manifold regularization a geometric framework for learning from labeled and unlabeled examples we propose a family of learning algorithms based on a new form of regularization that allows us to exploit the geometry of the marginal distribution we focus on a semi supervised framework that incorporates labeled and unlabeled data in a general purpose learner some transductive graph learning algorithms and standard methods including support vector machines and regularized least squares can be obtained as special cases we use properties of reproducing kernel hilbert spaces to prove new representer theorems that provide theoretical basis for the algorithms as a result in contrast to purely graph based approaches we obtain a natural out of sample extension to novel examples and so are able to handle both transductive and truly semi supervised settings we present experimental evidence suggesting that our semi supervised algorithms are able to use unlabeled data effectively finally we have a brief discussion of unsupervised and fully supervised learning within our general framework

eigenvalues of second order difference equations with periodic and antiperiodic boundary conditions this paper is concerned with periodic and antiperiodic boundary value problems for self adjoint second order difference equations existence of eigenvalues of these two different boundary value problems is proved numbers of their eigenvalues are calculated and their relationships are obtained in addition a representation of solutions of a nonhomogeneous linear equation with initial conditions is given

periodic solutions for nonlinear second order difference equations abstract this paper is devoted to introduce a new approach to investigate the existence of periodic solutions for nonlinear difference equations transforming a second order nonlinear difference equation on a finite discrete segment with periodic boundary conditions into a continuous system we obtain the existence conditions of periodic solutions of the equation the proof relies on coincidence degree theory and matrix theory without using green s function

conservation laws for some systems of nonlinear pdes via the symmetry adjoint symmetry pair method abstract this paper applies the symmetry adjoint symmetry pair method to find conservation laws for nonlinear partial differential equations as examples conservation laws are obtained for a combined potential system with nonlinear telegraph and nonlinear diffusion convection equations as special cases as well as for an extended nonlinear wave equation system in addition for both examples conservation laws are obtained for and nonlocally related equivalent systems the paper concludes with a posed open problem

transitivity bounds in additive fuzzy preference structures transitivity plays a crucial role in preference modeling and related fields in this paper we discuss this property in the general context of additive fuzzy preference structures of particular interest is the decomposition of a large preference relation r in its symmetric part i indifference relation and its asymmetric part p strict preference relation by means of a so called indifference generator i given the type of transitivity of a large preference relation r w r t a conjunctor and a generator we establish basic lower bounds and general upper bounds on the transitivity of p and i these bounds are due to the careful design of generic counterexamples moreover we identify the situations in which these bounds are effectively reached thereby establishing connections with interesting properties such as dominance bisymmetry the 1 lipschitz property and rotation invariance

on representing and generating kernels by fuzzy equivalence relations kernels are two placed functions that can be interpreted as inner products in some hilbert space it is this property which makes kernels predestinated to carry linear models of learning optimization or classification strategies over to non linear variants following this idea various kernel based methods like support vector machines or kernel principal component analysis have been conceived which prove to be successful for machine learning data mining and computer vision applications when applying a kernel based method a central question is the choice and the design of the kernel function this paper provides a novel view on kernels based on fuzzy logical concepts which allows to incorporate prior knowledge in the design process it is demonstrated that kernels mapping to the unit interval with constant one in its diagonal can be represented by a commonly used fuzzy logical formula for representing fuzzy rule bases this means that a great class of kernels can be represented by fuzzy logical concepts apart from this result which only guarantees the existence of such a representation constructive examples are presented and the relation to unlabeled learning is pointed out

general transitivity conditions for fuzzy reciprocal preference matrices a reciprocal fuzzy matrix relation is a non negative matrix q qij such that qij qji 1 for all i j 1 2 n we define general transitivity conditions named fg transitivities for fuzzy reciprocal preference relations and show that they generalize some well known transitivities we also study relationships of these conditions with two models of rational preferences the so called utility model and the multidimensional model

multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces abstract the existence of multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces is discussed in this paper our nonlinearity may be singular in its dependent variable and our analysis relies on a nonlinear alternative of leray schauder type and on a fixed point theorem in cones

transitivity of fuzzy preference relations an empirical study based on an experiment we show that human preferences when represented by fuzzy relations may violate transitivity we define an index of transitivity and compute the degree to which the transitivity is violated

