axiomatics for fuzzy rough sets abstract a fuzzy t rough set consists of a set x and a t similarity relation r on x where t is a lower semi continuous triangular norm we generalize the farinas prade definition for the upper approximation operator a i x i x of a fuzzy t rough set x r given originally for the special case t min to the case of arbitrary t we propose a new definition for the lower approximation operator a i x i x of x r our definition satisfies the two important identities a a a and a a a as well as a number of other interesting properties we provide axiomatics to fully characterize those upper and lower approximations

rough approximation operators on two universes of discourse and their fuzzy extensions based on a crisp binary relation between two universes of discourse one crisp covering and three crisp binary relations on the single universe of discourse are induced in this paper based on these induced notions four pairs of rough approximation operators are formulated so that the approximating sets and the approximated sets are on the same universe of discourse furthermore basic properties of the new approximation operators are investigated and the relationships among the operators are also examined subsequently conditions under which some or all of these approximation operators are equivalent are obtained on the other hand the generalizations of these rough approximation operators are also made in a fuzzy approximation space via a fuzzy implicator i and a triangular norm t consequently four pairs of fuzzy rough approximation operators are constructed basic properties and comparison of the fuzzy rough approximation operators are discussed if i is an r implicator with respect to a t norm t then it is shown that three pairs of the fuzzy rough approximation operators are identical iff the induced fuzzy covering is a fuzzy t partition

analytic treatment of a system with a vibro impact nonlinear energy sink abstract this communication reports on an approximate analytic procedure for analysis of transient damped response in a system comprising a primary linear oscillator and vibro impact nonlinear energy sink nes multiple scale expansion reduces the problem in the first approximation to low dimensional vibro impact system with forcing and damping the latter problem yields to direct analytic approach the results computed by this way fairly correlate with direct numeric simulations the analysis takes advantage of small ratio of the nes and the primary masses but does not require the restitution coefficient to be close to unity the proposed approach paves the way for efficient design of systems with vibro impact ness

dynamics of forced system with vibro impact energy sink abstract the paper treats forced response of primary linear oscillator with vibro impact energy sink this system exhibits some features of dynamics which resemble forced systems with other types of nonlinear energy sinks such as steady state and strongly modulated responses however the differences are crucial in the system with vibro impact sink the strongly modulated response consists of randomly distributed periods of resonant and non resonant motion this salient feature allows us to identify this type of dynamic behavior as chaotic strongly modulated response csmr it is demonstrated that the csmr exists due to special structure of a slow invariant manifold sim the latter is derived in a course of a multiple scale analysis of the system in the considered system this manifold has only one stable and one unstable branch this feature defines new class of universality for the nonlinear energy sinks very different physical system with topologically similar sim the oscillator with rotational energy sink also exhibits csmrs in the system with the vibro impact sink such responses are observed even for very low level of the external forcing this feature makes such system viable for possible energy harvesting applications

study of a class of subharmonic motions using a non smooth temporal transformation nstt abstract a nonlinear boundary value problem nbvp formulation for computing strongly nonlinear subharmonic orbits of a class of harmonically forced conservative systems is presented the formulation is based on a non smooth temporal transformation nstt to replace the independent temporal variable of the problem with two new piecewise linear periodic temporal variables as a result the problem of computing the subharmonic motions is reduced to a set of nbvps with homogeneous boundary conditions some numerical and analytical solutions of the reduced nbvps are given and the asymptotic behavior in the limit of large period of the derived analytical approximations is discussed in particular the formulation shows a definite connection between vibro impact oscillations considered in mechanics and strongly nonlinear oscillations close to a heteroclinic orbit an interesting feature of a family of subharmonic solutions analyzed in this work is that in the limit of large periods it degenerates to the perturbed stable and unstable manifolds of an unstable periodic orbit of the system

