weak focus limit cycles and bifurcations for bounded quadratic systems abstract we study the bounded quadratic systems with either two weak foci or a weak focus of order 2 from the first case we obtain 1 1 configuration of limit cycles and in the second case we prove that there is no limit cycle surrounding the weak focus of order 2 also we unfold the bounded quadratic system with a center and the maximum number of limit cycles in this case is two

closure of finite state automaton languages the relation of fuzzy 3 type grammars and fuzzy finite state automata was studied by the author in this paper the properties of fuzzy finite state automaton languages are studied i e under the operators of union concatenation and closure fuzzy finite state automaton languages are close

linear estimate for the number of limit cycles of a perturbed cubic polynomial differential system abstract perturbing the cubic polynomial differential systems x y a 1 x a 0 b 1 y b 0 y x a 1 x a 0 b 1 y b 0 having a center at the origin inside the class of all polynomial differential systems of degree n we obtain using the averaging theory of second order that at most 17 n 15 limit cycles can bifurcate from the periodic orbits of the center

support vector machine soft margin classifiers error analysis the purpose of this paper is to provide a pac error analysis for the q norm soft margin classifier a support vector machine classification algorithm it consists of two parts regularization error and sample error while many techniques are available for treating the sample error much less is known for the regularization error and the corresponding approximation error for reproducing kernel hilbert spaces we are mainly concerned about the regularization error it is estimated for general distributions by a k functional in weighted lq spaces for weakly separable distributions i e the margin may be zero satisfactory convergence rates are provided by means of separating functions a projection operator is introduced which leads to better sample error estimates especially for small complexity kernels the misclassification error is bounded by the v risk associated with a general class of loss functions v the difficulty of bounding the offset is overcome polynomial kernels and gaussian kernels are used to demonstrate the main results the choice of the regularization parameter plays an important role in our analysis

on the state minimization of fuzzy automata this paper investigates the minimization problem of fuzzy automata aiming to obtain a procedure for finding a minimal state fuzzy automaton equivalent to a given one the decision version of the minimization problem is as follows given a fuzzy automaton cal a and a natural number k i e a pair langle cal a k rangle is there a k state fuzzy automaton equivalent to cal a we prove that the above problem is decidable for fuzzy automata over totally ordered lattices and then obtain a procedure for minimizing a given fuzzy automaton to this end we introduce the concept of systems of fuzzy polynomial equations present a procedure for finding solutions of these systems and finally reduce the above decision problem to finding a solution of a system of fuzzy polynomial equations it is worth pointing out that although some algorithms in the literature were claimed to be minimization algorithms the term minimization there did not mean state minimization in our sense since these algorithms did not aim at a minimal fuzzy automaton but found reasonably small fuzzy automata

a categorical approach to minimal realization for a fuzzy language abstract in this study we provide a general theory of minimal realization for a given fuzzy language in a category theory setting we introduce categorical characterizations of a run map reachability map and observability map for a crisp deterministic fuzzy automaton finally we show that all minimal realizations for a given fuzzy language are isomorphic to the nerode realization for the same case

learning the kernel with hyperkernels this paper addresses the problem of choosing a kernel suitable for estimation with a support vector machine hence further automating machine learning this goal is achieved by defining a reproducing kernel hilbert space on the space of kernels itself such a formulation leads to a statistical estimation problem similar to the problem of minimizing a regularized risk functional we state the equivalent representer theorem for the choice of kernels and present a semidefinite programming formulation of the resulting optimization problem several recipes for constructing hyperkernels are provided as well as the details of common machine learning problems experimental results for classification regression and novelty detection on uci data show the feasibility of our approach

finite nondeterministic automata simulation and minimality abstract motivated by recent applications of finite automata to theoretical physics we study the minimization problem for nondeterministic automata with outputs but no initial states we use ehrenfeucht fraisse like games to model automata responses and simulations the minimal automaton is constructed and in contrast with the classical case proved to be unique up to an isomorphism finally we investigate the partial ordering induced by automata simulations for example we prove that with respect to this ordering the class of deterministic automata forms an ideal in the class of all automata

