application of sobolev gradient method to poisson boltzmann system the idea of a weighted sobolev gradient introduced and applied to singular differential equations in 1 is extended to a poisson boltzmann system with discontinuous coefficients the technique is demonstrated on fully nonlinear and linear forms of the poisson boltzmann equation in one two and three dimensions in a finite difference setting a comparison between the weighted gradient and fas multigrid is given for large jump size in the coefficient function

new results on global stability analysis of discontinuous cohen grossberg neural networks of neutral type in hale s form in this paper we aim to investigate the globally asymptotic stability of the equilibrium point for the discontinuous cohen grossberg neural networks of neutral type in hale s form to deal with th

a geometric approach to leveraging weak learners adaboost is a popular and effective leveraging procedure for improving the hypotheses generated by weak learning algorithms adaboost and many other leveraging algorithms can be viewed as performing a constrained gradient descent over a potential function at each iteration the distribution over the sample given to the weak learner is proportional to the direction of steepest descent we introduce a new leveraging algorithm based on a natural potential function for this potential function the direction of steepest descent can have negative components therefore we provide two techniques for obtaining suitable distributions from these directions of steepest descent the resulting algorithms have bounds that are incomparable to adaboost s the analysis suggests that our algorithm is likely to perform better than adaboost on noisy data and with weak learners returning low confidence hypotheses modest experiments confirm that our algorithm can perform better than adaboost in these situations

sobolev preconditioning for the poisson boltzmann equation abstract this paper presents an overview of least squares steepest descent using sobolev gradients for several prototype differential equations in the linear case the method is viewed as a very effective preconditioning strategy for the basic iterative method which arises from steepest descent in particular it acts to smooth the euclidean gradient results are given for the one dimensional possion boltzmann equation from semiconductor device modeling

learning by canonical smooth estimation i simultaneous estimation this paper examines the problem of learning from examples in a framework that is based on but more general than valiant s probably approximately correct pac model for learning in our framework the learner observes examples that consist of sample points drawn and labeled according to a fixed unknown probability distribution based on this empirical data the learner must select from a set of candidate functions a particular function or hypothesis that will accurately predict the labels of future sample points the expected mismatch between a hypothesis prediction and the label of a new sample point is called the hypothesis generalization error following the pioneering work of vapnik and chervonenkis others have attacked this sort of learning problem by finding hypotheses that minimize the relative frequency based empirical error estimate we generalize this approach by examining the simultaneous estimation problem we demonstrate how one can learn from such a simultaneous error estimate and propose a new class of estimators called smooth estimators we characterize the class of simultaneous estimation problems solvable by a smooth estimator

noise tolerant parallel learning of geometric concepts we present several efficient parallel algorithms for pac learning geometric concepts in a constant dimensional space the algorithms are robust even against malicious classification noise of any rate less than 1 2 we first give an efficient noise tolerant parallel algorithm to pac learn the class of geometric concepts defined by a polynomial number of d 1 dimensional hyperplanes against an arbitrary distribution where each hyperplane has a slope from a set of known slopes we then describe how boosting techniques can be used so that our algorithms dependence on and does not depend ond next we give an efficient noise tolerant parallel algorithm to pac learn the class of geometric concepts defined by a polynomial number of d 1 dimensional hyperplanes of unrestricted slopes against a uniform distribution we then show how to extend our algorithm to learn this class against any unknown product distribution next we define a complexity measure of any setsof d 1 dimensional surfaces that we call thevariantofsand prove that the class of geometric concepts defined by surfaces of polynomial variant can be efficiently learned in parallel under a product distribution even under malicious classification noise furthermore we show that the vc dimension of the class of geometric concepts defined by a single surface of variant one is finally we give an efficient parallel noise tolerant algorithm to pac learn any 2 dimensional geometric concept defined by a setsof 1 dimensional surfaces of polynomial length under a uniform distribution

monotone viable trajectories for functional differential inclusions this paper is a study on functional differential inclusions with memory which represent the multivalued version of retarded functional differential equations the main result gives a necessary and sufficient equations the main result gives a necessary and sufficient condition ensuring the existence of viable trajectories that means trajectories remaining in a given nonempty closed convex set defined by given constraints the system must satisfy to be viable some motivations for this paper can be found in control theory where f t f t u u u is the set of possible velocities of the system at time t depending on the past history represented by the function and on a control u ranging over a set u of controls other motivations can be found in planning procedures in microeconomics and in biological evolutions where problems with memory do effectively appear in a multivalued version all these models require viability constraints represented by a closed convex set

viability solutions to structured hamilton jacobi equations under constraints structured hamilton jacobi partial differential equations are hamilton jacobi equations where the time variable is replaced by a vector valued variable structuring the system it could be the time age pair hamilton jacobi mckendrick equations or candidates for initial or terminal conditions hamilton jacobi cournot equations among a manifold of examples here we define the concept of viability solution which always exists and can be computed by viability algorithms this is a constructive approach allowing us to derive from the tools of viability theory dealing with sets instead of functions some known as well as new properties of its solutions regarding functions as their epigraphs we bypass the regularity issues to arrive directly at the concept of barron jensen frankowska viscosity solutions above all we take into account viability constraints and extend classical boundary conditions to other internal conditions beyond that we use the fenchel legendre transform to associate a lagrangian with the hamiltonian and uncover an underlying variational problem it is proved that the viability solution to the structured hamilton jacobi equation is the valuation function of this variational principle the optimal evolutions of which are regulated by a regulation map which is constructed from the viability solution and can be computed by viability algorithms in other words the viability solution solves all at once a class of first order partial differential equations of intertemporal optimization problems and the regulation of their optimal evolutions