bayesian classification of tumours by using gene expression data summary precise classification of tumours is critical for the diagnosis and treatment of cancer diagnostic pathology has traditionally relied on macroscopic and microscopic histology and tumour morphology as the basis for the classification of tumours current classification frameworks however cannot discriminate between tumours with similar histopathologic features which vary in clinical course and in response to treatment in recent years there has been a move towards the use of complementary deoxyribonucleic acid microarrays for the classification of tumours these high throughput assays provide relative messenger ribonucleic acid expression measurements simultaneously for thousands of genes a key statistical task is to perform classification via different expression patterns gene expression profiles may offer more information than classical morphology and may provide an alternative to classical tumour diagnosis schemes the paper considers several bayesian classification methods based on reproducing kernel hilbert spaces for the analysis of microarray data we consider the logistic likelihood as well as likelihoods related to support vector machine models it is shown through simulation and examples that support vector machine models with multiple shrinkage parameters produce fewer misclassification errors than several existing classical methods as well as bayesian methods based on the logistic likelihood or those involving only one shrinkage parameter

fractional norm regularization learning with very few relevant features learning in the presence of a large number of irrelevant features is an important problem in high dimensional tasks previous studies have shown that l1 norm regularization can be effective in such cases while l2 norm regularization is not furthermore work in compressed sensing suggests that regularization by nonconvex e g fractional semi norms may outperform l1 regularization however for classification it is largely unclear when this may or may not be the case in addition the nonconvex problem is harder to solve than the convex l1 problem in this paper we provide a more in depth analysis to elucidate the potential advantages and pitfalls of nonconvex regularization in the context of logistic regression where the regularization term employs the family of lq semi norms first using results from the phenomenon of concentration of norms and distances in high dimensions we gain intuition about the working of sparse estimation when the dimensionality is very high second using the probably approximately correct pac bayes methodology we give a data dependent bound on the generalization error of lq regularized logistic regression which is applicable to any algorithm that implements this model and may be used to predict its generalization behavior from the training set alone third we demonstrate the usefulness of our approach by experiments and applications where the pac bayes bound is used to guide the choice of semi norm in the regularization term the results support the conclusion that the optimal choice of regularization depends on the relative fraction of relevant versus irrelevant features and a fractional norm with a small exponent is most suitable when the fraction of relevant features is very small

use of the zero norm with linear models and kernel methods we explore the use of the so called zero norm of the parameters of linear models in learning minimization of such a quantity has many uses in a machine learning context for variable or feature selection minimizing training error and ensuring sparsity in solutions we derive a simple but practical method for achieving these goals and discuss its relationship to existing techniques of minimizing the zero norm the method boils down to implementing a simple modification of vanilla svm namely via an iterative multiplicative rescaling of the training data applications we investigate which aid our discussion include variable and feature selection on biological microarray data and multicategory classification

dynamics of linear discrete systems connected to local essentially non linear attachments the dynamics of a linear periodic substructure weakly coupled to an essentially non linear attachment are studied the essential non linearizable non linearity of the attachment enables it to resonate with any of the linearized modes of the subtructure leading to energy pumping phenomena e g passive one way irreversible transfer of energy from the substructure to the attachment as a specific application the dynamics of a finite linear chain of coupled oscillators with a non linear end attachment is examined in the absence of damping it is found that the dynamical effect of the non linear attachment is predominant in neighborhoods of internal resonances between the attachment and the chain when damping exists energy pumping phenomena are realized in the system it is shown that energy pumping strongly depends on the topological structure of the non linear normal modes nnms of the underlying undamped system this is due to the fact that energy pumping is caused by the excitation of certain damped invariant nnm manifolds that are analytic continuations for weak damping of nnms of the underlying undamped system the bifurcations of the nnms of the undamped system help explain resonance capture cascades in the damped system this is a series of energy pumping phenomena occurring at different frequencies with sudden lower frequency transitions between sequential events the observed multi frequency energy pumping cascades are particularly interesting from a practical point of view since they indicate that non linear attachments can be designed to resonate and extract energy from an a priori specified set of modes of a linear structure in compatibility with the design objectives

