asymptotics and random sampling for bci and bck lambda terms this paper presents a bijection between combinatorial maps and a class of enriched trees corresponding to a class of expression trees in some logical systems constrained lambda terms starting from two alternative definitions of combinatorial maps the classical definition by gluing half edges and a definition by non ambiguous depth first traversal we derive non trivial asymptotic expansions and efficient random generation of logic formulae syntactic trees in the bci or bck systems

combinatorial approach to generalized bell and stirling numbers and boson normal ordering problem we consider the numbers arising in the problem of normal ordering of expressions in boson creation a and annihilation a operators a a 1 we treat a general form of a boson string a rnasn a r2as2 a r1as1 which is shown to be associated with generalizations of stirling and bell numbers the recurrence relations and closed form expressions dobinski type formulas are obtained for these quantities by both algebraic and combinatorial methods by extensive use of methods of combinatorial analysis we prove the equivalence of the aforementioned problem to the enumeration of special families of graphs this link provides a combinatorial interpretation of the numbers arising in this normal ordering problem

boosting the margin a new explanation for the effectiveness of voting methods one of the surprising recurring phenomena observed in experiments with boosting is that the test error of the generated classifier usually does not increase as its size becomes very large and often is observed to decrease even after the training error reaches zero in this paper we show that this phenomenon is related to the distribution of margins of the training examples with respect to the generated voting classification rule where the margin of an example is simply the difference between the number of correct votes and the maximum number of votes received by any incorrect label we show that techniques used in the analysis of vapnik s support vector classifiers and of neural networks with small weights can be applied to voting methods to relate the margin distribution to the test error we also show theoretically and experimentally that boosting is especially effective at increasing the margins of the training examples finally we compare our explanation to those based on the bias variance decomposition

polynomial functors and combinatorial dyson schwinger equations we present a general abstract framework for combinatorial dyson schwinger equations in which combinatorial identities are lifted to explicit bijections of sets and more generally equivalences of groupoids key features of combinatorial dyson schwinger equations are revealed to follow from general categorical constructions and universal properties rather than beginning with an equation inside a given hopf algebra and referring to given hochschild 1 cocycles our starting point is an abstract fixpoint equation in groupoids shown canonically to generate all the algebraic structures precisely for any finitary polynomial endofunctor p defined over groupoids the system of combinatorial dyson schwinger equations x 1 p x has a universal solution namely the groupoid of p trees the isoclasses of p trees generate naturally a connes kreimer like bialgebra in which the abstract dyson schwinger equation can be internalised in terms of canonical b operators the solution to this equation is a series the

a combinatorial approach to nonlinear functional expansions an introduction with an example in this paper we present a new approach to causal functionals we introduce combinatorial interpretations of the solutions of nonlinear differential equations with forcing terms this theory parallels the algebraic approach with formal power series in noncommutative variables developed by fliess lamnabhi and lamnabhi lagarrigue this theory makes use of certain combinatorial objects called weighted increasing trees weighted paths and histories we can deduce very efficient algorithms for the computation of the corresponding volterra kernels we present an introduction to our combinatorial theory an example with a nonlinear circuit gives the flavor of our approach the complete proofs and general theory will be discussed in a further paper

analytic functors between presheaf categories over groupoids the paper studies analytic functors between presheaf categories generalising results of a joyal 11 and r hasegawa 9 for analytic endofunctors on the category of sets we give two characterisations of analytic functors between presheaf categories over groupoids i as functors preserving filtered colimits quasi pullbacks and cofiltered limits and ii as functors preserving filtered colimits and wide quasi pullbacks the development establishes that small groupoids analytic functors between their presheaf categories and quasi cartesian natural transformations between them form a 2 category

robust stability analysis for discrete time stochastic neural networks systems with time varying delays in this paper the mean square exponential stability is investigated for a class of discrete time stochastic neural networks with time varying delays and norm bounded uncertainties based on lyapunov stability theory and stochastic approaches delay dependent criteria are derived to ensure the robust exponential stability in the mean square for the addressed system meantime by using the numerically efficient matlab lmi toolbox a example is presented to show the usefulness of the derived lmi based stability condition

sequence operators from groups abstract this paper is inspired by the paper some canonical sequences of integers by bernstein and sloane the main observation is that seven of the operators defined in that paper have natural interpretations in terms of counting orbits of groups providing a pattern which is completed by five further operators some of their eigen sequences also have group theoretic meaning

an optimal algorithm to generate rooted trivalent diagrams and rooted triangular maps a trivalent diagram is a connected two colored bipartite graph parallel edges allowed but not loops such that every black vertex is of degree 1 or 3 and every white vertex is of degree 1 or 2 with a cyclic order imposed on every set of edges incident to the same vertex a rooted trivalent diagram is a trivalent diagram with a distinguished edge its root we shall describe and analyze an algorithm giving an exhaustive list of rooted trivalent diagrams of a given size number of edges the list being non redundant in that no two diagrams of the list are isomorphic the algorithm will be shown to have optimal performance in that the time necessary to generate a diagram will be seen to be bounded in the amortized sense the bound being independent of the size of the diagrams we call this the cat property one objective of the paper is to provide a reusable theoretical framework for algorithms generating exhaustive lists of complex combinatorial structures with attention paid to the case of unlabeled structures and to those generators having the cat property

two bijective proofs for the arborescent form of the good lagrange formula and some applications to colored rooted trees and cacti goulden and kulkarni j combin theory ser a 80 2 1997 295 give a bijective proof of an arborescent form of the good lagrange multivariable inversion formula this formula was first stated explicitly by bender and richmond electron j combin 5 1 1998 4pp but is implicit in goulden and kulkarni 1997 in this paper we propose two new simple bijective proofs of this formula and we illustrate the interest of these proofs by applying them to the enumeration and random generation of colored rooted trees and rooted m ary cacti