fast rates for support vector machines using gaussian kernels for binary classification we establish learning rates up to the order of n 1 for support vector machines svms with hinge loss and gaussian rbf kernels these rates are in terms of two assumptions on the considered distributions tsybakov s noise assumption to establish a small estimation error and a new geometric noise condition which is used to bound the approximation error unlike previously proposed concepts for bounding the approximation error the geometric noise assumption does not employ any smoothness assumption 1 introduction in recent years support vector machines svms have been the subject of many theoretical considerations despite this effort their learning performance on restricted classes of distributions is still widely unknown in particular it is unknown under which nontrivial circumstances svms can guarantee fast learning rates the aim of this work is to use concepts like tsybakov s noise assumption and local rademacher averages to establish learning rates up to the order of n 1 for nontrivial distributions in addition to these concepts that are used to deal with the stochastic part of the analysis we also introduce a geometric assumption for distributions that allows us to estimate the approximation properties of gaussian rbf kernels unlike many other concepts introduced for bounding the approximation error our geometric assumption is not in terms of smoothness but describes the concentration and the noisiness of the data generating distribution near the decision boundary let us formally introduce the statistical classification problem to this end let us fix a subset x r d we write y 1 1 given a finite training set

on the chebyshev property for a new family of functions abstract we analyze whether a given set of analytic functions is an extended chebyshev system this family of functions appears studying the number of limit cycles bifurcating from some nonlinear vector field in the plane our approach is mainly based on the so called derivation division algorithm we prove that under some natural hypotheses our family is an extended chebyshev system and when some of them are not fulfilled then the set of functions is not necessarily an extended chebyshev system one of these examples constitutes an extended chebyshev system with high accuracy

simultaneous bifurcation of limit cycles from a cubic piecewise center with two period annuli abstract we study the number of periodic orbits that bifurcate from a cubic polynomial vector field having two period annuli via piecewise perturbations the cubic planar system x y y x 1 2 y 2 x x 1 2 y 2 has simultaneously a center at the origin and at infinity we study up to first order averaging analysis the bifurcation of periodic orbits from the two period annuli first separately and second simultaneously this problem is a generalization of 24 to the piecewise systems class when the polynomial perturbation has degree n we prove that the inner and outer abelian integrals are rational functions and we provide an upper bound for the number of zeros when the perturbation is cubic the same degree as the unperturbed vector field the maximum number of limit cycles up to first order perturbation from the inner and outer annuli is 9 and 8 respectively when the simultaneous bifurcation problem is considered 12 limit cycles exist these limit cycles appear in three types of configurations 9 3 6 6 and 4 8 in the non piecewise scenario only 5 limit cycles were found

bifurcation of limit cycles from an n dimensional linear center inside a class of piecewise linear differential systems abstract let n be an even integer we study the bifurcation of limit cycles from the periodic orbits of the n dimensional linear center given by the differential system x 1 x 2 x 2 x 1 x n 1 x n x n x n 1 perturbed inside a class of piecewise linear differential systems our main result shows that at most 4 n 6 n 2 1 limit cycles can bifurcate up to first order expansion of the displacement function with respect to a small parameter for proving this result we use the averaging theory in a form where the differentiability of the system is not needed

further improvements of determinization methods for fuzzy finite automata in this paper we provide further improvements of determinization methods for fuzzy finite automata these methods perform better than all previous determinization methods for fuzzy finite automata developed by b lohlavek 3 li and pedrycz 38 ignjatovic et al 25 and jancic et al 33 in the sense that they produce smaller automata while require the same computation time the only exception is the brzozowski type determinization algorithm developed recently by jancic and iric 34 which produces a minimal crisp deterministic fuzzy automaton but the algorithms created here can also be used within the brzozowski type algorithm and improve its performance

the number of limit cycles of a quintic polynomial system in this paper we consider the bifurcation of limit cycles of the system email protected y x 2 a 2 y 2 b 2 email protected x y email protected x x 2 a 2 y 2 b 2 email protected x y for e sufficiently small where a email protected r 0 and p q are polynomials of degree n we obtain that up to first order in e the upper bounds for the number of limit cycles that bifurcate from the period annulus of the quintic center given by e 0 are 3 2 n sin 2 email protected 2 1 if a b and n 1 if a b moreover there are systems with at least 3 2 n sin 2 email protected 2 1 if a b and n 1 limit cycles if a b

limit cycles appearing from the perturbation of a system with a multiple line of critical points abstract consider planar ordinary differential equations of the form x y c x y y x c x y where c x y is an algebraic curve we are interested in knowing whether the existence of multiple factors for c is important or not when we study the maximum number of zeros of the abelian integral m that controls the limit cycles that bifurcate from the period annulus of the origin when we perturb it with an arbitrary polynomial vector field with this aim we study in detail the case c x y 1 y m where m is a positive integer number and prove that m has essentially no impact on the number of zeros of m this result improves the known studies on m one of the key points of our approach is that we obtain a simple expression of m based on some successive reductions of the integrals appearing during the procedure

