equivalence of deterministic top down tree to string transducers is decidable we prove that equivalence of deterministic top down tree to string transducers is decidable thus solving a long standing open problem in formal language theory we also present efficient algorithms for subclasses for linear transducers or total transducers with unary output alphabet over a given top down regular domain language as well as for transducers with the single use restriction these results are obtained using techniques from multi linear algebra for our main result we introduce polynomial transducers and prove that for these validity of a polynomial invariant can be certified by means of an inductive invariant of polynomial ideals this allows us to construct two semi algorithms one searching for a certificate of the invariant and one searching for a witness of its violation via a translation into polynomial transducers we thus obtain that equivalence of general y dt transducers is decidable in fact our translation also shows that equivalence is decidable when the output is not in a free monoid but in a free group

applications and generalizations of fisher hartwig asymptotics fisher hartwig asymptotics refers to the large n form of a class of toeplitz determinants with singular generating functions this class of toeplitz determinants occurs in the study of the spin spin correlations for the two dimensional ising model and the ground state density matrix of the impenetrable bose gas amongst other problems in mathematical physics we give a new application of the original fisher hartwig formula to the asymptotic decay of the ising correlations above tc while the study of the bose gas density matrix leads us to generalize the fisher hartwig formula to the asymptotic form of random matrix averages over the classical groups and the gaussian and laguerre unitary matrix ensembles another viewpoint of our generalizations is that they extend to hankel determinants the fisher hartwig asymptotic form known for toeplitz determinants

determinants of laplacians the determinant of the laplacian on spinor fields on a riemann surface is evaluated in terms of the value of the selberg zeta function at the middle of the critical strip a key role in deriving this relation is played by the barnes double gamma function

adding nesting structure to words we propose the model of nested words for representation of data with both a linear ordering and a hierarchically nested matching of items examples of data with such dual linear hierarchical structure include executions of structured programs annotated linguistic data and html xml documents nested words generalize both words and ordered trees and allow both word and tree operations we define nested word automata finite state acceptors for nested words and show that the resulting class of regular languages of nested words has all the appealing theoretical properties that the classical regular word languages enjoys deterministic nested word automata are as expressive as their nondeterministic counterparts the class is closed under union intersection complementation concatenation kleene prefixes and language homomorphisms membership emptiness language inclusion and language equivalence are all decidable and definability in monadic second order logic corresponds exactly to finite state recognizability we also consider regular languages of infinite nested words and show that the closure properties mso characterization and decidability of decision problems carry over the linear encodings of nested words give the class of visibly pushdown languages of words and this class lies between balanced languages and deterministic context free languages we argue that for algorithmic verification of structured programs instead of viewing the program as a context free language over words one should view it as a regular language of nested words or equivalently a visibly pushdown language and this would allow model checking of many properties such as stack inspection pre post conditions that are not expressible in existing specification logics we also study the relationship between ordered trees and nested words and the corresponding automata while the analysis complexity of nested word automata is the same as that of classical tree automata they combine both bottom up and top down traversals and enjoy expressiveness and succinctness benefits over tree automata

on the τ function of the painleve equations abstract concerning the hamiltonian structure associated with painleve equations we define the function related to a hamiltonian and show that the correlation function given in the study of the two dimensional ising model is a function for the third painleve equation the list of the hamiltonians which are polynomials in the two canonical variables is given and we prove that the function related to each of them is holomorphic

the x ray problem revisited in this letter we revisit the x ray problem assuming point interaction between the conduction electrons and the instantaneously created core hole the latter s green s function can be represented as a fredholm determinant of certain wiener hopf operators acting on l2 0 t with discontinuous symbols here the symbols are the local conduction electron green s function in the frequency domain and t is the time the core hole spends in the system before removal in this situation the classical theory of singular integral equations usually employed in the literature to compute the large t asymptotics of the fredholm determinant ceased to be applicable a rigorous theory first put forward in the context of operator theory comes into play and universal constants are found in the asymptotics