combinatorial models for coalgebraic structures abstract we introduce a convenient category of combinatorial objects known as cell sets on which we study the properties of the appropriate free abelian group functor we obtain a versatile generalization of the notion of incidence coalgebra giving rise to an abundance of coalgebras hopf algebras and comodules all of whose structure constants are positive integers with respect to certain preferred bases our category unifies and extends existing constructions in algebraic combinatorics providing proper functorial descriptions it is inspired in part by the notion of cw complex and is also geared to future applications in algebraic topology and the theory of formal group laws

symmetric hilbert spaces arising from species of structures symmetric hilbert spaces such as the bosonic and the fermionic fock spaces over some lsquo one particle space mathcal k are formed by certain symmetrization procedures performed on the full fock space we investigate alternative ways of symmetrization by building on joyal s notion of a combinatorial species any such species f gives rise to an endofunctor gamma f of the category of hilbert spaces with contractions mapping a hilbert space mathcal k to a symmetric hilbert space gamma f mathcal k with the same symmetry as the species f a general framework for annihilation and creation operators on these spaces is developed and compared to the generalised brownian motions of r speicher and m bozejko as a corollary we find that the commutation relation a ia j a j a i f n delta ij with na i a i n a i admits a realization on a symmetric hilbert space whenever f has a power series with infinite radius of convergence and positive coefficients

automatic average case analysis of algorithms abstract many probabilistic properties of elementary discrete combinatorial structures of interest for the average case analysis of algorithms prove to be decidable this paper presents a general framework in which such decision procedures can be developed it is based on a combination of generating function techniques for counting and complex analysis techniques for asymptotic estimations we expose here the theory of exact analysis in terms of generating functions for four different domains the iterative recursive and unlabelled labelled data type domains we then present some major components of the associated asymptotic theory and exhibit a class of naturally arising functions that can be automatically analyzed a fair fragment of this theory is also incorporated into a system called lambda upsilon omega in this way using computer algebra one can produce automatically non trivial average case analyses of algorithms operating over a variety of decomposable combinatorial structures at a fundamental level this paper is part of a global attempt at understanding why so many elementary combinatorial problems tend to have elementary asymptotic solutions in several cases it proves possible to relate entire classes of elementary combinatorial problems whose structure is well defined with classes of elementary special functions and classes of asymptotic forms relative to counting probabilities or average case complexity

self similarity in the combinatorics of orthogonal polynomials resume dans cet article nous construisons des structures combinatoires invariantes sous l action d une permutation donnee dans certains cas le poids de ces dernieres correspon a des polynomes orthogonaux dont on a ajuste convenablement les parametres on examine ici de ces structures associees aux polynomes de hermite laguerre krawtchouk meixner meixner pollaczek et de jacobi

finite time stochastic input to state stability of impulsive switched stochastic nonlinear systems this paper mainly tends to investigate the finite time stochastic input to state stability ftsiss problem for a class of impulsive switched stochastic nonlinear systems an efficient lemma is firstly established to construct a generalized class kl function further a set of lyapunov based sufficient conditions are derived to check the finite time stability in probability ftsip and finite time input to state stability ftiss properties of switched stochastic nonlinear systems with or without impulses in particular some useful results in favor of simulations are obtained in terms of average dwell time technique finally a numerical example is provided to illustrate the effectiveness of proposed results

on the fourier spectrum of monotone functions in this paper monotone boolean functions are studied using harmonic analysis on the cube the main result is that any monotone boolean function has most of its power spectrum on its fourier coefficients of degree at most o n under any product distribution this is similar to a result of linial et al 1993 which showed that ac 0 functions have almost all of their power spectrum on the coefficients of degree at most log n o 1 under the uniform distribution as a consequence of the main result the following two corollaries are obtained for any e 0 monotone boolean functions are pac learnable with error e under product distributions in time 2 o 1 e n any monotone boolean function can be approximated within error e under product distributions by a non monotone boolean circuit of size 2 o 1 e n and depth o 1 e n the learning algorithm runs in time subexponential as long as the required error is 1 n log n it is shown that this is tight in the sense that for any subexponential time algorithm there is a monotone boolean function for which this algorithm cannot approximate with error better than o 1 n the main result is also applied to other problems in learning and complexity theory in learning theory several polynomial time algorithms for learning some classes of monotone boolean functions such as boolean functions with o log 2 n log log n relevant variables are presented in complexity theory some questions regarding monotone np complete problems are addressed

weak convergence of random p mappings and the exploration process of inhomogeneous continuum random trees we study the asymptotics of the p mapping model of random mappings on n as n gets large under a large class of asymptotic regimes for the underlying distribution p we encode these random mappings in random walks which are shown to converge to a functional of the exploration process of inhomogeneous random trees this exploration process being derived aldous miermont pitman 2004 from a bridge with exchangeable increments our setting generalizes previous results by allowing a finite number of attracting points to emerge

weighted derangements and the linearization coefficients of orthogonal sheffer polynomials the present paper is devoted to a systematic study of the combinatorial interpretations of the moments and the linearization coefficients of the orthogonal sheffer polynomials i e hermite charlier laguerre meixner and meixner pollaczek polynomials in particular we show that viennot s combinatorial interpretations of the moments can be derived directly from their classical analytical expressions and that the linearization coefficients of meixner pollaczek polynomials have an interpretation in the model of derangements analogous to those of laguerre and meixner polynomials