bifurcation of limit cycles from centers and separatrix cycles of planar analytic systems this paper presents a survey of results on the bifurcation of limit cycles from centers and separatrix cycles of perturbed planar analytic systems and contributes some new results on the bifurcation of multiple limit cycles from centers and on the multiplicity of separatrix cycles of such systems the basic theme throughout the paper is that the number positions and multiplicities of the limit cycles that bifurcate under perturbations are related to the number positions and multiplicities of the zeros of the melnikov function for the system the general theory is illustrated by a number of examples from the literature some of which are extended to include new results

limit cycles of cubic polynomial vector fields via the averaging theory abstract in this paper we study the maximum number of limit cycles that can bifurcate from the period annulus surrounding the origin of a class of cubic polynomial differential systems using the averaging method more precisely we prove that the perturbations of the period annulus of the center located at the origin of the cubic polynomial differential system x y f x y y x f x y where f x y 0 is a conic such that f 0 0 0 by arbitrary cubic polynomial differential systems provide at least six limit cycles bifurcating from the periodic orbits of the period annulus using only the first order averaging method

characterizations of complete residuated lattice valued finite tree automata this paper deals with the concept of complete residuated lattice valued referred to as l valued finite tree automata in this regard we first define an l valued regular tree language and then we prove a necessary and sufficient condition for the regularity of an l valued tree language furthermore we generalize the pumping lemma for l valued finite tree automata l fta afterwards the behavior of l fta is addressed and some theorems are provided moreover the existence of the minimal form of an l fta is considered finally a minimization algorithm of the l fta is presented and its time complexity is analyzed

computing behavior of finite fuzzy machines algorithm and its application to reduction and minimization we define finite fuzzy machines and investigate their behavior algorithm and software are proposed for computing behavior for establishing equivalence and redundancy of states and for solving reduction and minimization problems computational complexity of the algorithm is discussed testing examples are supplied the results are valid for finite max min min max and max product fuzzy machines

minimization of fuzzy finite automata abstract in this paper we show that a fuzzy finite automation m 1 has an equivalent minimal fuzzy finite automation m we also show that m can be chsen so that if m 2 is equivalent to m 1 then m is a homomorphic image of m 2

bifurcation of limit cycles from a two dimensional center inside rn abstract we study the bifurcation of limit cycles from the periodic orbits of a linear differential system in r n perturbed inside a class of piecewise linear differential systems which appears in a natural way in control theory our main result shows that at most one limit cycle can bifurcate up to first order expansion of the displacement function with respect to the small parameter this upper bound is reached for proving this result we use the averaging theory in a form where the differentiability of the system is not needed

minimization algorithm of fuzzy finite automata in this paper a class of fuzzy finite automata corresponding to the mealy type of ordinary automata is formulated and also two types of statewise equivalence relations are introduced from the equivalence relations a minimal form is defined and a minimization algorithm of the mealy type of fuzzy finite automata is obtained

products of mealy type fuzzy finite state machines we introduce seven ways of constructing products for mealy type fuzzy finite state machines as well as the concept of covering the properties of transition functions and output functions are discussed furthermore we investigate the covering properties and mutual relationship with regard to these products we also prove that the covering relationship holds in the product of factor machines

rademacher chaos complexities for learning the kernel problem we develop a novel generalization bound for learning the kernel problem first we show that the generalization analysis of the kernel learning problem reduces to investigation of the suprema of the rademacher chaos process of order 2 over candidate kernels which we refer to as rademacher chaos complexity next we show how to estimate the empirical rademacher chaos complexity by well established metric entropy integrals and pseudo dimension of the set of candidate kernels our new methodology mainly depends on the principal theory of u processes and entropy integrals finally we establish satisfactory excess generalization bounds and misclassification error rates for learning gaussian kernels and general radial basis kernels

statistical behavior and consistency of classification methods based on convex risk minimization we study how closely the optimal bayes error rate can be approximately reached using a classification algorithm that computes a classifier by minimizing a convex upper bound of the classification error function the measurement of closeness is characterized by the loss function used in the estimation we show that such a classification scheme can be generally regarded as a nonmaximum likelihood conditional in class probability estimate and we use this analysis to compare various convex loss functions that have appeared in the literature furthermore the theoretical insight allows us to design good loss functions with desirable properties another aspect of our analysis is to demonstrate the consistency of certain classification methods using convex risk minimization this study sheds light on the good performance of some recently proposed linear classification methods including boosting and support vector machines it also shows their limitations and suggests possible improvements

