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

alternative proof for the localization of sinai s walk we give an alternative proof of the localization of sinai s random walk in random environment under weaker hypothesis than the ones used by sinai moreover we give estimates that are stronger than the one of sinai on the localization neighborhood and on the probability for the random walk to stay inside this neighborhood

blowup solutions for a class of fluid dynamical equations in rn abstract in this paper we consider the blowup solutions for the euler poisson type equations such as isentropic euler poisson navier stokes poisson euler and navier stokes equations and the systems with frictional damping it is shown that these systems have similar blowup phenomenon which generalize the previous results of the blowup solution for the euler poisson equations for gaseous stars besides the blowup rate is given furthermore under the assumption r 0 some radial symmetric blowup solutions for the euler poisson type equations are given

laguerre type derivatives dobinski relations and combinatorial identities we consider properties of the operators d r m ar a a m which we call generalized laguerre type derivatives with r 1 2 m 0 1 where a and a are boson annihilation and creation operators respectively satisfying a a 1 we obtain explicit formulas for the normally ordered form of arbitrary taylor expandable functions of d r m with the help of an operator relation that generalizes the dobinski formula coherent state expectation values of certain operator functions of d r m turn out to be generating functions of combinatorial numbers in many cases the corresponding combinatorial structures can be explicitly identified

a numerical study of diffusion and conduction in a 2d random medium investigates numerically the diffusion and conduction properties of a two dimensional random lattice hopping model the results are compared with a recent field theoretical renormalisation group approach showing that two is the upper critical dimension for such models the predicted scaling law with a universal logarithmic behaviour of the velocity is clearly observed the finiteness of the diffusion constant are well as its regular dependence versus the strength of randomness are evenly in good agreement with analytical results the author also determines the frequency dependence of the ac conductivity tensor

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

critical random walk in random environment on trees we study the behavior of random walk in random environment rwre on trees in the critical case left open in previous work representing the random walk by an electrical network we assume that the ratios of resistances of neighboring edges of a tree are i i d random variables whose logarithms have mean zero and finite variance then the resulting rwre is transient if simple random walk on is transient but not vice versa we obtain general transience criteria for such walks which are sharp for symmetric trees of polynomial growth in order to prove these criteria we establish results on boundary crossing by tree indexed random walks these results rely on comparison inequalities for percolation processes on trees and on some new estimates of boundary crossing probabilities for ordinary mean zero finite variance random walks in one dimension which are of independent interest

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

random walks in asymmetric random environments we consider random walks on zd with transition ratesp x y given by a random matrix ifp is a small random perturbation of the simple random walk we show that the walk remains diffusive for almost all environmentsp ifd 2 the result also holds for a continuous time markov process with a random drift the corresponding path space measures converge weakly in the scaling limit to the wiener process for almost everyp

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

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

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

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 the combinatorics of normal ordering bosonic operators and deformations of it recently some combinatorial aspects for the normal ordering of powers of arbitrary monomials of boson operators were discussed in particular it was shown that the resulting formulae lead to generalizations of the usual bell and stirling numbers in this paper these considerations are generalized to the q deformed case in particular it is shown that the simplest example of this generalization leads to q deformed lah numbers the connection between generalized stirling and bell numbers and matrix elements of the above mentioned operators with respect to the usual fock space basis and coherent states is discussed

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

some useful combinatorial formulas for bosonic operators we give a general expression for the normally ordered form of a function f w a a where w is a function of boson creation and annihilation operators satisfying a a 1 the expectation value of this expression in a coherent state becomes an exact generating function of feynman type graphs associated with the zero dimensional quantum field theory defined by f w this enables one to enumerate explicitly the graphs of given order in the realm of combinatorially defined sequences we give several examples of the use of this technique including the applications to kerr type and superfluidity type hamiltonians