lagrange inversion in infinitely many variables this paper is concerned with the problem of finding an effective definition for the compositional inverse of formal power series in several variables with special regard to series in infinitely many variables the problem of defining analogs for several variables of the lagrange inversion formula has been open for a long time see e g 3 53 so far all indications point to the fact that an explicit lagrange inversion formula in infinitely many variables does not exist in the complete generality that we find for series in finitely many variables 4 our main result in this paper is a lagrange inversion formula for sets of power series in infinitely many variables c r xr x2 c x x2 where the formal power series x x x does not depend on the variables xl x27 7 x this situation occurs in several instances most notably in the composition for series in infinitely many variables which has come to be called plethysm we recall that the plethysm of a series x x2 with a series u x x2 is defined as follows set c x x2 a x2 the plethysm of p with a is then the series a a there are other applications as well as is well known a lagrange inversion formula requires the understanding of the embedding of the ring of formal power series into a ring of formal laurent series which turns out to be a field for series with coefficients in a field this is an old and thorny problem which also has not received to this day a satisfactory answer there are at least two possible definitions of formal laurent series in many variables 2 81 they can be required to be either series such that the set of their monomials which are always of finite length has an infimum or series for which the set of weights of their monomials has a minimum in this second case it is necessary to add a finiteness condition on monomials

on gammelgaard s formula for a star product with separation of variables we show that gammelgaard s formula expressing a star product with separation of variables on a pseudo kahler manifold in terms of directed graphs without cycles is equivalent to an inversion formula for an operator on a formal fock space we prove this inversion formula directly and thus offer an alternative approach to gammelgaard s formula which gives more insight into the question why the directed graphs in his formula have no cycles

a new approximate maximal margin classification algorithm a new incremental learning algorithm is described which approximates the maximal margin hyperplane w r t norm p 2 for a set of linearly separable data our algorithm called alma p approximate large margin algorithm w r t norm p takes o p 1 2 2 corrections to separate the data with p norm margin larger than 1 where g is the normalized p norm margin of the data alma p avoids quadratic or higher order programming methods it is very easy to implement and is as fast as on line algorithms such as rosenblatt s perceptron algorithm we performed extensive experiments on both real world and artificial datasets we compared alma 2 i e alma p with p 2 to standard support vector machines svm and to two incremental algorithms the perceptron algorithm and li and long s romma the accuracy levels achieved by alma 2 are superior to those achieved by the perceptron algorithm and romma but slightly inferior to svm s on the other hand alma 2 is quite faster and easier to implement than standard svm training algorithms when learning sparse target vectors alma p with p 2 largely outperforms perceptron like algorithms such as alma 2

exponential stability analysis for delayed neural networks with switching parameters average dwell time approach this paper is concerned with the problem of exponential stability analysis of continuous time switched delayed neural networks by using the average dwell time approach together with the piecewise lyapunov function technique and by combining a novel lyapunov krasovskii functional which benefits from the delay partitioning method with the free weighting matrix technique sufficient conditions are proposed to guarantee the exponential stability for the switched neural networks with constant and time varying delays respectively moreover the decay estimates are explicitly given the results reported in this paper not only depend upon the delay but also depend upon the partitioning which aims at reducing the conservatism numerical examples are presented to demonstrate the usefulness of the derived theoretical results

exponentiated gradient versus gradient descent for linear predictors we consider two algorithm for on line prediction based on a linear model the algorithms are the well known gradient descent gd algorithm and a new algorithm which we call eg they both maintain a weight vector using simple updates for the gd algorithm the update is based on subtracting the gradient of the squared error made on a prediction the eg algorithm uses the components of the gradient in the exponents of factors that are used in updating the weight vector multiplicatively we present worst case loss bounds for eg and compare them to previously known bounds for the gd algorithm the bounds suggest that the losses of the algorithms are in general incomparable but eg has a much smaller loss if only a few components of the input are relevant for the predictions we have performed experiments which show that our worst case upper bounds are quite tight already on simple artificial data

cyclic linear differential automata a simple class of hybrid dynamical systems abstract we introduce a special class of hybrid dynamical systems cyclic linear differential automata clda we show that any clda can be reduced to a linear discrete time system with periodic coefficients any clda has no equilibrium points therefore the simplest attractor in such system is a periodic trajectory we call a clda globally stable if it has a periodic trajectory which attracts all other trajectories of the system a necessary and sufficient condition for global stability of clda is given we apply our result to prove global stability of a flexible manufacturing system modelled as a switched server system

independent process approximations for random combinatorial structures abstract many random combinatorial objects have a component structure whose joint distribution is equal to that of a process of mutually independent random variables conditioned on the value of a weighted sum of the variables it is interesting to compare the combinatorial structure directly to the independent discrete process without renormalizing the quality of approximation can often be conveniently quantified in terms of total variation distance for functionals which observe part but not all of the combinatorial and independent processes among the examples are combinatorial assemblies e g permutations random mapping functions and partitions of a set multisets e g polynomials over a finite field mapping patterns and partitions of an integer and selections e g partitions of an integer into distinct parts and square free polynomials over finite fields we consider issues common to all the above examples including equalities and upper bounds for total variation distances existence of limiting processes heuristics for good approximations the relation to standard generating functions moment formulas and recursions for computing densities refinement to the process which counts the number of parts of each possible type the effect of further conditioning on events of moderate probability large deviation theory and nonuniform measures on combinatorial objects and the possibility of getting useful results by overpowering the conditioning

on combinatorial differential equations abstract we analyse the solution set of first order initial value differential problems of the form dy dx x y y 0 0 in the context of combinatorial species in the sense of a joyal adv in math 42 1981 1 82 it turns out that the situation is much richer than in the case of formal power series many non isomorphic combinatorial solutions are possible for a given problem although they all have the same underlying generating series we give many examples of this phenomenon and also elaborate a combinatorial newton raphson iterative scheme for the construction of the solutions the multidimensional case is treated explicitly

two applications of analytic functors we apply the theory of analytic functors to two topics related to theoretical computer science one is a mathematical foundation of certain syntactic well quasi orders and well orders appearing in graph theory the theory of term rewriting systems and proof theory the other is a new verification of the lagrange good inversion formula using several ideas appearing in semantics of lambda calculi especially the relation between categorical traces and fixpoint operators copyright 2002 elsevier science b v