convexity classification and risk bounds many of the classification algorithms developed in the machine learning literature including the support vector machine and boosting can be viewed as minimum contrast methods that minimize a convex surrogate of the 0 1 loss function the convexity makes these algorithms computationally efficient the use of a surrogate however has statistical consequences that must be balanced against the computational virtues of convexity to study these issues we provide a general quantitative relationship between the risk as assessed using the 0 1 loss and the risk as assessed using any nonnegative surrogate loss function we show that this relationship gives nontrivial upper bounds on excess risk under the weakest possible condition on the loss function that it satisfies a pointwise form of fisher consistency for classification the relationship is based on a simple variational transformation of the loss function that is easy to compute in many applications we also present a refined version of this result in the

melnikov theory for a class of planar hybrid systems abstract in this paper we extend the well known melnikov theory for smooth systems to a class of two zonal planar hybrid piecewise smooth systems in this class each zone of differentiability is separated by a straight line and in these zones the dynamics are governed by a smooth system when an orbit reaches the separation line then a reset map applies before entering the orbit in the other zone the family of non generic systems of this class which has a continuum of periodic orbits is considered then we study the periodic orbits of the continuum that persist under a perturbation to achieve this objective first we build an appropriate poincare map after that we introduce a function of one variable whose zeros provide us the periodic orbits that remain after the perturbation this function will be called the melnikov function because it is very close to the so called melnikov function for smooth systems finally we apply the obtained results to some piecewise linear systems continuous as well as discontinuous ones

automata theory based on complete residuated lattice valued logic reduction and minimization automata theory based on complete residuated lattice valued logic called l valued finite automata l vfas has been established by qiu recently in this paper we define a kind of mealy type of l vfas mlfas a generalization of l vfas two kinds of statewise equivalence relations are introduced and a minimal form is defined we study the existence of the minimal form of an mlfa then we show that any two states can be distinguished by some word with finite length also a minimization algorithm of the mlfas is presented in addition we obtain a minimization algorithm for l vfas as well finally we define l valued languages l vls and l valued regular languages l vrls recognized by l vfas and provide some properties of l vrls

the cyclicity of period annuli for a class of cubic hamiltonian systems with nilpotent singular points abstract this paper deals with the limit cycles of a class of cubic hamiltonian systems under polynomial perturbations we suppose that the corresponding hamiltonian system which has at least one center has finite singular points and is symmetrical with respect to both x axis and y axis and also the origin is a nilpotent singular point hence the hamiltonian h x y can be written as h x y x 2 a x 4 b x 2 y 2 c y 4 a b c r 3 c 0 for the above h x y the corresponding vector field is x 2 y b x 2 2 c y 2 y 2 x 1 2 a x 2 b y 2 we first obtain that the above vector fields with at least one center can be divided into 8 classes by its topological phase portraits for the following perturbed system x 2 y b x 2 2 c y 2 e f x y y 2 x 1 2 a x 2 b y 2 e g x y where 0 e 1 f x y and g x y are polynomials in x y with degree n we give an estimation of the number of isolated zeros of the corresponding abelian integral the number of limit cycles follows from poincare pontryagin theorem

on the abelian integrals of quadratic reversible centers with orbits formed by genus one curves of higher degree abstract in this study we determine the number of zeros for abelian integrals in four cases of quadratic reversible centers of genus one based on the results of li et al we prove that the upper bound of the associated number of zeros for abelian integrals with orbits formed by quartics quintics sextics and eight degree curves under polynomial perturbations of arbitrary degree n depends linearly on n

limit cycles bifurcating from piecewise quadratic systems separated by a straight line abstract in this paper we are concerned with limit cycles bifurcating from piecewise quadratic systems separated by a straight line by means of the first order averaging theory of discontinuous differential systems and the argument principle we present an estimate of the maximum number of limit cycles bifurcating from the period annulus around the center of piecewise smooth systems