pac bayesian generalisation error bounds for gaussian process classification approximate bayesian gaussian process gp classification techniques are powerful non parametric learning methods similar in appearance and performance to support vector machines based on simple probabilistic models they render interpretable results and can be embedded in bayesian frameworks for model selection feature selection etc in this paper by applying the pac bayesian theorem of mcallester 1999a we prove distribution free generalisation error bounds for a wide range of approximate bayesian gp classification techniques we also provide a new and much simplified proof for this powerful theorem making use of the concept of convex duality which is a backbone of many machine learning techniques we instantiate and test our bounds for two particular gpc techniques including a recent sparse method which circumvents the unfavourable scaling of standard gp algorithms as is shown in experiments on a real world task the bounds can be very tight for moderate training sample sizes to the best of our knowledge these results provide the tightest known distribution free error bounds for approximate bayesian gpc methods giving a strong learning theoretical justification for the use of these techniques

on the approximability of minimizing nonzero variables or unsatisfied relations in linear systems we investigate the computational complexity of two closely related classes of combinatorial optimization problems for linear systems which arise in various fields such as machine learning operations research and pattern recognition in the first class min ulr one wishes given a possibly infeasible system of linear relations to find a solution that violates as few relations as possible while satisfying all the others in the second class min rvls the linear system is supposed to be feasible and one looks for a solution with as few nonzero variables as possible for both min ulr and min rvls the four basic types of relational operators and are considered while min rvls with equations was mentioned to be np hard in garey and johnson 1979 we established in amaldi 1992 amaldi and kann 1995 that min ulr with equalities and inequalities are np hard even when restricted to homogeneous systems with bipolar coefficients the latter problems have been shown hard to approximate in arora et al 1993 in this paper we determine strong bounds on the approximability of various variants of min rvls and min ulr including constrained ones where the variables are restricted to take binary values or where some relations are mandatory while others are optional the various np hard versions turn out to have different approximability properties depending on the type of relations and the additional constraints but none of them can be approximated within any constant factor unless p np particular attention is devoted to two interesting special cases that occur in discriminant analysis and machine learning in particular we disprove a conjecture of van horn and martinez 1992 regarding the existence of a polynomial time algorithm to design linear classifiers or perceptrons that involve a close to minimum number of features

bifurcations of self excitation regimes in a van der pol oscillator with a nonlinear energy sink abstract the paper investigates regimes of self excitation in a van der pol oscillator with an attached nonlinear energy sink nes initial equations are reduced by averaging to a 3d system the small relative mass of the nes justifies analysis of this averaged system as singularly perturbed with two slow and one super slow variable such an approach in turn provides a complete analytic description of possible response regimes in addition to almost unperturbed limit cycle oscillations lcos the system can exhibit complete elimination of self excitation small amplitude lcos as well as excitation of a quasiperiodic strongly modulated response smr in the space of parameters the latter can be approached by three distinct bifurcation mechanisms canard explosion shil nikov bifurcation and heteroclinic bifurcation some of the above oscillatory regimes can co exist for the same values of the system parameters in this case it is possible to establish the basins of attraction for the co existing regimes direct numeric simulations demonstrate good coincidence with the analytic predictions

characterisation of main classes of fuzzy relations using fuzzy modal operators fuzzy modal operators express interactions between binary fuzzy relations and fuzzy sets each of these operators is determined by a binary fuzzy relation on a given universe and transforms one fuzzy set in this universe to another one in this paper we define several fuzzy modal operators and provide characterisations of main classes of binary fuzzy relations by means of these operators interpretation of these characterisations is presented in the context of fuzzy modal logics we show that these characterisations constitute the basis for determining characteristic axioms of particular classes of fuzzy modal logics