a set theoretic interpretation of the umbral calculus abstract through the notion of enriched species we develop a bijective calculus set theoretic counterpart of the theory of binomial enumeration

kernels for generalized multiple instance learning the multiple instance learning mil model has been successful in numerous application areas recently a generalization of this model and an algorithm for it were introduced showing significant advantages over the conventional mil model on certain application areas unfortunately that algorithm is not scalable to high dimensions we adapt that algorithm to one using a support vector machine with our new kernel k wedge this reduces the time complexity from exponential in the dimension to polynomial computing our new kernel is equivalent to counting the number of boxes in a discrete bounded space that contain at least one point from each of two multisets we show that this problem is p complete but then give a fully polynomial randomized approximation scheme fpras for it we then extend k by enriching its representation into a new kernel kmin and also consider a normalized version of k wedge that we call k which may or may not not be a kernel but whose approximation yielded positive semidefinite gram matrices in practice we then empirically evaluate all three measures on data from content based image retrieval biological sequence analysis and the musk data sets we found that our kernels performed well on all data sets relative to algorithms in the conventional mil model

a generalized cyclotomic identity abstract in this paper we prove a generalization of the cyclotomic identity for each group g satisfying suitable finiteness conditions we prove this generalized cyclotomic identity by counting coloured g set structures the cyclotomic identity corresponds to taking g the free abelian group of rank one for the free abelian group of rank two for example we get an analogous identity involving the generating function of the partition function interalia we show that studying g set structures gives a unified and conceptually simple approach to other classical combinatorial identities not to the author s knowledge previously linked in the literature

categorical aspects of generating functions i exponential formulas and krull schmidt categories abstract in this paper we study formal power series with exponents in a category for example the generating function of a category e with finite hom sets is defined by e t tx aut x where the summation is taken over all isomorphism classes of objects of e we can use such power series to enumerate the number of e structures along a faithful functor theorem 4 6 our theory is closely related to the theory of species joyal 1981 a species can be identified with a faithful functor from a groupoid to the category of finite sets theorem 3 6 we use mainly the concept of faithful functors with finite fibers instead of species so that we can separate the roles categories and functors play for example the exponential formula e t exp con e t means the unique coproduct decomposition property theorem 5 8 in the final section we give some applications of our theory to rather classical enumerations

brill gordan loci transvectants and an analogue of the foulkes conjecture abstract the hypersurfaces of degree d in the projective space p n correspond to points of p n where n n d d 1 now assume d 2 e is even and let x n d p n denote the subvariety of two e fold hyperplanes we exhibit an upper bound on the castelnuovo regularity of the ideal of x n d and show that this variety is r normal for r 2 the latter result is representation theoretic and says that a certain gl n 1 equivariant morphism s r s 2 e c n 1 s 2 s r e c n 1 is surjective for r 2 a statement which is reminiscent of the foulkes howe conjecture for its proof we reduce the statement to the case n 1 and then show that certain transvectants of binary forms are nonzero the latter part uses explicit calculations with feynman diagrams and hypergeometric series for ternary quartics and binary d ics we give explicit generators for the defining ideal of x n d expressed in the language of classical invariant theory

stability criteria for bam neural networks with leakage delays and probabilistic time varying delays this paper is concerned with the stability criteria for bidirectional associative memory bam neural networks with leakage time delay and probabilistic time varying delays by establishing a stochastic variable with bernoulli distribution the information of probabilistic time varying delay is transformed into the deterministic time varying delay with stochastic parameters based on the lyapunov krasovskii functional and stochastic analysis approach delay probability distribution dependent sufficient conditions are derived to achieve the globally asymptotically mean square stable of the considered bam neural networks the criteria are formulated in terms of a set of linear matrix inequalities lmis which can be checked efficiently by use of some standard numerical packages finally a numerical example and its simulations are given to demonstrate the usefulness and effectiveness of the proposed results

expansions of chromatic polynomials and log concavity in this paper we present several results and open problems about log concavity properties of sequences associated with graph colorings five polynomials intimately related to the chromatic polynomial of a graph are introduced and their zeros combinatorial and log concavity properties are studied four of these polynomials have never been considered before in the literature and some yield new expansions for the chromatic polynomial

behavioural differential equations a coinductive calculus of streams automata and power series we present a theory of streams infinite sequences automata and languages and formal power series in terms of the notions of homomorphism and bisimulation which are the cornerstones of the theory of universal coalgebra this coalgebraic perspective leads to a unified theory in which the observation that each of the aforementioned sets carries a so called final automaton structure plays a central role finality forms the basis for both definitions and proofs by coinduction the coalgebraic counterpart of induction coinductive definitions take the shape of what we have called behavioural differential equations after brzozowski s notion of input derivative a calculus is developed for coinductive reasoning about all of the afore mentioned structures closely resembling calculus from classical analysis

scaling limits of random graphs from subcritical classes we study the uniform random graph c n with n vertices drawn from a subcritical class of connected graphs our main result is that the resealed graph c n root n converges to the brownian continuum random tree t e multiplied by a constant scaling factor that depends on the class under consideration in addition we provide sub gaussian tail bounds for the diameter d c n and height h c n center dot of the rooted random graph c n center dot we give analytic expressions for the scaling factor in several cases including for example the class of outerplanar graphs our methods also enable us to study first passage percolation on c n where we also show the convergence to t e under an appropriate rescaling

groupoids and faa di bruno formulae for green functions in bialgebras of trees we prove a faa di bruno formula for the green function in the bialgebra of p trees for any polynomial endofunctor p the formula appears as relative homotopy cardinality of an equivalence of groupoids

