random walks on combs we develop techniques to obtain rigorous bounds on the behaviour of random walks on combs using these bounds we calculate exactly the spectral dimension of random combs with infinite teeth at random positions or teeth with random but finite length we also calculate exactly the spectral dimension of some fixed non translationally invariant combs we relate the spectral dimension to the critical exponent of the mass of the two point function for random walks on random combs and compute mean displacements as a function of walk duration we prove that the mean first passage time is generally infinite for combs with anomalous spectral dimension

the segal bargmann coherent state transform for compact lie groups abstract let k be an arbitrary compact connected lie group we describe on k an analog of the segal bargmann coherent state transform and we prove theorem 1 that this generalized coherent state transform maps l 2 k isometrically onto the space of holomorphic functions in l 2 g where g is the complexification of k and where is an appropriate heat kernel measure on g the generalized coherent state transform is defined in terms of the heat kernel on the compact group k and its analytic continuation to the complex group g we also define a k averaged version of the coherent state transform and we prove a similar result for it theorem 2 in the appendix we describe the coherent states which motivate the definitions of these transforms in section 9 we obtain inversion formulas for both the coherent state transform and the k averaged coherent state transform theorems 3 4 as a consequence we obtain formulas for certain bergman kernels on the complex group g theorems 5 6 in section 10 we derive a general paley wiener theorem for k and we exhibit a family of convolution transforms each of which is an isometric isomorphism of l 2 k onto a certain l 2 space of holomorphic functions on g these transforms include the k averaged coherent state transform as a special case we also discuss the analogous results for r 1 the results of section 10 are theorems 7 10 finally in section 11 we discuss the case of quotient spaces we prove theorem 11 that both the coherent state transform and the k averaged coherent state transform pass in a natural way from k to k h where h is a closed connected subgroup of k

some properties of a random walk on a comb structure abstract we analyze transport properties of a random walk on a comb structure which serves as a model for a random walk on the backbone of a percolation cluster it is shown that the random walk along the x axis which is the analog of the backbone exhibits anomalous diffusion in that x 2 n n 1 2 and the expected number of x sites visited is proportional to n 1 4 for large n the distribution function is found to be a two dimensional gaussian if a field in the x direction so that diffusion is asymmetric the expected displacement is found to be asymptotically proportional to n 1 2

biased random walks on random combs we develop rigorous analytic techniques to study the behaviour of biased random walks on combs this enables us to calculate exactly the spectral dimension of random comb ensembles for any bias scenario in the teeth or spine two specific examples of random comb ensembles are discussed the random comb with nonzero probability of an infinitely long tooth at each vertex on the spine and the random comb with a power law distribution of tooth lengths we also analyse transport properties along the spine for these probability measures

efficient algorithms for membership in boolean hierarchies of regular languages this paper provides efficient algorithms that decide membership for classes of several boolean hierarchies for which efficiency or even decidability were previously not known we develop new forbidden chain characterizations for the single levels of these hierarchies and obtain the following results the classes of the boolean hierarchy over level 1 of the dot depth hierarchy are decidable in nl previously only the decidability was known the same remains true if predicates mod d for fixed d are allowed if modular predicates for arbitrary d are allowed then the classes of the boolean hierarchy over level 1 are decidable for the restricted case of a two letter alphabet the classes of the boolean hierarchy over level 2 of the straubing therien hierarchy are decidable in nl this is the first decidability result for this hierarchy the membership problems for all mentioned boolean hierarchy classes are logspace many one hard for nl the membership problems for quasi aperiodic languages and for d quasi aperiodic languages are logspace many one complete for pspace

the inverse segal bargmann transform for compact lie groups in this paper i give a new inversion formula theorem 1 for the generalized segal bargmann transform introduced in b c hall j funct anal 122 1994 103 151 the inversion formula may be viewed as a formula for theinverseheat operator for a compact lie group i then use this formula to give a new direct proof of the unitary of thek invariant form of the segal bargmann transform theorem 2 the proof of the inversion formula relies on an identity theorem 5 which relates the laplacian for a compact lie groupkto the laplacian for the non compact dual symmetric space kc k

harmonic analysis with respect to heat kernel measure this paper surveys developments over the last decade in harmonic analysis on lie groups relative to a heat kernel measure these include analogs of the hermite expansion the segal bargmann transform and the taylor ex pansion some of the results can be understood from the standpoint of geo metric quantization others are intimately related to stochastic analysis

random walk on random infinite looptrees looptrees have recently arisen in the study of critical percolation on the uniform infinite planar triangulation here we consider random infinite looptrees defined as the local limit of the looptree associated with a critical galton watson tree conditioned to be large we study simple random walk on these infinite looptrees by means of providing estimates on volume and resistance growth we prove that if the offspring distribution of the galton watson process is in the domain of attraction of a stable distribution with index alpha in 1 2 then the spectral dimension of the looptree is 2 alpha alpha 1

the spectral dimension of random trees we present a simple yet rigorous approach to the determination of the spectral dimension of random trees based on the study of the massless limit of the gaussian model on such trees as a by product we obtain evidence in favour of a new scaling hypothesis for the gaussian model on generic bounded graphs and in favour of a previously conjectured exact relation between spectral and connectivity dimensions on more general tree like structures

