temperature and magnetic field dependent correlators of the exactly integrable 1 1 dimensional gas of impenetrable fermions the author shows that the finite temperature equal time field correlators of the exactly integrable 1 1 dimensional gas of impenetrable fermions in a magnetic field can be expressed as the first fredholm minor of a completely integrable linear operator this result enables the author to derive differential equations for quantum correlators and to obtain analytical formulae for long distance asymptotics in particular a formula for correlation length as a function of temperature and magnetic field is presented his analysis reveals the presence of two length scales which separate short distance intermediate and extreme asymptotic domains the importance of the cross over region is emphasized

a continuous analogue of the fisher hartwig formula for piecewise continuous symbols abstract the fisher hartwig formula alluded to describes the asymptotic behavior of large toeplitz determinants generated by piecewise continuous functions the purpose of the present paper is to establish an analogue of this formula for truncated wiener hopf integral operators under the assumption that these are of the form identity plus trace operator

feasible insertions in job shop scheduling short cycles and stable sets insertion problems arise in scheduling when additional activities have to be inserted into a given schedule this paper investigates insertion problems in a general disjunctive scheduling framework capturing a variety of job shop scheduling problems and insertion types first a class of scheduling problems is introduced characterized by disjunctive graphs with the so called short cycle property and it is shown that in such problems the feasible selections correspond to the stable sets of maximum cardinality in an associated conflict graph two types of insertion problems are then identified where the underlying disjunctive graph is through or bi connected for these cases it is shown that the short cycle property holds and the conflict graph is bipartite allowing to derive a polyhedral characterization of all feasible insertions an efficient method for deciding whether there exists a feasible insertion and a lower and upper bound procedure for the minimum makespan insertion problem are developed for bi connected graphs this procedure solves the insertion problem to optimality the obtained results are applied to three extensions of the classical job shop the multi processor task blocking and no wait job shop and two types of insertions job and block insertion

asymptotic solutions of nonlinear evolution equations and a painleve transcedent abstract two related problems are resolved one is to find the long time behavior of solutions of the korteweg devries modified korteweg devries mkdv or sine gordon equations that evolve from given initial data on

the tallest column optimality revisited abstract the problem of the optimal design of the tallest unloaded column under selfweight is revisited with a view towards clarifying the optimality of the design proposed by keller and niordson the tallest column j math mech 16 1966 pp 433 446 the height of the tallest column is related to the first eigenvalue of a sturm liouville operator since the operator associated with the keller niordson design does not possess a discrete spectrum nonsmooth analysis is used to establish necessary conditions of optimality of the design upon solving the optimality conditions of an appropriately tapered design for the design variable an iterative numerical scheme is developed for the optimal first eigenvalue the scheme is then modified since it contains divergent integrals and used to find the optimal design and height of the tallest column the heights associated with various materials are evaluated for comparision

asymptotics for the fredholm determinant of the sine kernel on a union of intervals in the bulk scaling limit of the gaussian unitary ensemble of hermitian matrices the probability that an interval of lengths contains no eigenvalues is the fredholm determinant of the sine kernel frac sin x y pi x y over this interval a formal asymptotic expansion for the determinant ass tends to infinity was obtained by dyson in this paper we replace a single interval of lengths bysj wherej is a union ofm intervals and present a proof of the asymptotics up to second order the logarithmic derivative with respect tos of the determinant equals a constant expressible in terms of hyperelliptic integrals timess plus a bounded oscillatory function ofs zero ifm 1 periodic ifm 2 and in general expressible in terms of the solution of a jacobi inversion problem pluso 1 also determined are the asymptotics of the trace of the resolvent operator which is the ratio in the same model of the probability that the set contains exactly one eigenvalue to the probability that it contains none the proofs use ideas from orthogonal polynomial theory

effective construction of a class of positive operators in hilbert space which do not admit triangular factorization to introduce the main notions of triangular factorization consider a hilbert space l 2 a b infty leq a b leq infty the orthogonal projectors p xi in l 2 a b are defined by the relations p xi f x f x for a x xi p xi f x 0 for xi x b f epsilon l 2 a b denote the identity operator by i

a polynomial activity insertion algorithm in a multi resource schedule with cumulative constraints and multiple modes abstract in this paper a polynomial activity insertion algorithm in a multi resource schedule with cumulative constraints general precedence relations and multiple modes is proposed insertion objective is to minimize the resulting impact on maximum lateness while keeping some essential characteristics of the initial schedule a new disjunctive arc based representation with multiple capacities associated to resource arcs is proposed under specific constraints some simple dominance rules allow to find an optimal insertion position in the digraph with low computational requirements