macro tree transducers attribute grammars and mso definable tree translations a characterization is given of the class of tree translations definable in monadic second order logic mso in terms of macro tree transducers the first main result is that the mso definable tree translations are exactly those tree translations realized by macro tree transducers mtts with regular look ahead that are single use restricted for this the single use restriction known from attribute grammars is generalized to mtts since mtts are closed under regular look ahead this implies that every mso definable tree translation can be realized by an mtt the second main result is that the class of mso definable tree translations can also be obtained by restricting mtts with regular look ahead to be finite copying i e to require that each input subtree is processed only a bounded number of times the single use restriction is a rather strong static restriction on the rules of an mtt whereas the finite copying restriction is a more liberal dynamic restriction on the derivations of an mtt

eigenvalues of hermitian toeplitz matrices with smooth simple loop symbols abstract the paper presents higher order asymptotic formulas for the eigenvalues of large hermitian toeplitz matrices with moderately smooth symbols which trace out a simple loop on the real line the formulas are established not only for the extreme eigenvalues but also for the inner eigenvalues the results extend and make more precise existing results which so far pertain to banded matrices or to matrices with infinitely differentiable symbols also given is a fixed point equation for the eigenvalues which may be solved numerically by an iteration method

toeplitz determinants with perturbations in the corners abstract the paper is devoted to exact and asymptotic formulas for the determinants of toeplitz matrices with perturbations by blocks of fixed size in the four corners if the norms of the inverses of the unperturbed matrices remain bounded as the matrix dimension goes to infinity then standard perturbation theory yields asymptotic expressions for the perturbed determinants this premise is not satisfied for matrices generated by so called fisher hartwig symbols in that case we establish formulas for pure single fisher hartwig singularities and for hermitian matrices induced by general fisher hartwig symbols

spectral problems of a class of non self adjoint one dimensional schrodinger operators abstract in this paper we investigate the one dimensional schrodinger operator l q with complex valued periodic potential q when q l 1 0 1 and q n 0 for n 0 1 2 where q n are the fourier coefficients of q with respect to the system e i 2 n x we prove that the bloch eigenvalues are 2 n t 2 for n z t c and find explicit formulas for the bloch functions then we consider the inverse problem for this operator

isophasal isopolar and isospectral schrodinger operators and elementary complex analysis we explicitly construct super exponentially decaying potentials on d so that the associated schrodinger operators are isophasal and isopolar we use similar techniques to construct isospectral schrodinger operators with periodic potentials on and isospectral schrodinger operators on

random matrix theory and entanglement in quantum spin chains we compute the entropy of entanglement in the ground states of a general class of quantum spin chain hamiltonians those that are related to quadratic forms of fermi operators between the first n spins and the rest of the system in the limit of infinite total chain length we show that the entropy can be expressed in terms of averages over the classical compact groups and establish an explicit correspondence between the symmetries of a given hamiltonian and those characterizing the haar measure of the associated group these averages are either toeplitz determinants or determinants of combinations of toeplitz and hankel matrices recent generalizations of the fisher hartwig conjecture are used to compute the leading order asymptotics of the entropy as n this is shown to grow logarithmically with n the constant of proportionality is determined explicitly as is the next constant term in the asymptotic expansion the logarithmic growth of the entropy was previously predicted on the basis of numerical computations and conformal field theoretic calculations in these calculations the constant of proportionality was determined in terms of the central charge of the virasoro algebra our results therefore lead to an explicit formula for this charge we also show that the entropy is related to solutions of ordinary differential equations of painleve type in some cases these solutions can be evaluated to all orders using recurrence relations

