the τ function of the ablowitz segur family of solutions to painleve ii as a widom constant the functions of certain painleve equations pvi pv and piii can be expressed as fredholm determinants furthermore the minor expansion of these determinants provides an interesting connection to random partitions this paper is a step toward understanding whether the function of painleve ii has a fredholm determinant representation the ablowitz segur family of solutions are special one parameter solutions of painleve ii and the corresponding function is known to be the fredholm determinant of the airy kernel we develop a formalism for open contour in parallel to the one formulated in terms of a suitable combination of toeplitz operators called the widom constant and verify that the widom constant for the ablowitz segur family of solutions is indeed the determinant of the airy kernel finally we construct a suitable basis and obtain the minor expansion of the ablowitz segur function the functions of certain painleve equations pvi pv and piii can be expressed as fredholm determinants furthermore the minor expansion of these determinants provides an interesting connection to random partitions this paper is a step toward understanding whether the function of painleve ii has a fredholm determinant representation the ablowitz segur family of solutions are special one parameter solutions of painleve ii and the corresponding function is known to be the fredholm determinant of the airy kernel we develop a formalism for open contour in parallel to the one formulated in terms of a suitable combination of toeplitz operators called the widom constant and verify that the widom constant for the ablowitz segur family of solutions is indeed the determinant of the airy kernel finally we construct a suitable basis and obtain the minor expansion of the ablowitz segur function

disturbed exploitation compact differential evolution for limited memory optimization problems this paper proposes a novel and unconventional memetic computing approach for solving continuous optimization problems characterized by memory limitations the proposed algorithm unlike employing an explorative evolutionary framework and a set of local search algorithms employs multiple exploitative search within the main framework and performs a multiple step global search by means of a randomized perturbation of the virtual population corresponding to a periodical randomization of the search for the exploitative operators the proposed memetic computing approach is based on a populationless compact evolutionary framework which instead of processing a population of solutions handles its statistical model this evolutionary framework is based on a differential evolution which cooperatively employs two exploitative search operators the first is based on a standard differential evolution mutation and exponential crossover and the second is the trigonometric mutation these two search operators have an exploitative action on the algorithmic framework and thus contribute to the rapid convergence of the virtual population towards promising candidate solutions the action of these search operators is counterbalanced by a periodical stochastic perturbation of the virtual population which has the role of disturbing the excessively exploitative action of the framework and thus inhibits its premature convergence the proposed algorithm namely disturbed exploitation compact differential evolution is a simple and memory wise cheap structure that makes use of the memetic computing paradigm in order to solve complex optimization problems the proposed approach has been tested on a set of various test problems and compared with state of the art compact algorithms and with some modern population based meta heuristics numerical results show that disturbed exploitation compact differential evolution significantly outperforms all the other compact algorithms present in literature and reaches a competitive performance with respect to modern population algorithms including some memetic approaches and complex modern differential evolution based algorithms in order to show the potential of the proposed approach in real world applications disturbed exploitation compact differential evolution has been implemented for performing the control of a space robot by simulating the implementation within the robot micro controller numerical results show the superiority of the proposed algorithm with respect to other modern compact algorithms present in literature

invariant functions on lie groups and hamiltonian flows of surface group representations si est le groupe fondamental d une surface orientee fermee s et g est un groupe de lie satisfaisant des conditions tres generales alors l espace hom g g des classes de conjugaison de representation g a une structure symplectique naturelle on etudie la geometrie de cette structure symplectique a l aide d une famille naturelle de fonctions sur hom g g

applying lehman s theorems to packing problems a 0 1 matrixa isideal if the polyhedronq a conv x qv a x 1 x 0 v denotes the column index set ofa is integral similarly a matrix isperfect ifp a conv x qv a x 1 x 0 is integral little is known about the relationship between these two classes of matrices we consider a transformation between the two classes which enables us to apply lehman s modified theorem about deletion minimal nonideal matrices to obtain new results about packing polyhedra this results in a polyhedral description for the stable set polytopes ofnear bipartite graphs the deletion of any neighbourhood produces a bipartite graph note that this class includes the complements of line graphs to date this is the only natural class besides the perfect graphs for which such a description is known for the graphs and their complements some remarks are also made on possible approaches to describing the stable set polyhedra of quasi line graphs and more generally claw free graphs these results also yield a new class oft perfect graphs

