph is planar and so the time complexity is linear in the number of variables

balanced forms canonicity and parameterization we show that balanced forms are canonical and continuously parameterize the set of systems of a given order which are asymptotically stable minimal and have distinct singular values the advantage of the balanced canonical form over other classical canonical forms lies precisely in the guaranteed stability and minimality of the parameterization we also derive formulas allowing the synthesis of systems using gradient methods in balanced form

the hardy inequality and the asymptotic behaviour of the heat equation with an inverse square potential we study the well posedness and describe the asymptotic behavior of solutions of the heat equation with inverse square potentials for the cauchy dirichlet problem in a bounded domain and also for the cauchy problem in rn in the case of the bounded domain we use an improved form of the so called hardy poincare inequality and prove the exponential stabilization towards a solution in separated variables in rn we first establish a new weighted version of the hardy poincare inequality and then show the stabilization towards a radially symmetric solution in self similar variables with a polynomial decay rate this work complements and explains well known work by baras and goldstein on the existence of global solutions and blow up for these equations in the present article the sign restriction on the data and solutions is removed the functional framework for well posedness is described and the asymptotic rates calculated examples of non uniqueness are also given

on the minimality and global consistency of row convex constraint networks constraint networks have been shown to be useful in formulating such diverse problems as scene labeling natural language parsing and temporal reasoning given a constraint network we often wish to i find a solution that satisfies the constraints and ii find the corresponding minimal network where the constraints are as explicit as possible both tasks are known to be np complete in the general case task 1 is usually solved using a backtracking algorithm and task ii is often solved only approximately by enforcing various levels of local consistency in this paper we identify a property of binary constraint called row convexity and show its usefulness in deciding when a form of local consistency called path consistency is sufficient to guarantee that a network is both minimal and globally consistent globally consistent networks have the property that a solution can be found without backtracking we show that one can test for the row convexity property efficiently and we show by examining applications of constraint networks discussed in the literature that our results are useful in practice thus we identify a class of binary constraint networks for which we can solve both tasks i and ii efficiently finally we generalize the results for binary constraint networks to networks with nonbinary constraints

constraint satisfaction neural networks for image segmentation abstract constraint satisfaction neural networks csnns are proposed for image recognition which is the process to identify objects of interest in an image the image recognition problem is formulated as a constraint satisfaction problem csp the regions resulting from a signal based image segmentation algorithm can be considered as objects and their entities as labels in the csp the proposed neural network is multi layered and the function of each layer is to identify one physical entity label from the input regions objects through cascading these layers each label can be assigned to the corresponding region s attention is focused on recognizing x ray ct computed tomography images of human heads e g anatomic structure in a medical image the proposed csnn is capable of identifying the components of a brain in a preliminarily segmented image the major components in a human head include bone ventricle and gyrus therefore three layers are needed for this task each layer is constructed based on the principles proposed in kohonen s self organizing feature maps and the optimal linear associative memories and their weights are trained by the features of each entity in the prestored model the experimental results show that the proposed approach is very promising

network based heuristics for constraint satisfaction problems abstract many ai tasks can be formulated as constraint satisfaction problems csp i e the assignment of values to variables subject to a set of constraints while some csps are hard those that are easy can often be mapped into sparse networks of constraints which in the extreme case are trees this paper identifies classes of problems that lend themselves to easy solutions and develops algorithms that solve these problems optimally the paper then presents a method of generating heuristic advice to guide the order of value assignments based on both the sparseness found in the constraint network and the simplicity of tree structured csps the advice is generated by simplifying the pending subproblems into trees counting the number of consistent solutions in each simplified subproblem and comparing these counts to decide among the choices pending in the original problem

generalising submodularity and horn clauses tractable optimization problems defined by tournament pair multimorphisms the submodular function minimization problem sfm is a fundamental problem in combinatorial optimization and several fully combinatorial polynomial time algorithms have recently been discovered to solve this problem the most general versions of these algorithms are able to minimize any submodular function whose domain is a set of tuples over any totally ordered finite set and whose range includes both finite and infinite values in this paper we demonstrate that this general form of sfm is just one example of a much larger class of tractable discrete optimization problems defined by valued constraints these tractable problems are characterized by the fact that their valued constraints have an algebraic property which we call a tournament pair multimorphism this larger tractable class also includes the problem of satisfying a set of horn clauses horn sat as well as various extensions of this problem to larger finite domains