periodic orbits for perturbations of piecewise linear systems abstract we consider the existence of periodic orbits in a class of three dimensional piecewise linear systems firstly we describe the dynamical behavior of a non generic piecewise linear system which has two equilibria and one two dimensional invariant manifold foliated by periodic orbits the aim of this work is to study the periodic orbits of the continuum that persist under a piecewise linear perturbation of the system in order to analyze this situation we build a real function of real variable whose zeros are related to the limit cycles that remain after the perturbation by using this function we state some results of existence and stability of limit cycles in the perturbed system as well as results of bifurcations of limit cycles the techniques presented are similar to the melnikov theory for smooth systems and the method of averaging

limit cycle bifurcations in perturbations of planar piecewise smooth systems with multiply lines of critical points abstract this paper investigates the limit cycle bifurcation problem of planar piecewise polynomial systems whose unperturbed system has a center at the origin and may have invariant multiply lines by melnikov function method this paper provides the upper and lower bounds for the poincare cyclicity in particular the exact cyclicities of two specific cases are given using analyzing method one of which has been discussed in the paper 7

construction of fuzzy automata from fuzzy regular expressions li and pedrycz have proved fundamental results that provide different equivalent ways to represent fuzzy languages with membership values in a lattice ordered monoid and generalize the well known results of the classical theory of formal languages in particular they have shown that a fuzzy language over an integral lattice ordered monoid can be represented by a fuzzy regular expression if and only if it can be recognized by a fuzzy finite automaton however they did not give any efficient method for constructing an equivalent fuzzy finite automaton from a given fuzzy regular expression in this paper we provide such an efficient method transforming scalars appearing in a fuzzy regular expression ainto letters of the new extended alphabet we convert the fuzzy regular expression ato an ordinary regular expression a r then starting from an arbitrary nondeterministic finite automaton a that recognizes the language a r represented by the regular expression a r we construct fuzzy finite automata a a and a a r with the same or even less number of states than the automaton a which recognize the fuzzy language a represented by the fuzzy regular expression a the starting nondeterministic finite automaton a can be obtained from a r using any of the well known constructions for converting regular expressions to nondeterministic finite automata such as glushkov mcnaughton yamada s position automaton brzozowski s derivative automaton antimirov s partial derivative automaton or ilie yu s follow automaton

periodic orbits from second order perturbation via rational trigonometric integrals abstract the second order poincare pontryagin melnikov perturbation theory is used in this paper to study the number of bifurcated periodic orbits from certain centers this approach also allows us to give the shape and the period up to the first order we address these problems for some classes of abel differential equations and quadratic isochronous vector fields in the plane we prove that two is the maximum number of hyperbolic periodic orbits bifurcating from the isochronous quadratic centers with a birational linearization under quadratic perturbations of second order in particular the configurations 2 0 and 1 1 are realizable when two centers are perturbed simultaneously the required computations show that all the considered families share the same iterated rational trigonometric integrals

refined generalization bounds of gradient learning over reproducing kernel hilbert spaces gradient learning gl initially proposed by mukherjee and zhou 2006 has been proved to be a powerful tool for conducting variable selection and dimensional reduction simultaneously this approach presents a nonparametric version of a gradient estimator with positive definite kernels without estimating the true function itself so that the proposed version has wide applicability and allows for complex effects between predictors in terms of theory however existing generalization bounds for gl depend on capacity independent techniques and the capacity of kernel classes cannot be characterized completely thus this letter considers gl estimators that minimize the empirical convex risk we prove generalization bounds for such estimators with rates that are faster than previous results moreover we provide a novel upper bound for rademacher chaos complexity of order two which also plays an important role in general pairwise type estimations including ranking and score problems

the number of limit cycles of a class of polynomial differential systems abstract in this paper we consider the bifurcation of limit cycles of a class of polynomial differential systems of the form x y 1 x 4 e p n x y y x 1 x 4 e q n x y where p n q n are arbitrary polynomials of degree n we prove that there is a system of the above form having at least 3 n 1 2 2 limit cycles in hopf bifurcation then applying the argument principle we obtain that up to first order in e the number of limit cycles that bifurcate from period annulus surrounding the origin of this system is at most 5 n 1 4 n 1 2 for n 5

on the number of limit cycles bifurcating from a non global degenerated center abstract we give an upper bound for the number of zeros of an abelian integral this integral controls the number of limit cycles that bifurcate by a polynomial perturbation of arbitrary degree n from the periodic orbits of the integrable system 1 x d h 0 where h is the quasi homogeneous hamiltonian h x y x 2 k 2 k y 2 2 the tools used in our proofs are the argument principle applied to a suitable complex extension of the abelian integral and some techniques in real analysis