asymptotics of toeplitz hankel and toeplitz hankel determinants with fisher hartwig singularities we study the asymptotics in n for n dimensional toeplitz determinants whose symbols possess fisher hartwig singularities on a smooth background we prove the general nondegenerate asymptotic behavior as conjectured by basor and tracy we also obtain asymptotics of hankel determinants on a nite interval as well as determinants of toeplitz hankel type our analysis is based on a study of the related system of orthogonal polynomials on the unit circle using the riemann hilbert approach

the onsager formula the fisher hartwig conjecture and their influence on research into toeplitz operators this is not primarily a paper about applications of mathematics to statistical physics but rather a report on how a particular problem of statistical physics has resulted in an extensive mathematical theory the problem alluded to is the computation of the spontaneous magnetizationm o t of the two dimensional ising model with nearest neighbor interactions whose solution for temperaturest below the curie pointt c was given by the famous formula of lars onsager in 1948 the theory grown out of this formula is the edifice of toeplitz determinants matrices and operators

the fisher hartwig conjecture and toeplitz eigenvalues abstract the conjecture of fisher and hartwig published in 1968 describes the asymptotic expansion of toeplitz determinants with singular generating functions for more than twenty years progress was made in extending the validity of the conjecture but recent computer experiments led to counterexamples that show the limits of the original conjecture and pointed the way to a revised conjecture this paper describes the history of the problem several numerical examples and the revised conjecture

asymptotics of toeplitz determinants and the emptiness formation probability for the xy spin chain we study an asymptotic behaviour of a special correlator known as the emptiness formation probability efp for the one dimensional anisotropic xy spin 1 2 chain in a transverse magnetic field this correlator is essentially the probability of formation of a ferromagnetic string of length n in the antiferromagnetic ground state of the chain and plays an important role in the theory of integrable models for the xy spin chain the correlator can be expressed as the determinant of a toeplitz matrix and its asymptotical behaviours for n throughout the phase diagram are obtained using known theorems and conjectures on toeplitz determinants we find that the decay is exponential everywhere in the phase diagram of the xy model except on the critical lines i e where the spectrum is gapless in these cases a power law prefactor with a universal exponent arises in addition to an exponential or gaussian decay the latter gaussian behaviour holds on the critical line corresponding to the isotropic xy model while at the critical value of the magnetic field the efp decays exponentially at small anisotropy one has a crossover from the gaussian to the exponential behaviour we study this crossover using the bosonization approach

the fisher hartwig formula and entanglement entropy toeplitz matrices have applications to different problems of statistical mechanics recently it was used for calculation of entanglement entropy in spin chains in the paper we review these recent developments we use the fisher hartwig formula as well as the recent results concerning the asymptotics of the block toeplitz determinants to calculate entanglement entropy of large block of spins in the ground state of xy spin chain

the spectral expansion for a nonself adjoint hill operator with a locally integrable potential abstract we construct the spectral expansion for the one dimensional schrodinger operator l d 2 dx 2 q x x in l 2 where q x is a 1 periodic lebesgue integrable on 0 1 and complex valued potential we obtain the asymptotic formulas for the eigenfunctions and eigenvalues of the operator l t for t 0 generated by this operation in l 2 0 1 and the t periodic boundary conditions using it we prove that the eigenfunctions and associated functions of l t form a riesz basis in l 2 0 1 for t 0 then we find the spectral expansion for the operator l

approximate singular values of the fractional difference and summation operators abstract we obtain sharp bounds on the singular values of the fractional difference and summation operators on r n these bounds allow us to establish that the convergence rates of the distributions of the singular values of these operators are o 1 n since the fractional difference operator of order is associated with the toeplitz matrix with fisher hartwig symbol 2 2 cos u 0 we are able to obtain similar bounds on the eigenvalues of this toeplitz matrix and a similar result on the convergence rate of the distribution of its eigenvalues