dynamics and energy exchanges between a linear oscillator and a nonlinear absorber with local and global potentials abstract the dynamical behavior of a two degree of freedom system made up of a linear oscillator and a coupled nonlinear energy sink with nonlinear global and local potentials is studied the nonlinear global potential of the energy sink performs direct interactions with the linear oscillator while its local potential depends only on its own behavior during vibratory energy exchanges between two oscillators a time multiple scale method around 1 1 1 resonance is used to detect slow invariant manifold of the system its equilibrium and singular points detected equilibrium points permit us to predict periodic regime s while singular points can lead the system to strongly modulated responses characterized by persistent bifurcations several possible scenarios occurring during these strongly modulated regimes are highlighted all analytical predictions are compared with those which are obtained by direct numerical integration of system equations

some properties of l fuzzy approximation spaces based on bounded integral residuated lattices in this paper we present a general framework for the study of l fuzzy rough sets based on bounded integral residuated lattices we introduce and discuss the notions of l fuzzy topological spaces left right lower and left right upper l fuzzy approximation operators based on a complete bounded integral residuated lattice investigate union intersection and composition operations of l fuzzy approximation spaces and study some properties of the l fuzzy topologies generated by these l fuzzy rough approximation operators and the l relations induced by an l fuzzy topology

algebras for galois style connections and their discrete duality bounded distributive lattices with pairs of operators forming galois dual galois residuation and dual residuation connections respectively are considered a discrete duality for these classes of algebras and the corresponding classes of relational systems is established these results are extended to the fuzzy logic mtl

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

one step sparse estimates in nonconcave penalized likelihood models fan and li propose a family of variable selection methods via penalized likelihood using concave penalty functions the nonconcave penalized likelihood estimators enjoy the oracle properties but maximizing the penalized likelihood function is computationally challenging because the objective function is nondifferentiable and nonconcave in this article we propose a new unified algorithm based on the local linear approximation lla for maximizing the penalized likelihood for a broad class of concave penalty functions convergence and other theoretical properties of the lla algorithm are established a distinguished feature of the lla algorithm is that at each lla step the lla estimator can naturally adopt a sparse representation thus we suggest using the one step lla estimator from the lla algorithm as the final estimates statistically we show that if the regularization parameter is appropriately chosen the one step lla estimates enjoy the oracle properties with good initial estimators computationally the one step lla estimation methods dramatically reduce the computational cost in maximizing the nonconcave penalized likelihood we conduct some monte carlo simulation to assess the finite sample performance of the one step sparse estimation methods the results are very encouraging

vibration reduction on beams subjected to moving loads using linear and nonlinear dynamic absorbers abstract the present work is focused on the analysis of the effectiveness of dynamic vibration absorbers applied to beams excited by moving loads the goal is to test the performance of nonlinear dampers in comparison with the classical linear damper simply supported beams are analysed using the euler bernoulli theory the partial differential equation governing the beam dynamics are reduced to an ordinary differential equation set by means of the galerkin bubnov method and a multimode expansion of the displacement field allows accurate analysis of the problem the performance of the dynamic dampers in vibration reduction is estimated through two indicators the maximum amplitude of vibration and the portion of energy dissipated by the dynamic damper the same indicators are used as objective functions for developing an optimisation approach two conservation laws are found for the optimal parameters and beam geometry for nonlinear cubic dynamic dampers

special properties closures and interiors of crisp and fuzzy relations this paper offers optimal definitions of the special properties which relations on a set may possess and which combine to define the classically important types of preorders both kinds of orders tolerances and equivalences the useful concept of local properties is introduced the definitions are shown to unify in a highly convenient way fuzzy and crisp considerations the concepts of closure and interior of a relation with respect to a property are defined necessary and sufficient conditions for the full existence of specific closures and interiors are given and formulae for all of them are presented and validated a widely important area of application of the closure and interior operators is discussed finally fast algorithms are given for the computer calculation of the more troublesome closures and a standardized procedure for the analysis of local orders is presented