on the number of limit cycles surrounding a unique singular point for polynomial differential systems of arbitrary degree abstract we study the number of limit cycles that bifurcate from the periodic orbits of the center x y r x y y x r x y where r is a convenient polynomial of degree 2 when we perturb it inside the class of all polynomial differential systems of degree n we use averaging theory for computing this number as a consequence of our study we provide the biggest number of limit cycles surrounding a unique singular point in terms of the degree of the system known up to now

on factorized l fuzzy automaton and its l fuzzy topological characterization abstract this paper is towards the study of l fuzzy automata via theory of factorizations where l is a complete residuated lattice we study the concept of factorization of l fuzzy sets and factorization of collection of l fuzzy sets such factorizations are used i to study the concept of minimal l fuzzy automaton and ii to introduce l fuzzy topologies for the study of concepts associated with an l fuzzy automaton specifically we introduce the notion of factorized l fuzzy automaton corresponding to a given l fuzzy automaton and study the minimality of such automaton further we associate l fuzzy topologies for factorized l fuzzy automaton and use these to characterize some algebraic concepts

on ranking and generalization bounds the problem of ranking is to predict or to guess the ordering between objects on the basis of their observed features in this paper we consider ranking estimators that minimize the empirical convex risk we prove generalization bounds for the excess risk of such estimators with rates that are faster than 1 n we apply our results to commonly used ranking algorithms for instance boosting or support vector machines moreover we study the performance of considered estimators on real data sets

bifurcation of limit cycles by perturbing a piecewise linear hamiltonian system with a homoclinic loop abstract in this paper we study limit cycle bifurcations for a kind of non smooth polynomial differential systems by perturbing a piecewise linear hamiltonian system with the center at the origin and a homoclinic loop around the origin by using the first melnikov function of piecewise near hamiltonian systems we give lower bounds of the maximal number of limit cycles in hopf and homoclinic bifurcations and derive an upper bound of the number of limit cycles that bifurcate from the periodic annulus between the center and the homoclinic loop up to the first order in e in the case when the degree of perturbing terms is low we obtain a precise result on the number of zeros of the first melnikov function

linear estimation of the number of zeros of abelian integrals for some cubic isochronous centers abstract this paper consists of two parts in the first part we study the relationship between conic centers all orbits near a singular point of center type are conics and isochronous centers of polynomial systems in the second part we study the number of limit cycles that bifurcate from the periodic orbits of cubic reversible isochronous centers having all their orbits formed by conics when we perturb such systems inside the class of all polynomial systems of degree n

reduction of fuzzy automata by means of fuzzy quasi orders abstract in our recent paper we have established close relationships between state reduction of a fuzzy automaton and resolution of a particular system of fuzzy relation equations in that paper we have also studied reductions by means of those solutions which are fuzzy equivalences in this paper we will see that in some cases better reductions can be obtained using the solutions of this system that are fuzzy quasi orders generally fuzzy quasi orders and fuzzy equivalences are equally good in the state reduction but we show that right and left invariant fuzzy quasi orders give better reductions than right and left invariant fuzzy equivalences we also show that alternate reductions by means of fuzzy quasi orders give better results than alternate reductions by means of fuzzy equivalences furthermore we study a more general type of fuzzy quasi orders weakly right and left invariant ones and we show that they are closely related to determinization of fuzzy automatons we also demonstrate some applications of weakly left invariant fuzzy quasi orders in conflict analysis of fuzzy discrete event systems

learning the kernel function via regularization we study the problem of finding an optimal kernel from a prescribed convex set of kernels k for learning a real valued function by regularization we establish for a wide variety of regularization functionals that this leads to a convex optimization problem and for square loss regularization we characterize the solution of this problem we show that although k may be an uncountable set the optimal kernel is always obtained as a convex combination of at most m 2 basic kernels where m is the number of data examples in particular our results apply to learning the optimal radial kernel or the optimal dot product kernel

learnability of gaussians with flexible variances gaussian kernels with flexible variances provide a rich family of mercer kernels for learning algorithms we show that the union of the unit balls of reproducing kernel hilbert spaces generated by gaussian kernels with flexible variances is a uniform glivenko cantelli ugc class this result confirms a conjecture concerning learnability of gaussian kernels and verifies the uniform convergence of many learning algorithms involving gaussians with changing variances rademacher averages and empirical covering numbers are used to estimate sample errors of multi kernel regularization schemes associated with general loss functions it is then shown that the regularization error associated with the least square loss and the gaussian kernels can be greatly improved when flexible variances are allowed finally for regularization schemes generated by gaussian kernels with flexible variances we present explicit learning rates for regression with least square loss and classification with hinge loss