toeplitz determinants with one fisher hartwig singularity letcbe a function defined on the unit circle with fourier coefficients cn n the fisher hartwig conjecture describes the asymptotic behaviour of the determinants of then ntoeplitz matricesdn c det ci j n 1i j 0for a certain class of functionsc in this paper we prove this conjecture in the case of functions with one singularity more precisely we consider functions of the formc ei b ei t ei 1 u ei 1 heret ei exp i 0 2 is a function with a jump discontinuity u ei 2 2 cos is a function which may have a zero a pole or a discontinuity of oscillating type andbis a sufficiently smooth nonvanishing function with winding number equal to zero the only restriction we impose on the parameters is that 2 is required not to be a negative integer in the case where re 1 2 i e where the corresponding functioncis not integrable we identifycin an appropriate way with a distribution

spectral functions special functions and the selberg zeta function the functional determinant of an eigenvalue sequence as defined by zeta regularization can be simply evaluated by quadratures we apply this procedure to the selberg trace formula for a compact riemann surface to find a factorization of the selberg zeta function into two functional determinants respectively related to the laplacian on the compact surface itself and on the sphere we also apply our formalism to various explicit eigenvalue sequences reproducing in a simpler way classical results about the gamma function and the barnesg function concerning the latter our method explains its connection to the selberg zeta function and evaluates the related glaisher kinkelin constanta

a new correlator in quantum spin chains we propose a new correlator in one dimensional quantum spin chains the s emptiness formation probability s efp this is a generalization of the emptiness formation probability efp which is the probability that the first n spins of the chain are all aligned downwards in the s efp we let the spins in question be separated by s sites the usual efp corresponds to the special case when s 1 taking s 1 allows us to quantify non local correlations we express the s efp for the anisotropic xy model in a transverse magnetic field a system with both critical and non critical regimes in terms of a toeplitz determinant for the isotropic xy model we find that the magnetic field induces an interesting length scale

regular transducer expressions for regular transformations abstract functional mso transductions deterministic two way transducers as well as streaming string transducers are all equivalent models for regular functions in this paper we show that every regular function either on finite words or on infinite words captured by a deterministic two way transducer can be described with a regular transducer expression rte for infinite words the two way transducer uses muller acceptance and regular look ahead rte s are constructed from constant functions using the combinators if then else deterministic choice hadamard product and unambiguous versions of the cauchy product the 2 chained kleene iteration and the 2 chained omega iteration our proof works for transformations of both finite and infinite words extending the result on finite words of alur et al in lics 14 the construction of an rte associated with a deterministic two way transducer is guided by a regular expression which is good wrt its transition monoid good expressions are unambiguous ensuring the functionality of the output computed moreover in good expressions iterations kleene plus or omega are restricted to subexpressions corresponding to idempotent elements of the transition monoid good expressions can be obtained with an unambiguous version of imre simon s famous forest factorization theorem to handle infinite words we introduce the notion of transition monoids for deterministic two way muller transducers with look ahead where the look ahead is captured by some backward deterministic buchi automaton this paper is an extended version of 15 presented at lics 18

the fisher hartwig conjecture and generalizations abstract we discuss the status of the fisher hartwig conjecture concerning the asymptotic expansion of a class of toeplitz determinants with singular generating functions a counterexample is given for a nonrational generating function and we formulate a generalized fisher hartwig conjecture

monodromy preserving deformation of linear ordinary differential equations with rational coefficients iii abstract a unified treatment of monodromy and spectrum preserving deformations is presented in particular a general procedure is described to reduce the latter into the former consistently the concept of the function previously introduced for the former 2 is extended to the isospectral context it is shown that the general monodromy and spectrum preserving deformation equations can be written as hirota s bilinear differential equations by using the functions as dependent variables