generalised brownian motion and second quantisation a new approach to the generalised brownian motion introduced by m bouzejko and r speicher is described based on symmetry rather than deformation the symmetrisation principle is provided by joyal s notions of tensorial and combinatorial species any such species v gives rise to an endofunctor fv of the category of hilbert spaces with contractions a generalised brownian motion is an algebra of creation and annihilation operators acting on fv h for arbitrary hilbert spaces h and having a prescription for the calculation of vacuum expectations in terms of a function t on pair partitions the positivity is encoded by a semigroup of broken pair partitions whose representation space with respect to t is v the existence of the second quantisation as functor t from hilbert spaces to noncommutative probability spaces is proved to be equivalent to the multiplicative property of the function t for a certain one parameter interpolation between the fermionic and the free brownian motion it is shown that the field algebras k are type ii1 factors when k is infinite dimensional

intersection theory on the moduli space of curves and the matrix airy function we show that two natural approaches to quantum gravity coincide this identity is nontrivial and relies on the equivalence of each approach to kdv equations we also investigate related mathematical problems

une theorie combinatoire des series formelles abstract this paper presents a combinatorial theory of formal power series the combinatorial interpretation of formal power series is based on the concept of species of structures a categorical approach is used to formulate it a new proof of cayley s formula for the number of labelled trees is given as well as a new combinatorial proof due to g labelle of lagrange s inversion formula polya s enumeration theory of isomorphism classes of structures is entirely renewed recursive methods for computing cycle index polynomials are described a combinatorial version of the implicit function theorem is stated and proved the paper ends with general considerations on the use of coalgebras in combinatorics

stability and l2 gain analysis for switched delay systems a delay dependent method in this paper we study stability and l 2 gain for a class of switched systems with time varying delays sufficient conditions for exponential stability and weighted l 2 gain are developed for a class of switching signals with average dwell time these conditions are delay dependent and are given in the form of linear matrix inequalities lmis as a special case of such switching signals we can obtain exponential stability and normal l 2 gain under arbitrary switching signals the state decay estimate is explicitly given two examples illustrate the effectiveness and applicability of the proposed method

stability analysis and design of fuzzy control systems abstract the stability analysis and the design technique of fuzzy control systems using fuzzy block diagrams are discussed first we show the concept of fuzzy blocks and consider the connection problems of fuzzy blocks diagrams we derive some theorems and corollaries with respect to two basic types of connections of fuzzy blocks in order to preserve some properties in a connection of fuzzy blocks continuous piecewise polynomial membership functions are defined secondly a sufficient condition which guarantees the stability of fuzzy systems is obtained in terms of lyapunov s direct method we give an important fact based on this condition thirdly we propose a new design technique of a fuzzy controller the fuzzy block diagrams and the stability analysis are applied to the design problems of a model based fuzzy controller

dwell time approach to input output stability properties for a class of discrete time dynamical systems abstract we consider a class of discrete time dynamical systems that consist of switching of many linear time invariant systems and it can be used to cover a sampled data version of switched linear systems which have been widely studied in the literature for this class of dynamical systems we establish the lmi linear matrix inequality formulation of analyzing their input output stability properties including both l 2 stability and passivity by constraining switching signals via the concept of dwell time the lmi formulation of l 2 stability is applied to evaluating the closed loop performance of an industrial refrigeration process that is regulated by several proportional integral pi controllers in a switching structure

learning decision trees using the fourier spectrum this work gives a polynomial time algorithm for learning decision trees with respect to the uniform distribution this algorithm uses membership queries the decision tree model that is considered is an extension of the traditional boolean decision tree model that allows linear operations in each node i e summation of a subset of the input variables over gf 2 this paper shows how to learn in polynomial time any function that can be approximated in norm l 2 by a polynomially sparse function i e a function with only polynomially many nonzero fourier coefficients the authors demonstrate that any function f whose l 1 norm i e the sum of absolute value of the fourier coefficients is polynomial can be approximated by a polynomially sparse function and prove that boolean decision trees with linear operations are a subset of this class of functions moreover it is shown that the functions with polynomial l 1 norm can be learned deterministically the algorithm can also exactly identi

combinatorial proofs of symmetry formulas for the generalized hypergeometric series abstract in this paper we introduce the notion of weakly isomorphic combinatorial models several symmetry formulas involving the generalized hypergeometric series for example the pfaff saalchutz identity which was first proved combinatorially by d foata are then proved by finding weak isomorphisms between models for the two sides of the identity this is applied to combinatorial models for hypergeometric orthogonal polynomials such as laguerre charlier meixner krawtchouk meixner pollaczek jacobi and hahn polynomials

context free grammars differential operators and formal power series abstract in this paper we introduce the concepts of a formal function over an alphabet and a formal derivative based on a set of substitution rules we call such a set of rules a context free grammar because these rules act like a context free grammar in the sense of a formal language given a context free grammar we can associate each formal function with an exponential formal power series in this way we obtain grammatical interpretations of addition multiplication and functional composition of formal power series a surprising fact about the grammatical calculus is that the composition of two formal power series enjoys a very simple grammatical representation we apply this method to obtain simple demonstrations of faa di bruno s formula and some identities concerning bell polynomials stirling numbers and symmetric functions in particular the lagrange inversion formula has a simple grammatical representation from this point of view one sees that cayley s formula on labeled trees is equivalent to the lagrange inversion formula

functors of white noise associated to characters of the infinite symmetric group the characters of the infinite symmetric group are extended to multiplicative positive definite functions on pair partitions by using an explicit representation due to versik and kerov the von neumann algebra generated by the fields with f in an infinite dimensional real hilbert space is infinite and the vacuum vector is not separating for a family depending on an integer n 1 an exclusion principle is found allowing at most identical particles on the same state the algebras are type factors functors of white noise are constructed and proved to be non equivalent for different values of n

model approximation for fuzzy switched systems with stochastic perturbation in this paper the model approximation problem is investigated for a takagi sugeno fuzzy switched system with stochastic disturbance for a high order considered system our attention is focused on the construction of a reduced order model which not only approximates the original system well with a hankel norm performance but translates it into a lower dimensional fuzzy switched system as well by using the average dwell time approach and the piecewise lyapunov function technique a sufficient condition is first proposed to guarantee the mean square exponential stability with a hankel norm error performance for the error system the model approximation is then converted into a convex optimization problem by using a linearization procedure finally simulations are provided to illustrate the effectiveness of the proposed theory