riemann hilbert approach to a generalised sine kernel and applications we investigate the asymptotic behavior of a generalized sine kernel acting on a finite size interval q q we determine its asymptotic resolvent as well as the first terms in the asymptotic expansion of its fredholm determinant further we apply our results to build the resolvent of truncated wiener hopf operators generated by holomorphic symbols finally the leading asymptotics of the fredholm determinant allows us to establish the asymptotic estimates of certain oscillatory multidimensional coupled integrals that appear in the study of correlation functions of quantum integrable models

irregular conformal blocks and connection formulae for painleve v functions we prove a fredholm determinant and short distance series representation of the painleve v tau function t associated with generic monodromy data using a relation of t to two different types of irregular c 1 virasoro conformal blocks and the confluence from painleve vi equation connection formulas between the parameters of asymptotic expansions at 0 and i are conjectured explicit evaluations of the connection constants relating the tau function asymptotics as t 0 i are obtained we also show that irregular conformal blocks of rank 1 for arbitrary central charge are obtained as confluent limits of the regular conformal blocks

finite size corrections in random matrix theory and odlyzko s dataset for the riemann zeros odlyzko has computed a dataset listing more than 109 successive riemann zeros starting from a zero number to beyond 1023 this dataset relates to random matrix theory as according to the montgomery odlyzko law the statistical properties of the large riemann zeros agree with the statistical properties of the eigenvalues of large random hermitian matrices moreover keating and snaith and then bogomolny and co workers have used n n random unitary matrices to analyse deviations from this law we contribute to this line of study in two ways first we point out that a natural process to apply to the dataset is to minimize it by deleting each member independently with some specified probability and we proceed to compute empirical two point correlation functions and nearest neighbour spacings in this setting second we show how to characterize the order 1 n 2 correction term to the spacing distribution for random unitary matrices in terms of a second order differential equation with coefficients that are painleve transcendents and where the thinning parameter appears only in the boundary condition this equation can be solved numerically using a power series method in comparison to the riemann zero data accurate agreement is exhibited

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

level spacing functions and the connection problem of a fifth painleve transcendent in the study of level spacing functions for the eigenvalues of random matrices mahoux and mehta 1993 studied the functions s t a t and b t related to certain fredholm determinants these functions can be expressed in terms of the fifth or the third painleve transcendents the asymptotic behaviour for t to infinity of these functions and their derivatives with respect to a parameter were derived by them except for an unknown constant using jimbo s 1982 method of monodromy preserving deformations we connect the behaviour of the fifth painleve transcendent at t 0 and at t infinity thus determining the unknown constant

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

feasible job insertions in the multi processor task job shop abstract the multi processor task job shop is an extension of the job shop problem where an operation of a job requires a set of machines instead of a single machine job insertion is the following given a feasible schedule of n 1 jobs find a feasible insertion of job n into the schedule such that makespan is minimized the problem is known to be np hard already for the job shop case in this note a polyhedral description of all feasible insertions is derived settling an open problem recently proposed by kis and hertz constrained feasible insertions satisfying additional constraints are introduced and a feasibility theorem is established a lower bound on the job insertion problem is derived and computed by repeatedly invoking the feasibility theorem numerical results show high quality of the bounds and short computation times

on the existence of monodromies for the rabi model we discuss the existence of monodromies associated with the singular points of the eigenvalue problem for the rabi model the complete control of the full monodromy data requires the taming of the stokes phenomenon associated with the unique irregular singular point the monodromy data in particular the composite monodromy are written in terms of the parameters of the model via the isomonodromy method and the function of the painleve v these data provide a systematic way to obtain the quantized spectrum of the rabi model

connection formulae for painleve v functions ii the d function bose gas problem abstract we continue our study of those painleve v equations which admit a one parameter family of solutions analytic at the origin and solve the global problem of connecting the 2 parameter expansion about t 0 with the 2 parameter expansion valid for t i a special case of our results is applied to the study of the one particle reduced density matrix of the one dimensional gas of impenetrable bosons we also use our results to obtain a 2 parameter connection formula for a class of painleve iii functions

deformations of the tracy widom distribution in random matrix theory the tracy widom tw distribution describes the behavior of the largest eigenvalue we consider here two models in which tw undergoes transformations in the first one disorder is introduced in the gaussian ensembles by superimposing an external source of randomness a competition between tw and a normal gaussian distribution results depending on the spreading of the disorder the second model consists of removing at random a fraction of correlated eigenvalues of a random matrix the usual formalism of fredholm determinants extends naturally a continuous transition from tw to the weilbull distribution characteristic of extreme values of an uncorrelated sequence is obtained