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

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

finite size scaling analysis of biased diffusion on fractals abstract diffusion on a t fractal lattice under the influence of topological biasing fields is studied by finite size scaling methods this allows to avoid proliferation and singularities which would arise in a renormalization group approach on infinite system as a consequence of logarithmic diffusion within the scheme logarithmic diffusion is proved on the basis of an analysis of various temporal scales such as first passage time moments and survival probability characteristic time this confirms and puts on firmer basis previous renormalization group results a careful study of the asymptotic occupation probabilities of different kinds of lattice points allows to elucidate the mechanism of trapping into dangling ends which is responsible of the logarithmic time dependence of average displacement

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

wick s theorem for q deformed boson operators in this paper combinatorial aspects of normal ordering of arbitrary words in the creation and annihilation operators of the q deformed boson are discussed in particular it is shown how by introducing appropriate q weights for the associated feynman diagrams the normally ordered form of a general expression in the creation and annihilation operators can be written as a sum over all q weighted feynman diagrams representing wick s theorem in the present context

rotation prevents finite time breakdown we consider a two dimensional 2d convection model augmented with the rotational coriolis forcing ut u xu 2ku w ith afi xed 2k being the inverse rossby number we ask whether the action of dispersive rotational forcing alone u prevents the generic finite time breakdown of the free nonlinear convection the answer provided in this work is a conditional yes namely we show that the rotating euler equations admit global smooth solutions for a subset of generic initial configurations with other configurations however finite time breakdown of solutions may and actually does occur thus global regularity depends on whether the initial configuration crosses an intrinsic o 1 critical threshold ct which is quantified in terms of the initial vorticity 0 u0 and the initial spectral gap associated with the 2 2 initial velocity gradient 0 2 0 1 0 j 0 j u0 specifically global regularity of the rotational euler equation is ensured if and only if 4k 0 2 4k 2 r 2 we also prove that the velocity field remains smooth if and only if it is periodic an equivalent lagrangian formulation reconfirms the ct and shows a global periodicity of velocity field as well as the associated particle orbits moreover we observe yet another remarkable periodic behavior exhibited by the gradient of the velocity field the spectral dynamics of the eulerian formulation siam j math anal 33 2001 930 reveals that the vorticity and the divergence of the flow evolve with their own path dependent period we conclude with a kinetic formulation of the rotating euler equation 2003 elsevier b v all rights reserved msc 35q35 35b30

analytical blowup solutions to the 2 dimensional isothermal euler poisson equations of gaseous stars ii in this article we construct analytical blowup solutions with non radial symmetry for the 2 dimensional euler poisson equations based on the previous solutions with radial symmetry for the 2 dimensional isothermal euler poisson equations some special blowup solutions with non radial symmetry are constructed by the separation method

global regularity of the 4d restricted euler equations we are concerned with the critical threshold phenomena in the restricted euler re equations using the spectral and trace dynamics we identify the critical thresholds for the 3d and 4d restricted euler equations it is well known that the 3d re solutions blow up projected on the 3 sphere the set of initial eigenvalues which give rise to bounded stable solutions is reduced to a single point which confirms that the 3d re blowup is generic in contrast we identify a surprisingly rich set of the initial spectrum on the 4 sphere which yields global smooth solutions thus 4d regularity is generic

on the interpretation of 1 f noise we propose a model of 1 f noise based on a random walk in a random potential numerical support for the model is given and physical applicability discussed

critical thresholds in 2d restricted euler poisson equations we provide a complete description of the critical threshold phenomenon for the two dimensional localized euler poisson equations introduced by the authors in comm math phys 228 2002 pp 435 466 here the questions of global regularity vs finite time breakdown for the two dimensional 2d restricted euler poisson solutions are classified in terms of precise explicit formulae describing a remarkable variety of critical threshold surfaces of initial configurations in particular it is shown that the 2d critical thresholds depend on the relative sizes of three quantities the initial density the initial divergence and the initial spectral gap that is the difference between the two eigenvalues of the 2 2 initial velocity gradient