new delay dependent robust stability criterion for uncertain neural networks with time varying delays in this paper the problem of delay dependent stability of uncertain neural networks with time varying delays is considered the parameter uncertainties are time varying and assumed to be bounded in magnitude by constructing a new augmented lyapunov functional a delay dependent stability criterion for the network is derived in terms of lmi linear matrix inequality which can be easily solved by various convex optimization algorithms three numerical examples are included to show the effectiveness of proposed criterion

new robust exponential stability results for discrete time switched fuzzy neural networks with time delays this paper provides a novel result on robust exponential stability for a class of uncertain discrete time switched fuzzy neural networks dsfnns with time varying delays and parameter uncertainties by implementing an average dwell time approach with a new lyapunov krasovskii functional we obtain some delay dependent sufficient conditions guaranteeing the robust exponential stability of the considered switched fuzzy neural networks in other words a class of switching signals specified by the average dwell time is identified to guarantee the exponential stability of the considered dsfnns the obtained conditions are formulated in terms of linear matrix inequalities lmis which can be easily verified via the lmi toolbox finally numerical examples with simulation results are provided to illustrate the applicability and usefulness of the obtained results

maximum margin algorithms with boolean kernels recent work has introduced boolean kernels with which one can learn linear threshold functions over a feature space containing all conjunctions of length up to k for any 1 k n over the original n boolean features in the input space this motivates the question of whether maximum margin algorithms such as support vector machines can learn disjunctive normal form expressions in the probably approximately correct pac learning model by using this kernel we study this question as well as a variant in which structural risk minimization srm is performed where the class hierarchy is taken over the length of conjunctions we show that maximum margin algorithms using the boolean kernels do not pac learn t n term dnf for any t n 1 even when used with such a srm scheme we also consider pac learning under the uniform distribution and show that if the kernel uses conjunctions of length n then the maximum margin hypothesis will fail on the uniform distribution as well our results concretely illustrate that margin based algorithms may overfit when learning simple target functions with natural kernels

object grammars and bijections a new systematic approach for the specification of bijections between sets of combinatorial objects is presented it is based on the notion of object grammars object grammars give recursive descriptions of objects and generalize context free grammars the study of a particular substitution in these object grammars confirms once more the key role of dyck words in the domain of enumerative and bijective combinatorics

decomposition spaces incidence algebras and mobius inversion i basic theory abstract this is the first in a series of papers devoted to the theory of decomposition spaces a general framework for incidence algebras and mobius inversion where algebraic identities are realised by taking homotopy cardinality of equivalences of groupoids a decomposition space is a simplicial groupoid satisfying an exactness condition weaker than the segal condition expressed in terms of active and inert maps in just as the segal condition expresses composition the new exactness condition expresses decomposition and there is an abundance of examples in combinatorics after establishing some basic properties of decomposition spaces the main result of this first paper shows that to any decomposition space there is an associated incidence coalgebra spanned by the space of 1 simplices and with coefficients in groupoids we take a functorial viewpoint throughout emphasising conservative ulf functors these induce coalgebra homomorphisms reduction procedures in the classical theory of incidence coalgebras are examples of this notion and many are examples of decalage of decomposition spaces an interesting class of examples of decomposition spaces beyond segal spaces is provided by hall algebras the waldhausen s construction of an abelian or stable infinity category is shown to be a decomposition space in the second paper in this series we impose further conditions on decomposition spaces to obtain a general mobius inversion principle and to ensure that the various constructions and results admit a homotopy cardinality in the third paper we show that the lawvere menni hopf algebra of mobius intervals is the homotopy cardinality of a certain universal decomposition space two further sequel papers deal with numerous examples from combinatorics note the notion of decomposition space was arrived at independently by dyckerhoff and kapranov 17 who call them unital 2 segal spaces our theory is quite orthogonal to theirs the definitions are different in spirit and appearance and the theories differ in terms of motivation examples and directions

constant depth circuits fourier transform and learnability in this paper boolean functions in 4c0 are studied using harmonic analysis on the cube the main result is that an aco boolean function has almost all of its power spectrum on the low order coefficients an important ingredient of the proof is hastad s switching lemma 8 this result implies several new properties of functions in 4c functions in ac have low average sensitivity they may be approximated well by a real polynomial of low degree and they cannot be pseudorandom function generators perhaps the most interesting application is an o n poiyiog n time algorithm for learning func tions in aco the algorithm observes the behavior of an ac function on o npo y og n randomly chosen inputs and derives a good approximation for the fourier transform of the function this approximation allows the algorithm to predict with high probability the value of the function on other randomly chosen inputs

stability analysis of impulsive stochastic cohen grossberg neural networks with mixed time delays abstract in this paper the problem of stability analysis for a class of impulsive stochastic cohen grossberg neural networks with mixed delays is considered the mixed time delays comprise both the time varying and infinite distributed delays by employing a combination of the m matrix theory and stochastic analysis technique a sufficient condition is obtained to ensure the existence uniqueness and exponential p stability of the equilibrium point for the addressed impulsive stochastic cohen grossberg neural network with mixed delays the proposed method which does not make use of the lyapunov functional is shown to be simple yet effective for analyzing the stability of impulsive or stochastic neural networks with variable and or distributed delays we then extend our main results to the case where the parameters contain interval uncertainties moreover the exponential convergence rate index is estimated which depends on the system parameters an example is given to show the effectiveness of the obtained results