global convergence of neural networks with mixed time varying delays and discontinuous neuron activations in this paper we investigate the dynamical behavior of a class of delayed neural networks with discontinuous neuron activations and general mixed time delays involving both time varying delays and distributed delays due to the presence of time varying delays and distributed delays the step by step construction of local solutions cannot be applied this difficulty can be overcome by constructing a sequence of solutions to delayed dynamical systems with high slope activations and show that this sequence converges to a desired filippov solution of the discontinuous delayed neural networks we then derive two sets of sufficient conditions for the global exponential stability and convergence of the neural networks in terms of linear matrix inequalities lmis and m matrix properties equivalently some diagonally dominant conditions respectively convergence behavior of both the neuron state and the neuron output are discussed the obtained results extend previous work on global stability of delayed neural networks with lipschitz continuous neuron activations and neural networks with discontinuous neuron activations and only constant delays

target problems under state constraints for nonlinear controlled impulsive systems we study the problem of reaching a target without leaving a prescribed constraint set for a dynamical system described by a controlled differential equation and a controlled instantaneous reset function we characterize all initial conditions from which the objective can be reached then we characterize the value function associated with the optimal reaching time problem

energy minimization related to the nonlinear schrodinger equation in this the window of the sobolev gradient technique to the problem of minimizing a schrodinger functional associated with a nonlinear schrodinger equation we show that gradients act in a suitably chosen sobolev space sobolev gradients can be used in finite difference and finite element settings in a computationally efficient way to find minimum energy states of schrodinger functionals

approximating time evolution related to ginzburg landau functionals via sobolev gradient methods in a finite element setting sobolev gradients have previously 1 been used to approximate time evolution related to a model a functional in a finite difference setting in this journal here a related approach in a finite element setting is discussed

playable differential games playability conditions of differential games are studied by using viability theory first the results on payability of time independent differential games are extended to time dependent games in fact time is introduced in the dynamics of the game in the state dependent constraints bearing on controls and in state constraints second some examples of pursuit games are studied necessary and sufficient conditions of playability of the game are provided here pursuit games are directly considered as games of kind in isaacs s sense and are not considered as games of degree the viability condition does not always provide the optimal strategy to be as close as possible to a certain goal but it supplies strategies allowing the system to reach a given goal

energy minimization using sobolev gradients application to phase separation and ordering a common problem in physics and engineering is the calculation of the minima of energy functionals the theory of sobolev gradients provides an efficient method for seeking the critical points of such a functional we apply the method to functionals describing coarse grained ginzburg landau models commonly used in pattern formation and ordering processes

the isaacs equation for differential games totally optimal fields of trajectories and related problems some aspects of the isaacs principle of transition and his main equation for differential games are considered to demonstrate that those equations contain some implicit assumptions and are valid under certain contiguity condition which is defined and analyzed for differential games the notion of total optimality is defined and the totally optimal fields of trajectories and control curves are introduced and studied in relation to the isaacs principle of transition bellman principle of optimality maximum principle of pontryagin and variational principles of mechanics it is demonstrated that the isaacs bellman and pontryagin theories are valid if and only if the optimal trajectories and optimal control curves generated by those methods are totally optimal in this context the hamilton jacobi partial differential equation can be used for sequential solution of multi games defined as n person games with m controls r cost functionals and multiple min max min max etc operators in fixed order of application and not creating multi objective game problems over totally optimal fields the structure of controls is invariant under time uncertainty parallel and series games are considered so the isaacs procedure can be reduced to the application of the bellman equations twice control systems with incomplete information or structural limitations on controls do not in general satisfy the contiguity condition thus are not totally optimal game problems for such systems may have optimal solutions which however cannot be obtained by the isaacs equations this fact is shown in an example of a widely used engineering system for which an optimal trajectory has all its parts non optimal and non contiguous to the optimal trajectory the paper presents theoretical justification of the isaacs equations for contiguous systems comparison of optimal control principles with variational principles of mechanics the consideration of total optimality and totally optimal fields of trajectories as necessary and sufficient conditions for validity of the three major optimal control theories and some other results important for applications

topological regularity of the set of solutions of a differential inclusion on etudie la regularite topologique de l ensemble solution de l inegalite variationnelle pour une fonction a valeur d ensemble x t tk x t f x t x 0 x ou x k k est un ensemble compact convexe non vide dans un espace de dimension finie m f est une correspondance semi continue superieurement t k x t est le cone tangent en x a k et l y est la projection de y dans l on etablit un theoreme d approximation pour la solution de l ensemble de l inegalite dans 0 t st x

on the sample complexity of weak learning in this paper we study the sample complexity of weak learning that is we ask how many data must be collected from an unknown distribution in order to extract a small but significant advantage in prediction we show that it is important to distinguish between those learning algorithms that output deterministic hypotheses and those that output randomized hypotheses we prove that in the weak learning model any algorithm using deterministic hypotheses to weakly learn a class of vapnik chervonenkis dimension d n requires formula examples in contrast when randomized hypotheses are allowed we show that 1 examples suffice in some cases we then show that there exists an efficient algorithm using deterministic hypotheses that weakly learns against any distribution on a set of size d n with only o d n 2 3 examples thus for the class of symmetric boolean functions over n variables where the strong learning sample complexity is n the sample complexity for weak learning using deterministic hypotheses is formula and o n2 3 and the sample complexity for weak learning using randomized hypotheses is 1 next we prove the existence of classes for which the distribution free sample size required to obtain a slight advantage in prediction over random guessing is essentially equal to that required to obtain arbitrary accuracy finally for a class of small circuits namely all parity functions of subsets of n boolean variables we prove a weak learning sample complexity of n this bound holds even if the weak learning algorithm is allowed to replace random sampling with membership queries and the target distribution is uniform on 0 1 n