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

thresholds in three dimensional restricted euler poisson equations abstract this work provides a description of the critical threshold phenomenon in multi dimensional restricted euler poisson rep equations introduced in h liu e tadmor spectral dynamics of the velocity gradient field in restricted fluid flows comm math phys 228 2002 435 466 for three dimensional rep equations we identified both upper thresholds for the finite time blow up of solutions and subthresholds for the global existence of solutions with the thresholds depending on the relative size of the eigenvalues of the initial velocity gradient matrix and the initial density for the attractive forcing case these one sided threshold conditions of the initial configurations are optimal and the corresponding results also hold for arbitrary n dimensions n 3

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

rigorous renormalisation group and disordered systems abstract we consider random walks in a non symmetric random environment we report a recent result based on a renormalisation group approach showing that for d 62 2 these walks are diffusive when the disorder is weak

spectral dynamics of the velocity gradient field in restricted flows we study the velocity gradients of the fundamental eulerian equation t u u u f which shows up in different contexts dictated by the different modeling of f s to this end we utilize a basic description for the spectral dynamics of u expressed in terms of the possibly complex eigenvalues u which are governed by the ricatti like equation t u 2 l fr we focus our investigation on four prototype models associated with different forcing f ranging from simple linear damping and viscous dusty medium models to the main thrust of the paper the restricted models of euler navier stokes equations and euler poisson equations in particular we address the question of the time regularity for these models that is whether they admit a finite time breakdown a global smooth solution or an intermediate scenario of critical threshold phenomena where global regularity depends on initial configurations using the spectral dynamics as our essential tool in these investigations we obtain a simple form of a critical threshold for the linear damping model and we identify the 2d vanishing viscosity limit for the viscous irrotational dusty medium model moreover for the n dimensional restricted euler equations we obtain n 2 1 global invariants interesting for their own sake which enable us to precisely characterize the local topology at breakdown time extending previous studies in the n 3 dimensional case finally as a fourth model we introduce the n dimensional restricted euler poisson rep system identifying a set of n 2 global invariants which in turn yield i sufficient conditions for finite time breakdown and ii characterization of a large class of 2 dimensional

repeated derivatives of composite functions and generalizations of the leibniz rule we use the properties of hermite and kampe de feriet polynomials to get closed forms for the repeated derivatives of functions whose argument is a quadratic or higher order polynomial these results are extended to product of functions of the above argument thus giving rise to expressions which can formally be interpreted as generalizations of the familiar leibniz rule finally examples of practical interest are discussed

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

solutions of euler poisson equations for gaseous stars in this paper we study the euler poisson equations governing gas motion under self gravitational force we are interested in the evolution of the gaseous stars for which the density function has compact support we establish existence theory for the stationary solutions and describe the behavior of the solutions near the vacuum boundary the boundary behavior thus obtained agrees with the physical boundary condition proposed and studied in l ly for both euler equations with damping and the euler poisson equations existence non existence uniqueness and instability of the stationary solutions with vacuum are also discussed in terms of the adiabatic exponent and the entropy function and the phenomena of the blowup that is the collapsing of the star to a single point with finite mass as well as the drifting of part of the star to infinity in space are also studied and shown to agree with the conjecture from the physical considerations

a hierarchical model for random walks in random media we show that the random walk generated by a hierarchical laplacian in d has standard diffusive behavior moreover we show that this behavior is stable under a class of random perturbations that resemble an off diagonal disordered lattice laplacian the density of states and its asymptotic behavior around zero energy are computed singularities appear in one and two dimensions

hierarchical dobinski type relations via substitution and the moment problem we consider the transformation properties of integer sequences arising from the normal ordering of exponentiated boson a a 1 monomials of the form exp a ra r 1 2 under the composition of their exponential generating functions they turn out to be of sheffer type we demonstrate that two key properties of these sequences remain preserved under substitutional composition a the property of being the solution of the stieltjes moment problem and b the representation of these sequences through infinite series dobinski type relations we present a number of examples of such composition satisfying properties a and b we obtain new dobinski type formulae and solve the associated moment problem for several hierarchically defined combinatorial families of sequences

