smoothness of transition maps in singular perturbation problems with one fast variable abstract this paper deals with the smoothness of the transition map between two sections transverse to the fast flow of a singularly perturbed vector field one fast multiple slow directions orbits connecting both sections are canard orbits i e they first move rapidly towards the attracting part of a critical surface then travel a distance near this critical surface even beyond the point where the orbit enters the repelling part of the critical surface and finally repel away from the surface we prove that the transition map is smooth in a transcritical situation however where orbits from an attracting part of one critical manifold follow the repelling part of another critical manifold the smoothness of the transition map may be limited due to resonance phenomena that are revealed by blowing up the turning point we present a polynomial example in r 3

exchange lemmas for singular perturbation problems with certain turning points abstract in this work singular perturbation problems with a certain type of turning point are studied from a geometric point of view we first describe the delay phenomenon initially studied by pontryagin of dynamics near the slow manifold possessing the turning points based on the local properties of such turning points and the center manifold theory for general invariant manifolds developed by chow liu and yi we extend the well known exchange lemma first formulated by jones and kopell for problems with normally hyperbolic slow manifolds to problems with this type of turning point applications to singular boundary value problems with turning points are discussed

a regularized monotonic fuzzy support vector machine model for data mining with prior knowledge incorporating prior knowledge into data mining is an interesting but challenging problem and this study proposes a novel fuzzy support vector machine svm model to explore this issue it considers the fact that in many applications each input point may not be exactly labeled as one particular class and thus it applies a fuzzy membership to each input point it also utilizes expert knowledge concerning the monotonic relations between the response and predictor variables which is represented in the form of monotonicity constraints we formulate the classification problem of a monotonically constrained fuzzy svm called a monotonic fsvm derive its dual optimization problem and theoretically analyze its monotonic property the tikhonov regularization method is further applied to ensure that the solution is unique and bounded a new measure i e the frequency monotonicity rate is proposed to evaluate the ability of the model to retain the monotonicity the results of the experiments on real world and synthetic datasets show that this method which considers different contributions of each data and the prior knowledge of the monotonicity has a number of advantages with regard to predictive ability and retaining monotonicity over the original fsvm and svm models when applied to classification problems

tracking invariant manifolds with differential forms in singularly perturbed systems abstract a technique is developed for finding homoclinic orbits in singularly perturbed systems the desired orbit lies near a singular solution that has both fast and slow time scales the orbit is to be constructed as the transverse intersection of a center stable and center unstable manifold which requires tracking the invariant manifolds near the singular solution the main technical advance is the exchange lemma which allows one to convert information about transversality of manifolds associated with the singular limit equations to information about behavior of the invariant manifold as it leaves a neighborhood of a slow segment of the singular solution the computations for this lemma are done with the use of differential forms and the lemma is applied to prove the existence of a homoclinic orbit in a class of singularly perturbed equations

recurrence and the asymptotic pseudo orbit tracing property abstract in this paper asymptotic pseudo orbits and the asymptotic pseudo orbit tracing property are considered for continuous maps from a compact metric space into itself we investigate transitive properties and recurrent properties for a map with the asymptotic pseudo orbit tracing property in particular a condition for which a homeomorphism is a non wandering homeomorphism is obtained

knowledge based proximal support vector machines we propose a proximal version of the knowledge based support vector machine formulation termed as knowledge based proximal support vector machines kbpsvms in the sequel for binary data classification the kbpsvm classifier incorporates prior knowledge in the form of multiple polyhedral sets and determines two parallel planes that are kept as distant from each other as possible the proposed algorithm is simple and fast as no quadratic programming solver needs to be employed effectively only the solution of a structured system of linear equations is needed

centered kernel alignment inspired fuzzy support vector machine abstract support vector machine svm is a theoretically well motivated algorithm developed from statistical learning theory which has shown impressive performance in many fields in spite of its success it still suffers from the noise sensitivity problem originating from the assumption that each training point has equal importance or weight in the training process to relax this problem the svm was extended to the fuzzy svm fsvm by applying a fuzzy membership to each training point such that different training points can make different contributions to the learning of the decision surface although well determined fuzzy memberships can improve classification performance there are no general guidelines for their construction in this paper inspired by the centered kernel alignment cka which measures the degree of similarity between two kernels or kernel matrices we propose a new fuzzy membership function calculation method in which a heuristic function derived from the cka is used to calculate the dependence between a data point and its associated label although the cka induced fsvm is similar to the kernel target alignment kta induced fsvm there is actually a critical difference without that centering the definition of alignment does not correlate well with the performance of learning machines extensive experiments are performed on real world data sets from the uci benchmark repository and the application domain of computational biology which validate the superiority of the proposed fsvm model in terms of several classification performance measures