characterising tractable constraints abstract finding solutions to a binary constraint satisfaction problem is known to be an np complete problem in general but may be tractable in cases where either the set of allowed constraints or the graph structure is restricted this paper considers restricted sets of constraints which are closed under permutation of the labels we identify a set of constraints which gives rise to a class of tractable problems and give polynomial time algorithms for solving such problems and for finding the equivalent minimal network we also prove that the class of problems generated by any set of constraints not contained in this restricted set is np complete

a geometric approach to the minimum sensitivity design problem in this paper a riemannian geometric framework is given for the minimum sensitivity design problem and its solution using a natural optimization criterion the theory is then applied to the case of linear systems to generate a class of minimum sensitivity realizations related to the so called balanced realizations in particular conditions are given which are applicable in the cases of fixed point and floating point implementations

bundle reduction and the alignment distance on spaces of state space lti systems this paper introduces a large class of differential geometric distances between finite dimensional linear dynamical systems collectively called the alignment distance contrary to the existing distances the alignment distance is based on the state space description of dynamical systems and is defined on the manifolds of systems of fixed order and fixed input output dimension under a matrix rank constraint e g minimality controllability or observability while the quotient topology and principal fiber bundle structure associated with such manifolds have been known since the early days of modern control theory distances natural to this structure have not been studied the starting point for defining such a distance is to identify a linear system of order n with its equivalence class of state space realizations all related by the so called similarity action i e state space change of basis under gl n the lie group of nonsingular n times n matrices the main idea of the alignment distance is to first find the best state space change of basis that brings a realization of a system as close as possible to a realization of another system the alignment step and then compare the aligned realizations a direct implementation of this idea due to noncompactness of gl n is complicated however using the notion of reduction of the structure group of a principal bundle we show that the change of basis can be restricted to an orthogonal change of basis provided one uses realizations in a reduced subbundle this key observation brings about significant computational benefits as a technical contribution possibly of independent interest we show that several forms of realization balancing available in the control literature have differential geometric significance and are indeed examples of reducing the structure group from gl n to its subgroup of orthogonal matrices o n the alignment distance can be defined for stable and unstable systems discrete or continuous time and stochastic systems

solutions to critical elliptic equations with multi singular inverse square potentials abstract let be an open bounded domain in r n n 3 with smooth boundary we are concerned with the multi singular critical elliptic problem u i 1 k i u x a i 2 q x u 2 2 u u h 0 1 where i r 2 2 n n 2 a i 1 i k and q x is a positive bounded function on using moser iteration we prove the asymptotic behavior of solutions for at points a i 1 i k by exploiting the effect of the coefficient of the critical nonlinearity we by means of a variational method establish the existence of positive and sign changing solutions for problem

balanced realizations canonical form parametrization model reduction abstract balanced realizations are used to construct a canonical form for the set of minimal and asymptotically stable mimo systems of a given order a parametrization resutt is derived that allows us to show a truncation property of this canonical form

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

existence of solutions for a nonlinear pde with an inverse square potential abstract via linking theorem and delicate energy estimates the existence of nontrivial solutions for a nonlinear pde with an inverse square potential and critical sobolev exponent is proved this result gives a partial positive answer to an open problem proposed in ferrero and gazzola j differential equations 177 2001 494

arc and path consistence revisited mackworth and freuder have analyzed the time complexity of several constraint satisfaction algorithms 5 we present here new algorithms for arc and path consistency and show that the arc consistency algorithm is optimal in time complexity and of the same order space complexity as the earlier algorithms a refined solution for the path consistency problem is proposed however the space complexity of the path consistency algorithm makes it practicable only for small problems these algorithms are the result of the synthesis techniques used in image a general constraint satisfaction system and local consistency methods 3