counting free fermions on a line a fisher hartwig asymptotic expansion for the toeplitz determinant in the double scaling limit we derive an asymptotic expansion for a wiener hopf determinant arising in the problem of counting one dimensional free fermions on a line segment at zero temperature this expansion is an extension of the result in the theory of toeplitz and wiener hopf determinants known as the generalized fisher hartwig conjecture the coefficients of this expansion are conjectured to obey certain periodicity relations which renders the expansion explicitly periodic in the counting parameter we present two methods to calculate these coefficients and verify the periodicity relations order by order the matrix riemann hilbert problem and the painleve v equation we show that the expansion coefficients are polynomials in the counting parameter and list explicitly first several coefficients

zero temperature correlation functions for the impenetrable fermion gas we calculate the long time and distance asymptotics of the one particle correlation functions in the model of impenetrable spin 1 2 fermions in 1 1 dimensions we consider the spin disordered zero temperature regime which occurs when the limit t 0 is taken at a positive chemical potential the asymptotic expressions are found from the asymptotic solution of the matrix riemann hilbert problem related to the determinant representation of the correlation functions

a fast taboo search algorithm for the job shop problem a fast and easily implementable approximation algorithm for the problem of finding a minimum makespan in a job shop is presented the algorithm is based on a taboo search technique with a specific neighborhood definition which employs a critical path and blocks of operations notions computational experiments up to 2 000 operations show that the algorithm not only finds shorter makespans than the best approximation approaches but also runs in shorter time it solves the well known 10 10 hard benchmark problem within 30 seconds on a personal computer

an isomonodromy cluster of two regular singularities we consider a linear 2 2 matrix ode with two coalescing regular singularities this coalescence is restricted with an isomonodromy condition with respect to the distance between the merging singularities in a way consistent with the ode in particular a zero distance limit for the ode exists the monodromy group of the limiting ode is calculated in terms of the original one this coalescing process generates a limit for the corresponding nonlinear systems of isomonodromy deformations in our main example the latter limit reads as p 6 p 5 where p n is the nth painleve equation we also discuss some general problems which arise while studying the above mentioned limits for the painleve equations

optimal shape of a strongest inverted column by using pontryagin s maximum principle we determine the shape of the strongest column positioned in a constant gravity field simply supported at the lower end and clamped at upper end with the possibility of axial sliding it is shown that the cross sectional area function is determined from the solution of a nonlinear boundary value problem a variational principle for this boundary value problem is formulated and two first integrals are constructed these integrals lead to an a priori estimate of the value of one the missing initial condition and to the reduction of the order of the system the optimal shape of a column is determined by numerical integration

zeros of the jimbo miwa ueno tau function we introduce a family of local deformations for meromorphic connections on p1 in the neighborhood of a higher rank simple singularity following the scheme in malgrange mathematique et physique progress in mathematics birkhauser boston 1983 vol 37 pp 381 400 ibid pp 401 426 ibid pp 427 438 we use these local models to prove that the zeros of the tau function introduced by jimbo miwa and ueno in their pioneering work on birkhoff deformations at irregular singular points physica d 2 306 352 2 407 448 1981 4 26 46 1983 publ rims kyoto univ 17 2 703 721 1981 occur at precisely those points in the deformation space at which a certain birkhoff riemann hilbert problem fails to have a solution

on the location of the least point of the essential spectrum a new criterion for the least point of the essential spectrum is given for a one term second order differential operator this criterion is sharper than the friedrichs criterion application is given to the singular eigenvalue problem associated with the buckling of a column under selfweight subjected to the constraint of fixed volume

a study of correlation functions for the delta function bose gas abstract the gel fand levitan equation for the quantum nonlinear schrodinger field theory is used to investigate the correlation functions of the delta function bose gas operator expressions are derived for the field and for nonlocal products of fields in terms of the quantized reflection operators which create and annihilate eigenstates of the hamiltonian for the two point function an explicit series expression is obtained in which the nth term is determined by well defined n body combinatorics in an infinite volume the inductive properties of this series are discussed and used to express the temperature and chemical potential dependence of the correlation functions entirely in terms of previously known thermodynamic functions the zero separation limit of the series for the two point function reproduces the thermodynamics derived by yang and yang while the infinite coupling limit gives the fredholm determinant result of schultz and lenard the latter is related to the painleve v equation by the monodromy arguments of jimbo miwa mori and sato the o 1 c correction to the large coupling limit is calculated from the gel fand levitan series and expressed in terms of solutions to painleve v the asymptotic behavior of the relevant painleve function is discussed and related to the long range behavior of the correlation function