gap probability of the circular unitary ensemble with a fisher hartwig singularity and the coupled painleve v system we consider the circular unitary ensemble with a fisher hartwig singularity of both jump type and root type at z 1 a rescaling of the ensemble at the fisher hartwig singularity leads to the confluent hypergeometric kernel by studying the asymptotics of the toeplitz determinants we show that the probability of there being no eigenvalues in a symmetric arc about the singularity on the unit circle for a random matrix in the ensemble can be explicitly evaluated via an integral of the hamiltonian of the coupled painleve v system in dimension four this leads to a painleve type representation of the confluent hypergeometric kernel determinant moreover the large gap asymptotics including the constant terms are derived by evaluating the total integral of the hamiltonian in particular we reproduce the large gap asymptotics of the confluent hypergeometric kernel determinant obtained by deift krasovsky and vasilevska and the sine kernel determinant as a special case including the constant term conjectured earlier by dyson

whittaker vectors of the virasoro algebra in terms of jack symmetric polynomial abstract we give an explicit formula of whittaker vector for virasoro algebra in terms of the jack symmetric functions our fundamental tools are the feigin fuchs bosonization and the split expression of the calogero sutherland model given by awata matsuo odake and shiraishi

tau functions as widom constants we define a tau function for a generic riemann hilbert problem posed on a union of non intersecting smooth closed curves with jump matrices analytic in their neighborhood the tau function depends on parameters of the jumps and is expressed as the fredholm determinant of an integral operator with block integrable kernel constructed in terms of elementary parametrices its logarithmic derivatives with respect to parameters are given by contour integrals involving these parametrices and the solution of the riemann hilbert problem in the case of one circle the tau function coincides with widom s determinant arising in the asymptotics of block toeplitz matrices our construction gives the jimbo miwa ueno tau function for riemann hilbert problems of isomonodromic origin painleve vi v iii garnier system etc and the sato segal wilson tau function for integrable hierarchies such as gelfand dickey and drinfeld sokolov

box total dual integrality box integrality and equimodular matrices box totally dual integral box tdi polyhedra are polyhedra described by systems which yield strong min max relations we characterize them in several ways involving the notions of principal box integer polyhedra and equimodular matrices a polyhedron is box integer if its intersection with any integer box ell le x le u is integer we define principally box integer polyhedra to be the polyhedra p such that kp is box integer whenever kp is integer a rational r times n matrix is equimodular if it has full row rank and its nonzero r times r determinants all have the same absolute value a face defining matrix is a full row rank matrix describing the affine hull of a face of the polyhedron our main result is that the following statements are equivalent along our proof we show that a polyhedral cone is box tdi if and only if it is box integer and that these properties are carried over to its polar we illustrate these charaterizations by reviewing well known results about box tdi polyhedra we also provide several applications the first one is a new perspective on the equivalence between two results about binary clutters secondly we refute a conjecture of ding zang and zhao about box perfect graphs thirdly we discuss connections with an abstract class of polyhedra having the integer caratheodory property finally we characterize the box tdiness of the cone of conservative functions of a graph and provide a corresponding box tdi system

integer round up property for the chromatic number of some h perfect graphs we prove that every h perfect line graph and every t perfect claw free graph g has the integer round up property for the chromatic number for every non negative integral weight function c on the vertices of g the weighted chromatic number of g c can be obtained by rounding up its fractional relaxation as a corollary we obtain that the weighted chromatic number can be computed in polynomial time for these graphs finally we show a new example of a graph operation which preserves the integer round up property for the chromatic number and use it to provide a first example of a t perfect 3 colorable graph which does not have this property