overlapping block balanced canonical forms and parametrizations the stable siso case the balanced canonical form and parametrization of ober for the case of siso stable systems are extended to block balanced canonical forms and related input normal forms and parametrizations they form an overlapping atlas of parametrizations of the manifold of stable siso systems of given order this extends the usefulness of these parametrizations e g in gradient algorithms for system identification as an implication of our construction it follows that each of the subsets of the parametrization of r ober internat j control 46 1987 pp 643 670 corresponding to a choice for the structural indices is in fact an imbedded submanifold of the manifold of stable siso systems of fixed order

system identification using balanced parametrizations some general issues in the black box identification of multivariable systems are first discussed it is then suggested that balanced parametrizations can be used to give identifiable forms a particular advantage is that balanced parametrizations are known for several useful classes of linear dynamic models including stable minimal models minimum phase models positive real models and normalized coprime factor models before optimizing the parameters of balanced parametrizations an initial model must be found we use realization based methods and so called subspace methods for this purpose these methods are very effective at finding accurate initial models without preliminary estimation of various structural indexes the paper ends with two simulation examples which compare the use of balanced parametrizations with more traditional ones and three real examples based on practical problems a distillation column an industrial dryer and the irrational spectrum or sea waves

reduction operations in fuzzy or valued constraint satisfaction in the constraint satisfaction problem csp knowledge about a relation on n variables is given in the form of constraints on subsets of the variables a solution to the csp is simply an instantiation of the n variables which satisfies all the constraints the fuzzy constraint satisfaction problem fcsp is a generalisation of the csp to the case in which the constraints are not categorical but represent preferences in the form of fuzzy sets the fcsp is then an optimisation problem in the space of all possible instantiations of the n variables it has potentially many more applications in real world problems than the csp when the aim is to maximise the value of the most violated constraint then the fcsp can be solved by solving a logarithmic number of csps fuzzy k consistency can even be established by an algorithm whose worst case complexity is identical to that of an optimal k consistency algorithm for csps when the operator used to aggregate constraint values is strictly monotonic as is the case in max csp for example the classic operation of arc consistency has to be completely redefined we show that an ordered version of arc consistency exists for all fcsps with a strictly monotonic aggregation operator and that in the case of max csp it can be established by an algorithm whose worst case complexity is identical to that of the optimal arc consistency algorithm for csps neighbourhood substitution in csps can be generalised to fuzzy neighbourhood substitution in fcsps whether the aggregation operator is strictly monotonic or idempotent we give an algorithm for applying fuzzy neighbourhood substitutions until convergence based on the corresponding algorithm for csps

on schrodinger operators with multipolar inverse square potentials abstract positivity essential self adjointness and spectral properties of a class of schrodinger operators with multipolar inverse square potentials are discussed in particular a necessary and sufficient condition on the masses of singularities for the existence of at least a configuration of poles ensuring the positivity of the associated quadratic form is established

an optimal k consistency algorithm abstract this paper generalizes the arc consistency algorithm of mohr and henderson 4 and the path consistency algorithm of han and lee 2 to a k consistency algorithm arc consistency and path consistency being 2 consistency and 3 consistency respectively the algorithm is a development of freuder s synthesis algorithm 1 it simultaneously establishes i consistency for each 1 i k it has worst case time and space complexity which is optimal when k is a constant and almost optimal for all other values of k in the case that all order i constraints exist for all 1 i n this algorithm is a solution to the consistent labeling problem with almost optimal worst case time and space complexity

constraint propogation techniques for the disjunctive scheduling problem abstract constraint propagation is an elementary method for reducing the search space of combinatorial search and optimization problems which has become more and more important in the last decades the basic idea of constraint propagation is to detect and remove inconsistent variable assignments that cannot participate in any feasible solution through the repeated analysis and evaluation of the variables domains and constraints describing a specific problem instance the contribution of this paper is twofold the first contribution is a description of efficient constraint propagation methods also known as consistency tests for the disjunctive scheduling problem dsp which is a generalization of the classical job shop scheduling problem jsp applying an elementary constraint based approach involving a limited number of search variables we will derive consistency tests that ensure 3 b consistency we will further present and analyze both new and classical consistency tests which to some extent are generalizations of the aforementioned consistency tests involving a higher number of variables but still can be implemented efficiently with a polynomial time complexity further the concepts of energetic reasoning and shaving are analyzed and discussed the other contribution is a classification of the consistency tests derived according to the domain reduction achieved the particular strength of using consistency tests is based on their repeated application so that the knowledge derived is propagated i e reused for acquiring additional knowledge the deduction of this knowledge can be described as the computation of a fixed point since this fixed point depends upon the order of the application of the tests we first derive a necessary condition for its uniqueness we then develop a concept of dominance which enables the comparison of different consistency tests as well as a simple method for proving dominance an extensive comparison of all consistency tests is given quite surprisingly we will find out that some apparently stronger consistency tests are subsumed by apparently weaker ones at the same time an open question regarding the effectiveness of energetic reasoning is answered