on the recovery of transport parameters in groundwater modelling the problem of recovering the coefficient functions in the groundwater transport equation from piezomet tic head and contaminant concentration measurements is solved by minimization of an associated convex functional

almost periodic dynamics of a class of delayed neural networks with discontinuous activations we use the concept of the filippov solution to study the dynamics of a class of delayed dynamical systems with discontinuous right hand side which contains the widely studied delayed neural network models with almost periodic self inhibitions interconnection weights and external inputs we prove that diagonal dominant conditions can guarantee the existence and uniqueness of an almost periodic solution as well as its global exponential stability as special cases we derive a series of results on the dynamics of delayed dynamical systems with discontinuous activations and periodic coefficients or constant coefficients respectively from the proof of the existence and uniqueness of the solution we prove that the solution of a delayed dynamical system with high slope activations approximates to the filippov solution of the dynamical system with discontinuous activations

a viability theorem for set valued states in a hilbert space abstract applications in robust control problems and shape evolution motivate the mathematical interest in control problems whose states are compact possibly non convex sets rather than vectors this leads to evolutions in a basic set which can be supplied with a metric like the well established pompeiu hausdorff distance but it does not have an obvious linear structure this article extends differential inclusions with state constraints to compact valued states in a separable hilbert space h the focus is on sufficient conditions such that a given constraint set of compact subsets is viable a k a weakly invariant our main result extends the tangential criterion in the well known viability theorem usually for differential inclusions in a vector space to the metric space of non empty compact subsets of h

sobolev gradient approach for the time evolution related to energy minimization of ginzburg landau functionals the sobolev gradient technique has been discussed previously in this journal as an efficient method for finding energy minima of certain ginzburg landau type functionals s sial j neuberger t lookman a saxena energy minimization using sobolev gradients application to phase separation and ordering j comput phys 189 2003 88 97 in this article a sobolev gradient method for the related time evolution is discussed

the minimum consistent dfa problem cannot be approximated within any polynomial the minimum consistent dfa problem is that of finding a dfa with as few states as possible that is consistent with a given sample a finite collection of words each labeled as to whether the dfa found should accept or reject assuming that p np it is shown that for any constant k no polynomial time algorithm can be guaranteed to find a consistent dfa with fewer than opt k states where opt is the number of states in the minimum state dfa consistent with the sample this result holds even if the alphabet is of constant size two and if the algorithm is allowed to produce an nfa a regular expression or a regular grammar that is consistent with the sample a similar nonapproximability result is presented for the problem of finding small consistent linear grammars for the case of finding minimum consistent dfas when the alphabet is not of constant size but instead is allowed to vary with the problem specification the slightly stronger lower bound on approximability of opt 1 e log log opt is shown for any e 0

approximate solutions to poisson boltzmann systems with sobolev gradients a weighted sobolev gradient approach 1 is presented to a nonlinear pbe 2 with discontinuous coefficient functions a comparison is given between the weighted and unweighted sobolev gradient in the finite element setting in two and three dimensions behavior of the various sobolev gradients is discussed for large jump size in the coefficient a comparison with newton s method is given where the failure of newton s method is demonstrated for a test problem

learning recursive functions a survey studying the learnability of classes of recursive functions has attracted considerable interest for at least four decades starting with gold s 1967 model of learning in the limit many variations modifications and extensions have been proposed these models differ in some of the following the mode of convergence the requirements intermediate hypotheses have to fulfill the set of allowed learning strategies the source of information available to the learner during the learning process the set of admissible hypothesis spaces and the learning goals a considerable amount of work done in this field has been devoted to the characterization of function classes that can be learned in a given model the influence of natural intuitive postulates on the resulting learning power the incorporation of randomness into the learning process the complexity of learning among others on the occasion of rolf wiehagen s 60th birthday the last four decades of research in that area are surveyed with a special focus on rolf wiehagen s work which has made him one of the most influential scientists in the theory of learning recursive functions

closure properties of uniform convergence of empirical means and pac learnability under a family of probability measures abstract open problem no 12 2 of vidyasagar a theory of learning and generalization with application to neural networks and control systems springer london 1997 asks are the properties of uniform convergence of empirical means and learnability preserved when the family of probability measures is replaced by its closure in this note the question is answered in the affirmative further it is shown that these properties are not preserved in general if the family of probability measures is replaced by its convex closure an open question is posed as to whether it is possible to replace the family of probability measures by its convex closure in case the family is compact

a harnack inequality approach to the regularity of free boundaries part ii flat free boundaries are lipschitz soit u une solution faible d un probleme aux limites libres du type u 0 sur u 0 et sur u 0 et soit le long de la frontiere libre f la relation u v g u x est satisfaite au sens faible alors si f est proche du graphe d une fonction de lipschitz dans la boule unite b 1 de r n f est le graphe d une fonction de lipschitz dans b 1 2

chaperoning state evolutions by variable durations mckendrick partial differential equations in demography provide the state of a population depending on chronological time t and age d and involve two time evolutions the chronological one t in 0 t mapsto t geq 0 and the calendar age t in t d t mapsto t t d geq 0 both with constant velocity equal to 1 the calendar age evolution chaperons so to speak the evolution t in t d t mapsto x t of the state of a system some physical biological and economic problems motivate the introduction of variable durations with variable velocities representing the fluidity of time offering mathematical metaphors of a subjective fleeting specious time passing more or less slowly variable durations are no longer prescribed but chosen among those available and regulated the joint evolution of the variable evolution and the state is assumed to be governed by a differential inclusion or control system and provides as a byproduct the unknown temporal windows on which they evolve toge

existence of slow monotone solutions to a differential inclusion la stabilisation des solutions des problemes de commande se pose en termes d obtention des commandes a reaction le long desquelles une fonction donnee de lyapunov decroit