a hybrid immune multiobjective optimization algorithm in this paper we develop a hybrid immune multiobjective optimization algorithm himo based on clonal selection principle in himo a hybrid mutation operator is proposed with the combination of gaussian and polynomial mutations gp hm operator the gp hm operator adopts an adaptive switching parameter to control the mutation process which uses relative large steps in high probability for boundary individuals and less crowded individuals with the generation running the probability to perform relative large steps is reduced gradually by this means the exploratory capabilities are enhanced by keeping a desirable balance between global search and local search so as to accelerate the convergence speed to the true pareto optimal front in the global space with many local pareto optimal fronts when comparing himo with various state of the art multiobjective optimization algorithms developed recently simulation results show that himo performs better evidently

balancing exploration and exploitation with adaptive variation for evolutionary multi objective optimization abstract although recent studies have shown that evolutionary algorithms are effective tools for solving multi objective optimization problems their performances are often bottlenecked by the suitability of the evolutionary operators with respect to the optimization problem at hand and their corresponding parametric settings to adapt the search dynamic of evolutionary operation in multi objective optimization this paper proposes an adaptive variation operator that exploits the chromosomal structure of binary representation and synergizes the function of crossover and mutation the overall search ability is deterministically tuned online to maintain a balance between extensive exploration and local fine tuning at different stages of the evolutionary search also the coordination between the two variation operators is achieved by means of an adaptive control that ensures an efficient exchange of information between the different chromosomal sub structures throughout the evolutionary search extensive comparative studies with several representative variation operators are performed on different benchmark problems and significant algorithmic performance improvements in terms of proximity uniformity and diversity are obtained with the incorporation of the proposed adaptive variation operator into the evolutionary multi objective optimization process

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

on claw free t perfect graphs a graph is called t perfect if its stable set polytope is defined by non negativity edge and odd cycle inequalities we characterise the class of all claw free t perfect graphs by forbidden t minors and show that they are 3 colourable moreover we determine the chromatic number of claw free h perfect graphs and give a polynomial time algorithm to compute an optimal colouring

how instanton combinatorics solves painleve vi v and iiis we elaborate on a recently conjectured relation of painlev transcendents and 2d conformal field theory general solutions of painlev vi v and iii are expressed in terms of c 1 conformal blocks and their irregular limits agt related to instanton partition functions in supersymmetric gauge theories with nf 0 1 2 3 4 the resulting combinatorial series representations of painlev functions provide an efficient tool for their numerical computation at finite values of the argument the series involves sums over bipartitions which in the simplest cases coincide with gessel expansions of certain toeplitz determinants considered applications include fredholm determinants of classical integrable kernels scaled gap probability in the bulk of the gaussian unitary ensemble and all order conformal perturbation theory expansions of correlation functions in the sine gordon field theory at the free fermion point

asymptotics of the airy kernel determinant the authors use riemann hilbert methods to compute the constant that arises in the asymptotic behavior of the airy kernel determinant of random matrix theory

dyson s constant in the asymptotics of the fredholm determinant of the sine kernel we prove that the asymptotics of the fredholm determinant of i k where k is the integral operator with the sine kernel open image in new window on the interval 0 are given by

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

the widom dyson constant for the gap probability in random matrix theory in the bulk scaling limit for the gaussian unitary ensemble in random matrix theory the probability that there are no eigenvalues in the interval 0 2s is given by p s det i k s where k s is the trace class operator with kernel k s x y sin x y p x y acting on l 2 0 2s in the analysis of the asymptotic behavior of p s as s there is particular interest in the constant term known as the widom dyson constant we present a new derivation of this constant which can be adapted to calculate similar critical constants in other problems arising in random matrix theory

asymptotics of tracy widom distributions and the total integral of a painleve ii function the tracy widom distribution functions involve integrals of a painleve ii function starting from positive infinity in this paper we express the tracy widom distribution functions in terms of integrals starting from minus infinity there are two consequences of these new representations the first is the evaluation of the total integral of the hastings mcleod solution of the painleve ii equation the second is the evaluation of the constant term of the asymptotic expansions of the tracy widom distribution functions as the distribution parameter approaches minus infinity for the gue tracy widom distribution function this gives an alternative proof of the recent work of deift its and krasovsky the constant terms for the goe and gse tracy widom distribution functions are new

long time asymptotics for integrable systems higher order theory the authors show how to obtain the full asymptotic expansion for solutions of integrable wave equations to all orders as t the method is rigorous and systematic and does not rely on an a priori ansatz for the form of the solution