asymptotics of a τ function arising in the two dimensional ising model the short distance assymptotics of the function associated to the 2 point function of the two dimensional ising model is computed as a function of the integration constant defined from the long distance behavior of the function the result is expressible in terms of the barnes double gamma function equivalently the barnesg function

quantum fluctuations of one dimensional free fermions and fisher hartwig formula for toeplitz determinants we revisit the problem of finding the probability distribution of a fermionic number of one dimensional spinless free fermions on a segment of a given length the generating function for this probability distribution can be expressed as a determinant of a toeplitz matrix we use the recently proven generalized fisher hartwig conjecture on the asymptotic behavior of such determinants to find the generating function for the full counting statistics of fermions on a line segment unlike the method of bosonization the fisher hartwig formula correctly takes into account the discreteness of charge furthermore we numerically check the precision of the generalized fisher hartwig formula find that it has a higher precision than rigorously proven so far and conjecture the form of the next order correction to the existing formula

non equilibrium 1d many body problems and asymptotic properties of toeplitz determinants non equilibrium bosonization technique facilitates the solution of a number of important many body problems out of equilibrium including the fermi edge singularity the tunneling spectroscopy and full counting statistics of interacting fermions forming a luttinger liquid we generalize the method to non equilibrium hard core bosons tonks girardeau gas and establish interrelations between all these problems the results can be expressed in terms of fredholm determinants of the toeplitz type we analyze the long time asymptotics of such determinants using szeg and fisher hartwig theorems our analysis yields dephasing rates as well as power law scaling behavior with exponents depending not only on the interaction strength but also on the non equilibrium state of the system

operator valued riemann hilbert problem for correlation functions of the xxz spin chain the generating functional of correlation functions for the xxz spin chain is considered in the thermodynamic limit we derive a system of integro difference equations that prescribe this functional on the basis of this system we establish the operator valued riemann hilbert problem for correlation functions of the xxz spin chain

randomly incomplete spectra and intermediate statistics by randomly removing a fraction of levels from a given spectrum a model is constructed that describes a crossover from this spectrum to a poisson spectrum the formalism is applied to the transitions towards poisson from random matrix theory rmt spectra and picket fence spectra it is shown that the fredholm determinant formalism of rmt extends naturally to describe incomplete rmt spectra

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

connection formulae for painleve v functions we study those painleve v equations which admit a one parameter family of solutions analytic at the origin and solve the global problem of connecting the two parameter asymptotic expansion at t with the two parameter expansion about t 0 this connection is the one needed to study the correlation functions of the transverse ising chain at the critical field strength from these connection formulae we derive a one parameter connection formula for the general painleve ii equation and a two parameter connection formula for a class of painleve iii equations

on the analysis of incomplete spectra in random matrix theory through an extension of the jimbo miwa ueno differential abstract several distribution functions in the classical unitarily invariant matrix ensembles are prime examples of isomonodromic tau functions as introduced by jimbo miwa and ueno jmu in the early 1980s 45 recent advances in the theory of tau functions 41 based on earlier works of b malgrange and m bertola have allowed to extend the original jimbo miwa ueno differential form to a 1 form closed on the full space of extended monodromy data of the underlying lax pairs this in turn has yielded a novel approach for the asymptotic evaluation of isomonodromic tau functions including the exact computation of all relevant constant factors we use this method to efficiently compute the tail asymptotics of soft edge hard edge and bulk scaled distribution and gap functions in the complex wishart ensemble provided each eigenvalue particle has been removed independently with probability 1 0 1

large time and distance asymptotics of field correlation function of impenetrable bosons at finite temperature abstract asymptotics of the temperature time dependent correlation function of two fields one dimensional bosons with infinite point like repulsion is calculated

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

one particle equal time correlation function for the spin incoherent infinite u hubbard chain we present a calculation of the one particle equal time correlation function x for the one dimensional 1d hubbard model in the infinite u limit we consider the zero temperature spin incoherent regime which is obtained by first taking the limit u infinity and then the limit t 0 using the determinant representation for x we derive analytical expressions for both large and small x at an arbitrary filling factor 0 rho 1 2 the large x asymptotics of rho x is found to be remarkably accurate starting from x sin 2 pi rho similar to 1 we find that the one particle momentum distribution function rho k is a smooth function of k and rho k is peaked at k 2k f in contrast to spin coherent liquid obeying the luttinger theorem

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