on the proper learning of axis parallel concepts we study the proper learnability of axis parallel concept classes in the pac learning and exact learning models these classes include union of boxes dnf decision trees and multivariate polynomials for constant dimensional axis parallel concepts c we show that the following problems have time complexities that are within a polynomial factor of each other c is properly exactly learnable with hypotheses of size at most times the target size from membership and equivalence queries c is properly pac learnable without membership queries under any product distribution there is an approximation algorithm for the minequic problem given a g c find a minimal size f c that is logically equivalent to g in particular if one has polynomial time complexity they all do using this we give the first proper learning algorithm of constant dimensional decision trees and the first negative results in proper learning from membership and equivalence queries for many classes for axis parallel concepts over a nonconstant dimension we show that with the equivalence oracle 1 3 we use this to show that binary decision trees are not properly learnable in polynomial time assuming p np and dnf is not se properly learnable e 1 in polynomial time even with an np oracle assuming 2p pnp

computationally efficient approach for the minimization of volume constrained vector valued ginzburg landau energy functional the minimization of volume constrained vector valued ginzburg landau energy functional is considered in the present study it has many applications in computational science and engineering like the conservative phase separation in multiphase systems such as the spinodal decomposition phase coarsening in multiphase systems color image segmentation and optimal space partitioning a computationally efficient algorithm is presented to solve the space discretized form of the original optimization problem the algorithm is based on the constrained nonmonotone l 2 gradient flow of ginzburg landau functional followed by a regularization step which is resulted from the tikhonov regularization term added to the objective functional that lifts the solution from the l 2 function space into h 1 space the regularization step not only improves the convergence rate of the presented algorithm but also increases its stability bound the step size selection based on the barzilai borwein approach is adapted to improve the convergence rate of the introduced algorithm the success and performance of the presented approach is demonstrated throughout several numerical experiments to make it possible to reproduce the results presented in this work the matlab implementation of the presented algorithm is provided as the supplementary material developing an efficient algorithm to solve conservative gradient flow of multiphase ginzburg landau functional performing the constrained minimization of energy functional on the h 1 space using non monotone optimization path by applying the barzilai borwein step size selection approach the matlab implementation of the presented algorithms is given as supplementary material

on constraint sampling in the linear programming approach to approximate dynamic programming in the linear programming approach to approximate dynamic programming one tries to solve a certain linear program the alp that has a relatively small numberk of variables but an intractable numberm of constraints in this paper we study a scheme that samples and imposes a subset ofm

learning by canonical smooth estimation ii learning and choice of model complexity this paper analyzes the properties of a procedure for learning from examples this canonical learner is based on a canonical error estimator developed in part i in learning problems one can observe data that consists of labeled sample points and the goal is to find a model or hypothesis from a set of candidates that will accurately predict the labels of new sample points the expected mismatch between a hypothesis prediction and the actual label of a new sample point is called the hypothesis generalization error we compare the canonical learner with the traditional technique of finding hypotheses that minimize the relative frequency based empirical error estimate it is shown that for a broad class of learning problems the set of cases for which such empirical error minimization works is a proper subset of the cases for which the canonical learner works we derive bounds to show that the number of samples required by these two methods is comparable we also address the issue of how to determine the appropriate complexity for the class of candidate hypotheses

some discriminant based pac algorithms a classical approach in multi class pattern classification is the following estimate the probability distributions that generated the observations for each label class and then label new instances by applying the bayes classifier to the estimated distributions that approach provides more useful information than just a class label it also provides estimates of the conditional distribution of class labels in situations where there is class overlap we would like to know whether it is harder to build accurate classifiers via this approach than by techniques that may process all data with distinct labels together in this paper we make that question precise by considering it in the context of pac learnability we propose two restrictions on the pac learning framework that are intended to correspond with the above approach and consider their relationship with standard pac learning our main restriction of interest leads to some interesting algorithms that show that the restriction is not stronger more restrictive than various other well known restrictions on pac learning an alternative slightly milder restriction turns out to be almost equivalent to unrestricted pac learning

impulse capture basins of sets under impulse control systems given a target contained in a constrained set and an impulse control system governing the evolutions of runs or executions that are hybrids of continuous and discrete evolutions this paper studies and provides several characterizations of the capture basin of the target viable in the constrained set it is the subset of initial runs from which start at least one run viable in the constrained set until it reaches the target in finite time it also provides algorithms and regulation rules governing the runs that reach the targets while obeying state constraints

exploiting random walks for learning in this paper we consider an approach to passive learning in contrast to the classical pac model we do not assume that the examples are independently drawn according to an underlying distribution but that they are generated by a time driven process we define deterministic and probabilistic learning models of this sort and investigate the relationships between them and with other models the fact that successive examples are related can often be used to gain additional information similar to the information gained by membership queries we show how this can be used to design on line prediction algorithms in particular we present efficient algorithms for exactly identifying boolean threshold functions and 2 term rse and for learning 2 term dnf when the examples are generated by a random walk on 0 1 n

sobolev gradient preconditioning for elliptic reaction diffusion problems with some nonsmooth nonlinearities abstract the sobolev gradient approach is an efficient way to construct preconditioned iterations for solving nonlinear problems we extend this technique to be applicable for elliptic equations describing stationary states of reaction diffusion problems if the nonlinearities have certain lack of differentiability we derive convergence results of the sobolev gradient method on an abstract level and then for our elliptic problem under different assumptions numerical tests show convergence as expected

learnability and the vapnik chervonenkis dimension valiant s learnability model is extended to learning classes of concepts defined by regions in euclidean space e n the methods in this paper lead to a unified treatment of some of valiant s results along with previous results on distribution free convergence of certain pattern recognition algorithms it is shown that the essential condition for distribution free learnability is finiteness of the vapnik chervonenkis dimension a simple combinatorial parameter of the class of concepts to be learned using this parameter the complexity and closure properties of learnable classes are analyzed and the necessary and sufficient conditions are provided for feasible learnability