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

evolving better population distribution and exploration in evolutionary multi objective optimization the aim of multi objective evolutionary optimization is to minimize the distance between the solution set and the true pareto front to distribute the solutions evenly and to maximize the spread of solution set this paper addresses these issues by presenting two features that enhance the optimization ability of multi objective evolutionary algorithms the first feature is a variant of the mutation operator that adapts the mutation rate along the evolution process to maintain a balance between the introduction of diversity and local fine tuning in addition this adaptive mutation operator adopts a new approach to strike a compromise between the preservation and disruption of genetic information the second feature is an enhanced exploration strategy that encourages the exploration towards less populated areas and hence achieves better discovery of gaps in the generated front the strategy also preserves non dominated solutions in the evolving population to achieve a good convergence for the optimization comparative studies of some well known diversity operators mutation operators and multi objective evolutionary algorithms are performed on different benchmark problems which illustrate the effectiveness and efficiency of the proposed features

the dependence on the monodromy data of the isomonodromic tau function the isomonodromic tau function defined by jimbo miwa ueno vanishes on the malgrange s divisor of generalized monodromy data for which a vector bundle is nontrivial or which is the same a certain riemann hilbert problem has no solution in their original work jimbo miwa ueno provided an algebraic construction of its derivatives with respect to isomonodromic times however the dependence on the generalized monodromy data i e monodromy representation and stokes parameters was not derived we fill the gap by providing a simpler and more general description in which all the parameters of the problem monodromy changing and monodromy preserving are dealt with at the same level we thus provide variational formulae for the isomonodromic tau function with respect to the generalized monodromy data the construction applies more generally given any sufficiently well behaved family of riemann hilbert problems rhp where the jump matrices depend arbitrarily on deformation parameters we can construct a one form not necessarily closed on the deformation space malgrange s differential defined off malgrange s divisor we then introduce the notion of discrete schlesinger transformation it means that we allow the solution of the rhp to have poles or zeros at prescribed point s even if is not closed its difference evaluated along the original solution and the transformed one is shown to be the logarithmic differential on the deformation space of a function as a function of the position of the points of the schlesinger transformation it yields a natural generalization of the sato formula for the baker akhiezer vector even in the absence of a tau function and it realizes the solution of the rhp as such ba vector some exemplifications in the setting of the painleve ii equation and finite toplitz hankel determinants are provided

fredholm determinant and nekrasov sum representations of isomonodromic tau functions we derive fredholm determinant representation for isomonodromic tau functions of fuchsian systems with n regular singular points on the riemann sphere and generic monodromy in gl n mathbb c the corresponding operator acts in the direct sum of n n 3 copies of l2 s1 its kernel has a block integrable form and is expressed in terms of fundamental solutions of n 2 elementary 3 point fuchsian systems whose monodromy is determined by monodromy of the relevant n point system via a decomposition of the punctured sphere into pairs of pants for n 2 these building blocks have hypergeometric representations the kernel becomes completely explicit and has cauchy type in this case fredholm determinant expansion yields multivariate series representation for the tau function of the garnier system obtained earlier via its identification with fourier transform of liouville conformal block or a dual nekrasov okounkov partition function further specialization to n 4 gives a series representation of the general solution to painleve vi equation

the np completeness of edge coloring we show that it is np complete to determine the chromatic index of an arbitrary graph the problem remains np complete even for cubic graphs