positive periodic solutions for nonlinear difference equations with periodic coefficients abstract we prove the existence of positive solutions of second order nonlinear difference equations on a finite discrete segment with periodic boundary conditions obtained results yield positive periodic solutions of the equation on the whole discrete axis provided that the coefficients are periodic

multiple solutions for discrete boundary value problems a recent multiplicity theorem for the critical points of a functional defined on a finite dimensional hilbert space established by ricceri is extended an application to dirichlet boundary value problems for difference equations involving the discrete p laplacian operator is presented

a compendium of fuzzy weak orders representations and constructions the present paper gives a state of the art overview of representation and construction results for fuzzy weak orders we do not assume that the underlying domain is finite instead we concentrate on results that hold in the most general case that the underlying domain is possibly infinite this paper presents three fundamental representation results each of which also provides a construction method i score function based representations ii inclusion based representations iii representations by decomposition into crisp linear orders and fuzzy equivalence relations which also facilitates a pseudo metric based construction

learning intransitive reciprocal relations with kernel methods abstract in different fields like decision making psychology game theory and biology it has been observed that paired comparison data like preference relations defined by humans and animals can be intransitive intransitive relations cannot be modeled with existing machine learning methods like ranking models because these models exhibit strong transitivity properties more specifically in a stochastic context where often the reciprocity property characterizes probabilistic relations such as choice probabilities it has been formally shown that ranking models always satisfy the well known strong stochastic transitivity property given this limitation of ranking models we present a new kernel function that together with the regularized least squares algorithm is capable of inferring intransitive reciprocal relations in problems where transitivity violations cannot be considered as noise in this approach it is the kernel function that defines the transition from learning transitive to learning intransitive relations and the kronecker product is introduced for representing the latter type of relations in addition we empirically demonstrate on two benchmark problems one in game theory and one in theoretical biology that our algorithm outperforms methods not capable of learning intransitive reciprocal relations

multiplicity results for nonlinear periodic fourth order difference equations with parameter dependence and singularities abstract this paper is devoted to the study of nonlinear singular and non singular fourth order difference equations coupled with periodic boundary value conditions in the paper some existence and nonexistence results are given the results are deduced from krasnoselskii s fixed point theorems in cones an exhaustive study of the green s function related to a linear fourth order problem is done

existence and uniqueness results for discrete second order periodic boundary value problems abstract we prove existence and uniqueness results for solutions of second order nonlinear difference equations on a finite discrete segment with periodic boundary conditions the results are based on the notion of upper and lower solutions

metrics and t equalities abstract the relationship between metrics and t equalities is investigated the latter are a special case of t equivalences a natural generalization of the classical concept of an equivalence relation it is shown that in the construction of metrics from t equalities triangular norms with an additive generator play a key role conversely in the construction of t equalities from metrics this role is played by triangular norms with a continuous additive generator or equivalently by continuous archimedean triangular norms these results are then applied to the biresidual operator e t of a triangular norm t it is shown that e t is a t equality on 0 1 if and only if t is left continuous furthermore it is shown that to any left continuous triangular norm t there correspond two particular t equalities on f x the class of fuzzy sets in a given universe x one of these t equalities is obtained from the biresidual operator e t t by means of a natural extension procedure these t equalities then give rise to interesting metrics on f x

multiple positive solutions of singular and nonsingular discrete problems via variational methods we employ the critical point theory to establish the existence of multiple solutions of some regular as well as singular discrete boundary value problems 2004 elsevier ltd all rights reserved msc 34b16 39a10

an existence theorem for fuzzy utility functions a new elementary proof first our paper consists in extending the ponsard s result to a convex set of objects by giving conditions under which a fuzzy complete preorder can be represented by a numerical function here we propose a new proof based on a fuzzy definition for topological separability basically the utility function we obtain is formally identical using homeomorphism of 0 1 and r and to a standard one but its foundations and thus the interpretation are different because of behavior fuzziness on the one hand and perfect separability property of a fuzzy preordered set on the other second we prove that it is always possible to exhibit an implicit fuzzy model that is hidden behind the standard one by means of the way the boolean standard models rank the elements of the quotient set

invariants for the time dependent harmonic oscillator i two equivalent families of linear invariants and two equivalent families of quadratic invariants are obtained for a harmonic oscillator with variable mass or with variable frequency explicit formulae are obtained for the linear and quadratic invariants in the case of a damped oscillator and the hamiltonian is seen as a special quadratic invariant