geometric quantization and multiplicities of group representations the heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum mechanical particle this can be rephrased as follows the smallest subsets of classical phase space in which the presence of a quantum mechanical particle can be detected are its lagrangian submanifolds for this reason it makes sense to regard the lagrangian submanifolds of phase space as being its true points see weinstein 17 now let g be a compact lie group and g x x x a hamiltonian action of g on x see w for definitions it is well known that the fixed points of this action form a symplectic submanifold of x see for instance guillemin and sternberg 5 however what can one say about the fixed points of g we will show that they are also the points of a symplectic manifold xc this manifold is the marsden weinstein reduction of x with respect to the zero orbit in g and will be described in sect 2 it was introduced in a completely different context fl om ours by marsden and weinstein 12 problems in classical mechanics can often be reduced to the study of hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on hilbert space this fact has inspired a number of efforts to quantize symplectic geometry by devising schemes for associating hilbert space to symplectic manifolds the no go theorems of groenwald and van hove impose some embarrassing limitations on all such schemes however it seems to be a useful idea heuristically to think of every symplectic manifold x a ic 1 as being symbiotically associated with a hilbert space xq a t in such a way that the classical observables on the first space correspond to quantum observables on the second space the heuristics further suggests that if g is a group of symmetries of x la i j it should also be a group of symmetries of xq a tum in this heuristic spirit we will state the main conjecture of this paper

the segal bargmann transform for the heat equation associated with root systems we study the heat equation associated to a multiplicity function on a root system where the corresponding laplace operator has been defined by heckman and opdam in particular we describe the image of the associated segal bargmann transform as a space of holomorphic functions in the case where the multiplicity function corresponds to a riemannian symmetric space g k of non compact type we obtain a description of the image of the space of k invariant l2 function on g k under the segal bargmann transform associated to the heat equation on g k thus generalizing and reproving a result of b hall and j mitchell for spaces of complex type

coherent state transforms and vector bundles on elliptic curves abstract we extend the coherent state transform cst of hall to the context of the moduli spaces of semistable holomorphic vector bundles with fixed determinant over elliptic curves we show that by applying the cst to appropriate distributions we obtain the space of level k rank n and genus one non abelian theta functions with the unitarity of the cst transform being preserved furthermore the shift in the level k k n appears in a natural way in this finite dimensional framework

coherent states for the quantum mechanics on a compact manifold we review our recent observations concerning quantum mechanics on such compact manifolds as a circle torus and sphere

the bargmann transform and canonical transformations this paper concerns a relationship between the kernel of the bargmann transform and the corresponding canonical transformation we study this fact for a bargmann transform introduced by thomas and wassell j math phys 36 5480 5505 1995 when the configuration space is the two sphere s2 and for a bargmann transform that we introduce for the three sphere s3 it is shown that the kernel of the bargmann transform is a power series in a function which is a generating function of the corresponding canonical transformation a classical analog of the bargmann transform we show in each case that our canonical transformation is a composition of two other canonical transformations involving the complex null quadric in c3 or c4 we also describe quantizations of those two other canonical transformations by dealing with spaces of holomorphic functions on the aforementioned null quadrics some of these quantizations have been studied by bargmann and todorov j math phys 18 1141 1148 1977 and the other qu

asymptotic behaviour of reproducing kernels of weighted bergman spaces let q be a domain in c f a nonnegative and g a positive function on q such that 1 g is locally bounded a2 the space of all holomorphic functions on q square integrable with respect to the measure f0gda where da is the 2n dimensional lebesgue measure and k x y the reproducing kernel for a2 it has been known for a long time that in some special situations such as on bounded symmetric domains q with g 1 and f the bergman kernel function the formula lim k x x 1 0 1 f x ae 00 holds true this fact even plays a crucial role in berezin s theory of quantization on curved phase spaces in this paper we discuss the validity of this formula in the general case the answer turns out to depend on loosely speaking how well the function log f can be approximated by certain pluriharmonic functions lying below it for instance holds if log f is convex and hence can be approximated from below by linear functions for any function g counterexamples are also given to show that in general may fail drastically or even be true for some x and fail for the remaining ones finally we also consider the question of convergence of k0 x y 1 0 for x y which leads to an unexpected result showing that the zeroes of the reproducing kernels are affected by the smoothness of f for instance if f is not real analytic at some point then kc x y must have zeroes for all at sufficiently large

uniform infinite planar triangulations the existence of the weak limit as n of the uniform measure on rooted triangulations of the sphere with n vertices is proved some properties of the limit are studied in particular the limit is a probability measure on random triangulations of the plane

first encounters on combs we consider two random walkers embedded in a finite two dimension comb and we study the mean first encounter time mfet evidencing mainly numerically different scalings with the linear size of the underlying network according to the initial position of the walkers if one of the two players is not allowed to move then the first encounter problem can be recast into a first passage problem mfpt for which we also obtain exact results for different initial configurations by comparing mfet and mfpt we are able to figure out possible search strategies and in particular we show that letting one player be fixed can be convenient to speed up the search as long as we can finely control the initial setting while for a random setting on average letting one player rest would slow down the search