the number of limit cycles of a quintic polynomial system with center abstract in this paper we study the global bifurcation of limit cycles for the system x y 1 x 4 y x 1 x 4 e y 2 m 1 j 0 l a j x j for e sufficiently small where l 2 n 2 or 2 n 3 m n are arbitrary positive integers and a 0 a 1 a l are real we have used argument principle to give estimate of an upper bound for the number of limit cycles that can bifurcate from period annulus of this system for e 0 furthermore we have shown that there exist a set of constant a 0 a 1 a l where the related abelian integral of this system has at least 3 m n 2 isolated zeros we have in order to prove our result applied the argument principle to a complex extension of the abelian integral

bisimulations for fuzzy automata bisimulations have been widely used in many areas of computer science to model equivalence between various systems and to reduce the number of states of these systems whereas uniform fuzzy relations have recently been introduced as a means to model the fuzzy equivalence between elements of two possible different sets here we use the conjunction of these two concepts as a powerful tool in the study of equivalence between fuzzy automata we prove that a uniform fuzzy relation between fuzzy automata a and b is a forward bisimulation if and only if its kernel and co kernel are forward bisimulation fuzzy equivalence relations on a and b and there is a special isomorphism between factor fuzzy automata with respect to these fuzzy equivalence relations as a consequence we get that fuzzy automata a and b are ufb equivalent i e there is a uniform forward bisimulation between them if and only if there is a special isomorphism between the factor fuzzy automata of a and b with respect to their greatest forward bisimulation fuzzy equivalence relations this result reduces the problem of testing ufb equivalence to the problem of testing isomorphism of fuzzy automata which is closely related to the well known graph isomorphism problem we prove some similar results for backward forward bisimulations and we point to fundamental differences because of the duality with the studied concepts backward and forward backward bisimulations are not considered separately finally we give a comprehensive overview of various concepts on deterministic nondeterministic fuzzy and weighted automata which are related to bisimulations

refined rademacher chaos complexity bounds with applications to the multikernel learning problem estimating the rademacher chaos complexity of order two is important for understanding the performance of multikernel learning mkl machines in this letter we develop a novel entropy integral for rademacher chaos complexities as compared to the previous bounds our result is much improved in that it introduces an adjustable parameter e to prohibit the divergence of the involved integral with the use of the iteration technique in steinwart and scovel 2007 we also apply our rademacher chaos complexity bound to the mkl problems and improve existing learning rates

fuzzy ω automata abstract in this paper we introduce the notion of fuzzy automata as an accepting device for fuzzy languages we define fuzzy finite state automata which accept fuzzy regular languages with different acceptance criteria we also define distributed fuzzy finite state automata which are a group of fuzzy finite state automata working in unison to accept a language with different modes of acceptance like the t mode mode k mode k mode and k mode we prove that distributed fuzzy finite state automata accept the same set of languages as fuzzy finite state automata analogously we define fuzzy pushdown automata and distributed fuzzy pushdown automata along with the different acceptance criteria and modes of acceptance for the distributed case the class of languages accepted by fuzzy pushdown automata is fuzzy cfl while the class of languages accepted by distributed fuzzy pushdown automata contains cfl we give proofs to show that all modes of acceptance are equal in the case of distributed fuzzy pushdown automata we also define l fuzzy automata and give some examples to show how it can be used to accept some non fuzzy languages which are not accepted by their classical counterparts

congruences and homomorphisms of fuzzy automata concepts of congruences and homomorphisms of fuzzy automata are defined and homomorphism theorem is proved lattices of congruences of fuzzy automata are described and an algorithm for computing the greatest congruence contained in an equivalence of a fuzzy automaton is presented it is used for minimization procedures for both fuzzy automata with and without outputs connections with previously known minimization algorithms are discussed

minimization of states in automata theory based on finite lattice ordered monoids in this study we consider a concept of complete l fuzzy matrix define complete lattice valued finite automata clfas and study their properties the definitions of statewise equivalence relations and automata equivalence relations of a clfa are given two algorithms are aimed at the minimization of states of a clfa