generalized halanay inequalities and their applications to neural networks with unbounded time varying delays in this brief we discuss some variants of generalized halanay inequalities that are useful in the discussion of dissipativity and stability of delayed neural networks integro differential systems and volterra functional differential equations we provide some generalizations of the halanay inequality which is more accurate than the existing results as applications we discuss invariant set dissipative synchronization and global asymptotic stability for the hopfield neural networks with infinite delays we also prove that the dynamical systems with unbounded time varying delays are globally asymptotically stable

razumikhin type theorems for hybrid system with memory in this paper we address the stability problem of hybrid systems with memory which are newly developed to model hybrid dynamic systems affected by delays for such systems stability criteria are developed based on razumikhin type conditions which are usually applied in the delay systems we prove that the hybrid system with memory is uniformly globally pre asymptotically stable by small gain theorem besides a relaxed razumikhin type stability criterion is also proposed which does not require strict decrease of the lyapunov razumikhin function during jumps several examples are given to illustrate the effectiveness and generality of the proposed results

viability in a non local population model structured by size and spatial position abstract motivated conceptually by multiscale models of cancer cell migration the focus is on a broad class of set valued population models structured both physiologically and spatially for time dependent sets of constraints given sufficient conditions are provided such that every permitted density function initializes at least one admissible weak solution this result about viability a k a weak invariance concerns non local hyperbolic differential inclusions of first order with memory and with one boundary condition for the minimal value of the scalar structure parameter

efficient algorithms for learning functions with bounded variation we show that the class fbv of 0 1 valued functions with total variation at most 1 can be agnostically learned with respect to the absolute loss in polynomial time from o 1 e2 log 1 examples matching a known lower bound to within a constant factor we establish a bound of o 1 m on the expected error of a polynomial time algorithm for learning fbv in the prediction model also matching a known lower bound to within a constant factor applying a known algorithm transformation to our prediction algorithm we obtain a polynomial time pac learning algorithm for fbv with a sample complexity bound of o 1 e log 1 this also matches a known lower bound to within a constant factor

macroscopic traffic models shifting from densities to celerities abstract the advent of techniques to measure velocities of gps equipped vehicles using satellite technology replacing the density of the road traffic sensors motivates and justifies the revision of conceptual mathematical algorithms and software based models this paper summarizes studies on the traffic evolutions achieving the minimum of a congestion function controlled macroscopic traffic velocities called celerities instead of founding traffic regulation on the measures of traffic densities the flux valued function is the fenchel transform of the fundamental diagram and is a convex decreasing function we use the properties of capture basins investigated in viability theory specifically a lax hopf formula characterizing them and the max plus morphism of capture basins for deriving the statements proved in this paper traffic conditions involve as well boundary conditions or conditions on trajectories inside the domain

generalized lyapunov approach for functional differential inclusions abstract this paper is concerned with the generalized lyapunov approach for functional differential inclusions fdi at first we prove a class of generalized halanay s inequalities then by applying the generalization of halanay s inequalities we investigate the uniformly ultimate boundedness and uniformly asymptotic stability of fdi

generalized lyapunov method for discontinuous systems abstract nonlinear dynamical systems described by differential equations with discontinuous right hand side and solutions intended in filippov sense are considered based on filippov differential inclusion and a new chain rule for differentiating regular functions along filippov solution trajectories different kinds of stability and convergence results are presented moreover we investigate the stability and convergence for the corresponding perturbation system of the discontinuous system and some new criteria are addressed

on saturation discontinuities and delays in iiss and iss feedback control redesign this paper addresses input to state stability iss and integral iss iiss redesign to enhance robustness of feedback control systems in the face of actuator disturbances the idea of the redesign is to modify control laws available for disturbance free systems a priori in order to achieve robustness against the disturbance in practice control system design often encounters time delays discontinuities and actuator limitations at the same time the purpose of this paper is to propose a unified treatment of these issues in the redesign for actuator disturbances for the new class of systems we pursue lgv type formulas which are widely used for ordinary differential equations in addition we incorporate saturation into the lgv type control for implementation in the presence of actuator limitations to derive such a redesign formula we develop a framework of iss iiss lyapunov krasovskii functionals allowing for the krasovskii solutions this paper demonstrates that the notion of invariantly differentiable functionals is useful for deriving lgv type control laws and dealing with discontinuities directly the iiss formulation which includes the iss as a special case allows us to address actuator limitations in the redesign problem in a flexible manner

generalized halanay inequalities for dissipativity of volterra functional differential equations abstract this paper is concerned with the dissipativity of theoretical solutions to nonlinear volterra functional differential equations vfdes at first we give some generalizations of halanay s inequality which play an important role in study of dissipativity and stability of differential equations then by applying the generalization of halanay s inequality the dissipativity results of vfdes are obtained which provides unified theoretical foundation for the dissipativity analysis of systems in ordinary differential equations odes delay differential equations ddes integro differential equations ides volterra delay integro differential equations vdides and vfdes of other type which appear in practice

equivalence of models for polynomial learnability abstract in this paper we consider several variants of valiant s learnability model that have appeared in the literature we give conditions under which these models are equivalent in terms of the polynomially learnable concept classes they define these equivalences allow comparisons of most of the existing theorems in valiant style learnability and show that several simplifying assumptions on polynomial learning algorithms can be made without loss of generality we also give a useful reduction of learning problems to the problem of finding consistent hypotheses and give comparisons and equivalences between valiant s model and the prediction learning models of haussler littlestone and warmuth in 29th annual ieee symposium on foundations of computer science 1988