fundamental properties of neighbourhood substitution in constraint satisfaction problems in combinatorial problems it is often worthwhile simplifying the problem using operations such as consistency before embarking on an exhaustive search for solutions neighbourhood substitution is such a simplification operation whenever a value x for a variable is such that it can be replaced in all constraints by another value y then x is eliminated this paper shows that neighbourhood substitutions are important whether the aim is to find one or all solutions it is proved that the result of a convergent sequence of neighbourhood substitutions is invariant modulo isomorphism an efficient algorithm is given to find such a sequence it is also shown that to combine consistency of any order and neighbourhood substitution we only need to establish consistency once

a note on the sign changing solutions to elliptic problems with critical sobolev and hardy terms let rn be a bounded domain such that 0 n 7 2 2nn 2 we obtain existence of sign changing solutions for the dirichlet problem u u x 2 u 2 2u u on u 0 on for suitable positive numbers and

on the elliptic problems involving multi singular inverse square potentials and critical sobolev exponent abstract assume that r n n 3 is a smooth bounded domain a i are different points in i 1 2 k k 2 0 2 q 2 n 2 2 2 2 2 n n 2 is the critical sobolev exponent we study the existence and asymptotic behavior of solutions to the multi singular critical elliptic problem u u i 1 k x a i 2 u 2 2 u u q 2 u with dirichlet boundary condition the results depend crucially on the parameters and q

cyclic consistency a local reduction operation for binary valued constraints valued constraint satisfaction provides a general framework for optimisation problems over finite domains it is a generalisation of crisp constraint satisfaction allowing the user to express preferences between solutions consistency is undoubtedly the most important tool for solving crisp constraints it is not only a family of simplification operations on problem instances it also lies at the heart of intelligent search techniques g kondrak p van beek artificial intelligence 89 1997 365 387 and provides the key to solving certain classes of tractable constraints p g jeavons d a cohen m c cooper artificial intelligence 101 1998 251 265 arc consistency was generalised to valued constraints by sacrificing the uniqueness of the arc consistency closure m c cooper t schiex artificial intelligence in press the notion of 3 cyclic consistency introduced in this paper again sacrifices the unique closure property in order to obtain a generalisation of path consistency to valued constraints which is checkable in polynomial time in max csp 3 cyclic consistency can be established in polynomial time and even guarantees a local form of optimality the space complexity of 3 cyclic consistency is optimal since it creates no new constraints

existence of solutions for elliptic equations with critical sobolev hardy exponents let omega subset of rn be a smooth bounded domain such that 0 is an element of omega n greater than or equal to 3 0 less than or equal to s 2 2 s 2 n s n 2 we prove the existence of nontrival solutions for the singular critical problem

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

solutions for semilinear elliptic equations with critical exponents and hardy potential abstract in this paper we answer affirmatively an open problem cf theorem 4 in ferrero and gazzola j differential equations 177 2001 494 let 0 be an open bounded domain r n n 5 and assume that 0 n 2 2 2 n 2 n 2 then for all 0 there exists a nontrivial solution with critical level in the range 0 1 n s n 2 for the problem u u x 2 u u 2 2 u in u 0 on

comments on mohr and henderson s path consistency algorithm abstract mohr and henderson have presented new algorithms for arc and path consistency in 1 though the underlying ideas of their algorithms are correct the path consistency algorithm pc 3 is in error in this paper we point out the errors in this algorithm and give a correct one the time complexity and space complexity of the revised algorithm are also analyzed