brown measure support and the free multiplicative brownian motion abstract the free multiplicative brownian motion b t is the large n limit of brownian motion b t n on the general linear group gl n c we prove that the brown measure for b t which is an analog of the empirical eigenvalue distribution for matrices is supported on the closure of a certain domain t in the plane the domain t was introduced by biane in the context of the large n limit of the segal bargmann transform associated to gl n c we also consider a two parameter version b s t the large n limit of a related family of diffusion processes on gl n c introduced by the second author we show that the brown measure of b s t is supported on the closure of a certain planar domain s t generalizing t introduced by ho in the process we introduce a new family of spectral domains related to any operator in a tracial von neumann algebra the l p n spectrum for n n and p 1 a subset of the ordinary spectrum defined relative to potentially unbounded inverses we show that in general the support of the brown measure of an operator is contained in its l 2 2 spectrum

going higher in first order quantifier alternation hierarchies on words we investigate quantifier alternation hierarchies in first order logic on finite words levels in these hierarchies are defined by counting the number of quantifier alternations in formulas we prove that one can decide membership of a regular language in the levels b 2 finite boolean combinations of formulas having only one alternation and 3 formulas having only two alternations and beginning with an existential block our proofs work by considering a deeper problem called separation which once solved for lower levels allows us to solve membership for higher levels

on shannon entropies in μ deformed segal bargmann analysis we consider a deformation of the segal bargmann transform which is a unitary map from a deformed quantum configuration space onto a deformed quantum phase space the deformed segal bargmann space both of these hilbert spaces have canonical orthonormal bases we obtain explicit formulas for the shannon entropy of some of the elements of these bases we also consider two reverse log sobolev inequalities in the deformed segal bargmann space which have been proved in a previous work and show that a certain known coefficient in them is the best possible

descriptional and computational complexity of finite automata a survey finite automata are probably best known for being equivalent to right linear context free grammars and thus for capturing the lowest level of the chomsky hierarchy the family of regular languages over the last half century a vast literature documenting the importance of deterministic nondeterministic and alternating finite automata as an enormously valuable concept has been developed in the present paper we tour a fragment of this literature mostly we discuss developments relevant to finite automata related problems like for example i simulation of and by several types of finite automata ii standard automata problems such as fixed and general membership emptiness universality equivalence and related problems and iii minimization and approximation we thus come across descriptional and computational complexity issues of finite automata we do not prove these results but we merely draw attention to the big picture and some of the main ideas involved

short time behavior of hermite functions on compact lie groups abstract let p t x be the gaussian heat kernel on r n at time t the classical hermite polynomials at time t may be defined by a rodriguez formula given by h x t p t x p t x where is a constant coefficient differential operator on r n recent work of gross 1993 and hijab 1994 has led to the study of a new class of functions on a general compact lie group g in analogy with the r n case these hermite functions on g are obtained by the same formula wherein p t x is now the heat kernel on the group x is replaced by x 1 and is a right invariant differential operator let g be the lie algebra of g composing a hermite function on g with the exponential map produces a family of functions on g we prove that these functions scaled appropriately in t approach the classical hermite polynomials at time 1 as t tends to 0 both uniformly on compact subsets of g and in l p g where 1 p is a gaussian measure on g similar theorems are established when g is replaced by g k where k is some closed connected subgroup of g

multigraph models for causal quantum gravity and scale dependent spectral dimension we study random walks on ensembles of a specific class of random multigraphs which provide an effective graph ensemble for the causal dynamical triangulation cdt model of quantum gravity in particular we investigate the spectral dimension of the multigraph ensemble for recurrent as well as transient walks we investigate the circumstances in which the spectral dimension and hausdorff dimension are equal and show that this occurs when the exponent for anomalous behaviour of the resistance to infinity is zero the concept of scale dependent spectral dimension in these models is introduced we apply this notion to a multigraph ensemble with a measure induced by a size biased critical galton watson process which has a scale dependent spectral dimension of two at large scales and one at small scales we conclude by discussing a specific model related to four dimensional cdt which has a spectral dimension of four at large scales and two at small scales

phase space bounds for quantum mechanics on a compact lie group let k be a compact connected lie group and k bbb c its complexification i consider the hilbert space cal h l 2 left k bbb c nu t right of holomorphic functions introduced in h1 where the parameter t is to be interpreted as planck s constant in light of l s the complex group k bbb c may be identified canonically with the cotangent bundle of k using this identification i associate to each f in cal h l 2 left k bbb c nu t right a phase space probability density the main result of this paper is theorem 1 which provides an upper bound on this density which holds uniformly over all f and all points in phase space specifically the phase space probability density is at most a t left 2 pi t right n where n dim k and a t is a constant which tends to one exponentially fast as t tends to zero at least for small t this bound cannot be significantly improved

mesoscopic description of random walks on combs combs are a simple caricature of various types of natural branched structures which belong to the category of loopless graphs and consist of a backbone and branches we study continuous time random walks on combs and present a generic method to obtain their transport properties the random walk along the branches may be biased and we account for the effect of the branches by renormalizing the waiting time probability distribution function for the motion along the backbone we analyze the overall diffusion properties along the backbone and find normal diffusion anomalous diffusion and stochastic localization diffusion failure respectively depending on the characteristics of the continuous time random walk along the branches and compare our analytical results with stochastic simulations