some contributions to fixed distribution learning theory we consider some problems in learning with respect to a fixed distribution we introduce two new notions of learnability these are probably uniformly approximately correct puac learnability which is a stronger requirement than the widely studied pac learnability and minimal empirical risk mer learnability which is a stronger requirement than the previously defined notions of solid or potential learnability it is shown that although the motivations for defining these two notions of learnability are entirely different these two notions are in fact equivalent to each other and in turn equivalent to a property introduced here referred to as the shrinking width property it is further shown that if the function class to be learned has the property that empirical means converge uniformly to their true values then all of these learnability properties hold in the course of proving conditions for these forms of learnability we also obtain a new estimate for the vc dimension of a collection of sets obtained by performing boolean operations on a given collection this result is of independent interest we consider both the case in which there is an underlying target function as well as the case of model free or agnostic learning finally we consider the issue of representation of a collection of sets by its subcollection of equivalence classes it is shown by example that by suitably choosing representatives of each equivalence class it is possible to affect the property of uniform convergence of empirical probabilities

preconditioning operators and sobolevgradients for nonlinear elliptic problems a preconditioning framework is presented for the iterative solution of nonlinear elliptic problems based on the preconditioning operator approach various fixed preconditioning operators are used in the iteration which can be interpreted as a weighted sobolev gradient method

lower semicontinuous solutions of hamilton jacobi bellman equations the value function of mayer s problem arising in optimal control is investigated and lower semicontinuous solutions of the associated hamilton jacobi bellman equation are defined in three equivalent ways under quite weak assumptions about the control system the value function is the unique solution moreover it is stable with respect to perturbations of the control system and the cost it coincides with the viscosity solution whenever it is continuous

a liouville type theorem for a variational problem with free boundary in three dimensions abstract we consider the minimum problem for the functional e u d u 2 2 u 0 in three dimensional space where 0 is a constant we will establish a liouville type theorem for this variational problem if u c r 3 is a nonnegative and nonzero global minimizer then u x x x 0 for some point x 0 and some unit vector

numerical solution of burgers equation by the sobolev gradient method abstract burgers equation is solved numerically with sobolev gradient methods a comparison is shown with other numerical schemes presented in this journal such as modified adomian method mam 1 and by a variational method vm which is based on the method of discretization in time 2 it is shown that the sobolev gradient methods are highly efficient while at the same time retaining the simplicity of steepest descent algorithms

the optimal sample complexity of pac learning this work establishes a new upper bound on the number of samples sufficient for pac learning in the realizable case the bound matches known lower bounds up to numerical constant factors this solves a long standing open problem on the sample complexity of pac learning the technique and analysis build on a recent breakthrough by hans simon

slow and quasi slow solutions of differential inclusions on donne des conditions suffisantes pour l existence d une solution viable lente pour le probleme de cauchy x r x t p pent 0 x 0 x 0 k x t kt 0 ou k est un sous ensemble convexe ferme d un espace vectoriel de dimension finie x et f k x est une application multiforme semi continue bornee superieurement a images compactes convexes

sobolev gradient preconditioning for the electrostatic potential equation sobolev gradient type preconditioning is proposed for the numerical solution of the electrostatic potential equation a constructive representation of the gradients leads to efficient laplacian preconditioners in the iteration thanks to the available fast poisson solvers convergence is then verified for the corresponding sequence in sobolev space implying mesh independent convergence results for the discretized problems a particular study is devoted to the case of a ball due to the radial symmetry of this domain a direct realization without discretization is feasible the simplicity of the algorithm and the fast linear convergence are finally illustrated in a numerical test example

unconditionally energy stable time stepping scheme for cahn morral equation an unconditionally energy stable time stepping scheme is introduced to solve cahn morral like equations in the present study it is constructed based on the combination of david eyre s time stepping scheme and schur complement approach although the presented method is general and independent of the choice of homogeneous free energy density function term logarithmic and polynomial energy functions are specifically considered in this paper the method is applied to study the spinodal decomposition in multi component systems and optimal space tiling problems a penalization strategy is developed in the case of later problem to avoid trivial solutions extensive numerical experiments demonstrate the success and performance of the presented method according to the numerical results the method is convergent and energy stable independent of the choice of time stepsize its matlab implementation is included in the appendix for the numerical evaluation of algorithm and reproduction of the presented results extension of eyre s convex concave splitting scheme to multiphase systems efficient solution of spinodal decomposition in multi component systems efficient solution of least perimeter periodic space partitioning problem developing a penalization strategy to avoid trivial solutions presentation of matlab implementation of the introduced algorithm

a unified framework for hybrid control model and optimal control theory we propose a very general framework that systematizes the notion of a hybrid system combining differential equations and automata governed by a hybrid controller that issues continuous variable commands and makes logical decisions we first identify the phenomena that arise in real world hybrid systems then we introduce a mathematical model of hybrid systems as interacting collections of dynamical systems evolving on continuous variable state spaces and subject to continuous controls and discrete transitions the model captures the identified phenomena subsumes previous models yet retains enough structure to pose and solve meaningful control problems we develop a theory for synthesizing hybrid controllers for hybrid plants in all optimal control framework in particular we demonstrate the existence of optimal relaxed and near optimal precise controls and derive generalized quasi variational inequalities that the associated value function satisfies we summarize algorithms for solving these inequalities based on a generalized bellman equation impulse control and linear programming

nagumo type condition for partial differential inclusions on etudie un probleme de la forme x t ax t g xc h x o x 0 k t o t x t k ou a est un operateur elliptique et k n est pas necessairement compact on trouve des conditions tangentielles de type nagumo