backtracking techniques for the job shop scheduling constraint satisfaction problem abstract this paper studies a version of the job shop scheduling problem in which some operations have to be scheduled within non relaxable time windows i e earliest latest possible start time windows this problem is a well known np complete constraint satisfaction problem csp a popular method for solving this type of problems involves using depth first backtrack search in our earlier work we focused on the development of consistency enforcing techniques and variable value ordering heuristics that improve the efficiency of this search procedure in this paper we combine these techniques with new look back schemes that help the search procedure recover from so called deadend search states i e partial solutions that cannot be completed without violating some constraints more specifically we successively describe three intelligent backtracking schemes 1 dynamic consistency enforcement dynamically identifies critical subproblems and determines how far to backtrack by selectively enforcing higher levels of consistency among variables participating in these critical subproblems 2 learning ordering from failure dynamically modifies the order in which variables are instantiated based on earlier conflicts and 3 incomplete backjumping heuristic abandons areas of the search space that appear to require excessive computational efforts these schemes are shown to 1 further reduce the average complexity of the backtrack search procedure 2 enable our system to efficiently solve problems that could not be solved otherwise due to excessive computation cost and 3 be more effective at solving job shop scheduling problems than other look back schemes advocated in the literature

balanced canonical forms for system identification parameterizations of linear state space systems by balanced canonical forms are considered the structure in particular the topological properties of these parameterizations are investigated based on these results an algorithm for identification obtained by optimizing a likelihood type criterion function is proposed and analyzed in the algorithm both the estimation of real valued parameters and of integer parameters defining dynamic specification is incorporated the analysis of the algorithm emphasizes consistency results

quasilinear elliptic problems with critical exponents and hardy terms abstract let 0 be an open bounded domain r n n p 2 we are concerned with the multiplicity of positive solutions of p u u p 2 u x p u p 2 u q x u p 2 u u w 0 1 p where p u div u p 2 u 1 p n p np n p 0 n p p p 0 and q x is a nonnegative function on by investigating the effect of the coefficient of the critical nonlinearity we by means of variational method prove the existence of multiple positive solutions

an elliptic equation with combined critical sobolev hardy terms let ohm be a bounded domain in r n n 3 with the origin 0 is an element of ohm mu 0 and q x 0 are two smooth functions on ohm in this paper we investigate the singular elliptic equation delta u mu u vertical bar x vertical bar 2 k x u 2 s 1 vertical bar x vertical bar s q x u 2 t 1 vertical bar x x 0 vertical bar t f x u with dirichlet boundary conditions we study the limit behavior of the rs sequence of the corresponding energy functional and give a global compactness theorem and then give some existence results c 2005 elsevier ltd all rights reserved

on the asymptotic complexity of matrix multiplication the main results of this paper have the following flavor given one algorithm for multiplying matrices there exists another better algorithm a consequence of these results is that omega the exponent for matrix multiplication is a limit point that is it cannot be realized by any single algorithm we also use these results to construct a new algorithm which shows that omega 2 495548

multi agent oriented constraint satisfaction this paper presents a multi agent oriented method for solving csps constraint satisfaction problems in this method distributed agents represent variables and a two dimensional grid like environment in which the agents inhabit corresponds to the domains of the variables thus the positions of the agents in such an environment constitute the solution to a csp in order to reach a solution state the agents will rely on predefined local reactive behaviors namely better move least move and random move while presenting the formalisms and algorithm we will analyze the correctness and complexity of the algorithm and demonstrate the proposed method with two benchmark csps i e n queen problems and coloring problems in order to further determine the effectiveness of different reactive behaviors we will examine the performance of this method in deriving solutions based on behavior prioritization and different selection probabilities

an analysis of the parametrization by data driven local coordinates for multivariable linear systems in this paper we study a novel parametrization for state space systems namely data driven local coordinates ddlc which have recently been introduced and applied even though ddlc has meanwhile become the default parametrization used in the system identification toolbox of the software package matlab an analysis of properties of ddlc which are relevant to identification has not been performed up to now in this paper we provide insights into the geometry and topology of the ddlc construction and show a number of results which are important for actual identification such as maximum likelihood type estimation