generic ising trees the ising model on a class of infinite random trees is defined as a thermodynamic limit of finite systems a detailed description of the corresponding distribution of infinite spin configurations is given as an application we study the magnetization properties of such systems and prove that they exhibit no spontaneous magnetization furthermore the values of the hausdorff and spectral dimensions of the underlying trees are calculated and found to be respectively and

hermite functions on compact lie groups ii abstract the usual formula for hermite polynomials on rd is extended to a compact lie group g yielding an isometry of l2 g p1 where p1 is the heat kernel measure at time one with a natural completion of the universal enveloping algebra of g the existence of such an isometry was first established by l gross here we present a simplified proof as well as the explicit form of the isometry

coherent state transforms for spaces of connections abstract the segal bargmann transform plays an important role in quantum theories of linear fields recently hall obtained a non linear analog of this transform for quantum mechanics on lie groups given a compact connected lie groupgwith its normalized haar measure h the hall transform is an isometric isomorphism froml2 g h to h g c l2 g c whereg c the complexification ofg h g c the space of holomorphic functions ong c and an appropriate heat kernel measure ong c we extend the hall transform to the infinite dimensional context of non abelian gauge theories by replacing the lie groupgby a certain extension of the space a g of connections modulo gauge transformations the resulting coherent state transform provides a holomorphic representation of the holonomyc algebra of real gauge fields this representation is expected to play a key role in a non perturbative canonical approach to quantum gravity in 4 dimensions

complexity of some problems from the theory of automata 1 1 given a finite alphabet l the regular events over x are those accepted by a finite state automaton by kleene s theorem a subset w of s is a regular event if and only if it can be constructed from the finiteword sets by boolean operations together with concatenation and operation in some sense the regular events are quite simple because they are accepted by a machine with no storage capacity nevertheless in recent years much attention has been paid to special subclasses of the class of regular events in this paper we shall be concerned with star free events events of dot depth one and piecewise testable events star free events are constructed like regular events from the finite word sets but with the restriction that the operation is not allowed events of dot depth one and piecewise testable events are star free of a very simple form and will be defined below star free events have been characterized in the work of schiitzenberger 1965 in terms of their syntactic monoid algebraic characterizations of the other two classes have been given simon 1975 knast 1983 from an algorithmic point of view these algebraic characterizations do not yield efficient procedures because computing the syntactic monoid of a regular event given e g by an automaton is obviously time consuming in this paper we investigate the complexity of three problems which we now describe in the style of garey and johnson 1979 we refer the reader to aho hopcroft and ullman 1983 garey and johnson 1979 for standard concepts of complexity theory

languages recognized by finite aperiodic groupoids we study the context free languages recognized by a groupoid g in terms of the algebraic properties of the multiplication monoid m g of g concentrating on the case where m g is group free we show that all regular languages can be recognized by groupoids for which m g is j trivial and that all groupoids for which m g belongs to the larger variety da recognize only regular languages further we give an example of a groupoid such that m g is in the smallest variety outside of da and which recognizes all context free languages not containing the empty word

extensions to barrington s m program model abstract barrington s polynomal length program over a monoid is a model of computation which has been studied intensively in connection with the structure of the complexity class nc 1 barrington 1986 barrington and therien 1987 1988 mckenzie and therien 1989 peladeau 1989 here two extensions of the model are considered first with the use of nonassociative structures hence groupoids instead of associative monoids polynomial length program characterizations of complexity classes tc 0 nl and logcfl as well as new characterizations of nc 1 are given new word problems complete for logcfl for nl and for nc 1 under dlogtime reductions are obtained as corollaries second using monoids but permitting the use of a different monoid to handle each input length new complexity classes are defined combinatorial arguments are then developed to resolve the relationships between various such classes defined in terms of polynomial length programs over growing abelian monoid sequences then the orders of growing abelian group and monoid sequences required to accept specific languages defined in terms of the presence of a given substring are investigated finally the two extensions are combined to obtain characterizations of l and nl in terms of polynomial length programs defined over polynomially growing groupoid sequences it is further argued that such programs are generally no more powerful than logcfl

the complex time segal bargmann transform abstract we introduce a new form of the segal bargmann transform for a connected lie group k of compact type we show that the heat kernel t x t 0 x k has a space time analytic continuation to a holomorphic function c z re 0 z k c where k c is the complexification of k the new transform is defined by the integral b f z k c z k 1 f k d k z k c if s 0 and d s s the disk of radius s centered at s this integral defines a holomorphic function on k c for each f l 2 k s we construct a heat kernel density s on k c such that for all s as above b s b l 2 k s is an isometric isomorphism from l 2 k s onto the space of holomorphic functions in l 2 k c s when t s the transform b t t coincides with the one introduced by the second author for compact groups and extended by the first author to groups of compact type when t 0 2 s the transform b s t coincides with the one introduced by the first two authors

coherent state transforms and the mackey stone von neumann theorem mackey showed that for a compact lie group k the pair k c 0 k has a unique non trivial irreducible covariant pair of representations we study the relevance of this result to the unitary equivalence of quantizations for an infinite dimensional family of k k invariant polarizations on t k the kahler polarizations in the family are generated by complex time hamiltonian flows applied to the schrodinger vertical real polarization the unitary equivalence of the corresponding quantizations of t k is then studied by considering covariant pairs of representations of k defined by geometric prequantization and of representations of c 0 k defined via heisenberg time evolution followed by time geometric quantization induced evolution we show that in the semiclassical and large imaginary time limits the unitary transform whose existence is guaranteed by mackey s theorem can be approximated by composition of the time geometric quantization induced evolution with the time evolution