multivariate bell polynomials and their applications to powers and fractionary iterates of vector power series and to partial derivatives of composite vector functions if a power series is raised to a power the coefficients of the new power series are called the partial bell polynomials they are useful for obtaining the inverse power series iterations possibly fractional of the original power series logs and exponents of power series the derivatives of a composite function and for solving various problems in partition theory multivariate versions of these diverse and useful results are given in terms of multivariate bell polynomials introduced here for the first time

existence and non linear stability of rotating star solutions of the compressible euler poisson equations we prove the existence of rotating star solutions which are steady state solutions of the compressible isentropic euler poisson euler poisson equations in three spatial dimensions with prescribed angular momentum and total mass this problem can be formulated as a variational problem of finding a minimizer of an energy functional in a broader class of functions having less symmetry than those functions considered in the classical auchmuty beals paper we prove the non linear dynamical stability of these solutions with perturbations having the same total mass and symmetry as the rotating star solution we also prove finite time stability of w 1 infty mathbb r 3 solutions where the perturbations are entropy weak solutions of the euler poisson equations finally we give a uniform in time a priori estimate for entropy weak solutions of the euler poisson equations

a dirichlet principle for non reversible markov chains and some recurrence theorems we extend the dirichlet principle to non reversible markov processes on countable state spaces we present two variational formulas for the solution of the poisson equation or equivalently for the capacity between two disjoint sets as an application we prove a some recurrence theorems in particular we show the recurrence of two dimensional cycle random walks under a second moment condition on the winding numbers

heisenberg weyl algebra revisited combinatorics of words and paths the heisenberg weyl algebra which underlies virtually all physical representations of quantum theory is considered from the combinatorial point of view we provide a concrete model of the algebra in terms of paths on a lattice with some decomposition rules we also discuss the rook problem on the associated ferrers board this is related to the calculus in the normally ordered basis from this starting point we explore combinatorial underpinning of the heisenberg weyl algebra which offers novel perspectives methods and applications

dobinski type relations some properties and physical applications we introduce a generalization of the dobinski relation through which we define a family of bell type numbers and polynomials for all these sequences we find the weight function of the moment problem and give their generating functions we provide a physical motivation of this extension in the context of the boson normal ordering problem and its relation to an extension of the kerr hamiltonian

formation of singularities in three dimensional compressible fluids presented are several results on the formation of singularities in solutions to the three dimensional euler equations for a polytropic ideal fluid under various assumptions on the initial data in particular it is shown that a localized fluid which is initially compressed and outgoing on average will develop singularities regardless of the size of the initial disturbance

multidimensional random walks in random environments with subclassical limiting behavior in this paper we will describe and analyze a class of multidimensional random walks in random environments which contain the one dimensional nearest neighbor situation as a special case and have the pleasant feature that quite a lot can be said about them our results make rigorous a heuristic argument of marinari et al 1983 and show that in anyd we can have a xn is recurrent and b xn logn 2

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

rotating fluids with self gravitation in bounded domains in this paper we study the steady solutions of euler poisson equations in bounded domains with prescribed angular velocity this models a rotating newtonian star consisting of a compressible perfect fluid with given equation of state p es when the domain is a ball and the angular velocity is constant we obtain both existence and non existence theorems depending on the adiabatic gas constant in addition we obtain some interesting properties of the solutions e g monotonicity of the radius of the star with both angular velocity and central density we also prove that the radius of a rotating spherically symmetric star with given constant angular velocity and constant entropy is uniformly bounded independent of the central density this is physically striking and in sharp contrast to the case of the non rotating star for general domains and variable angular velocities both an existence result for the isentropic equations of state and non existence result for the non isentropic equation of state are also obtained

properties and possibilities of quantum shapelets quantum shapelets arise as the solution of a d dimensional harmonic oscillator or d dimensional coulomb problem and may be obtained by requiring scale space invariance these functions have application to image processing in conventional or quantum contexts we recall the scale space based derivation of shapelets and present novel properties of these functions including integral relations infinite series and finite convolution sums many of these relations also have application to the combinatorics of zero dimensional quantum field theory

formation of singularities to the euler poisson equations abstract in this paper the finite time blow up of smooth solutions to the cauchy problem for multi dimensional euler poisson equations is proved for a large class of initial data which contains the cases of full euler poisson equations and isentropic euler poisson equations with repulsive forces or attractive it does not require that the initial data have compact support or contain vacuum in any finite region