on the greatest solutions to weakly linear systems of fuzzy relation inequalities and equations in this paper we study systems of fuzzy relation inequalities and equations of the form u v i v i u i i where u is an unknown and v i i i are given fuzzy relations the dual systems v i u u v i i i their conjunctions the systems of the form u v i v i u i i and certain special types of these systems we call them weakly linear systems for each weakly linear system with a complete residuated lattice as the underlying structure of truth values we prove the existence of the greatest solution and we provide an algorithm for computing the greatest solution which works whenever the underlying complete residuated lattice is locally finite otherwise we determine some sufficient conditions under which the algorithm works the algorithm is iterative and each its single step can be viewed as solving of a particular linear system weakly linear systems emerged from the fuzzy automata theory but we show that they also have important applications in other fields e g in the concurrency theory and social network analysis

myhill nerode type theory for fuzzy languages and automata the myhill nerode theory is a branch of the algebraic theory of languages and automata in which formal languages and deterministic automata are studied through right congruences and congruences on a free monoid in this paper we develop a general myhill nerode type theory for fuzzy languages with membership values in an arbitrary set with two distinguished elements 0 and 1 which are needed to take crisp languages in consideration we establish connections between extensionality of fuzzy languages w r t right congruences and congruences on a free monoid and recognition of fuzzy languages by deterministic automata and monoids and we prove the myhill nerode type theorem for fuzzy languages we also prove that each fuzzy language possess a minimal deterministic automaton recognizing it we give a construction of this automaton using the concept of a derivative automaton of a fuzzy language and we give a method for minimization of deterministic fuzzy recognizers in the second part of the paper we introduce and study nerode s and myhill s automata assigned to a fuzzy automaton with membership values in a complete residuated lattice the obtained results establish nice relationships between fuzzy languages fuzzy automata and deterministic automata

error analysis of coefficient based regularized algorithm for density level detection in this letter we consider a density level detection dld problem by a coefficient based classification framework with regularizer and data dependent hypothesis spaces although the data dependent characteristic of the algorithm provides flexibility and adaptivity for dld it leads to difficulty in generalization error analysis to overcome this difficulty an error decomposition is introduced from an established classification framework on the basis of this decomposition the estimate of the learning rate is obtained by using rademacher average and stepping stone techniques in particular the estimate is independent of the capacity assumption used in the previous literature

minimization of lattice finite automata and its application to the decomposition of lattice languages in this paper we study the minimization of lattice valued finite automata with membership values in a distributive lattice and its relationships to the decomposition of lattice valued regular languages first we establish the equivalence of nondeterministic lattice finite automata la and deterministic lattice finite automata dla furthermore we provide some characterization of lattice valued regular languages and regular operations on family of lattice valued regular languages in the sequel we introduce some notions that help clarify the concept of minimal dlas and present an effective algorithm to obtain a minimal dla from a given la using the construction of minimal dla we introduce some simple classes of lattice valued regular languages such as l unitary and l prefix ones we demonstrate that any lattice valued regular language can be decomposed as disjoint joins of such kinds of simple languages

learning the kernel matrix with semidefinite programming kernel based learning algorithms work by embedding the data into a euclidean space and then searching for linear relations among the embedded data points the embedding is performed implicitly by specifying the inner products between each pair of points in the embedding space this information is contained in the so called kernel matrix a symmetric and positive semidefinite matrix that encodes the relative positions of all points specifying this matrix amounts to specifying the geometry of the embedding space and inducing a notion of similarity in the input space classical model selection problems in machine learning in this paper we show how the kernel matrix can be learned from data via semidefinite programming sdp techniques when applied to a kernel matrix associated with both training and test data this gives a powerful transductive algorithm using the labeled part of the data one can learn an embedding also for the unlabeled part the similarity between test points is inferred from training points and their labels importantly these learning problems are convex so we obtain a method for learning both the model class and the function without local minima furthermore this approach leads directly to a convex method for learning the 2 norm soft margin parameter in support vector machines solving an important open problem

covering number bounds of certain regularized linear function classes recently sample complexity bounds have been derived for problems involving linear functions such as neural networks and support vector machines in many of these theoretical studies the concept of covering numbers played an important role it is thus useful to study covering numbers for linear function classes in this paper we investigate two closely related methods to derive upper bounds on these covering numbers the first method already employed in some earlier studies relies on the so called maurey s lemma the second method uses techniques from the mistake bound framework in online learning we compare results from these two methods as well as their consequences in some learning formulations