three positive solutions of boundary value problems for p laplacian difference equations in this paper existence criteria for three positive solutions of p laplacian difference equation f p u t 1 a t f u t 0 t 1 t 1 u 0 u t 2 0 oru 0 u t 1 0 are established by using the five functionals fixed point theorem as an application one example is given to illustrate the main result in this paper

a discrete fourth order lidstone problem with parameters various existence multiplicity and nonexistence results for nontrivial solutions to a nonlinear discrete fourth order lidstone boundary value problem with dependence on two parameters are given using a symmetric green s function approach an existence result is also given for a related semipositone problem thus relaxing the usual assumption of nonnegativity on the nonlinear term

connection between classical and quantum harmonic oscillators for a time dependent harmonic oscillator we show a connection between the integrals of classical equations of motion and the lewis riesenfeld lr invariant and express the lr invariant in terms of the integrals of classical equations of motion as applications we find the lr invariant for the guth pi hamiltonian and the asymptotic form of the lr invariant for a general time dependent harmonic oscillator

multiplicity results for periodic solutions to delay differential equations via critical point theory abstract by the critical point theory we study the existence and multiplicity of periodic solutions to the following system of delay differential equations x t f x t r where x r n f c r n r n and r 0 is a given constant a sufficient condition on the existence and multiplicity of periodic solutions for eq is obtained to the authors knowledge this is the first time the existence of periodic solutions to systems of delay differential equations is dealt with by using variational approaches directly

positive solutions for nonlinear discrete periodic boundary value problems we are concerned with the nonlinear discrete periodic boundary value problem p d p t 1 du t 1 q t u t rg t f u t t 1 t z u 0 u t p 0 du 0 p t du t where r is a positive parameter optimal interval will be given to the parameter r to ensure that p has at least one positive solution

positive periodic solutions of functional differential equations abstract we consider the existence multiplicity and nonexistence of positive periodic solutions for the periodic equation x t a t g x x t b t f x t t where a b c r 0 are periodic 0 a t dt 0 0 b t dt 0 f g c 0 0 and f u 0 for u 0 g x is bounded t is a continuous periodic function define f 0 lim u 0 f u u f lim u f u u i0 number of zeros in the set f 0 f and i number of infinities in the set f 0 f we show that the equation has i0 or i positive periodic solution s for sufficiently large or small 0 respectively

contact symmetries of the harmonic oscillator the symmetry group of contact transformations for the one dimensional harmonic oscillator is determined its generators contain two arbitrary functions i e it is an infinite parameter lie group

adjoint symmetries for time dependent second order equations the authors extend part of their previous work on autonomous second order systems sarlet et al 1987 to time dependent differential equations the main subject of the paper concerns the notion of adjoint symmetries they are introduced as a particular type of 1 form whose leading coefficients satisfy the adjoint equations of the equations determining symmetry vector fields it is shown that all interesting properties of adjoint symmetries known from the autonomous theory have their counterparts in the present framework of particular interest is a result establishing that lagrangian systems seem to be the only ones for which there is a natural duality between symmetries and adjoint symmetries a number of examples illustrate how the construction of adjoint symmetries of a given system can be explored in a systematic way

some remarks on the lie group of point transformations for the harmonic oscillator some aspects of the symmetry group sl 3 r of the simple harmonic oscillator are revisited a realisation of the general element of the group in newtonian spacetime is obtained with the eight parameters included the group is shown to be sl 3 r in fact without recourse to the lie algebra finally in this way the group law of binary composition of the parameters is calculated easily in the group manifold

periodic solutions of second order discrete convex systems involving the p laplacian some existence theorems are obtained for periodic solutions of second order discrete convex systems involving the p laplacian by using the dual least action principle

periodic solutions of second order nonlinear difference equations with discrete o laplacian we use brouwer degree to prove existence and multiplicity results for the periodic solutions of some nonlinear second order difference equations involving discrete phi laplacian we obtain in particular upper and lower solutions theorems ambrosetti prodi type multiplicity results sharp existence conditions for nonlinearities which are bounded from below or from above and necessary and sufficient conditions for the existence of positive periodic solutions when the nonlinearity is singular at 0 c 2006 elsevier inc all rights reserved