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

blowup of smooth solutions to the compressible navier stokes equation with compact density we present a sufficient condition on the blowup of smooth solutions to the compressible navier stokes equations in arbitrary space dimensions with initial density of compact support as an immediate application it is shown that any smooth solutions to the compressible navier stokes equations for polytropic fluids in the absence of heat conduction will blow up in finite time as long as the initial densities have compact support and an upper bound which depends only on the initial data on the blowup time follows from our elementary analysis immediately another implication is that there is no global small decay in time or even bounded in the case that all the viscosity coefficients are positive smooth solutions to the compressible navier stokes equations for polytropic fluids no matter how small the initial data are as long as the initial density is of compact support this is in contrast to the classical theory of global existence of small solutions to the same system with initial data being a small perturbation of a constant state that is not a vacuum the blowup of smooth solutions to the compressible euler system with initial density and velocity of compact support is a simple consequence of our argument 1998 john wiley sons inc

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

random walk in random environment a counterexample we describe a family of random walks in random environments which have exponentially decaying correlations nearest neighbor transition probabilities which are bounded away from 0 and yet are subdiffusive in any dimensiond

critical phenomena for spitzer s reversible nearest particle systems motivated by several results and open problems concerning harris basic contact process we consider the relationship between the critical behavior of the finite and infinite versions of spitzer s reversible nearest particle systems we show that the critical values for the finite and infinite systems agree but that the behavior of the two systems at the common critical value can differ the nash williams recurrence criterion for reversible markov chains is an important tool used in the proofs of the main results and we give a new treatment of that theory finally we compute several critical exponents for the nearest particle systems

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

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

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

normal ordering for nonlinear deformed ladder operators and the f generalization of the stirling and bell numbers we resolve the normal ordering problem for symmetric d d n and asymmetric d rd n strings of the nonlinear deformed ladder operators d for supersymmetric and shape invariant potential systems where r and n are positive integers we provide exact and explicit expressions for their normal forms n d d n and n d rd n where in n all d are at the right side we find that the solutions involve sequence of expansion coefficients which for r n 1 corresponds to the f deformed generalization of the classical stirling and bell numbers of the second kind we apply the general formalism for the translational shape invariant potential systems as well as for the particular case of the harmonic oscillator potential system we show that these numbers are obtained for families of polynomial expressions generated with the deformations parameters and the parameters related to the forms of the supersymmetric partner potentials

formation of singularities in the euler and euler poisson equations abstract we establish finite time loss of smoothness for several variants of the one dimensional euler equations we consider the euler poisson equations in one dimension and the euler and euler poisson equations in three dimensions in the presence of spherical or cylindrical symmetry we show that no matter how smooth the initial data are solutions of these equations can be forced to lose smoothness in finite time by using an interesting variant of lax s method for showing the finite time loss of smoothness of solutions of pairs of hyperbolic conservation laws we show the finite time loss of smoothness of solutions of the one dimensional euler poisson equations using our method we also find a sufficient condition for the existence of a globally smooth solution of the one dimensional euler poisson equations for the other cases we use a characteristic based method in all cases we give local conditions for the finite time loss of smoothness of solutions

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

quantum field theory of partitions given a sequence of numbers an it is always possible to find a set of feynman rules that reproduce that sequence for the special case of the partitions of the integers the appropriate feynman rules give rise to graphs that represent the partitions in a clear pictorial fashion these feynman rules can be used to generate the bell numbers b n and the stirling numbers s n k that are associated with the partitions of the integers

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

representing elements of the weyl algebra by labeled trees it is shown how arbitrary elements of the weyl algebra can be represented by labeled plane trees using normal ordering several examples are treated and combinatorial aspects are discussed also possible avenues for future research are described

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