the q t gaussian process the q t fock space fq t h introduced in this paper is a deformation of the q fock space of bozejko and speicher the corresponding creation and annihilation operators now satisfy the commutation relation aq t f aq t g qaq t g aq t f f g htn turn mathjax off a defining relation of the chakrabarti jagannathan deformed quantum oscillator algebra the moments of the deformed gaussian element sq t h aq t h aq t h are encoded by the joint statistics of crossings and nestings in pair partitions the q 0 t specialization yields a natural single parameter deformation of the full boltzmann fock space of free probability with the corresponding semicircular measure variously encoded via the rogers ramanujan continued fraction the t airy function the t catalan numbers of carlitz riordan and the first order statistics of the reduced wigner process

decomposable functors and the exponential principle abstract we develop a combinatorial theory for the operation of forming the logarithm of generating functions associated with covariant functors on the category of finite sets and bijective maps this leads to a rather general and flexible exponential principle for labeled combinatorial structures we also introduce the concept of a topos like category which turns out to be a convenient framework for the discussion of a wide range of applications a few of which are studied in detail

a physicist s proof of the lagrange good multivariable inversion formula we provide a new proof inspired by quantum field theory of the lagrange good multivariable inversion formula

scaling limits of random polya trees polya trees are rooted trees considered up to symmetry we establish the convergence of large uniform random polya trees with arbitrary degree restrictions to aldous continuum random tree with respect to the gromov hausdorff metric our proof is short and elementary and it is based on a novel decomposition it shows that the global shape of a random polya tree is essentially dictated by a large galton watson tree that it contains we also derive sub gaussian tail bounds for both the height and the width which are optimal up to constant factors in the exponent

a calculus for the random generation of labelled combinatorial structures a systematic approach to the random generation of labelled combinatorial objects is presented it applies to structures that are decomposable i e formally specifiable by grammars involving set sequence and cycle constructions a general strategy is developed for solving the random generation problem with two closely related types of methods for structures of size n the boustrophedonic algorithms exhibit a worst case behaviour of the form o n log n the sequential algorithms have worst case o n2 while offering good potential for optimizations in the average case the complexity model is in terms of arithmetic operations and both methods appeal to precomputed numerical table of linear size that can be computed in time o n2 a companion calculus permits systematically to compute the average case cost of the sequential generation algorithm associated to a given specification using optimizations dictated by the cost calculus several random generation algorithms of the sequential type are developed most of them have expected complexity 1 2n log n and are thus only slightly superlinear the approach is exemplified by the random generation of a number of classical combinatorial structures including cayley trees hierarchies the cycle decomposition of permutations binary trees functional graphs surjections and set partitions

h_ infty controller synthesis via switched pdc scheme for discrete time t s fuzzy systems this paper is concerned with the problem of designing switched state feedback h infin controllers for discrete time takagi sugeno t s fuzzy systems new types of state feedback controllers namely switched parallel distributed compensation pdc controllers are proposed which are switched based on the values of membership functions switched quadratic lyapunov functions are exploited to derive a new method for designing switched pdc controllers to guarantee the stability and h infin performances of closed loop nonlinear systems the design conditions are given in terms of solvability of a set of linear matrix inequalities it is shown that the new method provides better or at least the same results of the existing design methods via the pure pdc scheme with a quadratic lyapunov function or switched constant controller gain scheme numerical examples are given to illustrate the effectiveness of the proposed method

cayley continuants in 1858 cayley considered a particular kind of tridiagonal determinants or continuants by a direct inspection of the first cases he conjectured an identity expressing these determinants in terms of certain other determinants considered by sylvester in 1854 then cayley proved the conjectured identity by induction but as he wrote he felt unsatisfied with his proof the main aim of this paper is to give a straightforward proof of cayley s identity using the method of formal series moreover we use this method and umbral calculus techniques to obtain several other identities cayley continuants appear in several contexts and in particular in enumerative combinatorics mittag leffler polynomials meixner polynomials of the first kind the falling and the raising factorials are just few instances of these continuants they can be interpreted in terms of weighted permutations moreover as we prove in this paper they also appear in the context of hankel determinants generated by certain catalan like numbers

the euler characteristic of the moduli space of curves let fg 1 g 1 be the mapping class group consisting of all isotopy classes of base point and orientation preserving homeomorphisms of a closed oriented surface f of genus g let 1 be its euler characteristic in the sense of wall that is z f i fgl f l z e f where f is any torsion free subgroup of finite index in f 1 and e is a contractible space on which f acts freely and properly discontinuously an example of such a space is the teichmiiller space 1 and g f can be interpreted as the orbifold euler characteristic of f i join the moduli space of curves of genus g with base point the purpose of this paper is to prove the following formula for f i

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

aperiodic rings necklace rings and witt vectors in n metropolis and gian carlo rota witt vectors and the algebra of necklaces adv in math 50 1983 95 125 metropolis and rota show that the cyclotomic identity provides a linkage between the necklace ring and the ring of witt vectors in this paper we study this linkage let a be the class of rings a with the property that the additive group of a is torsion free we will prove that the ring of witt vectors over a for any a a is isomorphic to a suitably defined subring of the necklace ring of the rationalization a q of a alternatively to a subring of the aperiodic ring of a a major part of this paper is concerned with a development of the properties and relationships involving the aperiodic ring which parallels the treatment of the necklace ring given in n metropolis and gian carlo rota witt vectors and the algebra of necklaces adv in math 50 1983 95 125 in addition we translate the combinatorial proof of the cyclotomic identity given by metropolis and rota n metropolis and gian carlo rota the cyclotomic identity contemp math 34 1984 19 27 into the language of species in the sense of joyal thereby making clear the naturality of the identity

on learning monotone dnf formulae under uniform distributions abstract we show how to learn in polynomial time monotone d term dnf formulae formulae in disjunctive normal form with at most d terms using positive examples drawn from a distribution that is a generalization of the uniform distribution

truncation of unitary operads abstract we introduce truncation ideals of a k linear unitary symmetric operad and use them to study ideal structure growth property and to classify operads of low gelfand kirillov dimension