optical tomography reconstruction algorithm with the finite element method an optimal approach with regularization tools optical tomography is mathematically treated as a non linear inverse problem where the optical properties of the probed medium are recovered through the minimization of the errors between the experimental measurements and their predictions with a numerical model at the locations of the detectors according to the ill posed behavior of the inverse problem some regularization tools must be performed and the tikhonov penalization type is the most commonly used in optical tomography applications this paper introduces an optimized approach for optical tomography reconstruction with the finite element method an integral form of the cost function is used to take into account the surfaces of the detectors and make the reconstruction compatible with all finite element formulations continuous and discontinuous through a gradient based algorithm where the adjoint method is used to compute the gradient of the cost function an alternative inner product is employed for preconditioning the reconstruction algorithm moreover appropriate re parameterization of the optical properties is performed these regularization strategies are compared with the classical tikhonov penalization one it is shown that both the re parameterization and the use of the sobolev cost function gradient are efficient for solving such an ill posed inverse problem

numerical approximation of time evolution related to ginzburg landau functionals using weighted sobolev gradients sobolev gradients have been discussed in sial et al 2003 as a method for energy minimization related to ginzburg landau functionals in this article a weighted sobolev gradient approach for the time evolution of a ginzburg landau functional is presented for different values of k a comparison is given between the weighted and unweighted sobolev gradients in a finite element setting it is seen that for small values of k the weighted sobolev gradient method becomes more and more efficient compared to using the unweighted sobolev gradient a comparison with newton s method is given where the failure of newton s method is demonstrated for a test problem

generalized lyapunov approach for convergence of neural networks with discontinuous or non lipschitz activations the paper considers a class of additive neural networks where the neuron activations are modeled by discontinuous functions or by continuous non lipschitz functions some tools are developed which enable us to apply a lyapunov like approach to differential equations with discontinuous right hand side modeling the neural network dynamics the tools include a chain rule for computing the time derivative along the neural network solutions of a nondifferentiable lyapunov function and a comparison principle for this time derivative which yields conditions for exponential convergence or convergence in finite time by means of the lyapunov like approach a general result is proved on global exponential convergence toward a unique equilibrium point of the neural network solutions moreover new results on global convergence in finite time are established which are applicable to neuron activations with jump discontinuities or neuron activations modeled by means of continuous non lipschitz holder functions

stability theorems for delay differential inclusions this technical note addresses stability problems of delay differential inclusions a chain rule is proposed for a wide class of lyapunov krasovskii functionals which are not necessarily invariantly differentiable we also obtain an invariance like theorem where the derivative of the candidate functional is bounded above by a continuous negative semidefinite function several examples are given to show the effectiveness of the results

viability kernels and capture basins of sets under differential inclusions this paper provides a characterization of viability kernels and capture basins of a target viable in a constrained subset as a unique closed subset between the target and the constrained subset satisfying tangential conditions or by duality normal conditions it is based on a method devised by helene frankowska for characterizing the value function of an optimal control problem as generalized contingent or viscosity solutions to hamilton jacobi equations these abstract results interesting by themselves can be applied to epigraphs of functions or graphs of maps and happen to be very efficient for solving other problems such as stopping time problems dynamical games boundary value problems for systems of partial differential equations and impulse and hybrid control systems which are the topics of other companion papers

rbf nets for approximating an object s boundary by image random sampling abstract our aim is to discuss the problem of approximating the boundary of an object in a binary image in contrast to earlier papers on similar topics we avoid finding contours and then their approximations instead we sample an image in a relatively small number of points 1 3 of pixels at random the collected data is used for imposing constraints on parameters of a radial basis functions rbf neural net it is proved that if the rbf net structure is sufficiently rich to approximate an object boundary then estimates of rbf net parameters tend to their true values as the number of samples approaches to infinity then a seemingly linear algorithm of estimating linear weights is proposed and its consistency is also proved its implementation which is based on an iterative use of the linear programming is briefly discussed and the results of its testing are shortly reported

decision theoretic generalizations of the pac model for neural net and other learning applications we describe a generalization of the pac learning model that is based on statistical decision theory in this model the learner receives randomly drawn examples each example consisting of an instance x in x and an outcome y in y and tries to find a hypothesis h x a where h in h that specifies the appropriate action a in a to take for each instance x in order to minimize the expectation of a loss l y a here x y and a are arbitrary sets l is a real valued function and examples are generated according to an arbitrary joint distribution on x times y special cases include the problem of learning a function from x into y the problem of learning the conditional probability distribution on y given x regression and the problem of learning a distribution on x density estimation we give theorems on the uniform convergence of empirical loss estimates to true expected loss rates for certain hypothesis spaces h and show how this implies learnability with bounded sample size disregarding computational complexity as an application we give distribution independent upper bounds on the sample size needed for learning with feedforward neural networks our theorems use a generalized notion of vc dimension that applies to classes of real valued functions adapted from pollard s work and a notion of capacity and metric dimension for classes of functions that map into a bounded metric space supersedes 89 30 and 90 52 also in information and computation vol 100 no 1 september 1992

a finite element method with mesh adaptivity for computing vortex states in fast rotating bose einstein condensates numerical computations of stationary states of fast rotating bose einstein condensates require high spatial resolution due to the presence of a large number of quantized vortices in this paper we propose a low order finite element method with mesh adaptivity by metric control as an alternative approach to the commonly used high order finite difference or spectral approximation methods the mesh adaptivity is used with two different numerical algorithms to compute stationary vortex states an imaginary time propagation method and a sobolev gradient descent method we first address the basic issue of the choice of the variable used to compute new metrics for the mesh adaptivity and show that refinement using simultaneously the real and imaginary part of the solution is successful mesh refinement using only the modulus of the solution as adaptivity variable fails for complicated test cases then we suggest an optimized algorithm for adapting the mesh during the evolution of the solution towards the equilibrium state considerable computational time saving is obtained compared to uniform mesh computations the new method is applied to compute difficult cases relevant for physical experiments large nonlinear interaction constant and high rotation rates