regular articlehermite functions on compact lie groups i the usual formula for hermite polynomials on rd is extended to a compact lie group g yielding an isometry of l2 g p1 where p1 is the heat kernel measure at time one with a natural completion of the universal enveloping algebra of g the existence of such an isometry was first established by l gross here we present a simplified proof as well as the explicit form of the isometry

the bargmann representation for the quantum mechanics on a sphere the bargmann representation is constructed corresponding to the coherent states for a particle on a sphere introduced in kowalski and rembielinski j phys a 33 6035 2000 the connection is discussed between the introduced formalism and the standard approach based on the hilbert space of square integrable functions on a sphere s2

the spectral dimension of random brushes we consider a class of random graphs called random brushes which are constructed by adding linear graphs of random lengths to the vertices of viewed as a graph we prove that for d 2 all random brushes have spectral dimension ds 2 for d 3 we have and for d 4 we have 3 ds d

the segal bargmann transform on a symmetric space of compact type abstract we study the segal bargmann transform on a symmetric space x of compact type mapping l 2 x into holomorphic functions on the complexification x c we invert this transform by integrating against a dual heat kernel measure in the fibers of a natural fibration of x c over x we prove that the segal bargmann transform is an isometry from l 2 x onto the space of holomorphic functions on x c which are square integrable with respect to a natural measure these results extend those of b hall in the compact group case

yang mills theory and the segal bargmann transform we use a variant of the segal bargmann transform to study canonically quantized yang mills theory on a space time cylinder with a compact structure group k the non existent lebesgue measure on the space of connections is approximated by a gaussian measure with large variance the segal bargmann transform is then a unitary map from the l 2 space over the space of connections to a holomorphicl 2 space over the space of complexified connections with a certain gaussian measure this transform is given roughly bye t1a 2 followed by analytic continuation here 1a is the laplacian on the space of connections and is the hamiltonian for the quantized theory on the gauge trivial subspace consisting of functions of the holonomy around the spatial circle the segal bargmann transform becomes e t1 k 2 followed by analytic continuation where 1k is the laplacian for the structure group k this result gives a rigorous meaning to the idea that 1a reduces to 1k on functions of the holonomy by letting the variance of the gaussian measure tend to infinity we recover the standard realization of the quantized yang mills theory on a space time cylinder namely 1 1k is the hamiltonian and l 2 k is the hilbert space as a byproduct of these considera tions we find a new one parameter family of unitary transforms from l 2 k to certain holomorphicl 2 spaces over the complexification of k this family of transformations interpolates between the two previously known unitary transformations our work is motivated by results of landsman and wren and uses probabilistic techniques similar to those of gross and malliavin

exact calculation of the tortuosity in disordered linear pores in the knudsen regime the squared reciprocal tortuosity ensuremath kappa ensuremath 2 d d 0 for linear diffusion on lattices and in pores in the knudsen regime is calculated analytically for a large variety of disordered systems here d 0 and d are the self diffusion coefficients of the smooth and the corresponding disordered system respectively to this end a building block principle is developed that composes the systems into substructures without cross correlations between them it is shown how the solutions of the different building blocks can be combined to gain d d 0 for pores of high complexity from the geometrical properties of the systems i e from the volumes of the different substructures as a test numerical simulations are performed that agree perfectly with the theory

the segal bargmann transform for path groups letkbe a connected lie group of compact type and letw k denote the set of continuous paths ink starting at the identity and with time interval 0 1 thenw k forms an infinite dimensional group under the operation of pointwise multiplication let denote the wiener measure onw k we construct an analog of the segal bargmann transform forw k letkcbe the complexification ofk w kc the set of continuous paths inkcstarting at the identity and the wiener measure onw kc our transform is a unitary map ofl2 w k onto the holomorphic subspace ofl2 w kc by analogy with the classical transform our transform is given by convolution with the wiener measure followed by analytic continuation we prove that the transform forw k is nicely related by means of the ito map to the classical segal bargmann transform for the path space in the lie algebra ofk

on boolean combinations forming piecewise testable languages abstract a regular language is k piecewise testable k pt if it is a boolean combination of languages of the form l a 1 a 2 a n a 1 a 2 a n where a i and 0 n k given a finite automaton a if the language l a is piecewise testable we want to express it as a boolean combination of languages of the above form the idea is as follows if the language is k pt then there exists a congruence k of finite index such that l a is a finite union of k classes every such class is characterized by an intersection of languages of the from l u for u k and their complements to represent the k classes we make use of the k canonical dfa we identify the states of the k canonical dfa whose union forms the language l a and use them to construct the required boolean combination we study the computational and descriptional complexity of related problems

from cascade decompositions to bit vector algorithms a vector algorithm is an algorithm that applies a bounded number of vector operations to an input vcctor regardless of the length of the input in this paper we describe the links between the existence of vector algorithms and the cascade decompositions of counter free automata we show that any computation that can be carried out with a counter free automaton can be recast as a vector algorithm moreover we show that for a class of automata that is closely related to algorithms in bio computing the complexity of the resulting algorithms is linear in the number of transitions of the original automaton