complicated dynamics of a linear oscillator with a light essentially nonlinear attachment abstract we study the dynamics of a two degree of freedom dof nonlinear system consisting of a grounded linear oscillator coupled to a light mass by means of an essentially nonlinear nonlinearizable stiffness we consider first the undamped system and perform a numerical study based on non smooth transformations to determine its periodic solutions in a frequency energy plot it is found that there is a sequence of periodic solutions bifurcating or emanating from the main backbone curve of the plot we then study analytically the periodic orbits of the undamped system using a complexification averaging technique in order to determine the frequency contents of the various branches of solutions and to understand the types of oscillation performed by the system at the different regimes of the motion the transient responses of the weakly damped system are then examined and numerical wavelet transforms are used to study the time evolutions of their harmonic components we show that the structure of periodic orbits of the undamped system greatly influences the damped dynamics as it causes complicated transitions between modes in the damped transient motion in addition there is the possibility of strong passive energy transfer energy pumping from the linear oscillator to the nonlinear attachment if certain periodic orbits of the undamped dynamics are excited by the initial conditions

communication between information systems using fuzzy rough sets communication between information systems is a basic problem in granular computing and the concept of homomorphism is a useful mathematical tool to study this problem in this paper some properties of communication between information systems based on fuzzy rough sets are investigated the concepts of fuzzy relation mappings between universes are first proposed in order to construct a fuzzy relation of one universe according to the given fuzzy relation on the other universe the main properties of the mappings are studied the notions of homomorphism of information systems based on fuzzy rough sets are then proposed and it is proved that properties of relation operations in the original information system and structural features of the system such as approximations of arbitrary fuzzy sets and attribute reductions are guaranteed in its image system under the condition of homomorphism

the relationship between l fuzzy rough set and l topology various fuzzy generalizations of rough approximations have been proposed in the literature this paper is devoted to the discussion of the relationship between l fuzzy rough sets and l topologies on an arbitrary universe finally one to one correspondence between the set of all reflexive transitive l relations and the set of all alexandrov l topologies is obtained

fuzzy neighborhood operators based on fuzzy coverings in many data mining processes neighborhood operators play an important role as they are generalizations of equivalence classes which were used in the original rough set model of pawlak in this article we introduce the notion of fuzzy neighborhood system of an object based on a given fuzzy covering as well as the notion of the fuzzy minimal and maximal descriptions of an object moreover we extend the definition of four covering based neighborhood operators as well as six derived coverings discussed by yao and yao to the fuzzy setting we combine these fuzzy neighborhood operators and fuzzy coverings and prove that only sixteen different fuzzy neighborhood operators are obtained moreover we study the partial order relations between those sixteen operators

vibration reduction of beams under successive traveling loads by means of linear and nonlinear dynamic absorbers abstract the goal of the present work is to assess the performances of dynamic vibration absorbers dva in suppressing the vibrations of a simply supported beam subjected to an infinite sequence of regularly spaced concentrated moving loads in particular several types of dva are considered linear cubic higher odd order monomials and piecewise linear stiffness linear cubic and linear quadratic viscous damping the purpose is to clarify if nonlinear dvas show improvements with respect to the classical linear devices the dynamic scenario is deeply investigated in a wide range of operating conditions spanning the parameter space of the dva damping stiffness nonlinear stiffness can lead to complex dynamics such as quasi periodic chaotic and sub harmonic responses moreover acting on the stiffness nonlinearity no improvement is found with respect to the linear dva a nonlinear non symmetric dissipation in the dva leads to a great reduction of the beam response the reduction is larger with respect to the linear dva

targeted energy transfer in systems with non polynomial nonlinearity abstract targeted energy transfer tet in a 2dof system consisting of primary linear oscillator and nonlinear energy sink nes with non polynomial potential is investigated use of non polynomial and even non analytic potential functions is motivated by needs of practical design it is demonstrated that the complexification averaging technique of analysis developed before can be successfully extended for these cases with proper modifications the procedure is illustrated by examples involving softening non polynomial and piecewise linear non analytic nes