fuzzy support vector machines a support vector machine svm learns the decision surface from two distinct classes of the input points in many applications each input point may not be fully assigned to one of these two classes in this paper we apply a fuzzy membership to each input point and reformulate the svms such that different input points can make different contributions to the learning of decision surface we call the proposed method fuzzy svms fsvms

ergodic properties of systems with asymptotic average shadowing property abstract in this paper we explore a topological system f m m with asymptotic average shadowing property and extend sigmund s results from bowen s specification case we show that every non empty compact and connected subset v m inv f coincides with some v f y moreover we show that the set m v y m v f y v is dense in v v supp in particular if max m inv f supp coincides with m then m max y v f y m inv f is residual in m as consequences we have several corollaries one is that every invariant measure has generic points another is that the set consisting of those points for which the birkhoff ergodic average does not exist called irregular set is either dense in max residual provided that max m or empty in particular we give an uncountable division of irregular set and obtain a refined characterization which can be as a substantial generalization of 18 with a different method remark that for systems with asymptotic average shadowing property this article studies birkhoff ergodic average from topological viewpoint but it is still unknown how about the theory from the perspective of dimension theory

on bifurcation delay an alternative approach using geometric singular perturbation theory abstract to explain the phenomenon of bifurcation delay which occurs in planar systems of the form x f x z z g x z z where f x 0 0 0 and g x 0 0 changes sign at least once on the x axis we use the exchange lemma in geometric singular perturbation theory to track the limiting behavior of the solutions using the trick of extending dimension to overcome the degeneracy at the turning point we show that the limiting attracting and repulsion points are given by the well known entry exit function and the minimum of z on the trajectory is of order exp 1 also we prove smoothness of the return map up to arbitrary finite order in

morse theory for lagrange multipliers and adiabatic limits abstract given two morse functions f on a compact manifold m we study the morse homology for the lagrange multiplier function on m r which sends x to f x x take a product metric on m r and rescale its r component by a factor 2 we show that generically for large the morse smale witten chain complex is isomorphic to the one for f and the metric restricted to 1 0 with grading shifted by one on the other hand in the limit 0 we obtain another chain complex which is geometrically quite different but has the same homology as the singular homology of 1 0 the isomorphism between the chain complexes is provided by the homotopy obtained by varying our proofs use both the implicit function theorem on banach manifolds and geometric singular perturbation theory

monotonic smooth takagi sugeno fuzzy systems with fuzzy sets with compact support in this paper sufficient conditions on monotonicity of takagi sugeno t s fuzzy systems with linear consequent parts with smooth membership functions with compact support i e equal to zero outside a compact set are derived the presented conditions considerably reduce conservatism of those known for t s fuzzy systems utilizing membership functions with the support that is not compact besides two illustrative examples will demonstrate that the monotonicity requirement imposed on fuzzy systems may significantly improve their extrapolation capability

viscous singular shock profiles for a system of conservation laws modeling two phase flow abstract this paper is concerned with singular shocks for a system of conservation laws via the dafermos regularization u t f u x t u x x for a system modeling incompressible two phase fluid flow the existence of viscous profiles is proved using geometric singular perturbation theory the weak convergence and the growth rate of the viscous solution are also derived the weak limit is the sum of a piecewise constant function and a measure supported on a shock line and the maximum value of the viscous solution is of order exp 1

a new fuzzy support vector machine to evaluate credit risk due to recent financial crises and regulatory concerns financial intermediaries credit risk assessment is an area of renewed interest in both the academic world and the business community in this paper we propose a new fuzzy support vector machine to discriminate good creditors from bad ones because in credit scoring areas we usually cannot label one customer as absolutely good who is sure to repay in time or absolutely bad who will default certainly our new fuzzy support vector machine treats every sample as both positive and negative classes but with different memberships by this way we expect the new fuzzy support vector machine to have more generalization ability while preserving the merit of insensitive to outliers as the fuzzy support vector machine svm proposed in previous papers we reformulate this kind of two group classification problem into a quadratic programming problem empirical tests on three public datasets show that it can have better discriminatory power than the standard support vector machine and the fuzzy support vector machine if appropriate kernel and membership generation method are chosen