on the kakutani ito segal gross and segal bargmann hall isomorphisms recently gross has shown that the kakutani ito segal isomorphism theorem has an extension from the setting of gaussian measure on a vector space to heat kernel measure pt on a simply connected lie group g of compact type the isomorphism relates l2 pt to a certain completion of the universal enveloping algebra of g lie g gross proves this result using the kakutani ito segal theorem and an infinite dimensional calculus associated to g valued brownian motion hijab has greatly simplified and clarified gross proof hiiab s proof avoids most but not all of the infinite dimensional analysis in the original proof in this paper we will build on hijab s proof to give a completely finite dimensional non probabilistic proof of gross isomorphism theorem the proof given here relies heavily on hall s beautiful extension of the segal bargmann transformation to the setting of compact lie groups this theorem relating l2 pt to a certain l2 space of holomorphic functions l2 gc h on the complexified lie group gc will also be generalized to lie groups of compact type in the process it is shown how to characterize in terms of summability conditions on all the derivatives at the identity in gc those holomorphic functions which are in l2 gc

isometry theorem for the segal bargmann transform on a noncompact symmetric space of the complex type we consider the segal bargmann transform on a noncompact symmetric space of the complex type we establish isometry and surjectivity theorems for the transform in a form as parallel as possible to the results in the dual compact case the isometry theorem involves integration over a tube of radius r in the complexification followed by analytic continuation with respect to r a cancellation of singularities allows the relevant integral to have a nonsingular extension to large r even though the function being integrated has singularities

transport properties on a random comb we study the random walk of a particle in a random comb structure both in the presence of a biasing field and an the field free case we show that the mean field treatment of the quenched disorder can be exactly mapped on to a continuous time random walk ctrw on the backbone of the comb with a definite waiting time density we find an exact expression for this central quantity the green function for the ctrw is then obtained its first and second moments determine the drift and diffusion at all times we show that the drift velocity v vanishes asymptotically for power law and stretched exponential distributions of branch lengths on the comb whatever be the biasing field strength for an exponential branch length distribution v is a nonmonotonic function of the bias increasing initially to a maximum and then decreasing to zero at a critical value in the field free case anomalous diffusion occurs for a range of power law distributions of the branch length the corresponding exponent for the mean square displacement is obtained as is the asymptotic form of the positional probability distribution for the random walk we show that normal diffusion occurs whenever the mean branch length is finite and present a simple formula for the effective diffusion constant these results are extended to regular nonrandom combs as well the physical reason for anomalous drift or diffusion is traced to the properties of the distribution of a first passage time on a finite chain that controls the effective waiting time density of the ctrw

some complexity results for polynomial rational expressions it is known that any languages of level 3 2 of the straubing therien hierarchy can be written as a polynomial expression that is a finite union of languages of the form a0 b1a 1 b kak where the ai and bi are subsets of the alphabet in this paper we prove that such a polynomial expression for a 3 2 level language can be of exponential size more precisely we exhibit an n state minimal automaton recognizing a level 3 2 language for which the shortest polynomial expresion has exponential size in n

the membership problem in aperiodic transformation monoids the problem of testing membership in aperiodic or group free transformation monoids is the natural counterpart to the well studied membership problem in permutation groups the class a of all finite aperiodic monoids and the class g of all finite groups are two examples of varieties the fundamental complexity units in terms of which finite monoids are classified the collection of all varieties v forms an infinite lattice under the inclusion ordering with the subfamily of varieties that are contained in a forming an infinite sublattice for each v a the associated problem memb v of testing membership in transformation monoids that belong to v is considered remarkably the computational complexity of each such problem turns out to look familiar moreover only five possibilities occur as v ranges over the whole aperiodic sublattice with one family of np hard exceptions whose exact status is still unresolved any such memb v is either pspace complete np complete p complete or in ac 0 these results thus uncover yet another surprisingly tight link between the theory of monoids and computational complexity theory

geometric quantization and the generalized segal bargmann transform for lie groups of compact type let k be a connected lie group of compact type and let t k be its cotangent bundle this paper considers geometric quantization of t k first using the vertical polarization and then using a natural kahler polarization obtained by identifying t k with the complexified group k the first main result is that the hilbert space obtained by using the kahler polarization is naturally identifiable with the generalized segal bargmann space introduced by the author from a different point of view namely that of heat kernels the second main result is that the pairing map of geometric quantization coincides with the generalized segal bargmann transform introduced by the author this means that the pairing map in this case is a constant multiple of a unitary map for both results it is essential that the half form correction be included when using the kahler polarization

appearance of vertices of infinite order in a model of random trees we study an equilibrium statistical mechanical model of tree graphs which are made up of a linear subgraph the spine to which leaves are attached we prove that the model has two phases a generic phase where the spine becomes infinitely long in the thermodynamic limit and all vertices have finite order and a condensed phase where the spine is finite with probability 1 and a single vertex of infinite order appears in the thermodynamic limit we calculate the spectral dimension of the graphs in both phases and prove the existence of a gibbs measure we discuss generalizations of this model and the relationship with models of nongeneric random trees