new result on exponential stability for neutral stochastic linear system with time varying delay abstract this paper mainly considers the problem for the exponential stability of neutral stochastic linear system with time varying delay in the light of a coupled delay integral inequality and using the stochastic analytic technique the exponential stability criteria for such systems are given in term of the algebraic inequalities regardless of whether the time varying delay is differentiable or not an illustrative numerical example is given to show the effectiveness of the obtained result

separating distribution free and mistake bound learning models over the boolean domain two of the most commonly used models in computational learning theory are the distribution free model in which examples are chosen from a fixed but arbitrary distribution and the absolute mistake bound model in which examples are presented in an arbitrary order over the boolean domain 0 1 n it is known that if the learner is allowed unlimited computational resources then any concept class learnable in one model is also learnable in the other in addition any polynomial time learning algorithm for a concept class in the mistake bound model can be transformed into one that learns the class in the distribution free model this paper shows that if one way functions exist then the mistake bound model is strictly harder than the distribution free model for polynomial time learning specifically given a one way function it is shown how to create a concept class over 0 1 n that is learnable in polynomial time in the distribution free model but not in the absolute mistake bound model in addition the concept class remains hard to learn in the mistake bound model even if the learner is allowed a polynomial number of membership queries the concepts considered are based upon the goldreich goldwasser and micali random function construction goldreich goldwasser and micali journal acm 33 1986 pp 792 807 and involve creating the following new cryptographic object an exponentially long sequence of strings sigma 1 sigma 2 ldots sigma r over 0 1 n that is hard to compute in one direction given sigma i one cannot compute sigma j for j i one can compute sigma j even if j is exponentially larger than i similar sequences considered previously blum blum and shub siam j comput 15 1986 pp 364 383 blum and micali siam j comput 13 1984 pp 850 863 did not allow random access jumps forward without knowledge of a seed allowing one to compute backwards as well

higher dimensional algebra iii n categories and the algebra of opetopes abstract we give a definition of weak n categories based on the theory of operads we work with operads having an arbitrary set s of types or s operads and given such an operad o we denote its set of operations by elt o then for any s operad o there is an elt o operad o whose algebras are s operads over o letting i be the initial operad with a one element set of types and defining i 0 i i i 1 i i we call the operations of i n 1 the n dimensional opetopes opetopes form a category and presheaves on this category are called opetopic sets a weak n category is defined as an opetopic set with certain properties in a manner reminiscent of street s simplicial approach to weak categories similarly starting from an arbitrary operad o instead of i we define n coherent o algebras which are n times categorified analogs of algebras of o examples include monoidal n categories stable n categories virtual n functors and representable n prestacks we also describe how n coherent o algebra objects may be defined in any n 1 coherent o algebra

on coding labeled trees we consider the problem of coding labeled trees by means of strings of node labels different codes have been introduced in the literature by prufer neville and deo and micikevicius for all of them we show that both coding and decoding can be reduced to integer radix sorting closing several open problems within a unified framework that can be applied both in a sequential and in a parallel setting our sequential coding and decoding schemes require optimal o n time when applied to n node trees yielding the first linear time decoding algorithm for a code presented by neville these schemes can be parallelized on the erew pram model so as to work in o logn time with cost o n or o nlogn depending on the code and on the operation in all cases they either match or improve the performances of the best ad hoc approaches known so far

plethysm and cohomology representations of external and symmetric products abstract we prove refined generating series formulae for characters of virtual cohomology representations of external products of suitable coefficients e g complexes of constructible or coherent sheaves or complexes of mixed hodge modules on spaces such as possibly singular complex quasi projective varieties these formulae generalize our previous results for symmetric and alternating powers of such coefficients and apply also to other schur functors the proofs of these results are reduced via an equivariant kunneth formula to a more general generating series identity for abstract characters of tensor powers v n of an element v in a suitable symmetric monoidal category a this abstract approach applies directly also in the equivariant context for spaces with additional symmetries e g finite group actions finite order automorphisms resp endomorphisms as well as for introducing an abstract plethysm calculus for symmetric sequences of objects in a

dual operators and lagrange inversion in several variables the original lagrange inversion formula which gives explicitly the inverse under composition of a formal series was obtained by lagrange in 1770 through formal computations involving logarithms of inlmite products see lagrange 121 there exists an extensive literature on various versions of the inversion formula obtained by algebraic analytic and combinatorial methods comtet 3 gives a survey and an ample bibliography during the last ten years there have appeared several papers on generalizations and new proofs of the inversion formula in 1974 abhyankar obtained an inversion formula using algebraic methods see bass connell and wright 11 several authors have related lagrange inversion with rota s finite operator calculus 151 for example joni 8 garsia and joni 4 hofhauer 7 and roman and rota 14 joyal and labelle used a combinatorial theory of formal series to obtain inversion formulas see 9 l i and viskov 161 used ideas from lie algebra theory to prove formulas similar to abhyankar s in the present paper we use operator methods to obtain inversion formulas in several variables that generalize the formulas of abhyankar joni and viskov we get our results studying an algebra with involution of linear operators on the algebra of formal laurent series in several indeterminates we also show that by means of an anti isomorphism of operator algebras which we call the operator bore1 transform the study of finite operator calculus can be reduced to the study of certain groups of operators on the algebra of formal laurent series our approach may be considered as an algebraic analogue of the complex variable methods used by good 5 to obtain the lagrange inversion formula as a consequence of the change of variables theorem for the

a hopf laboratory for symmetric functions an analysis of symmetric function theory is given from the perspective of the underlying hopf and bi algebraic structures these are presented explicitly in terms of standard symmetric function operations particular attention is focused on laplace pairing sweedler cohomology for 1 and 2 cochains and twisted products rota cliffordizations induced by branching operators in the symmetric function context the latter are shown to include the algebras of symmetric functions of orthogonal and symplectic type a commentary on related issues in the combinatorial approach to quantum field theory is given