partially monotonic decision trees abstract in multicriteria decision tasks certain features are linearly ordered according to the decision and are called criteria whereas others called regular attributes are not in practice regular attributes and criteria coexist in most classification tasks in this paper we propose a rank inconsistent rate that distinguishes attributes from criteria furthermore it represents the directions of the monotonic relationships between criteria and decisions we design a partially monotonic decision tree algorithm to extract decision rules for partially monotonic classification tasks experimental results show that the proposed algorithm is effective and efficient

geometric singular perturbation theory with real noise abstract we prove the existence of families of random invariant manifolds for singularly perturbed systems of ordinary differential equations with sufficiently small real noise we use these invariant manifolds to prove a random version of the inclination theorem or exchange lemma

the asymptotic average shadowing property and transitivity abstract we introduce the notion of the asymptotic average shadowing property abbrev aasp and investigate the relation between the aasp and transitivity it is shown that a continuous map on x with the aasp is chain transitive and a l hyperbolic homeomorphism on x with the aasp is topologically transitive where x is a compact metric space

pseudo orbit shadowing in the family of tent maps on montre que les applications en tente ont la propriete d ombrage pour presque tous les parametres s mais elles n ont pas cette propriete pour un ensemble dense non denombrable des parametres on montre que pour une application tente chaque pseudo orbite peut etre approchee par une orbite actuelle d une application en tente avec une boucle peut etre legerement plus grande

asymptotic average shadowing property on compact metric spaces abstract we prove that if for a continuous map f on a compact metric space x the chain recurrent set r f has more than one chain component then f does not satisfy the asymptotic average shadowing property we also show that if a continuous map f on a compact metric space x has the asymptotic average shadowing property and if a is an attractor for f then a is the single attractor for f and we have a r f we also study diffeomorphisms with asymptotic average shadowing property and prove that if m is a compact manifold which is not finite with dim m 2 then the c 1 interior of the set of all c 1 diffeomorphisms with the asymptotic average shadowing property is characterized by the set of stable diffeomorphisms

global optimization algorithms for linearly constrained indefinite quadratic problems abstract in this paper we consider the problem of finding the constrained global optimum of an indefinite quadratic function since such a function may have many local optima finding the global optimum is a computationally difficult problem we give an overview of the most important methods used which include benders decomposition concave programming approaches enumerative techniques and bilinear programming

on monotonicity of takagi sugeno fuzzy systems with ellipsoidal regions this paper presents sufficient conditions on monotonicity of a takagi sugeno t s fuzzy system with linear submodels in the consequents of the rules where the input space is considered to be partitioned to ellipsoidal regions such regions commonly arise in practice if clustering algorithms are used to identify a fuzzy model from measured data by the monotonicity it is meant that the partial derivatives of the output of the mapping represented by the fuzzy model with respect to all inputs are nonnegative on a given universe of discourse the conditions are given in the form of linear matrix inequalities with respect to the parameters of the submodels that may be useful in solving associated optimization problems via efficient semidefinite programming techniques the proposed conditions reduce conservatism of the existing ones from two reasons first the conservatism is introduced only once since the conditions are not separated between antecedent and consequent parts of fuzzy rules second the domain of interest where monotonicity is enforced may be restricted the proposed algorithm is illustrated on least squares approximation of a multivariate function by a monotonic t s fuzzy system

invariant manifolds and singularly perturbed boundary value problems boundary value problems for singularly perturbed equations often have singular solutions with internal and boundary layers this paper addresses the question of when singular solutions to such prob

exchange lemmas 1 deng s lemma abstract deng s lemma gives estimates on the behavior of solutions of ordinary differential equations in the neighborhood of a partially hyperbolic equilibrium we prove a generalization in which partially hyperbolic equilibrium is replaced by normally hyperbolic invariant manifold

exchange lemmas 2 general exchange lemma abstract exchange lemmas are used in geometric singular perturbation theory to track flows near normally hyperbolic invariant manifolds we prove a general exchange lemma and show that it implies versions of existing exchange lemmas for rectifiable slow flows and loss of stability turning points