a polynomial time algorithm for the local testability problem of deterministic finite automata the local testability problem of deterministic finite automata is investigated a locally testable language is a language with the property that for some nonnegative integer k whether or not a word w is in the language depends on 1 the prefix and suffix of w of length k and 2 the set of substrings of w length k 1 without regard to the order in which these substrings occur the local testability problem is given a deterministic finite automation to decide whether or not it accepts a locally testable language the authors present an o n sup 2 time algorithm for the local testability problem based on two simple properties that characterize locally testable automata

characterizations of some classes of regular events abstract we study two special classes of regular events the piecewise testable events and the events of dot depth one several characterizations of these events are given especially in terms of their minimal automaton

condensation in nongeneric trees we study nongeneric planar trees and prove the existence of a gibbs measure on infinite trees obtained as a weak limit of the finite volume measures it is shown that in the infinite volume limit there arises exactly one vertex of infinite degree and the rest of the tree is distributed like a subcritical galton watson tree with mean offspring probability m 1 we calculate the rate of divergence of the degree of the highest order vertex of finite trees in the thermodynamic limit and show it goes like 1 m n where n is the size of the tree these trees have infinite spectral dimension with probability one but the spectral dimension calculated from the ensemble average of the generating function for return probabilities is given by 2 2 if the weight wn of a vertex of degree n is asymptotic to n

holomorphic sobolev spaces and the generalized segal bargmann transform we consider the generalized segal bargmann transform ct for a compact group k introduced in b c hall j funct anal 122 1994 103 151 let kc denote the complexification of k we give a necessary and sufficient pointwise growth condition for a holomorphic function on kc to be in the image under ct of c 1 k we also characterize the image under ct of sobolev spaces on k the proofs make use of a holomorphic version of the sobolev embedding theorem

heat kernel analysis and cameron martin subgroup for infinite dimensional groups abstract the heat kernel measure t is constructed on gl h the group of invertible operators on a complex hilbert space h this measure is determined by an infinite dimensional lie algebra g and a hermitian inner product on it the cameron martin subgroup g cm is defined and its properties are discussed in particular there is an isometry from the l 2 t closure of holomorphic polynomials into a space h t g cm of functions holomorphic on g cm this means that any element from this l 2 t closure of holomorphic polynomials has a version holomorphic on g cm in addition there is an isometry from h t g cm into a hilbert space associated with the tensor algebra over g the latter isometry is an infinite dimensional analog of the taylor expansion as examples we discuss a complex orthogonal group and a complex symplectic group

pspace complete problems for subgroups of free groups and inverse finite automata we investigate the complexity of algorithmic problems on finitely generated subgroups of free groups margolis and meakin showed how a finite monoid synt h can be canonically and effectively associated with such a subgroup h we show that h is pure that is closed under radical if and only if synt h is aperiodic we also show that testing for this property of h is pspace complete in the process we show that certain problems about finite automata which are pspace complete in general remain pspace complete when restricted to injective and inverse automata with single accept state whereas they are known to be in nc for permutation automata with single accept state

segal bargmann transform functional calculus on matrix spaces and the theory of semi circular and circular systems letmd c be the space ofd dcomplex matrices and xdbe the subspace of hermitian matrices we study the segal bargmann transform of functions on xd with values inmd c which are given by functional calculus we show that whend the transform of such a map becomes close to the space of holomorphic functional calculus onmd c and that this yields in the limit an isometry between thel2space of wigner s semi circle distribution and the hardy space of the disk we relate this isometry to voiculescu s theory of circular and semi circular systems and we study its analogue when the segal bargmann transform is replaced by the hall transform on unitary groups of large dimensions

coherent states on spheres we describe a family of coherent states and an associated resolution of the identity for a quantum particle whose classical configuration space is the d dimensional sphere sd the coherent states are labeled by points in the associated phase space t sd these coherent states are not of perelomov type but rather are constructed as the eigenvectors of suitably defined annihilation operators we describe as well the segal bargmann representation for the system the associated unitary segal bargmann transform and a natural inversion formula although many of these results are in principle special cases of the results of hall and stenzel we give here a substantially different description based on ideas of thiemann and of kowalski and rembielinski all of these results can be generalized to a system whose configuration space is an arbitrary compact symmetric space we focus on the sphere case in order to carry out the calculations in a self contained and explicit way

berezin toeplitz quantization on lie groups let k be a connected compact semisimple lie group and kc its complexification the generalized segal bargmann space for kc is a space of square integrable holomorphic functions on kc with respect to a k invariant heat kernel measure this space is connected to the schrodinger hilbert space l 2 k by a unitary map the generalized segal bargmann transform this paper considers certain natural operators on l 2 k namely mul tiplication operators and differential operators conjugated by the general ized segal bargmann transform the main results show that the resulting operators on the generalized segal bargmann space can be represented as toeplitz operators the symbols of these toeplitz operators are expressed in terms of a certain subelliptic heat kernel on kc i also examine some of the results from an infinite dimensional point of view based on the work of l gross and p malliavin

the segal bargmann transform for noncompact symmetric spaces of the complex type we consider the generalized segal bargmann transform defined in terms of the heat operator for a noncompact symmetric space of the complex type for radial functions we show that the segal bargmann transform is a unitary map onto a certain l 2 space of meromorphic functions for general functions we give an inversion formula for the segal bargmann transform involving integration against an unwrapped version of the heat kernel for the dual compact symmetric space both results involve delicate cancellations of singularities