existence and multiplicity of solutions for nonlinear second order difference boundary value problems in the present paper the following nonlinear second order difference boundary value problems d p n 1 du n 1 q n u n f n u n 0 n z 1 n u 0 u n p 0 du 0 p n du n are studied by using the critical theory and three new multiple results of solutions are obtained where z 1 n 1 2 n f z 1 n xr r is a continuous function in the second variable p q z 0 n r and p n 0

on the symmetries and invariants of the harmonic oscillator a general formulation of noether s theorem is applied to the equation of a harmonic oscillator the definition of symmetry includes the usual lie invariance as a special case and unlike standard formulations generates the full set of invariants i e gives closure under functional composition the analysis for a time dependent oscillator casts doubt on the importance of a known class of invariants the existence of a lagrangian function is shown to be inessential to the analysis

a kernel based framework for learning graded relations from data driven by a large number of potential applications in areas such as bioinformatics information retrieval and social network analysis the problem setting of inferring relations between pairs of data objects has recently been investigated intensively in the machine learning community to this end current approaches typically consider datasets containing crisp relations so that standard classification methods can be adopted however relations between objects like similarities and preferences are often expressed in a graded manner in real world applications a general kernel based framework for learning relations from data is introduced here it extends existing approaches because both crisp and graded relations are considered and it unifies existing approaches because different types of graded relations can be modeled including symmetric and reciprocal relations this framework establishes important links between recent developments in fuzzy set theory and machine learning its usefulness is demonstrated through various experiments on synthetic and real world data the results indicate that incorporating domain knowledge about relations improves the predictive performance

pseudo symmetries noether s theorem and the adjoint equation pseudo symmetries were introduced by sarlet and cantrijn 1984 for time dependent non conservative systems they are reconsidered in the context of general autonomous second order systems relying on the new approach to such systems which was presented by sarlet et al they further introduce the notion of adjoint symmetries of a second order system as being associated to invariant 1 forms and show how they may be related to first integrals or to lagrangians under appropriate circumstances their results enable them to clarify a rather unusual account of noether s theorem which was recently given by gordon 1986

positive solutions for nonlinear discrete second order boundary value problems with parameter dependence in this paper we consider the nonlinear discrete boundary value problem p t 1 u t 1 q t u t f t u t t z 1 t u 0 u t p 0 u 0 p t u t where is a positive parameter by using the fixed point index theory the criteria of the existence multiplicity and nonexistence of positive solutions are established in terms of different values of

transitivity frameworks for reciprocal relations cycle transitivity versus fg transitivity for a reciprocal relation q on a set of alternatives a two transitivity frameworks which generalize both t transitivity and stochastic transitivity are compared the framework of cycle transitivity introduced by the present authors soc choice welf to appear and which is based upon the ordering of the numbers q a b q b c and q c a for all a b c a 3 and the framework of fg transitivity introduced by switalski fuzzy sets and systems 137 2003 85 as an immediate generalization of stochastic transitivity the rules that enable to express fg transitivity in the form of cycle transitivity and cycle transitivity in the form of fg transitivity illustrate that for reciprocal relations the concept of cycle transitivity provides a framework that can cover more types of transitivity than does the concept of fg transitivity

on the existence of positive solutions for nonlinear differential equations with periodic boundary conditions we prove the existence of positive solutions of second order nonlinear differential equations on a finite interval with periodic boundary conditions and give upper and lower bounds for these positive solutions obtained results yield positive periodic solutions of the equation on the whole real axis provided that the coefficients are periodic

on the influence of the kernel on the consistency of support vector machines in this article we study the generalization abilities of several classifiers of support vector machine svm type using a certain class of kernels that we call universal it is shown that the soft margin algorithms with universal kernels are consistent for a large class of classification problems including some kind of noisy tasks provided that the regularization parameter is chosen well in particular we derive a simple sufficient condition for this parameter in the case of gaussian rbf kernels on the one hand our considerations are based on an investigation of an approximation property the so called universality of the used kernels that ensures that all continuous functions can be approximated by certain kernel expressions this approximation property also gives a new insight into the role of kernels in these and other algorithms on the other hand the results are achieved by a precise study of the underlying optimization problems of the classifiers furthermore we show consistency for the maximal margin classifier as well as for the soft margin svm s in the presence of large margins in this case it turns out that also constant regularization parameters ensure consistency for the soft margin svm s finally we prove that even for simple noise free classification problems svm s with polynomial kernels can behave arbitrarily badly