isomonodromic tau functions from liouville conformal blocks the goal of this note is to show that the riemann hilbert problem to find multivalued analytic functions with rm sl 2 mathbb c valued monodromy on riemann surfaces of genus zero with n punctures can be solved by taking suitable linear combinations of the conformal blocks of liouville theory at c 1 this implies a similar representation for the isomonodromic tau function in the case n 4 we thereby get a proof of the relation between tau functions and conformal blocks discovered in gamayun et al j high energy phys 10 038 2012 we briefly discuss a possible application of our results to the study of relations between certain mathcal n 2 supersymmetric gauge theories and conformal field theory

tracy widom distributions in critical unitary random matrix ensembles and the coupled painleve ii system we study fredholm determinants of the painleve ii and painleve xxxiv kernels in certain critical unitary random matrix ensembles these determinants describe special gap probabilities of eigenvalues we obtain tracy widom formulas for the fredholm determinants which are explicitly given in terms of integrals involving a family of distinguished solutions to the coupled painleve ii system in dimension four moreover the large gap asymptotics for these fredholm determinants are derived where the constant terms are given explicitly in terms of the riemann zeta 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

monodromy dependence and connection formulae for isomonodromic tau functions we discuss an extension of the jimbo miwa ueno differential 1 form to a form closed on the full space of extended monodromy data of systems of linear ordinary differential equations with rational coefficients this extension is based on the results of m bertola generalizing a previous construction by b malgrange we show how this 1 form can be used to solve a long standing problem of evaluation of the connection formulae for the isomonodromic tau functions which would include an explicit computation of the relevant constant factors we explain how this scheme works for fuchsian systems and in particular calculate the connection constant for the generic painleve vi tau function the result proves the conjectural formula for this constant proposed by iorgov lisovyy and tykhyy we also apply the method to non fuchsian systems and evaluate constant factors in the asymptotics of the painleve ii tau function

irregular conformal blocks with an application to the fifth and fourth painleve equations we develop the theory of irregular conformal blocks of the virasoro algebra in previous studies expansions of irregular conformal blocks at regular singular points were obtained as degeneration limits of regular conformal blocks however such expansions at irregular singular points were not clearly understood this is because precise definitions of irregular vertex operators had not been provided previously in this paper we present precise definitions of irregular vertex operators of two types and we prove that one of our vertex operators exists uniquely then we define irregular conformal blocks with at most two irregular singular points as expectation values of given irregular vertex operators our definitions provide an understanding of expansions of irregular conformal blocks and enable us to obtain expansions at irregular singular points as an application we propose conjectural formulas of series expansions of the tau functions of the fifth and fourth painleve equations using expansions of i

on integrals of the tronquee solutions and the associated hamiltonians for the painleve ii equation abstract we consider a family of tronquee solutions of the painleve ii equation q s 2 q s 3 s q s 2 1 2 1 2 which is characterized by the stokes multipliers s 1 e 2 i s 2 s 3 e 2 i with being a free parameter these solutions include the well known generalized hastings mcleod solution as a special case if 0 we derive asymptotics of integrals of the tronquee solutions and the associated hamiltonians over the real axis for 1 2 and 0 with the constant terms evaluated explicitly our results agree with those already known in the literature if the parameters and are chosen to be special values some applications of our results in random matrix theory are also discussed

computing clique and chromatic number of circular perfect graphs in polynomial time a main result of combinatorial optimization is that clique and chromatic number of a perfect graph are computable in polynomial time grotschel et al in combinatorica 1 2 169 197 1981 perfect graphs have the key property that clique and chromatic number coincide for all induced subgraphs we address the question whether the algorithmic results for perfect graphs can be extended to graph classes where the chromatic number of all members is bounded by the clique number plus one we consider a well studied superclass of perfect graphs satisfying this property the circular perfect graphs and show that for such graphs both clique and chromatic number are computable in polynomial time as well in addition we discuss the polynomial time computability of further graph parameters for certain subclasses of circular perfect graphs all the results strongly rely upon lovasz s theta function

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

scattering theory for the korteweg de vries kdv equation and its hamiltonian interpretation abstract it is shown that the scattering transform for the kdv equation is not canonical in the naive sense an explanation of this phenomenon is found and a correct hamiltonian formulation of the scattering theory is proposed