on the spectral dimension of causal triangulations we introduce an ensemble of infinite causal triangulations called the uniform infinite causal triangulation and show that it is equivalent to an ensemble of infinite trees the uniform infinite planar tree it is proved that in both cases the hausdorff dimension almost surely equals 2 the infinite causal triangulations are shown to be almost surely recurrent or equivalently their spectral dimension is almost surely less than or equal to 2 we also establish that for certain reduced versions of the infinite causal triangulations the spectral dimension equals 2 both for the ensemble average and almost surely the triangulation ensemble we consider is equivalent to the causal dynamical triangulation model of two dimensional quantum gravity and therefore our results apply to that model

some results on the generalized star height problem abstract we prove some results related to the generalized star height problem in this problem as opposed to the restricted star height problem complementation is considered as a basic operator we first show that the class of languages of star height n is closed under certain operations left and right quotients inverse alphabetic morphisms injective star free substitutions it is known that languages recognized by a commutative group are of star height 1 we extend this result to nilpotent groups of class 2 and to the groups that divide a semidirect product of a commutative group by z 2 z n in the same direction we show that one of the languages that were conjectured to be of star height 2 during the past ten years is in fact of star height 1 next we show that if a rational language l is recognized by a monoid of the variety generated by wreath products of the form m g n where m and n are aperiodic monoids and g is a commutative group the l is of star height 1 finally we show that every rational language is the inverse image under some morphism between free monoids of a language of resticted star height 1

the statistical geometry of scale free random trees the properties of random trees galton watson trees with scale free power like probability distribution of coordinations are investigated in the thermodynamic limit the scaling form of volume probability is found and the connectivity dimensions are determined and compared with other exponents which describe the growth the local spectral dimension is also determined through the study of the massless limit of the gaussian model on such trees

uniqueness of ground states for schrodinger operators over loop groups abstract over the space l of based loops in a compact connected lie group g there is a natural gradient operator induced by the left action on l of the hilbert lie group k 0 which consists of finite energy elements of l denote by pinned brownian motion measure on l the schrodinger operator v acting in l 2 l is shown to have a unique ground state over each homotopy class in l the proof of uniqueness is reduced to the proof of ergodicity of the left action of k 0 on l for a simply connected group g of compact type ergodicity in turn is proved by first characterizing the ito expansion coefficients of k 0 invariant functions and then regularizing such functions via a smooth approximation of multiple stratonovich integrals as a byproduct of this method one obtains an isometric isomorphism of l 2 g heat kernel measure at time one with a natural completion of the universal enveloping algebra of g thus extending to the noncommutative case the well known isometry between l 2 r d gauss measure and the space of symmetric tensors over r d

geometric quantization complex structures and the coherent state transform abstract it is shown that the heat operator in the hall coherent state transform for a compact lie group k j funct anal 122 1994 103 151 is related with a hermitian connection associated to a natural one parameter family of complex structures on t k the unitary parallel transport of this connection establishes the equivalence of geometric quantizations of t k for different choices of complex structures within the given family in particular these results establish a link between coherent state transforms for lie groups and results of hitchin comm math phys 131 1990 347 380 and axelrod et al j differential geom 33 1991 787 902

validity of the mean field approximation for diffusion on a random comb we consider unbiased diffusion on a random comb structure an infinitely long backbone with loopless branches of arbitrary length emanating from it if t ifmmode pm else textpm fi j 0 mathrm mathit w mathit t 0 is the mean time averaged over all random walks for first passage from an arbitrary origin 0 on the backbone to either of the sites j or j on it in a given realization of the structure the exact diffusion constant for the problem is defined as k mathrm lim mathit j ensuremath rightarrow mathrm ensuremath infty mathit j 2 1 mathit t 0 mathrm mathit c where mathrm mathit c stands for the configuration average over the realizations of the random comb the diffusion constant in the mean field approximation is given by mathit k mathrm mf mathrm lim mathit j ensuremath rightarrow mathrm ensuremath infty mathit j 2 mathit t 0 mathrm mathit c we compute mathit t 0 exactly for an arbitrary realization of the comb and then show rigorously that owing to the suppression of the relative fluctuations in mathit t 0 in the thermodynamic limit j ensuremath rightarrow ensuremath infty we have mathit k mathrm mf k whenever the moments of certain random variables ensuremath gamma l ensuremath alpha ensuremath beta are finite here the site dependent random variables l ensuremath alpha and ensuremath beta are respectively the branch length stay probability at the tip of a branch and the backbone to branch jump probability finally we discuss different situations in which k will not be equal to mathit k mathrm mf although the transport remains diffusive as opposed to those in which anomalous diffusion occurs textcopyright 1996 the american physical society

the spectral dimension of generic trees we define generic ensembles of infinite trees these are limits as n of ensembles of finite trees of fixed size n defined in terms of a set of branching weights among these ensembles are those supported on trees with vertices of a uniformly bounded order the associated probability measures are supported on trees with a single spine and hausdorff dimension dh 2 our main result is that the spectral dimension of the ensemble average is ds 4 3 and that the critical exponent of the mass defined as the exponential decay rate of the two point function along the spine is 1 3