viability theorems for a class of differential operator inclusions abstract this paper discusses the following viability problem of a differential inclusion x t ax t f x t x 0 x0 k t 0 x t k where k is a compact subset of a banach space x a is the infinitesimal generator of a compact differentiable semigroup of bounded linear operators on x and f k x is upper semicontinuous with compact convex values we show that a natural tanegential condition is necessary and sufficient for the existence of a global solution to this problem

how to use expert advice we analyze algorithms that predict a binary value by combining the predictions of several prediction strategies called experts our analysis is for worst case situations i e we make no assumptions about the way the sequence of bits to be predicted is generated we measure the performance of the algorithm by the difference between the expected number of mistakes it makes on the bit sequence and the expected number of mistakes made by the best expert on this sequence where the expectation is taken with respect to the randomization in the predictins we show that the minimum achievable difference is on the order of the square root of the number of mistakes of the best expert and we give efficient algorithms that achieve this our upper and lower bounds have matching leading constants in most cases we then show how this leads to certain kinds of pattern recognition learning algorithms with performance bounds that improve on the best results currently know in this context we also compare our analysis to the case in which log loss is used instead of the expected number of mistakes

impulse differential inclusions a viability approach to hybrid systems impulse differential inclusions are introduced as a framework for modeling hybrid phenomena connections to standard problems in the area of hybrid systems are discussed conditions are derived that allow one to determine whether a set of states is viable or invariant under the action of an impulse differential inclusion for sets that violate these conditions methods are developed for approximating their viability and invariance kernels that is the largest subset that is viable or invariant under the action of the impulse differential inclusion the results are demonstrated on examples

improved razumikhin type theorem for input to state stability of nonlinear time delay systems this note describes a lyapunov razumikhin function with an indefinite derivative that can be used to determine the input to state stability iss of a nonlinear time delay system first an extended comparison principle for estimating the solution of a differential inequality with an upper bound that is not semi negative definite is presented then that principle and a lyapunov razumikhin function with an indefinite derivative are employed to derive an iss criterion finally two examples show the effectiveness of our method and its superiority over others

finite time stabilization of delayed memristive neural networks discontinuous state feedback and adaptive control approach in this paper a general class of delayed memristive neural networks dmnns system described by functional differential equation with discontinuous right hand side is considered under the extended filippov framework we investigate the finite time stabilization problem for dmnns by using the famous finite time stability theorem and the generalized lyapunov functional method to do so we design two classes of novel controllers including discontinuous state feedback controller and discontinuous adaptive controller without assuming the boundedness and monotonicity of the activation functions several sufficient conditions are given to stabilize the states of this class of dmnns in finite time moreover the upper bounds of the settling time for stabilization are estimated finally numerical examples are provided to demonstrate the effectiveness of the developed method and the theoretical results

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

learning halfspaces with malicious noise we give new algorithms for learning halfspaces in the challenging malicious noise model where an adversary may corrupt both the labels and the underlying distribution of examples our algorithms can tolerate malicious noise rates exponentially larger than previous work in terms of the dependence on the dimension n and succeed for the fairly broad class of all isotropic log concave distributions we give poly n 1 e time algorithms for solving the following problems to accuracy e learning origin centered halfspaces in rn with respect to the uniform distribution on the unit ball with malicious noise rate e2 log n e the best previous result was e n log n e 1 4 learning origin centered halfspaces with respect to any isotropic log concave distribution on rn with malicious noise rate e3 log2 n e this is the first efficient algorithm for learning under isotropic log concave distributions in the presence of malicious noise we also give a poly n 1 e time algorithm for learning origin centered halfspaces under any isotropic log concave distribution on rn in the presence of adversarial label noise at rate e3 log 1 e in the adversarial label noise setting or agnostic model labels can be noisy but not example points themselves previous results could handle e but had running time exponential in an unspecified function of 1 e our analysis crucially exploits both concentration and anti concentration properties of isotropic log concave distributions our algorithms combine an iterative outlier removal procedure using principal component analysis together with smooth boosting

dynamical behaviors of delayed neural network systems with discontinuous activation functions in this letter without assuming the boundedness of the activation functions we discuss the dynamics of a class of delayed neural networks with discontinuous activation functions a relaxed set of sufficient conditions is derived guaranteeing the existence uniqueness and global stability of the equilibrium point convergence behaviors for both state and output are discussed the constraints imposed on the feedback matrix are independent of the delay parameter and can be validated by the linear matrix inequality technique we also prove that the solution of delayed neural networks with discontinuous activation functions can be regarded as a limit of the solutions of delayed neural networks with high slope continuous activation functions

computational limitations on learning from examples the computational complexity of learning boolean concepts from examples is investigated it is shown for various classes of concept representations that these cannot be learned feasibly in a distribution free sense unless r np these classes include a disjunctions of two monomials b boolean threshold functions and c boolean formulas in which each variable occurs at most once relationships between learning of heuristics and finding approximate solutions to np hard optimization problems are given

boundary value problems for systems of hamilton jacobi bellman inclusions with constraints we study in this paper boundary value problems for systems of hamilton jacobi bellman first order partial differential equations and variational inequalities the solutions of which are constrained to obey viability constraints they are motivated by some control problems such as impulse control and financial mathematics we shall prove the existence and uniqueness of such solutions in the class of closed set valued maps by giving a precise meaning to what a solution means in this case we shall also provide explicit formulas for this problem when we deal with hamilton jacobi bellman equations we obtain the existence and uniqueness of frankowska contingent episolutions we shall deduce these results from the fact that the graph of the solution is the viable capture basin of the graph of the boundary conditions under an auxiliary system and then from their properties and their characterizations proved in j p aubin siam j control optim 40 2001 pp 853 881