toeplitz matrices and determinants with fisher hartwig symbols abstract this paper is concerned with toeplitz matrices generated by symbols of the form a t r 1 r t t r 2 r t t r r b t t 1 where t 1 t r are pairwise distinct points on the unit circle b is sufficiently smooth b t 0 t 1 and ind b 0 t t r r is defined as exp i r arg t t r with arg t t r and r r are complex numbers satisfying re r 1 2 re r 1 2 weighted hardy spaces l 2 1 and l 2 2 are defined such that the toeplitz operator t a l 2 1 l 2 2 is bounded and invertible and that the finite section method is applicable to t a considered as acting from l 2 1 onto l 2 2 these results are then applied to prove a conjecture of fisher and hartwig adv in chem phys 15 1968 333 353 which asserts that the determinants of the n n sections of t a are asymptotically equal to g b n n q e a n where q r 2 r 2 and g b e a are certain nonzero constants under quite general conditions on the smoothness of b these constants are completely identified

some results on complex toeplitz eigenvalues we consider the asymptotics of the eigenvalues of toeplitz matrices generated by a complex valued symbol f which is essentially bounded we prove that if the essential range of f has empty interior and does not separate the complex plane then the corresponding toeplitz eigenvalues have the canonical distribution in the sense of widom the assumptions on the symbol only concern the topological structure of its essential range and are completely independent of its smoothness properties we also prove some clustering properties of eigenvalues and some new distribution results

emergence of a singularity for toeplitz determinants and painleve v we obtain asymptotic expansions for toeplitz determinants corresponding to a family of symbols depending on a parameter t for t positive the symbols are regular so that the determinants obey szeg s strong limit theorem if t 0 the symbol possesses a fisher hartwig singularity letting t 0 we analyze the emergence of a fisher hartwig singularity and a transition between the two different types of asymptotic behavior for toeplitz determinants this transition is described by a special painleve v transcendent a particular case of our result complements the classical description of wu mccoy tracy and barouch of the behavior of a 2 spin correlation function for a large distance between spins in the two dimensional ising model as the phase transition occurs

toeplitz and wiener hopf determinants with piecewise continuous symbols abstract the asymptotics for determinants of toeplitz and wiener hopf operators with piecewise continuous symbols are obtained in this paper if w is the wiener hopf operator defined on l2 0 with piecewise continuous symbol having a finite number of discontinuities at r then under appropriate conditions it is shown that det w g r2k where g exp log 0 r 1 2 log r r and k is a completely determined constant an analogous result is obtained for toeplitz operators the main point of the paper is to obtain a result in the wiener hopf case since the toeplitz case had been treated earlier in the toeplitz case it was discovered that one could obtain asymptotics fairly easily for symbols with several singularities if for each singularity one could find a single example of a symbol with a singularity of that kind whose associated asymptotics were known fortunately in the toeplitz case such asymptotics were known the difficulty in the wiener hopf case is that there was not a single singular case where the determinant was explicitly known this problem was overcome by using the fact that wiener hopf determinants when discretized become toeplitz determinants whose entries depend on the size of the matrix no theorem on toeplitz matrices can be applied directly but these theorems are modified to obtain the desired results

streaming tree transducers the theory of tree transducers provides a foundation for understanding expressiveness and complexity of analysis problems for specification languages for transforming hierarchically structured data such as xml documents we introduce streaming tree transducers as an analyzable executable and expressive model for transforming unranked ordered trees and forests in a single pass given a linear encoding of the input tree the transducer makes a single left to right pass through the input and computes the output in linear time using a finite state control a visibly pushdown stack and a finite number of variables that store output chunks that can be combined using the operations of string concatenation and tree insertion we prove that the expressiveness of the model coincides with transductions definable using monadic second order logic mso existing models of tree transducers either cannot implement all mso definable transformations or require regular look ahead that prohibits single pass implementation we show a variety of analysis problems such as type checking and checking functional equivalence are decidable for our model

thin structure of eigenvalue clusters for non hermitian toeplitz matrices in contrast to the hermitian case the unfair behavior of non hermitian toeplitz eigenvalues is still to be unravelled we propose a general technique for this which reveals the eigenvalue clusters for symbols from l moreover we study a thin structure of those clusters in the terms of properly defined subclusters in some cases this leads to as much as the szego like formulas